Abstract:

A method to solve the statically indeterminate force problem of the general passive overconstrained parallel mechanisms (PMs) is proposed by using the generalized inverse of weighted Moore-Penrose of the force mapping matrix. The general passive overconstrained PMs refer to the PMs in which each limb supplies one or more constraint forces/couples (including driving forces/torques) to the moving platform. The general expression for the stiffness matrix of the limb’s constraint wrenches is derived based on the superposition principle of micro-deformation and screw theory. The mapping relationship between the magnitudes of all limbs’ constraint wrenches and the external wrench is obtained, which just is the generalized inverse of weighted Moore-Penrose of the force mapping matrix. The weighted factor matrix is the inverse matrix of a block diagonal matrix composed of the stiffness matrices of each limb’s constraint wrenches. Taking 3-RRC overconstrained PM and a parallel vibration platform as examples, the magnitudes of all limbs’ constraint wrenches are solved through the generalized inverse of weighted Moore-Penrose of the force mapping matrix. The correctness of the method is verified by the simulation work. The solving process of the statically indeterminate force problem of passive overconstrained PMs is greatly simplified, and the solution is obtained with a unified and simple form by using the generalized inverse of weighted Moore-Penrose.

Key words: force analysis, passive overconstrained PM, static indeterminacy, stiffness matrix of constraint wrenches, generalized inverse of weighted Moore-Penrose