Forward position analysis of the SP-PS-RS architectures

A wide family of parallel manipulators (PMs) groups all the PMs with three legs where the legs become kinematic chains consisting 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. There are 10 topologies of the structures generated by these manipulators, when the actuators are locked. One 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) that generates an 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 there are at most 12 solutions of the forward position analysis of any parallel manipulator that generates a SP-PS-RS structure when the actuators are locked.