Unified Solving Jacobian/Hessian Matrices of Some Parallel Manipulators With n SPS Active Legs and a Passive Constrained Leg

摘要

Some parallel manipulators with n spherical joint-prismatic joint-spherical joint (SPS)-type active legs and a passive constrained leg possess a larger capability of load bearing and are simple in structure of the active leg. In this paper, a unified and simple approach is proposed for solving Jacobian/Hessian matrices and inverse/forward velocity and acceleration of this type of parallel manipulators. First, a general parallel manipulator with n SPS-type active legs and one passive constrained leg in various possible serial structure is synthesized, and some formulae for solving the poses of constrained force/ torque and active/constrained force matrix are derived. Second, the formulae for solving extension of active legs, the auxiliary velocity/acceleration equation are derived. Third, the formulae for solving inverse/forward velocity and acceleration and a Jacobian matrix without the first-order partial differentiation and a Hessian matrix without the second-order partial differentiation are derived. Finally, the procedure is applied to three parallel manipulators with four and five SPS-type active legs and one passive constrained leg in different serial structures and to illustrate.