Implementable Accommodation Matrices for Passive Force Control

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. KEY WORDS—automated assembly, passive programmable wrist, RCC, Stewart platform, hydraulic network, accommodation matrix ∗This work was done at Northwestern University. The International Journal of Robotics Research Vol. 18, No. 8, August 1999, pp. 830-844, ©1999 Sage Publications, Inc. 1. Motivation and Background In precision tasks such as robotic assembly, force control seems to be the natural choice and is widely believed to be superior to pure position control (Ang and Andeen 1995; Trong, Betemps, and Jutard 1995; Hogan 1985; Whitney 1987). In a typical force-control scheme, the motion of a robot is guided, according to a predefined control law, by the forces the robot encounters while interacting with the environment. The performance of such a scheme depends on the particular forcecontrol law and the nature of its implementation. Force-control laws may be broadly classified into two types: passive laws and active laws. A passive law describes a force-motion behavior that may, in principle, be exhibited by some passive physical system. Active laws, on the other hand, require the presence of a power source in the system. Since a passive system is guaranteed stable (Desoer and Kuh 1969), a passive control law, mimicking a passive system, is also stable. While an actively controlled system may certainly be stable, it is only a passive system that remains stable at all frequencies while interacting with arbitrary passive environments (Colgate and Hogan 1988) that are typical in robotic assembly. A passive force-control law may be implemented either by a software algorithm or by an unpowered mechanical system. In a software-controlled system, active components (such as motors) are controlled in such a way that the overall system emulates a passive behavior (Anderson 1990; Chapel and Su 1992; Newman and Dohring 1991; Wang and Vidyasagar 1990). Unfortunately, the speed and performance of such a system is limited by the control-system bandwidth (Whitney 1987), force-feedback gain (Hogan and Colgate 1989), response time of the actuators, and noncollocation of the sensors and actuators (Eppinger 1988).