The weighted Moore–Penrose generalized inverse and the force analysis of overconstrained parallel mechanisms

Abstract

This paper reveals the relationship between the weighted Moore–Penrose generalized inverse and the force analysis of overconstrained parallel mechanisms (PMs), including redundantly actuated PMs and passive overconstrained PMs. The solution for the optimal distribution of the driving forces/torques of redundantly actuated PMs is derived in the form of a weighted Moore–Penrose inverse. Therefore, different force distributions can be achieved simply by changing the value of the weighted factor matrix in terms of different optimization goals, and this approach greatly improves computational efficiency in solving such problems. In addition, the explicit expression is deduced between the weighted Moore–Penrose generalized inverse and the constraint wrenches solution of general passive overconstrained PMs (in which each supporting limb may supply single or multiple constraint wrenches). In this expression, the weighted factor matrix is composed of the stiffness matrices of each limb’s constraint wrenches. As numerical examples, the driving forces/torques or the constraint forces/couples for two kinds of overconstrained PMs are solved directly by the weighted Moore–Penrose generalized inverse. The verification results show the correctness of the relationship obtained in this paper between the weighted Moore–Penrose generalized inverse and the force analysis of overconstrained PMs. Using the Moore–Penrose generalized inverse to solve the driving forces/torques or constraint forces/couples of overconstrained PMs provides solutions of a unified, simple form and improves computational efficiency.