A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms

Ying Zhang,Xin Liu,Shimin Wei,Xiaodong Zhang,Changchun Liang,Pei Zhang,Yaobing Wang

doi:10.1155/2018/6245341

Abstract

A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that (1) the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and (2) the size of the resultant matrix is smaller than those existed.

Highlights

The Stewart platform [1] is a fully parallel, six-degree-of-freedom manipulator that generally consists of a base platform, a moving platform, and six limbs connected to each other in parallel
A closed-form solution provides more information about the geometric and kinematic behavior over a numerical solution, and the closed-form univariate polynomial equation has significant theoretical values as it is fundamental to many other kinematic problems
Obtaining a closed-form solution to the direct kinematic analysis is clearly preferred in most cases

Summary

Introduction

The Stewart platform [1] is a fully parallel, six-degree-of-freedom manipulator that generally consists of a base platform, a moving platform, and six limbs connected to each other in parallel. Liao et al (1995) [5] formulated this problem based on the vector method from equivalent mechanisms and obtained all the 32 solutions by constructed 10 by 10 resultant matrix. We will formulate the direct kinematic analysis problem of 6-4 Stewart platform using CGA and construct a 7 by 7 resultant using Grobner-Sylvester hybrid method [23, 24] which leads to a 32-degree univariate polynomial equation without extraneous roots.

Fundamentals of Conformal

The CGA Representation of the Positions of Two Spherical

Solution Procedure

Numerical Example

Conclusion