Motion commands

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, $\mu$, 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 $u \in U$, let $M_u = \{u,u_T\}$, in which $u_T$ is the termination action. A motion command is a feedback plan, ${\mu}: {\cal I}_{hist}\rightarrow M_u$, in which ${\cal I}_{hist}$ 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 $t=0$, ${\mu}({\eta}_0) = u$. Using a history I-state ${\eta}$ gathered during execution, the motion command will eventually yield ${\mu}({\eta}) = u_T$, which terminates it. If the goal was not achieved, then the high-level module can apply another motion command.