##

15.4.3 Determining Controllability

Determining complete integrability is the first step toward
determining whether a driftless control-affine system is
STLC. The Lie bracket
attempts to produce motions in directions that do not seem to be
allowed by the system distribution. At each , a velocity not in
may be produced by the Lie bracket. By working further
with Lie brackets, it is possible to completely characterize *all*
of the directions that are possible from each . So far, the Lie
brackets have only been applied to the system vector fields ,
, . It is possible to proceed further by applying Lie
bracket operations on Lie brackets. For example,
can be computed. This might generate a vector field that is linearly
independent of all of the vector fields considered in Section
15.4.2 for the Frobenius theorem. The main idea in this
section is to apply the Lie bracket recursively until no more
independent vector fields can be found. The result is called the Lie
algebra. If the number of independent vector fields obtained in this
way is the dimension of , then it turns out that the system is
STLC.

**Subsections**
Steven M LaValle
2020-08-14