Approximating Displacement with the Body Velocity Integral

Abstract

In this paper, we present a technique for approximating the net displacement of a locomoting system over a gait without directly integrating its equations of motion. The approximation is based on a volume integral, which, among other benefits, is more open to optimization by algorithm or inspection than is the full displacement integral. Specifically, we develop the concept of a body velocity integral (BVI), which is computable over a gait as a volume integral via Stokes’s theorem. We then demonstrate that, given an appropriate choice of coordinates, the BVI for a gait approximates the displacement of the system over that gait. This consideration of coordinate choice is a new approach to locomotion problems, and provides significantly improved results over past attempts to apply Stokes’s theorem to gait analysis.