  This paper studies the reduction by symmetry of a variational obstacle
avoidance problem. We derive the reduced necessary conditions in the case of
Lie groups endowed with a left-invariant metric, and for its corresponding
Riemannian homogeneous spaces by considering an alternative variational problem
written in terms of a connection on the horizontal bundle of the Lie group. A
number of special cases where the obstacle avoidance potential can be computed
explicitly are studied in detail, and these ideas are applied to the obstacle
avoidance task for a rigid body evolving on SO$(3)$ and for the unit sphere
$S^2$.
