Reconfiguration and Locomotion with Joint Movements in the Amoebot Model

Abstract

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 $Ω(D)$ 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: reconfiguration 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 reconfiguration algorithms of various modular robot systems. Finally, we construct three amoebot structures capable of locomotion by rolling, crawling, and walking, respectively.

Figures (35)

• Figure 1: Amoebot structure. The amoebots are shown in black. Red edges indicate bonds.
• Figure 2: Movements in the classical model. Red edges indicate bonds. Blue amoebots are expanding. Green amoebots are contracting.
• Figure 3: Movements in the extension. Red edges indicate bonds. Blue amoebots are expanding. Green amoebots are contracting. The figures show the movements in $0.5$ time steps.
• Figure 4: Structural conflict. We are not able to expand the blue amoebots horizontally without tearing up the amoebot structure. Hence, the expansions cause a structural conflict and the amoebot structure transitions into an undefined state.
• Figure 5: Collision. Initially, we have a valid amoebot structure given ($t = 0$). We expand the blue amoebots horizontally. Note that the amoebot structure is also valid at $t = 1$. However, for $t \in [0.25, 0.75]$, parts of the amoebot structure collide. Hence, the amoebot structure transitions into an undefined state.

Theorems & Definitions (31)

• Remark 1
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10
• Lemma 11