The Geometry of Optimal Gaits for Inertia-Dominated Kinematic Systems

Isolated mechanical systems—e.g., those floating in space, free-fall, or on a frictionless surface—are able to achieve net rotation by cyclically changing their shape, even if they have no angular momentum. Similarly, swimmers immersed “perfect fluids” are use cyclic shape changes both translate and rotate the swimmer-fluid system has linear Finally, systems fully constrained direct nonholonomic constraints (e.g., passive wheels) can push against these move through world. Previous work demonstrated that displacement induced corresponds amount of constraint curvature gaits enclose. Properly assessing optimizing utility gait also requires considering time resources required execute it: A produces small per cycle, but be executed short time, may produce faster average velocity than more displacement, takes longer complete cycle at same instantaneous effort. In this paper, we consider under two measures For each costs, demonstrate fixing cost unit value allows us transform effort costs into time-to-execute for any given cycle. We then illustrate how interaction between constraint curvature leads characteristic geometries optimal cycles, which trajectories resemble elastic hoops distended from within internal pressures.