The tools have now been provided to solve motion planning problems using value iteration. The configuration space is a continuous state space; let . The action space is also continuous, . For motion planning problems, is only obtained only when 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:

- The action space must be bounded.
- A discrete-time approximation must be made to derive a state transition equation that works over stages.
- The cost functional must be discretized.
- The obstacle region, , must be taken into account.
- At least some interpolation region must yield , which represents the goal region.

- Bounding the action space
- Obtaining a state transition equation
- Approximating the cost functional
- Handling obstacles
- Handling the goal region
- Using as a navigation function
- Topological considerations

Steven M LaValle 2020-08-14