Jacobian Matrix of General Parallel and Hybrid Mechanisms With Rigid and Flexible Links: A Software-Oriented Approach

This paper first presents a procedure for the determination of the Jacobian matrix exact to the first order of general parallel manipulators (GPMs) with rigid and flexible links. The formulation is then extended to the class of general hybrid parallel manipulators (GHPMs) with rigid and flexible links. The acronym GHPM refers to as a parallel mechanism in which the base is connected to the platform by serial subchains and/or hybrid subchains containing closed-chain modules, and with an arbitrarily located subset of actuated joints. The model of each flexible link includes bending in two perpendicular directions as well as torsion around its longitudinal axis. The proposed procedures involve the symbolic generation of a set of augmented matrices combined with a numerical postprocessing and are believed to be especially dedicated to the automatic software generation of the instantaneous kinematic equations of most types of complex closed-chain mechanisms, including all types of in-parallel and hybrid parallel manipulators, cooperative robots, and multifingered hands. A hybrid manipulator, pertaining to the class of GHPMs, is finally considered as an example.