Overview

Every Robot system is subject to a variety of motion constraints, but not all of these can be expressed as configuration constraints. A familiar example of such a system is a car. At low speeds, the rear wheels of the car roll freely in the direction they are pointing, but they prevent slipping motion in the perpendicular direction. This constraint implies that the car cannot translate directly to the side. We know by experience, however, that this velocity constraint does not imply a constraint on configurations; the car can reach any position and orientation in the obstacle-free plane. In fact, the prevented sideways translation can be approximated by parallel- parking maneuvers.

This no-slip constraint is a nonholonomic constraint, a constraint on the velocity. In addition to rolling without slipping, conservation of angular momentum is a common source of nonholonomic constraints in mechanical systems.

If, instead of viewing the car as a system subject to a motion constraint, we considered the fact that there are only two inputs (speed and steering angle) to control the car's three degrees of freedom, we might call the system underactuated. Under- actuated systems have fewer controls than degrees of freedom. For second-order mechanical systems, such as those described in the previous chapter, underactuation implies equality constraints on the possible accelerations of the system.

In this section we study motion planning for systems that are underactuated or subject to motion constraints. Our first task is to determine if the constraints actually limit the...