We consider a swarm of $n$ autonomous mobile robots, distributed on a 2-dimensional grid. A basic task for such a swarm is the gathering process: All robots have to gather at one (not predefined) place. A common local model for extremely simple robots is the following: The robots do not have a common compass, only have a constant viewing radius, are autonomous and indistinguishable, can move at most a constant distance in each step, cannot communicate, are oblivious and do not have flags or states. The only gathering algorithm under this robot model, with known runtime bounds, needs $\mathcal{O}(n^2)$ rounds and works in the Euclidean plane. The underlying time model for the algorithm is the fully synchronous $\mathcal{FSYNC}$ model. On the other side, in the case of the 2-dimensional grid, the only known gathering algorithms for the same time and a similar local model additionally require a constant memory, states and "flags" to communicate these states to neighbors in viewing range. They gather in time $\mathcal{O}(n)$.
In this paper we contribute the (to the best of our knowledge) first gathering algorithm on the grid that works under the same simple local model as the above mentioned Euclidean plane strategy, i.e., without memory (oblivious), "flags" and states. We prove its correctness and an $\mathcal{O}(n^2)$ time bound in the fully synchronous $\mathcal{FSYNC}$ time model. This time bound matches the time bound of the best known algorithm for the Euclidean plane mentioned above. We say gathering is done if all robots are located within a $2\times 2$ square, because in $\mathcal{FSYNC}$ such configurations cannot be solved.
Conservatoire national des arts et métiers（法国国立工艺学院）；Collège de France（法兰西公学院）；Sorbonne Universités, UPMC Univ Paris 06（索邦大学，巴黎第六大学（UPMC））；Institut Universitaire de France（法国大学研究院）；École Nat. Sup. d’Informatique pour l’Industrie et l’Entreprise (ENSIIE)（法国国立工业与企业信息学高等学院（ENSIIE））；Université Paris-Sud（巴黎第十一大学）
We present a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite number of steps the same position, not known beforehand) without relying on a common chirality. We give very strong guaranties on the correctness of our algorithm by proving formally that it is correct, using the COQ proof assistant. To our knowledge, this is the first certified positive (and constructive) result in the context of oblivious mobile robots. It demonstrates both th…
Cédric – Conservatoire national des arts et métiers（法国国立工艺学院）；Collège de France（法兰西公学院）；UPMC Sorbonne Universités（巴黎第六大学索邦大学）；LIP6-CNRS 7606（法国国家科学研究中心LIP6实验室（7606号混合研究单位））；Institut Universitaire de France（法国大学研究院）；École Nat. Sup. d’Informatique pour l’Industrie et l’Entreprise (ENSIIE)（法国国立工业企业高等计算机学院（ENSIIE））；LRI, CNRS UMR 8623（法国国家科学研究中心LRI实验室（8623号混合研究单位））；Université Paris-Sud（巴黎南大学）；Université Paris-Saclay（巴黎萨克雷大学）
We present a unified formal framework for expressing mobile robots models, protocols, and proofs, and devise a protocol design/proof methodology dedicated to mobile robots that takes advantage of this formal framework. As a case study, we present the first formally certified protocol for oblivious mobile robots evolving in a two-dimensional Euclidean space. In more details, we provide a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite numb…
Nagoya Institute of Technology（名古屋工业大学）；Japan Advanced Institute of Science and Technology（日本北陆先端科学技术大学院大学）；Kyushu University（九州大学）
Anonymous mobile robots are often classified into synchronous, semi-synchronous and asynchronous robots when discussing the pattern formation problem. For semi-synchronous robots, all patterns formable with memory are also formable without memory, with the single exception of forming a point (i.e., the gathering) by two robots. However, the gathering problem for two semi-synchronous robots without memory is trivially solvable when their local coordinate systems are consistent, and the impossibility proof essentially uses the inconsistencies in their coordinate systems. Motivated by this, this …
Aix-Marseille University（艾克斯-马赛大学）；University of Ottawa（渥太华大学）；Carleton University（卡尔顿大学）；JAIST（北陆先端科学技术大学院大学）；Kyushu University（九州大学）
The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $σ$ (where $σ=1$ corresponds to no rotational…
Gathering is a fundamental coordination problem in cooperative mobile robotics. In short, given a set of robots with arbitrary initial locations and no initial agreement on a global coordinate system, gathering requires that all robots, following their algorithm, reach the exact same but not predetermined location. Gathering is particularly challenging in networks where robots are oblivious (i.e., stateless) and direct communication is replaced by observations on their respective locations. Interestingly any algorithm that solves gathering with oblivious robots is inherently self-stabilizing i…
Nagoya Institute of Technology（名古屋工业大学）；Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；CNRS（法国国家科学研究中心）
This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show s…
University Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；LIP6-CNRS 7606（法国国家科学研究中心LIP6实验室（7606））；BGU（本-古里安大学）；Technion-Israel Institute of Technology（以色列理工学院）
The problem of gathering multiple mobile robots to a single location, is one of the fundamental problems in distributed coordination between autonomous robots. The problem has been studied and solved even for robots that are anonymous, disoriented, memoryless and operate in the semi-synchronous (ATOM) model. However all known solutions require the robots to be faulty-free except for the results of [Agmon and Peleg 2006] who solve the gathering problem in presence of one crash fault. This leaves open the question of whether gathering of correct robots can be achieved in the presence of multiple…
Institut für Algebra, TU Dresden（德累斯顿工业大学代数研究所）；University of Ottawa（渥太华大学）
A fundamental problem in Distributed Computing is the Pattern Formation problem, where some independent mobile entities, called robots, have to rearrange themselves in such a way as to form a given figure from every possible (non-degenerate) initial configuration. In the present paper, we consider robots that operate in the Euclidean plane and are dimensionless, anonymous, oblivious, silent, asynchronous, disoriented, non-chiral, and non-rigid. For this very elementary type of robots, the feasibility of the Pattern Formation problem has been settled, either in the positive or in the negative, …
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE（法国国家信息与自动化研究所）；VERIMAG UMR 5104（VERIMAG实验室（法国国家科研中心联合研究单位5104））；Université Joseph Fourier（约瑟夫·傅里叶大学）；INRIA（法国国家信息与自动化研究所）；LIP UMR 5668（LIP实验室（法国国家科研中心联合研究单位5668））；Université de Lyon（里昂大学）；ENS Lyon（里昂高等师范学院）；Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；LIP6（LIP6实验室）
We consider a team of $k$ identical, oblivious, asynchronous mobile robots that are able to sense (\emph{i.e.}, view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by \emph{deterministic} robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the \emph{exploration} problem of anonymous unoriented rings of any size. It is known that $Θ(\log n)$ robots are necessary and sufficient t…
Aix-Marseille University（艾克斯-马赛大学）；University of Ottawa（渥太华大学）；Carleton University（卡尔顿大学）；JAIST（北陆先端科学技术大学院大学）
In this paper we investigate the computational power of a set of mobile robots with limited visibility. At each iteration, a robot takes a snapshot of its surroundings, uses the snapshot to compute a destination point, and it moves toward its destination. Each robot is punctiform and memoryless, it operates in $\mathbb{R}^m$, it has a local reference system independent of the other robots' ones, and is activated asynchronously by an adversarial scheduler. Moreover, robots are non-rigid, in that they may be stopped by the scheduler at each move before reaching their destination (but are guarant…
Università di Pisa（比萨大学）；University of Ottawa（渥太华大学）
In this paper we study the Near-Gathering problem for a finite set of dimensionless, deterministic, asynchronous, anonymous, oblivious and autonomous mobile robots with limited visibility moving in the Euclidean plane in Look-Compute-Move (LCM) cycles. In this problem, the robots have to get close enough to each other, so that every robot can see all the others, without touching (i.e., colliding with) any other robot. The importance of solving the Near-Gathering problem is that it makes it possible to overcome the restriction of having robots with limited visibility. Hence it allows to exploit…
In this paper we address the complexity issues of two agreement problems in oblivious robot networks namely gathering and scattering. These abstractions are fundamental coordination problems in cooperative mobile robotics. Moreover, their oblivious characteristics makes them appealing for self-stabilization since they are self-stabilizing with no extra-cost. Given a set of robots with arbitrary initial location and no initial agreement on a global coordinate system, gathering requires that all robots reach the exact same but not predetermined location while scattering aims at scatter robots su…
IRIF - Univ Paris Diderot（巴黎狄德罗大学 IRIF）；Sorbonne Universités, UPMC Univ Paris 06（索邦大学 皮埃尔和玛丽·居里大学）
We study verification problems for autonomous swarms of mobile robots that self-organize and cooperate to solve global objectives. In particular, we focus in this paper on the model proposed by Suzuki and Yamashita of anonymous robots evolving in a discrete space with a finite number of locations (here, a ring). A large number of algorithms have been proposed working for rings whose size is not a priori fixed and can be hence considered as a parameter. Handmade correctness proofs of these algorithms have been shown to be error-prone, and recent attention had been given to the application of fo…
\textsc{Arbitrary Pattern Formation} is a fundamental problem in autonomous mobile robot systems. The problem asks to design a distributed algorithm that moves a team of autonomous, anonymous and identical mobile robots to form any arbitrary pattern $F$ given as input. In this paper, we study the problem for robots whose movements can be inaccurate. Our movement model assumes errors in both direction and extent of the intended movement. Forming the given pattern exactly is not possible in this setting. So we require that the robots must form a configuration which is close to the given pattern …
Consider a system of autonomous mobile robots initially randomly deployed on the nodes of an anonymous finite grid. A gathering algorithm is a sequence of moves to be executed independently by each robot so that all robots meet at a single node after finite time. The robots operate in Look-Compute-Move cycles. In each cycle, a robot takes a snapshot of the current configuration of the grid in terms of occupied nodes (\emph{Look}), then based on the perceived configuration, decides whether to stay put or to move to an adjacent node (\emph{Compute}), and in the later case makes an instantaneous …
A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm? We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., a…
We propose a new probabilistic pattern formation algorithm for oblivious mobile robots that operates inthe ASYNC model. Unlike previous work, our algorithm makes no assumptions about the local coordinatesystems of robots (the robots do not share a common "North" nor a common "Right"), yet it preserves theability from any initial configuration that contains at least 5 robots to form any general pattern (and not justpatterns that satisfy symmetricity predicates). Our proposal also gets rid of the previous assumption (in thesame model) that robots do not pause while moving (so, our robots really …
We propose an information transmission scheme by a swarm of anonymous oblivious mobile robots on a graph. The swarm of robots travel from a sender vertex to a receiver vertex to transmit a symbol generated at the sender. The codeword for a symbol is a pair of an initial configuration at the sender and a set of terminal configurations at the receiver. The set of such codewords forms a code. We analyze the performance of the proposed scheme in terms of its code size and transmission delay. We first demonstrate that a lower bound of the transmission delay depends on the size of the swarm, and the…
A team of anonymous mobile agents represented by points freely moving in the plane have to gather at a single point and stop. Agents start at different points of the plane and at possibly different times chosen by the adversary. They are equipped with compasses, a common unit of distance and clocks. They execute the same deterministic algorithm and travel at speed 1. When agents are at distance at most $ε$, for some positive constant $ε$ unknown to them, they can exchange all information. Due to the anonymity of the agents and the symmetry of the plane, gathering is impossible, e.g., if agents…
In this paper, we propose a distributed algorithm to uniformly scatter the robots along a grid, with robots on alternate nodes of this grid distribution. These homogeneous, autonomous mobile robots place themselves equidistant apart on the grid, which can be required for guarding or covering a geographical area by the robots. The robots operate by executing cycles of the states "look-compute-move". In the look phase, it looks to see the position of the other robots; in the compute phase, it computes a destination to move to; and then in the move phase, it moves to that computed destination. Th…
This paper revisits the widely researched \textit{gathering} problem for two robots in a scenario which allows randomization in the asynchronous scheduling model. The scheduler is considered to be the adversary which determines the activation schedule of the robots. The adversary comes in two flavors, namely, oblivious and adaptive, based on the knowledge of the outcome of random bits. The robots follow \textit{wait-look-compute-move} cycle. In this paper, we classify the problems based on the capability of the adversary to control the parameters such as wait time, computation delay and the sp…
We propose a gathering protocol for an even number of robots in a ring-shaped network that allows symmetric but not periodic configurations as initial configurations, yet uses only local weak multiplicity detection. Robots are assumed to be anonymous and oblivious, and the execution model is the non- atomic CORDA model with asynchronous fair scheduling. In our scheme, the number of robots k must be greater than 8, the number of nodes n on a network must be odd and greater than k+3. The running time of our protocol is O(n2) asynchronous rounds.
The exploration problem in the discrete universe, using identical oblivious asynchronous robots without direct communication, has been well investigated. These robots have sensors that allow them to see their environment and move accordingly. However, the previous work on this problem assume that robots have an unlimited visibility, that is, they can see the position of all the other robots. In this paper, we consider deterministic exploration in an anonymous, unoriented ring using asynchronous, oblivious, and myopic robots. By myopic, we mean that the robots have only a limited visibility. We…
Min-Sum Uniform Coverage Problem by Autonomous Mobile Robots
Department of Mathematics（数学系）；Birla Institute of Technology, Mesra（比拉理工学院，梅斯拉）；Advanced Computing and Microelectronics Unit, Indian Statistical Institute（印度统计研究所高级计算与微电子单位）
In the gathering problem, n autonomous robots have to meet on a single point. We consider the gathering of a closed chain of point-shaped, anonymous robots on a grid. The robots only have local knowledge about a constant number of neighboring robots along the chain in both directions. Actions are performed in the fully synchronous time model FSYNC. Every robot has a limited memory that may contain one timestamp of the global clock, also visible to its direct neighbors. In this synchronous time model, there is no limited view gathering algorithm known to perform better than in quadratic runtime…
We consider a swarm of mobile robots evolving in a bidimensional Euclidean space. We study a variant of the crash-tolerant gathering problem: if no robot crashes, robots have to meet at the same arbitrary location, not known beforehand, in finite time; if one or several robots crash at the same location, the remaining correct robots gather at the crash location to rescue them. Motivated by impossibility results in the semi-synchronous setting, we present the first solution to the problem for the fully synchronous setting that operates in the vanilla Look-Compute-Move model with no additional h…
We consider the gathering problem for asynchronous and oblivious robots that cannot communicate explicitly with each other, but are endowed with visibility sensors that allow them to see the positions of the other robots. Most of the investigations on the gathering problem on the discrete universe are done on ring shaped networks due to the number of symmetric configuration. We extend in this paper the study of the gathering problem on torus shaped networks assuming robots endowed with local weak multiplicity detection. That is, robots cannot make the difference between nodes occupied by only …
Jadavpur University（贾达普大学）；University of North Florida（北佛罗里达大学）；Indian Statistical Institute（印度统计研究所）
In this paper, we study the circle formation problem by multiple autonomous and homogeneous disc-shaped robots (also known as fat robots). The goal of the robots is to place themselves on the periphery of a circle. Circle formation has many real-world applications, such as boundary surveillance. This paper addresses one variant of such problem { uniform circle formation, where the robots have to be equidistant apart. The robots operate by executing cycles of the states wait-look-compute-move. They are oblivious, indistinguishable, anonymous, and do not communicate via message passing. First, w…
The gathering over meeting nodes problem requires the robots to gather at one of the pre-defined meeting nodes. This paper investigates the problem with respect to the objective function that minimizes the total number of moves made by all the robots. In other words, the sum of the distances traveled by all the robots is minimized while accomplishing the gathering task. The robots are deployed on the nodes of an anonymous two-dimensional infinite grid which has a subset of nodes marked as meeting nodes. The robots do not agree on a global coordinate system and operate under an asynchronous sch…
The Arbitrary Pattern Formation problem asks to design a distributed algorithm that allows a set of autonomous mobile robots to form any specific but arbitrary geometric pattern given as input. The problem has been extensively studied in literature in continuous domains. This paper investigates a discrete version of the problem where the robots are operating on a two dimensional infinite grid. The robots are assumed to be autonomous, identical, anonymous and oblivious. They operate in Look-Compute-Move cycles under a fully asynchronous scheduler. The robots do not agree on any common global co…
In this paper, we consider the partial gathering problem of mobile agents in asynchronous unidirectional rings equipped with whiteboards on nodes. The partial gathering problem is a new generalization of the total gathering problem. The partial gathering problem requires, for a given integer $g$, that each agent should move to a node and terminate so that at least $g$ agents should meet at the same node. The requirement for the partial gathering problem is weaker than that for the (well-investigated) total gathering problem, and thus, we have interests in clarifying the difference on the move …
Université Pierre et Marie Curie - Paris 6（巴黎第六大学（皮埃尔和玛丽·居里大学））；Ben Gurion University of the Negev（内盖夫本·古里安大学）
This paper introduces the \emph{RoboCast} communication abstraction. The RoboCast allows a swarm of non oblivious, anonymous robots that are only endowed with visibility sensors and do not share a common coordinate system, to asynchronously exchange information. We propose a generic framework that covers a large class of asynchronous communication algorithms and show how our framework can be used to implement fundamental building blocks in robot networks such as gathering or stigmergy. In more details, we propose a RoboCast algorithm that allows robots to broadcast their local coordinate syste…
Consider a set of $n$ mobile entities, called robots, located and operating on a continuous circle, i.e., all robots are initially in distinct locations on a circle. The \textit{gathering} problem asks to design a distributed algorithm that allows the robots to assemble at a point on the circle. Robots are anonymous, identical, and homogeneous. Robots operate in a deterministic Look-Compute-Move cycle within the circular path. Robots agree on the clockwise direction. The robot's movement is rigid and they have limited visibility $π$, i.e., each robot can only see the points of the circle which…
We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering requires that all robots reach the same, not predefined position. In the related near-gathering task, they must reach distinct positions such that every robot sees the entire swarm. In the considered setting, gathering can be solved in $\mathcal{O}(n + Δ^2)$ synchronous rounds both in two and three dimensions, where $Δ$ denotes the initial maximal distance …
VERIMAG（VERIMAG实验室）；Université Joseph Fourier（约瑟夫·傅里叶大学）；MIS（MIS实验室）；Université de Picardie Jules Verne（皮卡第儒勒·凡尔纳大学）；LIP6（LIP6实验室）；UPMC Sorbonne Universités（巴黎第六大学索邦大学）
We consider a team of {\em autonomous weak robots} that are endowed with visibility sensors and motion actuators. Autonomous means that the team cannot rely on any kind of central coordination mechanism or scheduler. By weak we mean that the robots are devoid of (1) any (observable) IDs allowing to differentiate them (anonymous), (2) means of communication allowing them to communicate directly, and (3) any way to remember any previous observation nor computation performed in any previous step (oblivious). Robots asynchronously operate in cycles of three phases: Look, Compute, and Move. Further…
This work deals with the problem of gathering $n$ oblivious mobile entities, called robots, at a point (not known beforehand) placed on an infinite triangular grid. The robots are considered to be myopic, i.e., robots have limited visibility. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is by Cicerone et al. on arbitrary pattern formation where full visibility is considered. The other one by Shibata …
We consider a set of k autonomous robots that are endowed with visibility sensors (but that are otherwise unable to communicate) and motion actuators. Those robots must collaborate to reach a sin- gle vertex that is unknown beforehand, and to remain there hereafter. Previous works on gathering in ring-shaped networks suggest that there exists a tradeoff between the size of the set of potential initial configurations, and the power of the sensing capabilities of the robots (i.e. the larger the initial configuration set, the most powerful the sensor needs to be). We prove that there is no such t…
This paper proposes a distributed algorithm which deterministically gathers n (n > 4) asynchronous, fat robots. The robots are assumed to be transparent and they have full visibility. The robots are initially considered to be stationary. A robot is visible in its motion. The robots do not store past actions. They are anonymous and can not be distinguished by their appearances and do not have common coordinate system or chirality. The robots do not communicate through message passing. In the proposed gathering algorithm one robot moves at a time towards its destination. The robot which moves, i…
JAIST（北陆先端科学技术大学院大学）；Kyushu University（九州大学）；Aix-Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；University of Rome “Sapienza”（罗马第一大学）；University of Ottawa（渥太华大学）；Carleton University（卡尔顿大学）
We consider a distributed system of n identical mobile robots operating in the two dimensional Euclidian plane. As in the previous studies, we consider the robots to be anonymous, oblivious, dis-oriented, and without any communication capabilities, operating based on the Look-Compute-Move model where the next location of a robot depends only on its view of the current configuration. Even in this seemingly weak model, most formation problems which require constructing specific configurations, can be solved quite easily when the robots are fully synchronized with each other. In this paper we int…
This work focuses on the following question related to the Gathering problem of $n$ autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of robots that do not agree on any axis of their coordinate systems (disoriented robots) and see other robots only up to a constant distance (limited visibility) in $o(n^2)$ fully synchronous rounds? The best known algorithm that solves Gathering of disoriented robots with limited visibility assuming oblivious robots needs $Θ(n^2)$ rounds [SPAA'11]. The lower bound for this algorithm even holds in a simplified closed chain model…
An oblivious mobile robot is a stateless computational entity located in a spatial universe, capable of moving in that universe. When activated, the robot observes the universe and the location of the other robots, chooses a destination, and moves there. The computation of the destination is made by executing an algorithm, the same for all robots, whose sole input is the current observation. No memory of all these actions is retained after the move. When the universe is a graph, distributed computations by oblivious mobile robots have been intensively studied focusing on the conditions for fea…
We consider n robots with limited visibility: each robot can observe other robots only up to a constant distance denoted as the viewing range. The robots operate in discrete rounds that are either fully synchronous (FSync) or semi-synchronized (SSync). Most previously studied formation problems in this setting seek to bring the robots closer together (e.g., Gathering or Chain-Formation). In this work, we introduce the Max-Line-Formation problem, which has a contrary goal: to arrange the robots on a straight line of maximal length. First, we prove that the problem is impossible to solve by robo…
