Analytic form solution of the forward position analysis of three-legged parallel mechanisms generating SR–PS–RS structures

Spatial parallel mechanisms (SPMs) become parallel structures when the actuators are locked. Parallel structures are constituted by two rigid bodies (platform and base) connected by a number of kinematic chains (legs) with only passive kinematic pairs. A set of SPMs is the one collecting the mechanisms which become SR–PS–RS structures. Such structures have three legs: one leg of type SR, another leg of type PS and the remaining one 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 SR–PS–RS structures has not been presented in the literature, yet. This paper presents an algorithm that analytically determines the assembly modes of the SR–PS–RS structures (i.e. that analytically solves the forward position analysis (FPA) of the SPMs that become SR–PS–RS structures when the actuators are locked). In particular, the closure equation system of a generic SR–PS–RS 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 twelfth-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 FPA of all these SPMs are at most twelve and can be analytically determined through the proposed algorithm.