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:
Steven M LaValle 2012-04-20