基于Groebner-Sylvester法的一般6-6型台体并联机构位置正解

Forward Kinematics of General 6-6 Stewart Mechanisms Based on Groebner-Sylvester Approach

为获得一般6-6型台体并联机构位置正解的解析解,使用分次字典序Groebner-Sylvester法的代数方法对该问题进行了研究.利用计算代数系统计算该机构位置封闭方程的分次字典序Groebner基,从得出的65个基中选取18个基,构造18阶Sylvester结式.通过分析符号形式方程组的变量次数,得出位置正解的一元高次方程的次数为40且该机构位置正解最多有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.研究结果表明,该计算方法简洁,求解速度快,并从理论上阐明了存在多个不同的结式可以获得该机构的位置正解.最后,从给出的数字实例中,验证了所有解均满足原始方程且无增根,从而为并联机构位置正解的研究提供了一种有效算法.

To obtain the closed-form solutions of forward kinematics ot a general 6-6 Stewart mechanism, the Groebner-Sylevester hybrid approach is presented. The reduced Groebner basis under degree lexicographic ordering for the closed-form equations is evaluated with computer algebra. Selecting 18 Groebner basis from 65 ones, an 18 × 18 Sylvester matrix can be constructed, which is relatively small in size. A 40th degree univariate equation is achieved from the determinate of the matrix. The proposed algorithm operates simply and rapidly to lead to the same univariate equation via various resultants. It is verified by a numerical example that the obtained roots satisfy the original equations.