A discrete cost functional must be derived from the continuous cost functional, (8.39). The final term is just assigned as . The cost at each stage is

and is used in the place of in (8.56). For many problems, the integral does not need to be computed repeatedly. To obtain Euclidean shortest paths, can be safely assigned for all and . A reasonable approximation to (8.65) if is small is .

Steven M LaValle 2020-08-14