Stewart台体并联机器人位置正解

Direct kinematics of Stewart mechanism robot

为获得Stewart台体并联机器人位置正解的解析解,使用分次字典序Groeb-ner基和Sylvester结式相结合的代数方法对该问题进行研究.利用计算代数中的分次字典序Groebner基算法,计算该机器人位置正解封闭数学模型的分次字典序Groebner基;从得出的51个基中选取20个基,构造20阶Sylvester结式,分析符号形式方程组的变量次数,可以直接得出该机器人位置正解一元高次方程的次数为40且最多有40组解的结论,其结果与前人的完全一致,但结式的尺寸却小得多.为了对结果进行验证,使用同伦连续法对同一个数字算例进行计算,两种方法得到结果一致.为并联机构位置正解的研究提供了一种新的有效算法.

The Groebner-Sylevester hybrid approach was presented to obtain the closed-form solutions of the direct kinematics of the Stewart mechanism robot. The reduced Groebner basis under degree lexicographic ordering for the closed-form equations was presented based on computer algebra method. And 20 Groebner basis were picked out from the 51 Groebner basis. Then a 20 × 20 dimensional Sylvester＇s matrix was construeted, which is relatively small in size. And a 40th degree univariate equation was presented from the determinate of the matrix by analyzing the numbers of the variables in the equations. To verify the proposed algorithm, a same numerical example was calculated based on the continuation method as well as the the proposed algorithm. It shows that the proposed algorithm calculates simply and rapidly to lead to the univariate equation. The results both by using the continuation method and based on the proposed algorithm are same. Then a novel and practical algorithm to solve the direct kinematics of the parallel mechanism robot was presented.