A distribution^{15.8} 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

is called the

(15.66) |

Furthermore, it

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

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

(15.68) |

The distribution is nonsingular because for any in the coordinate neighborhood, the resulting vector space is two-dimensional.

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 2012-04-20