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)
There are two interesting ways to interpret (13.2):
To obtain the first interpretation of , hold fixed; for each , a velocity vector is obtained. The set of all allowable velocity vectors at is
To obtain the second interpretation, hold fixed. For example, let
. The vector field
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
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 2020-08-14