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 implementable force control laws. We quantify the versatility of mechanically implementable force control laws by comparing this space to the space of all matrices.
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 Jutard, 1995; Hogan, 1985; Whitney, 1987). In a typical force control scheme, the motion of a robot is guided, according to a pre-defined control law, by the forces the robot encounters while interacting with the environment. The performance of such a scheme depends on the particular force control 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), 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 (Whi87), force-feedback gain (Hogan and Colgate, 1989) , response time of the actuators, and non-collocation of the sensors and actuators (Eppinger, 1988).
Unpowered devices with fixed mechanical properties lack the versatility offered by software controllers. An attractive alternative for implementing force control laws is the use of passive mechanical devices with user-programmable properties. Such a device is able to regain some of the versatility of its active counterpart. Rather than involving the whole robot arm for the fine positionings necessary for the completion of most assembly tasks, using a low inertia robotic wrist mounted at the end of the robot arm will have the advantage of higher mechanical bandwidth (Sharon, Hogan and Hardt, 1989). This was demonstrated by the success of the remote center of compliance (RCC) wrist in peg-in-hole assembly (Drake, 1977; Whitney, 1982). Biological evolution also seems to have taken notice of this fact as is apparent in human manipulation. High power tasks that do not a require high bandwidth or a high dexterity (e.g. pushing a heavy table, swinging a baseball bat) generally directly involve the powerful muscles of the upper arm. Low power tasks requiring a high bandwidth (such as typing) and/or a high dexterity (such as writing) tend to decouple the heavier upper arm and, instead, use the low inertia fingers (Cutkosky and Wright, 1986.).
Recently we have noted a renewed interest in passivity (Charles, 1994; Davis, 1996) in such diverse areas as haptic displays (Peshkin, Colgate and Moore, 1996), medical robotics (Troccaz and Lavalle, 1993), and exercise machines (Li and Horowitz, 1995), in addition to applications in automated assembly (Gershon, 1994; Ang and Andeen, 1995). This work falls in the general category of research that seeks to quantitatively characterize passive devices in terms of their limitations and utilities. In this paper we focus on the use of the programmable passive mechanical robot wrist which, by virtue of its inherent mechanical properties, allow a simple and robust implementation of stable and fast force control laws.