Static Stiffness Property Analyses of a Novel PRRR+PURU+S Spherical Parallel Humanoid Robotic Ankle Mechanism

Abstract

Abstract: A novel PRRR+PURU+S spherical parallel humanoid robotic ankle mechanism with a 8 order statically indeterminate mechanism was proposed, and the calculations of its stiffness were conducted. Firstly, by means of small deformation superposition principle, the relationship between the angle displacements of moving platform and linear displacements of spherical point and the forces acted on the components were derived out. Then, based on the results of static analyses including the relational equations between applied forces and the forces acted on the components, the relationship between the applied forces, the line displacements and the angle displacements of the mechanism were set up. Finally, the static stiffness matrix was obtained. The six principal stiffnesses of the mechanism and the corresponding directions were achieved by the orthogonal transformation. The results show that the torsion stiffness and the line deformation stiffness are all increased compared to the UP+R ankle mechanism, and the line deformation stiffness sharply increased. The stiffness of the novel ankle mechanism basically achieve to balance.

Key words: humanoid robot, ankle mechanism, static stiffness, deformation equation of compatibility

摘要： 提出了一种新型PRRR+PURU+S球面并联人形机器人踝关节机构，对该8次超静定机构进行了静刚度分析。首先基于各构件的弹性变形，利用小变形叠加原理，推导出机构动平台角位移、球心点线位移与各作用力的解析关系式；然后利用该机构静力学分析的结果，即各构件上的力与机构外载荷的关系方程，建立机构外载荷与机构动平台角位移、球心点线位移的解析关系，进而得到机构的静刚度矩阵；最后采用正交变换方法，得到机构的6个主刚度及其所在方向。研究结果表明，与UP+R踝关节机构相比，新型踝关节机构的角位移刚度、线位移刚度均有增大，其中线位移刚度大幅增大，基本实现刚度均衡。

关键词: 人形机器人, 踝关节机构, 静刚度, 变形协调方程

CLC Number:

TP24

ZHOU Yulin1;YANG Long1, 2;XIAO Chao1. Static Stiffness Property Analyses of a Novel PRRR+PURU+S Spherical Parallel Humanoid Robotic Ankle Mechanism[J]. China Mechanical Engineering.

周玉林1;杨龙1,2;肖超1. 新型PRRR+PURU+S球面并联人形机器人踝关节机构静刚度性能分析[J]. 中国机械工程.

References

[1]徐东涛，孙志礼，于晓光. 考虑杆件弹性变形的改进型Delta并联机构刚度特性分析[J]. 农业机械学报,2013,44(5)：294-298.
XU Dongtao, SUN Zhili, YU Xiaoguang. Stiffness Characteristic Analysis of Modified Delta Parallel Mechanism Based on Bars' Elastic Deformation[J]. Transactions of the Chinese Society for Agricultural Machinery,2013,44(5):294-298.
[2]曲海波，梁艺瀚，方跃法，等. 4-RRS冗余球面并联机构的静力学与刚度分析[J]. 机械工程学报，2015，51(11)：8-15.
QU Haibo, LIANG Yihan, FANG Yuefa, et al. Statics and Stiffness Analysis of 4-RRS Redundant Spherical Parallel Mechanism[J]. Journal of Mechanical Engineering,2015,51(11):8-15.
[3]CECCARELLI M, CARBONE G. A Stiffness Analysis for CaPaMan (Cassino Parallel Manipulator)[J]. Mechanism and Machine Theory,2002,37(5):427-439.
[4]TASI L W, JOSHI S. Comparison Study of Architecture of Four 3 degree-of-freedom translational parallel manipulators [C]//Proceedings of the 2011 IEEE International Conference on Robots and Automation. Seoul: IEEE，2001：1283-1288.
[5]CHAKAROV D. Study of the Antagonistic Stiffness of Parallel Manipulators with Actuation Redundancy[J]. Mechanism and Machine Theory,2004,39(6):583-601.
[6]王友渔，黄田，CHETWYND D G，等. Tricept机械手静刚度解析建模方法[J]. 机械工程学报，2008，44(8)：13-19.
WANG Youyu, HUANG Tian, CHETWYND D G, et al. Analytical Method for Stiffness Modeling of the Tricept Robot[J]. Chinese Journal of Mechanical Engineering,2008,44(8),13-19.
[7]汪满新，王攀峰，宋轶民，等.4自由度混联机器人静刚度分析[J].机械工程学报，2011，47(15)：9-16.
WANG Manxin, WANG Panfeng, SONG Yimin, et al. Stiffness Analysis of a 4-DOF Hybrid Robot[J]. Chinese Journal of Mechanical Engineering,2011,47(15):9-16.
[8]李树军，GOSSELIN C.3-RRR平面并联机构的刚度特性分析[J].东北大学学报(自然科学版)，2007，28(1)：91-94.
LI Shujun, GOSSELIN C. Stiffness Analysis and Stiffness Mapping of 3-RRR Planar Parallel Mechanism Based on CCT Stiffness Matrix[J]. Journal of Northeastern University (Natural Science),2007,28(1):91-94.
[9]赵铁石，赵延治，边辉，等.空间并联机构连续刚度非线性映射[J].机械工程学报，2008，44(8)：20-25.
ZHAO Tieshi, ZHAO Yanzhi, BIAN Hui, et al. Continuous Stiffness Nonlinear Mapping of Spatial Parallel Mechanism[J]. Journal of Mechanical Engineering,2008,44(8):20-25.
[10]胡波，路懿.求解3-RPS并联机构刚度的新方法[J].机械工程学报，2010，46(1)：24-29.
HU Bo, LU Yi. New Approach for Analyzing the Stiffness of 3-RPS Parallel Manipulator[J]. Journal of Mechanical Engineering,2010,46(1):24-29.
[11]ZHOU Yulin, YANG Long, LI Bo, et al. Stiffness Analysis of a Spherical Parallel Mechanism UP+R with 2-DOF[J]. Journal of Harbin Institute of Technology,2013,20(4)：41-47.

[1]	CHEN Guangyu, WENG Zhiwei, ZHANG Song, . Static Stiffness Analysis of Double-nut Ball Screws Considering Dimensional Errors and Radial Loads [J]. China Mechanical Engineering, 2023, 34(24): 2952-2962.
[2]	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.
[3]	LIANG Yi, WANG Xudong, WANG Xiaoyi, FENG Jianwen, FENG Benzhe, OU Yi, FENG Hutian. Relationship of Raceway Center Distance Errors and Load-bearing Performance of Rolling Linear Guide Pairs [J]. China Mechanical Engineering, 2021, 32(18): 2254-2261,2267.
[4]	ZHANG Wentao1;WANG Zhenhu2;FANG Xiangdong3;YANG Xuyue1;LI Luoxing2;WANG Wanlin2. A Calculation Method of Car Body in White's Static Stiffness Based on Modal Theory [J]. China Mechanical Engineering, 2018, 29(05): 511-518.
[5]	QU Mengke1,2,3;WANG Hongbo1,3;RONG Yu2. Static Stiffness Modeling and Experiments of Polishing Manipulator Arm [J]. China Mechanical Engineering, 2017, 28(20): 2395-2401.
[6]	XU Yiliu1;YANG Long2,3;YANG Zhongyuan2;XIAO Chao2;ZHOU Yulin2. Dynamics Property Analyses of a Novel PURU+RR+S Spherical Parallel Humanoid Robotic Ankle Mechanisms [J]. China Mechanical Engineering, 2017, 28(16): 1971-1976.
[7]	YANG Long, ZHOU Yulin, MA Qiuming. Statics Property Analyses of a Novel Spherical Parallel Humanoid Robotic Ankle Mechanism [J]. China Mechanical Engineering, 2017, 28(07): 835-841.
[8]	CHEN Guangsheng, ZHENG Qingzhen. A Rapid Identification Method for Stiffness among Ends of Multi-axis CNC Machine Tools [J]. China Mechanical Engineering, 2017, 28(03): 322-326.
[9]	Yu Huijie, Liu Wenhui, Wang Yasu. Research on Stiffness Characteristics and Mechanics Model of Metal Rubbers [J]. China Mechanical Engineering, 2016, 27(23): 3167-3171.
[10]	Wang Jiaxing, Ni Yanbing, Dong Na, Wu Nan. Working Load Prediction and Stiffness Analysis of Large Friction Welding Machine Tools [J]. China Mechanical Engineering, 2016, 27(02): 258-264.
[11]	Chang Dingyong;Fang Yuefa. Design and Analysis of a Multiple Output 3D Printing Redundant Parallel Manipulator [J]. China Mechanical Engineering, 2015, 26(12): 1595-1602.
[12]	Hu Wei, Jiang Zhengfeng, Zeng Rong, Chen Lei, Wang Liwei. Study on Static Stiffness Model of Long Circumferential Spring Dual Mass Flywheel [J]. China Mechanical Engineering, 2014, 25(24): 3278-3281,3288.
[13]	ZHOU Yu-Lin-1, LI Bei-2, YANG Long-1, GAO Feng-3. Statics Analysis of a Spherical Parallel Mechanism UP+R with 2-DOF [J]. China Mechanical Engineering, 2013, 24(15): 2081-2087.
[14]	LIU Xiao-Jing, HU Ye-Fa, ZHANG Wei-Wei, WEI Jian. Measuring Experiments on Construction Static and Dynamic Stiffness of Magnetic Suspension System [J]. China Mechanical Engineering, 2010, 21(8): 908-912.
[15]	Xu Qiang;Liao Qizheng;Wei Shimin;Huang Xiguang. Construction Principle and Static Torque of Translational  Meshing Motor Integrated with Compliant Mechanism [J]. J4, 2009, 20(03): 0-369.