After the cost-to-go values stabilize, the resulting cost-to-go function, can be used as a navigation function. Even though is defined only over , the value of the navigation function can be obtained using interpolation over any point in . The optimal action is selected as the one that satisfies the in (8.6). This means that the state trajectory does not have to visit the grid points as in the Manhattan model. A trajectory can visit any point in , which enables trajectories to converge to the true optimal solution as and the grid spacing tend to zero.

Steven M LaValle 2012-04-20