### 15.4.2.2 Distributions

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.

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 2012-04-20