The evolution of the backward P. Hall coordinates is therefore given
by the Chen-Fliess-Sussmann (CFS) equation:
Example 15..23 (The CFS Equation for the Nonholonomic Integrator)
The extended system for the nonholonomic integrator
was given in
(
15.115). The differential equation (
15.128)
for the Lie group is
|
(15.133) |
because
.
There are two expressions for its solution. The Chen-Fliess series
(15.129) becomes
|
(15.134) |
The initial condition
is satisfied if
for
from
to
. The second expression for
is
(
15.130), which in the case of the nonholonomic
integrator
becomes
|
(15.135) |
Note that
|
(15.136) |
Equating (
15.133) and (
15.135) yields
|
(15.137) |
in which the time dependencies have been suppressed to shorten the
expression. The formal Lie series expansions, appropriately for the
exponentials, are now used. For
,
|
(15.138) |
and
|
(15.139) |
Also,
|
(15.140) |
and
|
(15.141) |
The truncation is clearly visible in (
15.140) and
(
15.141). The
terms are absent because
is
a polynomial of degree two, and its square would be of degree four.
Substitution into (15.137), performing noncommutative
multiplication, and applying the Lie bracket definition yields
|
(15.142) |
Equating like terms yields the Chen-Fliess-Sussmann equation
|
(15.143) |
Recall that
is given. By integrating (
15.143) from
, the backward P. Hall coordinate
trajectory
is obtained.