Time-optimal Trajectory Planning of 6-RUS Parallel Robots Based on Grid Search Algorithm

doi:10.3969/j.issn.1004-132X.2023.13.008

Abstract

Abstract: Aiming at the problems of uneven reproducing trajectory and low trajectory planning efficiency of 6-RUS parallel painting robots， a trajectory planning method of optimal time of 6-RUS parallel painting robots was proposed based on optimizing the position of Bézier curve nodes. Firstly， the preprocessed trajectory was discretized to grid points， the node parameters were updated and the arc length of Bézier curve was optimized， and the path of small line segment was further fitted to obtain the optimal geometric path. Then， the optimal velocity corresponding to different coarse grid points and the solution time were calculated， and the appropriate coarse grid points were selected. The paths between grid points were further densified by small steps， and the forward and backward maximum velocity were iteratively solved. The maximum feasible velocity curve of the path was searched to obtain the optimal running time of the 6-RUS parallel robots. Finally， the experiments were carried out on a self-developed 6-RUS parallel robot platform. The results show that under the same teaching trajectory， the path length of the improved Bessel curve algorithm herein is as 8.12 m， which is better than that of traditional Bézier curve and G2CBS（G2 continuous cubic Bézier spiral）aigorithm. Meanwhile， the improved time-optimal path parameterization algorithm herein was used for the optimized teaching path. The solution time of the optimal velocity curve of the algorithm herein is as 416.4 ms， which is 244.7 ms less than that of TOPP-RA algorithm. Moreover， the time-optimal trajectory planning time of the algorithm herein is also better than that of TOPP-RA algorithm. The method improves the solving speed of the optimal velocity， shortens the track reproduction time of 6-RUS parallel robots， and improves the working efficiency.

Key words: 6-RUS parallel robot, improved time-optimal path parameterization, Bézier curve, grid search

摘要： 针对6-RUS并联喷涂机器人再现轨迹不平滑、轨迹规划效率低等问题，提出了基于优化贝塞尔曲线节点位置的6-RUS并联机器人时间最优轨迹规划方法。首先，将预处理的轨迹离散化为网格点，更新节点参数并优化贝塞尔曲线弧长，进一步拟合小线段路径获取最优几何路径；然后，计算不同粗网格点对应的最佳速度以及求解时间，选择合适的粗网格点，进一步以较小步长密化网格点间路径，迭代求解正反向最大速度，搜索路径的最佳速度曲线，获取6-RUS并联机器人的最佳运行时间。最后，在自研的6-RUS并联机器人平台上进行实验。结果表明，在相同示教轨迹条件下，基于所提的改进贝塞尔曲线算法得到的路径长为8.12 m，优于传统贝塞尔曲线算法以及G2CBS算法的结果；同时将改进的时间最优轨迹规划算法（TOPP）用于优化后的示教路径，所提算法的最优速度曲线的求解时间为416.4 ms，与TOPP-RA算法的最优速度曲线的求解时间相比缩短了244.7 ms，而且该算法下最优轨迹规划时间也优于TOPP-RA算法，该方法提高了最佳速度的求解速率，缩短了6-RUS并联机器人轨迹再现时间，提高了工作效率。

关键词: 6-RUS并联机器人, 改进的时间最优轨迹规划, 贝塞尔曲线, 网格搜索

CLC Number:

TH112

LIU Dongcai, DONG Guangyu, DU Yuhong, LI Wenpeng, . Time-optimal Trajectory Planning of 6-RUS Parallel Robots Based on Grid Search Algorithm[J]. China Mechanical Engineering, 2023, 34(13): 1589-1598.

刘栋财, 董广宇, 杜玉红, 李文鹏, . 基于网格搜索算法的6-RUS并联机器人时间最优轨迹规划[J]. 中国机械工程, 2023, 34(13): 1589-1598.

References

［1］BOBROW J， DUBOWSKY S， GIBSON J. Time-optimal Control of Robotic Manipulators along Specified Paths［J］. The International Journal of Robotics Research， 1985， 4（3）：3-17.
［2］SHIN K， MCKAY N. Selection of Near-minimum Time Geometric Paths for Robotic Manipulators［J］. IEEE Transactions on Automatic Control， 1986， 31（6）：501-511.
［3］PHAM Q C. Characterizing and Addressing Dynamic Singularities in the Time-optimal Path Parameterization Algorithm［C］∥2013 IEEE/RSJ International Conference on Intelligent Robots and Systems. Tokyo， 2013：2357-2363.
［4］KUNZ T， STILMAN M. Time-optimal Trajectory Generation for Path Following with Bounded Acceleration and Velocity［M］∥AGARWAL P， KUMAR S， RYDE J， et al. Robotics：Science and Systems Ⅷ. Cambridge：MIT Press， 2013：209-216.
［5］PHAM Q C. A General， Fast， and Robust Implementation of the Time-optimal Path Parameterization Algorithm［J］. IEEE Transactions on Robotics， 2014， 30（6）：1533-1540.
［6］OBERHERBER M， GATTRINGER H， MLLER A. Successive Dynamic Programming and Subsequent Spline Optimization for Smooth Time Optimal Robot Path Tracking［J］. Mechanical Sciences， 2015， 6（2）：245-254.
［7］CONSTANTINESCU D， CROFT E A. Smooth and Time-optimal Trajectory Planning for Industrial Manipulators along Specified Paths［J］. Journal of Robotic Systems， 2000， 17（5）：233-249.
［8］KASERER D， GATTRINGER H， MULLER A. Nearly Optimal Path Following with Jerk and Torque Rate Limits Using Dynamic Programming［J］. IEEE Transactions on Robotics， 2019， 35（2）：521-528.
［9］VERSCHEURE D， DEMEULENAERE B， SWEVERS J， et al. Time-optimal Trajectory Planning for Robots：a Convex Optimization Approach［J］. IEEE Transactions on Automatic Control， 2009， 54（10）：2318-2327.
［10］HAUSER K. Fast Interpolation and Time-optimization with Contact［J］. The International Journal of Robotics Research， 2014， 33（9）：1231-1250.
［11］PHAM H， PHAM Q C. A New Approach to Time-optimal Path Parameterization Based on Reachability Analysis［J］. IEEE Transactions on Robotics， 2018， 34（3）：645-659.
［12］HOBBYJ D. Smooth， Easy to Compute Interpolating Splines［J］. Discrete & Computational Geometry， 1986， 1（2）：123-140.
［13］CLAPHAM C， NICHOLSON J， NICHOLSON J R. The Concise Oxford Dictionary of Mathematics［M］. Oxford：Oxford University Press， 2014.
［14］KIRKPATRICK S， GELATT JR C D， VECCHI M P. Optimization by Simulated Annealing［J］. Science， 1983， 220（4598）：671-680.
［15］YANGK， JUNG D， SUKKARIEH S. Continuous Curvature Path-smoothing Algorithm Using Cubic Bezier Spiral Curves for Non-holonomic Robots［J］. Advanced Robotics， 2013， 27（4）：247-258.
［16］SEIDEL R. Small-dimensional Linear Programming and Convex Hulls Made Easy［J］. Discrete & Computational Geometry， 1991， 6：423-434.

[1]	HU Bo, AN Jinyun, YIN Lairong, ZHOU Changjiang. Calculation Method of Time-varying Meshing Stiffness of Small Module Gear Transmissions [J]. China Mechanical Engineering, 2024, 35(01): 74-82.
[2]	WANG Qiliang, LIU Tong, LI Yongqi, WEI Jianming, XU Meijuan, HONG Yongfeng. Topology Optimization Design of Flexible Parallel Microgrippers with Low Parasitic Displacements [J]. China Mechanical Engineering, 2023, 34(21): 2577-2584,2591.
[3]	PU Zhixin, GUO Jianwei, PAN Yuqi, BAI Yangxi. Performance Analysis and Optimization Design of 2PPaPaR Parallel Mechanism [J]. China Mechanical Engineering, 2023, 34(19): 2304-2312.
[4]	SHEN Huiping, LI XiaLI Ju, LI Tao, MENG Qingmei, WU Guanglei. Effect of Constrained or Unconstrained Branches on Kinematics and Stiffness Performance of PMs —Design，Analysis and Optimization of Two Novel Three-translation PMs with Partial Motion Decoupling Being Examples#br# [J]. China Mechanical Engineering, 2023, 34(13): 1533-1549.
[5]	DU Zhongqiu, SHEN Huiping, MENG Qingmei, LI Tao, YANG Tingli. Design and Performance Analysis of 8R Two-translational Spatial Parallel Mechanism with Motion Decoupling and Symbolic Positive Solutions [J]. China Mechanical Engineering, 2023, 34(12): 1425-1435.
[6]	SUN Liangbo, HONG Xixi, LIU Xiaocui, LIU Xin. New Definition of Basic Loop Set of Kinematic Chain and Its Generation Method [J]. China Mechanical Engineering, 2023, 34(09): 1061-1066，1076.
[7]	JI Jialu, YAO Yanan, ZHANG Ying, LAN Guorong. Design of Reconstruction Closed-chain Walking Platforms and Research on Obstacle-overcoming Strategy [J]. China Mechanical Engineering, 2023, 34(05): 524-532，542.
[8]	CHEN Xin, LIU Jiangnan, LONG Wangpeng, LYU Jianwen. Modelling and Analysis of Multiple-cross-springs Flexure Pivots with Arc Curved Beams [J]. China Mechanical Engineering, 2023, 34(03): 269-276，286.
[9]	TIAN Qihua, SHU Zhengtao, FU Junjian, DU Yixian, ZHOU Xiangman, TIAN Lei. Optimization Design Method for Graded Hollow Cellular Structures [J]. China Mechanical Engineering, 2022, 33(23): 2867-2878.
[10]	SUN Peng, LI Yanbiao, ZHANG Cong, SHAN Xiaohang, YUE Yi, WEI Baochen. Scale Optimization of Humanoid Robotic Arms Based on Kinematic Performance Analysis [J]. China Mechanical Engineering, 2022, 33(19): 2331-2340.
[11]	LIU Wenlan, YU Haixia, QU Huaiyu, MA Pengfei, GAO Xiaojun, WANG Jiansen. Configuration Synthesis of Working Mechanisms of Face-shovel Hydraulic Excavators by Adding Ⅱ-class Basic Bar Group [J]. China Mechanical Engineering, 2022, 33(16): 1919-1927.
[12]	XIE Zhijiang, CHENG Qing, DING Jun, HE Miao, FAN Naiji, WU Xiaoyong. Dimensional Design and Performance Analysis of a 6-DOF Parallel Manipulator with Two Limbs [J]. China Mechanical Engineering, 2022, 33(14): 1680-1690.
[13]	WANG Yixi, SHEN Huiping, CHEN Pu, WU Guanglei. Three-level Orderly Proceeding Optimization Design and Its Applications for Parallel Mechanisms Based on Kinematics，Stiffness and Dynamics [J]. China Mechanical Engineering, 2022, 33(13): 1560-1575,1621.
[14]	WANG Xianye, LIU Haitao, HUANG Tian. Dimensional Parameter Optimization of Planar Closed-loop Legged Mechanisms [J]. China Mechanical Engineering, 2022, 33(11): 1261-1268.
[15]	HUANG Kaiwei, SHEN Huiping, LI Ju, ZHU Zhongqi, YANG Tingli. Topological Design and Dynamics Modeling of a Spatial 2T1R Parallel Mechanism with Partially Motion Decoupling and Symbolic Forward Kinematics [J]. China Mechanical Engineering, 2022, 33(02): 160-169.