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

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

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

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

Generalized Stewart Platforms and their Direct

Abstract. In this paper, we introduce the generalized Stewart platform consisting of two rigid bodies connected with six distance and/or angular constraints between six pairs of points, lines and/or planes in the base and moving platform respectively, which could be considered as the most general form of parallel manipulators with six DOF in certain sense. We prove that there exist 3850 possible forms of generalized Stewart platforms which could provide more practical six DOF parallel manipulators. We show that by using combinations of various joints, all these GSPs can be realized as real mechanisms. We give the upper bounds for the number of solutions of the direct kinematics for all the generalized Stewart platforms. We also obtain analytical solutions and the best upper bounds of real solutions of the direct kinematics for a class of 1120 generalized Stewart platforms