Geometric Swimming on a Granular Surface

Abstract

Snake robots can contact their environments along their whole bodies. This distributed contact makes them versatile and robust locomotors, but also makes controlling them a challenging problem involving high-dimensional configuration spaces, with no direct way to break their motion down into “driving” and “steering” actions. In this paper, we use concepts from geometric mechanics — e.g., expanded Lie bracket analysis — to simplify the problem of controlling a snake robot moving across a granular surface. Without needing force laws that model the interaction of the snake robot with the granular surface, the relationship between shape and body velocities can be experimentally derived by perturbing the robot’s shape from a sampling of initial configurations, which allows us to: 1. Generate an intuitive and visualizable relationship between gait cycles and the motion they induce; 2. Make accurate predictions about the most efficient gaits available to the robot; and 3. Identify an effective turning gait for the robot that to the best of our knowledge has not previously appeared in the snake robot literature. This geometric analysis of snake robot locomotion serves as a demonstration of how differential-geometric tools can provide insight into the motion of systems that do not have the analytic models often associated with such approaches.