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:缩回映射被用于将构形流形的切空间离散化为两个构形流形的副本,在那里动力学发生。这样的离散化映射可以方便地提升到高阶切空间,以构造高阶欧拉-拉格朗日方程的几何积分器。给定一种代价函数,全力作用的机械系统的最优控制问题可以理解为一种高阶变分问题。在本文中,我们引入了与缩回映射相关的高阶离散化映射的概念,以构造机械系统最优控制的几何积分器。特别是,我们研究了在避开平面刚体的障碍物时的路径规划应用。