Certified solving of the forward kinematics problem with an exact algebraic method for the general parallel manipulator

Abstract

This article introduces an exact method to solve the forward kinematics problem (FKP) specifically applied to spatial parallel manipulators. The majority can modeled by the 6-6 parallel manipulator. This manipulator is a hexapod made up of a fixed base and a mobile platform attached to six kinematics chains with linear (prismatic) actuators located between two ball or Cardan joints. In order to implement algebraic methods, the parallel manipulator kinematics will be formulated as polynomial equations systems where the equation number is equal to the unknown numbers. One position-based kinematics model will be identified to solve the difficult FKP. The selected proven algebraic method implements Gröbner bases and constructs an equivalent univariate polynomial system. The exact resolution of this last system determines the real solution which exactly corresponds to the manipulator postures. The FKP resolution of the general 6-6 parallel manipulator outputs 40 complex solutions. We provide several examples of various hexapod types yielding eight real solutions. This algebraic method is exact and computes certified results.