Motion planning deals with the problem of finding trajectory between two configurations in some space under some constraints. The science of motion planning deals with solving this problem efficiently in different spaces and under varying sets of constraints. This thesis provides an extended analysis of the PRM algorithm by using tools from measure theory to obtain greater generality. We then apply the results to planning in finite CW complexes and detail an application to planning with a polygonal robot in a polygonal workspace when contact is allowed. The final part of thesis deals with an application of motion planning to untangling mathematical knots. This problem is particularly interesting as it addresses motion planning with large number of degrees of freedom in a reconfigurable space. Planning applications with flexible robots, reconfigurable robots, molecular biology and other instances that share similar characteristics are some of the most challenging in planning today.
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