Synthesis of the forward kinematics problem algebraic modeling for the general parallel manipulator: displacement-based equations

This paper closes a triptic to address the issue of the forward kinematics problem (FKP) aimed at certified solving with an exact algebraic method. This solving method was described in the first article published in Advanced Robotics. The second one investigated the formulation specifically applied to the planar parallel manipulators. This third paper is the logical one in the footsteps of the formersones, since it continues the formulation analyses and brings them to the general spatial parallel manipulator. Hence, this paper focuses on the displacement-based equation systems. This paper is the first one to present a synthesis on forward kinematics modeling focusing on finding an optimal mathematical formulation based on the displacement-based equation systems. The majority of parallel manipulators in applications can be modeled by the 6-6 hexapod or so-called Gough platform which is constituted by a fixed base and a mobile platform attached to six kinematics chains with linear (prismatic) actuators located between two revolute or Cardan joints. Again, in order to implement algebraic methods, the parallel manipulator kinematics shall be formulated as polynomial equations systems where the equation number is at least equal to the unknown numbers. Six geometric formulations were derived. The selected algebraic proven method is implementing Grobner bases from which it constructs an equivalent univariate polynomial system. The resolution of this last system exactly determines the real solutions which correspond to the manipulator postures. The FKP resolution of the general 6-6 parallel manipulator outputs 40 complex solutions. Several instantiations shall be computed in order to select the model which leads to the FKP resolution with the lowest response times and smaller file sizes. It was possible to reject three modelings leading to bad performances or resolution failure. It was possible to determine one formulation where the solving computations were definitely better than the others.