We are considering the geometric amoebot model where a set of $n$ amoebots is placed on the triangular grid. An amoebot is able to send information to its neighbors, and to move via expansions and contractions. Since amoebots and information can only travel node by node, most problems have a natural lower bound of $\Omega(D)$ where $D$ denotes the diameter of the structure. Inspired by the nervous and muscular system, Feldmann et al. have proposed the reconfigurable circuit extension and the joint movement extension of the amoebot model with the goal of breaking this lower bound. In the joint movement extension, the way amoebots move is altered. Amoebots become able to push and pull other amoebots. Feldmann et al. demonstrated the power of joint movements by transforming a line of amoebots into a rhombus within $O(\log n)$ rounds. However, they left the details of the extension open. The goal of this paper is therefore to formalize and extend the joint movement extension. In order to provide a proof of concept for the extension, we consider two fundamental problems of modular robot systems: shape formation and locomotion. We approach these problems by defining meta-modules of rhombical and hexagonal shape, respectively. The meta-modules are capable of movement primitives like sliding, rotating, and tunneling. This allows us to simulate shape formation algorithms of various modular robot systems. Finally, we construct three amoebot structures capable of locomotion by rolling, crawling, and walking, respectively.

翻译：我们考虑几何阿米巴模型，其中一组$n$个阿米巴体放置在三角形网格上。阿米巴体能够向邻居发送信息，并通过膨胀和收缩进行移动。由于阿米巴体和信息只能逐节点传播，大多数问题的自然下界为$\Omega(D)$，其中$D$表示结构的直径。受神经系统和肌肉系统的启发，Feldmann等人提出了阿米巴模型的可重构电路扩展和关节运动扩展，旨在突破这一下界。在关节运动扩展中，阿米巴体的移动方式被改变，它们能够推拉其他阿米巴体。Feldmann等人通过在对数时间$O(\log n)$轮内将一条阿米巴线转化为菱形，展示了关节运动的能力。然而，他们未明确给出扩展的细节。因此，本文的目标是形式化并扩展关节运动扩展。为提供该扩展的概念验证，我们考虑模块化机器人系统的两个基本问题：形状形成和移动。我们分别通过定义菱形和六边形元模块来解决这些问题。这些元模块能够实现滑动、旋转和隧道穿越等基本移动原语，从而使得我们能够模拟多种模块化机器人系统的形状形成算法。最后，我们构建了三种分别通过滚动、爬行和步行实现移动的阿米巴结构。