Effective inspection of pipelines is of significant importance in the industry. In order to reduce human labour, risk and expenses in this area, robots can be applied. The pipe robot should perform not only the linear motion in the axial direction but also the rotary motion in the circumferential direction while working. Meanwhile, in order to achieve reasonable performance and efficiency, these robots should have high enough normal linear motion velocity. They should be able to cross a pipe in one direction to another pipe in a different direction to achieve the overobstacle function. In this study, a doublejoint wheeled robot model is established to fulfil the actions of fast walking, rotation around the pipe and overobstacle on spatial pipes. Based on the expected position on the pipe for the robot constructed by the proposed model, the precise starting position and control synthesis of the axial and circumferential velocities are initially determined. In order to study the velocity synthesis more accurately, the pipe is unfolded along the axial direction to transform the spatial motion into a plane motion. A novel spatial distribution form of driving wheels is proposed. In addition, based on the initial position of the straight, Tshaped, Lshaped, twoaxis ‘+’shaped and spatial threeaxis ‘+’shaped pipes, the posture adjustment of the robot prior, during and after the overobstacle is investigated. Furthermore, the corresponding planning algorithms are established.
Keywords
 pipe climbing robot
 spatial velocity synthesis
 overobstacle algorithm
 path planning
 spatial pipe
At present, the inspection and maintenance of pipelines are mainly performed by staff who climb up these pipes. The majority of the assigned tasks are repetitive with high labour intensity, low work efficiency and extremely high risk. Moreover, many industrial transport pipes have hazardous features such as high temperature, high pressure, high toxicity and radiant fluids, which add to the risk and hazard of the job. If these pipes are not regularly inspected and maintained, significant economic loss, injuries or even human casualty may be caused once a leakage occurs. Therefore, developing climbing robots for outdoor pipelines with specific capabilities, such as climbing and passing over obstacles, is in high demand. It is expected that these robots can replace humans in the task of regular inspection and maintenance of pipes.
Takahiro Fukui et al. [1] proposed autonomous gait transition and galloping over unperceived obstacles of a quadruped robot with central pattern generators (CPG) modulated by vestibular feedback. Laura Hak et al. [2] applied stability margins to better investigate variations of the obstacle crossing strategy. Moreover, Heeseung Hong et al. [3] presented a composite locomotive strategy by estimating the contact angle for a stairclimbing mobile platform. Since then, this strategy has been widely applied for designing and analysing the corresponding performance of many wallclimbing robots [4,5,6,7,8]. Subsequently, the optimal design and kinetic analysis were proposed for stairclimbing mobile robots in different structures [9,10]. Morales et al. [11] studied the obstacle surpassing and posture control of a stairclimbing robotic mechanism. Sorin and Nitulescu [12] studied locomotion over common types of obstacles for a legged mobile robot. Furthermore, Ryosuke Yajima et al. [13] studied the tracked vehicle from different aspects, including the ability to traverse obstacles. Fengyu Xu et al. [14] proposed the obstaclenegotiation capability of rodclimbing robots and improved the design mechanisms. Du Qiaoling et al. [15] designed a microrobot that enables the robot to climb a straight pole with a fivebar linkage. Pfotzer et al. [16] proposed an autonomous navigation system for reconfigurable snakelike robots in challenging unknown environments. Adrián Peidró et al. [17] designed compact switchable magnetic grippers for the HyReCRo structureclimbing robot. Ma and Mao [18] used global path planning and local path adjustment for the coal mine robot to avoid obstacles in gaspolluted areas. Recently, Lili Bykerk [19] proposed a method for identifying beam members in truss structures based on the tactile data. Klein and Haugland [20] performed the obstacleaware optimisation for cable layouts of offshore wind farms.
The literature review performed indicates that numerous investigations have been conducted so far on different aspects of pipeline robots; however, only a few of them are dedicated to the velocity synthesis of robots when walking on threedimensional (3D) pipes. In addition, posture adjustments of the robot prior to the overobstacle, processing the overobstacle and the robot posture after the overobstacle are rarely studied for 3D pipes.
The overobstacle outpipe climbing robot can not only walk fast on the straight pipe but can also can rotate around the pipe. Therefore, these types of robots can be effectively applied in diverse configurations of pipes, including the plane twoaxis, Tshaped, ‘+’shaped, Lshaped and spatial threeaxis ‘+’shaped pipes. It is worth noting that these robots require special structural theory models. To this end, it is intended to design a novel robot based on previous achievements and new requirements.
Figure 1 presents the configuration of the outpipe climbing robot. It indicates that the robot is generally divided into two parts (A and B), where these parts are hung with an axis. Each part has two legs that can be opened and closed. Moreover, there is a driving assembly at the end of each leg. Figure 1 shows that each part contains two guiding wheels.
In order to study the synthetic velocity direction of the robot walking on the pipe more accurately, the outer circumference of the pipe is expanded along the axis direction. This is schematically illustrated in Figure 2, where W_{A1} and W_{A2} denote the contact points between the guiding wheel and the pipe on part A of the robot, while W_{A3} and W_{A4} are the contact points between the driving wheel and the pipe on part A of the robot. All the points are located on the same circumferential line. Similarly, W_{B1} and W_{B2} are the contact points between the guiding wheel and the pipe on part B of the robot. Moreover, W_{B3} and W_{B4} are the contact points between the guiding wheel and the driving wheel, and the pipe on part B, respectively. All the points are located on the same circumferential line. Points W_{A1}, W_{A2}, W_{B1}, and W_{B2} are collinear and parallel to the axis. The angle between the lines W_{A1}W_{B2} and W_{A3}W_{B3} is
Figure 3 shows the rotation of the driving wheels around the pipe axis. When the velocities of all the driving wheels W_{A3}, W_{A4}, W_{B3} and W_{B4} are to the right, the resultant velocity will also be to the right so that the entire machine walks towards the right, and it rotates forward around the pipe. On the contrary, when the velocities of all driving wheels W_{A3}, W_{A4}, W_{B3} and W_{B4} are to the left, the entire machine walks to the left and performs a reverse rotation around the pipe. The robot stops moving when all the driving wheels stop their movement. Based on the performed analysis, the proposed robot can only rotate around the pipe. In other words, it cannot move straight along the pipe axis with no rotation.
Figure 4 presents the driving wheel axis perpendicular to the pipe axis. When all the velocities of driving wheels W_{A3}, W_{A4}, W_{B3} and W_{B4} are upward, the resultant velocity of the whole robot machine is upward, so that the whole machine walks upwards. On the contrary, when all the velocities of the driving wheels W_{A3}, W_{A4}, W_{B3}, and W_{B4} are downward, the whole machine goes down. Furthermore, the resultant velocity for stationary driving wheels is zero, so that under this circumstance, the robot stops moving. Based on the performed analysis, the proposed robot can only go straight along the pipe axis. In other words, it cannot rotate around the pipe axis.
Figure 4(a) shows the paths of the driving wheels along the double inverted Vdirection. It is worth noting that in this scheme, driving wheels W_{A3}, W_{A4}, W_{B3}, and W_{B4} are controlled by the electronic control system. When all the front driving wheels (W_{A3} and W_{A4}) and the rear driving wheels (W_{B3} and W_{B4}) move upwards in accordance with the inverted ‘V’shaped direction, then the resultant velocity will be upward and the whole machine will move upward. On the other hand, when all driving wheels move downwards along the inverted ‘V’shaped direction, the resultant velocity is downward, and the whole machine moves downwards accordingly. Finally, the whole machine stops for stationary driving wheels. It should be indicated that there are different methods to analyse the resultant velocity. Based on the performed analysis, it is found that the robot can move upward, downward or stay stationary in a certain place, but it cannot rotate around the pipe axis.
This case is similar to that of the previous part, where the driving wheels walk along the inverted ‘V’shaped direction. Based on the performed analysis of the resultant velocity shown in Figure 6(a) and (b), it is concluded that the robot can only move upward, downward or stay stationary in a certain place. In other words, it cannot perform a complete rotation around the pipe axis.
In this case, the axis of the driving wheels in part A is arranged in a ‘V’shape, while the axis of the driving wheels in part B is arranged in an inverted ‘V’shape. Accordingly, the upper driving wheels (W_{A3} and W_{A4}) and the lower driving wheels (W_{B3} and W_{B4}) walk along the ‘V’shaped and inverted ‘V’shaped directions, respectively. Moreover, the angles between the upper and lower driving wheels and the axes are equal. Under these circumstances, the resultant velocity of the whole machine is upward, so that the whole machine moves upwards.
Figure 7(b) illustrates that when the upper driving wheels (W_{A3} and W_{A4}) and the lower driving wheels (W_{B3} and W_{B4}) walk downward, the resultant velocity of the whole machine is downward, so that the whole robot moves downward.
As shown in Figure 7(c), the upper driving wheel W_{A3} is downward, while the upper driving wheel W_{A4} moves upward at a constant velocity. The lower driving wheel W_{B3} is upward, while the lower driving wheel W_{B4} walks downward at a constant velocity. Subsequently, the resultant velocity of the whole machine is to the right, so that the whole robot moves to the right along the pipe circumference. Under this circumstance, the robot rotates to the right around the pipe axis.
As shown in Figure 7(d), when the upper driving wheel W_{A3} moves upward, the upper driving wheel W_{A4} moves downward at a constant velocity, the lower driving wheel W_{B3} is downward and the lower driving wheel W_{B4} walks upward at a constant velocity. Therefore, the resultant velocity of the whole machine is to the left, so that the whole robot walks to the left along the circumference. In this case, the robot rotates to the left around the pipe axis.
Based on the performed analysis in different working conditions, it is found that the robot can achieve upward and downward movements, as well as rotation around the pipe axis. Therefore, this scheme is feasible to be applied as a powerful method to design overobstacle outpipe climbing robots.
Figure 8(a) indicates that in this scheme, the axes of the driving wheels in parts A and B are set as inverted ‘V’shape and ‘V’shape, respectively. Accordingly, the upper driving wheels (W_{A3} and W_{A4}) walk along the inverted ‘V’shaped direction, while the lower driving wheels (W_{B3} and W_{B4}) move along the ‘V’shaped direction. Moreover, the angles between the upper and lower driving wheels and the axis direction are equal. When all the upper and lower driving wheels move upward, the resultant velocity of the whole machine is upward, so that the whole robot walks upwards.
Figure 8(b) indicates that when all the upper and lower driving wheels walk downward, the resultant velocity of the whole machine is downward, so that the entire robot moves downward.
Figure 8(c) shows that when the upper driving wheel W_{A3} moves downward, the upper driving wheel W_{A4} walks upward at a constant velocity, while the lower driving wheel W_{B3} moves upward and the lower driving wheel W_{B4} walks downward at a constant velocity. Subsequently, the resultant velocity of the whole machine is to the left, so that the whole robot walks to the left along the circumference. In this case, the robot rotates to the left around the pipe axis.
According to Figure 8(d), it is found that when the upper driving wheel W_{A3} moves upward, the upper driving wheel W_{A4} walks downward at a constant velocity. The lower driving wheel W_{B3} moves downward, while the lower driving wheel W_{B4} walks upward at a constant velocity. Subsequently, the resultant velocity of the whole machine is to the right, so that the whole robot walks to the right along the circumference. Under this circumference, the robot rotates to the right around the pipe axis.
Based on the performed analysis in this section, it is concluded that the robot can achieve downward and upward movements, as well as rotation around the pipe axis. Therefore, this scheme is feasible to be applied as a powerful method to design overobstacle outpipe climbing robots.
Based on the above analysis for the driving wheel velocity synthesis, two feasible schemes are proposed. Considering the control constraints, driving wheels are selected to walk along the ‘V’shaped and inverted ‘V’shaped models.
Figure 8 shows the design model for investigating the overobstacle performance of the robot. Figure 8(a) illustrates that the robot adjusts its posture by walking and rotating on the horizontal rod to reach the vicinity of the vertical rod. The following is the gait analysis of the robot:
Adjust the posture by walking and rotating to the plane defined by the horizontal rod axis and the vertical rod axis, as shown in (a).
Release the pipeholding mechanism of part
Part
Part
After reaching the position shown in (c), the pipeholding mechanism of part
Starting from (c), the driving wheels on part
Part
The pipeholding mechanism of part
When the entire robot runs on the pipe, all parts of the robot are moving. Moreover, relative positions among the robot body components change due to the rotation. In order to determine the position of the robot in the 3D space and provide a reference for the pose of the robot body joints, a coordinate origin should be determined. Through analysing the structure of the robot without considering the pose of the robot, it is observed that there is only one axis with a constant distance from the pipe centre, which is the axis of the hinge joint of parts
Definition of parameters.
R  Outer radius of the pipe, 
r  Minimum distance from the coordinate origin 
a  Initial movement distance from the origin 
b  Initial movement distance from the origin 
c  Initial movement distance from the origin 
The initial position angle (rad) around the 

The initial position angle (rad) around the 

The initial position angle (rad) around the 

A1  Positive trajectory of the robot coordinate origin 
B1  Inverse trajectory of the robot coordinate origin. 
According to the robot theory, the corresponding translation and rotation matrices are defined as follows:
The translation matrix
Rotation matrix around the
Rotation matrix around the
Rotation matrix around the
The homogeneous vector corresponding to the coordinate origin
Considering the objective existence of the pipe radius and the robot spatial size and the achievement of the mutual overobstacle function between the pipes, the initial positions on different axes are determined as follows:
The initial position with xaxis as the pipe axis
Based on the hypothetical stepbystep method of the robot particle on the pipe, [21] the initial position of the
The initial position of the
The initial position of the
There are many types of pipe combinations in real applications. Therefore, several typical types are considered as the research objects. These research objects include straight pipes, ‘+’shaped pipes with twoaxes in the same plane, Tshaped pipes with twoaxes in the same plane, Lshaped pipes with twoaxes in the same plane and spatial threeaxis ‘+’shaped pipes.
The complete overobstacle process of the robot has four aspects, including the posture adjustment before the overobstacle, robot position after the posture adjustment, overobstacle process and the robot position after the overobstacle. In order to simplify the expression of the motion planning, the posture adjustment algorithm is defined as Matrix
Fig. 10 shows that straight pipes can be placed horizontally, vertically and inclined in space. There are three forms of robot movement on the pipe, which include linear movements along the pipe axis, rotation around the pipe and spiral movement around the pipe axis. Since the submovement of the robot studied in the present article is the movement along the pipe axis and rotation around the pipe, the spiral movement around the pipe axis is a stepcombined movement of these two movements. The movement on the horizontal pipe is taken as an example to study the different planning methods.
The linear movement on the horizontal pipe is the translational movement from the initial point Stpx(
The new position when the initial point Stpx(
There are two paths for the step spiral movement. One is to rotate Δ
The other type of step spiral movement initially performs translation
The results obtained from Equations (11) and (12) are the same, although the process can be very different. When approaching the overobstacle position, the robot should first rotate and then perform translation to adjust the posture. However, in order to prevent the overobstacle position, the translation should be initially performed and then the robot should rotate to adjust the posture. This can prevent the robot from interfering with the pipe during the rotation process. Therefore, the robot cannot achieve the expected action.
Straight pipe planning includes vertical, horizontal and inclined pipe planning and is the basis of complicated planning. When the robot is climbing on the pipe, all motion states are synthesised from the basic motions.
The ‘+’shaped pipe with twoaxes in the same plane means that the axis of the ‘+’shaped pipe has two axes and the two axes are perpendicular to each other in a plane. It should be indicated that planning mainly includes the transition process planning and overobstacle planning. The structure of the plane ‘+’shaped pipe is relatively complicated in the plane planning with multiple paths and full of information. Therefore, it is also the emphasis of the research.
It should be indicated that the overobstacle on the twoaxis plane ‘+’shaped pipe is more complicated than walking on a straight pipe. The outpipe climbing robot should adjust its posture to perform an overobstacle at a specific location when climbing the pipe. The specific position refers to the plane defined by the two pipe axes. Figure 11 shows that there are no pipes in the
On each projection diagram, the clockwise motion is defined as the forward motion, while the counterclockwise motion is defined as the reverse motion. The overobstacle from the horizontal pipe to the vertical pipe in the first quadrant is taken as an example. The corresponding motion planning and algorithm are as follows:
The initial position is the result of Equation (7). This can be expressed as stpy(
Starting from the initial
Stretch legs of part
Part
Part
The legs of part
The legs of part
Part
Part
The legs of part
Move on the vertical pipe.
It should be indicated that the overobstacle planning and algorithms in other quadrants are consistent with the abovementioned principles.
According to Equations (1) and (3), the posture adjustment matrix before the overobstacle can be expressed as follows:
According to Equations (1) and (13), the overobstacle process matrix approaching from the yaxis to the zaxis is as follows:
According to Equations (1) and (13), the position after the posture adjustment is as follows:
According to Equations and (14), the position algorithm after the overobstacle is as follows:
The structure of the Tshaped pipe is also a part of the twoaxis ‘+’shaped structure. Moreover, its motion planning is included in the ‘+’shaped pipe planning. The differences are studied as follows.
Figure 12 shows the planning diagram of the Tshaped pipe. Compared with the ‘+’shaped pipe, there is one less branch pipe. Comparing Figure 12 with Figure 11 shows that most of the planning details are the same, while they are different only in some ways. The different parts are introduced in detail in Figure 12.
Since Pipe 4 does not exist, the second quadrant, the third quadrant, ⑦ and ⑧ of the planning path do not exist. The planning between the second and third quadrants is the same as the planning is in the right view. However, the angles around one or three axes are different.
The structure of the Lshaped pipe is a part of the twoaxis ‘+’shaped structure. Therefore, the motion planning is included in the ‘+’shaped pipe planning. However, they have differences that will be shown in this study.
Figure 13 shows the planning diagram of the Lshaped pipe. Compared with the ‘+’shaped pipe, it only has two branch pipes. Comparing Figure 13 with Figure 11, it is observed that many of planning is the same, while they are only partly different. The different parts are introduced in detail as follows:
Since Pipes 3 and 4 do not exist, the second, third and fourth quadrants, ⑤, ⑥, ⑦, and ⑧ of the planning path do not exist.
The spatial threeaxis ‘+’shaped pipe means that there are three axes of the pipe, which are perpendicular to each other. This is similar to the three coordinate axes of the spatial coordinate system. The motion planning of this structure is the most complicated planning among all, which should be supported by a plane planning.
Figure 14 illustrates a projection diagram of a spatial threeaxis ‘+’shaped pipe. It is observed that there are six pipes, which are indicated by the corresponding symbols. The front view, the top view and the left view are represented as
In the spatial threeaxis ‘+’shaped overobstacle process, the path planning from one pipe to an adjacent pipe is Ltype planning. There are 24 planning routes in total. The Ltype planning has been studied above, and will not be discussed here.
Figure 14 shows that the overobstacle planning is complicated between Pipes 2 and 4, 1 and 3, as well as 5 and 6, while these three methods are the same. Therefore, only one of them is studied. For example, planning from Pipes 2–4 is investigated, in which there are numerous motion plannings.
Figure 15 shows the detailed path scheme. According to the scheme, there are four planned paths from Pipes 2–4. Since the methods are the same, only the planning from Pipes 2–1 and subsequently from Pipes 1–4 is studied. It should be indicated that if the planning from Pipes 2–1 is an Ltype positive planning, then the planning from Pipes 14 is an Ltype inverse planning. It is necessary to rotate 180° around the axis of Pipe 1 between the two ‘L’type plannings. The algorithm of the overobstacle from Pipes 2–1 and then from 1–4 is described as follows:
The following conclusions are drawn from the present study:
The equivalent method of spatial pipeline expansion is proposed. Therefore, the robot motion is transformed from a space problem study to a plane problem study, which simplifies the analysis process.
The model of pipe climbing robot with obstacle surmounting function is established. Based on the model, the posture analysis of the driving wheel in space and the principle of the velocity synthesis solve the difficult problems of the robot moving straight along the pipe, rotating around the pipe and overobstacle. Finally, the model showing that the driving wheels walk along the combination of the ‘V’shaped and inverted ‘V’shaped axes is determined.
The present study determines the initial position of the robot on the pipe based on the hypothetical stepbystep method, which provides the theoretical foundation for planning and obstacle crossing analysis.
The present study initially studies the planning methods of the robot on a straight pipe, such as upward, downward, hovering and rotating along the pipe. Then, based on the twoaxis ‘’shaped pipe, it is focused on the path planning, posture adjustment algorithm, overobstacle algorithm, state algorithm after posture adjustment, and state algorithm after overobstacle in the four quadrants of the coordinate system and the process of overobstacle from the side. Moreover, the overobstacle plannings of the Tshaped pipe and Lshaped pipe are studied, respectively. Furthermore, combined with the doubleaxis ‘+’shaped pipe planning principles and algorithms, the threeaxis ‘+’shaped spatial pipe planning and algorithm is studied.
Definition of parameters.
R  Outer radius of the pipe, 
r  Minimum distance from the coordinate origin 
a  Initial movement distance from the origin 
b  Initial movement distance from the origin 
c  Initial movement distance from the origin 
The initial position angle (rad) around the 

The initial position angle (rad) around the 

The initial position angle (rad) around the 

A1  Positive trajectory of the robot coordinate origin 
B1  Inverse trajectory of the robot coordinate origin. 
Regarding new wave distributions of the nonlinear integropartial Ito differential and fifthorder integrable equations Nonlinear Mathematical Modelling of Bone Damage and Remodelling Behaviour in Human Femur Value Creation of Real Estate Company Spinoff Property Service Company Listing Entrepreneur's Passion and Entrepreneurial Opportunity Identification: A Moderated Mediation Effect Model Applications of the extended rational sinecosine and sinhcosh techniques to some nonlinear complex models arising in mathematical physics Study on the Classification of Forestry Infrastructure from the Perspective of Supply Based on the Classical Quartering Method A Modified Iterative Method for Solving Nonlinear Functional Equation New Principles of NonLinear Integral Inequalities on Time Scales Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Analysis of the agglomeration of Chinese manufacturing industries and its effect on economic growth in different regions after entering the new normal Study on the social impact Assessment of Primary Land Development: Empirical Analysis of Public Opinion Survey on New Town Development in Pinggu District of Beijing Possible Relations between Brightest Central Galaxies and Their Host Galaxies Clusters and Groups Attitude control for the rigid spacecraft with the improved extended state observer An empirical investigation of physical literacybased adolescent health promotion MHD 3dimensional nanofluid flow induced by a powerlaw stretching sheet with thermal radiation, heat and mass fluxes The research of power allocation algorithm with lower computational complexity for nonorthogonal multiple access Research on the normalisation method of logging curves: taking XJ Oilfield as an example A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ Tcells model On the interaction of species capable of explosive growth Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator Combustion stability control of gasoline compression ignition (GCI) under lowload conditions: A review Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method The Comprehensive Diagnostic Method Combining Rough Sets and Evidence Theory Study on Establishment and Improvement Strategy of Aviation Equipment Design of softwaredefined network experimental teaching scheme based on virtualised Environment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode System dynamics model of output of ball mill The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Overlimit Interpolation Equation Optimal allocation of microgrid using a differential multiagent multiobjective evolution algorithm About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Cartesian space robot manipulator clamping movement in ROS simulation and experiment Effects of internal/external EGR and combustion phase on gasoline compression ignition at lowload condition Research of urban waterfront space planning and design based on childrenfriendly idea Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Impact of Web Page House Listing Cues on Internet Rental Research on management and control strategy of Ebikes based on attribute reduction method A study of aerial courtyard of super highrise building based on optimisation of space structure Exact solutions of (2 + 1)AblowitzKaupNewellSegur equation