The Minimum-Time Trajectories for an Omni-Directional Vehicle

One common mobile robot design consists of three ‘omniwheels’ arranged at the vertices of an equilateral triangle, with wheel axles aligned with the rays from the center of the triangle to each wheel. Omniwheels, like standard wheels, are driven by the motors in a direction perpendicular to the wheel axle, but unlike standard wheels, can slip in a direction parallel to the axle. Unlike a steered car, a vehicle with this design can move in any direction without needing to rotate first, and can spin as it does so. We show that if there are independent bounds on the speeds of the wheels, the fastest trajectories for this vehicle contain only spins in place, circular arcs, and straight lines parallel to the wheel axles. We classify optimal trajectories by the order and type of the segments; there are four such classes, and there are no more than 18 control switches in any optimal trajectory.