#### Forward projections under a fixed plan

Forward projections can now be defined under the constraint that a particular plan is executed. The specific expression of actions is replaced by . Each time an action is needed from a state , it is obtained as . In this formulation, a different may be used for each , assuming that is correctly defined to use whatever actions are actually available in for each .

First we will consider the nondeterministic case. Suppose that the initial state and a plan are known. This means that , which can be substituted into (10.10) to compute the one-stage forward projection. To compute the two-stage forward projection, is determined from for use in (10.11). A recursive formulation of the nondeterministic forward projection under a fixed plan is (10.28)

The probabilistic forward projection in (10.10) can be adapted to use , which results in (10.29)

The basic idea can be applied times to compute .

A state transition matrix can be used once again to express the probabilistic forward projection. In (10.15), all columns correspond to the application of the action . Let , be the forward projection due to a fixed plan . Each column of may represent a different action because each column represents a different state . Each entry of is (10.30)

The resulting defines a Markov process that is induced under the application of the plan .

Steven M LaValle 2020-08-14