  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.
