Dynamic interpolation for obstacle avoidance on Riemannian manifolds

This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem. We illustrate the results with examples of rigid bodies, both planar and spatial, and underactuated vehicles including a unicycle and an underactuated unmanned vehicle.