This section introduces a simple and effective way to sample the space of action trajectories. Section 14.2.3 covers the more general case. Under differential constraints, sampling-based motion planning algorithms all work by sampling the space of action trajectories. This results in a reduced set of possible action trajectories. To ensure some form of completeness, a motion planning algorithm should carefully construct and refine the sample set. As in Chapter 5, the qualities of a sample set can be expressed in terms of dispersion and denseness. The main difference in the current setting is that the algorithms here work with a sample sequence over , as opposed to over as in Chapter 5. This is required because solution paths can no longer be expressed directly on (or ).
The discrete-time model is depicted in Figure 14.5 and is characterized by three aspects:
For some problems, may already be finite. Imagine, for example, a model of firing one of several thrusters (turn them on or off) on a free-floating spacecraft. In this case no discretization of is necessary. In the more general case, may be a continuous set. The sampling methods of Section 5.2 can be applied to determine a finite subset .
Any action trajectory in can be conveniently expressed as an action sequence , in which each gives the action to apply from time to time . After stage , it is assumed that the termination action is applied.