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 $q$, a velocity not in ${\triangle}(q)$ 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 $q$. So far, the Lie brackets have only been applied to the system vector fields $h_1$, $\ldots$, $h_m$. It is possible to proceed further by applying Lie bracket operations on Lie brackets. For example, $[h_1,[h_1,h_2]]$ 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 $X$, then it turns out that the system is STLC.