Controllability

In addition to integrability, another important property of a state transition equation is controllability. Intuitively, controllability implies that the robot is able to overcome its differential constraints by using Lie brackets to compose new motions. The controllability concepts assume that there are no obstacles.

Two kinds of controllability will be considered. A point, $x^\prime$,is reachable from x, if there exists an input that can be applied to bring the state from x to $x^\prime$. Let R(x) denote the set of all points reachable from x. A system is locally controllable if for all $x \in X$, R(x) contains an open set that contains x. This implies that any state can be reached from any other state.

Let $R(x,\Delta t)$ denote the set of all points reachable in time $\Delta t$. A system is small-time controllable if for all $x \in X$ and any $\Delta t$, then $R(x,\Delta t)$ contains an open set that contains x.

The Dubins car is an example of a system that is locally controllable, but not small-time controllable. If there are no obstacles, it is possible to bring the car to any desired configuration from any initial configuration. This implies that the car is locally controllable. Suppose one would like to move the car to a position that would be obtained by the Reeds-Shepp car by moving a small amount in reverse. Because the Dubins car must drive forward to reach this configuration, it could require time larger than some small $\Delta t$. Hence, the Dubins care is not small-time controllable.

However, a substantial amount of time might be required to drive the care

Chow's theorem is used to determine small-time controllability.

Theorem 788 ((Chow))

A system is small-time controllable if and only if the dimension of $CLA(\triangle)$ is n, the dimension of X.