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.