Randomized potential fields

The randomized potential field method of Section 5.4.3 can be easily adapted to handle differential constraints. Instead of moving in any direction to reduce the potential value, motion primitives are applied and integrated to attempt to reduce the value. For example, under the discrete-time model, each $u \in U_d$ can be applied over $\Delta t$, and the one for which the next state has the lowest potential value should be selected as part of the descent. Random walks can be tried whenever no such action exists, but once again, motion in any direction is not possible. Random actions can be chosen instead. The main problems with the method under differential constraints are 1) it is extremely challenging to design a good potential function, and 2) random actions do not necessarily provide motions that are similar to those of a random walk. Section 15.1.2 discusses Lyapunov functions, which serve as good potential functions in the presence of differential constraints (but usually neglect obstacles). In the place of random walks, other planning methods, such as an RDT, could be used to try to escape local minima.