  Retractions maps are used to define a discretization of the tangent bundle of
the configuration manifold as two copies of the configuration manifold where
the dynamics take place. Such discretization maps can be conveniently lifted to
a higher-order tangent bundle to construct geometric integrators for the
higher-order Euler-Lagrange equations. Given a cost function, an optimal
control problem for fully actuated mechanical systems can be understood as a
higher-order variational problem. In this paper we introduce the notion of a
higher-order discretization map associated with a retraction map to construct
geometric integrators for the optimal control of mechanical systems. In
particular, we study applications to path planning for obstacle avoidance of a
planar rigid body.
