Over the last few years, Probabilistic Roadmaps (PRMs) have emerged as a powerful approach for solving complex motion planning problems in robotics. Even beyond robotics, PRMs can be used to predict motions of biological macro-molecules such as proteins and synthesize motions for digital actors. Current PRM-based research focuses on challenges that arise as PRMs are being applied to motion planning problems in various scenarios. In response to some of those challenges, the following four contributions are being made in this thesis: (1) a dynamic checker for PRMs that exactly determines whether a path lies in free space, (2) a sampling strategy, called "small-step retraction" (SSR), that allows a PRM planner to efficiently construct roadmaps in free spaces with narrow passages, (3) an efficient multi-goal PRM planner, and (4) a PRM planner that can compute the motions and (re-)grasp operations of a two-arm system in order to tie self-knots of deformable linear objects (DLOs), as well as knots around simple static objects.

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