arXiv 2005.074952020-05-13cs.CGcs.RO

三维移动机器人的局部聚集

Local Gathering of Mobile Robots in Three Dimensions

Heinz Nixdorf Institute（海因茨·尼克斯多夫研究所）；Paderborn University（帕德博恩大学）

Michael Braun, Jannik Castenow, Friedhelm Meyer auf der Heide

摘要：

在这项工作中，我们开始研究的聚集问题的机器人有限的视野范围内的三维欧氏空间。在集合问题中，一组最初分散的机器人需要聚集在同一位置。这些机器人的能力非常有限--它们不同意任何坐标系统或指南针，视野有限，没有过去的记忆，也不能交流。我们研究了两种不同的时间模型，在FSYNC（完全同步离散轮）和连续时间模型的问题。对于FSYNC，我们引入了3D-Go-To-The-Center-strategy，并证明了$Θ(n^2)$的运行时间与欧几里得平面中相同模型的当前最佳运行时间边界相匹配[SPAA'11]。我们的主要结果是从[Algosensors'17]到三维的合同策略（连续时间）的推广。在收缩策略中，位于所有机器人位置的全局凸包上的每个机器人以全速向凸包的内部移动。我们证明了运行时界的$O(Δ\cdot n^{3/2})$的任何三维收缩策略，其中$Δ$表示的直径的初始配置。这导致$\sqrt{n}$的因子接近$Ω(Δ\cdot n)$的下限，这在二维中已经是正确的。在一般情况下，它可能是很难的机器人有限的视野范围，以决定他们是否位于全球凸包和运动保持群体的连接性，使具体的收缩策略的设计具有挑战性的任务。我们证明了3D-Go-To-The-Center的连续变体是收缩的，并且保持了群体的连接。此外，我们给出了一个简单的设计标准的三维收缩策略，保持了群体的连通性，并介绍了一个示例性的策略，基于这个标准。

英文摘要：

In this work, we initiate the research about the Gathering problem for robots with limited viewing range in the three-dimensional Euclidean space. In the Gathering problem, a set of initially scattered robots is required to gather at the same position. The robots' capabilities are very restricted -- they do not agree on any coordinate system or compass, have a limited viewing range, have no memory of the past and cannot communicate. We study the problem in two different time models, in FSYNC (fully synchronized discrete rounds) and the continuous time model. For FSYNC, we introduce the 3D-Go-To-The-Center-strategy and prove a runtime of $Θ(n^2)$ that matches the currently best runtime bound for the same model in the Euclidean plane [SPAA'11]. Our main result is the generalization of contracting strategies (continuous time) from [Algosensors'17] to three dimensions. In contracting strategies, every robot that is located on the global convex hull of all robots' positions moves with full speed towards the inside of the convex hull. We prove a runtime bound of $O(Δ\cdot n^{3/2})$ for any three-dimensional contracting strategy, where $Δ$ denotes the diameter of the initial configuration. This comes up to a factor of $\sqrt{n}$ close to the lower bound of $Ω(Δ\cdot n)$ which is already true in two dimensions. In general, it might be hard for robots with limited viewing range to decide whether they are located on the global convex hull and which movement maintains the connectivity of the swarm, rendering the design of concrete contracting strategies a challenging task. We prove that the continuous variant of 3D-Go-To-The-Center is contracting and keeps the swarm connected. Moreover, we give a simple design criterion for three-dimensional contracting strategies that maintains the connectivity of the swarm and introduce an exemplary strategy based on this criterion.

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摘要：

英文摘要：

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