Minimization of Energy in Quasi-Static Manipulation

Quasi-static mechanical systems are those in which mass or acceleration are sufficiently small that the inertial term ma in F = ma is negligible compared to dissipative forces. Many instances of robotic manipulation can be well approximated as quasi-static systems, with the dissipative force being dry friction. Energetic formulations of Newton’s laws are valuable for mechanics problems involving multiple constraints. The following energetic principle for quasi-static systems seems intuitively appealing, or perhaps even obvious: A quasi-static system chooses that motion, from among all motions satisfying the constraints, which minimizes the instantaneous power. Roughly speaking, the above minimum power principle states that a system chooses at every instant the lowest energy, or “easiest,” motion in conformity with the constraints. Surprisingly, the principle is in general false. For example, if viscous forces act, the motion predicted by the minimum power principle will be incorrect. We prove that the principle is correct if there are no forces with velocity-dependent magnitude. This allows its application to many systems with Coulomb friction.