Example 15.17 considered a special case of a
*chained-form system*. The system in (15.102) can be
generalized to any as

This can be considered as a system with

- Apply any action trajectory for and that brings and to their goal values. The evolution of the other states is ignored in this stage.
- This phase is repeated for each from to . Steer
to its desired value by applying
and (15.161)

in which and are chosen to satisfy the constraint(15.162)

Each execution of this phase causes the previous state variables to return to their previous values.

For a proof of the correctness of the second phase, and more
information in general, see [727,846]. It may appear that
very few systems fit the forms given in this section; however, it is
sometimes possible to transform systems to fit this form. Recall that
the original simple car model in (13.15) was simplified to
(15.54). Transformation methods for putting systems into
chained form have been developed. For systems that still cannot be
put in this form, Fourier techniques can be used to obtain approximate
steering methods that are similar in spirit to the methods in this
section. When the chained-form system is expressed using Pfaffian
constraints, the result is often referred
to as the *Goursat normal form*. The method can be extended even
further to *multi-chained-form systems*.

Steven M LaValle 2020-08-14