Geometric Construction of Time Optimal Trajectories for Differential Drive Robots Devin J. Balkcom, Carnegie Mellon University, Pittsburgh PA 15213 Matthew T. Mason, Carnegie Mellon University, Pittsburgh PA 15213 Abstract We consider a differential drive mobile robot: two unsteered coaxial wheels are independently actuated. Each wheel has bounded velocity, but no bound on torque or acceleration. Pontryagin’s Maximum Principle gives an elegant descrip- tion of the extremal trajectories, which are a superset of the time optimal trajectories. Further analysis gives an enumer- ation of the time optimal trajectories, and methods for iden- tifying the time optimal trajectories between any two con- figurations. This paper recapitulates and refines the results of [1] and [2] and presents a simple graphical technique for constructing time optimal trajectories. 1 Introduction This paper addresses the time optimal paths for differen- tial drive mobile robots with bounded velocity. Differential drive (or diff drive) means there are two unsteered indepen- dently actuated coaxial wheels. Bounded velocity means that each wheel is independently bounded in velocity, but acceleration is not bounded. Even discontinuities in wheel velocity are permitted. The environment is planar and free of obstacles. Under these assumptions, we will see that the time op- timal paths are composed of straight lines alternating with turns “in place”, i.e. turns about the center of the robot. Optimal paths contain at most three straights and two turns. There are a number of other restrictions, leading to a set of 40 different combinations arranged in 9 different symmetry classes. The simplest nontrivial motions are turn-straight- turn motions: turn to face the goal (or away from the goal); roll straight forward (or backward) to the goal; turn to the goal orientation. In some instances the optimal path passes through an intermediate “via” point. See Figure 1 for exam- ple motions from seven of the nine classes. Figure 1: The seven simplest optimal trajectory classes. To derive the optimal paths we will use Pontryagin’s maximum principle to obtain a geometric program for the extremal trajectories, which are a superset of the optimal trajectories. We then derive some additional necessary con- ditions, leading to a complete enumeration of optimal tra- jectories and a planning algorithm. Finally we reformulate the analysis to give a more intuitive geometric procedure for constructing optimal trajectories. Previous Work This paper expands on the results presented in [1] and [2]. Other work on diff drive robots has assumed bounds on ac- celeration rather than on velocity; for example see papers by Reister and Pin [6] and Renaud and Fourquet [7]. For the bounded acceleration model, the time optimal trajectories have been found numerically, and there is current work to