Time-Optimal Trajectories for Mobile Robots With Two Independently Driven Wheels

Abstract

This article addresses the problem of time-optimal motions for a mobile platform in a planar environment. The platform has two nonsteerable, independently driven wheels. The overall mission of the robot is expressed in terms of a sequence of via points at which the platform must be at rest in a given configuration (position and orientation). The objective is to plan time-optimal trajectories between these configurations, assuming an unobstructed environment. Using Pontryagin's maximum principle (PMP), we formally demonstrate that all time-optimal motions of the platform for this problem occur for bang-bang controls on the wheels (at each instant, the acceleration on each wheel is at either its upper or its lower limit). The PMP, however, provides only the conditions necessary for time optimality. To find the time- optimal robot trajectories, we first parameterize the bang-bang trajectories using the switch times on the wheels (the times at which the wheel accelerations change sign). With this param eterization, we can fully search the robot trajectory space and find the switch times that will produce particular paths to a desired final configuration of the platform. We show numer ically that robot trajectories with three switch times (two on one wheel and one on the other) can reach any position, while trajectories with four switch times can reach any configuration. By numerical comparison with other trajectories involving sim ilar or greater numbers of switch times, we then identify the sets of time-optimal trajectories. These are uniquely defined using ranges of the parameters and consist of subsets of trajec tories with three switch times (for the problem when the final orientation of the robot is not specified) or four switch times (when a full final configuration is specified). We conclude with a description of the use of the method for trajectory planning for one of our robots and discuss some comparisons of sample time-optimal paths with minimum length paths.