### 13.1.1.2 Parametric constraints

The parametric way of expressing velocity constraints gives a different interpretation to . Rather than directly corresponding to a velocity, each is interpreted as an abstract action vector. The set of allowable velocities is then obtained through a function that maps an action vector into . This yields the configuration transition equation (or system)

 (13.2)

in which is a continuous-time version of the state transition function that was developed in Section 2.1. Note that (13.2) actually represents scalar equations, in which is the dimension of . The system will nevertheless be referred to as a single equation in the vector sense. Usually, is fixed for all . This will be assumed unless otherwise stated. In this case, the fixed action set is denoted as .

There are two interesting ways to interpret (13.2):

1. Subspace of the tangent space: If is fixed, then maps from into . This parameterizes the set of allowable velocities at because a velocity vector, , is obtained for every .
2. Vector field: If is fixed, then can be considered as a function that maps each into . This means that defines a vector field over for every fixed .

Example 13..1 (Two Interpetations of )   Suppose that , which yields a two-dimensional velocity vector space at every . Let , and be defined as and .

To obtain the first interpretation of , hold fixed; for each , a velocity vector is obtained. The set of all allowable velocity vectors at is

 (13.3)

Suppose that . In this case, any vector of the form for any is allowable.

To obtain the second interpretation, hold fixed. For example, let . The vector field over is obtained.

It is important to specify when defining the configuration transition equation. We previously allowed, but discouraged, the action set to depend on . Any differential constraints expressed as for any can be alternatively expressed as by defining

 such that (13.4)

for each . In this definition, is not necessarily a subset of . It is usually more convenient, however, to use the form and keep the same for all . The common interpretation of is that it is a set of fixed actions that can be applied from any point in the C-space.

In the context of ordinary motion planning, a configuration transition equation did not need to be specifically mentioned. This issue was discussed in Section 8.4. Provided that contains an open subset that contains the origin, motion in any direction is allowed. The configuration transition equation for basic motion planning was simply . Since this does not impose constraints on the direction, it was not explicitly mentioned. For the coming models in this chapter, constraints will be imposed on the velocities that restrict the possible directions. This requires planning algorithms that handle differential models and is the subject of Chapter 14.

Steven M LaValle 2012-04-20