Suzuki, I., and Yamashita, M. 1999. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4):1347--1363.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....should not rely on an ordering of the robots, especially, there is no leader among the robots. The question is the following: Which distributed algorithm can the robots use to complete the task Besides practical studies, the point formation problem has been investigated in a more theoretical way [SY93, SS96, SY99, Pre01, FPSW99, FPSW00]. The solutions presented in these publications are complex, and some of them are hard to understand or verify. We present a new approach whose most distinguishing feature is its simplicity. A rough outline of our solution is as follows: Every robot takes a snapshot and detects the multi set of ....

.... if p = p n if p = p 1 we obtain that C(X) C(X # ) c.Letc # : p n p n 1 ) rotation by 180 around c # transforms X # into itself. Therefore, c = c # . But since c # is not invariant under movement towards it, X # has no contraction point. This Remark: In contrast to [SY99], contraction functions do not solve the problem for an even number of collinear points without median. But the problem can be solved with a little extension and one assumption from [SY99] Initially, the positions of the robots are di#erent. If the number of robots is bigger than 2, then the ....

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I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing, 28(4):1347--1363, 1999.

.... is to identify the factors which in uence solvability of a given problem (the task) These questions have been extensively studied both experimentally and theoretically in the unlimited visibility setting, that is assuming that the robots are capable to sense ( see ) the entire space (e.g. see [4, 6, 10, 12]) In general and more realistically, robots can sense only a surrounding with a radius of bounded size. This setting, called the limited visibility case, is understandably more dicult, and only few algorithmic results are known [1, 11] In this paper we are interested in gathering: the basic ....

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. SIAM J. Comput., 28(4):1347-1363, 1999.

....of the study. An investigation with an algorithmic flavor has been undertaken within the AI community by Durfee [13] who argues in favor of limiting the knowledge that an intelligent robot must possess in order to be able to coordinate its behavior with others. The work of Suzuki and Yamashita [1, 26, 27] is the closest to our study (and, with this focus, a rarity in the mobile robots literature) it gives a nice and systematic account on the algorithmics of pattern formation for robots, under several assumptions on the power of the individual robot. Our model, however, differs with respect to the ....

....is that they do not deal with formal correctness and they are only analyzed empirically. Algorithmic aspects were somehow implicitly an issue, but clearly not a major concern. We aim to identify the algorithmic limitations of what autonomous, mobile robots can do. The work of Suzuki and Yamashita [27] is the closest to our study. The model that we use differ from those of [27] in that our agents are as weak as possible in every single aspect of their behavior. The reason is that we want to identify the role of the robots common knowledge of the world for performing a task. Other significant ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347--1363, 1999.

....dynamical systems for optimization purposes, and to [4] for gradient descent flows in distributed computation in settings with fixed communication topologies. Recent years have witnessed a large research e#ort focused on motion planning and formation control problems for multi vehicle systems [12, 18, 19, 20, 24, 30, 31]. Within the literature on behavior based robotics, heuristic approaches to the design of interaction rules and emerging behaviors have been investigated (see [2] and references therein) Along this specific line of research, no formal results guaranteeing the correctness of the proposed ....

....correctness. A key aspect of our treatment is the inherent complexity of studying networks whose communication topology changes along the system evolution, as opposed to networks with fixed communication topologies. This key aspect is present in the analysis of distributed control laws in [18, 30, 31] and of agreement protocols in [24] Statement of contributions. We consider two facility location functions from geometric optimization that characterize coverage performance criteria. A collection of sites provides optimal service to a domain of interest if (i) it minimizes the largest distance ....

I. Suzuki and M. Yamashita, Distributed anonymous mobile robots: Formation of geometric patterns, SIAM Journal on Computing, 28 (1999), pp. 1347--1363.

....the leader in di#erent positions) that the vehicles share the same coordinate system; that the vehicles have observable identities; that they know beforehand their destination point; and . that they act synchronously. Following the motivations that prompted previous studies ([14 17]) we adopt simple units to study the problem: the units are completely anonymous, identical (no identities are used during the computation) asynchronous, memoryless, and with no means of direct communication. We describe an algorithm (the same for all the vehicles) that allows the followers to ....

I. Suzuki, M. Yamashita, Distributed Anonymous Mobile Robots: Formation of Geometric Patterns, Siam Journal of Computing 28 (4) (1999) 1347--1363. 32

....robots can or cannot do. An investigation with this flavor has been undertaken within the AI community by Durfee [2] who argues in favor of limiting the knowledge that an intelligent agent must possess in order to be able to coordinate its behavior with others. The work of Suzuki and Yamashita [7, 8, 9] is closest to our study (and, with this focus, a rarity in the mobile robots literature) it gives a nice and systematic account on the algorithmics of pattern formation for robots, under several assumptions on the power of the indivdual robot. The models that we use differ from those of [7, 8, ....

....[7, 8, 9] is closest to our study (and, with this focus, a rarity in the mobile robots literature) it gives a nice and systematic account on the algorithmics of pattern formation for robots, under several assumptions on the power of the indivdual robot. The models that we use differ from those of [7, 8, 9] in the fact that our robots are as weak as possible in every single aspect of their behavior. The reason is that we want to identify the role of the robots common knowledge of the world for performing a task. In contrast with [7, 8, 9] we do not assume that on a move, we know ahead of time the ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347--1363, 1999.

....the capabilities of the robots employed, and the robots knowledge of the environment are given. Recently, concerns on computability and complexity of the coordination problem have motivated algorithmic investigations, and the problem has also been approached from a computational point of view [1, 4, 5, 8, 9, 10]. In [1] and [10] any action of the robots, including moving, is instantaneous, while in [4, 5, 8, 9] as well as in this paper, there is no such assumption. We consider a very weak model of robots: The robots are anonymous, have no common knowledge, no central coordination, and no means of ....

....employed, and the robots knowledge of the environment are given. Recently, concerns on computability and complexity of the coordination problem have motivated algorithmic investigations, and the problem has also been approached from a computational point of view [1, 4, 5, 8, 9, 10] In [1] and [10], any action of the robots, including moving, is instantaneous, while in [4, 5, 8, 9] as well as in this paper, there is no such assumption. We consider a very weak model of robots: The robots are anonymous, have no common knowledge, no central coordination, and no means of direct communication. ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam Journal of Computing, 28(4):1347-- 1363, 1999.

....observing the leader in di erent positions) that the vehicles share the same coordinate system; that the vehicles have observable identities; that they know beforehand their destination point; and that they act synchronously. Following the motivations that prompted previous studies ([14 17]) we adopt simple units to study the problem: the units are completely anonymous, identical (no identities are used during the computation) asynchronous, memoryless, and with no means of direct communication. We describe an algorithm (the same for all the vehicles) that allows the followers to ....

I. Suzuki, M. Yamashita, Distributed Anonymous Mobile Robots: Formation of Geometric Patterns, Siam Journal of Computing 28 (4) (1999) 1347-1363. 32

....studies have focused on trying to formally model an environment constituted by mobile units, studying which kind of capabilities they must have in order to achieve their goals; in other words, to study the problem from a computational point of view. This paper focuses on two of these studies [1, 6, 14] (the only ones, to our knowledge, that analyze the problem of coordinating and controlling a set of autonomous, mobile units from this point of view) First, their main features are described. Then, the main di erences are highlighted, showing the relationship between the class of problems ....

....the problem from a computational point of view. This paper deals with two studies leading in this direction (the only ones, to our knowledge, that analyze the problem of coordinating and controlling a set of autonomous, mobile units from this point of view) The rst study is by Suzuki et al. [1, 13, 14]. It gives a nice and systematic account on the algorithmics of pattern formation for robots, operating under several assumptions on the power of the individual robot. The second is by Flocchini et al. 6, 8] they present a model (that we will refer to as Corda Coordination and control of a set ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347-1363, 1999.

....is analyzed: can the robots form an arbitrary geometric pattern if they have a compass Can they gather in a point Which information each robot must have about its fellows in order for them to collectively achieve their goal The first work proposed under this light is that of I. Suzuki et al. [1, 20, 21] (and, with this focus, a rarity in the mobile robots literature) It approaches the algorithmic issues related to the pattern formation for robots, under several assumptions on the power of the individual robot. The only other study, to our knowledge, that approaches the problem of coordinating ....

....A brief discussion of open issues in the field, together with some conclusion, complete the paper. 2 Computational models In this section we present two approaches to model the control and coordination of a set of autonomous mobile robots. The first one, the SYm model, was originally proposed in [21]; the second one, our CORDA model, was introduced in [9] As already stated, these are the only two studies, to our knowledge, that tackle the problem of controlling an environment populated by a set of totally autonomous mobile robots from a computational viewpoint. 2.1 Common Features The two ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347--1363, 1999.

....the capabilities of the robots employed, and the robots knowledge of the environment are given. Recently, concerns on computability and complexity of the coordination problem have motivated algorithmic investigations, and the problem has also been approached from a computational point of view [1, 5, 6, 10, 11, 13]. In [1] and [13] any action of the robots, including moving, is instantaneous, while in [5, 6, 10, 11] as well as in this paper, there is no such assumption. An extended abstract of this paper has been published in [3] We consider a very weak model of robots: the robots are anonymous, have no ....

....employed, and the robots knowledge of the environment are given. Recently, concerns on computability and complexity of the coordination problem have motivated algorithmic investigations, and the problem has also been approached from a computational point of view [1, 5, 6, 10, 11, 13] In [1] and [13], any action of the robots, including moving, is instantaneous, while in [5, 6, 10, 11] as well as in this paper, there is no such assumption. An extended abstract of this paper has been published in [3] We consider a very weak model of robots: the robots are anonymous, have no common ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam Journal of Computing, 28(4):1347--1363, 1999. 31

....is analyzed: can the robots form an arbitrary geometric pattern if they have a compass Can they gather in a point Which information each robot must have about its fellows in order for them to collectively achieve their goal The first work proposed under this light is that of I. Suzuki et al. [1, 20, 21] (and, with this focus, a rarity in the mobile robots literature) It approaches the algorithmic issues related to the pattern formation for robots, under several assumptions on the power of the individual robot. The only other study, to our knowledge, that approaches the problem of coordinating ....

....A brief discussion of open issues in the field, together with some conclusion, complete the paper. 2 Computational models In this section we present two approaches to model the control and coordination of a set of autonomous mobile robots. The first one, the SYm model, was originally proposed in [21]; the second one, our CORDA model, was introduced in [9] As already stated, these are the only two studies, to our knowledge, that tackle the problem of controlling an environment populated by a set of totally autonomous mobile robots from a computational viewpoint. 2.1 Common Features The two ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347--1363, 1999.

....various geometrical patterns. They propose an algorithm for the formation of an approximation of a circle, based on heuristics. In some cases, the shape obtained with their algorithm is a Reuleaux triangle (a hybrid shape, between a triangle and a circle) rather than a circle. Suzuki and Yamashita [15] propose a non oblivious algorithm for the formation of a regular polygon. In other words, the robots eventually reach a configuration in which they are arranged at regular intervals on the boundary of a circle. To achieve this, they must however require the robots to be able to remember all past ....

....if two robots with the same coordinate system happen to have the same initial position. This is because it would be impossible to separate them in a deterministic manner. 2. SYSTEM MODEL AND DEFINITIONS 2. 1 System Model The system model considered in this paper, defined by Suzuki and Yamashita [15], considers a collection of anonymous mobile robots evolving asynchronously on the Euclidean plane, with no common origin, unit distance, or sense of direction. More precisely, the model is defined as follows. Each robot r i is modeled as a mobile processor with infinite memory, and a sensor to ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal of Computing, 28(4):1347--1363, 1999.

....they must have in order to achieve their goals; in other words, to study the problem from a computational point of view. This paper focuses on one of these studies, based on a model rst introduced in [6] First, its main features are described. Then, the main di erences from a previous model [1, 14] (the only one, to our knowledge, that analyzes the problem of coordinating and controlling a set of autonomous, mobile units from this point of view) are highlighted, showing the novelty of this approach. 1 Introduction In a system consisting of a set of totally distributed agents the goal ....

....motivations) An investigation with an algorithmic avor has been undertaken within the AI community by Durfee [5] who argues in favor of limiting the knowledge that an intelligent robot must possess in order to be able to coordinate its behavior with others. The work of Suzuki and Yamashita [1, 13, 14], however, is the closest to our study (and, with this focus, a rarity in the mobile robots literature) It gives a nice and systematic account on the algorithmics of pattern formation for robots, operating under several assumptions on the power of the individual robot. Although the model of ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347-1363, 1999. 19

....its behavior with others. However, the main studies, to our knowledge, that aim to better understand the power and the limitations of the distributed coordination and control of a set of autonomous, mobile robots from an algorithmic and computational point of view, are by Suzuki and Yamashita [1, 13, 14], and by Flocchini et al. 6, 7] In order to analyze the problem in this light, they present formal models focused on the understanding of the algorithmics of several problems, in particular pattern formation, under several assumptions on the power of the individual robot. The main objective of ....

....robot. The main objective of these studies is to highlight 2 which kind of capabilities features (available amount of memory, need or kind of common knowledge, sensors power) the units must have in order to eciently coordinate and reach some common goal they are required to achieve. In both [1, 14] and [6, 7] are de ned models to formally describe the mobile units and the environment in which they operate. There are, however, substantial di erences between the two models, which are analyzed in depth in [12] In this paper we will study the pattern formation problem under the model of [6, ....

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347-1363, 1999. 14

....the impact of the knowledge of the environment: can the robots form an arbitrary geometric pattern if they share a compass Can they gather in a point Which information each robot must have about its fellows in order for them to collectively achieve their goal The work of I. Suzuki et al. [1, 19] is the closest to our study (and, with this focus, a rarity in the mobile robots literature) It approaches the algorithmic issues related to the pattern formation for robots, under several assumptions on the power of the individual robot. Our model, however, di ers with respect to the ....

....The main motivations that prompted us to study the problem in this new perspective can be found in [10] 3 Main Results As already mentioned, the work of I. Suzuki et al. is the closest to ours. There are, however, some aspects that render our approach and theirs quite di erent. In particular, in [1, 19] instantaneous action of the robots is modeled: every robots execute their cycle atomically. One consequence of this approach, is that it is not possible to model di erent motorial and computational speed of the robots. Moreover, a robot can not be seen while it is moving by robots that are ....

[Article contains additional citation context not shown here]

I. Suzuki and M. Yamashita. Distributed Anonymous Mobile Robots: Formation of Geometric Patterns. Siam J. Comput., 28(4):1347-1363, 1999.

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Suzuki, I., and Yamashita, M. 1999. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4):1347--1363.

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I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal of Computation, 28(4):1347--1363, 1999. 47

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I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal of Computation, 28(4):1347--1363, 1999.

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I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing, 28(4):1347-1363, 1999.

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I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing, 28(4):1347-1363, 1999.

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