车型机器人的时间最优轨迹

基于几何最优控制理论, 本文提出了一种新的几何推理方法, 这种方法能够有效地推导出车型机器人给定两点间的时间最优轨迹类型的充分集合. 同时也为此类非线性问题提供了一种新的思路. 首先根据旁氏极大值原理和李代数推导出切换函数的结构特性, 根据这种结构特性建立一个切换坐标系并引入一个新的向量, 该向量在此坐标系中的旋转轨迹与时间最优轨迹具有一一对应关系. 进而得到一个结论: 如果该向量的始末旋转位置和方向一致, 那么将唯一的确定一条最优轨迹. 这是第一次得到一个能够直接应用于计算一条精确的最优轨迹的结论.

Time-optimal Trajectories for a Car-like Robot

This paper provides a new geometric method for achieving the sufficient family of the time-optimal trajectories to connect any two configurations of the robot in a 3-dimensional manifold based on the geometric optimal control theory.We provide a new perspective for analyzing this special type of nonlinear problems.Based on the structural characteristics of the switching functions and their derivatives from the Pontryagin's minimum principle(PMP)and the Lie algebra,we build a special coordinate system and introduce a new vector.We discover the one-to-one mapping between the rotation trajectory of this new vector and the optimal control trajectory.Furthermore,we define a switching vector that denotes the position and rotation direction of this vector,and reach a conclusion that the specified initial and final switching vectors can uniquely determine an optimal trajectory.In addition,it is the first time a condition that can be used directly for selecting a time-optimal trajectory is provided.