Navigation of a Quadratic Potential with Ellipsoidal Obstacles

Successful navigation of a convex quadratic potential in a space with ellipsoidal obstacles can be attained with Rimon-Koditschek artificial potentials in spaces where the ellipsoids are not too eccentric (flat). This paper proposes a modification to gradient dynamics that allows successful navigation of an environment with a quadratic cost and ellipsoidal obstacles regardless of their eccentricity. This is accomplished by altering gradient dynamics with the addition of a second order curvature correction that is intended to imitate worlds with spherical obstacles in which Rimon-Koditschek potentials are known to work. Convergence to the goal is proven for all environments with a single obstacle. In worlds with multiple obstacles convergence is guaranteed in cases when the obstacles are not tightly packed around the agent's target. Results are numerically verified with a discretized version of the proposed flow dynamics.