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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 considerably faster than previous path planners and solves problems 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 minima of this function. The most powerful of these techniques is a Monte Carlo technique that escapes local minima by executing Brownian motions. The overall approach is made possible by the systematic use of distributed representations (bitmaps) for the robot's work space and configuration space. We have experimented with the planner using several computer-simulated robots, including rigid objects with 3 DOFs (in 2D work space) and 6 DOFs (in 3D work space) and manipulator arms with 8, 10, and 31 DOFs (in 2D and 3D work spaces). Some of the most significant experiments are reported in this article..

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an acknowledgement of the authors and individual contributors to the work; and all applicable portions of the copyright notice. Copying, reproducing, or republishing for any other purpose shall require a license with payment of fee to the Paris Research Laboratory. All rights reserved. ii AbstractWe consider the problem of pricing an American contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the maximum over all possible early exercise strategies of the discounted expected cash flows under the modified risk-neutral information process.Several efficient numerical techniques exist for pricing American securities depending on one or few (up to 3) risk sources. They are either lattice-based techniques or finite difference approximations of the Black-Scholes diffusion equation. However, these methods cannot be used for high-dimensional problems, since their memory requirement is exponential in the number of risk sources.In this paper, we present an efficient numerical technique that combines Monte Carlo simulation with a particular partitioning method of the underlying assets space, which we call Stratified State Aggregation (SSA). Using this technique we can compute accurate approximations of prices of American securities with an arbitrary number of underlying assets. Our numerical experiments show that the method is practical for pricing American claims depending on up to 400 risk sources. On all problems for which we could compare the method with known optimal solutions, the price computed through stratified state aggregation was indistinguishable from the optimal theoretical price. Several numerical examples are presented and discussed.iii Acknowledgements This research benefited from discussions with Bruno Langlois. We wish to thank Thierry Pudet for reviewing an earlier draft.iv