基于共形几何代数的两种6自由度串并混联机构位置逆解分析

基于共形几何代数（Conformal geometric algebra，CGA）基本理论，解决了（3-PRS）+（3-SPR）和（3-RPS）+（3-SPS+UP）两种典型串并混联机构的位置逆解问题。首先根据机构的几何结构特征选择合适的未知参数，基于共形几何代数的基本运算建立了与该参数相关的机构中间平台某一顶点的数学表达式；然后根据该顶点坐标表达式和机构中存在的几何和尺寸约束，构造若干相关的平面或球面几何体，并对其进行外积求交运算得出中间平台其他两个顶点坐标的共形几何表达式；最后结合得到的顶点坐标与共形几何代数中的矢量内积运算，推导出只含有一个未知参数的多项式方程，由此得到机构的全部位置反解。该方法具有计算简明、高效、几何直观性强的特点，避免了传统方法求解串并混联机构位置逆解时冗长复杂的计算过程，为此类机构位置逆解的研究提供了参考。

Inverse Position Analysis of Two Kinds of 6-DOF Serial-parallel Manipulators Based on Conformal Geometric Algebra

Based on the basic theory of conformal geometric algebra (CGA), the inverse displacement of (3-PRS)+(3-SPR) and (3-RPS)+(3-SPS+UP) typical serial-parallel manipulators are solved. First, a single unknown is selected according to the geometrical structure of the manipulators, and based on the basic operation of CGA, the mathematical expression of a vertex of the intermediate platform of the manipulators related to this parameter is established. Secondly, according to the coordinate expression of the this vertex and the geometric and dimensional constraints in the manipulators, some basic geometric elements including planes and spheres are constructed, and the conformal geometric expression of the coordinates of other two vertices of the intermediate platform are obtained using the outer product intersection operation. At last, combined with the expressions of the obtained vertices and the inner product operation in CGA, the input-output polynomial equation with a single unknown can be derived and all the inverse position solutions of the manipulators are obtained. The method has the characteristics of simple calculation, high efficiency and remarkable geometric intuition. It avoids the tedious and complicated calculation process compared with traditional methods, and provides a reference for the study of the inverse position analysis of serial-parallel manipulators.