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

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

The driftless case will be used throughout most of this section. The set , , , is referred to as the

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.

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

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

In the driftless case, this yields

as an equivalent way to express (15.53)

It is sometimes convenient to work with Pfaffian constraints,

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

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

Steven M LaValle 2020-08-14