Navigation Functions for Convex Potentials in a Space with Convex Obstacles

TL;DR

This paper establishes conditions under which artificial potential functions reliably guide navigation in convex spaces with obstacles, ensuring convergence to the goal without local minima interference.

Contribution

It derives theoretical conditions guaranteeing a single minimum for artificial potentials in convex obstacle environments, enhancing navigation reliability.

Findings

Artificial potentials succeed when the Hessian's condition number is low.

Navigation is effective when the destination isn't near obstacle borders.

Numerical analysis supports the theoretical conditions for practical scenarios.

Abstract

Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with potentials that repel the agent from the border of the obstacles. This is a popular approach to navigation problems because it can be implemented with spatially local information that is acquired during operation time. Artificial potentials can, however, have local minima that prevent navigation to the minimum of the natural potential. This paper derives conditions that guarantee artificial potentials have a single minimum that is arbitrarily close to the minimum of the natural potential. The qualitative implication is that artificial potentials succeed when either the condition number-- the ratio of the maximum over the minimum eigenvalue-- of the Hessian of the natural potential is not large and the obstacles are not too flat or when the destination is not close to the border of an obstacle. Numerical analyses explore the practical value of these theoretical conclusions.