Harmonic Functions and Collision Probabilities

There is a close relationship between harmonic functions— which have recently been proposed for path planning—and hitting probabilities for random processes. The hitting proba bilities for random walks can be cast as a Dirichlet problem for harmonic functions, in much the same way as in path plan ning. This equivalence has implications both for uncertainty in motion planning and in the application of machine-learning techniques to some robot problems. In particular, Erdmann's method can directly incorporate such hitting probabilities. In addition, the value functions obtained by reinforcement learn ing algorithms can be rapidly reconstructed by relaxation or resistive networks, once the extrema for such functions are known.