Efficient Nearest-Neighbor Search for Dynamical Systems with Nonholonomic Constraints

Nearest-neighbor search dominates the asymptotic complexity of sampling-based motion planning algorithms and is often addressed with k-d tree data structures. While it is generally believed that the expected complexity of nearest-neighbor queries is O(log(N)) in the size of the tree, this paper reveals that when a classic k-d tree approach is used with sub-Riemannian metrics, the expected query complexity is in fact Θ(N log(N)) for a number p ∈ [0, 1) determined by the degree of nonholonomy of the system. These metrics arise naturally in nonholonomic mechanical systems, including classic wheeled robot models. To address this negative result, we propose novel k-d tree build and query strategies tailored to sub-Riemannian metrics and demonstrate significant improvements in the running time of nearest-neighbor search queries.