Let denote the set of permissible action trajectories for the system, as considered in Section 14.1.1. By default, this is taken as any for which (14.1) can be integrated. A system is called controllable if for all , there exists a time and action trajectory such that upon integration from , the result is . Controllability can alternatively be expressed in terms of the reachable sets of Section 14.2.1. The system is controllable if for all .
A system is therefore controllable if a solution exists to any motion planning problem in the absence of obstacles. In other words, a solution always exists to the two-point boundary value problem (BVP).
Many methods for determining controllability of a system are covered in standard textbooks on control theory. If the system is linear, as given by (13.37) with dimensions and , then it is controllable if and only if the controllability matrix
For fully actuated systems of the form , controllability can be determined by converting the system into double-integrator form, as considered in Section 14.4.1. Let the system be expressed as , in which . If contains an open neighborhood of the origin of , and the same neighborhood can be used for any , then the system is controllable. If a nonlinear system is underactuated, as in the simple car, then controllability issues become considerably more complicated. The next concept is suitable for such systems.
Steven M LaValle 2020-08-14