碰撞检测还是最近邻搜索？基于采样的运动规划计算瓶颈研究
Collision detection or nearest-neighbor search? On the computational bottleneck in sampling-based motion planning
摘要：
最近邻搜索的复杂性决定了许多基于采样的运动规划算法的渐近运行时间。然而，碰撞检测往往被认为是在实践中的计算瓶颈。检查各种渐近最优规划算法，我们表征设置，我们称之为NN敏感，其中最近邻搜索的实际计算作用远非可以忽略不计，即，最近邻搜索所占用的运行时间部分与冲突检测所占用的时间部分相当，或者有时甚至更大。这加强并证实了运动规划算法可以显著受益于高效且可能专门定制的最近邻数据结构的主张。这些算法的渐近（近）最优性依赖于规定的连接半径，在配置$q$周围定义一个球，使得$q$需要连接到该球中的所有其他配置。为了方便我们的研究，我们展示了如何适应这个半径的非欧几里德空间，这是普遍的运动规划。该技术结果具有独立的意义，因为它使得能够将径向连接方法与常见的替代方案进行比较，即，将每个配置连接到其$k$最近的邻居（$k$ -NN）。事实上，正如我们所证明的，在某些情况下，使用径向连接方案，产生特定成本的解决方案路径比使用$k$ -NN快十倍（甚至更多）。
英文摘要：
The complexity of nearest-neighbor search dominates the asymptotic running time of many sampling-based motion-planning algorithms. However, collision detection is often considered to be the computational bottleneck in practice. Examining various asymptotically optimal planning algorithms, we characterize settings, which we call NN-sensitive, in which the practical computational role of nearest-neighbor search is far from being negligible, i.e., the portion of running time taken up by nearest-neighbor search is comparable, or sometimes even greater than the portion of time taken up by collision detection. This reinforces and substantiates the claim that motion-planning algorithms could significantly benefit from efficient and possibly specifically-tailored nearest-neighbor data structures. The asymptotic (near) optimality of these algorithms relies on a prescribed connection radius, defining a ball around a configuration $q$, such that $q$ needs to be connected to all other configurations in that ball. To facilitate our study, we show how to adapt this radius to non-Euclidean spaces, which are prevalent in motion planning. This technical result is of independent interest, as it enables to compare the radial-connection approach with the common alternative, namely, connecting each configuration to its $k$ nearest neighbors ($k$-NN). Indeed, as we demonstrate, there are scenarios where using the radial connection scheme, a solution path of a specific cost is produced ten-fold (and more) faster than with $k$-NN.
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