Navigation of a Quadratic Potential with Ellipsoidal Obstacles

TL;DR

This paper introduces a Hessian correction to gradient dynamics for navigating environments with quadratic potentials and ellipsoidal obstacles, enabling successful navigation regardless of obstacle eccentricity, with proven convergence and empirical validation.

Contribution

It proposes a novel modification to gradient-based navigation that ensures obstacle avoidance even with highly eccentric ellipsoids, extending the applicability of Rimon-Koditschek potentials.

Findings

Successful navigation with arbitrary eccentricity obstacles.

Almost global convergence to the goal.

Empirical validation through numerical simulations.

Abstract

Given a convex quadratic potential of which its minimum is the agent's goal and a Euclidean space populated with ellipsoidal obstacles, one can construct a Rimon-Koditschek (RK) artificial potential to navigate. Its negative gradient attracts the agent toward the goal and repels the agent away from the boundary of the obstacles. This is a popular approach to navigation problems since it can be implemented with local spatial information that is acquired during operation time. However, navigation is only successful in situations where the obstacles 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 a Hessian correction that is intended to imitate worlds with spherical obstacles in which RK potentials are known to work. The resulting dynamics simplify by the quadratic form of the obstacles. Convergence to the goal and obstacle avoidance is established from almost every initial position (up to a set of measure one) in the free space, with mild conditions on the location of the target. Results are corroborated empirically with numerical simulations.