摘要
提出一种求解一般6-6型平台并联机构位置正解的代数消元法。通过变量替换将9个约束方程中的6个转换为线性方程组,采用线性消元消去9个变量中的6个。基于计算机符号计算,运用计算机代数系统中的分次字典序Groebner基算法,推导出15个只含剩余3个变量最高次数为4次的多项式,应用推导出的多项式构造Sylvester结式,获得一般6-6型平台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20次且该机构位置正解最多有40组解的结论。通过改变单项式的分次字典序,在理论上阐明存在多个不同的结式都可以获得该机构的位置正解。推导出的15个符号形式的多项式可直接用于求解一般6-6型平台并联机构位置正解,从而实现该问题的数学机械化求解。最后给出数字实例,经反解验证所有解满足原始方程,且无增根。
A new algebraic elimination method for the forward kinematics analysis of the general 6-6 platform parallel mechanism is presented. Six in nine constraint quadratic equations are transformed into linear equations by introducing substitution variables and six in nine variables are eliminated by using Cramer algorithm. The reduced Groebner basis under degree lexicographic ordering for the substitution equations and the remainder closed-form equations are obtained. A univariate equation of higher degree is derived from the determinant of the Sylvester＇s matrix, constructed by the 4th degree subset of the Groebner basis, the size of which is 15 x 15 Based on computer symbolic manipulating, it can be concluded that the degree of the univariate polynomial equation is at most 20 and the number of closed-form solutions is at most 40. It is proved theoretically that there are many completely different resultants which can derive all closed-form solutions in terms of different term orderings through changing the degree lexicographic order. The direct kinematics of the general 6-6 Stewart platform can be solved directly by the 15 derived equations. And the mathematical mechanized solution of the problem can be realized. The result is verified by a numerical example, whose solutions agree with the original equations without extraneous roots.
出处
《机械工程学报》
EI
CAS
CSCD
北大核心
2009年第1期56-61,共6页
Journal of Mechanical Engineering
基金
国家重点基础研究发展计划(973计划
2004CB31800)
国家自然科学基金(50475161
50705010
50775012)
国家高技术研究发展计划(863计划
2007AA042200)
北京市自然科学基金(3053017
3082014)资助项目。
