Preimages

Now consider how to construct motion commands. Using the hierarchical approach, the main task of terminating in the goal region can be decomposed into achieving intermediate subgoals. The preimage $P({\mu},G)$ of a motion command ${\mu}$ and subgoal $G \subset \operatorname{cl}({\cal C}_{free})$ is the set of all history I-states from which ${\mu}$ is guaranteed to be achieved in spite of all interference from nature. Each motion command must recognize that the subgoal has been achieved so that it can apply its termination action. Once a subgoal is achieved, the resulting history I-state must lie within the required set of history I-states for the next motion command in the plan. Let ${\cal M}$ denote the set of all allowable motion commands that can be defined. This can actually be considered as an action space for the high-level module.