Forward Position Analysis of the SP-PS-RS Architectures

A wide family of parallel manipulators (PMs) is the one that groups all the PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. One out of these topologies is the SP-PS-RS topology. This paper presents an algorithm that determines all the assembly modes of the structures with topology SP-PS-RS in analytical form. The presented algorithm can be applied without changes to solve, in analytical form, the forward position analysis of any parallel manipulator (SP-PS-RS architecture) which generates a SP-PS-RS structure when the actuators are locked. In particular, the closure equations of a generic SP-PS-RS structure are written. The eliminant of this system of equations is determined and the solution procedure is presented. Finally, the proposed procedure is applied to a real case. This work demonstrates that the solutions of the forward position analysis of any parallel manipulator which generates a SP-PS-RS structure when the actuators are locked are at most twelve.