  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.
