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Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers
Algorithmica ( IF 1.1 ) Pub Date : 2021-01-02 , DOI: 10.1007/s00453-020-00784-6
Hugo A. Akitaya , Esther M. Arkin , Mirela Damian , Erik D. Demaine , Vida Dujmović , Robin Flatland , Matias Korman , Belen Palop , Irene Parada , André van Renssen , Vera Sacristán

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses $$O(n^2)$$ O ( n 2 ) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.

中文翻译:

通过枢轴对连接面的模块化机器人进行通用重构:O(1) 火枪手

我们提出了第一个通用重新配置算法,用于使用枢轴移动在任何两个小平面连接的方形网格配置之间转换模块化机器人。更准确地说,我们展示了五个额外的“助手”模块(“火枪手”)足以在任何两个给定配置之间重新配置剩余的 n 个模块。我们的算法使用 $$O(n^2)$$ O ( n 2 ) 支点移动,这是最坏情况下的最优。以前的重新配置算法要么需要较少限制的“滑动”移动,不保留小平面连接性,要么对于我们考虑的设置,只能处理由局部禁止模式定义的配置的一小部分子集。具有禁止模式的配置确实具有断开连接的重新配置图(离散配置空间),并且我们确实表明它们可以具有指数数量的连接组件。但是在整个配置中禁止局部模式远没有必要,因为我们表明,只需添加恒定数量的模块(放置为可自由重新配置)就足以实现通用可重新配置。我们还对保留小平面连接性的三种不同的自然枢轴移动模型进行了分类,并显示了这些模型之间的分离。
更新日期:2021-01-02
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