The Control of Natural Motion in Mechanical Systems

This paper concerns a simple extension of Lord Kelvin’s observation that energy decays in a dissipative mechanical system. The global limit behavior ofsuch systems can be made essentially equivalent to that of much simpler gradient systems by the introduction of a “navigationfunction” in the role of an artificial field. This recourse to the mechanical system’s natural motion helps transform the open-ended problem of autonomous machine design into the more structured problem of finding an appropriate “cost function” in the many situations that the goal may be encoded as a setpoint problem with configuration constraints.

This paper offers a unified exposition of some recent results [13, 12, 15] heretofore scattered throughout a more mathematically oriented literature that strengthen our original suggestion [8, 9] concerning the utility of controlling natural motion as a means of simultaneously encoding, planning and effecting tasks in mechanical systems. The chief theoretical insight, Theorem 2, is a global global version of Lord Kelvin’s century old result on the dissipation of total energy. Establishing this extension yields a rather general design principlethe notion of a navigation function-that seems to have useful application in a variety of settings. Roughly speaking, it offers a checklist of criteria for achieving the strongest possible convergence properties allowed on a configuration space by a smooth and bounded force/torque control strategy. Some simple examples introduced here may aid the exposition of these ideas. A sequel [10] to this paper illustrates how the ideas may be applied in more realistic settings.