To facilitate the notation, suppose in this section that for all . In Section 10.1.3 this will be lifted.

Suppose that the initial state, , is known. If the action is applied, then the set of possible next states is

which is just a special version of (10.5). Now suppose that an action will be applied. The forward projection must determine which states could be reached from by applying followed by . This can be expressed as

This idea can be repeated for any number of iterations but becomes quite cumbersome in the current notation. It is helpful to formulate the forward projection recursively. Suppose that an action history is fixed. Let denote the forward projection at stage , given that is the forward projection at stage . This can be computed as

This may be applied any number of times to compute from an initial condition .

Steven M LaValle 2012-04-20