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.

翻译：收缩映射用于将构型流形的切束离散化为两个构型流形的副本，动力学行为在其中发生。此类离散化映射可被方便地提升至高阶切束，从而构造高阶欧拉-拉格朗日方程的几何积分器。给定代价函数后，全驱动机械系统的最优控制问题可被理解为高阶变分问题。本文引入与收缩映射相关联的高阶离散化映射概念，用于构造机械系统最优控制的几何积分器。特别地，我们研究了该理论在平面刚体避障路径规划中的应用。