TIME-OPTIMAL TRAJECTORIES FOR MOBILE ROBOTS WITH 2 INDEPENDENTLY DRIVEN WHEELS

This article addresses the problem of time-optimal motions for a mobil e platform in a planar environment. The platform has two nonsteerable, independently driven wheels. The overall mission of the robot is expr essed in terms of a sequence of via points at which the platform must be at rest in a given configuration (position and orientation). The ob jective is to plan time-optimal trajectories between these configurati ons, 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 i ts upper or its lower limit). The PMP, however provides only the condi tions necessary for time optimality. To find the time-optimal robot tr ajectories 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 ro bot trajectory space and find the switch times that will produce parti cular paths to a desired final configuration of the platform. We show numerically that robot trajectories with three switch times (two on on e wheel and one on the other) can reach any position, while trajectori es with four switch times can reach any configuration. By numerical co mparison with other trajectories involving similar or greater numbers of switch times, we then identify the sets of time-optimal trajectorie s. These are uniquely defined using ranges of the parameters and consi st of subsets of trajectories with three switch times Vbr the problem when the final orientation of the robot is not specified) or four swit ch limes (when a full final configuration is specified). We conclude w ith 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.