Generalizing the analysis of PRM

This paper presents it novel analysis of the probabilistic roadmap method (PRM) for path planning. We formulate the problem in terms of computing the transitive closure of a relation over a probability space and give a bound in terms of the number of intermediate points for some path and the probability of choosing a point from a certain set. Explicit geometric assumptions are not necessary to complete this analysis and consequently it provides some unification of the previous work as well as generalizing new path planning problems, two of which, 2k-DOF kinodynamic point robots and deformable robots with force field control, are presented in this paper.