In this paper, we present a technique for approx- imating 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. I. INTRODUCTION Locomotion is everywhere. Snakes crawl, fish swim, birds fly, and all manner of creatures walk. The facility with which animals use internal joint motions to move through their environments far exceeds that which has been achieved in artificial systems; consequently there is much interest in raising the locomotion capabilities of such systems to match or surpass those of their biological counterparts. A fundamental aspect of animal locomotion is that it is primarily composed of gaits - cyclic shape motions which efficiently transport the animal. Examples of such gaits include a horse's walking, trotting, and galloping, a fish's translation and turning strokes, and a snake's slithering and sidewinding. The efficacy of these motions, along with the abstraction that they allow from shape to position changes, suggests that gaits will form an equally important part of artificial locomotion. Here, we are specifically interested in producing tools for designing gaits for mechanical systems which result in desired net position changes. Much prior work in gait design has taken the approach of choosing parameterized basis functions for gaits and simulating the motion of the system while executing the gaits, optimizing the input parameters to find gaits which meet the design requirements. Such optimization with forward simulation is computationally expensive and vulnerable to local minima. Therefore, there is growing interest in using curvature analysis tools, such as Stokes's theorem, to replace the simulation step with a simple volume integration, which is more amenable to optimization. Unfortunately, these Stokes's theorem methods as previously developed are not completely applicable to most interesting systems; either they are restricted to designing small, inefficient motions, or they provide incomplete information about the actual displacement of the system over the course of a gait. In this paper we address these limitations by developing the concept of a body velocity integral (BVI), which provides an expanded and physically meaningful interpretation of previous locomotion work based on Stokes's theorem. We then identify conditions under which this body velocity integral is a good estimate of the true displacement resulting from a gait. We finish by introducing the notion that rather than being intrinsic to the system, the presence of these conditions is dependent on the choice of parameterization of the system, and demonstrat- ing that this choice of parameterization can be manipulated to ensure their existence.