8.5.2 Dynamic Programming with Interpolation

This section concludes Part II by solving the motion planning problem with value iteration, which was introduced in Section 2.3. It has already been applied to obtain discrete feedback plans in Section 8.2. It will now be adapted to continuous spaces by allowing interpolation first over a continuous state space and then by additionally allowing interpolation over a continuous action space. This yields a numerical approach to computing optimal navigation functions and feedback plans for motion planning. The focus will remain on backward value iteration; however, the interpolation concepts may also be applied in the forward direction. The approach here views optimal feedback motion planning as a discrete-time optimal control problem [28,84,151,583].