Performance Evaluation and Dimensional Synthesis of a 3PRRR Parallel Manipulator

doi:10.3901/JME.260409

Abstract

Abstract: To solve the dynamics and stiffness modeling difficulties caused by the multi-closed loop and multi-constraint characteristics of parallel mechanisms (PMs), an integrated method of elastodynamic and stiffness modeling for PMs that is applicable to both the pre-design and the prototype design stages is proposed. The model first establishes the element stiffness and mass matrices of each free body based on the matrix structure analysis method, and then extracts the global independent generalized displacement coordinates of the mechanism in the global coordinate system by combining multi-point constraint theory and kinematic constraint equations. The elastodynamic equation of the mechanism is established through Lagrange equation, and the elastostatic stiffness model corresponding to the exit node is established through static condensation technology. Based on the Lagrange equation and overall stiffness matrix, an ellipsoid index for the extreme stiffness and isotropy of the mechanism is established. A multi-objective genetic algorithm is used to establish the Pareto front of the multi-objective optimization problem. Further, the cooperative equilibrium point method is combined to determine a set of optimal solutions on the Pareto front. On this basis, the prototype of the 3PRRR PM is designed. The stiffness and mass matrices of components involving irregular cross-sections are extracted using finite element software to replace those of beam elements in the pre-design stage to achieve rapid performance evaluation of the three-dimensional model of the 3PRRR PM. The maximum error between the proposed theoretical model and the finite element model is within 1.48%. The performance comparison of the mechanism before and after optimization shows that the fundamental frequency, isotropic of the linear stiffness and angular stiffness have been improved by 2.3%, 19.05%, and 43.75%, respectively. The reasonable cross-section design of the links based on the screw theory further improves the performance indices of the mechanism, verifying the correctness and effectiveness of the model proposed in this work.

Key words: parallel manipulator, elastodynamic, stiffness modeling, performance evaluation, dimensional synthesis

CLC Number:

TG156

ZHU Gaoyong, LU Yebo, ZHANG Lingling, YE Wei, YANG Chao. Performance Evaluation and Dimensional Synthesis of a 3PRRR Parallel Manipulator[J]. Journal of Mechanical Engineering, 2026, 62(9): 88-103.

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: http://www.cjmenet.com.cn/EN/10.3901/JME.260409

http://www.cjmenet.com.cn/EN/Y2026/V62/I9/88

References

[1] LIANG Dong，MAO Yi，SONG Yimin，et al. Kinematics，dynamics and multi-objective optimization based on singularity-free task workspace for a novel SCARA parallel manipulator[J]. Journal of Mechanical Science and Technology，2024，38(1)：423-438.
[2] ZHAO Kai，LIU Zhihua，CAI Chenguang，et al. Design and calibration of the 6-DOF motion tracking system integrated on the Stewart parallel manipulator[J]. Optics Express，2024，32(1)：287-300.
[3] CHENG Hongtai，ZHANG Zhifei，LI Wei，et al. Dynamic error modeling and compensation in high speed delta robot pick-and-place process[C]// 2015 IEEE International Conference on Cyber Technology in Automation，Control，and Intelligent Systems，2015：36-41.
[4] KAZEZKHAN G，XU Qian，WANG Na，et al. Performance analysis and optimization of a modified stewart platform for the qitai radio telescope[J]. Research in Astronomy and Astrophysics，2023，23(9)：095022.
[5] 董成林，岳巍，刘海涛，等. TriMule与Tricept机器人刚度解析与比对[J]. 机械工程学报，2021，57(19)：30-38. DONG Chenglin，YUE Wei，LIU Haitao，et al. Stiffness analysis and comparison of trimule and tricept robots[J]. Journal of Mechanical Engineering，2021，57(19)：30-38.
[6] 张俊，赵艳芹. Exechon并联模块的静刚度建模与分析[J]. 机械工程学报，2016，52(19)：34-41. ZHANG Jun，ZHAO Yanqin. Stiffness modeling and evaluation for exechon parallel kinematic machine module[J]. Journal of Mechanical Engineering，2016，52(19)：34-41.
[7] CHEN Xiang，LIU Xinjun，XIE Fugui，et al. A comparison study on motion/force transmissibility of two typical 3-DOF parallel manipulators：The sprint z3 and a3 tool heads[J]. International Journal of Advanced Robotic Systems，2014，11：5.
[8] GERMAIN C，BRIOT S，CARO S，et al. Natural frequency computation of parallel robots[J]. Journal of Computational and Nonlinear Dynamics，2015，10(2)：021004.
[9] PAN Lanqing，ZHANG J W，ZHANG Dan，et al. Modeling and analysis of a novel 3R parallel compliant mechanism[J]. Machines，2023，11(3)：375.
[10] SHENG Lianchao，LI Wei，WANG Yuqiao，et al. Rigid-flexible coupling dynamic model of a flexible planar parallel robot for modal characteristics research[J]. Advances in Mechanical Engineering，2019，11(1)：1-10.
[11] LIU Zhen，YANG Song，DING Tao，et al. Dynamic modeling and performance analysis of a lower-mobility parallel robot with a rotatable platform[J]. Mathematical Biosciences and Engineering，2023，20(2)：3918-3943.
[12] ZHANG Jun，ZHAO Y Q，CECCARELLI M. Elastodynamic model-based vibration characteristics prediction of a three prismatic-revolute-spherical parallel kinematic machine[J]. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME，2016，138(4)：041009.
[13] LIAN Binbin，WANG Lihui，WANG X V. Elastodynamic modeling and parameter sensitivity analysis of a parallel manipulator with articulated traveling plate[J]. International Journal of Advanced Manufacturing Technology，2019，102(5-8)：1583-1599.
[14] CHEBILI Z，ALEM S，DJEFFAL S. Comparative analysis of stiffness in redundant co-axial spherical parallel manipulator using matrix structural analysis and vjm method[J]. Iranian Journal of Science and Technology-Transactions of Mechanical Engineering，2023，47(4)：2225-2241.
[15] HU Bo，HUANG Zhen. Kinetostatic model of overconstrained lower mobility parallel manipulators[J]. Nonlinear Dynamics，2016，86(1)：309-322.
[16] 许允斗，姚建涛，金林茹，等. 考虑杆件空间复合弹性变形的过约束并联机构受力分析方法[J]. 机械工程学报，2015，51(7)：53-60. XU Yundou，YAO Jiantao，JIN Linru，et al. Method for force analysis of the overconstrained parallel mechanism considering the link’s spatial composite elastic deformations[J]. Journal of Mechanical Engineering，2015，51(7)：53-60.
[17] DEBLAISE D，HERNOT X，MAURINE P. A systematic analytical method for PKM stiffness matrix calculation[C]// ICRA 2006. Proceedings 2006 IEEE International Conference on Robotics and Automation，2006：4213-4219.
[18] KLIMCHIK A，PASHKEVICH A，CHABLAT D. Fundamentals of manipulator stiffness modeling using matrix structural analysis[J]. Mechanism and Machine Theory，2019，133：365-394.
[19] YANG Chao，LI Qinchuan，CHEN Qiaohong，et al. Elastostatic stiffness modeling of overconstrained parallel manipulators[J]. Mechanism and Machine Theory，2018，122：58-74.
[20] GUO Jinwei，ZHAO Yongsheng，CHEN Bo，et al. Force analysis of the overconstrained mechanisms based on equivalent stiffness considering limb axial deformation[J]. Chinese Journal of Mechanical Engineering，2023，36(1)：23.
[21] HAN Bo，JIANG Yuan，YANG Wei，et al. Kinematics characteristics analysis of a 3-UPS/S parallel airborne stabilized platform[J]. Aerospace Science and Technology，2023，134：108163.
[22] BEN-HORIN P，SHOHAM M. Application of grassmann-cayley algebra to geometrical interpretation of parallel robot singularities[J]. International Journal of Robotics Research，2009，28(1)：127-141.
[23] LIU Xinjun，WU Chao，WANG Jinsong，et al. A new index for the performance evaluation of parallel manipulators：a study on planar parallel manipulators[C]// Proceedings of the 7th World Congress on Intelligent Control and Automation，June 25- 27，2008，Chongqing，China. 2008：353-357.
[24] GONG Yaping，LOU Junbin. Natural frequency prediction of the 3-RPS parallel manipulator using the substructure synthesis technology[J]. Robotica，2023，41(9)：2850-2860.
[25] CARBONE G，CECCARELLI M. Comparison of indices for stiffness performance evaluation[J]. Frontiers of Mechanical Engineering，2010，5(3)：270-278.
[26] LI Yangmin，XU Qingsong. Stiffness analysis for a 3-PUU parallel kinematic machine[J]. Mechanism and Machine Theory，2008，43(2)：186-200.
[27] COURTEILLE E，DEBLAISE D，MAURINE P，et al. Design optimization of a delta-like parallel robot through global stiffness performance evaluation[C]// 2009 IEEE-RSJ International Conference on Intelligent Robots and Systems，2009：5159-5166.
[28] YAN S J，ONG S K，NEE A Y C. Stiffness analysis of parallelogram-type parallel manipulators using a strain energy method[J]. Robotics and Computer-Integrated Manufacturing，2016，37：13-22.
[29] GORGULU I，DEDE M I C. A new stiffness performance index：Volumetric isotropy index[J]. Machines，2019，7(2)：44.
[30] WANG Ruizhou，ZHANG Xianmin. Optimal design of a planar parallel 3-DOF nanopositioner with multi-objective[J]. Mechanism and Machine Theory，2017，112：61-83.
[31] XU Peng，LI Bing，CHEUNG C F，et al. Stiffness modeling and optimization of a 3-dof parallel robot in a serial-parallel polishing machine[J]. International Journal of Precision Engineering and Manufacturing，2017，18(4)：497-507.
[32] 杨超，叶伟，陈巧红. 并联机构敏感性分析和多目标优化设计方法[J]. 机械工程学报，2022，58(19)：229-241. YANG Chao，YE Wei，CHEN Qiaohong. Sensitivity analysis and multi-objective optimization design of parallel manipulators[J]. Journal of Mechanical Engineering，2022，58(19)：229-241.
[33] VILLARREAL-CERVANTES M G. Approximate and widespread pareto solutions in the structure-control design of mechatronic systems[J]. Journal of Optimization Theory and Applications，2017，173(2)：628-657.
[34] YANG Chao，LI Qinchuan，CHEN Qiaohong. Natural frequency analysis of parallel manipulators using global independent generalized displacement coordinates[J]. Mechanism and Machine Theory，2021，156：104145.
[35] CAMMARATA A，SINATRA R，MADDIO P D. Static condensation method for the reduced dynamic modeling of mechanisms and structures[J]. Archive of Applied Mechanics，2019，89(10)：2033-2051.