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.
Highlights
Introduction and Calculation Method for ForwardThe parallel mechanism (PM) [1,2,3,4] is composed of a moving platform and a fixed platform, which is connected by at least two independent kinematic chains and has two or more degrees of freedom (DOF)
A new geometric modeling and calculation method for closed-form solution of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework which contributes to a 16-degree univariate input-output pol-ynomial equation without algebraic elimination and superfluous roots
A novel geometric modeling and calculation method for closed-form solution of 6-3 Stewart platforms based on the framework of conformal geometric algebra is presented
Summary
The parallel mechanism (PM) [1,2,3,4] is composed of a moving platform and a fixed platform, which is connected by at least two independent kinematic chains and has two or more degrees of freedom (DOF). For the elimination of equations, the Newton–Raphson method [6] and polynomial constraints solver [7] are commonly used In this discourse, we will revisit the forward displacement analysis of the 6-3 Stewart platforms (6-3SPS mechanisms), which has been researched in [8,9,10,11,12,13,14]. Zhang et al [12] and Wei et al [13] committed to an inves-tigation into forward displacement analysis of the 6-3 Stewart platforms using the resultant method and the CGA theory. A new geometric modeling and calculation method for closed-form solution of the 6-3 Stewart platforms is proposed by using the CGA framework which contributes to a 16-degree univariate input-output pol-ynomial equation without algebraic elimination and superfluous roots.
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