Time-optimal Trajectories for a Car-like Robot
车型机器人的时间最优轨迹

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.