Forward Kinematics Analysis of the General 6-6 Platform Parallel Mechanism Based on Algebraic Elimination

Abstract

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 15×15. 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.