Algebraic Solution for the Forward Displacement Analysis of the General 6-6 Stewart Mechanism  

作　　者：WEI Feng WEI Shimin ZHANG Ying LIAO Qizheng 

机构地区：[1]Automation School,Beijing University of Posts and Telecommunications [2]School of Mechanical and Power Engineering,Henan Polytechnic University

出　　处：《Chinese Journal of Mechanical Engineering》2016年第1期56-62,共7页中国机械工程学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant No.51375059);National Hi-tech Research and Development Program of China(863 Program,Grant No.2011AA040203);Special Fund for Agro-scientific Research in the Public Interest of China(Grant No.201313009-06);National Key Technology R&D Program of the Ministry of Science and Technology of China(Grant No.2013BAD17B06)

摘　　要：The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.

关 键 词：general 6-6 Stewart mechanism forward displacement analysis （FDA） conformal geometric algebra （CGA） Gr6bner basis Sylvester resultant 

分 类 号：TH112[机械工程—机械设计及理论]