Solving geometric constraint problems through Monte Carlo optimization

Geometric constraint problems appear in many situations, including CAD systems, robotics, and computational biology. The complexity of these problems inspires the search for efficient solutions. We have developed a method to solve geometric constraint problems in the areas of geometric computation and robot path planning using configuration space subdivision. In this approach the configuration space, or parameter space, is subdivided and conservatively tested to find collision-free regions, which are then numerically searched for specific path solutions. This thesis presents a new more general approach to this last solution search step, using Monte Carlo optimization. In this new search approach, within a single subdivided area of configuration space, space is randomly sampled and then iteratively resampled based on importance weighting, until convergence to a solution with an acceptable error. We show that by using Monte Carlo optimization to extend configuration space subdivision we can solve higher dimensional problems more efficiently than configuration space subdivision by itself.