15.4.1 Control-Affine Systems
Nonholonomic system theory is restricted to a special class of
nonlinear systems. The techniques of Section 15.4 utilize
ideas from linear algebra. The main concepts will be formulated in
terms of linear combinations of vector fields on a smooth manifold
. Therefore, the formulation is restricted to control-affine
systems, which were briefly introduced in Section
13.2.3. For these systems,
is of the
form
|
(15.52) |
in which each is a vector field on .
The vector fields are expressed using a coordinate neighborhood of
. Usually, , in which is the dimension of . Unless
otherwise stated, assume that
. In some cases,
may be restricted.
Each action variable
can be imagined as a coefficient
that determines how much of is blended into the result
. The drift term always remains
and is often such a nuisance that the driftless case will be the main focus. This means that
for
all , which yields
|
(15.53) |
The driftless case will be used throughout most of this section. The
set , , , is referred to as the system
vector fields. It is essential that contains at least an open
set that contains the origin of
. If the origin is not
contained in , then the system is no longer
driftless.15.7
Control-affine systems arise in many mechanical systems. Velocity
constraints on the C-space frequently are of the Pfaffian
form (13.5). In Section
13.1.1, it was explained that under such constraints, a
configuration transition equation (13.6) can be
derived that is linear if is fixed. This is precisely the
driftless form (15.53) using
. Most of the
models in Section 13.1.2 can be expressed in this form.
The Pfaffian constraints on configuration
are often called kinematic constraints because they arise due to
the kinematics of bodies in contact, such as a wheel rolling. The
more general case of (15.52) for a phase space arises
from dynamic constraints that are obtained from
Euler-Lagrange equation (13.118) or
Hamilton's equations (13.198) in the formulation of
the mechanics. These constraints capture conservation laws, and the
drift term usually appears due to momentum.
Example 15..5 (A Simplified Model for Differential Drives and Cars)
Both the simple-car and the differential-drive models of Section
13.1.2 can be expressed in the form (
15.53)
after making simplifications. The simplified model,
(
15.48), can be adapted to conveniently express versions of
both of them by using different restrictions to define
. The third
equation of (
15.48) can be reduced to
without affecting the set of velocities that can be achieved. To
conform to (
15.53), the equations can then be written in
a linear-algebra form as
|
(15.54) |
This makes it clear that there are two system vector fields, which can
be combined by selecting the scalar values
and
. One
vector field allows pure translation, and the other allows pure
rotation. Without restrictions on
, this system behaves like a
differential drive because the simple car cannot execute pure
rotation. Simulating the simple car with (
15.54) requires
restrictions on
(such as requiring that
be
or
, as
in Section
15.3.2). This is equivalent to the unicycle from
Figure
13.5 and (
13.18).
Note that (15.54) can equivalently be expressed as
|
(15.55) |
by organizing the vector fields into a matrix.
In (15.54), the vector fields were written as column
vectors that combine linearly using action variables. This suggested
that control-affine systems can be alternatively expressed using
matrix multiplication in (15.55). In general, the vector
fields can be organized into an
matrix as
|
(15.56) |
In the driftless case, this yields
|
(15.57) |
as an equivalent way to express (15.53)
It is sometimes convenient to work with Pfaffian
constraints,
|
(15.58) |
instead of a state transition equation. As indicated in Section
13.1.1, a set of independent Pfaffian
constraints can be converted into a state
transition equation with
action variables. The
resulting state transition equation is a driftless control-affine
system. Thus, Pfaffian constraints
provide a dual way of specifying driftless control-affine systems.
The Pfaffian constraints can be
expressed in matrix form as
|
(15.59) |
which is the dual of (15.57), and is a
matrix formed from the coefficients of each Pfaffian
constraint. Systems with drift can be
expressed in a Pfaffian-like form by constraints
|
(15.60) |
Steven M LaValle
2020-08-14