The Geometry of Optimal Gaits for Drag-Dominated Kinematic Systems

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.