Variational obstacle avoidance problem on Riemannian manifolds

We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and covariant acceleration, among a set of admissible curves, and also depending on a navigation function used to avoid an obstacle on the workspace, a Riemannian manifold. We study two different scenarios, a general one on a Riemannian manifold and, a sub-Riemannian problem. By introducing a left-invariant metric on a Lie group, we also study the variational obstacle avoidance problem on a Lie group. We apply the results to the obstacle avoidance problem of a planar rigid body and an unicycle.