Forward Displacement Analysis of a General 6-3 Stewart Platform Using Conformal Geometric Algebra

Feng Wei,Ying Zhang,Shimin Wei,Qizheng Liao

doi:10.1155/2017/4687638

Abstract

In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform (6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm.

Highlights

Stewart platforms [1] are six-degree-of-freedom parallel manipulators that generally consist of a base platform, a moving platform, and six legs connected to each other in parallel
The basic geometric structure of the 6-3SPS consists of a moving platform connected to the adjacent links at three distinct points Bi, i = 1, 2, 3, by spherical kinematic pairs, which is shown in Figure 1. 6-3SPSs can be divided into either platform mechanisms or general mechanisms according to whether the six spherical points on the base platform are restricted to lie in plane or not
This paper proposes a new process for obtaining the closedform solution of the forward displacement analysis (FDA) of the 6-3SPS using conformal geometric algebra (CGA)

Summary

Introduction

Stewart platforms [1] are six-degree-of-freedom parallel manipulators that generally consist of a base platform, a moving platform, and six legs connected to each other in parallel They were first applied to animate flight simulator platforms. Griffis and Duffy [3] obtained a closed-form solution to the platform mechanisms of the 6-3SPS in 1989, but their method is not suitable for the solution of the general mechanisms. Cheng et al [9] obtained an eighth-order input-output equation based on the orthogonal complement method. Their method introduces errors due to rounding off the coefficient of the high-order term in the process of elimination. This paper proposes a new process for obtaining the closedform solution of the FDA of the 6-3SPS using CGA.

Conformal Geometric Algebra

Constraint Equations

Solution Procedure

G5 M2 M3 0

Numerical Verification

Conclusions