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The Rapidly-exploring Random Tree (RRT) algorithm has been one of the mostprevalent and popular motion-planning techniques for two decades now.Surprisingly, in spite of its centrality, there has been an active debate underwhich conditions RRT is probabilistically complete. We provide two new proofsof probabilistic completeness (PC) of RRT with a reduced set of assumptions.The first one for the purely geometric setting, where we only require that thesolution path has a certain clearance from the obstacles. For the kinodynamiccase with forward propagation of random controls and duration, we only considerin addition mild Lipschitz-continuity conditions. These proofs fill a gap inthe study of RRT itself. They also lay sound foundations for a variety of morerecent and alternative sampling-based methods, whose PC property relies on thatof RRT.  Our original publication contains an error in the analysis of the case of thekinodynamic RRT. Here, we rectify the problem by modifying the proof of Theorem2, which, in particular, necessitated a revision of Lemma 3. Briefly, theoriginal (and erroneous) proof of Theorem 2 used a sequence of equal-sizeballs. The correction uses a sequence of balls of increasing radii. Weemphasize that the correction is in Lemma 3 and the proof of Theorem 2 only.The main results remain unchanged.

Probabilistic completeness of RRT for geometric and kinodynamic planning with forward propagation