This paper presents the time optimal trajectories for differential drive vehicles in the unobstructed plane. The wheel angular velocities are bounded, but may be discontinuous. The paper proves the existence of optimal controls, derives the structure of optimal trajectories, and develops an algorithm for producing a time optimal trajectory between any two configurations. Every nontrivial optimal trajectory is composed of straight segments alternating with turns about the robot's center. Optimal trajectories may have as many as five actions, but four actions are sufficient-for every optimal trajectory of five actions, there is an equally fast trajectory with four actions.

Origami, the art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example of the many difficulties and general manipulation problems faced in robotics. This paper describes our initial exploration, and highlights key problems in the manipulation, modeling, and design of foldable structures. Results include the design of the first origami-folding robot, a complete fold-sequence planner for a simple class of origami, and analysis of the kinematics of more complicated folds, including the common paper shopping bag.

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.

The shortest paths for a mobile robot are a fundamental property of the mechanism, and may also be used as a family of primitives for motion planning in the presence of obstacles. This paper characterizes shortest paths for differential-drive mobile robots, with the goal of classifying solutions in the spirit of Dubins curves and Reeds—Shepp curves for car-like robots. To obtain a well-defined notion of shortest, the total amount of wheel-rotation is optimized. Using the Pontryagin Maximum Principle and other tools, we derive the set of optimal paths, and we give a representation of the extremals in the form of finite automata. It turns out that minimum time for the Reeds—Shepp car is equal to minimum wheel-rotation for the differential-drive, and minimum time curves for the convexified Reeds—Shepp car are exactly the same as minimum wheel-rotation paths for the differential-drive. It is currently unknown whether there is a simpler proof for this fact.

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