This work introduces the foundation of a new class of input shaping algorithm, designed based on the particle swarm optimisation. This algorithm is utilised to control the residual vibrations in the crane system. The motivation is the development of simple algorithms and architecture for controlling the motion in under-actuated nonlinear systems with minimal modelling effort. By recording the payload swing signal of the crane only once, the approach can automatically calculate the optimisation amplitude and time locations of the impulses required by a common zero-vibration-derivative (ZVD) technique. In this work, we use this algorithm to design a ZVD shaper for controlling the motion of an under-actuated nonlinear crane model system. We validate the approach using experiments. If this algorithm is implanted into the embedded system and applied to the actual crane, which will solve the problem of the traditional ZVD parameter adjustment and improve the vibration suppression effect of the ZVD Algorithm.

#### Keywords

- Crane
- Vibration suppression
- Zero-vibration-derivative
- Particle swarm optimisation

#### MSC 2010

- 70Q05

Cranes have been a critical component of many industries. There have been dramatic improvements in crane payload capacity during their history [1], but one major problem is shared between today's cranes and the cranes of centuries ago: payload oscillation. To solve the problem, some efficient vibration suppression algorithms have been developed. As one of the most efficient algorithms, input shaping had been used for vibration suppression of crane from 2000s [2, 3]. The algorithm was proposed to reduce vibration by slightly modifying the reference command through convolution of it with a series of impulses. Until the early 1990s, Signer et al. improved the robustness of this method, which allowed the algorithm to be applied. After that, Singhose, Seering and Singer proposed an extremely insensitive method to improve the robustness of the system [2]. Since then, input shaping has gradually formed a variety of methods. These include negative unity-magnitude (UM) shaping, specified-negative-amplitude (SNA) shaping, negative zero-vibration (ZV)shaping [4], negative zero-vibration-derivative (ZVD) [5] and shaping and negative zero-vibration-derivative–derivative (ZVDD) shaping.

Input shaping has been applied to a wide variety of cranes [6, 7], including bridge, tower and boom crane [8, 9]. As an open-loop algorithm, the input shaping algorithm is simple and efficient, and no additional sensors (or estimation algorithms) are needed. But the parameters of input shaping are difficult to determine. For example, there are five parameters to be determined in the algorithm of ZVD shaping, and they are coupled with each other. The best way to find the parameters of ZVD shaping is to build the model of crane system [10, 11]. But as an under-actuated system, the dynamics are highly nonlinear and complex [12]. Especially in the actual hoisting, the payload parameters and states are therefore difficult to measure. As a result the difficulty in measuring the payload parameters becomes an even larger problem.

In order to decrease the difficulty of parameter selection, many scholars focus on improving the robustness of input shaping algorithm [13, 14]. The robustness of the algorithm is increased by adding constraints, such as ZVDD. However, by contrast, the sensitivity of the algorithm is reduced and the dynamic performance of the system is sacrificed. Therefore, it is more important to find the parameters accurately than to improve the robustness of the input shaping algorithm. Recently, scholars have paid attention to the method of parameter selection of ZVD shaping [10, 15]. For instance, Vaughan J. proposed a method of designing input shaping based on the length of suspension cable only, and studied the accuracy of these estimates by simulation and experiment. Ha M. estimated the effect of the natural frequency error for residual vibration of flexible beam in ZV, ZVD and ZVDD shaping. Nonetheless so far, no efficient and accurate method has been proposed.

In this work we use an under-actuated nonlinear crane as the model system, and propose an automatic parameter selection ZVD (APS-ZVD) shaping algorithm based on particle swarm optimisation (PSO). Then, we verified the effectiveness of this algorithm by experiments, where the vibration has been suppressed by 89.85%. The vibration suppression frequency calculated by the APS-ZVD shaping algorithm agreed very well with the resonant frequency of the model system. The APS-ZVD shaping algorithm is simple and efficient, no modelling is required and it is easy to be implemented in engineering.

The principle of the traditional ZVD [16], [17] shaping algorithm is shown in Figure 1. In this algorithm, the speed commands are shaped through convolution of it with three different impulses.

The amplitude of these three impulses are defined as _{1}, _{2}, and _{3}, and the time location of these three impulses are defined as _{1}, _{2}, and _{3}, which are shown in Equations 1–7.

According to the difficulty of parameters selection in the traditional ZVD shaping algorithm, we proposed an APS-ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation. The APS-ZVD shaping algorithm comprises three parts: signal acquisition, evaluation and particle swarm optimisation, as shown in Figure 1. This algorithm comprises the following steps:

A square-wave speed command

A(t) is entered to the transfer function of ZVD input shaper _{i}

The suppressed pendulum angles _{i}_{e}

The e(t) are evaluated by transfer function of evaluation

The evaluated results are feedback to the PSO as the fitness values;

According to the fitness values, PSO adjusts the parameters (_{i}_{i}

Repeat steps 2–6 to find the optimal parameters of

Finally, the optimal

We input a square-wave speed signal to the system to let the trolley move for a short time, and collect the pendulum angle during the movement. The speed and acceleration of the trolley's movement should be as large as possible so as to do not damage the system. Because a short-time motion with a large acceleration contains enormous low-frequency vibration energy, under this circumstance the low-order resonance frequency of the system can be fully stimulated. In this way, the pendulum angle can be used to translate the characteristics more accurately.

As shown in Equations (1–6), there are five parameters to be optimised, _{1}, _{2}, _{3}, _{2} and _{3}. We assume that

The evaluation function should be able to evaluate the impact of adjustment time, amplitude and overshooting on the crane system. Considering the above factors, the design evaluation function can be written as Equation 8:
_{i}_{e}_{p}_{1} and _{2} are the weighted values, and _{p}_{1} and _{2} are used to reflect the relation between _{p}_{2}, but the response time will be extended.

The particle swarm optimisation algorithm is described as:
_{1} and c_{2} are study factors that represent the information exchange between each particle in the whole population; r_{1} and r_{2} are acceleration coefficients, which are random numbers that are uniformly distributed in the range of [0,1] and are used to increase the randomness of particle movement. Equation 17 is the fitness function of the PSO. By finding the minimum value of Equation 17, the optimal values of K and T can be reached, and then by putting them back into Equations 7–12, the optimal Ai and ti can be obtained. Finally, the ZVD shaper C(t) can be structured by the optimised Ai and ti, and then be put behind the speed command to suppress vibration by shaping the speed command.

In order to simulate the real crane structure, an experimental platform was built, as shown in Figure 8. The platform comprises a servo motor, a pendulum (14.5 mm) and a payload (0.05 kg). The encoder records the swing angle of the pendulum.

Figure 3 shows the swing angles of pendulum recorded at the trolley's speed of 480 mm/s. It can be seen that the swing angle of the pendulum was very large (35 o) when the trolley stops at the beginning (at 4.5 s), then it decreased gradually until 0 o at 12.4 s.

The recorded swing angles of pendulum were input to APS-ZVD (as described in Figure 2). The key parameters of APS-ZVD algorithm

Initial parameters of PSO

1 | 0.6 | |

2 | _{1} |
2 |

3 | c_{2} |
2 |

4 | Dimension | 2 |

5 | _{1} |
1 |

6 | _{2} |
1 |

Figure 5 shows the fitness curve of PSO. After 10 iterations, the final fitness value stabilised at 26.68. Compared with 44.6 in the first iteration, the fitness value decreased by 40.17%.

The optimisation curves of parameter

Putting _{1} = 0, t_{2} = 0.3626, t_{3} = 0.7251, and then the ZVD shaper equation can be written as:

We put the achieved C(t) in front of the model system to verify the effect of APS-ZVD, the swing angles of pendulum with APS-ZVD and without APS-ZVD as shown in Figure 8. With APS-ZVD, the time to stop swinging is shortened from 7.9 s to 1.26 s, and the maximum swing angle is decreased from 35° to 6.6°.

We next quantified the swing energy of the pendulum with and without the APS-ZVD using Equation 13:

We did the Fourier transform for the swing of the pendulum and found the resonance frequency to be 1.4 Hz (shown in Figure 9, blue). We also detected the sensitivity curve of APS-ZVD (shown in Figure 9, green), and found the optimal frequency of the suppression is 1.38 Hz, which is matched very well with the resonance frequency of the pendulum. The Fourier transform for the swing of the pendulum with APS-ZVD is also plotted in Figure 7, shown in the red, where the swing amplitude has been sharply reduced by the order of three.

In the actual hoisting, usually the position of trolley and the length of cable are changed frequently, which affects the vibration suppression effect. Thus, we changed the pendulum length to detect the suppression effect of optimal parameter found by APS-ZVD, and the results are as shown in Table 2. In Table 2, Pendulum 1 is the original one which is used to generate the optimal parameters. The length of Pendulum 2 is extended by 6.9% compared to 1, and the length of Pendulum 3 is shortened by 13.8% compared to 1. The parameters found by the APS-ZVD algorithm based on the length of Pendulum 1 were used in all three experiments (_{1} = 0.3708, _{2} = 0.4985, _{3} = 0.2231, _{1} = 0, _{2} = 0.3626, _{3} = 0.7251), and the vibration suppression results are obtained.

Vibration suppression effect

1 | 14.5 | 89.85 | 5.63 | 1.90 |

2 | 15.5 | 86.87 | 5.889 | 1.72 |

3 | 12.5 | 62.46 | 7.56 | 2.26 |

In the case of Pendulum 3, the pendulum length changed by 13.8% compared to 1; the damping percentage of APS-ZVD shaper can still achieve 62%, and the peak swing angle and the adjusting time of residual vibration did not change significantly. The comparison shows that the robustness of the selected parameters can be ensured by our designed APS-ZVD algorithm.

In this work, an APS-ZVD shaper based on particle swarm optimization algorithm has been put forward. The key advantage of this algorithm is that by collecting the swing angle of the crane system once, the algorithm can calculate the ZVD parameters automatically. The experimental results show that the central suppression frequency of APS-ZVD matched very well with the resonance frequency of the model crane system, and vibration suppression effect was obtained and the maximum reduction can reach 89.85%. The accuracy and robustness of the selected parameters of the APS-ZVD algorithm have been verified in both time and frequency domains. We believe the proposed APS-ZVD algorithm will provide a new idea for the application of ZVD shaper in the crane industry.

#### Initial parameters of PSO

1 | 0.6 | |

2 | _{1} |
2 |

3 | c_{2} |
2 |

4 | Dimension | 2 |

5 | _{1} |
1 |

6 | _{2} |
1 |

#### Vibration suppression effect

1 | 14.5 | 89.85 | 5.63 | 1.90 |

2 | 15.5 | 86.87 | 5.889 | 1.72 |

3 | 12.5 | 62.46 | 7.56 | 2.26 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model