The time-optimal trajectories for an omni-directional vehicle

A 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. The shortest paths for this vehicle are straight lines. However, the vehicle can move more quickly in some directions than in others. What are the fastest trajectories? We consider a kinematic model of the vehicle and place independent bounds on the speeds of the wheels, but do not consider dynamics or bound accelerations. We derive the analytical fastest trajectories between configurations. The time-optimal trajectories 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.