We propose a new approach to robot path planning that consists of building and searching a graph connecting the local minima of a potential function defined over the robot’s configuration space. A planner based on this approach has been implemented. This planner is consider-ably faster than previous path planners and solves prob-lems for robots with many more degrees of freedom (DOFs). The power of the planner derives both from the "good " properties of the potential function and from the efficiency of the techniques used to escape the local min-ima of this function. The most powerful of these tech-niques is a Monte Carlo technique that escapes local min-ima by executing Brownian motions. The overall approach is made possible by t...