Synthesis on forward kinematics problem algebraic modeling for the planar parallel manipulator: displacement-based equation systems

Abstract

Based on a proven exact method which solves the forward kinematics problem (FKP) this article investigates the FKP formulation specifically applied to planar parallel manipulators. It focuses on the displacement-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator, which is a tripod constituted by a fixed base and a triangular mobile platform attached to three kinematics chains with linear (prismatic) actuators located between two revolute joints. In order to implement the algebraic method, the parallel manipulator kinematics are formulated as polynomial equation systems where the number of equations is equal to or exceeds the number of unknowns. Three geometrical formulations are derived to model the difficult FKP. The selected proven algebraic method uses Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots are isolated using this last system. Each real solution exactly corresponds to one manipulator assembly mode, which is also called a manipulator posture. The FKP resolution of the planar 3-RPR parallel manipulator outputs six complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are given. It is then possible to compare the models and to reject one. Moreover, a number of real solutions are obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.