• CN:11-2187/TH
  • ISSN:0577-6686

机械工程学报 ›› 2018, Vol. 54 ›› Issue (19): 27-33.doi: 10.3901/JME.2018.19.027

• 机构学及机器人 • 上一篇    下一篇

平面并联机构正运动学分析的几何建模和免消元计算

张英, 魏世民, 李端玲, 廖启征   

  1. 北京邮电大学自动化学院 北京 100876
  • 收稿日期:2017-10-19 修回日期:2018-03-30 出版日期:2018-10-05 发布日期:2018-10-05
  • 通讯作者: 张英(通信作者),女,1987年出生,博士,讲师。主要研究方向为机器人机构学。E-mail:graduate_yingzh@bupt.edu.cn
  • 作者简介:魏世民,男,1965年出生,博士,教授,博士研究生导师。主要研究方向为机器人机构学。E-mail:wsmly@bupt.edu.cn;李端玲,女,1974年出生,博士,教授,博士研究生导师。主要研究方向为机器人机构学。E-mail:liduanling@163.com;廖启征,男,1947年出生,博士,教授。主要研究方向为机器人机构学。E-mail:qzliao@bupt.edu.cn
  • 基金资助:
    国家自然科学基金(51605036,51775052,51375059)、北京市自然科学基金-海淀原始创新联合基金(L172031)和中央高校基本科研业务费专项(2016RCGD23)资助项目。

Geometric Modeling and Free-elimination Computing Method for the Forward Kinematics Analysis of Planar Parallel Manipulators

ZHANG Ying, WEI Shimin, LI Duanling, LIAO Qizheng   

  1. School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876
  • Received:2017-10-19 Revised:2018-03-30 Online:2018-10-05 Published:2018-10-05

摘要: 为了解决三自由度平面并联机构的正运动学分析建模和求解时,需要建立坐标系和消元的问题,基于共形几何代数(Conformal geometric algebra,CGA)提出一种脱离坐标系的几何建模和免消元计算方法。在共形几何代数框架下通过基本几何体的相交、分离和对偶运算,表示出动平台上的两个铰链位置;根据动平台三角形面积的有向性,并经过一系列的几何代数运算和化简推导出该问题的特征多项式方程;通过半角正切变换、欧拉变换或不需要任何变换可直接获得任意构型的平面并联机构正运动学分析的一元高次方程。特征多项式方程的推导脱离了坐标系,不需要经过消元,且不需要任何前提条件。数字实例求解表明提出的方法对于特殊构型和一般构型的平面并联机构都是适用的,验证了算法的正确性,结果表明算法数值鲁棒性好,为平面并联机构运动学求解理论提供了一种新思路。

关键词: 共形几何代数, 几何建模, 免消元计算, 平面并联机构, 正运动学

Abstract: In order to cope with the requirements of the coordinate establishment and elimination process, in the process of the modelling and computing for the forward kinematic analysis of general planar parallel manipulators, a geometric modeling and free-elimination computing method for the forward kinematics of planar parallel manipulators is proposed using conformal geometric algebra (CGA). Under the frame of CGA, two of the three coordinates in the moving platform are formulated by the intersecting, dissecting and dual of the basic geometric entities; in terms with the area sign of the moving triangular platform, the characteristic polynomial equation is derived by a sequence of geometric algebra operation and simplification; a high-degree polynomial equation for planar parallel manipulators of any link parameters is deduced by tangent-half-angle substitution, Euler-angle substitution or no substitution. The derivation of the characteristic polynomial is free of coordinate and no elimination process and no assumption are required. Numerical examples are given to validate the correctness of the procedure and that the proposed algorithm is feasible to all cases of planar parallel manipulators including the special and general structures. At last, the results show that the proposed algorithm has a readily numerical robustness and provides a new sight for the theoretical solution to the kinematics of planar parallel manipulators.

Key words: conformal geometric algebra, forward kinematics, free-elimination computing, geometric modeling, planar parallel manipulators

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