1. bookVolume 14 (2021): Edition 1 (January 2021)
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Development of Low Cost Autonomous Underwater Vehicle Platform

Publié en ligne: 15 Jul 2021
Volume & Edition: Volume 14 (2021) - Edition 1 (January 2021)
Pages: 1 - 22
Reçu: 27 Dec 2020
Détails du magazine
License
Format
Magazine
eISSN
1178-5608
Première parution
01 Jan 2008
Périodicité
1 fois par an
Langues
Anglais
<p>Over the years, AUVs have gained attention as specialized tools for carrying out various underwater operations, eventually leading to a dramatic increase in the number of scientific studies undertaken (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_005">Blidberg, 2001</a>). There are four separate sub-classes of unmanaged vehicles (UVs) systems. Submersibles towed UVs behind ships are the first category, serving as ideal platforms for adding different sensors. The second category is the remotely operated vehicle (ROV), which gets directly controlled by the surface ship, with power and connectivity (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_027">Roberts, 2008</a>). The unmanned untethered vehicle (UUV) is the third group, with its own onboard control, but is controlled remotely through a means of wireless communication protocol. <a ref-type="bibr" href="#j_ijssis-2021-005_ref_006">Budiyono (2009)</a>. Nevertheless, the requirement for a networking link and control platform restricts ROV and UUV usage and their capability (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_011">Gonzalez, 2004</a>). The fourth category of UVs, the AUVs, are fully autonomous underwater platforms capable of conducting underwater operations and activities and have their own sensor, control and payload equipment (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_005">Blidberg, 2001)</a>. The key benefit of an AUV is that a human operator is not required, is cheaper than a manually driven vehicle, and is able to work that is too risky for a human being (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_002">Alt, 2003</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_007">Caccia, 2006</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_031">Side and Junku, 2005</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_033">Smallwood David and Whitcomb, 2004</a>) like monitoring of rising levels of seas (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_013">Hadi et al., 2020</a>). In the 1980s, innovation and software developments enhanced the research to facilitate the design and implementation of sophisticated autonomous system (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_005">Blidberg, 2001</a>). Nowadays, technicians, engineers, and computer programmers play an important role in developing technologies (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_019">Innella and Rodgers, 2021</a>) and the developed tools can be used in various applications. The enthusiasm in AUVs in academic science has therefore been revived and several universities have built their own AUVs. The Australian Centre for Field Robotics (ACFR) of the University of Sydney runs an ocean-going AUV labeled Sirius, Designed at WHOI in the form of an Integrated Marine Observation System (IMOS), AUV facility supports competitively to facilitate deployment in Australia as a contributor to marine studies (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_032">Singh et al., 2004</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_040">Williams et al., 2009</a>). There are many AUV contests in which a vehicle has a certain duty to carry out autonomously, which are mostly organized by colleges or other educational institutions (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_011">Gonzalez, 2004</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_001">Akhtman et al., 2008</a>).</p><p>Power and autonomy are the two most critical technical obstacles in AUV design, since the former sets limitations on mission times and the latter decides the degree to which an AUV may be independent (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_016">Holtzhausen, 2010</a>). While AUVs are more precise due to new technology, the creation of a completely autonomous one is an incredibly challenging problem as precise and robust controllers are required. (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_029">Salgado-Jimenez et al., 2004</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_030">Salman et al., 2011)</a>. <a ref-type="bibr" href="#j_ijssis-2021-005_ref_016">Holtzhausen (2010)</a> claimed that the structure of the AUV is one of the most critical aspects of an AUV. There are a number of ways to approach the construction that must take into account factors such as the pressure and/or depth needed, height, operating temperature ranges, conditions of impact, and water permeability along with corrosion and chemical resistance. The hydrodynamic coefficients that decide the dynamics of the AUV are affected not only by the structure of the AUV but also by the water current and the vehicle speed and an innovative method to evaluate velocity and heading is proposed in <a ref-type="bibr" href="#j_ijssis-2021-005_ref_026">Rezaali and Ardalan (2016)</a>. Moreover, a hull is built to minimize the drag for reducing the propulsive force (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_038">Wang et al., 2009a</a>). Wang stated that the first shape proposed for a hull was a spherical design, while this could withstand pressure, it influenced stability. So a circular cylindrical design, which has the benefits of being a good framework to resist the effects of hydrostatic pressure, can be expanded to provide extra room internally, seems to be a better hydrodynamic shape than a spherical one with the same volume (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_028">Ross 2006</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_039">Wang et al., 2009b</a>). The cavitations (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_022">Kondoa and Ura, 2004</a>) and instability (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_028">Ross, 2006</a>) are, however, the drawbacks of a cylindrical hull. Most of the existing AUVs have a cylindrical circular hull (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_009">Evans and Nahon, 2004</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_017">Hsu et al., 2005</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_043">Woods Hole Oceanographic Institution, 2012</a>). Similarly, a spherical shaped nose has enhanced stability and cavitation (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_039">Wang et al., 2009b</a>).</p><p>The dynamics of an AUV are determined mainly by propulsion and buoyancy. Four kinds of propulsion systems are possible. Most common form of propulsion is via thrusters with dynamic diving technology. The other three come from static diving technology, they are piston-type ballast tank, a hydraulic pump-based ballast system and air compressor-based system (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_023">Krieg and Mohseni, 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_042">Wolf, 2003</a>).</p><p>AUV can use a single thruster for both horizontal and vertical movements with diving planes (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_022">Kondoa and Ura, 2004</a>). The thruster technique allows the AUV to be near-neutrally buoyant with the major benefit that the vehicle can hover without propulsion. Even then, once the vehicle is in motion, the thrusters must remain ON, since it has to drive forward to remain submerged. For propulsion, most AUVs use propeller-type thrusters (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_008">Cavallo et al., 2005</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_037">Von Alt, 2003</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_043">Woods Hole Oceanographic Institution, 2012</a>).</p><p>Jet propulsion is another method influenced by the natural locomotion of squids and other cephalopods, whereby water is pulled into a wide cavity and expelled through a nozzle at a high momentum to propel the vehicle forward (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_012">Griffiths et al., 2000</a>). This gives the potential of an AUV to carry out low-speed maneuvering without impacting the forward drag on its structure. In this research, this type of propulsion is used to maneuver the vehicle while the propeller is being used for forward motion in the horizontal and vertical planes.</p><p>Power demands for all the equipment, such as on-board controls, motor controllers, propulsion systems, sensors and instruments, and navigation systems, must be fulfilled and supported by the electrical system of the AUV. While some AUVs are operated by fuel cells and a few uses solar power, the most popular ones are operated by batteries (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_018">Hyakudome et al., 2009</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_020">Jalbert et al., 2003</a>). Silent operation, ease of speed regulation and simplicity are the benefits of utilizing electrical over thermal propulsion. Since lithium-ion battery technological advancements have made them an attractive alternative to silver-zinc batteries (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_041">Wilson and Bales, 2006</a>), they are employed in our research.</p><p>A pressure sensor is the most common sensor used in vehicle depth measurement. Strain gauges and quartz crystals are the most common pressure sensor technologies for deep-ocean applications (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_021">Kinsey et al., 2006</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_038">Wang et al., 2009a</a>).</p><p>GPS can provide superior 3-D navigation functionality. Since the water absorbs the radio signals in underwater conditions, it can be used only when a vehicle surfaces periodically to correct the readings (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_024">Nicholson and Healey, 2008</a>). This implies that other sensors, such as the Long Baseline (LBL), Doppler Velocity Log (DVL), and magnetometer or compass are needed for the tracking and navigation of underwater vehicles for GPS fixes (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_021">Kinsey et al., 2006</a>); e.g., Bluefin-21 AUV surfaces are needed periodically for GPS fixes. The echo sounder (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_011">Gonzalez, 2004</a>) was used in the MAKO AUV project as another method to assess vertical depth, but only if the depth of the water is known.</p><p>The LBL, Ultra-short Baseline (USBL), and Short Baseline (SBL) systems (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_021">Kinsey et al., 2006</a>) are known as underwater acoustic positioning systems used for depth measurement. The LBL device is essentially a method of triangulation which is used when a vehicle triangulates the acoustic position of the vehicle in the network of transponders (beacons) used on the seabed with known locations (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_035">The International Marine Contractors Association IMCA, 2009</a>). A complete system comprises of a transceiver placed under a ship and a vehicle transponder. The time between initial acoustic pulse transmission and reaction detection is estimated and transformed into a range.</p><p>A DVL is a sensor that uses high frequency Doppler beam sonar for calculating the Doppler changes of sonar signals reflected in the ground (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_039">Wang et al., 2009b</a>). This navigation method is only useful if the vehicle is above the sea level (18–100 m), since it is more precise at low speed and not affected by sea currents (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_024">Nicholson and Healey, 2008</a>).</p><p>The most popular sensors used in marine equipment are magnetic sensors or electronic compass modules. Vasilijevic (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_036">Vasilijevic et al., 2012</a>) reported that a magnetometer calculates the magnetic field of the Earth in the X, Z coordinates position and returns these three measures separately to represent the magnetic field vector values. There are a wide number of single-axis (heading only) and three-axis flux-gate magnetometers available commercially. The overall efficiency of a navigation device is mostly based on the accuracy of its magnetic sensor, which is believed to be the leading cause of error. There are three different causes of errors: the magnetic interference of the vehicle with itself or the environment (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_044">Ye et al., 2009</a>), the compensation of the roll and pitch dependent on gravity, and finally, the orientation of the compass mounting within the vehicle.</p><p>Stutters indicated that the Inertial Navigation Device (INS) or Inertial Management Unit (IMU) utilizes accelerometers and gyroscopic sensors to detect vehicle acceleration of the three axes (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_034">Stutters et al., 2008</a>). A gyroscope measures rotation levels, and linear acceleration is determined by an accelerometer. IMU sensors are not prone to magnetic fluctuations and drift with time, leading to erroneous measurements. Laser or fiber-optic gyroscopes do not have moving components and are included in the new INSs (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_039">Wang et al., 2009b</a>).</p><p>A mixture of two or more of the above systems has often established some of the finest AUV navigation systems. Multi-sensor integration can be characterized as the synergistic use of multiple sensory device information that can minimize navigational errors to assist a system in carrying out its tasks (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_016">Holtzhausen, 2010</a>).</p><p>In brief, the main objective of navigation sensors is to acquire the data and parameters of an AUV system in real time from its surrounding environment. These data are then collected by the AUV control system through its processing unit to control and maneuver AUVs effectively in order to perform a pre-identified operation.</p><p>The control accuracy provided by guidance and control systems is the basis for the successful completion of AUV missions as the autonomous control of an AUV poses serious challenges due to its complex, inherently nonlinear, and time-varying dynamics. In addition, its hydrodynamic coefficients are difficult to model accurately because of their variations under different navigational conditions and when manoeuvring in uncertain environments which expose an AUV to unpredictable external disturbances. Hence, there is a need to design and develop a robust and stable control system which can achieve AUV’s desired positions and velocities, satisfy the commands generated by its guidance system and maintain steady conditions during its mission. In order to design an adaptive controller for an AUV, suitable models of the nonlinear plant are necessary as the controllers have to cope with uncertainties due to discrepancies in modeling the unknown dynamics of the plant.</p><p>Thus, the development of an AUV for research or commercial purpose involves a number of criteria to be satisfied along with the cost. This paper proposes the development of an AUV. The main goal of this study is to develop an in-house project, find low-cost solutions for AUV navigation problems, and develop a small-sized, low-cost AUV. The platform is expected to demonstrate autonomous manoeuvre for changing conditions and under the influence of external disturbances in a controlled environment such as a swimming pool.</p><p>The rest of the paper structure is organized as follows. The detailed specifications of different AUV subsystems considered for the work in this study are described in the second section. The third section discusses the AUV complete system and its integration. The kinematics and the nonlinear mathematical modeling of an AUV in 6-DOF is discussed in the fourth section that includes the Simulink/Matlab model of the AUV along with the investigation of its open-loop characteristics. The main features of the Hardware-in-Loop (HIL) simulation are explained in the fifth section. The sixth section presents the experimental results of the proposed model-based control algorithm on the AUV. The study is concluded in the final section.</p></sec><sec id="j_ijssis-2021-005_s_002"><div>Platform design</div><p>The integration on the UNSW Canberra AUV with electrical and electronic systems, which include prototypes, specifications, power distribution system and storage, actuators and sensors, is discussed. The instrumentation required to accommodate on board the AUV in order the completely collect the data associated with different maneauvers, the interface and integration of all subsystems with the PC 104 device are addressed.</p><p>This paper presents the development of an AUV experimental platform with adaptive control capabilities aimed at achieving the mission requirements. A detailed block diagram indicating different stages in this project is shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_001">Fig. 1</a>.</p><figure id="j_ijssis-2021-005_fig_001" fig-type="figure"><h2>Figure 1:</h2><figCaption><p>Block diagram of various stages of project.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_001.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b55eb309511a6949ac10a5f04a4901707e5b3f2547b4c0ce21d0f02c21c8f2d3" class="mw-100"></img></figure><sec id="j_ijssis-2021-005_s_002_s_001"><div>UNSW Canberra AUV platform</div><p>In the UNSW Canberra workshop, a torpedo-shaped underwater vehicle prototype is constructed as a demonstration of the concepts of the underwater vehicle, which comprises of three segments; nose cone, body (middle section), and tail cone. <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al. (2011</a>, <a ref-type="bibr" href="#j_ijssis-2021-005_ref_015">2013)</a>. The nose and tail cone moulds are made up of Stayfoam, then wrapped with fiberglass and those are wet sections to decrease the buoyancy force. The body is made of a 255 mm PVC pipe and is known to be the dry section where the batteries are located along with the circuitry, sensors, and control unit. The dry portion is segregated by the bulkheads for providing waterproof environment. An on-board data logger and control unit have been equipped with the AUV platform to allow sensor and actuator data to be recorded and processed to implement advanced system identification and control techniques. Necessary computers and sensors are chosen, considering the need for low cost, compact, simplicity of program and processor capabilities. The UNSW Canberra AUV model is shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_002">Fig. 2</a>.</p><figure id="j_ijssis-2021-005_fig_002" fig-type="figure"><h2>Figure 2:</h2><figCaption><p>UNSW Autonomous Underwater Vehicle platform (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_038">Hassanein et al., 2011</a>).</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_002.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_002.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=69ef9a0f85bf0848e41eeacca67e5e0c998fad696ac4fd321eeb1680601aa2ff" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_002_s_002"><div>Actuators</div><p>The AUV employs two forms of actuators, the electrical propeller for thrust and secondly, four bilge pumps to operate in the horizontal and vertical axis.</p><p>The thrusters that use propeller systems and a small DC motor wrapped in a watertight enclosure to spin the propeller that produce torque are the key method of actuation in the AUV. The existing thruster is a 12Vdc electric motor, typically employed for small river vessels. The ‘Endura C2’ thruster was obtained from a ‘MINN KOTA’ company shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_003">Fig. 3</a>. An ‘ET-OPTO RELAY4’ made by the ‘ETT’ company is used since the electric motor requires a controller or driver shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_003">Fig. 3</a>. According to the function and operating order, this has two inputs to drive the motor, the power given by the battery and a control signal from the I/O module in the processor unit.</p><figure id="j_ijssis-2021-005_fig_003" fig-type="figure"><h2>Figure 3:</h2><figCaption><p>Forward thruster with electrical motor driver.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_003.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_003.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=54ef71f66924c2920e9db0c0f2485aedf146390ee9125ac90eedc3db7ddedcc4" class="mw-100"></img></figure><p>In order to maneuver in horizontal and vertical planes (pitch and lake angle), submersible bilge pumps are obtained from a company named ‘Rule Mate.’ These are shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_004">Fig. 4</a>. There are three distinct pump configurations and all the pumps are battery operated by 12 Vdc. The left and right pumps used to control the AUV in the horizontal plane have an output of 750 GPH and a force of 0.0022 N. The pump used to push down this vehicle is rated by 0.0033 N at 1,000 GPH and by 0.0044 N at 1,500 GPH. The pumps are operated in on/off mode using the control unit’s PWM signals. The motor driver for such pumps is an ET-OPTO DC-OUT4 manufactured by ETT as shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_004">Fig. 4</a>.</p><figure id="j_ijssis-2021-005_fig_004" fig-type="figure"><h2>Figure 4:</h2><figCaption><p>Submersible bilge pump with the motor driver.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_004.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=84d97478cd47a19680c08c9e47917d98c02ea42eb6157dfcbd6166a8bdcf0e38" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_002_s_003"><div>Inertial measurements UNIT (IMU)</div><p>The IMU is the key sensor used for AUV navigation and comprises three gyroscopes and three accelerometers, each mounted along the <italic>X</italic>, <italic>Y</italic>, and <italic>Z</italic> axes for the calculation of rotational speeds and linear accelerations. The gyroscope data was used to calculate the rotations of the vehicle along the three axes indicated by roll, pitch, and yaw. In terms of these three DOFs, the rotational position of the vehicle is calculated by the determination of the rotational velocity integral over the period of the measurement. In order to determine the location of the vehicle in a three-dimensional space, one must continuously convert from the local coordination system of the vehicle to the coordination system of the earth. This transformation is addressed later in this paper. The ADIS16367 shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_005">Fig. 5</a> is the unique IMU used in this project, which is a 3-axis system acquired from Analog Devices Co. The ADIS16367 combines industry-leading and signal conditioning IMEMS technology that optimizes dynamic efficiency and is a comparatively cheap device. IMU outputs are compared with the calibrated digital compass outputs with no major errors amongst them.</p><figure id="j_ijssis-2021-005_fig_005" fig-type="figure"><h2>Figure 5:</h2><figCaption><p>ADIS16367 unit.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_005.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_005.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=f104ed817846a3e38f73e02e2ad566ba78423d51a10f57296b76ec3184fc2b97" class="mw-100"></img></figure><p>A serial peripheral interface (SPI) card was designed and developed as part of this project as shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_006">Fig. 6</a>. An I/O board is necessary for dealing with the lower level interface between the IMU and the processing device. The I/O module is on the ATmega168-based Arduino Pro-Mini microcontroller board consisting of 14 digital input/output pins, 6 analog inputs, an on-board resonator, a reset button and pin header mounting holes. To provide communication to the board and USB power, a six-pin header can be plugged into an FTDI cable or Sparkfun breakout board. The serial peripheral interface sends the data to the PC104 at a rate of 115 kbps through an RS232 serial port.</p><figure id="j_ijssis-2021-005_fig_006" fig-type="figure"><h2>Figure 6:</h2><figCaption><p>SPI interface for IMU.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_006.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_006.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=1fff07b55678674f81749704e819ae11b68365def5b4cb7e74a183b5e92c8755" class="mw-100"></img></figure><p>A ‘burst data read collection’ is the best way to retrieve data from the IMU, which is a process-efficient form of obtaining ADIS16367 data in which all output registers are clocked on DOUT (SPI Data Output), 16 bits at a time, in sequential data cycles (each divided by a single SCLK interval (SPI Serial Clock)). DIN (SPI data input) is set to 0x3E00 to start a burst read sequence, then the contents of every output register is transferred to DOUT, that is from SUPPLY OUT to AUX ADC, as seen in <a ref-type="fig" href="#j_ijssis-2021-005_fig_006">Figs. 6</a> and <a ref-type="fig" href="#j_ijssis-2021-005_fig_007">7</a>. The data are being sent to a PC104 on-board COM port which contains RS232 driver blocks to handle serial communications provided by the Matlab xPC goal toolbox.</p><figure id="j_ijssis-2021-005_fig_007" fig-type="figure"><h2>Figure 7:</h2><figCaption><p>Burst read sequence (ADIS16367 Data Sheet).</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_007.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_007.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=d9b3cfb12fc6e8803258894edfd3680f68cde78cd167c0e705d98c9b226c2614" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_002_s_004"><div>Electronic Digital Compass</div><p>A magnetometer or electronic digital compass is also a sensor used for the AUV navigation, which measures the magnet field of the Earth in the direction of the <italic>X</italic>, <italic>Y</italic>, and <italic>Z</italic> axes. It returns the three measurements separately to provide the magnetic field vector values, which are then used for determining the heading of the vehicle. It does not need to be transformed since this heading calculation is taken in the global reference frame or coordinates. The magnetometer is susceptible to other disturbance in the magnetic field of the Earth. To stop incorrect readings, it can be recalibrated before any dive. The magnetometer can often be disturbed by other magnetic objects in close vicinity of the sensor as well.</p><p>As shown in the <a ref-type="fig" href="#j_ijssis-2021-005_fig_008">Fig. 8</a>, an Oceansever OS5000-USD with a depth sensor, a 3-axis tilt-compensated compass with both Serial & USB direct interfaces and connections on both sides of the module, is the digital compass used in this project. It comprises a 3-axis AMR magnetic tracker, a 3-axis STM accelerometer, a 50 MIPS microprocessor that facilitates precise tilt correction floating point IEEE operations and 24-bit differential input Sigma-Delta AD converters.</p><figure id="j_ijssis-2021-005_fig_008" fig-type="figure"><h2>Figure 8:</h2><figCaption><p>Wiring from pressure sensor to compass.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_008.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_008.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=adc2c7d8b701c5d780f430fa45d958005c4434df3381ee184c9ea7b4dd0ea975" class="mw-100"></img></figure><p>For operation, the compass needs power supply of 5 V, when attached to the pressure sensor, it measures the depth and its outputs include a heading, pitch and roll in degrees, a 3-axis magnetic field, a 3-axis acceleration, gyro reading in 2 axes and temperature. Its built-in calibration involves rotation around the <italic>Z</italic>, <italic>Y</italic>, and <italic>X</italic> axes that takes place after installation in the AUV and before the experiments begin.</p></sec><sec id="j_ijssis-2021-005_s_002_s_005"><div>Water pressure sensor</div><p>The water pressure is a variation in the surface pressure and it is calculated in Pascals (Pa). A water pressure sensor used in this project for measuring the water pressure on the exterior of the vehicle and for determining the vehicle’s depth. As there is a linear relationship between the water pressure and the vehicle’s depth, this can be easily determined from Pascals law, <disp-formula id="j_ijssis-2021-005_eq_001"><label>(1)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>Δ</mi><mi>P</mi><mo>=</mo><mi>ρ</mi><mi>g</mi><mo stretchy="false">(</mo><mi>Δ</mi><mi>h</mi><mo stretchy="false">)</mo><mo>,</mo></math><tex-math/></alternatives></disp-formula>(1)where <italic>ρ</italic> is the water density in kg/m3, Δh is the depth of the vehicle in meters (m) and <italic>g</italic> is the gravitational acceleration (9.81 m/s<sup>2</sup>).</p><p>The water pressure sensor used in this project is an LM series low-pressure media-isolated sensor with an analogue interface. It uses three wires, power, ground and signal. The output signal range is between 0.5 and 4.5 V depending on the pressure. The relationship between the measured voltage and water pressure is linear. As this sensor has a measuring range from 0 to 15 PSI, it can be used to measure depths of up to 10 m. It is connected directly to the digital compass to calculate the depth, as shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_008">Fig. 8</a>.</p><p>A 50 cm tube with a tape measure is placed on the sensor to calibrate the sensor. The water is then fed into the tube every 2 cm and the values obtained were recorded. The error of depth measurement is less than 0.14 mm. <a ref-type="fig" href="#j_ijssis-2021-005_fig_009">Fig. 9</a> shows the acquired precision of the measurement.</p><figure id="j_ijssis-2021-005_fig_009" fig-type="figure"><h2>Figure 9:</h2><figCaption><p>Depth measurement accuracy from water pressure sensor.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_009.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_009.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=9f68299f37f7e92b51e074a7e8a09308a9f289af0bdf1aba71f0f3f2ebb69db6" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_002_s_006"><div>On-board computer</div><p>AUV is managed by an on-board processing unit, which involves fast processors and broad memories. The PC104 architecture is one of those widely used processors for embedded systems. A great feature of the architecture of the PC104 is that it has a defined form factor that enables to have several add-on boards compatible with the main processor card. The other benefits of the PC104 system are its compact size and high processing capacity. There is an Industry Standard Architecture (ISA) bus for these PC104 boards that runs through all the interconnected boards. On the ISA bus there are 104 pins. The PC104-plus comes with an ISA bus and a peripheral part interface (PCI). The PCI bus works at lower levels of voltage and is much faster than the ISA bus. In contrast to the ISA bus width of 16 bits, the PCI bus is 32 bits or 64 bits wide.</p><p>Compared to many other architectures, the advantages of a PC 104-based embedded device have contributed to the usage of this AUV design in UNSW Canberra with a PC104-plus, which comprise the core of the AUV control unit. It incorporates an Intel Atom N450 processor with 1 GB RAM, 250 GB hard drive, speed of 1.66 GHz, and it uses a compact flash disk for real-time applications as data storage.</p><p>It consists four USB ports, one Ethernet port and two serial COM ports. The serial ports are either RS-232 or RS-485, chosen by jumpers on the chip-set. <a ref-type="fig" href="#j_ijssis-2021-005_fig_010">Fig. 10</a> shows the analog and digital I/O boards and a DC-DC supply card; these are also included in the modules on the PC104 Stack.</p><figure id="j_ijssis-2021-005_fig_010" fig-type="figure"><h2>Figure 10:</h2><figCaption><p>PC104 computer system stack.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_010.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_010.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=3fe8ed7628046366cad33ab1989cb92f00b02a0e6b01e732a40d52caf2978d4b" class="mw-100"></img></figure><p>The dedicated PC104 architecture power supply board PCM-3910, is used to provide constant voltages of various amplitudes. As compared to other standard methods, this board provides reliable voltage regulation. This board has a 10–30 V input range and provides on-board 5, 12, ‒5, and ‒12 V outputs for the stack and various sensors. PC104 stack also contains an Analog-to-Digital (A/D) conversion board, DMM-32X-AT by Diamond Systems. The resolution of A/D is 16-bit with 32 analog channels.</p><p>The PC104’s real-time environment is an xPC target, a Matlab toolbox that offers the Matlab® Simulink® real-time kernel and application development environment. It instantly produces code via the Real-Time Workshop (RTW), which can be downloaded to a second computer utilizing the xPC target real-time kernel for simplification and minimization of development time and debugging. Using the Simulink® libraries available on Matlab®, the Simulink model is initially developed and then compiled using the xPC target option which creates an executable embedded code that can be downloaded to the target device.</p></sec><sec id="j_ijssis-2021-005_s_002_s_007"><div>Power distribution</div><p>The whole power distribution system is fitted with a wireless power switch that enables the hardware to be connected and disconnected from the distance. There are four batteries in the AUV’s battery system; two 12 V 20 Ah batteries and two 12 V 2.5 Ah batteries. The power specifications of on-board modules are as follows; one 12 Vdc 20 Ah battery supplies the thruster and the second one supplies the bilge pumps. A 12V 2.5 Ah is required by the PC104 and has its own power module to supply the processor and I/O modules. Another power distribution system has been designed that connects directly to the fourth battery and it provides 5 V to water pressure sensors, IMU, magnetometers, and digital compass sensors.</p></sec></sec><sec id="j_ijssis-2021-005_s_003"><div>System integration</div><p>The PC104 is the system that performs all of the key control tasks. It is then link to the IO module, which includes low-level interfaces for both sensors and actuators. <a ref-type="fig" href="#j_ijssis-2021-005_fig_011">Fig. 11</a> displays the whole system and its integration, showing all interfaces between the AUV hardware and component interfaces. The integrated system used to control the AUV maneuvering is shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_012">Fig. 12</a>. Eventually, the AUV parameter values are measured and indicated in <a ref-type="table" href="#j_ijssis-2021-005_tab_001">Table 1</a> (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_015">Hassanein et al., 2013</a>).</p><table-wrap id="j_ijssis-2021-005_tab_001" position="float"><label>Table 1.</label><caption><p>Specifications of AUV model.</p></caption><table frame="hsides"><colgroup span="1"><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/></colgroup><thead><tr><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">Mass of the AUV</td><td align="center" rowspan="1" colspan="1">10.4102</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">CG (original)</td><td align="center" rowspan="1" colspan="1">[‒10.7498‒3.49512428.206]</td><td align="center" rowspan="1" colspan="1">mm</td></tr><tr><td align="left" rowspan="1" colspan="1">Payload capacity</td><td align="center" rowspan="1" colspan="1">21.1408</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">Total length (L)</td><td align="center" rowspan="1" colspan="1">1.064</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">Mass of AUV + Payload</td><td align="center" rowspan="1" colspan="1">31.551</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">Diameter (d)</td><td align="center" rowspan="1" colspan="1">0.250</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">Ix</td><td align="center" rowspan="1" colspan="1">0.6384</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1">Speed</td><td align="center" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">m/s</td></tr><tr><td align="left" rowspan="1" colspan="1">Iy</td><td align="center" rowspan="1" colspan="1">6.4110</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1000</td><td align="center" rowspan="1" colspan="1">Kg/m<sup>3</sup></td></tr><tr><td align="left" rowspan="1" colspan="1">Iz</td><td align="center" rowspan="1" colspan="1">6.4110</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1">BC</td><td align="center" rowspan="1" colspan="1">[0 0 0]</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">CG</td><td align="center" rowspan="1" colspan="1">[0 0 0]</td><td align="center" rowspan="1" colspan="1">m</td><td align="left" rowspan="1" colspan="1">BC(original)</td><td align="center" rowspan="1" colspan="1">[0 0 21.306]</td><td align="center" rowspan="1" colspan="1">mm</td></tr></tbody></table></table-wrap><figure id="j_ijssis-2021-005_fig_011" fig-type="figure"><h2>Figure 11:</h2><figCaption><p>Complete hardware diagram of AUV.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_011.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_011.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=25dea38f644611c47f557fa1eb50e48f2652e46df8b259f4b197b15c54aeb7f3" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_012" fig-type="figure"><h2>Figure 12:</h2><figCaption><p>Complete AUV hardware.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_012.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_012.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=bdd41c1953dc5506d31d7cd27221ead84c75113f0ad34540b04e18b1cfd8c058" class="mw-100"></img></figure><p>Buoyancy would be a problem that requires a lot of consideration in order to submerge the vehicle. The power supply along with the power distribution system, actuators and other instruments, all sensors, and PC104 computer are located in the dry section of the vehicle. Initially, the entire vehicle was placed in the water, a strap test rig is wrapped around the vehicle and a hanger is then added to carry weights. This allows one to add weight until the buoyancy of the vehicle is resolved, enabling one to see how much weight is needed in the vehicle. By taking into consideration of the gravity and buoyancy centers, this weight is placed in the dry area. In addition, the nose and tail cones are equipped with an air tube in order to fine-tune the vehicle’s buoyancy at the start of each examination. The neutral vehicle buoyancy in the UNSW Canberra swimming pool is shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_013">Fig. 13</a>.</p><figure id="j_ijssis-2021-005_fig_013" fig-type="figure"><h2>Figure 13:</h2><figCaption><p>Buoyancy adjustment of the AUV.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_013.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_013.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=780e974d2b54f2008fb8e60f075013aee078f35079dc5e557ca1e196e06ad024" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_004"><div>AUV dynamic modeling</div><p>The AUVs is modeled via the analysis of statics and dynamics. The first is for the equilibrium of a body at rest or moving at constant speed and the second is the body experiencing accelerated movement (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>, <a ref-type="bibr" href="#j_ijssis-2021-005_ref_017">2013</a>). The coordinate system and specifications of the motion parameters should be given first in order to derive a 6-DOF nonlinear mathematical model of the AUV.</p><sec id="j_ijssis-2021-005_s_004_s_001"><div>Coordinate system</div><p>The position, orientation, linear velocity, and angular velocity of an underwater vehicle are defined in two frames of reference. The first frame is the <italic>X</italic><sub><italic>O</italic></sub><italic>Y</italic><sub><italic>O</italic></sub><italic>Z</italic><sub><italic>O</italic></sub> fixed frame, typically chosen to match the body’s CG and represented in relation to a gravitational reference or the Earth-set frame, E-xyz as seen in <a ref-type="fig" href="#j_ijssis-2021-005_fig_014">Fig. 14</a>. As with underwater vehicles, it is believed that the acceleration of a position on the surface of the Earth may be ignored (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_039">Hassanein et al., 2013</a>), the Earth-fixed frame is known to be inertial in this work. The position of the vehicle and its orientation in relation to the inertial reference frame, and linear and angular velocities are described to be the body reference frame. The typical evaluations for SNAME (1950) underwater vehicles as described on Nomenclature appendix (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli, 2018</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli et al., 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>, <a ref-type="bibr" href="#j_ijssis-2021-005_ref_017">2013</a>).</p><figure id="j_ijssis-2021-005_fig_014" fig-type="figure"><h2>Figure 14:</h2><figCaption><p>Body-fixed and earth-fixed coordinate system (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_015">Hassanein et al., 2013</a>).</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_014.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_014.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b34171e52b483aeb3e2ec2257c9ee9b41fce039e61d16e26c9a3f1758165ba20" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_004_s_002"><div>Kinematics</div><p>The position and orientation of the vehicle, as mentioned earlier, is defined with regard to the inertial reference system in which <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>ℜ</mi></mrow><mn>3</mn></msup></math><tex-math/></alternatives></inline-formula> determines the position of the body (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli, 2018</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli et al., 2008</a>), <disp-formula id="j_ijssis-2021-005_eq_002"><label>(2)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(2)where the vector <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mi>η</mi><mo>̇</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math><tex-math/></alternatives></inline-formula> is the corresponding derivative time and is also connected to the inertial frame as (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli, 2018</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli et al., 2008</a>), <disp-formula id="j_ijssis-2021-005_eq_003"><label>(3)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>ν</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mi>u</mi></mtd></mtr><mtr><mtd><mi>v</mi></mtd></mtr><mtr><mtd><mi>w</mi></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(3)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>ν</mi></mrow><mrow><mn>1</mn></mrow></msub></math><tex-math/></alternatives></inline-formula> is the linear velocity of the body-fixed frame of origin, which can be defined as the body-fixed frame with regard to the inertial reference frame (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli, 2018</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli et al., 2008</a>),</p><p><disp-formula id="j_ijssis-2021-005_eq_004"><label>(4)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_004.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>I</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>−</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(4)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>I</mi></mrow><mrow><mi>B</mi></mrow></msubsup></math><tex-math/></alternatives></inline-formula> is the rotational matrix that describes the inertial frame transformation and can be described as the fixed frame, <disp-formula id="j_ijssis-2021-005_eq_005"><label>(5)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_005.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mrow><mi>R</mi></mrow><mrow><mi>I</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>ψ</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>φ</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(5)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_006.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>c</mi></mrow><mrow><mi>θ</mi></mrow></msub><mspace width=".25em"/><mi mathvariant="normal">and</mi><mspace width=".25em"/><msub><mrow><mi>s</mi></mrow><mrow><mi>θ</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> are short notations for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>cos</mi><mspace width=".25em"/><mi>θ</mi><mspace width=".25em"/><mi mathvariant="normal">and</mi><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>θ</mi></math><tex-math/></alternatives></inline-formula>, respectively. Similarly, to define the underwater vehicle orientation, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_008.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mspace width=".25em"/><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>ℜ</mi></mrow><mn>3</mn></msup></math><tex-math/></alternatives></inline-formula> is described by the body’s co-ordinates in the inertial frame (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli et al., 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>), <disp-formula id="j_ijssis-2021-005_eq_006"><label>(6)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mi>φ</mi></mtd></mtr><mtr><mtd><mi>θ</mi></mtd></mtr><mtr><mtd><mi>ψ</mi></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(6)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_009.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub></math><tex-math/></alternatives></inline-formula> is the body Euler-angle coordinates in the inertial frame, commonly called roll, pitch and yaw, respectively, <disp-formula id="j_ijssis-2021-005_eq_007"><label>(7)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_007.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mi>q</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>,</mo></math><tex-math/></alternatives></disp-formula>(7)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_010.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub></math><tex-math/></alternatives></inline-formula> is the angular velocity vector of the origin of the body-fixed frame in order to attain that, with respect to the inertial frame of reference described in the body-fixed frame, <disp-formula id="j_ijssis-2021-005_eq_008"><label>(8)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_008.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mover accent="true"><mi>η</mi><mo>̇</mo></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></math><tex-math/></alternatives></disp-formula>(8)where the vector describing the angular velocity of the body-fixed frame relative to the inertial frame is <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_011.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mspace width=".25em"/><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub></math><tex-math/></alternatives></inline-formula>. It must be observed that the <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_012.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mover accent="true"><mi>η</mi><mo>̇</mo></mover></mrow><mrow><mn>2</mn></mrow></msub></math><tex-math/></alternatives></inline-formula> vector has no physical significance and its relationship to the body-fixed frame is through the proper Jacobian matrix <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_013.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub></math><tex-math/></alternatives></inline-formula>, and the Jacobian matrix <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_014.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>ℜ</mi></mrow><mrow><mn>3</mn><mi>Χ</mi><mn>3</mn></mrow></msup></math><tex-math/></alternatives></inline-formula>, is within the Euler angles expressed as, <disp-formula id="j_ijssis-2021-005_eq_009"><label>(9)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_009.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mi>cos</mi><mspace width=".25em"/><mi>θ</mi></mrow></mtd><mtd><mrow><mi>cos</mi><mspace width=".25em"/><mi>θ</mi><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>φ</mi></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>φ</mi></mrow></mtd><mtd><mrow><mi>cos</mi><mspace width=".25em"/><mi>θ</mi><mspace width=".25em"/><mi>cos</mi><mspace width=".25em"/><mi>φ</mi></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mo>.</mo></math><tex-math/></alternatives></disp-formula></p></sec><sec id="j_ijssis-2021-005_s_004_s_003"><div>Equations of motion</div><p>Equations of Motion for a rigid body can be obtained from Newton’s second law and a rigid body’s dynamic behavior is generally written in the notation of SNAME (1950) as described on Nomenclature appendix (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli, 2018</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli et al., 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>, <a ref-type="bibr" href="#j_ijssis-2021-005_ref_017">2013</a>). The rigid body dynamics can be written in component form as:<list id="j_ijssis-2021-005_list1" list-type="bullet"><list-item><p>For translational motion: <disp-formula id="j_ijssis-2021-005_ueq_001"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ueq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>X</mi><mo>=</mo><mi>m</mi><mo stretchy="false">[</mo><mover accent="true"><mi>u</mi><mo>̇</mo></mover><mo>‒</mo><mi>v</mi><mi>r</mi><mo>+</mo><mi>w</mi><mi>q</mi><mo>‒</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>q</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>r</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mi>q</mi><mo>‒</mo><mover accent="true"><mi>r</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mi>r</mi><mo>+</mo><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math><tex-math/></alternatives></disp-formula><disp-formula id="j_ijssis-2021-005_ueq_002"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ueq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>Y</mi><mo>=</mo><mi>m</mi><mo stretchy="false">[</mo><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mo>−</mo><mi>w</mi><mi>p</mi><mo>+</mo><mi>u</mi><mi>r</mi><mo>‒</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>r</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mi>r</mi><mo>‒</mo><mover accent="true"><mi>p</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mi>p</mi><mo>+</mo><mover accent="true"><mi>r</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math><tex-math/></alternatives></disp-formula><disp-formula id="j_ijssis-2021-005_ueq_003"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ueq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>Z</mi><mo>=</mo><mi>m</mi><mo stretchy="false">[</mo><mover accent="true"><mi>w</mi><mo>̇</mo></mover><mo>‒</mo><mi>u</mi><mi>q</mi><mo>+</mo><mi>v</mi><mi>p</mi><mo>‒</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mi>p</mi><mo>‒</mo><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mi>q</mi><mo>+</mo><mover accent="true"><mi>p</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math><tex-math/></alternatives></disp-formula></p></list-item><list-item><p>For the rotational motion</p></list-item></list><disp-formula id="j_ijssis-2021-005_eq_010"><label>(10)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_010.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable><mtr><mtd><maligngroup/><mspace width=".25em"/><mspace width=".25em"/><mi>K</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi></mrow></msub><mover accent="true"><mi>p</mi><mo>̇</mo></mover><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">)</mo><mi>q</mi><mi>r</mi><mo>‒</mo><mo stretchy="false">(</mo><mover accent="true"><mi>r</mi><mo>̇</mo></mover><mo>+</mo><mi>p</mi><mi>q</mi><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi>r</mi></mrow><mn>2</mn></msup><mo>‒</mo><msup><mrow><mi>q</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mi>p</mi><mi>r</mi><mo>‒</mo><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><maligngroup/><mo>+</mo><mi>m</mi><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>w</mi><mo>̇</mo></mover><mo>‒</mo><mi>u</mi><mi>q</mi><mo>+</mo><mi>v</mi><mi>p</mi><mo stretchy="false">)</mo><mo>‒</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mo>‒</mo><mi>w</mi><mi>p</mi><mo>+</mo><mi>u</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><maligngroup/><mi>M</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi></mrow></msub><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mo stretchy="false">)</mo><mi>r</mi><mi>p</mi><mo>‒</mo><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo>+</mo><mi>q</mi><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo>‒</mo><msup><mrow><mi>r</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mi>q</mi><mi>p</mi><mo>‒</mo><mover accent="true"><mi>r</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub></mtd></mtr><mtr><mtd><maligngroup/><mo>+</mo><mi>m</mi><mo stretchy="false">[</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>u</mi><mo>̇</mo></mover><mo>‒</mo><mi>v</mi><mi>r</mi><mo>+</mo><mi>w</mi><mi>q</mi><mo stretchy="false">)</mo><mo>‒</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>w</mi><mo>̇</mo></mover><mo>‒</mo><mi>u</mi><mi>q</mi><mo>+</mo><mi>v</mi><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><maligngroup/><mi>N</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mover accent="true"><mi>r</mi><mo>̇</mo></mover><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi></mrow></msub><mo stretchy="false">)</mo><mi>p</mi><mi>q</mi><mo>‒</mo><mo stretchy="false">(</mo><mover accent="true"><mi>q</mi><mo>̇</mo></mover><mo>+</mo><mi>r</mi><mi>p</mi><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi>q</mi></mrow><mn>2</mn></msup><mo>‒</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mi>r</mi><mi>q</mi><mo>‒</mo><mover accent="true"><mi>p</mi><mo>̇</mo></mover><mo stretchy="false">)</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><maligngroup/><mspace width=".25em"/><mo>+</mo><mi>m</mi><mo stretchy="false">[</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mo>−</mo><mi>w</mi><mi>p</mi><mo>+</mo><mi>u</mi><mi>r</mi><mo stretchy="false">)</mo><mo>‒</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>u</mi><mo>̇</mo></mover><mo>−</mo><mi>v</mi><mi>r</mi><mo>+</mo><mi>w</mi><mi>q</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></math><tex-math/></alternatives></disp-formula></p><p>The above equations represented in a compact form as, <disp-formula id="j_ijssis-2021-005_eq_011"><label>(11)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_011.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mover accent="true"><mi>ν</mi><mo>̇</mo></mover><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><mi>ν</mi><mo>=</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub></math><tex-math/></alternatives></disp-formula></p><p>where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_015.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> is the rigid-body inertia matrix and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_016.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> the rigid-body Coriolis and Centripetal matrix.</p><p>Property 3.1 The parameterization of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_017.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> is unique and satisfies, <disp-formula id="j_ijssis-2021-005_eq_012"><label>(12)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_012.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msubsup><mo>></mo><mn>0</mn><mo>;</mo><mspace width=".25em"/><msub><mrow><mover accent="true"><mi>M</mi><mo>̇</mo></mover></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo>=</mo><mn>0</mn></math><tex-math/></alternatives></disp-formula><disp-formula id="j_ijssis-2021-005_eq_013"><label>(13)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_013.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mi>m</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mi>m</mi><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>m</mi></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mi>m</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>m</mi></mtd><mtd><mrow><mi>m</mi><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><mi>m</mi><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>x</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>x</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mrow><mi>m</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi><mi>x</mi></mrow></msub></mrow></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi><mi>y</mi></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow></math><tex-math/></alternatives></disp-formula></p><p>Property 3.2, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_018.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> can always be parameterized such that it is skew-symmetrical, that is, <disp-formula id="j_ijssis-2021-005_eq_014"><label>(14)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_014.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mo>‒</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width=".25em"/><mo>∀</mo><mspace width=".25em"/><mi>ν</mi><mo>∈</mo><msup><mrow><mi>ℜ</mi></mrow><mn>6</mn></msup></math><tex-math/></alternatives></disp-formula><disp-formula id="j_ijssis-2021-005_eq_015"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_015.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>p</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>p</mi><mo>‒</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>‒</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mrow></math><tex-math/></alternatives></disp-formula><disp-formula id="j_ijssis-2021-005_eq_015a"><label>(15)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_015a.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mo>‒</mo><mi>m</mi><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd><mtd><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi>y</mi><mi>z</mi></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>x</mi><mi>z</mi></mrow></msub><mi>p</mi><mo>‒</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>z</mi></mrow></msub><mi>r</mi></mrow></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></math><tex-math/></alternatives></disp-formula></p><p>An underwater vehicle’s general equation of motion can be written as (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>), <disp-formula id="j_ijssis-2021-005_eq_016"><label>(16)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_016.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>R</mi><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mi>v</mi><mspace width=".25em"/><mo>+</mo><mspace width=".25em"/><msub><mrow><mi>M</mi></mrow><mrow><mi>A</mi></mrow></msub><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mo>+</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mi>v</mi><mo>+</mo><mi>D</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mi>v</mi><mspace width=".25em"/><mo>+</mo><mspace width=".25em"/><mi>g</mi><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi></math><tex-math/></alternatives></disp-formula>(16)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_019.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>τ</mi></math><tex-math/></alternatives></inline-formula> vector of control inputs. The terminology of the rigid-body describe the rigid body’s equation of motion in an empty space. Nevertheless, hydrodynamic theories have been applied to the equation because ships and underwater vehicles require the presence of accelerations caused by fluid to be taken in to account, whereas hydrostatic concepts reflect the gravitational force and buoyancy that exist while a rigid body is completely or partially immersed in a fluid. The added mass and inertia and damping effects compose the hydrodynamic terms.</p><p>The added inertia of the fluid around the body is accelerated by the motion of the body which must be included in the equations of motion when a rigid body is submerged which travels in a fluid. Fossen has stated that, as the air density is significantly lower than that of a moving mechanical system, it is possible to ignore the impact of the additional mass and inertia in industrial robotics (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>). In the case of an underwater vehicle application, the water and vehicle densities are comparable; for example, at 0<sup>o</sup>C, the density of fresh water is 1,002.68 and 1,028.48 kg/m<sup>3</sup> of seawater with 3.5% salinity. The fluid exerts a reaction force which is equal in the magnitude and opposite direction as a moving body accelerates the fluid around.</p><p>This is the extra mass contribution consisting of the addition of mass inertia and the Coriolis and Centripetal matrices, <italic>M</italic><sub><italic>A</italic></sub> and <italic>C</italic><sub><italic>A</italic></sub> matrices, respectively. As stated in <a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen (1994)</a>, <a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli et al. (2008)</a>, and <a ref-type="bibr" href="#j_ijssis-2021-005_ref_003">Antonelli (2018</a>) owing to the body’s XZ-plane symmetry, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_020.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>A</mi></mrow></msub></math><tex-math/></alternatives></inline-formula> can be generalized as, <disp-formula id="j_ijssis-2021-005_eq_017"><label>(17)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_017.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>M</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>u</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>p</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mover accent="true"><mi>u</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>K</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>K</mi></mrow><mrow><mover accent="true"><mi>p</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>K</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>M</mi></mrow><mrow><mover accent="true"><mi>u</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>M</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>M</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>N</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>N</mi></mrow><mrow><mover accent="true"><mi>p</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>N</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow></math><tex-math/></alternatives></disp-formula></p><p>For example, because of the acceleration ν·, the hydrodynamic added mass force along the y-axis is expressed as, <disp-formula id="j_ijssis-2021-005_eq_018"><label>(18)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_018.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>Y</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub><mover accent="true"><mi>v</mi><mo>̇</mo></mover><mspace width=".25em"/><mi mathvariant="normal">where</mi><mspace width=".25em"/><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub><mo>≈</mo><mfrac><mrow><mi>∂</mi><mi>Y</mi></mrow><mrow><mi>∂</mi><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></mfrac></math><tex-math/></alternatives></disp-formula></p><p>Likewise, the matrix of Coriolis and Centripetal <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_021.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>C</mi></mrow><mrow><mi>A</mi></mrow></msub></math><tex-math/></alternatives></inline-formula>, can be extracted as, <disp-formula id="j_ijssis-2021-005_eq_019"><label>(19)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_019.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>C</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mo>‒</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mtd><mtd><mrow><mi>b</mi><mmultiscripts><mn>1</mn><none/><none/><mprescripts/><none/></mmultiscripts></mrow></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow></math><tex-math/></alternatives></disp-formula></p><p>Where, <disp-formula id="j_ijssis-2021-005_ueq_004"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ueq_004.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>u</mi><mo>̇</mo></mover></mrow></msub><mi>u</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub><mi>v</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub><mi>w</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>p</mi><mo>̇</mo></mover></mrow></msub><mi>p</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub><mi>q</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub><mi>r</mi><mspace width=".25em"/></math><tex-math/></alternatives></disp-formula></p><p>The damping of an underwater vessel traveling at high speed in 6-DOFs can usually be highly nonlinear and coupled. Even so, a conservative calculation that might be considered because of the vehicle symmetry is that terms above the second order are insignificant, indicating a diagonal structure with only linear and quadratic damping constraints (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli et al., 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>) as, <disp-formula id="j_ijssis-2021-005_eq_020"><label>(20)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_020.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi><mo>|</mo><mi>u</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>u</mi><mo stretchy="true">|</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>v</mi><mo>|</mo><mi>v</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>v</mi><mo stretchy="true">|</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>w</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>w</mi><mo>|</mo><mi>w</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>w</mi><mo stretchy="true">|</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>|</mo><mi>p</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>P</mi><mo stretchy="true">|</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>q</mi><mo>|</mo><mi>q</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>q</mi><mo stretchy="true">|</mo></mrow></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>|</mo><mi>r</mi><mo>|</mo></mrow></msub><mrow><mo stretchy="true">|</mo><mi>r</mi><mo stretchy="true">|</mo></mrow></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow></math><tex-math/></alternatives></disp-formula></p><p>The gravitational and buoyant forces are considered restoring forces in hydrodynamics terminology, which behave on the vehicle’s CG and have components around the axes of the body. The <italic>z</italic>-axis is considered to be positive downwards whereas the restore force and moment vector are defined as (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_004">Antonelli et al., 2008</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>), <disp-formula id="j_ijssis-2021-005_eq_021"><label>(21)</label><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_eq_021.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><msub><mrow><mi>g</mi></mrow><mrow><mi>R</mi><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo></mrow></mrow></msub><mo>=</mo><mspace width=".25em"/><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>W</mi><mo>‒</mo><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mrow><mo stretchy="false">(</mo><mrow><mi>W</mi><mo>‒</mo><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>s</mi><mi>i</mi><mi>n</mi><mi>∅</mi></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mrow><mo stretchy="false">(</mo><mrow><mi>W</mi><mo>‒</mo><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>c</mi><mi>o</mi><mi>s</mi><mi>∅</mi></mrow></mtd></mtr><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mo>‒</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>cos</mi><mspace width=".25em"/><mi>∅</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>∅</mi><mspace width=".25em"/></mrow></mtd></mtr><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi><mo>+</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>cos</mi><mspace width=".25em"/><mi>∅</mi></mrow></mtd></mtr><mtr><mtd><mrow><mo>‒</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mspace width=".25em"/><mi>sin</mi><mspace width=".25em"/><mi>∅</mi><mo>‒</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>G</mi></mrow></msub><mi>W</mi><mo>‒</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>B</mi></mrow></msub><mi>B</mi></mrow><mo stretchy="false">)</mo></mrow><mi>s</mi><mi>i</mi><mi>n</mi><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow></math><tex-math/></alternatives></disp-formula>(21)where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_022.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>W</mi><mo>=</mo><mi>m</mi><mi>g</mi></math><tex-math/></alternatives></inline-formula> is the submerged weight of the body, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_023.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>=</mo><mi>ρ</mi><mo>∇</mo><mi>g</mi></math><tex-math/></alternatives></inline-formula> the buoyancy, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_024.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math><tex-math/></alternatives></inline-formula> the fluid density, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_025.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∇</mo></math><tex-math/></alternatives></inline-formula> the volume of the body and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_026.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mn>9.81</mn></mrow></mtd></mtr></mtable></mrow><mo stretchy="false">]</mo></mrow><mi>T</mi></msup></math><tex-math/></alternatives></inline-formula> the acceleration of gravity.</p></sec><sec id="j_ijssis-2021-005_s_004_s_004"><div>Simulink model</div><p><a ref-type="fig" href="#j_ijssis-2021-005_fig_015">Fig. 15</a> depicts the UNSW Canberra AUV mathematical model designed as a Simulink/Matlab model for the behavior of AUV (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_010">Fossen, 1994</a>; <a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>). To understand and analyses the device dynamics of an AUV that involve hydrodynamic uncertainty, highly nonlinear, time varying and coupled, a simulink/Matlab analysis is used to analyses the UNSW Canberra model (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_014">Hassanein et al., 2011</a>). The AUV hydrodynamic coefficients are calculated using the Computational Fluid Dynamics (CFD) approach in the Autonomous System Lab at UNSW Canberra, as tabulated in <a ref-type="table" href="#j_ijssis-2021-005_tab_002">Table 2</a>, to obtain the nonlinear mathematical model in different conditions (<a ref-type="bibr" href="#j_ijssis-2021-005_ref_025">Osama et al., 2016</a>).</p><table-wrap id="j_ijssis-2021-005_tab_002" position="float"><label>Table 2.</label><caption><p>Hydrodynamic coefficient of UNSW Canberra AUV model.</p></caption><table frame="hsides"><colgroup span="1"><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/></colgroup><thead><tr><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th><th align="center" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_027.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi><mo>|</mo><mi>u</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒7.365</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_028.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.12</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_029.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>v</mi><mo>|</mo><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.737</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_030.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.31</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_031.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi><mo>|</mo><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.737</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_032.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.31</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_033.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>q</mi><mo>|</mo><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.065</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_034.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.51</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_035.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>r</mi><mo>|</mo><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.065</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_036.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.51</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_037.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mi>v</mi><mo>|</mo><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒112.2</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_038.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒62.45</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_039.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mi>r</mi><mo>|</mo><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.250</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_040.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.12</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_041.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mi>w</mi><mo>|</mo><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒112.2</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_042.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒62.45</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_043.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi><mo>|</mo><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.250</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_044.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.12</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_045.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>|</mo><mi>p</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.5975</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_046.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.3125</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_047.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>w</mi><mo>|</mo><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">2.244</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_048.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>w</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.2</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_049.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>q</mi><mo>|</mo><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒119.5</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_050.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mi>q</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒59.75</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_051.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mi>v</mi><mo>|</mo><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.244</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_052.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mi>v</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.2</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_053.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>|</mo><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒59.75</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_054.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mi>r</mi></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒31.25</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_055.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>X</mi></mrow><mrow><mover accent="true"><mi>u</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.17</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_056.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>K</mi></mrow><mrow><mover accent="true"><mi>p</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_057.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒34.834</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_058.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_059.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Y</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_060.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>M</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.659</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_061.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mover accent="true"><mi>w</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒34.834</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_062.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mover accent="true"><mi>v</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_063.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>Z</mi></mrow><mrow><mover accent="true"><mi>q</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_064.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>N</mi></mrow><mrow><mover accent="true"><mi>r</mi><mo>̇</mo></mover></mrow></msub></math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.659</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr></tbody></table></table-wrap><figure id="j_ijssis-2021-005_fig_015" fig-type="figure"><h2>Figure 15:</h2><figCaption><p>Simulink block diagram.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_015.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_015.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=5374724cd224a7fba2168db3d88dde25445715be96e1de21864d23e57e465055" class="mw-100"></img></figure><p>The step sources are to be configured as temporary inputs for the three input variables in order to determine their control on the different position outputs (these are <italic>x</italic>, <italic>y</italic>, and <italic>z</italic> in meters, and the roll, the pitch, and the latter in degrees).</p><p>As seen in <a ref-type="fig" href="#j_ijssis-2021-005_fig_016">Figs. 16</a> and <a ref-type="fig" href="#j_ijssis-2021-005_fig_017">17</a> the surge input is a step from 0 to 10 at time = 30 seconds and no input for other variables, pitch and yaw. <a ref-type="fig" href="#j_ijssis-2021-005_fig_018">Fig. 18</a> reveals that the only hydrodynamic influence relating to the x motion is in the x direction, it can also be seen that there is no hydrodynamic coupling between x and y, yaw, z and pitching motions. Since the AUV is a cylindrical hull vehicle, the rolling motion has no hydrodynamic impact.</p><figure id="j_ijssis-2021-005_fig_016" fig-type="figure"><h2>Figure 16:</h2><figCaption><p>Surge input Force.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_016.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_016.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=4724782f2e0933b4bc3bf52e662f178bc7f6171efdca181fa3ba0394249c29ce" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_017" fig-type="figure"><h2>Figure 17:</h2><figCaption><p>Pitch and yaw input Force.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_017.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_017.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b8afab3b3565e4bb6b5a869a8b6e2775033896e1c7443a539d259ddf411e2266" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_018" fig-type="figure"><h2>Figure 18:</h2><figCaption><p>Surge input responses.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_018.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_018.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=4ce9e5ca63d8c9d254580658f65b670a52654848bad7aea542a69f70b658aea4" class="mw-100"></img></figure><p>A step input was then entered into the pitch motion as shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_017">Fig. 17</a>, and the surge was set at its usual speed state (1 m/sec) so that the effects on the angles could actually be observed. The findings of the simulation in <a ref-type="fig" href="#j_ijssis-2021-005_fig_019">Figs. 19</a> and <a ref-type="fig" href="#j_ijssis-2021-005_fig_020">20</a> demonstrate that hydrodynamic coupling occur between the pitch angle and the motion in x and z directions. There is no change experienced in y and yaw, this is a result of the vehicle symmetry.</p><figure id="j_ijssis-2021-005_fig_019" fig-type="figure"><h2>Figure 19:</h2><figCaption><p>Pitch input response in AUV translational and orientational movement.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_019.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_019.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=90b2791daf50a375f056401bed8ad3460ce1a177ea917331adc77769f03c17f0" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_020" fig-type="figure"><h2>Figure 20:</h2><figCaption><p>Pitch input response in AUV velocities.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_020.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_020.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=1c68c33e25fa903529a94a5162abb5d1869d4743eb6cd67d8a138f8792ac0611" class="mw-100"></img></figure><p>In the yaw movement, the step input as in <a ref-type="fig" href="#j_ijssis-2021-005_fig_017">Fig. 17</a> was applied and the surge remained in its usual working condition. The simulation results shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_021">Figs. 21</a> and <a ref-type="fig" href="#j_ijssis-2021-005_fig_022">22</a> prove that a hydrodynamic coupling occurs between the yaw angle and the motion in the direction x and y. y is calculated as positive value in the right direction. There is also no change in z value and pitch angle. Such initial findings are used for fuzzy and Hybrid Neural Fuzzy Network (HNFN) techniques in the identification approach.</p><figure id="j_ijssis-2021-005_fig_021" fig-type="figure"><h2>Figure 21:</h2><figCaption><p>Yaw input response in translational and orientational movement.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_021.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_021.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=17999&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b972c55c014fbbf08feb5283c0742a7b017433f959196d3ee92d1e771938296f" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_022" fig-type="figure"><h2>Figure 22:</h2><figCaption><p>Yaw input response in AUV velocities.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_022.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_022.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=7374dcd637e5205f0e77afb2af53ee16d41f51aed29409117308145ce560a734" class="mw-100"></img></figure></sec></sec><sec id="j_ijssis-2021-005_s_005"><div>Hardware in loop simulation (HIL) structure</div><p>In order to test the identification and control algorithms before and after AUV testing, a real-time simulation is established. A basic test-bed built in-house with the serial port data communication capability is being used for validation. The autopilot systems of the UNSW Canberra AUV platform is made up of the PC104 microcontroller with separate I/O boards. There are block libraries in the xPC-Target toolbox that support certain PC104 peripheral cards with drivers. For collecting data from the encoder, the target block DM6814 xPC is used and the board’s base address is set to x320. Along with Matlab, a Null modem serial RS-232 cable communicates between the PC104 device and computer. This medium supports floating point numbers, integers and decimal values, Also data is transmitted and received at a constant baud rate. <a ref-type="fig" href="#j_ijssis-2021-005_fig_023">Fig. 23</a> shows an overview of the HIL simulation where the PC104 transmits and receives data from the Simulink model in real time, using RS-232 serial communication.</p><figure id="j_ijssis-2021-005_fig_023" fig-type="figure"><h2>Figure 23:</h2><figCaption><p>Hardware In loop (HIL) simulation.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_023.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_023.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=f297f2d9db3d0d0cbb1d601f13af02312bbd534bf474508ccd87b1f6e9422dbc" class="mw-100"></img></figure></sec><sec id="j_ijssis-2021-005_s_006"><div>Experimental results</div><p>The real-tests have been explicitly undertaken to model the dynamics system and to excite the related dynamics. Many experiments have been conducted at UNSW Canberra to collect a range of data under various operating conditions. All such data are used by the identification algorithm to generate, tuning and identify the AUV dynamics model. The swimming pool at UNSW and the shallow water branch of Molonglo River in Canberra are chosen as test environments for the AUV to validate the effectiveness and the robustness of the algorithm. For coupled dynamics, a suitable manoeuvre is conducted such that all the control inputs are used and the coupled dynamics are excited over a period of time. The electronics, instruments, sensory system and on-board processing unit that used in the AUV are described previously in platform design section. <a ref-type="fig" href="#j_ijssis-2021-005_fig_024">Fig. 24</a> demonstrates the real-time manoeuvring of the AUV in a separate environments during the data collection process.</p><figure id="j_ijssis-2021-005_fig_024" fig-type="figure"><h2>Figure 24:</h2><figCaption><p>UNSW Canberra AUV during experimental teat data collection.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_024.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_024.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=8130e7ddd904d4f77dd975e12136d4d567e7ae3dff72f37e044baf0a32b3517d" class="mw-100"></img></figure><p>The ultimate goal of any controller design is to prove its validity in real conditions with unknown disturbances and the associated difficulties with real time implementation. This is a particularly challenging task for a complex, nonlinear and time-varying system like an AUV. Real-time tests are necessary to test the ability of the controller in the presence of unknown weather conditions and noise in the system. Due to the wind conditions and a strong water flow during the real time tests, there were fluctuations in the controlled value. To prevent vehicle fluctuations on the water surface, there had been a ± 2.5° dead-band across the pitch controller set point.</p><p>In the present work, an indirect adaptive control system is applied to the AUV control problem where the controller parameters are adjusted based not only on the error between the reference input and the process output but also on the process sensitivity which can be approximately derived from the identification model of the process, The control system is based on the fuzzy system and HNFN (Hybrid Neural Fuzzy Network) techniques (Osama Hassanein et al., 2016).</p><p><a ref-type="fig" href="#j_ijssis-2021-005_fig_025">Figs 25</a> and <a ref-type="fig" href="#j_ijssis-2021-005_fig_026">26</a> demonstrate the related behavior of pumps for pitch and yaw motions. The required pitch angle is 0 for the pitch controller as shown in <a ref-type="fig" href="#j_ijssis-2021-005_fig_025">Fig. 25</a>. The controller controls all the pumps in a pulse manner utilizing the PWM that is embedded into the controller programming. When the AUV began moving, it was going down, and then the controller instructed the UP (Up-direction) pump to drive the vehicle in the upward direction. However, the vehicle went behind the controller set-point; consequently, the DN (down-direction) pump was commanded by the controller to force the vehicle down.</p><figure id="j_ijssis-2021-005_fig_025" fig-type="figure"><h2>Figure 25:</h2><figCaption><p>1st set of AUV control based fuzzy experimental results: coupled dynamics.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_025.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_025.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=6a81f0aab10ebb784f5e9b759d1596301fef1ab6a0e8038e581161415a0ac9a8" class="mw-100"></img></figure><figure id="j_ijssis-2021-005_fig_026" fig-type="figure"><h2>Figure 26:</h2><figCaption><p>2nd set of AUV control based HNFN experimental results: coupled dynamics.</p></figCaption><img xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_fig_026.jpg" src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_026.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=5b92e3ea462175f47a901fa46e26860a0c35195f284c1896f13c3092c7bbfe02" class="mw-100"></img></figure><p>In the yaw controller, the same criterion is used as it has the same pitch controller actions except with the left (LT) and right (RT) directions and it is obvious that the AVU follows the command given by the controllers. In the below figures, the blue line indicates the real AUV response and the desired trajectory is described by the red line.</p></sec><sec id="j_ijssis-2021-005_s_007"><div>Conclusion</div><p>This paper addressed development and testing of a compact, cost-effective AUV model. A prototype for underwater vehicles has been developed at the UNSW Canberra Workshop and low-cost sensors, actuators and other instruments have been mounted on it to have six degrees of freedom during the operation. The system is configured with an on board data logger and control unit to monitor and process sensor and actuator data and to apply advanced system identification and control techniques. The fundamental AUV principle is used throughout the UNSW Canberra AUV simulation model, nonlinear effects and coupling factors correlated with AUV dynamics are examined. The model based adaptive control system utilizes intelligent techniques, considered as a perfect choice for the AUV built at UNSW Canberra on the account of non-linearity and uncertainty related to modeling. To validate the identification and control for the AUV platform, multiple experiment trials lasting over a number of days were performed. The swimming pool at UNSW Canberra, the shallow water branch of the Molonglo River and Gungahlin Lake in Canberra were selected to be the test operating conditions for the AUV to evaluate the effectiveness of the proposed control algorithm in order to build confidence in employing this controller for real life tasks. The test data demonstrate the usefulness of the developed AUV platform for identification and model-based control to accomplish autonomous operation in various environments.</p></sec></div></div></div></div><div id="pane-3" class="SeriesTab_card__26XnC SeriesTab_tab-pane__3pc7y card tab-pane" role="tabpanel" aria-labelledby="tab-3"><div class="SeriesTab_card-header__1DTAS card-header d-md-none pl-0" role="tab" id="heading-3"><h4 class="mb-0"><a data-toggle="collapse" href="#collapse-3" data-parent="#content" aria-expanded="false" aria-controls="collapse-3" style="padding:24px 0">Figures et tableaux<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" class="svg-inline--fa fa-chevron-down fa-w-14 " role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M207.029 381.476L12.686 187.132c-9.373-9.373-9.373-24.569 0-33.941l22.667-22.667c9.357-9.357 24.522-9.375 33.901-.04L224 284.505l154.745-154.021c9.379-9.335 24.544-9.317 33.901.04l22.667 22.667c9.373 9.373 9.373 24.569 0 33.941L240.971 381.476c-9.373 9.372-24.569 9.372-33.942 0z"></path></svg></a></h4></div><div id="collapse-3" class="SeriesTab_seriesTabCollapse__2csiF collapse" role="tabpanel" aria-labelledby="heading-3" data-parent="#content"><div class="SeriesTab_series-tab-body__1tZ1H SeriesTab_card-body__31JEh card-body Article_figures-tables__2SC5X"><figure><h4 class="mb-4">Figure 1:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b55eb309511a6949ac10a5f04a4901707e5b3f2547b4c0ce21d0f02c21c8f2d3" alt="Block diagram of various stages of project." class="mw-100"/><figcaption class="fw-500">Block diagram of various stages of project.</figcaption></figure><figure><h4 class="mb-4">Figure 2:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_002.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=69ef9a0f85bf0848e41eeacca67e5e0c998fad696ac4fd321eeb1680601aa2ff" alt="UNSW Autonomous Underwater Vehicle platform (Hassanein et al., 2011)." class="mw-100"/><figcaption class="fw-500">UNSW Autonomous Underwater Vehicle platform (Hassanein et al., 2011).</figcaption></figure><figure><h4 class="mb-4">Figure 3:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_003.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=54ef71f66924c2920e9db0c0f2485aedf146390ee9125ac90eedc3db7ddedcc4" alt="Forward thruster with electrical motor driver." class="mw-100"/><figcaption class="fw-500">Forward thruster with electrical motor driver.</figcaption></figure><figure><h4 class="mb-4">Figure 4:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=84d97478cd47a19680c08c9e47917d98c02ea42eb6157dfcbd6166a8bdcf0e38" alt="Submersible bilge pump with the motor driver." class="mw-100"/><figcaption class="fw-500">Submersible bilge pump with the motor driver.</figcaption></figure><figure><h4 class="mb-4">Figure 5:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_005.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=f104ed817846a3e38f73e02e2ad566ba78423d51a10f57296b76ec3184fc2b97" alt="ADIS16367 unit." class="mw-100"/><figcaption class="fw-500">ADIS16367 unit.</figcaption></figure><figure><h4 class="mb-4">Figure 6:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_006.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=1fff07b55678674f81749704e819ae11b68365def5b4cb7e74a183b5e92c8755" alt="SPI interface for IMU." class="mw-100"/><figcaption class="fw-500">SPI interface for IMU.</figcaption></figure><figure><h4 class="mb-4">Figure 7:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_007.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=d9b3cfb12fc6e8803258894edfd3680f68cde78cd167c0e705d98c9b226c2614" alt="Burst read sequence (ADIS16367 Data Sheet)." class="mw-100"/><figcaption class="fw-500">Burst read sequence (ADIS16367 Data Sheet).</figcaption></figure><figure><h4 class="mb-4">Figure 8:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_008.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=adc2c7d8b701c5d780f430fa45d958005c4434df3381ee184c9ea7b4dd0ea975" alt="Wiring from pressure sensor to compass." class="mw-100"/><figcaption class="fw-500">Wiring from pressure sensor to compass.</figcaption></figure><figure><h4 class="mb-4">Figure 9:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_009.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=9f68299f37f7e92b51e074a7e8a09308a9f289af0bdf1aba71f0f3f2ebb69db6" alt="Depth measurement accuracy from water pressure sensor." class="mw-100"/><figcaption class="fw-500">Depth measurement accuracy from water pressure sensor.</figcaption></figure><figure><h4 class="mb-4">Figure 10:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_010.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=3fe8ed7628046366cad33ab1989cb92f00b02a0e6b01e732a40d52caf2978d4b" alt="PC104 computer system stack." class="mw-100"/><figcaption class="fw-500">PC104 computer system stack.</figcaption></figure><figure><h4 class="mb-4">Figure 11:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_011.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=25dea38f644611c47f557fa1eb50e48f2652e46df8b259f4b197b15c54aeb7f3" alt="Complete hardware diagram of AUV." class="mw-100"/><figcaption class="fw-500">Complete hardware diagram of AUV.</figcaption></figure><figure><h4 class="mb-4">Figure 12:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_012.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=bdd41c1953dc5506d31d7cd27221ead84c75113f0ad34540b04e18b1cfd8c058" alt="Complete AUV hardware." class="mw-100"/><figcaption class="fw-500">Complete AUV hardware.</figcaption></figure><figure><h4 class="mb-4">Figure 13:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_013.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=780e974d2b54f2008fb8e60f075013aee078f35079dc5e557ca1e196e06ad024" alt="Buoyancy adjustment of the AUV." class="mw-100"/><figcaption class="fw-500">Buoyancy adjustment of the AUV.</figcaption></figure><figure><h4 class="mb-4">Figure 14:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_014.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b34171e52b483aeb3e2ec2257c9ee9b41fce039e61d16e26c9a3f1758165ba20" alt="Body-fixed and earth-fixed coordinate system (Hassanein et al., 2013)." class="mw-100"/><figcaption class="fw-500">Body-fixed and earth-fixed coordinate system (Hassanein et al., 2013).</figcaption></figure><figure><h4 class="mb-4">Figure 15:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_015.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=5374724cd224a7fba2168db3d88dde25445715be96e1de21864d23e57e465055" alt="Simulink block diagram." class="mw-100"/><figcaption class="fw-500">Simulink block diagram.</figcaption></figure><figure><h4 class="mb-4">Figure 16:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_016.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=4724782f2e0933b4bc3bf52e662f178bc7f6171efdca181fa3ba0394249c29ce" alt="Surge input Force." class="mw-100"/><figcaption class="fw-500">Surge input Force.</figcaption></figure><figure><h4 class="mb-4">Figure 17:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_017.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b8afab3b3565e4bb6b5a869a8b6e2775033896e1c7443a539d259ddf411e2266" alt="Pitch and yaw input Force." class="mw-100"/><figcaption class="fw-500">Pitch and yaw input Force.</figcaption></figure><figure><h4 class="mb-4">Figure 18:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_018.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=4ce9e5ca63d8c9d254580658f65b670a52654848bad7aea542a69f70b658aea4" alt="Surge input responses." class="mw-100"/><figcaption class="fw-500">Surge input responses.</figcaption></figure><figure><h4 class="mb-4">Figure 19:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_019.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=90b2791daf50a375f056401bed8ad3460ce1a177ea917331adc77769f03c17f0" alt="Pitch input response in AUV translational and orientational movement." class="mw-100"/><figcaption class="fw-500">Pitch input response in AUV translational and orientational movement.</figcaption></figure><figure><h4 class="mb-4">Figure 20:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_020.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=1c68c33e25fa903529a94a5162abb5d1869d4743eb6cd67d8a138f8792ac0611" alt="Pitch input response in AUV velocities." class="mw-100"/><figcaption class="fw-500">Pitch input response in AUV velocities.</figcaption></figure><figure><h4 class="mb-4">Figure 21:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_021.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=17999&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=b972c55c014fbbf08feb5283c0742a7b017433f959196d3ee92d1e771938296f" alt="Yaw input response in translational and orientational movement." class="mw-100"/><figcaption class="fw-500">Yaw input response in translational and orientational movement.</figcaption></figure><figure><h4 class="mb-4">Figure 22:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_022.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=7374dcd637e5205f0e77afb2af53ee16d41f51aed29409117308145ce560a734" alt="Yaw input response in AUV velocities." class="mw-100"/><figcaption class="fw-500">Yaw input response in AUV velocities.</figcaption></figure><figure><h4 class="mb-4">Figure 23:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_023.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=f297f2d9db3d0d0cbb1d601f13af02312bbd534bf474508ccd87b1f6e9422dbc" alt="Hardware In loop (HIL) simulation." class="mw-100"/><figcaption class="fw-500">Hardware In loop (HIL) simulation.</figcaption></figure><figure><h4 class="mb-4">Figure 24:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_024.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=8130e7ddd904d4f77dd975e12136d4d567e7ae3dff72f37e044baf0a32b3517d" alt="UNSW Canberra AUV during experimental teat data collection." class="mw-100"/><figcaption class="fw-500">UNSW Canberra AUV during experimental teat data collection.</figcaption></figure><figure><h4 class="mb-4">Figure 25:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_025.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=6a81f0aab10ebb784f5e9b759d1596301fef1ab6a0e8038e581161415a0ac9a8" alt="1st set of AUV control based fuzzy experimental results: coupled dynamics." class="mw-100"/><figcaption class="fw-500">1st set of AUV control based fuzzy experimental results: coupled dynamics.</figcaption></figure><figure><h4 class="mb-4">Figure 26:</h4><img src="https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_026.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230202T121249Z&X-Amz-SignedHeaders=host&X-Amz-Expires=18000&X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=5b92e3ea462175f47a901fa46e26860a0c35195f284c1896f13c3092c7bbfe02" alt="2nd set of AUV control based HNFN experimental results: coupled dynamics." class="mw-100"/><figcaption class="fw-500">2nd set of AUV control based HNFN experimental results: coupled dynamics.</figcaption></figure><h4 class="mb-4 mt-4">Specifications of AUV model.</h4><table frame="hsides"><colgroup span="1"><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/></colgroup><thead><tr><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1">Mass of the AUV</td><td align="center" rowspan="1" colspan="1">10.4102</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">CG (original)</td><td align="center" rowspan="1" colspan="1">[‒10.7498‒3.49512428.206]</td><td align="center" rowspan="1" colspan="1">mm</td></tr><tr><td align="left" rowspan="1" colspan="1">Payload capacity</td><td align="center" rowspan="1" colspan="1">21.1408</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">Total length (L)</td><td align="center" rowspan="1" colspan="1">1.064</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">Mass of AUV + Payload</td><td align="center" rowspan="1" colspan="1">31.551</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="left" rowspan="1" colspan="1">Diameter (d)</td><td align="center" rowspan="1" colspan="1">0.250</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">Ix</td><td align="center" rowspan="1" colspan="1">0.6384</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1">Speed</td><td align="center" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">m/s</td></tr><tr><td align="left" rowspan="1" colspan="1">Iy</td><td align="center" rowspan="1" colspan="1">6.4110</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ρ</mml:mi></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1000</td><td align="center" rowspan="1" colspan="1">Kg/m<sup>3</sup></td></tr><tr><td align="left" rowspan="1" colspan="1">Iz</td><td align="center" rowspan="1" colspan="1">6.4110</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup></td><td align="left" rowspan="1" colspan="1">BC</td><td align="center" rowspan="1" colspan="1">[0 0 0]</td><td align="center" rowspan="1" colspan="1">m</td></tr><tr><td align="left" rowspan="1" colspan="1">CG</td><td align="center" rowspan="1" colspan="1">[0 0 0]</td><td align="center" rowspan="1" colspan="1">m</td><td align="left" rowspan="1" colspan="1">BC(original)</td><td align="center" rowspan="1" colspan="1">[0 0 21.306]</td><td align="center" rowspan="1" colspan="1">mm</td></tr></tbody></table><h4 class="mb-4 mt-4">Hydrodynamic coefficient of UNSW Canberra AUV model.</h4><table frame="hsides"><colgroup span="1"><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/><col align="left"/></colgroup><thead><tr><th align="left" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th><th align="center" rowspan="1" colspan="1">Parameter</th><th align="center" rowspan="1" colspan="1">Calculated value</th><th align="center" rowspan="1" colspan="1">Unit</th></tr></thead><tbody><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_027.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>|</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒7.365</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_028.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.12</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_029.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>|</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.737</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_030.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.31</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_031.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi><mml:mo>|</mml:mo><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.737</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_032.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.31</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_033.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>|</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.065</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_034.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.51</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_035.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>|</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.065</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_036.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.51</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_037.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>|</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒112.2</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_038.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒62.45</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_039.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>|</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.250</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_040.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.12</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_041.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi><mml:mo>|</mml:mo><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒112.2</td><td align="center" rowspan="1" colspan="1">Kg/m</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_042.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒62.45</td><td align="center" rowspan="1" colspan="1">Kg/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_043.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>|</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.250</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_044.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0.12</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_045.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>|</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.5975</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_046.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒0.3125</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_047.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi><mml:mo>|</mml:mo><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">2.244</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_048.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.2</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_049.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>|</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒119.5</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_050.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒59.75</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_051.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>|</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.244</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_052.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.2</td><td align="center" rowspan="1" colspan="1">Kg.m/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_053.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>|</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒59.75</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/rad2</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_054.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒31.25</td><td align="center" rowspan="1" colspan="1">Kg.m<sup>2</sup>/sec</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_055.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.17</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_056.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">0</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_057.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒34.834</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_058.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>w</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_059.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Y</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>r</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_060.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>q</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.659</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_061.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>w</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒34.834</td><td align="center" rowspan="1" colspan="1">Kg</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_062.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr><tr><td align="left" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_063.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>q</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒1.042</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td><td align="center" rowspan="1" colspan="1"><inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ijssis-2021-005_ieq_064.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>r</mml:mi><mml:mo>̇</mml:mo></mml:mover></mml:mrow></mml:msub></mml:math><tex-math/></alternatives></inline-formula></td><td align="center" rowspan="1" colspan="1">‒2.659</td><td align="center" rowspan="1" colspan="1">Kg.m/rad</td></tr></tbody></table></div></div></div><div id="reference" class="SeriesTab_card__26XnC SeriesTab_tab-pane__3pc7y card tab-pane" role="tabpanel" aria-labelledby="tab-4"><div class="SeriesTab_card-header__1DTAS card-header d-md-none pl-0" role="tab" id="heading-4"><h4 class="mb-0"><a data-toggle="collapse" href="#collapse-4" data-parent="#content" aria-expanded="false" 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Please read \u003cA href=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/IJSSIS/Guidelines_and_Manuscript_Preparation.pdf\"\u003eGuidelines and Manuscript Preparation\u003c/A\u003e to find out more.\u003cA href=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/IJSSIS/Open_Access_Agreement.pdf\"\u003e\u003c/A\u003e \u003c/P\u003e"},{"type":"editorial","language":"English","textformat":null,"content":"\u003cP\u003e\u003cSTRONG\u003eEditor-in-Chief\u003c/STRONG\u003e\u003cBR\u003eDr. Subhas Chandra Mukhopadhyay - Macquarie University, Sydney, Australia \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eAssociate Editors\u003c/STRONG\u003e\u003cBR\u003eEshrat E. Alahi - Chinese University of Sciences, Shenzhen, China\u003cBR\u003eCesare Alippi - Politecnico Di Milano, Italy\u003cBR\u003eFrancisco J. Arregui - Public University of Navarre, Spain\u003cBR\u003eMohsen Asadnia - Macquarie University, Australia\u003cBR\u003eTakehito Azuma - Utsunomiya University, Japan\u003cBR\u003eThierry Bosch - Institut National Polytechnique de Toulouse, France\u003cBR\u003eGraham Brooker - University of Sydney, Australia\u003cBR\u003eGoutam Chakraborty - Iwate Prefectural University, Japan\u003cBR\u003eBasabi Chakraborty - Pattern Recognition \u0026amp; Machine Learning Laboratory, Faculty of Software and Information Science, Iwate Prefectural University, Japan\u003cBR\u003eGoutam Chattopadhyay - JPL, CALTECH, USA\u003cBR\u003eSuming Chen - National Taiwan University, Taiwan\u003cBR\u003eBhaskar Choubey - Oxford University, United Kingdom\u003cBR\u003ePau-Choo Chung - National Cheng Kung University, Taiwan\u003cBR\u003eWan-Young Chung - Pukyong National University, Korea\u003cBR\u003eYeon-Ho Chung - Pukyong National University, Korea\u003cBR\u003eAndrea Cusano - University of Sannio, Italy\u003cBR\u003eKourtiche Djilali - French National Centre for Scientific Research · Institut Jean Lamour · Nanomaterials, Electronics and Living Matter, France\u003cBR\u003eAnton Fuchs - Graz University of Technology, Austria\u003cBR\u003eElena Gaura - Coventry University, United Kingdom\u003cBR\u003eBoby George - Indian Institute of Technology Madras, India\u003cBR\u003eBoris Ginzburg - Soreq NRC, Israel\u003cBR\u003eChinthaka Gooneratne - Sensors and Instrumentation, EXPEC Advanced Research Center, Saudi Arabia\u003cBR\u003eNgo Ha-Duong - Microsystems Engineering, University of Applied Sciences Berlin and Group Leader Microsensors Technologies at Fraunhofer Institute IZM, Germany\u003cBR\u003eMaki Habib - The American University in Cairo, Egypt\u003cBR\u003eMichael J. Haji-Sheikh - Northern Illinois University, USA\u003cBR\u003eTarikul Islam - Jamia Islami University, India\u003cBR\u003eWisnu Jatmiko - Universitas Indonesia, Indonesia\u003cBR\u003eK.P. Jayasundera - Massey University, New Zealand\u003cBR\u003eNemai Karmakar - Monash University, Australia\u003cBR\u003eFakhri Karray - University of Waterloo, Canada\u003cBR\u003eJohn V Kennedy - National Isotope Centre, GNS Science, New Zealand\u003cBR\u003eTan Kok Kiong - National University of Singapore, Singapore\u003cBR\u003eJürgen Kosel - King Abdullah University of Science, Saudi Arabia\u003cBR\u003eJoyanta Kumar Roy - MCKV Institute of Engineering, Liluah, India\u003cBR\u003eIvan Laktionov - SHEE 'Donetsk National Technical University', Ukraine\u003cBR\u003eAimé Lay-Ekuakille - Università del Salento, Italy\u003cBR\u003eWarren Huang-Chen Lee - National Chung Cheng University, Taiwan\u003cBR\u003eHenry Leung - University of Calgary, Canada\u003cBR\u003eElfed Lewis - University of Limerick, Ireland\u003cBR\u003eZhi Liu - Shandong University, China\u003cBR\u003eAndreas Loizos - National Technical University of Athens(NTUA), Greece\u003cBR\u003eIliana Marinova - Technical University of Sofia, Bulgaria\u003cBR\u003eLuiz de Siqueira Martins-Filho - Universidade federal do ABC-UFABC, Brazil\u003cBR\u003eIgnacio Matias - Professor, Universidad Pública de Navarra, Spain\u003cBR\u003eRosario Morello - University Mediterranea of Reggio Calabria, Italy\u003cBR\u003eMustapha Nadi - Université de Lorraine, France\u003cBR\u003eAndrew Nafalski - University of South Australia, Adelaide, Australia\u003cBR\u003eAnindya Nag - King Abdullah University of Science, Saudi Arabia\u003cBR\u003eVisconti Paolo - University of Salento, Italy\u003cBR\u003eOctavian Postolache - Instituto de Telecomunicações, Lisboa/IT, Portugal\u003cBR\u003eD. M. G. Preethichandra - Central Queensland University, Australia\u003cBR\u003eKonandur Rajanna - IISc, Bangalore, India\u003cBR\u003ePavel Ripka - Czech Technical University, Czech Republic\u003cBR\u003eAbed El Saddik - University of Ottawa, Canada\u003cBR\u003eYahaya Md. Sam - University Technology of Malaysia, Malaysia\u003cBR\u003eFrode Eika Sandnes - Oslo University, Norway\u003cBR\u003eNorbert Schwesinger - Technische Universität München, Germany\u003cBR\u003eSiddhartha Sen - IIT, Kharagpur, India\u003cBR\u003eSMN Arosha Senanayake - Brueni Darusalem University, Brunei\u003cBR\u003eLakmal Seneviratne - Kings College, London, United Kingdom\u003cBR\u003eMohamed Serry - Princeton University, USA\u003cBR\u003eNitinipun Sharma - BITS, Pilani, India\u003cBR\u003eLei Shu - Osaka University, Japan\u003cBR\u003eValery Anatolevitch Sklyarov - University of Aveiro, Portugal\u003cBR\u003eQingquan Sun - Oakland University, USA\u003cBR\u003eNagender Kumar Suryadevara - Geethanjali College of Engineering and Technology, India\u003cBR\u003eKay Chen Tan - National University of Singapore, Singapore\u003cBR\u003eGui Yun Tian - University of Newcastle upon Tyne, United Kingdom\u003cBR\u003eAthanasios Vasilakos - National Technical University of Athens, Greece\u003cBR\u003eJoseph Walsh - Institute of Technology, Tralee, Ireland\u003cBR\u003eYuhao Wang - Nanchang University, China\u003cBR\u003eYoke-San Wong - National University of Singapore, Singapore\u003cBR\u003eWanqing Wu - Shenzhen Institute of Advanced Technology, Chinese Academcy of Sciences, China\u003cBR\u003ePeter Xu - The University of Auckland, New Zealand\u003cBR\u003eSotoshi Yamada - Kanazawa University, Japan\u003cBR\u003eRuqiang Yan - Southeast University, China\u003cBR\u003eJize Yan - University of Southampton, United Kingdom\u003cBR\u003eJar-Ferr Yang - National Cheng Kung University, Taiwan\u003cBR\u003eAndrew Yeh - National Tsing Hua University, Taiwan\u003cBR\u003eWuliang Yin - The University of Manchester, United Kingdom\u003cBR\u003eMehmet Yuce - Monash University, Australia\u003cBR\u003eHeye Zhang - Sun Yat-Sen University, China\u003cBR\u003eArcady Zhukov - Basque Country University, UPV/EHU, Spain \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eHonorary Editor\u003c/STRONG\u003e\u003cBR\u003eToshio Fukuda - Nagoya University, Japan\u003cBR\u003eEmil Petriu - University of Ottawa, Canada\u003cBR\u003ePhilip Sallis - Auckland University of Technology, New Zealand\u003cBR\u003eMel Siegel - Carnegie Mellon University, USA\u003cBR\u003eShoogo Ueno - Tokyo University, Japan \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eDeputy Editor\u003c/STRONG\u003e\u003cBR\u003eRay (Yuef-Min) Huang - National Cheng-Kung University, Taiwan \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eManaging Editor\u003c/STRONG\u003e\u003cBR\u003eKevin Yen-Hung Kuo - Fusions360, Taiwan\u003cBR\u003eLisa Lightband - Massey University, New Zealand \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eContact\u003c/STRONG\u003e\u003cBR\u003e\u003cA href=\"mailto:Subhas.Mukhopadhyay@mq.edu.au\"\u003eSubhas.Mukhopadhyay@mq.edu.au\u003c/A\u003e \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003ePublisher\u003c/STRONG\u003e\u003cBR\u003eDe Gruyter Poland\u003cBR\u003eBogumiła Zuga 32A Str.\u003cBR\u003e01-811 Warsaw, Poland\u003cBR\u003eT: +48 22 701 50 15 \u003c/P\u003e"},{"type":"advantages","language":"English","textformat":null,"content":"\u003cP\u003e\u003cSTRONG\u003e\u003cEM\u003eThe International Journal on Smart Sensing and Intelligent Systems (S2IS)\u003c/EM\u003e\u003c/STRONG\u003e is an on-line journal to be available to everybody at no cost. From volume 2018 the journal is published in a \u003cU\u003econtinous format\u003c/U\u003e. \u003c/P\u003e \u003cP\u003eThe journal publishes refereed papers quarterly on any topic in the fields of Smart Sensing, Intelligent Sensing, Smart Systems, and/or Intelligent Systems. \u003c/P\u003e \u003cP\u003eResearchers working on any areas of sensing, sensors and systems which they consider as Smart and/or Intelligent may consider submitting their manuscript for possible publication in the journal. The submitted papers will go through a double-blind peer review process and authors will be notified of the outcome in the shortest possible time. \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eAims and Scope\u003c/STRONG\u003e \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eInternational Journal on Smart Sensing and Intelligent Systems (S2IS)\u003c/STRONG\u003e is a rapid and high-quality international forum wherein academics, researchers and practitioners may publish their high-quality, original, and state-of-the-art papers describing theoretical aspects, system architectures, analysis and design techniques, and implementation experiences in intelligent sensing technologies. The journal publishes articles reporting substantive results on a wide range of smart sensing approaches applied to variety of domain problems, including but not limited to: \u003c/P\u003e \u003cUL\u003e \u003cLI\u003eAmbient Intelligence and Smart Environment \u003c/LI\u003e \u003cLI\u003eAnalysis, Evaluation, and Test of Smart Sensors \u003c/LI\u003e \u003cLI\u003eIntelligent Management of Sensors \u003c/LI\u003e \u003cLI\u003eFundamentals of Smart Sensing Principles and Mechanisms \u003c/LI\u003e \u003cLI\u003eMaterials and its Applications for Smart Sensors \u003c/LI\u003e \u003cLI\u003eSmart Sensing Applications, Hardware, Software, Systems, and Technologies \u003c/LI\u003e \u003cLI\u003eSmart Sensors in Multidisciplinary Domains and Problems \u003c/LI\u003e \u003cLI\u003eSmart Sensors in Science and Engineering \u003c/LI\u003e \u003cLI\u003eSmart Sensors in Social Science and Humanity \u003c/LI\u003e\u003c/UL\u003e \u003cP\u003e\u003cSTRONG\u003eRejection rate\u003c/STRONG\u003e: 50% \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eOpen Access Policy\u003c/STRONG\u003e \u003c/P\u003e \u003cP\u003eThis journal provides immediate open access to its content under the \u003cA href=\"https://creativecommons.org/licenses/by-nc-nd/4.0/\"\u003eCreative Commons CC BY-NC-ND 4.0 license\u003c/A\u003e on the principle that making research freely available to the public supports a greater global exchange of knowledge. Under the \u003cA href=\"https://creativecommons.org/licenses/by-nc-nd/4.0/\"\u003eCC BY-NC-ND 4.0 license\u003c/A\u003e users are free to share the work (copy and redistribute the material in any medium or format), if the contribution is properly attributed and used for non-commercial purposes. The material published in the journal may not be altered or build upon. \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eABOUT SOCIETY\u003c/STRONG\u003e \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eInternational Journal on Smart Sensing and Intelligent Systems\u003c/STRONG\u003e was created and is lead by \u003cSTRONG\u003e\u003cEM\u003eProfessor Subhas Chandra Mukhopadhyay\u003c/EM\u003e\u003c/STRONG\u003e. \u003c/P\u003e \u003cP\u003eProfessor Subhas Chandra Mukhopadhyay is Professor f Mechanical and Electronics Engineering affiliated with FIEEE (USA), FIEE (UK), FIETE (India), Distinguished Lecturer, IEEE Sensors Council. \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003eArea of research interest:\u003c/STRONG\u003e \u003c/P\u003e \u003cUL\u003e \u003cLI\u003eSmart Sensors \u003c/LI\u003e \u003cLI\u003eSensing Technology \u003c/LI\u003e \u003cLI\u003eWireless Sensor Networks \u003c/LI\u003e \u003cLI\u003eInternet of Things \u003c/LI\u003e \u003cLI\u003ePower Electronics \u003c/LI\u003e \u003cLI\u003eMotor Drives \u003c/LI\u003e \u003cLI\u003eMagnetic Bearing \u003c/LI\u003e \u003cLI\u003eElectrical and Electronics Engineering \u003c/LI\u003e \u003cLI\u003eMechatronics \u003c/LI\u003e\u003c/UL\u003e \u003cP\u003e\u003cSTRONG\u003eArchiving\u003c/STRONG\u003e \u003c/P\u003e \u003cP\u003eSciendo archives the contents of this journal in \u003cA href=\"https://www.portico.org/\"\u003ePortico\u003c/A\u003e - digital long-term preservation service of scholarly books, journals and collections. \u003c/P\u003e \u003cP\u003e\u003cSTRONG\u003ePlagiarism Policy\u003c/STRONG\u003e \u003c/P\u003e \u003cP\u003eThe editorial board is participating in a growing community of \u003cA href=\"https://www.crossref.org/services/similarity-check/\"\u003eSimilarity Check System's\u003c/A\u003e users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.\u003c/P\u003e"}]}],"metrics":"","pricing":null,"publicationFrequency":{"frequency":"1","period":"YEAR"},"permissions":null,"contributors":"","serial":null,"publishMonth":"1","publishYear":"2021","tableCount":null,"figureCount":null,"refCount":null,"keywords":[],"figures":null,"tables":null,"planPubDates":[],"epubLink":null,"pdfLink":null,"coverImage":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/cover-image.jpg","coverImageOriginal":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/cover-image-original.jpg","pdfFiles":[],"parentObjectId":"62242a2c0d198124537c32f6","isParentConference":false,"relatedTitles":null,"forAuthors":null,"nextPackageId":"62242a390d198124537c3300","prevPackageId":"62249acd0d198124537c4567","parentName":"Volume 14 (2021): Issue 1 (January 2021)","grandParentId":"61df788b1792e62a88ea438a","grandParentName":"International Journal on Smart Sensing and Intelligent Systems","isGrandParentConference":false,"publisherName":"Sciendo","publisherLocation":null,"nextMap":{"id":{"timestamp":1646537273,"date":"2022-03-06T03:27:53.000+00:00"},"doi":"10.21307/ijssis-2021-006"},"prevMap":{"id":{"timestamp":1646566093,"date":"2022-03-06T11:28:13.000+00:00"},"doi":"10.21307/ijssis-2021-004"},"counter":0,"apaString":"Hassanein,O.,Sreenatha,G.,Aboobacker,S. \u0026 Ali,S.(2021).\u003carticle-title\u003eDevelopment of Low Cost Autonomous Underwater Vehicle Platform\u003c/article-title\u003e. 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For research, industrial and military underwater applications, AUVs are generally used, which modeling, system identification and control of these vehicles pose serious challenges due to the vehicles’ complex, inherently nonlinear, and time-varying dynamics. Here, the AUV is considered to have 6-DOF for the development of the electrical, electronics, power distribution, sensors, and actuators. A low-cost IMU is used along with other reasonably low-cost detectors, such as a magnetometer and a water pressure sensor for depth evaluation. This study addresses the configuration and selection of the onboard instruments required to collect data using a processing unit (PC104) based on-board data logger to record complete manoeuvring data obtained from various sensors and process it based on the experiment. Real-time validations using Hardware-in-Loop (HIL) simulations are carried out. HIL simulations help to simulate the behavior of the developed model for surge, pitch and yaw movement, and also it makes clear that the used identification methods are feasible for real time control. Real time experiments are carried out with the developed 6-DOF instrumented AUV platform in various conditions and environments to validate its dynamics identification with adaptive controller and the results are presented for surge, the control of pitch, and yaw. The results revealed that the adaptive controller can effectively control the developed AUV and show its robust properties in the real world.\u003c/p\u003e\u003c/abstract\u003e"}],"figures":[{"label":"Figure 1:","caption":"Block diagram of various stages of project.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=b55eb309511a6949ac10a5f04a4901707e5b3f2547b4c0ce21d0f02c21c8f2d3"},{"label":"Figure 2:","caption":"UNSW Autonomous Underwater Vehicle platform (Hassanein et al., 2011).","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_002.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=69ef9a0f85bf0848e41eeacca67e5e0c998fad696ac4fd321eeb1680601aa2ff"},{"label":"Figure 3:","caption":"Forward thruster with electrical motor driver.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_003.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=54ef71f66924c2920e9db0c0f2485aedf146390ee9125ac90eedc3db7ddedcc4"},{"label":"Figure 4:","caption":"Submersible bilge pump with the motor driver.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=84d97478cd47a19680c08c9e47917d98c02ea42eb6157dfcbd6166a8bdcf0e38"},{"label":"Figure 5:","caption":"ADIS16367 unit.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_005.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=f104ed817846a3e38f73e02e2ad566ba78423d51a10f57296b76ec3184fc2b97"},{"label":"Figure 6:","caption":"SPI interface for IMU.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_006.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=1fff07b55678674f81749704e819ae11b68365def5b4cb7e74a183b5e92c8755"},{"label":"Figure 7:","caption":"Burst read sequence (ADIS16367 Data Sheet).","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_007.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=d9b3cfb12fc6e8803258894edfd3680f68cde78cd167c0e705d98c9b226c2614"},{"label":"Figure 8:","caption":"Wiring from pressure sensor to compass.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_008.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=adc2c7d8b701c5d780f430fa45d958005c4434df3381ee184c9ea7b4dd0ea975"},{"label":"Figure 9:","caption":"Depth measurement accuracy from water pressure sensor.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_009.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=9f68299f37f7e92b51e074a7e8a09308a9f289af0bdf1aba71f0f3f2ebb69db6"},{"label":"Figure 10:","caption":"PC104 computer system stack.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_010.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=3fe8ed7628046366cad33ab1989cb92f00b02a0e6b01e732a40d52caf2978d4b"},{"label":"Figure 11:","caption":"Complete hardware diagram of AUV.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_011.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=25dea38f644611c47f557fa1eb50e48f2652e46df8b259f4b197b15c54aeb7f3"},{"label":"Figure 12:","caption":"Complete AUV hardware.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_012.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=bdd41c1953dc5506d31d7cd27221ead84c75113f0ad34540b04e18b1cfd8c058"},{"label":"Figure 13:","caption":"Buoyancy adjustment of the AUV.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_013.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=780e974d2b54f2008fb8e60f075013aee078f35079dc5e557ca1e196e06ad024"},{"label":"Figure 14:","caption":"Body-fixed and earth-fixed coordinate system (Hassanein et al., 2013).","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_014.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=b34171e52b483aeb3e2ec2257c9ee9b41fce039e61d16e26c9a3f1758165ba20"},{"label":"Figure 15:","caption":"Simulink block diagram.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_015.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=5374724cd224a7fba2168db3d88dde25445715be96e1de21864d23e57e465055"},{"label":"Figure 16:","caption":"Surge input Force.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_016.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=4724782f2e0933b4bc3bf52e662f178bc7f6171efdca181fa3ba0394249c29ce"},{"label":"Figure 17:","caption":"Pitch and yaw input Force.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_017.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=b8afab3b3565e4bb6b5a869a8b6e2775033896e1c7443a539d259ddf411e2266"},{"label":"Figure 18:","caption":"Surge input responses.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_018.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=4ce9e5ca63d8c9d254580658f65b670a52654848bad7aea542a69f70b658aea4"},{"label":"Figure 19:","caption":"Pitch input response in AUV translational and orientational movement.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_019.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=90b2791daf50a375f056401bed8ad3460ce1a177ea917331adc77769f03c17f0"},{"label":"Figure 20:","caption":"Pitch input response in AUV velocities.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_020.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=1c68c33e25fa903529a94a5162abb5d1869d4743eb6cd67d8a138f8792ac0611"},{"label":"Figure 21:","caption":"Yaw input response in translational and orientational movement.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_021.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=17999\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=b972c55c014fbbf08feb5283c0742a7b017433f959196d3ee92d1e771938296f"},{"label":"Figure 22:","caption":"Yaw input response in AUV velocities.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_022.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=7374dcd637e5205f0e77afb2af53ee16d41f51aed29409117308145ce560a734"},{"label":"Figure 23:","caption":"Hardware In loop (HIL) simulation.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_023.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=f297f2d9db3d0d0cbb1d601f13af02312bbd534bf474508ccd87b1f6e9422dbc"},{"label":"Figure 24:","caption":"UNSW Canberra AUV during experimental teat data collection.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_024.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=8130e7ddd904d4f77dd975e12136d4d567e7ae3dff72f37e044baf0a32b3517d"},{"label":"Figure 25:","caption":"1st set of AUV control based fuzzy experimental results: coupled dynamics.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_025.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=6a81f0aab10ebb784f5e9b759d1596301fef1ab6a0e8038e581161415a0ac9a8"},{"label":"Figure 26:","caption":"2nd set of AUV control based HNFN experimental results: coupled dynamics.","imageLink":"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_026.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026X-Amz-Date=20230202T121249Z\u0026X-Amz-SignedHeaders=host\u0026X-Amz-Expires=18000\u0026X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026X-Amz-Signature=5b92e3ea462175f47a901fa46e26860a0c35195f284c1896f13c3092c7bbfe02"}],"tableContent":{"Specifications of AUV model.":"\u003ctable frame=\"hsides\"\u003e\u003ccolgroup span=\"1\"\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eMass of the AUV\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e10.4102\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eCG (original)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[‒10.7498‒3.49512428.206]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003emm\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003ePayload capacity\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e21.1408\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eTotal length (L)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.064\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eMass of AUV + Payload\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e31.551\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eDiameter (d)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.250\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIx\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.6384\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eSpeed\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em/s\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIy\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e6.4110\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_001.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:mi\u003eρ\u003c/mml:mi\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1000\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003csup\u003e3\u003c/sup\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIz\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e6.4110\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eBC\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 0]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eCG\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 0]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eBC(original)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 21.306]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003emm\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e","Hydrodynamic coefficient of UNSW Canberra AUV model.":"\u003ctable frame=\"hsides\"\u003e\u003ccolgroup span=\"1\"\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_027.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eu\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003eu\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒7.365\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_028.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eu\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_029.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.737\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_030.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.31\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_031.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.737\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_032.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.31\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_033.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.065\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_034.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.51\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_035.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.065\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_036.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.51\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_037.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒112.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_038.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒62.45\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_039.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.250\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_040.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_041.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒112.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_042.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒62.45\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_043.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.250\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_044.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_045.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eK\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ep\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ep\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.5975\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_046.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eK\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ep\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.3125\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_047.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e2.244\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_048.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_049.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒119.5\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_050.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒59.75\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_051.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.244\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_052.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_053.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003cmml:mo\u003e|\u003c/mml:mo\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒59.75\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_054.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒31.25\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_055.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eX\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003eu\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.17\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_056.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eK\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003ep\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_057.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒34.834\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_058.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_059.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eY\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_060.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eM\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.659\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_061.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003ew\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒34.834\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_062.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003ev\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_063.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eZ\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003eq\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_064.png\"/\u003e\u003cmml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmml:msub\u003e\u003cmml:mrow\u003e\u003cmml:mi\u003eN\u003c/mml:mi\u003e\u003c/mml:mrow\u003e\u003cmml:mrow\u003e\u003cmml:mover accent=\"true\"\u003e\u003cmml:mi\u003er\u003c/mml:mi\u003e\u003cmml:mo\u003ė\u003c/mml:mo\u003e\u003c/mml:mover\u003e\u003c/mml:mrow\u003e\u003c/mml:msub\u003e\u003c/mml:math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.659\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e"},"tables":null,"articleContent":"\n\u003cdiv\u003e\u003csec id=\"j_ijssis-2021-005_s_001\"\u003e\u003ctitle/\u003e\u003cp\u003eOver the years, AUVs have gained attention as specialized tools for carrying out various underwater operations, eventually leading to a dramatic increase in the number of scientific studies undertaken (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_005\"\u003eBlidberg, 2001\u003c/a\u003e). There are four separate sub-classes of unmanaged vehicles (UVs) systems. Submersibles towed UVs behind ships are the first category, serving as ideal platforms for adding different sensors. The second category is the remotely operated vehicle (ROV), which gets directly controlled by the surface ship, with power and connectivity (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_027\"\u003eRoberts, 2008\u003c/a\u003e). The unmanned untethered vehicle (UUV) is the third group, with its own onboard control, but is controlled remotely through a means of wireless communication protocol. \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_006\"\u003eBudiyono (2009)\u003c/a\u003e. Nevertheless, the requirement for a networking link and control platform restricts ROV and UUV usage and their capability (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_011\"\u003eGonzalez, 2004\u003c/a\u003e). The fourth category of UVs, the AUVs, are fully autonomous underwater platforms capable of conducting underwater operations and activities and have their own sensor, control and payload equipment (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_005\"\u003eBlidberg, 2001)\u003c/a\u003e. The key benefit of an AUV is that a human operator is not required, is cheaper than a manually driven vehicle, and is able to work that is too risky for a human being (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_002\"\u003eAlt, 2003\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_007\"\u003eCaccia, 2006\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_031\"\u003eSide and Junku, 2005\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_033\"\u003eSmallwood David and Whitcomb, 2004\u003c/a\u003e) like monitoring of rising levels of seas (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_013\"\u003eHadi et al., 2020\u003c/a\u003e). In the 1980s, innovation and software developments enhanced the research to facilitate the design and implementation of sophisticated autonomous system (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_005\"\u003eBlidberg, 2001\u003c/a\u003e). Nowadays, technicians, engineers, and computer programmers play an important role in developing technologies (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_019\"\u003eInnella and Rodgers, 2021\u003c/a\u003e) and the developed tools can be used in various applications. The enthusiasm in AUVs in academic science has therefore been revived and several universities have built their own AUVs. The Australian Centre for Field Robotics (ACFR) of the University of Sydney runs an ocean-going AUV labeled Sirius, Designed at WHOI in the form of an Integrated Marine Observation System (IMOS), AUV facility supports competitively to facilitate deployment in Australia as a contributor to marine studies (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_032\"\u003eSingh et al., 2004\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_040\"\u003eWilliams et al., 2009\u003c/a\u003e). There are many AUV contests in which a vehicle has a certain duty to carry out autonomously, which are mostly organized by colleges or other educational institutions (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_011\"\u003eGonzalez, 2004\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_001\"\u003eAkhtman et al., 2008\u003c/a\u003e).\u003c/p\u003e\u003cp\u003ePower and autonomy are the two most critical technical obstacles in AUV design, since the former sets limitations on mission times and the latter decides the degree to which an AUV may be independent (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_016\"\u003eHoltzhausen, 2010\u003c/a\u003e). While AUVs are more precise due to new technology, the creation of a completely autonomous one is an incredibly challenging problem as precise and robust controllers are required. (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_029\"\u003eSalgado-Jimenez et al., 2004\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_030\"\u003eSalman et al., 2011)\u003c/a\u003e. \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_016\"\u003eHoltzhausen (2010)\u003c/a\u003e claimed that the structure of the AUV is one of the most critical aspects of an AUV. There are a number of ways to approach the construction that must take into account factors such as the pressure and/or depth needed, height, operating temperature ranges, conditions of impact, and water permeability along with corrosion and chemical resistance. The hydrodynamic coefficients that decide the dynamics of the AUV are affected not only by the structure of the AUV but also by the water current and the vehicle speed and an innovative method to evaluate velocity and heading is proposed in \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_026\"\u003eRezaali and Ardalan (2016)\u003c/a\u003e. Moreover, a hull is built to minimize the drag for reducing the propulsive force (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_038\"\u003eWang et al., 2009a\u003c/a\u003e). Wang stated that the first shape proposed for a hull was a spherical design, while this could withstand pressure, it influenced stability. So a circular cylindrical design, which has the benefits of being a good framework to resist the effects of hydrostatic pressure, can be expanded to provide extra room internally, seems to be a better hydrodynamic shape than a spherical one with the same volume (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_028\"\u003eRoss 2006\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_039\"\u003eWang et al., 2009b\u003c/a\u003e). The cavitations (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_022\"\u003eKondoa and Ura, 2004\u003c/a\u003e) and instability (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_028\"\u003eRoss, 2006\u003c/a\u003e) are, however, the drawbacks of a cylindrical hull. Most of the existing AUVs have a cylindrical circular hull (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_009\"\u003eEvans and Nahon, 2004\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_017\"\u003eHsu et al., 2005\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_043\"\u003eWoods Hole Oceanographic Institution, 2012\u003c/a\u003e). Similarly, a spherical shaped nose has enhanced stability and cavitation (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_039\"\u003eWang et al., 2009b\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eThe dynamics of an AUV are determined mainly by propulsion and buoyancy. Four kinds of propulsion systems are possible. Most common form of propulsion is via thrusters with dynamic diving technology. The other three come from static diving technology, they are piston-type ballast tank, a hydraulic pump-based ballast system and air compressor-based system (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_023\"\u003eKrieg and Mohseni, 2008\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_042\"\u003eWolf, 2003\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eAUV can use a single thruster for both horizontal and vertical movements with diving planes (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_022\"\u003eKondoa and Ura, 2004\u003c/a\u003e). The thruster technique allows the AUV to be near-neutrally buoyant with the major benefit that the vehicle can hover without propulsion. Even then, once the vehicle is in motion, the thrusters must remain ON, since it has to drive forward to remain submerged. For propulsion, most AUVs use propeller-type thrusters (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_008\"\u003eCavallo et al., 2005\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_037\"\u003eVon Alt, 2003\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_043\"\u003eWoods Hole Oceanographic Institution, 2012\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eJet propulsion is another method influenced by the natural locomotion of squids and other cephalopods, whereby water is pulled into a wide cavity and expelled through a nozzle at a high momentum to propel the vehicle forward (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_012\"\u003eGriffiths et al., 2000\u003c/a\u003e). This gives the potential of an AUV to carry out low-speed maneuvering without impacting the forward drag on its structure. In this research, this type of propulsion is used to maneuver the vehicle while the propeller is being used for forward motion in the horizontal and vertical planes.\u003c/p\u003e\u003cp\u003ePower demands for all the equipment, such as on-board controls, motor controllers, propulsion systems, sensors and instruments, and navigation systems, must be fulfilled and supported by the electrical system of the AUV. While some AUVs are operated by fuel cells and a few uses solar power, the most popular ones are operated by batteries (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_018\"\u003eHyakudome et al., 2009\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_020\"\u003eJalbert et al., 2003\u003c/a\u003e). Silent operation, ease of speed regulation and simplicity are the benefits of utilizing electrical over thermal propulsion. Since lithium-ion battery technological advancements have made them an attractive alternative to silver-zinc batteries (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_041\"\u003eWilson and Bales, 2006\u003c/a\u003e), they are employed in our research.\u003c/p\u003e\u003cp\u003eA pressure sensor is the most common sensor used in vehicle depth measurement. Strain gauges and quartz crystals are the most common pressure sensor technologies for deep-ocean applications (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_021\"\u003eKinsey et al., 2006\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_038\"\u003eWang et al., 2009a\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eGPS can provide superior 3-D navigation functionality. Since the water absorbs the radio signals in underwater conditions, it can be used only when a vehicle surfaces periodically to correct the readings (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_024\"\u003eNicholson and Healey, 2008\u003c/a\u003e). This implies that other sensors, such as the Long Baseline (LBL), Doppler Velocity Log (DVL), and magnetometer or compass are needed for the tracking and navigation of underwater vehicles for GPS fixes (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_021\"\u003eKinsey et al., 2006\u003c/a\u003e); e.g., Bluefin-21 AUV surfaces are needed periodically for GPS fixes. The echo sounder (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_011\"\u003eGonzalez, 2004\u003c/a\u003e) was used in the MAKO AUV project as another method to assess vertical depth, but only if the depth of the water is known.\u003c/p\u003e\u003cp\u003eThe LBL, Ultra-short Baseline (USBL), and Short Baseline (SBL) systems (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_021\"\u003eKinsey et al., 2006\u003c/a\u003e) are known as underwater acoustic positioning systems used for depth measurement. The LBL device is essentially a method of triangulation which is used when a vehicle triangulates the acoustic position of the vehicle in the network of transponders (beacons) used on the seabed with known locations (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_035\"\u003eThe International Marine Contractors Association IMCA, 2009\u003c/a\u003e). A complete system comprises of a transceiver placed under a ship and a vehicle transponder. The time between initial acoustic pulse transmission and reaction detection is estimated and transformed into a range.\u003c/p\u003e\u003cp\u003eA DVL is a sensor that uses high frequency Doppler beam sonar for calculating the Doppler changes of sonar signals reflected in the ground (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_039\"\u003eWang et al., 2009b\u003c/a\u003e). This navigation method is only useful if the vehicle is above the sea level (18–100 m), since it is more precise at low speed and not affected by sea currents (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_024\"\u003eNicholson and Healey, 2008\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eThe most popular sensors used in marine equipment are magnetic sensors or electronic compass modules. Vasilijevic (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_036\"\u003eVasilijevic et al., 2012\u003c/a\u003e) reported that a magnetometer calculates the magnetic field of the Earth in the X, Z coordinates position and returns these three measures separately to represent the magnetic field vector values. There are a wide number of single-axis (heading only) and three-axis flux-gate magnetometers available commercially. The overall efficiency of a navigation device is mostly based on the accuracy of its magnetic sensor, which is believed to be the leading cause of error. There are three different causes of errors: the magnetic interference of the vehicle with itself or the environment (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_044\"\u003eYe et al., 2009\u003c/a\u003e), the compensation of the roll and pitch dependent on gravity, and finally, the orientation of the compass mounting within the vehicle.\u003c/p\u003e\u003cp\u003eStutters indicated that the Inertial Navigation Device (INS) or Inertial Management Unit (IMU) utilizes accelerometers and gyroscopic sensors to detect vehicle acceleration of the three axes (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_034\"\u003eStutters et al., 2008\u003c/a\u003e). A gyroscope measures rotation levels, and linear acceleration is determined by an accelerometer. IMU sensors are not prone to magnetic fluctuations and drift with time, leading to erroneous measurements. Laser or fiber-optic gyroscopes do not have moving components and are included in the new INSs (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_039\"\u003eWang et al., 2009b\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eA mixture of two or more of the above systems has often established some of the finest AUV navigation systems. Multi-sensor integration can be characterized as the synergistic use of multiple sensory device information that can minimize navigational errors to assist a system in carrying out its tasks (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_016\"\u003eHoltzhausen, 2010\u003c/a\u003e).\u003c/p\u003e\u003cp\u003eIn brief, the main objective of navigation sensors is to acquire the data and parameters of an AUV system in real time from its surrounding environment. These data are then collected by the AUV control system through its processing unit to control and maneuver AUVs effectively in order to perform a pre-identified operation.\u003c/p\u003e\u003cp\u003eThe control accuracy provided by guidance and control systems is the basis for the successful completion of AUV missions as the autonomous control of an AUV poses serious challenges due to its complex, inherently nonlinear, and time-varying dynamics. In addition, its hydrodynamic coefficients are difficult to model accurately because of their variations under different navigational conditions and when manoeuvring in uncertain environments which expose an AUV to unpredictable external disturbances. Hence, there is a need to design and develop a robust and stable control system which can achieve AUV’s desired positions and velocities, satisfy the commands generated by its guidance system and maintain steady conditions during its mission. In order to design an adaptive controller for an AUV, suitable models of the nonlinear plant are necessary as the controllers have to cope with uncertainties due to discrepancies in modeling the unknown dynamics of the plant.\u003c/p\u003e\u003cp\u003eThus, the development of an AUV for research or commercial purpose involves a number of criteria to be satisfied along with the cost. This paper proposes the development of an AUV. The main goal of this study is to develop an in-house project, find low-cost solutions for AUV navigation problems, and develop a small-sized, low-cost AUV. The platform is expected to demonstrate autonomous manoeuvre for changing conditions and under the influence of external disturbances in a controlled environment such as a swimming pool.\u003c/p\u003e\u003cp\u003eThe rest of the paper structure is organized as follows. The detailed specifications of different AUV subsystems considered for the work in this study are described in the second section. The third section discusses the AUV complete system and its integration. The kinematics and the nonlinear mathematical modeling of an AUV in 6-DOF is discussed in the fourth section that includes the Simulink/Matlab model of the AUV along with the investigation of its open-loop characteristics. The main features of the Hardware-in-Loop (HIL) simulation are explained in the fifth section. The sixth section presents the experimental results of the proposed model-based control algorithm on the AUV. The study is concluded in the final section.\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002\"\u003e\u003cdiv\u003ePlatform design\u003c/div\u003e\u003cp\u003eThe integration on the UNSW Canberra AUV with electrical and electronic systems, which include prototypes, specifications, power distribution system and storage, actuators and sensors, is discussed. The instrumentation required to accommodate on board the AUV in order the completely collect the data associated with different maneauvers, the interface and integration of all subsystems with the PC 104 device are addressed.\u003c/p\u003e\u003cp\u003eThis paper presents the development of an AUV experimental platform with adaptive control capabilities aimed at achieving the mission requirements. A detailed block diagram indicating different stages in this project is shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_001\"\u003eFig. 1\u003c/a\u003e.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_001\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 1:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eBlock diagram of various stages of project.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_001.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=b55eb309511a6949ac10a5f04a4901707e5b3f2547b4c0ce21d0f02c21c8f2d3\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_001\"\u003e\u003cdiv\u003eUNSW Canberra AUV platform\u003c/div\u003e\u003cp\u003eIn the UNSW Canberra workshop, a torpedo-shaped underwater vehicle prototype is constructed as a demonstration of the concepts of the underwater vehicle, which comprises of three segments; nose cone, body (middle section), and tail cone. \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al. (2011\u003c/a\u003e, \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_015\"\u003e2013)\u003c/a\u003e. The nose and tail cone moulds are made up of Stayfoam, then wrapped with fiberglass and those are wet sections to decrease the buoyancy force. The body is made of a 255 mm PVC pipe and is known to be the dry section where the batteries are located along with the circuitry, sensors, and control unit. The dry portion is segregated by the bulkheads for providing waterproof environment. An on-board data logger and control unit have been equipped with the AUV platform to allow sensor and actuator data to be recorded and processed to implement advanced system identification and control techniques. Necessary computers and sensors are chosen, considering the need for low cost, compact, simplicity of program and processor capabilities. The UNSW Canberra AUV model is shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_002\"\u003eFig. 2\u003c/a\u003e.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_002\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 2:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eUNSW Autonomous Underwater Vehicle platform (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_038\"\u003eHassanein et al., 2011\u003c/a\u003e).\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_002.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_002.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=69ef9a0f85bf0848e41eeacca67e5e0c998fad696ac4fd321eeb1680601aa2ff\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_002\"\u003e\u003cdiv\u003eActuators\u003c/div\u003e\u003cp\u003eThe AUV employs two forms of actuators, the electrical propeller for thrust and secondly, four bilge pumps to operate in the horizontal and vertical axis.\u003c/p\u003e\u003cp\u003eThe thrusters that use propeller systems and a small DC motor wrapped in a watertight enclosure to spin the propeller that produce torque are the key method of actuation in the AUV. The existing thruster is a 12Vdc electric motor, typically employed for small river vessels. The ‘Endura C2’ thruster was obtained from a ‘MINN KOTA’ company shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_003\"\u003eFig. 3\u003c/a\u003e. An ‘ET-OPTO RELAY4’ made by the ‘ETT’ company is used since the electric motor requires a controller or driver shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_003\"\u003eFig. 3\u003c/a\u003e. According to the function and operating order, this has two inputs to drive the motor, the power given by the battery and a control signal from the I/O module in the processor unit.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_003\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 3:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eForward thruster with electrical motor driver.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_003.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_003.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=54ef71f66924c2920e9db0c0f2485aedf146390ee9125ac90eedc3db7ddedcc4\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eIn order to maneuver in horizontal and vertical planes (pitch and lake angle), submersible bilge pumps are obtained from a company named ‘Rule Mate.’ These are shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_004\"\u003eFig. 4\u003c/a\u003e. There are three distinct pump configurations and all the pumps are battery operated by 12 Vdc. The left and right pumps used to control the AUV in the horizontal plane have an output of 750 GPH and a force of 0.0022 N. The pump used to push down this vehicle is rated by 0.0033 N at 1,000 GPH and by 0.0044 N at 1,500 GPH. The pumps are operated in on/off mode using the control unit’s PWM signals. The motor driver for such pumps is an ET-OPTO DC-OUT4 manufactured by ETT as shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_004\"\u003eFig. 4\u003c/a\u003e.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_004\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 4:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eSubmersible bilge pump with the motor driver.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_004.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_004.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=84d97478cd47a19680c08c9e47917d98c02ea42eb6157dfcbd6166a8bdcf0e38\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_003\"\u003e\u003cdiv\u003eInertial measurements UNIT (IMU)\u003c/div\u003e\u003cp\u003eThe IMU is the key sensor used for AUV navigation and comprises three gyroscopes and three accelerometers, each mounted along the \u003citalic\u003eX\u003c/italic\u003e, \u003citalic\u003eY\u003c/italic\u003e, and \u003citalic\u003eZ\u003c/italic\u003e axes for the calculation of rotational speeds and linear accelerations. The gyroscope data was used to calculate the rotations of the vehicle along the three axes indicated by roll, pitch, and yaw. In terms of these three DOFs, the rotational position of the vehicle is calculated by the determination of the rotational velocity integral over the period of the measurement. In order to determine the location of the vehicle in a three-dimensional space, one must continuously convert from the local coordination system of the vehicle to the coordination system of the earth. This transformation is addressed later in this paper. The ADIS16367 shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_005\"\u003eFig. 5\u003c/a\u003e is the unique IMU used in this project, which is a 3-axis system acquired from Analog Devices Co. The ADIS16367 combines industry-leading and signal conditioning IMEMS technology that optimizes dynamic efficiency and is a comparatively cheap device. IMU outputs are compared with the calibrated digital compass outputs with no major errors amongst them.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_005\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 5:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eADIS16367 unit.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_005.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_005.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=f104ed817846a3e38f73e02e2ad566ba78423d51a10f57296b76ec3184fc2b97\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eA serial peripheral interface (SPI) card was designed and developed as part of this project as shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_006\"\u003eFig. 6\u003c/a\u003e. An I/O board is necessary for dealing with the lower level interface between the IMU and the processing device. The I/O module is on the ATmega168-based Arduino Pro-Mini microcontroller board consisting of 14 digital input/output pins, 6 analog inputs, an on-board resonator, a reset button and pin header mounting holes. To provide communication to the board and USB power, a six-pin header can be plugged into an FTDI cable or Sparkfun breakout board. The serial peripheral interface sends the data to the PC104 at a rate of 115 kbps through an RS232 serial port.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_006\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 6:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eSPI interface for IMU.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_006.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_006.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=1fff07b55678674f81749704e819ae11b68365def5b4cb7e74a183b5e92c8755\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eA ‘burst data read collection’ is the best way to retrieve data from the IMU, which is a process-efficient form of obtaining ADIS16367 data in which all output registers are clocked on DOUT (SPI Data Output), 16 bits at a time, in sequential data cycles (each divided by a single SCLK interval (SPI Serial Clock)). DIN (SPI data input) is set to 0x3E00 to start a burst read sequence, then the contents of every output register is transferred to DOUT, that is from SUPPLY OUT to AUX ADC, as seen in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_006\"\u003eFigs. 6\u003c/a\u003e and \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_007\"\u003e7\u003c/a\u003e. The data are being sent to a PC104 on-board COM port which contains RS232 driver blocks to handle serial communications provided by the Matlab xPC goal toolbox.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_007\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 7:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eBurst read sequence (ADIS16367 Data Sheet).\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_007.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_007.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=d9b3cfb12fc6e8803258894edfd3680f68cde78cd167c0e705d98c9b226c2614\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_004\"\u003e\u003cdiv\u003eElectronic Digital Compass\u003c/div\u003e\u003cp\u003eA magnetometer or electronic digital compass is also a sensor used for the AUV navigation, which measures the magnet field of the Earth in the direction of the \u003citalic\u003eX\u003c/italic\u003e, \u003citalic\u003eY\u003c/italic\u003e, and \u003citalic\u003eZ\u003c/italic\u003e axes. It returns the three measurements separately to provide the magnetic field vector values, which are then used for determining the heading of the vehicle. It does not need to be transformed since this heading calculation is taken in the global reference frame or coordinates. The magnetometer is susceptible to other disturbance in the magnetic field of the Earth. To stop incorrect readings, it can be recalibrated before any dive. The magnetometer can often be disturbed by other magnetic objects in close vicinity of the sensor as well.\u003c/p\u003e\u003cp\u003eAs shown in the \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_008\"\u003eFig. 8\u003c/a\u003e, an Oceansever OS5000-USD with a depth sensor, a 3-axis tilt-compensated compass with both Serial \u0026amp; USB direct interfaces and connections on both sides of the module, is the digital compass used in this project. It comprises a 3-axis AMR magnetic tracker, a 3-axis STM accelerometer, a 50 MIPS microprocessor that facilitates precise tilt correction floating point IEEE operations and 24-bit differential input Sigma-Delta AD converters.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_008\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 8:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eWiring from pressure sensor to compass.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_008.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_008.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=adc2c7d8b701c5d780f430fa45d958005c4434df3381ee184c9ea7b4dd0ea975\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eFor operation, the compass needs power supply of 5 V, when attached to the pressure sensor, it measures the depth and its outputs include a heading, pitch and roll in degrees, a 3-axis magnetic field, a 3-axis acceleration, gyro reading in 2 axes and temperature. Its built-in calibration involves rotation around the \u003citalic\u003eZ\u003c/italic\u003e, \u003citalic\u003eY\u003c/italic\u003e, and \u003citalic\u003eX\u003c/italic\u003e axes that takes place after installation in the AUV and before the experiments begin.\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_005\"\u003e\u003cdiv\u003eWater pressure sensor\u003c/div\u003e\u003cp\u003eThe water pressure is a variation in the surface pressure and it is calculated in Pascals (Pa). A water pressure sensor used in this project for measuring the water pressure on the exterior of the vehicle and for determining the vehicle’s depth. As there is a linear relationship between the water pressure and the vehicle’s depth, this can be easily determined from Pascals law,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_001\"\u003e\u003clabel\u003e(1)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_001.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmi\u003eΔ\u003c/mi\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eρ\u003c/mi\u003e\u003cmi\u003eg\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eΔ\u003c/mi\u003e\u003cmi\u003eh\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(1)where \u003citalic\u003eρ\u003c/italic\u003e is the water density in kg/m3, Δh is the depth of the vehicle in meters (m) and \u003citalic\u003eg\u003c/italic\u003e is the gravitational acceleration (9.81 m/s\u003csup\u003e2\u003c/sup\u003e).\u003c/p\u003e\u003cp\u003eThe water pressure sensor used in this project is an LM series low-pressure media-isolated sensor with an analogue interface. It uses three wires, power, ground and signal. The output signal range is between 0.5 and 4.5 V depending on the pressure. The relationship between the measured voltage and water pressure is linear. As this sensor has a measuring range from 0 to 15 PSI, it can be used to measure depths of up to 10 m. It is connected directly to the digital compass to calculate the depth, as shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_008\"\u003eFig. 8\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eA 50 cm tube with a tape measure is placed on the sensor to calibrate the sensor. The water is then fed into the tube every 2 cm and the values obtained were recorded. The error of depth measurement is less than 0.14 mm. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_009\"\u003eFig. 9\u003c/a\u003e shows the acquired precision of the measurement.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_009\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 9:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eDepth measurement accuracy from water pressure sensor.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_009.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_009.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=9f68299f37f7e92b51e074a7e8a09308a9f289af0bdf1aba71f0f3f2ebb69db6\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_006\"\u003e\u003cdiv\u003eOn-board computer\u003c/div\u003e\u003cp\u003eAUV is managed by an on-board processing unit, which involves fast processors and broad memories. The PC104 architecture is one of those widely used processors for embedded systems. A great feature of the architecture of the PC104 is that it has a defined form factor that enables to have several add-on boards compatible with the main processor card. The other benefits of the PC104 system are its compact size and high processing capacity. There is an Industry Standard Architecture (ISA) bus for these PC104 boards that runs through all the interconnected boards. On the ISA bus there are 104 pins. The PC104-plus comes with an ISA bus and a peripheral part interface (PCI). The PCI bus works at lower levels of voltage and is much faster than the ISA bus. In contrast to the ISA bus width of 16 bits, the PCI bus is 32 bits or 64 bits wide.\u003c/p\u003e\u003cp\u003eCompared to many other architectures, the advantages of a PC 104-based embedded device have contributed to the usage of this AUV design in UNSW Canberra with a PC104-plus, which comprise the core of the AUV control unit. It incorporates an Intel Atom N450 processor with 1 GB RAM, 250 GB hard drive, speed of 1.66 GHz, and it uses a compact flash disk for real-time applications as data storage.\u003c/p\u003e\u003cp\u003eIt consists four USB ports, one Ethernet port and two serial COM ports. The serial ports are either RS-232 or RS-485, chosen by jumpers on the chip-set. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_010\"\u003eFig. 10\u003c/a\u003e shows the analog and digital I/O boards and a DC-DC supply card; these are also included in the modules on the PC104 Stack.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_010\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 10:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003ePC104 computer system stack.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_010.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_010.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=3fe8ed7628046366cad33ab1989cb92f00b02a0e6b01e732a40d52caf2978d4b\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eThe dedicated PC104 architecture power supply board PCM-3910, is used to provide constant voltages of various amplitudes. As compared to other standard methods, this board provides reliable voltage regulation. This board has a 10–30 V input range and provides on-board 5, 12, ‒5, and ‒12 V outputs for the stack and various sensors. PC104 stack also contains an Analog-to-Digital (A/D) conversion board, DMM-32X-AT by Diamond Systems. The resolution of A/D is 16-bit with 32 analog channels.\u003c/p\u003e\u003cp\u003eThe PC104’s real-time environment is an xPC target, a Matlab toolbox that offers the Matlab® Simulink® real-time kernel and application development environment. It instantly produces code via the Real-Time Workshop (RTW), which can be downloaded to a second computer utilizing the xPC target real-time kernel for simplification and minimization of development time and debugging. Using the Simulink® libraries available on Matlab®, the Simulink model is initially developed and then compiled using the xPC target option which creates an executable embedded code that can be downloaded to the target device.\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_002_s_007\"\u003e\u003cdiv\u003ePower distribution\u003c/div\u003e\u003cp\u003eThe whole power distribution system is fitted with a wireless power switch that enables the hardware to be connected and disconnected from the distance. There are four batteries in the AUV’s battery system; two 12 V 20 Ah batteries and two 12 V 2.5 Ah batteries. The power specifications of on-board modules are as follows; one 12 Vdc 20 Ah battery supplies the thruster and the second one supplies the bilge pumps. A 12V 2.5 Ah is required by the PC104 and has its own power module to supply the processor and I/O modules. Another power distribution system has been designed that connects directly to the fourth battery and it provides 5 V to water pressure sensors, IMU, magnetometers, and digital compass sensors.\u003c/p\u003e\u003c/sec\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_003\"\u003e\u003cdiv\u003eSystem integration\u003c/div\u003e\u003cp\u003eThe PC104 is the system that performs all of the key control tasks. It is then link to the IO module, which includes low-level interfaces for both sensors and actuators. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_011\"\u003eFig. 11\u003c/a\u003e displays the whole system and its integration, showing all interfaces between the AUV hardware and component interfaces. The integrated system used to control the AUV maneuvering is shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_012\"\u003eFig. 12\u003c/a\u003e. Eventually, the AUV parameter values are measured and indicated in \u003ca ref-type=\"table\" href=\"#j_ijssis-2021-005_tab_001\"\u003eTable 1\u003c/a\u003e (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_015\"\u003eHassanein et al., 2013\u003c/a\u003e).\u003c/p\u003e\u003ctable-wrap id=\"j_ijssis-2021-005_tab_001\" position=\"float\"\u003e\u003clabel\u003eTable 1.\u003c/label\u003e\u003ccaption\u003e\u003cp\u003eSpecifications of AUV model.\u003c/p\u003e\u003c/caption\u003e\u003ctable frame=\"hsides\"\u003e\u003ccolgroup span=\"1\"\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eMass of the AUV\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e10.4102\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eCG (original)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[‒10.7498‒3.49512428.206]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003emm\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003ePayload capacity\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e21.1408\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eTotal length (L)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.064\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eMass of AUV + Payload\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e31.551\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eDiameter (d)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.250\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIx\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.6384\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eSpeed\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em/s\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIy\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e6.4110\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_001.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eρ\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1000\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003csup\u003e3\u003c/sup\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eIz\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e6.4110\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eBC\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 0]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eCG\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 0]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003em\u003c/td\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eBC(original)\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e[0 0 21.306]\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003emm\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/table-wrap\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_011\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 11:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eComplete hardware diagram of AUV.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_011.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_011.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=25dea38f644611c47f557fa1eb50e48f2652e46df8b259f4b197b15c54aeb7f3\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_012\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 12:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eComplete AUV hardware.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_012.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_012.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=bdd41c1953dc5506d31d7cd27221ead84c75113f0ad34540b04e18b1cfd8c058\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eBuoyancy would be a problem that requires a lot of consideration in order to submerge the vehicle. The power supply along with the power distribution system, actuators and other instruments, all sensors, and PC104 computer are located in the dry section of the vehicle. Initially, the entire vehicle was placed in the water, a strap test rig is wrapped around the vehicle and a hanger is then added to carry weights. This allows one to add weight until the buoyancy of the vehicle is resolved, enabling one to see how much weight is needed in the vehicle. By taking into consideration of the gravity and buoyancy centers, this weight is placed in the dry area. In addition, the nose and tail cones are equipped with an air tube in order to fine-tune the vehicle’s buoyancy at the start of each examination. The neutral vehicle buoyancy in the UNSW Canberra swimming pool is shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_013\"\u003eFig. 13\u003c/a\u003e.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_013\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 13:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eBuoyancy adjustment of the AUV.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_013.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_013.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=780e974d2b54f2008fb8e60f075013aee078f35079dc5e557ca1e196e06ad024\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_004\"\u003e\u003cdiv\u003eAUV dynamic modeling\u003c/div\u003e\u003cp\u003eThe AUVs is modeled via the analysis of statics and dynamics. The first is for the equilibrium of a body at rest or moving at constant speed and the second is the body experiencing accelerated movement (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e, \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_017\"\u003e2013\u003c/a\u003e). The coordinate system and specifications of the motion parameters should be given first in order to derive a 6-DOF nonlinear mathematical model of the AUV.\u003c/p\u003e\u003csec id=\"j_ijssis-2021-005_s_004_s_001\"\u003e\u003cdiv\u003eCoordinate system\u003c/div\u003e\u003cp\u003eThe position, orientation, linear velocity, and angular velocity of an underwater vehicle are defined in two frames of reference. The first frame is the \u003citalic\u003eX\u003c/italic\u003e\u003csub\u003e\u003citalic\u003eO\u003c/italic\u003e\u003c/sub\u003e\u003citalic\u003eY\u003c/italic\u003e\u003csub\u003e\u003citalic\u003eO\u003c/italic\u003e\u003c/sub\u003e\u003citalic\u003eZ\u003c/italic\u003e\u003csub\u003e\u003citalic\u003eO\u003c/italic\u003e\u003c/sub\u003e fixed frame, typically chosen to match the body’s CG and represented in relation to a gravitational reference or the Earth-set frame, E-xyz as seen in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_014\"\u003eFig. 14\u003c/a\u003e. As with underwater vehicles, it is believed that the acceleration of a position on the surface of the Earth may be ignored (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_039\"\u003eHassanein et al., 2013\u003c/a\u003e), the Earth-fixed frame is known to be inertial in this work. The position of the vehicle and its orientation in relation to the inertial reference frame, and linear and angular velocities are described to be the body reference frame. The typical evaluations for SNAME (1950) underwater vehicles as described on Nomenclature appendix (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli, 2018\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli et al., 2008\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen, 1994\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e, \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_017\"\u003e2013\u003c/a\u003e).\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_014\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 14:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eBody-fixed and earth-fixed coordinate system (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_015\"\u003eHassanein et al., 2013\u003c/a\u003e).\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_014.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_014.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=b34171e52b483aeb3e2ec2257c9ee9b41fce039e61d16e26c9a3f1758165ba20\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_004_s_002\"\u003e\u003cdiv\u003eKinematics\u003c/div\u003e\u003cp\u003eThe position and orientation of the vehicle, as mentioned earlier, is defined with regard to the inertial reference system in which \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_002.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eℜ\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e determines the position of the body (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli, 2018\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli et al., 2008\u003c/a\u003e),\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_002\"\u003e\u003clabel\u003e(2)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_002.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(2)where the vector \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_003.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eη\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the corresponding derivative time and is also connected to the inertial frame as (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli, 2018\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli et al., 2008\u003c/a\u003e),\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_003\"\u003e\u003clabel\u003e(3)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_003.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(3)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_004.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the linear velocity of the body-fixed frame of origin, which can be defined as the body-fixed frame with regard to the inertial reference frame (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli, 2018\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli et al., 2008\u003c/a\u003e),\u003c/p\u003e\u003cp\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_eq_004\"\u003e\u003clabel\u003e(4)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_004.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsubsup\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msubsup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(4)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_005.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsubsup\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msubsup\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the rotational matrix that describes the inertial frame transformation and can be described as the fixed frame,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_005\"\u003e\u003clabel\u003e(5)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_005.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsubsup\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msubsup\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(5)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_006.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi mathvariant=\"normal\"\u003eand\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e are short notations for \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_007.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi mathvariant=\"normal\"\u003eand\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e, respectively. Similarly, to define the underwater vehicle orientation, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_008.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eℜ\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/msup\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is described by the body’s co-ordinates in the inertial frame (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli et al., 2008\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e),\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_006\"\u003e\u003clabel\u003e(6)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_006.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003eψ\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(6)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_009.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the body Euler-angle coordinates in the inertial frame, commonly called roll, pitch and yaw, respectively,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_007\"\u003e\u003clabel\u003e(7)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_007.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(7)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_010.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the angular velocity vector of the origin of the body-fixed frame in order to attain that, with respect to the inertial frame of reference described in the body-fixed frame,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_008\"\u003e\u003clabel\u003e(8)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_008.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eJ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eη\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e,\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(8)where the vector describing the angular velocity of the body-fixed frame relative to the inertial frame is \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_011.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eν\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e. It must be observed that the \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_012.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eη\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e vector has no physical significance and its relationship to the body-fixed frame is through the proper Jacobian matrix \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_013.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eJ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e, and the Jacobian matrix \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_014.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eJ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e∈\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eℜ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi\u003eΧ\u003c/mi\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e, is within the Euler angles expressed as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_009\"\u003e\u003clabel\u003e(9)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_009.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eJ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eη\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003ecos\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eφ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmo\u003e.\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_004_s_003\"\u003e\u003cdiv\u003eEquations of motion\u003c/div\u003e\u003cp\u003eEquations of Motion for a rigid body can be obtained from Newton’s second law and a rigid body’s dynamic behavior is generally written in the notation of SNAME (1950) as described on Nomenclature appendix (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli, 2018\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli et al., 2008\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen, 1994\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e, \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_017\"\u003e2013\u003c/a\u003e). The rigid body dynamics can be written in component form as:\u003clist id=\"j_ijssis-2021-005_list1\" list-type=\"bullet\"\u003e\u003clist-item\u003e\u003cp\u003eFor translational motion:\n\u003cdisp-formula id=\"j_ijssis-2021-005_ueq_001\"\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ueq_001.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmi\u003eX\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_ueq_002\"\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ueq_002.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmi\u003eY\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_ueq_003\"\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ueq_003.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003c/list-item\u003e\u003clist-item\u003e\u003cp\u003eFor the rotational motion\u003c/p\u003e\u003c/list-item\u003e\u003c/list\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_eq_010\"\u003e\u003clabel\u003e(10)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_010.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmaligngroup/\u003e\u003cmspace width=\".25em\"/\u003e\u003cmspace 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accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmaligngroup/\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmaligngroup/\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo 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accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmaligngroup/\u003e\u003cmi\u003eN\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/msup\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmaligngroup/\u003e\u003cmspace width=\".25em\"/\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e−\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eThe above equations represented in a compact form as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_011\"\u003e\u003clabel\u003e(11)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_011.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eν\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eν\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003eν\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eτ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_015.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the rigid-body inertia matrix and \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_016.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e the rigid-body Coriolis and Centripetal matrix.\u003c/p\u003e\u003cp\u003eProperty 3.1 The parameterization of \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_017.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is unique and satisfies,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_012\"\u003e\u003clabel\u003e(12)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_012.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsubsup\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/mrow\u003e\u003c/msubsup\u003e\u003cmo\u003e\u0026gt;\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo\u003e;\u003c/mo\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eM\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_eq_013\"\u003e\u003clabel\u003e(13)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_013.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo 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stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eProperty 3.2, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_018.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e can always be parameterized such that it is skew-symmetrical, that is,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_014\"\u003e\u003clabel\u003e(14)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" 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stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003cdisp-formula id=\"j_ijssis-2021-005_eq_015a\"\u003e\u003clabel\u003e(15)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_015a.png\"\u003e\u003c/graphic\u003e\u003cmath 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stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eI\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ez\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eAn underwater vehicle’s general equation of motion can be written as (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e),\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_016\"\u003e\u003clabel\u003e(16)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_016.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eR\u003c/mi\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003eg\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003eη\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eτ\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(16)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_019.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eτ\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e vector of control inputs. The terminology of the rigid-body describe the rigid body’s equation of motion in an empty space. Nevertheless, hydrodynamic theories have been applied to the equation because ships and underwater vehicles require the presence of accelerations caused by fluid to be taken in to account, whereas hydrostatic concepts reflect the gravitational force and buoyancy that exist while a rigid body is completely or partially immersed in a fluid. The added mass and inertia and damping effects compose the hydrodynamic terms.\u003c/p\u003e\u003cp\u003eThe added inertia of the fluid around the body is accelerated by the motion of the body which must be included in the equations of motion when a rigid body is submerged which travels in a fluid. Fossen has stated that, as the air density is significantly lower than that of a moving mechanical system, it is possible to ignore the impact of the additional mass and inertia in industrial robotics (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen, 1994\u003c/a\u003e). In the case of an underwater vehicle application, the water and vehicle densities are comparable; for example, at 0\u003csup\u003eo\u003c/sup\u003eC, the density of fresh water is 1,002.68 and 1,028.48 kg/m\u003csup\u003e3\u003c/sup\u003e of seawater with 3.5% salinity. The fluid exerts a reaction force which is equal in the magnitude and opposite direction as a moving body accelerates the fluid around.\u003c/p\u003e\u003cp\u003eThis is the extra mass contribution consisting of the addition of mass inertia and the Coriolis and Centripetal matrices, \u003citalic\u003eM\u003c/italic\u003e\u003csub\u003e\u003citalic\u003eA\u003c/italic\u003e\u003c/sub\u003e and \u003citalic\u003eC\u003c/italic\u003e\u003csub\u003e\u003citalic\u003eA\u003c/italic\u003e\u003c/sub\u003e matrices, respectively. As stated in \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen (1994)\u003c/a\u003e, \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli et al. (2008)\u003c/a\u003e, and \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_003\"\u003eAntonelli (2018\u003c/a\u003e) owing to the body’s XZ-plane symmetry, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_020.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e can be generalized as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_017\"\u003e\u003clabel\u003e(17)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_017.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eFor example, because of the acceleration ν·, the hydrodynamic added mass force along the y-axis is expressed as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_018\"\u003e\u003clabel\u003e(18)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_018.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi mathvariant=\"normal\"\u003ewhere\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e≈\u003c/mo\u003e\u003cmfrac\u003e\u003cmrow\u003e\u003cmi\u003e∂\u003c/mi\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003e∂\u003c/mi\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eLikewise, the matrix of Coriolis and Centripetal \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_021.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e, can be extracted as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_019\"\u003e\u003clabel\u003e(19)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_019.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eC\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmi\u003eb\u003c/mi\u003e\u003cmmultiscripts\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cnone/\u003e\u003cnone/\u003e\u003cmprescripts/\u003e\u003cnone/\u003e\u003c/mmultiscripts\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eWhere,\n\u003cdisp-formula id=\"j_ijssis-2021-005_ueq_004\"\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ueq_004.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmn\u003e1\u003c/mn\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eThe damping of an underwater vessel traveling at high speed in 6-DOFs can usually be highly nonlinear and coupled. Even so, a conservative calculation that might be considered because of the vehicle symmetry is that terms above the second order are insignificant, indicating a diagonal structure with only linear and quadratic damping constraints (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_004\"\u003eAntonelli et al., 2008\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen, 1994\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e) as,\n\u003cdisp-formula id=\"j_ijssis-2021-005_eq_020\"\u003e\u003clabel\u003e(20)\u003c/label\u003e\u003calternatives\u003e\u003cgraphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_eq_020.png\"\u003e\u003c/graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"\u003e\u003cmi\u003eD\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmrow\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmrow\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmrow\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo 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stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003eP\u003c/mi\u003e\u003cmo 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stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmo\u003e+\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmrow\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo stretchy=\"true\"\u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e\u003c/p\u003e\u003cp\u003eThe gravitational and buoyant forces are considered restoring forces in hydrodynamics terminology, which behave on the vehicle’s CG and have components around the axes of the body. 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stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eW\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmi\u003ec\u003c/mi\u003e\u003cmi\u003eo\u003c/mi\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003esin\u003c/mi\u003e\u003cmspace width=\".25em\"/\u003e\u003cmi\u003e∅\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eG\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eW\u003c/mi\u003e\u003cmo\u003e‒\u003c/mo\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003ey\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003cmi\u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmi\u003es\u003c/mi\u003e\u003cmi\u003ei\u003c/mi\u003e\u003cmi\u003en\u003c/mi\u003e\u003cmi\u003eθ\u003c/mi\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/disp-formula\u003e(21)where \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_022.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eW\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003em\u003c/mi\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e is the submerged weight of the body, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_023.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eB\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmi\u003eρ\u003c/mi\u003e\u003cmo\u003e∇\u003c/mo\u003e\u003cmi\u003eg\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e the buoyancy, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_024.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eρ\u003c/mi\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e the fluid density, \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_025.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmo\u003e∇\u003c/mo\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e the volume of the body and \u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_026.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmi\u003eg\u003c/mi\u003e\u003cmo\u003e=\u003c/mo\u003e\u003cmsup\u003e\u003cmrow\u003e\u003cmo stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmrow\u003e\u003cmtable\u003e\u003cmtr\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mtd\u003e\u003cmtd\u003e\u003cmrow\u003e\u003cmn\u003e9.81\u003c/mn\u003e\u003c/mrow\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/mrow\u003e\u003cmo stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mrow\u003e\u003cmi\u003eT\u003c/mi\u003e\u003c/msup\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e the acceleration of gravity.\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_004_s_004\"\u003e\u003cdiv\u003eSimulink model\u003c/div\u003e\u003cp\u003e\u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_015\"\u003eFig. 15\u003c/a\u003e depicts the UNSW Canberra AUV mathematical model designed as a Simulink/Matlab model for the behavior of AUV (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_010\"\u003eFossen, 1994\u003c/a\u003e; \u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e). To understand and analyses the device dynamics of an AUV that involve hydrodynamic uncertainty, highly nonlinear, time varying and coupled, a simulink/Matlab analysis is used to analyses the UNSW Canberra model (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_014\"\u003eHassanein et al., 2011\u003c/a\u003e). The AUV hydrodynamic coefficients are calculated using the Computational Fluid Dynamics (CFD) approach in the Autonomous System Lab at UNSW Canberra, as tabulated in \u003ca ref-type=\"table\" href=\"#j_ijssis-2021-005_tab_002\"\u003eTable 2\u003c/a\u003e, to obtain the nonlinear mathematical model in different conditions (\u003ca ref-type=\"bibr\" href=\"#j_ijssis-2021-005_ref_025\"\u003eOsama et al., 2016\u003c/a\u003e).\u003c/p\u003e\u003ctable-wrap id=\"j_ijssis-2021-005_tab_002\" position=\"float\"\u003e\u003clabel\u003eTable 2.\u003c/label\u003e\u003ccaption\u003e\u003cp\u003eHydrodynamic coefficient of UNSW Canberra AUV model.\u003c/p\u003e\u003c/caption\u003e\u003ctable frame=\"hsides\"\u003e\u003ccolgroup span=\"1\"\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003ccol align=\"left\"/\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eParameter\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eCalculated value\u003c/th\u003e\u003cth align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eUnit\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_027.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒7.365\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_028.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eu\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_029.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.737\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_030.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.31\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_031.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.737\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_032.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.31\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_033.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.065\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_034.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.51\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_035.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.065\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_036.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.51\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_037.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒112.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_038.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" 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display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_041.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒112.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/m\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_042.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒62.45\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_043.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.250\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_044.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0.12\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_045.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.5975\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_046.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ep\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒0.3125\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_047.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e2.244\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_048.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ew\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_049.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒119.5\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_050.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003eq\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒59.75\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_051.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.244\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_052.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003ev\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_053.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003e|\u003c/mo\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒59.75\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/rad2\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_054.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmi\u003er\u003c/mi\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒31.25\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m\u003csup\u003e2\u003c/sup\u003e/sec\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_055.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eX\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eu\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.17\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_056.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eK\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ep\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e0\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_057.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒34.834\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_058.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_059.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eY\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_060.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eM\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.659\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_061.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ew\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒34.834\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_062.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003ev\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_063.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eZ\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003eq\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒1.042\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e\u003cinline-formula\u003e\u003calternatives\u003e\u003cinline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_ieq_064.png\"\u003e\u003c/inline-graphic\u003e\u003cmath xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"\u003e\u003cmsub\u003e\u003cmrow\u003e\u003cmi\u003eN\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow\u003e\u003cmover accent=\"true\"\u003e\u003cmi\u003er\u003c/mi\u003e\u003cmo\u003ė\u003c/mo\u003e\u003c/mover\u003e\u003c/mrow\u003e\u003c/msub\u003e\u003c/math\u003e\u003ctex-math/\u003e\u003c/alternatives\u003e\u003c/inline-formula\u003e\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003e‒2.659\u003c/td\u003e\u003ctd align=\"center\" rowspan=\"1\" colspan=\"1\"\u003eKg.m/rad\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/table-wrap\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_015\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 15:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eSimulink block diagram.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_015.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_015.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=5374724cd224a7fba2168db3d88dde25445715be96e1de21864d23e57e465055\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eThe step sources are to be configured as temporary inputs for the three input variables in order to determine their control on the different position outputs (these are \u003citalic\u003ex\u003c/italic\u003e, \u003citalic\u003ey\u003c/italic\u003e, and \u003citalic\u003ez\u003c/italic\u003e in meters, and the roll, the pitch, and the latter in degrees).\u003c/p\u003e\u003cp\u003eAs seen in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_016\"\u003eFigs. 16\u003c/a\u003e and \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_017\"\u003e17\u003c/a\u003e the surge input is a step from 0 to 10 at time = 30 seconds and no input for other variables, pitch and yaw. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_018\"\u003eFig. 18\u003c/a\u003e reveals that the only hydrodynamic influence relating to the x motion is in the x direction, it can also be seen that there is no hydrodynamic coupling between x and y, yaw, z and pitching motions. Since the AUV is a cylindrical hull vehicle, the rolling motion has no hydrodynamic impact.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_016\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 16:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eSurge input Force.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_016.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_016.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=4724782f2e0933b4bc3bf52e662f178bc7f6171efdca181fa3ba0394249c29ce\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_017\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 17:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003ePitch and yaw input Force.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_017.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_017.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=b8afab3b3565e4bb6b5a869a8b6e2775033896e1c7443a539d259ddf411e2266\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_018\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 18:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eSurge input responses.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_018.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_018.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=4ce9e5ca63d8c9d254580658f65b670a52654848bad7aea542a69f70b658aea4\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eA step input was then entered into the pitch motion as shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_017\"\u003eFig. 17\u003c/a\u003e, and the surge was set at its usual speed state (1 m/sec) so that the effects on the angles could actually be observed. The findings of the simulation in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_019\"\u003eFigs. 19\u003c/a\u003e and \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_020\"\u003e20\u003c/a\u003e demonstrate that hydrodynamic coupling occur between the pitch angle and the motion in x and z directions. There is no change experienced in y and yaw, this is a result of the vehicle symmetry.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_019\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 19:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003ePitch input response in AUV translational and orientational movement.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_019.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_019.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=90b2791daf50a375f056401bed8ad3460ce1a177ea917331adc77769f03c17f0\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_020\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 20:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003ePitch input response in AUV velocities.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_020.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_020.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=1c68c33e25fa903529a94a5162abb5d1869d4743eb6cd67d8a138f8792ac0611\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eIn the yaw movement, the step input as in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_017\"\u003eFig. 17\u003c/a\u003e was applied and the surge remained in its usual working condition. The simulation results shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_021\"\u003eFigs. 21\u003c/a\u003e and \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_022\"\u003e22\u003c/a\u003e prove that a hydrodynamic coupling occurs between the yaw angle and the motion in the direction x and y. y is calculated as positive value in the right direction. There is also no change in z value and pitch angle. Such initial findings are used for fuzzy and Hybrid Neural Fuzzy Network (HNFN) techniques in the identification approach.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_021\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 21:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eYaw input response in translational and orientational movement.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_021.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_021.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=17999\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=b972c55c014fbbf08feb5283c0742a7b017433f959196d3ee92d1e771938296f\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_022\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 22:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eYaw input response in AUV velocities.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_022.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_022.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=7374dcd637e5205f0e77afb2af53ee16d41f51aed29409117308145ce560a734\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_005\"\u003e\u003cdiv\u003eHardware in loop simulation (HIL) structure\u003c/div\u003e\u003cp\u003eIn order to test the identification and control algorithms before and after AUV testing, a real-time simulation is established. A basic test-bed built in-house with the serial port data communication capability is being used for validation. The autopilot systems of the UNSW Canberra AUV platform is made up of the PC104 microcontroller with separate I/O boards. There are block libraries in the xPC-Target toolbox that support certain PC104 peripheral cards with drivers. For collecting data from the encoder, the target block DM6814 xPC is used and the board’s base address is set to x320. Along with Matlab, a Null modem serial RS-232 cable communicates between the PC104 device and computer. This medium supports floating point numbers, integers and decimal values, Also data is transmitted and received at a constant baud rate. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_023\"\u003eFig. 23\u003c/a\u003e shows an overview of the HIL simulation where the PC104 transmits and receives data from the Simulink model in real time, using RS-232 serial communication.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_023\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 23:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eHardware In loop (HIL) simulation.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_023.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_023.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=f297f2d9db3d0d0cbb1d601f13af02312bbd534bf474508ccd87b1f6e9422dbc\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_006\"\u003e\u003cdiv\u003eExperimental results\u003c/div\u003e\u003cp\u003eThe real-tests have been explicitly undertaken to model the dynamics system and to excite the related dynamics. Many experiments have been conducted at UNSW Canberra to collect a range of data under various operating conditions. All such data are used by the identification algorithm to generate, tuning and identify the AUV dynamics model. The swimming pool at UNSW and the shallow water branch of Molonglo River in Canberra are chosen as test environments for the AUV to validate the effectiveness and the robustness of the algorithm. For coupled dynamics, a suitable manoeuvre is conducted such that all the control inputs are used and the coupled dynamics are excited over a period of time. The electronics, instruments, sensory system and on-board processing unit that used in the AUV are described previously in platform design section. \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_024\"\u003eFig. 24\u003c/a\u003e demonstrates the real-time manoeuvring of the AUV in a separate environments during the data collection process.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_024\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 24:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003eUNSW Canberra AUV during experimental teat data collection.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_024.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_024.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=8130e7ddd904d4f77dd975e12136d4d567e7ae3dff72f37e044baf0a32b3517d\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eThe ultimate goal of any controller design is to prove its validity in real conditions with unknown disturbances and the associated difficulties with real time implementation. This is a particularly challenging task for a complex, nonlinear and time-varying system like an AUV. Real-time tests are necessary to test the ability of the controller in the presence of unknown weather conditions and noise in the system. Due to the wind conditions and a strong water flow during the real time tests, there were fluctuations in the controlled value. To prevent vehicle fluctuations on the water surface, there had been a ± 2.5° dead-band across the pitch controller set point.\u003c/p\u003e\u003cp\u003eIn the present work, an indirect adaptive control system is applied to the AUV control problem where the controller parameters are adjusted based not only on the error between the reference input and the process output but also on the process sensitivity which can be approximately derived from the identification model of the process, The control system is based on the fuzzy system and HNFN (Hybrid Neural Fuzzy Network) techniques (Osama Hassanein et al., 2016).\u003c/p\u003e\u003cp\u003e\u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_025\"\u003eFigs 25\u003c/a\u003e and \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_026\"\u003e26\u003c/a\u003e demonstrate the related behavior of pumps for pitch and yaw motions. The required pitch angle is 0 for the pitch controller as shown in \u003ca ref-type=\"fig\" href=\"#j_ijssis-2021-005_fig_025\"\u003eFig. 25\u003c/a\u003e. The controller controls all the pumps in a pulse manner utilizing the PWM that is embedded into the controller programming. When the AUV began moving, it was going down, and then the controller instructed the UP (Up-direction) pump to drive the vehicle in the upward direction. However, the vehicle went behind the controller set-point; consequently, the DN (down-direction) pump was commanded by the controller to force the vehicle down.\u003c/p\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_025\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 25:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003e1st set of AUV control based fuzzy experimental results: coupled dynamics.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_025.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_025.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=6a81f0aab10ebb784f5e9b759d1596301fef1ab6a0e8038e581161415a0ac9a8\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cfigure id=\"j_ijssis-2021-005_fig_026\" fig-type=\"figure\"\u003e\u003ch2\u003eFigure 26:\u003c/h2\u003e\u003cfigCaption\u003e\u003cp\u003e2nd set of AUV control based HNFN experimental results: coupled dynamics.\u003c/p\u003e\u003c/figCaption\u003e\u003cimg xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ijssis-2021-005_fig_026.jpg\" src=\"https://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/62242a2c0d198124537c32f6/j_ijssis-2021-005_fig_026.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256\u0026amp;X-Amz-Date=20230202T121249Z\u0026amp;X-Amz-SignedHeaders=host\u0026amp;X-Amz-Expires=18000\u0026amp;X-Amz-Credential=AKIA6AP2G7AKP25APDM2%2F20230202%2Feu-central-1%2Fs3%2Faws4_request\u0026amp;X-Amz-Signature=5b92e3ea462175f47a901fa46e26860a0c35195f284c1896f13c3092c7bbfe02\" class=\"mw-100\"\u003e\u003c/img\u003e\u003c/figure\u003e\u003cp\u003eIn the yaw controller, the same criterion is used as it has the same pitch controller actions except with the left (LT) and right (RT) directions and it is obvious that the AVU follows the command given by the controllers. In the below figures, the blue line indicates the real AUV response and the desired trajectory is described by the red line.\u003c/p\u003e\u003c/sec\u003e\u003csec id=\"j_ijssis-2021-005_s_007\"\u003e\u003cdiv\u003eConclusion\u003c/div\u003e\u003cp\u003eThis paper addressed development and testing of a compact, cost-effective AUV model. A prototype for underwater vehicles has been developed at the UNSW Canberra Workshop and low-cost sensors, actuators and other instruments have been mounted on it to have six degrees of freedom during the operation. The system is configured with an on board data logger and control unit to monitor and process sensor and actuator data and to apply advanced system identification and control techniques. The fundamental AUV principle is used throughout the UNSW Canberra AUV simulation model, nonlinear effects and coupling factors correlated with AUV dynamics are examined. The model based adaptive control system utilizes intelligent techniques, considered as a perfect choice for the AUV built at UNSW Canberra on the account of non-linearity and uncertainty related to modeling. To validate the identification and control for the AUV platform, multiple experiment trials lasting over a number of days were performed. The swimming pool at UNSW Canberra, the shallow water branch of the Molonglo River and Gungahlin Lake in Canberra were selected to be the test operating conditions for the AUV to evaluate the effectiveness of the proposed control algorithm in order to build confidence in employing this controller for real life tasks. The test data demonstrate the usefulness of the developed AUV platform for identification and model-based control to accomplish autonomous operation in various environments.\u003c/p\u003e\u003c/sec\u003e\u003c/div\u003e","keywords":[{"title":"Keywords","language":null,"keywords":["Underwater vehicle","Autonomous","Modeling","AUV modeling","System integration"]}],"recentIssues":{"10.2478/ijssis-2022-0008":"\u003carticle-title\u003eEigen-structure problem optimization for multirate, multi-input multi-output systems applied to a roll rate autopilot\u003c/article-title\u003e","10.2478/ijssis-2022-0009":"\u003carticle-title\u003eThe ordinary negative changing refractive index for estimation of optical confinement factor\u003c/article-title\u003e","10.2478/ijssis-2022-0006":"\u003carticle-title\u003eModel based on the principles of smart agriculture to mitigate the effects of frost and improve agricultural production in the Cundiboyacense 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The role of the university is to select and channel books and book proposals for this publishing co-operation, as well as to promote this publishing opportunity to its faculty.","conferenceServices.first":"If you would like to learn more about these services, please contact Sales \u0026 Publishing Specialist — Services for conference organizers: \u003c1\u003ealexandru.vlad@sciendo.com\u003c/1\u003e or call directly \u003c3\u003e+44 2086388130\u003c/3\u003e.","conferenceServices.second":"Sciendo is the only company in the world that meets the two most important needs of an academic conference organizer. As well as publishing conference proceedings, we can also provide the organizer with one of the world's best event management systems. We have partnered with Cvent and Converia.","conferenceServices.third":"We can publish your conference proceedings and optionally provide you with the event management systems. We publish conference proceedings online using theOpen Access model. 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