Analytic Form Solution of the Direct Position Analysis of the SP-2RS Architectures

Abstract

When the actuators are locked, parallel manipulators (PMs) become parallel structures, that are structures constituted by two rigid bodies (platform and base) connected by a number of kinematic chains (limbs) with only passive kinematic pairs. A set of PMs is the one collecting the manipulators (SP-2RS architectures) which become structures with one limb of type SP and two limbs of type RS (P, R and S stand for prismatic pair, revolute pair and spherical pair respectively). The analytic determination of the assembly modes of the SP-2RS structures (i.e. the solution in analytic form of the direct position analysis of the SP-2RS architectures) has not been presented in the literature yet. This paper presents the solution in analytic form of the DPA of the SP-2RS architectures. In particular, the closure equation system of a generic SP-2RS structure is written in the form of three non-linear equations in three unknowns. The solution of the non-linear system is reduced to the determination of the roots of a sixteenth-degree univariate polynomial equation plus a simple back substitution procedure. The proposed solution algorithm is applied to a real case. The result of this study is that the solutions of the direct position analysis of all the SP-2RS architectures are at most sixteen and can be analytically determined through the proposed algorithm.