8.5.2.2 The connection to feedback motion planning

The tools have now been provided to solve motion planning problems using value iteration. The configuration space is a continuous state space; let $X = {\cal C}_{free}$. The action space is also continuous, $U(x) = T_x(X)$. For motion planning problems, $0 \in T_x(X)$ is only obtained only when $u_T$ is applied. Therefore, it does not need to be represented separately. To compute optimal cost-to-go functions for motion planning, the main concerns are as follows:

1. The action space must be bounded.
2. A discrete-time approximation must be made to derive a state transition equation that works over stages.
3. The cost functional must be discretized.
4. The obstacle region, ${\cal C}_{obs}$, must be taken into account.
5. At least some interpolation region must yield $G^*(x) = 0$, which represents the goal region.

We now discuss each of these.