A Novel Geometric Modeling and Calculation Method for Forward Displacement Analysis of 6-3 Stewart Platforms

Abstract

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

Keywords

• 6-3 Stewart platform
• conformal geometric algebra
• forward displacement analysis