The Lie bracket is the only tool needed to determine whether a system
is completely integrable (holonomic) or nonholonomic (not integrable).
Suppose that a system of the form (15.53) is given.
Using the system vector fields , , there are
Lie brackets of the form for that
can be formed. A distribution
is called involutive [133] if for each of
these brackets there exist coefficients
such that
|
(15.86) |
In other words, every Lie bracket can be expressed as a linear
combination of the system vector fields, and therefore it already belongs
to
. The Lie brackets are unable to escape
and
generate new directions of motion. We did not need to consider all
possible Lie brackets of the system vector fields because it
turns out that
and consequently
. Therefore, the definition of involutive is not altered by looking
only at the
pairs.
If the system is smooth and the distribution is nonsingular, then the
Frobenius theorem immediately characterizes integrability:
A system is completely integrable if and only if it is
involutive.
Proofs of the Frobenius theorem appear in numerous differential
geometry and control theory books [133,156,478,846].
There also exist versions that do not require the distribution to be
nonsingular.
Determining integrability involves performing Lie brackets and
determining whether (15.86) is satisfied. The search for
the coefficients can luckily be avoided by using linear algebra tests
for linear independence. The
matrix , which was
defined in (15.56), can be augmented into an
matrix by adding as a new column. If the rank
of is for any pair and , then it is
immediately known that the system is nonholonomic. If the rank of
is for all Lie brackets, then the system is completely
integrable. Driftless linear systems, which are expressed as
for a fixed matrix , are completely integrable because all Lie
brackets are zero.
Example 15..11 (The Differential Drive Is Nonholonomic)
For the
differential drive model in (
15.54), the Lie bracket
was determined in Example
15.9 to be
. The matrix
, in which
, is
|
(15.87) |
The rank of
is
for all
(the determinant of
is
). Therefore, by the Frobenius theorem, the system is
nonholonomic.
Example 15..12 (The Nonholonomic Integrator Is Nonholonomic)
We would hope that the nonholonomic integrator is nonholonomic. In
Example
15.10, the Lie bracket was determined to be
. The matrix
is
|
(15.88) |
which clearly has full rank for all
.
Example 15..13 (Trapped on a Sphere)
Suppose that the following system is given:
|
(15.89) |
for which
and
. Since the vector fields are
linear, the Jacobians are constant (as in Example
15.10):
|
(15.90) |
Using (
15.80),
|
(15.91) |
This yields the matrix
|
(15.92) |
The determinant is zero for all
, which means that
is never linearly independent of
and
. Therefore, the
system is completely integrable.
15.10
The system can actually be constructed by differentiating the equation
of a sphere. Let
|
(15.93) |
and differentiate with respect to time to obtain
|
(15.94) |
which is a Pfaffian constraint.
A
parametric representation of the set of vectors that satisfy
(
15.94) is given by (
15.89). For
each
, (
15.89) yields a vector
that satisfies (
15.94). Thus, this was an example
of being trapped on a sphere, which we would expect to be completely
integrable. It was difficult, however, to suspect this using only
(
15.89).
Steven M LaValle
2020-08-14