The game against nature involves two decision makers: nature and the
robot. Once the plan is formulated, the decisions of the robot become
fixed, which leaves nature as the only remaining decision maker.
Using this interpretation, a directed graph can be defined in the same
way as in Section 2.1, except nature actions are used
instead of the robot's actions. It can even be imagined that nature
itself faces a discrete feasible planning problem as in Formulation
2.1, in which
replaces , and
there is no goal set. Let
denote a *plan-based
state transition graph*, which arises under the constraint that
is executed. The vertex set of
is . A
directed edge in
exists from to if
there exists some
such that
. Thus, from each vertex in
,
the set of outgoing edges represents all possible transitions to next
states that are possible, given that the action is applied according
to . In the case of probabilistic uncertainty,
becomes a weighted graph in which each edge is
assigned the probability
. In
this case,
corresponds to the graph representation
commonly used to depict a Markov chain.

A nondeterministic forward projection can easily be derived from by following the edges outward from the current state. The outward edges lead to the states of the one-stage forward projection. The outward edges of these states lead to the two-stage forward projection, and so on. The probabilistic forward projection can also be derived from .

Steven M LaValle 2020-08-14