Robot stiffness evaluability problem: Solution by Schur complements and collinear stiffness values

An engineering tool for the robot stiffness evaluation – the collinear stiffness values (CSV) – is applied to an elastically-supported system, whose stiffness is defined by a dimensionally inhomogeneous stiffness matrix (DISM). There are two formal representations of the DISM: as a matrix of the simultaneous linear equations and as a matrix of the quadratic form in single twist entities. The solution of the former and optimization of the latter models are shown to result in identical solutions: the direct stiffness evaluation expressing through eigenvalues of two Schur complements, which are constructed from the blocks of the initial DISM. Both formalizations have a simple engineering interpretation in terms of equilibrium conditions and a virtual work, respectively. The applied solutions are carried out by the wrench transformation technique and the unit twist transformation technique. The eigenvalues of the Schur complements present the translation and rotation stiffness values generically-named as the CSV. The necessary and sufficient conditions of the robot singular configurations are defined. The stiffness-related parameters of the Gough-Stewart platform and serial RRR planar manipulator, including their singularity problem, are formulated through the Schur complements.