The planning problem can now be described. It may be tempting to express the model using continuous time, as opposed to discrete stages. This is a viable approach, but leads to planning under differential constraints, which is the topic of Part IV and is considerably more complicated. In the preimage-planning framework, a hierarchical approach is taken. A restricted kind of plan called a motion command, , will be defined, and the goal is achieved by constructing a sequence of motion commands. This has the effect of converting the continuous-time decision-making problem into a planning problem that involves discrete stages. Each time a motion command is applied, the robot must apply a termination action to end it. At that point another motion command can be issued. Thus, imagine that a high-level module issues motion commands, and a low-level module executes each until a termination condition is met.
For some action , let , in which is the termination action. A motion command is a feedback plan, , in which is the standard history I-space, based on initial conditions, the action history, and the sensing history. The motion command is executed over continuous time. At , . Using a history I-state gathered during execution, the motion command will eventually yield , which terminates it. If the goal was not achieved, then the high-level module can apply another motion command.
Steven M LaValle 2012-04-20