Fundamentals of manipulator stiffness modeling using matrix structural analysis

Abstract

The paper generalizes existing contributions to the stiffness modeling of robotic manipulators using Matrix Structural Analysis. It presents a unified and systematic approach that is suitable for serial, parallel and hybrid architectures containing closed-loops, flexible links, and rigid connections, passive and elastic joints, flexible and rigid platforms, taking into account external loadings and preloadings. The proposed approach can be applied to both under-constrained, fully-constrained and over-constrained manipulators in generic and singular configurations, it is able to produce either non-singular or rank-deficient Cartesian stiffness matrices in a semi-analytical manner. It is based on a unified mathematical formulation that presents the manipulator stiffness model as a set of two groups of matrix equations describing elasticity of separate links and connections between the links in the form of constraints. Its principal advantage is the simplicity of the model generation that includes straightforward aggregation of link/joint equations without conventional merging of rows and columns in the global stiffness matrix. The advantages of this method and its application are illustrated by an example that deals with the stiffness analysis of NaVaRo parallel manipulator where the efficiency of the proposed technique was confirmed by the FEA modeling. For this example, the computational expenses for stiffness modeling were estimated on the level of 10 ms per a manipulator configuration (in Matlab environment on the conventional laptop).