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Varricchio, Valerio, Frazzoli, Emilio.  2018.  Asymptotically Optimal Pruning for Nonholonomic Nearest-Neighbor Search. 2018 IEEE Conference on Decision and Control (CDC). :4459—4466.
Nearest-Neighbor Search (NNS) arises as a key component of sampling-based motion planning algorithms and it is known as their asymptotic computational bottleneck. Algorithms for exact Nearest-Neighbor Search rely on explicit distance comparisons to different extents. However, in motion planning, evaluating distances is generally a computationally demanding task, since the metric is induced by the minimum cost of steering a dynamical system between states. In the presence of driftless nonholonomic constraints, we propose efficient pruning techniques for the k-d tree algorithm that drastically reduce the number of distance evaluations performed during a query. These techniques exploit computationally convenient lower and upper bounds to the geodesic distance of the corresponding sub-Riemannian geometry. Based on asymptotic properties of the reachable sets, we show that the proposed pruning techniques are optimal, modulo a constant factor, and we provide experimental results with the Reeds-Shepp vehicle model.