A distribution15.8 expresses a set of vector fields on a smooth manifold.
Suppose that a driftless control-affine system (15.53) is
given. Recall the vector space definition from Section
8.3.1 or from linear algebra. Also recall that a state
transition equation can be interpreted as a vector field if the
actions are fixed and as a vector space if the state is instead fixed.
For
and a fixed , the state transition equation
defines a vector space in which each evaluated at is a basis
vector and each is a coefficient. For example, in
(15.54), the vector fields and evaluated at
become
and
,
respectively. These serve as the basis vectors. By selecting values
of
, a 2D vector space results. Any vector of the form
can be represented by setting and
. More generally, let
denote the vector space
obtained in this way for any .
The dimension of a vector space is the number of independent basis
vectors. Therefore, the dimension of
is the rank of
from (15.56) when evaluated at the particular . Now consider defining
for every . This
yields a parameterized family of vector spaces, one for each . The result could just as well be interpreted as a parameterized
family of vector fields. For example, consider actions for from
to of the form and for all
.
If the action is held constant over all , then it selects a
single vector field from the collection of vector fields:
|
(15.63) |
Using constant actions, an -dimensional vector space can be defined
in which each vector field is a basis vector (assuming the
are linearly independent), and the
are the
coefficients:
|
(15.64) |
This idea can be generalized to allow the to vary over .
Thus, rather than having constant, it can be interpreted as a
feedback plan
, in which the action at
is given by
. The set of all vector fields that can be
obtained as
|
(15.65) |
is called the distribution of the set
of
vector fields and is denoted as
. If
is obtained
from an control-affine system, then
is called the system
distribution. The resulting set of vector
fields is not quite a vector space because the nonzero coefficients
do not necessarily have a multiplicative inverse. This is
required for the coefficients of a vector field and was satisfied by
using
in the case of constant actions. A distribution is
instead considered algebraically as a module [469]. In
most circumstances, it is helpful to imagine it as a vector space
(just do not try to invert the coefficients!). Since a distribution
is almost a vector space, the
notation from linear algebra is
often used to define it:
|
(15.66) |
Furthermore, it is actually a vector space with respect to
constant actions
. Note that for each fixed ,
the vector space
is obtained, as defined earlier. A
vector field is said to belong to a distribution
if it can be expressed using (15.65). If for all , the dimension of
is , then
is called a
nonsingular distribution (or regular
distribution).
Otherwise,
is called a singular
distribution, and the
points for which the dimension of
is less than
are called singular points. If the dimension of
is a constant
over all , then is called the dimension of the
distribution and is denoted by
. If the vector fields
are smooth, and if is restricted to be smooth, then a
smooth distribution is obtained, which is a subset of the
original distribution.
Figure 15.15:
The distribution
can be
imagined as a slice of the tangent bundle . It restricts the
tangent space at every .
|
As depicted in Figure 15.15, a nice interpretation of
the distribution can be given in terms of the tangent bundle of a
smooth manifold. The tangent bundle was defined for
in (8.9) and generalizes to any smooth manifold
to obtain
|
(15.67) |
The tangent bundle is a -dimensional manifold in which
is the dimension of . A phase space for which
is
actually
. In the current setting, each element of
yields a state and a velocity,
. Which pairs are possible
for a driftless control-affine system? Each
indicates the
set of possible values for a fixed . The point is
sometimes called the base and
is called the fiber over
in . The distribution
simply specifies a subset
of for every . For a vector field to belong to
, it must satisfy
for all .
This is just a restriction to a subset of . If and the
system vector fields are independent, then any vector field is
allowed. In this case,
includes any vector field that can be
constructed from the vectors in .
Example 15..7 (The Distribution for the Differential Drive)
The system in (
15.54) yields a two-dimensional
distribution:
|
(15.68) |
The distribution is nonsingular because for any
in the
coordinate neighborhood, the resulting vector space
is two-dimensional.
Example 15..8 (A Singular Distribution)
Consider the following system, which is given in [
478]:
|
(15.69) |
The distribution is
|
(15.70) |
The first issue is that for any
,
,
which implies that the vector fields are linearly dependent over all
of
. Hence, this distribution is singular because
and
the dimension of
is
if
. If
,
then the dimension of
drops to
. The dimension of
is not defined because the dimension of
depends
on
.
A distribution can alternatively be defined directly from Pfaffian
constraints. Each
is called
an annihilator because enforcing the constraint eliminates many
vector fields from consideration. At each ,
is
defined as the set of all velocity vectors that satisfy all
Pfaffian constraints. The constraints
themselves can be used to form a codistribution, which is a kind
of dual to the distribution. The codistribution can be interpreted as
a vector space in which each constraint is a basis vector.
Constraints can be added together or multiplied by any
,
and there is no effect on the resulting distribution of allowable
vector fields.
Steven M LaValle
2020-08-14