Trajectory planning of a spatial flexible manipulator for vibration suppression

doi:10.1016/j.robot.2019.103316

Robotics and Autonomous Systems

Volume 123, January 2020, 103316

https://doi.org/10.1016/j.robot.2019.103316 Get rights and content

Abstract

Vibration decreases operational accuracy and productivity of flexible manipulators, and trajectory planning is an effective way to suppress vibration. However, the existing trajectory planning methods can only suppress vibration of planar flexible manipulators. In this paper, a trajectory planning method is proposed to suppress vibration of spatial flexible manipulators. Firstly, the dynamic models of each link of the flexible manipulator are established separately. Then with the constraint equations, the dynamic model of the flexible manipulator is established as a Differential Algebraic Equation (DAE). Secondly, the trajectory functions are designed as quintic polynomials, and the conditions are deduced to satisfy acceleration limits of each joint. Finally, the trajectory planning problem is transferred to an optimal problem. Particle Swam Optimization (PSO) is adopted to solve the optimal problem. Numerical simulation is conducted to demonstrate the good performance of the proposed method.

Introduction

Compared with traditional rigid manipulators, flexible manipulators are light weight and energy-saving. Therefore, they have important application value in many fields, such as aerospace, nuclear plants, and construction sites. In these scenarios, flexible manipulators have to move from one point to the other point in order to pick or release, scan and weld objects, which can be classified as point-to-point motions. However, due to the low stiffness of flexible links, a flexible manipulator would vibrates severely at the end of each point-to-point motion. Vibration not only influences operational accuracy in a negative way, but also accelerates fatigue damage of flexible links. Therefore, how to suppress residual vibration of flexible manipulators at the end of a point-to-point motion is a significant research topic. Robot motion planning can be divided into three levels: path planning, trajectory planning and path tracking [1]. Trajectory planning is an effective way to suppress residual vibration.

There are countless possible trajectories that can be used to move a flexible manipulator between two points. The residual vibrations of flexible links depend on the trajectory used to move a flexible manipulator between the start and the end point. The task of trajectory planning is to find the optimal trajectory among the possible trajectories, which can suppress residual vibrations effectively. The existing trajectory planning methods can be classified into two categories [2]. The first is called direct method. For direct method, after defining trajectories as specific functions, the trajectory planning problem is transferred to a parameter optimization problem. Then many heuristic algorithms can solve the optimization problem. For instance, in [3], assumed mode method (AMM) is applied to establish the dynamic model of a two-link planar flexible manipulator. Then the trajectories are defined as fourth order polynomials and genetic algorithm is employed to solve the optimization problem. In [4], for a single link flexible manipulator, a neural network is adopted as the trajectory function and PSO is used to optimize the parameters of the neural network in order to minimize operational energy and suppress residual vibration. Based on neural networks and genetic algorithm, Abe also proposes a trajectory planning method for a rigid manipulator mounted on a flexible base [5]. For a flexible space robot, Wu et al. establish the dynamic model via AMM. Then the trajectory functions are defined as B spline curves, of which the control points are optimized by PSO [6]. Kojima et al. use four cubic polynomials to describe angular velocities of flexible manipulator’s two joints. Then genetic algorithm is adopted to optimize the parameters of the cubic polynomials [7]. In [8], for a planar flexible–rigid manipulator, the dynamic model is built by AMM, the trajectories are defined as cubic splines, and PSO is used to optimize the control points of the splines. In [9], a direct method is also proposed for planar flexible manipulators, and the trajectory functions are defined as high order polynomials.

The other trajectory planning method is called indirectmethod. For indirect method, the trajectory planning problem is transferred to an optimal control problem, of which the performance index is to minimize operational energy and suppress vibration. Then based on Pontryagin Minimum Principle, the optimal control problem is transferred to a two-point boundary value problem (TPBVP). Via solving the TPBVP, the optimal trajectory can be determined. For instance, Korayem et al. establish the dynamic model of a two-link planar flexible manipulator via AMM [2]. Then an indirect method is proposed to generate an optimal trajectory. In [10], the dynamic model of a flexible manipulator mounted on a mobile base is established by AMM. Then the indirect method is applied to minimize vibration and operational energy. Besides, the torques of each joint are bounded. Boscariol et al. also proposed an indirect method to generate a trajectory for flexible multi-body planar manipulators [11]. However, for a spatial flexible manipulator, because it is difficult to determine the mode functions for each spatial flexible link, AMM is not applicable to this situation. Therefore, the aforementioned trajectory planning methods can only suppress vibration of planar flexible manipulators. Besides, due to the same reason, the aforementioned methods also cannot suppress vibration of flexible manipulators with curved links.

In this paper, a trajectory planning method is proposed to suppress vibration of spatial flexible manipulators with curved links. In Section 2 , based on Absolute Nodal Coordinate Formulation (ANCF) and Lagrange equation, the dynamic models of each link of the spatial flexible manipulator are established separately. Then the constraint equations of each joint are deduced. With the combination of flexible links’ dynamic models and constraint equations, the dynamic model of the spatial flexible manipulator is established as DAE. In Section 3, the trajectory functions are defined as quintic polynomials. In order to satisfy acceleration limits of each joint, the sufficient and necessary conditions are deduced. Then, the trajectory planning problem is transferred to a parameter optimization problem, and PSO is adopted to find the optimal parameters. The simulation is conducted in Section 4 to demonstrate the effectiveness of the proposed trajectory planning algorithm. The conclusion is made in Section 5.

Section snippets

Problem definition

As shown in Fig. 1, one end of the curved flexible beam is fixed with the base and the other end is articulated with a rigid manipulator which has N links. $O_{A}$ $- X$ $_{A}$ $Y_{A}$ $Z_{A}$ denotes the base frame. $O_{B}$ $- X$ $_{B}$ $Y_{B}$ $Z_{B}$ is the frame fixed with the end of the flexible beam. $O_{i}$ $- X$ $_{i}$ $Y_{i}$ $Z_{i}$ (i $=$ 1 $\sim$ N) is the body frame of the i $th$ rigid link. The direction of gravity is alone the opposite direction of $O_{A}$ $Z_{A}$ . $θ_{i}$ is the angle of the i $th$ joint, and $θ$ $=$ [ $θ_{1}$ $, \dots,$ $θ_{N}$ ] $^{T}$ . The angular acceleration of the i

Design of trajectory function

For each joint, the trajectory function $θ_{d, i}$ is defined as $θ_{d, i} (t) = a_{i, 0} + a_{i, 1} t + a_{i, 2} t^{2} + a_{i, 3} t^{3} + a_{i, 4} t^{4} + a_{i, 5} t^{5} (i = 1, \dots, N)$

Because the spatial flexible manipulator will complete the point-to-point motion defined in Section 2, the trajectory function should satisfy the following conditions $θ_{d, i} (0) = θ_{0, i}$ ${\dot{θ}}_{d, i} (0) = v_{0, i}$ $θ_{d, i} (t_{f}) = θ_{f, i}$ ${\dot{θ}}_{d, i} (t_{f}) = v_{f, i} (i = 1, \dots, N),$ where $θ_{0, i}$ is the initial angle of joint i and $θ_{f, i}$ is the final angle of joint i. $v_{0, i}$ is the initial velocity of joint i and $v_{f, i}$ is the final velocity

Simulation

Numerical simulations are conducted in this section to demonstrate the effectiveness of the proposed trajectory planning algorithm. Here the spatial flexible manipulator with a three-link rigid manipulator mounted on a curved flexible beam is adopted. Three rigid links are the same. The parameters of the spatial flexible manipulator is shown in Table 1, and the parameters of the proposed trajectory planning algorithm is shown in Table 2. In order to demonstrate that the proposed trajectory

Conclusion

For spatial flexible manipulators, a trajectory planning algorithm is proposed in this paper in order to suppress residual vibration and satisfy the acceleration limits of each joint. The dynamic model of the spatial flexible manipulator is established as DAE based on ANCF and Lagrange equation. The trajectory functions of each joint are designed as quintic polynomials and the sufficient and necessary conditions for acceleration limits of each joint are deduced. Then the trajectory planning

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant 61722309 and U1613218.

Leilei Cui received the B.Eng. degree in automation from Northwestern Polytechnical University, Xian, China, in 2016, and the M.S. degree in control engineering at Shanghai Jiao Tong University, Shanghai, China, in 2019. He is currently a Ph.D. candidate in the Control and Networks Lab, Tandon School of Engineering, New York University. His research interests include robot control, reinforcement learning, adaptive dynamic programming (ADP), and optimal control.

References (21)

BenosmanM. et al.
Rest-to-rest motion for planar multi-link flexible manipulator through backward recursion
IFAC Proc. Vol.
(2003)
KorayemM.H. et al.
Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints
Appl. Math. Model.
(2012)
BoscariolP. et al.
Model-based trajectory planning for flexible-link mechanisms with bounded jerk
Robot. Comput. Integr. Manuf.
(2013)
R. Haschke, Erik Weitnauer, Helge Ritter, On-Line Planning of Time-Optimal, Jerk-Limited Trajectories, in: IEEE...
KorayemM.H. et al.
Trajectory optimization of flexible link manipulators in point-to-point motion
Robotica
(2008)
FarisW.F. et al.
Energy minimization approach for a two-link flexible manipulator
J. Vib. Control
(2009)
AbeA.
Minimum energy trajectory planning for vibration control of a flexible manipulator using a multi-objective optimisation approach
Int. J. Mechatron. Autom.
(2012)
A. Abe, Minimum energy trajectory planning method for robot manipulator mounted on flexible base, in: IEEE Asian...
WuH. et al.
Optimal trajectory planning of a flexible dual-arm space robot with vibration reduction
J. Intell. Robot. Syst.
(2004)
H. Kojima, T. Kibe, Optimal trajectory planning of a two-link flexible robot arm based on genetic algorithmfor residual...

There are more references available in the full text version of this article.

Cited by (50)

Vibration-polynomial-based optimal trajectory planning for mobile concrete pumping equipment with state constraints
2024, Automation in Construction
To improve the operating stabilities of the flexible hydraulic manipulator in concrete pumping equipment, a trajectory planning method based on optimal vibration energy is presented. First, predefined trajectories are generated by the quintic polynomial method, and kinematic inversions are established by unifying the velocity and angle constraints into the velocity level. Then, relationships between the boom vibration and the vibration energy are solved in modal space. Specifically, hydraulic actuators are regarded as spring-damping systems to calculate generalized forces into the vibration equation. The vibration equation can solve the modal coordinate and vibration model function, such that the elastic potential energy to cause vibration is completely described. Next, floating values are captured by a genetic algorithm to minimize vibration energy, and then introduced to the predefined trajectory to weaken the boom vibration. Last, the proposed trajectory planning method is verified by a concrete pumping spreader with a 13 m hydraulic manipulator.
Three link flexible arm robust regulation via proportional retarded control scheme
2024, Applied Mathematical Modelling
In this paper, a delayed distributed proportional control is developed for a class of multiple-input-multiple-output systems in order to reduce the effects of external disturbances and uncertain dynamics. The system under consideration is a three degrees-of-freedom planar robot, consisting of an active joint driving the first link and passive flexible joints for the second and third links. The control challenge for this system is to drive the flexible arm around the desired setpoint or trajectory by controlling the first link. Actually, by using a specific Lyapunov-Krasovskii storage functional associated with a constrained optimization problem, the robustness of the proposed proportional retarded control action is achieved. This analysis, together with the attractive ellipsoid concept allows to conclude with Ultimately-Uniformly-Bounded stability. The closed-loop control algorithm is tested in numerical simulation and a comparative study with some existing recent control schemes based on the performance error is presented.
Hybrid vision/strain-based control strategy for a parallel manipulator with flexible links
2024, Mechanism and Machine Theory
Parallel manipulators may present higher speed/acceleration ratios and energy efficiency when compared with serial manipulators. In addition, reducing their components’ inertia can further improve performance. However, this design alternative might yield vibrations requiring novel joint and task space control strategies. While the former involves precise models, the latter might require adequate computation vision schemes. These requirements impose critical challenges. This manuscript proposes a hybrid control strategy and experimentally investigates using a 3RRR prototype with flexible links. This hybrid strategy comprises two control loops: position-based visual servo and strain-based feedback control schemes. The strain-based loop uses a sliding mode control and the time derivative of the signals obtained by strain gauges installed in the flexible links. Compared with the position-based visual servo scheme, the hybrid strategy can considerably attenuate the vibrations since the approach reduced the overshoot response by 36% and the maximum value of the Euclidean error on position by 38%, on average.
Robust deflection control and analysis of a fishing rod-type flexible robotic manipulator for collaborative robotics
2023, Robotics and Autonomous Systems
Citation Excerpt :
Various researcher have used different techniques for modelling flexible manipulators. Using the Euler–Lagrange approach [19,20] and Lagrange [21,22] proposed dynamic modelling of flexible manipulators with several flexible linkages and joints. Heidari et al. [23] have come up with a new nonlinear finite element model that is intended for use in the dynamic analysis of flexible link manipulators.
Due to high-speed operation at low inertia, flexible manipulators are becoming more and more popular in today’s world. These manipulators produce excessive vibration, which must be reduced using an efficient control technique for the manipulator to function well. In the present study, a flexible manipulator model with a single link is built in such a manner that it can be considered as a flexible fishing rod. The free end of the flexible rod has a provision for applying a payload, which serves to give a deflection of the flexible rod. A lumped parameter method is adopted for modelling the flexible rod as well as the string using the Sim-Mechanics tool in MATLAB. Simulations were carried out for sudden and sinusoidal loading. It has been found that the flexible link produces excessive vibration under both sudden and sinusoidal loading. A proportional integral derivative (PID) controller is used to suppress the excessive vibration generated in the simulation model. Four different locations (Location 1: 15 cm; Location 2: 30 cm; Location 3: 45 cm; and Location 4: 60 cm) are selected for controller positioning. Simulation revealed that the minimum deflection was observed at location 4, i.e., at the tip for both sudden and sinusoidal loading. The developed model is validated using two loading conditions, viz., the beam’s self-weight and a point load of 30 N at the free end. It has been found that the simulation results resemble the analytical results with an error of 0.44% and 0.36% for both the loading conditions.
Research on vibration suppression and trajectory tracking control strategy of a flexible link manipulator
2022, Applied Mathematical Modelling
Citation Excerpt :
In [18], trajectory planning is studied for a mobile manipulator system consisting of flexible links, which converts the optimal control problem into a two-point boundary value problem by employing Pontryagin's minimum principle and deriving the optimality conditions, and the effectiveness and capability of the trajectory planning method is verified using computer simulations. Cui et al. [19] deduced the dynamics model of the flexible linkage robot arm based on the absolute coordinate method, while using the joint acceleration constraints and boundary conditions to establish the optimization objective of minimizing the residual vibration, and completed the joint trajectory optimization with the help of the traditional PSO algorithm. In [20], the dynamics equation is derived based on assumed modes method (AMM) for a planar flexible-link manipulator, while the open-loop optimal control principle is used to transform the trajectory planning into the general form of the optimal control problem, and the two-point edge value problem is solved based on the Pontryagin's minimum principle to obtain the optimal path to carry out the problem satisfying the minimum vibration, maximum load, etc.
Residual vibrations caused by point-to-point motions reduce the positioning accuracy and dynamic performance of flexible link manipulators, and the trajectory planning method, a well-known open-loop control strategy, has been proven to effectively suppress the residual vibration in point-to-point motion. However, this open-loop control method has limitations in dealing with model uncertainty and disturbance problems. In this paper, a control strategy based on dynamics-based trajectory planning and fuzzy self-tuning PD (FST PD) controller is proposed, and applied to achieve residual vibration suppression and trajectory tracking of a flexible link manipulator. To improve the trajectory tracking performance under external disturbances, this control strategy is combined with a linear extended state observer (LESO). Firstly, the dynamic model of the system is deduced using the floating reference frame and Lagrange equation. On this basis, the objective function to minimize the lateral deformation of the tip is constructed, and then the optimal trajectory is obtained by using the Crossbreeding PSO (CBPSO) algorithm. Secondly, the optimal trajectory is used as the desired trajectory to suppress residual vibration, and the FST PD controller combined with the LESO is used to achieve high-precision trajectory tracking control of the flexible link manipulator. Finally, the effectiveness and feasibility of the composite control strategy is verified through multiple simulations. The proposed method can provide support for vibration suppression of flexible link manipulators; meanwhile, the controller structure of this method is simple and easy to implement, and the use of sensors/actuators and other equipment in the system is reduced, which effectively reduces the economic cost.
Generalized Input Preshaping Vibration Control Approach for Multi-Link Flexible Manipulators using Machine Intelligence
2022, Mechatronics
Citation Excerpt :
There are three main categories of open-loop input-shaping vibration control methods. These include input preshaping with an S-curve or B-spline profile [7–13], preprocessing filters [14–19], and tuning time parameters [23–27]. The input preshaping method is a viable option for resolving the problem.
In this study, an open-loop vibration control through parameter tuning method for multi-link flexible manipulators is generalized using the machine learning paradigm. The experimental studies part of the study consists of two stages. In the first stage, a decision tree model is created using the C4.5 algorithm to predict transient (during motion) and residual (stationary) vibration levels, and decision rules are derived from this model. The paper utilizes the C4.5 (decision tree) algorithm on a broad experimental dataset and compares its performance to that of other four well-known traditional machine learning algorithms (Naive Bayes (NB), Multilayer Perceptron (MLP), Support Vector Machine (SVM), and k-Nearest Neighbor (kNN)). The reason for choosing the decision tree method is that it has a transparent decision-making mechanism that works in tandem with the Explainable Artificial Intelligence (XAI) perspective. The experiments showed that the C4.5 algorithm achieved the best classification performance for predicting both root mean square (RMS) values of residual and transient vibrations (RMSres and RMStrans) with 92.25% and 83.75% accuracy rates. In the second stage, the generalized coefficients for use in parameter tuning were found by interpreting the rules obtained from this model. Furthermore, the rules derived from the tree in this study were applied to different systems. Control applications showed that an average of 93.9% and 84.5% suppression ratios in residual and transient vibrations could be achieved with this method. In addition, vibration settling times were shortened by an average of 41.4 seconds. Following the experimental studies, the simulation was used to compare results with the aid of Matlab Simscape-Multibody. The results prove that the rules obtained from the tree in this study generate successful results in several different systems.

View all citing articles on Scopus

Hesheng Wang (SM’15) received the B.Eng. degree in electrical engineering from the Harbin Institute of Technology, Harbin, China, in 2002, and the M.Phil. and Ph.D. degrees in automation and computer-aided engineering from the Chinese University of Hong Kong, Hong Kong, in 2004 and 2007, respectively. From 2007 to 2009, he was a Postdoctoral Fellow and a Research Assistant in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong. He has been with Shanghai Jiao Tong University, Shanghai, China, since 2009, where he is currently a Professor with the Department of Automation. He was a Visiting Professor at the University of Zurich, Zurich, Switzerland. His research interests include visual servoing, service robot, robot control, and computer vision.

Dr. Wang is an Associate Editor of Robotics and Biomimetics, Assembly Automation, the International Journal of Humanoid Robotics, and the IEEE TRANSACTIONS ON ROBOTICS . He served as an Associate Editor on the Conference Editorial Board of the IEEE Robotics and Automation Society from 2011 to 2015. He is actively involved in organizing international conferences. He served as an Organizing Committee Member for many international conferences such as the IEEE International Conference on Robotics and Automation (ICRA) and the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). He was the General Chair of the 2016 IEEE International Conference on Real-time Computing and Robotics, and the Program Chair of the 2014 IEEE International Conference on Robotics and Biomimetics. He is the Program Chair of the 2019 IEEE/American Society of Mechanical Engineering International Conference on Advanced Intelligent Mechatronics.

Weidong Chen received the B.S. and M.S. degrees in control engineering, and the Ph.D. degree in mechatronics from the Harbin Institute of Technology, Harbin, China, in 1990, 1993, and 1996, respectively.

Since 1996, he has been with Shanghai JiaoTong University, Shanghai, China, where he is currently the Head and a Professor with the Department of Automation, and is the Director of the Institute of Robotics and Intelligent Processing. He is the Founder of the Autonomous Robot Laboratory with Shanghai Jiao Tong University. He was a Visiting Professor in the Artificial Intelligence Laboratory, University of Zurich, Zurich, Switzerland, in 2012. He is a Visiting Professor with the Brain Science Life Support Research Center, University of ElectroCommunications, Japan, from 2016 to 2017. His current research interests include autonomous robotics, assistive robotics, and medical robotics.

View full text

Trajectory planning of a spatial flexible manipulator for vibration suppression

Abstract

Introduction

Section snippets

Problem definition

Design of trajectory function

Simulation

Conclusion

Declaration of Competing Interest

Acknowledgments

IFAC Proc. Vol.

Appl. Math. Model.

Robot. Comput. Integr. Manuf.

Trajectory optimization of flexible link manipulators in point-to-point motion

Robotica

Energy minimization approach for a two-link flexible manipulator

J. Vib. Control

Minimum energy trajectory planning for vibration control of a flexible manipulator using a multi-objective optimisation approach

Int. J. Mechatron. Autom.

Optimal trajectory planning of a flexible dual-arm space robot with vibration reduction

J. Intell. Robot. Syst.