Mechanically Implementable Accommodation Matrices for Passive Force Control

Abstract

Robot force control implemented by means of passive mechanical devices has inherent advantages over active implementations with regard to stability, response rapidity, and physical robustness. The class of devices considered in this paper consists of a Stewart platform-type mechanism interconnected with a network of adjustable mechanical elements such as springs and dampers. The control law repertoire of such a device, imagined as a robot wrist, is given by the range of admittance matrices that it may be programmed to possess. This paper focuses on wrists incorporating damper networks for which the admittance matrices reduce to accommodation or inverse-damping matrices. We show that a hydraulic network of fully adjustable damper elements may attain any diagonally dominant accommodation matrix. We describe the technique of selecting the individual damping coefficients to design a desired matrix. We identify the set of dominant matrices as a polyhedral convex cone in the space of matrix entries, and show that each dominant matrix can be composed of a positive linear combination of a fixed set of basis matrices. The overall wrist-accommodation matrix is obtained by projecting the accommodation matrix of the damper network through the wrist kinematics. The linear combination of the dominant basis matrices projected through the wrist kinematics generates the entire space of mechanically implementable force-control laws. We quantify the versatility of mechanically implementable force-control laws by comparing this space to the space of all matrices.