Navigation of a Quadratic Potential with Ellipsoidal Obstacles

Abstract

Given a convex quadratic potential of which its minimum is the agent's goal and a space populated with ellipsoidal obstacles, one can construct a Rimon-Koditschek artificial potential to navigate. These potentials are such that they combine the natural attractive potential of which its minimum is the destination of the agent with potentials that repel the agent from the boundary of the obstacles. This is a popular approach to navigation problems since it can be implemented with spatially local 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 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 and obstacle avoidance is established from every initial position in the free space. Results are numerically verified with a discretized version of the proposed flow dynamics.

Avoiding Dense and Dynamic Obstacles in Enclosed Spaces: Application to Moving in Crowds

“…The modulation matrix in (15) consists of two parts. The diagonal eigenvalue matrix defined in (17), which can be evaluated using the Inverted distance function from (22). Conversely, the basis matrix is constant along radial direction, hence it is defined within the free space of enclosing walls except the reference point.…”

“…These algorithm often create (topologically) avoidable local minimum [16]. Using quadratic potential functions allowed to obtain full convergence around ellipse obstacles [17]. Learning methods have be used to tune the hyper-parameters of potential fields to obtain human-inspired behavior for obstacle avoidance of learned motion [18].…”