The geometry of time-optimal trajectories for an omni-directional robot

The optimal trajectories are known analytically for only a few ground vehicles: steered cars (Dubins 1957; Reeds & Shepp 1990), and wheel-chair-like differential-drive vehicles (Balkcom & Mason 2002). This paper presents the analytical time-optimal trajectories for a kinematic model of a vehicle of a new class, shown in figure 1(a). The wheels are omniwheels; unlike regular wheels, omniwheels slip freely in the direction perpendicular to the controlled direction. The arrangement of wheels allows this robot to drive in any direction without the need to turn first, and to spin as it does so. We model the vehicle as a rigid body in the unobstructed plane, with configuration (x, y, θ) ∈ SE. We assume that the controls are the angular velocities of the wheels in the powered directions, (v1, v2, v3), and assume that the controls are independently bounded: v{1,2,3} ∈ [−1, 1]. Although the vehicle can move in arbitrary directions in the configuration space, some directions are faster than others. The optimal trajectories consist of sequences of three types of constant controls: spins in place (all three wheels spin at maximum speed in the same direction), arcs of circles, (all wheels spin at maximum speed, but not all in the same direction), and straight lines (two wheels spin at maximum speed in opposite directions, and the third does not spin). We assign a symbol to each constant control: P, C , S, respectively. No optimal trajectory contains more than 18 control switches, and only certain sequences are ever optimal. There are four classes of optimal trajectory, which we call spin, roll, shuffe, and tangent; figure 3 shows an example of each class. This classification is complete in the sense that every optimal trajectory must be one of these types, and minimal in the sense that for each class there exists a pair of configurations for which only a trajectory of that class is optimal. One focus of QR research is reasoning about continuous systems with some known structure. Optimal trajectories are a fundamental characteristic of a system. Robots expend resources to achieve goals; the simplest resource is time, and the simplest goal is for the system to reach a desired configuration. The time-optimal trajectories are therefore a basic property of the mechanism, and are a basis for complex reasoning about mechanism design, planning among obstacles, or vehicle control. The trajectories also provide a natural metric on the configuration space that is independent of the (a) Photograph. v 1