15.5.1 Using the P. Hall Basis

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*

in which each is an action variable, and is a vector field in , the P. Hall basis. For every , each term of (15.114) is , which comes from the original system. For , each represents a Lie product in , and is a

The first two terms correspond to the original system. The last term arises from the Lie bracket . Only one fictitious action variable is needed because the three P. Hall vector fields are independent at every .

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:

- Determine an action trajectory for the extended system, for which and for some .
- Convert into an action trajectory that eliminates the fictitious variables and uses the actual action variables , , .