Dynamic interpolation for obstacle avoidance on Riemannian manifolds
This work provides a theoretical framework for motion planning with obstacle avoidance in geometric settings, but it is incremental as it extends known variational methods to include potential functions.
The authors derive first-order necessary conditions for optimal dynamic interpolation with obstacle avoidance on Riemannian manifolds, using artificial potential functions. They illustrate the results with examples of rigid bodies and underactuated vehicles.
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.