一般6-4台体型并联机构位置正解分析

Forward Displacement Analysis of a General 6-4 In-parallel Platform

提出一种求解一般6-4台体型并联机构(6-4 in-parallel platform,6-4SPS)位置正解的代数消元法。将6-4SPS型机构进行构型变换,转换成2RPS-2SPS型等效机构,使用由重心坐标推导出的含有一些Cayley-Menger行列式的三边测量法公式对等效机构建模。通过矢量回路关系和变量替换将8个约束方程转换为含有5个变量的5个基本约束方程;用矢量消元法对其中4个(含有3个相同变量)约束方程进行消元,推导出一个含有其余两个变量的方程;将消元后得到的方程与余下的一个约束方程联立,构造一个10×10的Sylvester结式,获得一般6-4台体型并联机构位置正解的一元高次方程,通过分析符号形式方程组变量的次数,得出该一元高次方程的次数为32次。给出了数字实例,经反解验证所有解满足原始方程,且无增根。

A new algebraic elimination algorithm for the forward displacement analysis of a general 6-4 in-parallel platform（6-4SPS） is presented.A 6-4SPS structure is changed into an equivalent 2RPS-2SPS structure and a mathematical modeling is established based on the trilateration formula containing a few number of Cayley-Menger determinants derived by using barycentric coordinates;five basic constraint equations with five variables are obtained from eight equations with eight variables by introducing vector loop relationships and substitution variables;the equation containing two variables is derived from the four equations containing other three variables by the vector elimination;a 32th-degree polynomial equation in a single variable is derived from a 10×10 Sylvester resultant which is constructed from the equation by vector elimination and the last of the five.A numerical example is given to confirm the results,whose solutions agree with the original equations without extraneous roots.