The controllability of a driftless control-affine system
(15.53) can be characterized using the *Lie algebra
rank condition* (or *LARC*). Recall the definition of STLC from
Section 15.1.3. Assume that either
or
at least contains an open set that contains the origin of
.
The *Chow-Rashevskii theorem* [112,156,846] states:

*A driftless control-affine system,
(15.53), is small-time locally controllable (STLC) at a
point if and only if
, the
dimension of .*

If the condition holds for every , then the whole system is STLC. Integrability can also be expressed in terms of . Assume as usual that . The three cases are:

(15.110) |

Steven M LaValle 2020-08-14