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

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.