Geometric Motion Planning for a System on the Cylindrical Surface

Abstract

Traditional geometric mechanics models used in locomotion analysis rely heavily on systems having symmetry in SE(2) (i.e., the dynamics and constraints are invariant with respect to a system’s position and orientation) to simplify motion planning. As a result, the symmetry assumption prevents locomotion analysis on non-flat surfaces because the system dynamics may vary as a function of position and orientation. In this paper, we develop geometric motion planning strategies for a mobile system moving on a position space whose manifold structure is a cylinder: constant non-zero curvature in one dimension and zero curvature in another. To handle this non-flat position space, we adapt conventional geometric mechanics tools - in particular the system connection and the constraint curvature function - to depend on the system orientation. In addition, we introduce a novel constraint projection method to a variational gait optimizer and demonstrate how to design gaits that allow the example system to move on the cylinder with optimal efficiency.