The steering method presented in this section is due to Lafferriere and Sussmann [574]. It is assumed here that a driftless control-affine system is given, in which is a Lie group, as introduced in Example 15.15. Furthermore, the system is assumed to be STLC. The steering method sketched in this section follows from the Lie algebra . The idea is to apply piecewise-constant motion primitives to move in directions given by the P. Hall basis. If the system is nilpotent, then this method reaches the goal state exactly. Otherwise, it leads to an approximate method that can be iterated to get arbitrarily close to the goal. Furthermore, some systems are nilpotentizable by using feedback [442].
The main idea is to start with (15.53) and construct an extended system
It is straightforward to move this system along a grid-based path in
. Motions in the and directions are obtained by
applying and , respectively. To move the
system in the direction, the commutator motion in
(15.71) should be performed. This corresponds to applying
. The steering method described in this section yields a
generalization of this approach. Higher degree Lie products can be
used, and motion in any direction can be achieved.
Suppose some and are given. There are two phases to the steering method: