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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. 

In this paper, we present a set of geometric princi- ples for understanding and optimizing the gaits of drag-dominated kinematic locomoting systems. For systems with two shape vari- ables, the dynamics of gait optimization are analogous to the pro- cess by which internal pressure and surface tension combine to produce the shape and size of a soap bubble. The internal pres- sure 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 ex- ecuting the gait, which when executed at optimal (constant-power) pacing is proportional to its pathlength measured under a Rie- mannian metric. We extend these principles to work on systems with three and then more than three shape variables. We demon- strate these principles on a variety of system geometries (including Purcell’s swimmer) and for optimization criteria that include max- imizing 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.