A key problem in robotic locomotion is in finding optimal shape changes to effectively displace systems through the world. Variational techniques for gait optimization require estimates of body displacement per gait cycle; however, these estimates introduce error due to unincluded high order terms. In this paper, we formulate existing estimates for displacement, and describe the contribution of low order terms to these estimates. We additionally describe the magnitude of higher (third) order effects, and identify that choice of body coordinate, gait diameter, and starting phase influence these effects. We demonstrate that variation of such parameters on two example systems (the differential drive car and Purcell swimmer) effectively manages third order contributions.

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

In this paper, we present a set of geometric principles for understanding and optimizing the gaits of drag-dominated kinematic locomoting systems. For systems with two shape variables, the dynamics of gait optimization are analogous to the process by which internal pressure and surface tension combine to produce the shape and size of a soap bubble. The internal pressure on the gait curve is provided by the flux of the curvature of the system constraints passing through the surface bounded by the gait, and surface tension is provided by the cost associated with executing the gait, which when executed at optimal (constant-power) pacing is proportional to its pathlength measured under a Riemannian metric. We extend these principles to work on systems with three and then more than three shape variables. We demonstrate these principles on a variety of system geometries (including Purcell's swimmer) and for optimization criteria that include maximizing displacement and efficiency of motion for both translation and turning motions. We also demonstrate how these principles can be used to simultaneously optimize a system's gait kinematics and physical design.