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

Higher-order retraction maps and construction of numerical methods for optimal control of mechanical systems