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.


Steven M LaValle 2020-08-14