Synthesis on the Forward Kinematics Problem algebraic modeling for the planar parallel manipulator, Part 1. Position-based equation systems

Based on a proven exact method to solve the forward kinematics problem , this article investigates the forward kinematics problem formulation specifically applied to planar parallel manipulators. This first part focuses on the position-based equation systems. The majority of planar tripods can modeled by the 3-RPR parallel manipulator. This manipulator 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 algebraic methods, the parallel manipulator kinematics shall be formulated as polynomial equations systems where the number of equations are equal to the unknown numbers. Two kinematics position-based formulation were derived to model the difficult forward kinematics problem . The selected algebraic proven method is implementing Gröbner bases from which it constructs an equivalent univariate system. Then, the real roots isolation is performed using this last system. Each real solution exactly corresponds to one manipulator assembly mode. The forward kinematics problem resolution of the planar 3-RPR parallel manipulator outputs 6 complex solutions which become a proven real solution number upper bound. In several typical examples, the resolution performances (computation times and memory usage) are compared and the number of real solutions were obtained and the corresponding postures drawn. The algebraic method is exact and produces certified results.