A geometric algebra algorithm for the closed-form forward displacement analysis of 3-PPS parallel mechanisms

Abstract

Abstract This paper presents an efficient graphical representation geometric algebra approach for the forward displacement analysis (FDA) of 3-PPS parallel mechanisms (PMs). Our approach based on conformal geometric algebra (CGA) reveals interesting intuitions and insight of the modelling and solution for FDA of this type PMs and reduces considerably computational burden and time. In this work, we first deduce the position of an arbitrary joint among three spherical joints on the moving platform through motors and represent it in terms of only one variable. Then, we formulate the position of either one joint of two remained spherical joints using the intersection of a sphere and two planes under the CGA framework. To obtain the geometric formulation of the third spherical joint position, we construct the outer product of two spatial balls with centers at two formulated spherical joints centers and two known planes. The positions of three spherical joints are all explicit geometric formulations of geometric entities of the parallel mechanisms. We achieve a coordinate-invariant equation expressed in terms of geometric entities via inner product of the last joint by itself. The symbolic 8th-degree univariate polynomial input-output equation is first proposed and all 8 sets of closed-form solutions can be obtained. The algorithm is the geometric algebra computation in a clear and coordinates-free way that avoids the use of rotation matrices, and complex algebraic modelling and nonlinear and multivariable elimination computations as most current approaches do. Finally, two numerical examples have been applied to demonstrate the efficiency of the algorithm comparing with the Dixon resultant method. The results show that this algorithm can simplify complexity of the problem and reduce computation time dramatically with strong geometrical intuition. The achieved symbolic univariate polynomial input-output equation can be applied directly to solve the DPA for all types of 3-PPS PMs instead of using any other methods. This work provides a novel geometric modeling and solution approach for the forward displacement analysis of spatial parallel mechanism and be of greatly useful for roboticists or engineers to design and develop 3-PPS PMs for kinds of applications.