Shape formation by programmable particles

Abstract

Shape formation (or pattern formation) is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter, where entities are assumed to be small and with severely limited capabilities. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins to elect a leader).In this paper we obtain several results that, among other things, provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. The characterization is constructive: we provide a universal shape formation algorithm that, for each feasible pair of shapes (S 0 , S F ), allows the particles to form the final shape S F (given in input) starting from the initial shape S 0 , unknown to the particles. The final configuration will be an appropriate scaled-up copy of S F depending on n.If randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that there are enough particles.Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential.We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of O(n 2 ) rounds and moves: this number of moves is also asymptotically optimal.

Provable self-organizing pattern formation by a swarm of robots with limited knowledge

“…Moreover, when they are all functionally homogeneous, no robot can be assigned as the seed. Without communication, they cannot elect one themselves either, as otherwise explored by Yamauchi and Yamashita (2014), Derakhshandeh et al (2016), andDi Luna et al (2017), where a swarm could self-elect a leader/seed robot.…”

A survey of autonomous self-reconfiguration methods for robot-based programmable matter

“…Since then, another approach to a general SOPS shape formation framework with an equilateral triangle approximation has been proposed by Di Luna et al (2018b), that relaxes some of the assumptions made in the previous algorithm by Derakhshandeh et al, albeit with an O(n 2 ) moves and rounds complexity. An interesting finding is that predetermining the orientation of movement is not a necessary condition.…”

Building a Nest by an Automaton

“…We also point out, that feasibility questions different from ours arise in the programmable matter models, including those of a possibility of formations of certain patterns [ 36 ]. In [ 20 ], shape formation algorithms have been proposed in a distributed multi-particle model, where the complexity measured as the number of rounds is . The latter is due to intermediate line formations but also to leader election protocol used by the algorithm.…”