Topology optimisation and singularity analysis of a 3-SPS parallel manipulator with a passive constraining spherical joint

This study focuses on topology optimisation and singularity analysis of a 3-SPS parallel manipulator with three identical limbs made of spherical + prismatic + spherical joints, and a passive spherical joint connecting a moving platform to a fixed base. First, the number synthesis for spatial parallel manipulators with three limbs is reviewed and the existence conditions for the parallel manipulator considered in this study are presented. Second, the inverse position formulation is established; a constrained optimisation algorithm based on the condition number of the manipulator Jacobian matrix over the entire workspace of the manipulator is employed to determine design parameters. A set of optimisation trials is accomplished to prove that the algorithm is valid for determining the optimum values of the manipulator design parameters. Third, forward and inverse Jacobian matrices are derived, and then the singularity loci of the manipulator with different values of the design parameters are generated using a methodology not requiring the explicit expressions for the roots of the determinant of the Jacobians. The singularity loci have been presented to show that this methodology is an efficient and versatile technique in the singularity determination of manipulators, and that a well-conditioned workspace has the least number of singular configurations.