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

• Published in: Chinese journal of mechanical engineering 2016, Vol.29 (1), p.56-62
• Main Authors: Wei, Feng, Wei, Shimin, Zhang, Ying, Liao, Qizheng
• Format: Article
• Language: English
• Subjects: Electrical Machines and Networks; Electronics and Microelectronics; Engineering; Engineering Thermodynamics; Grobner基; Heat and Mass Transfer; Instrumentation; Machines; Manufacturing; Mechanical Engineering; Mechanism and Robotics; Power Electronics; Processes; Theoretical and Applied Mechanics; 代数解; 位移分析; 共形几何代数; 反对称矩阵; 多项式方程; 机构; 约束方程
• ISSN: 1000-9345; 2192-8258
• DOI: 10.3901/CJME.2015.1015.122

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.