北京航空航天大学学报 ›› 2017, Vol. 43 ›› Issue (12): 2377-2381.doi: 10.13700/j.bh.1001-5965.2016.0917

• 论文 • 上一篇    下一篇

基于共形几何代数的空间并联机构位置正解

黄昔光1,2, 黄旭1   

  1. 1. 北方工业大学 机械与材料工程学院, 北京 100144;
    2. 北方工业大学 柔性变截面辊弯成形北京市工程技术研究中心, 北京 100144
  • 收稿日期:2016-12-06 修回日期:2017-03-06 出版日期:2017-12-20 发布日期:2017-05-10
  • 通讯作者: 黄昔光 E-mail:marchbupt@126.com
  • 作者简介:黄昔光,男,博士,副教授,硕士生导师。主要研究方向:机构学与机器人学。
  • 基金资助:
    国家自然科学基金(51105003);北京市自然科学基金(3172010)

Direct kinematics of a spatial parallel mechanism based on conformal geometric algebra

HUANG Xiguang1,2, HUANG Xu1   

  1. 1. School of Mechanical and Materials Engineering, North China University of Technology, Beijing 100144, China;
    2. Beijing Engineering Research Center of Flexible Roll Forming, North China University of Technology, Beijing 100144, China
  • Received:2016-12-06 Revised:2017-03-06 Online:2017-12-20 Published:2017-05-10
  • Supported by:
    National Natural Science Foundation of China (51105003); Beijing Natural Science Foundation (3172010)

摘要: 将共形几何代数(CGA)引入空间并联机构位置正解中,提出了一种空间3-RPS并联机构位置正解新算法。以任意一条支链轴线与静平台平面的夹角为待求变量,基于点的CGA表达方法建立了该支链与动平台连接的铰接点关于待求变量的数学表达式;通过2次构造2个空间球和1个平面的外积,分别获得动平台其余2个铰接点的点对;利用距离公式,只需简单的平方运算可直接推导出该问题关于待求变量的一元16次输入输出方程,进而获得了该机构的全部16组解析解,无增无漏。该方法没有繁琐的坐标变换和矩阵计算,以及复杂的多元高次非线性方程组消元求解。通过数字实例计算表明,求解过程较清晰地揭示出机构运动的几何特点,几何直观性好。

关键词: 共形几何代数(CGA), 空间并联机构, 位置正解, 输入输出方程, 解析解

Abstract: An algorithm is proposed for the direct kinematics analysis of a spatial general 3-RPS parallel mechanism based on conformal geometric algebra (CGA). The angle between the axis of an arbitrary kinematic chain and the plane of the fixed platform can be regarded as the unknown variable. The mathematical expression of the position of the spherical joint connecting the moving platform with the kinematic chain can be expressed in the unknown variable based on CGA. The outer product of two space balls and a flat surface are constructed two times, and the corresponding points of the remaining two vertices of the moving platform are obtained respectively. The 16th degree input-output polynomial equation in the unknown variable is straightforwardly obtained by distance formula and all 16 sets of closed-form solutions can be achieved. The algorithm avoids the use of rational angles or matrices, and complex computations for nonlinear and multivariable equations. A numerical example is given to demonstrate geometric characteristics of the motion and the algorithm is intuitive.

Key words: conformal geometric algebra (CGA), spatial parallel mechanism, direct kinematics, input and output equation, analytical solution

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