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The critical radius in sampling-based motion planning
The International Journal of Robotics Research ( IF 9.2 ) Pub Date : 2019-07-01 , DOI: 10.1177/0278364919859627
Kiril Solovey 1 , Michal Kleinbort 2
Affiliation  

We develop a new analysis of sampling-based motion planning in Euclidean space with uniform random sampling, which significantly improves upon the celebrated result of Karaman and Frazzoli and subsequent work. In particular, we prove the existence of a critical connection radius proportional to Θ ( n − 1 / d ) for n samples and d dimensions: below this value the planner is guaranteed to fail (similarly shown by Karaman and Frazzoli). More importantly, for larger radius values the planner is asymptotically (near-)optimal. Furthermore, our analysis yields an explicit lower bound of 1 − O ( n − 1 ) on the probability of success. A practical implication of our work is that asymptotic (near-)optimality is achieved when each sample is connected to only Θ ( 1 ) neighbors. This is in stark contrast to previous work that requires Θ ( log n ) connections, which are induced by a radius of order ( log n n ) 1 / d . Our analysis applies to the probabilistic roadmap method (PRM), as well as a variety of “PRM-based” planners, including RRG, FMT*, and BTT. Continuum percolation plays an important role in our proofs. Lastly, we develop similar theory for all the aforementioned planners when constructed with deterministic samples, which are then sparsified in a randomized fashion. We believe that this new model, and its analysis, is interesting in its own right.

中文翻译:

基于采样的运动规划中的临界半径

我们对欧几里得空间中基于采样的运动规划进行了统一随机采样的新分析,这显着改进了 Karaman 和 Frazzoli 以及后续工作的著名结果。特别是,我们证明了对于 n 个样本和 d 个维度,存在与 Θ ( n − 1 / d ) 成正比的临界连接半径:低于此值,规划器肯定会失败(Karaman 和 Frazzoli 类似地展示了这一点)。更重要的是,对于较大的半径值,规划器是渐近(接近)最优的。此外,我们的分析得出了成功概率的明确下限 1 - O ( n - 1 )。我们工作的一个实际含义是,当每个样本仅连接到 Θ ( 1 ) 邻居时,就会实现渐近(近)最优。这与之前需要 Θ ( log n ) 连接的工作形成鲜明对比,这是由阶数 (log nn) 1 / d 引起的。我们的分析适用于概率路线图方法 (PRM),以及各种“基于 PRM”的规划器,包括 RRG、FMT* 和 BTT。连续渗透在我们的证明中起着重要作用。最后,当使用确定性样本构建时,我们为所有上述规划器开发了类似的理论,然后以随机方式稀疏化。我们相信这个新模型及其分析本身就很有趣。当使用确定性样本构建时,我们为所有上述规划器开发了类似的理论,然后以随机方式稀疏化。我们相信这个新模型及其分析本身就很有趣。当使用确定性样本构建时,我们为所有上述规划器开发了类似的理论,然后以随机方式稀疏化。我们相信这个新模型及其分析本身就很有趣。
更新日期:2019-07-01
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