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In this paper parameter-free concepts for exact motion planning are investigated. With the proposed RDT+ approach the collision detection parameters of a Rapidly exploring Dense Tree (RDT) are automatically adjusted until an exact solution can be found. For efficient planning discrete collision detection routines are used within the RDT planner and by verifying the results with exact collision detection methods, the RDT+ concept allows to compute motions that are guaranteed collision-free. We show the probabilistic completeness of the proposed planner and present an extension for handling narrow passages. The algorithms are evaluated in different experiments, including narrow passages and high-dimensional planning problems, that are solved in simulation and on the humanoid robot ARMAR-III.