## Coordination of groups of mobile autonomous agents using nearest neighbor rules (2003)

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Venue: | IEEE Transactions on Automatic Control |

Citations: | 704 - 47 self |

### BibTeX

@ARTICLE{Jadbabaie03coordinationof,

author = {Ali Jadbabaie and Jie Lin and A. Stephen Morse},

title = {Coordination of groups of mobile autonomous agents using nearest neighbor rules},

journal = {IEEE Transactions on Automatic Control},

year = {2003},

volume = {48},

pages = {988--1001}

}

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### Abstract

et al. propose a simple but compelling discrete-time model of autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent’s heading is updated using a local rule based on the average of its own heading plus the headings of its “neighbors. ” In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent’s set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. Index Terms—Cooperative control, graph theory, infinite products, multiagent systems, switched systems. I.

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Citation Context ...tive matrix is meant a matrix whose entries are all non-negative. Each Fp also has the property that its row sums all equal 1 {i.e., Fp1 = 1}. Matrices with these two properties are called stochastic =-=[28]. -=-The Fp have the additional property that their diagonal elements are all non-zero. For the case when p ∈ Q {i.e., when Gp is connected}, it is known that (I + Ap) m becomes a matrix with all positiv... |

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Citation Context ...gents which are either in or on a circle of pre-specified radius centered at agent ’s current position. The Vicsek model turns out to be a special version of a model introduced previously by Reynolds =-=[2]-=- for simulating visually satisfying flocking and schooling behaviors for the animation industry. In their paper, Vicsek et al. provide a variety of interesting simulation results which demonstrate tha... |

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Citation Context ...ors over time. Changing nearest neighbor sets is an inherent property of the Vicsek model and in the other models we consider. To analyze such models, it proves useful to appeal to well-known results =-=[19, 20]-=- characterizing the convergence of infinite products of certain types of non-negative matrices. The study of infinite matrix products is ongoing [21, 22, 23, 24, 25, 26] and is undoubtedly producing r... |

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Citation Context ... increased, i.e., there is a critical density after which all agents eventually become aligned. It is likely that this phenomenon can be adequately explained using percolation theory of random graphs =-=[35]-=-. The results of this paper have been extended to the case where there are inter-agent forces due to attraction, repulsion and alignment [36]. The new result indicate that the convergence arguments us... |

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Citation Context ...tical explanation for this observed behavior. There is a large and growing literature concerned with the coordination of groups of mobile autonomous agents. Included here is the work of Czirok et al. =-=[3] who p-=-ropose one-dimensional models which exhibit the same type of behavior as Vicsek’s. In [4, 5], Toner and Tu construct a continuous ”hydrodynamic” model of the group of agents, while other authors... |

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Citation Context ... with the coordination of groups of mobile autonomous agents. Included here is the work of Czirok et al. [3] who propose one-dimensional models which exhibit the same type of behavior as Vicsek’s. I=-=n [4, 5], To-=-ner and Tu construct a continuous ”hydrodynamic” model of the group of agents, while other authors such as Mikhailov and Zanette [6] consider the behavior of populations of self propelled particle... |

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Citation Context ... with the coordination of groups of mobile autonomous agents. Included here is the work of Czirok et al. [3] who propose one-dimensional models which exhibit the same type of behavior as Vicsek’s. In =-=[4]-=- and [5], Toner and Tu construct a continuous ”hydrodynamic" model of the group of agents, while other authors such as Mikhailov and Zanette [6] consider the behavior of populations of self propelled ... |

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Citation Context ... proves useful to appeal to well-known results [19, 20] characterizing the convergence of infinite products of certain types of non-negative matrices. The study of infinite matrix products is ongoing =-=[21, 22, 23, 24, 25, 26] a-=-nd is undoubtedly producing results which will find application in the theoretical study of emergent behaviors. Vicsek’s model is set up in Section 2 as a system of n simultaneous, one-dimensional r... |

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Citation Context ... proves useful to appeal to well-known results [19, 20] characterizing the convergence of infinite products of certain types of non-negative matrices. The study of infinite matrix products is ongoing =-=[21, 22, 23, 24, 25, 26] a-=-nd is undoubtedly producing results which will find application in the theoretical study of emergent behaviors. Vicsek’s model is set up in Section 2 as a system of n simultaneous, one-dimensional r... |

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Citation Context ... with the coordination of groups of mobile autonomous agents. Included here is the work of Czirok et al. [3] who propose one-dimensional models which exhibit the same type of behavior as Vicsek’s. I=-=n [4, 5], To-=-ner and Tu construct a continuous ”hydrodynamic” model of the group of agents, while other authors such as Mikhailov and Zanette [6] consider the behavior of populations of self propelled particle... |

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Citation Context ...ms on a finite-dimensional space are equivalent. On the other hand, a “tight” sufficient condition for the existence of a common quadratic Lyapunov function for the matrices in M, is ρM ∈ [0, 1=-= √ m ) [24]. Th-=-is condition is tight in the sense that one can find a finite set of m × m matrices with joint spectral radius ρ = 1 √ m , whose infinite products converge to zero despite the fact that there does... |

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Citation Context ...ch papers focusing on the detailed mathematical analysis of emergent behaviors are beginning to appear. For example, Lui et al. [13] use Lyapunov methods and Leonard et al. [14] and Olfati and Murray =-=[15] -=-use potential function theory to understand flocking behavior, and Ögren et al. [16] uses Control Lyapunov function-based ideas to analyze formation stability, while Fax and Murray [17] and Desai et ... |

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Citation Context ...aws of this form can lead to chattering because neighbor relations can change abruptly with changes in agents’ positions. One way to avoid this problem is to introduce dwell time, much as was done i=-=n [32]-=-. What this means in the present context is that each agent is constrained to change its control law only at discrete times. In particular, instead of using (54), to avoid chatter agent i would use a ... |

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Citation Context ...dels which exhibit the same type of behavior as Vicsek’s. In [4, 5], Toner and Tu construct a continuous ”hydrodynamic” model of the group of agents, while other authors such as Mikhailov and Za=-=nette [6]-=- consider the behavior of populations of self propelled particles with long range interactions. Schenk et al. determined interactions between individual self-propelled spots from underlying reaction-d... |

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Citation Context ...vior of populations of self propelled particles with long range interactions. Schenk et al. determined interactions between individual self-propelled spots from underlying reaction-diffusion equation =-=[7].-=- Meanwhile in modelling biological systems, Grünbaum and Okubo use statistical methods to analyze group behavior in animal aggregations [8]. This paper and for example, the work reported in [9, 10, 1... |

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Citation Context ...uately explained using percolation theory of random graphs [35]. The results of this paper have been extended to the case where there are inter-agent forces due to attraction, repulsion and alignment =-=[36]-=-. The new result indicate that the convergence arguments used in this paper also apply to the more general problem considered in [36] under similar assumptions on the connectivity of the graph represe... |

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Modeling social animal aggregations,” Frontiers Theoret
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Citation Context ...propelled spots from underlying reaction-diffusion equation [7]. Meanwhile, in modeling biological systems, Grünbaum and Okubo use statistical methods to analyze group behavior in animal aggregations =-=[8]-=-. This paper and, for example, the work reported in [9]–[12] are part of a large literature in the biological sciences focusing on many aspects of aggregation behavior in different species. In additio... |

3 |
A bound for the disturbance-to-tracking error gain of a supervised set-point control system
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Citation Context ... will converge to θ01, no matter what the value of τD, so long as τD is greater than zero. This is in sharp contrast to other convergence results involving dwell time switching such as those given =-=in [33], which ho-=-ld only for sufficiently large values of τD. Theorem 5 is a more or less obvious consequence of the following lemma. Lemma 7 Let Then Mp ∆ = −(I + Dp + Bp) −1 (Lp + Bp), p ∈ ¯ P (61) ||e Mpt... |

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2 |
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Citation Context ...ms on a finite-dimensional space are equivalent. On the other hand, a “tight” sufficient condition for the existence of a common quadratic Lyapunov function for the matrices in M, is ρM ∈ [0, 1=-= √ m ) [30]. Th-=-is condition is tight in the sense that one can find a finite set of m × m matrices with joint spectral radius ρ = 1 √ m , whose infinite products converge to zero despite the fact that there does... |

2 |
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Citation Context ...question might also be approached by replacing the point models implicitly used in this paper, with the model of a bumper-like “virtual shell” within which each agent vehicle is constrained to rem=-=ain [34]-=-. While the analysis in this paper is deterministic and does not address the noise issue, the results obtained suggest that to understand the effect of additive noise, one should focus on how noise in... |

2 |
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Citation Context ...ncreased, i.e., there is a critical density after which all agents eventually become aligned. It is possible that this phenomenon can be adequately explained using percolation theory of random graphs =-=[35]-=-. The results of this paper have been extended to the case where there are inter-agent forces due to attraction, repulsion and alignment [36]. The new results indicate that the convergence arguments u... |

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1 |
Non-negative Matrices andMarkov CHains
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(Show Context)
Citation Context ...ors over time. Changing nearest neighbor sets is an inherent property of the Vicsek model and in the other models we consider. To analyze such models, it proves useful to appeal to well-known results =-=[19, 20]-=- characterizing the convergence of infinite products of certain types of non-negative matrices. The study of infinite matrix products is ongoing [21, 22, 23, 24, 25, 26] and is undoubtedly producing r... |

1 |
range order in a two dimensional �� model: How birds fly together,” Phys
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Citation Context ...e coordination of groups of mobile autonomous agents. Included here is the work of Czirok et al. [3] who propose one-dimensional models which exhibit the same type of behavior as Vicsek’s. In [4] and =-=[5]-=-, Toner and Tu construct a continuous ”hydrodynamic" model of the group of agents, while other authors such as Mikhailov and Zanette [6] consider the behavior of populations of self propelled particle... |

1 |
bound for the disturbance-to-tracking error gain of a supervised set-point control system
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Citation Context ...tes that will converge to , no matter what the value of , so long as is greater than zero. This is in sharp contrast to other convergence results involving dwell time switching such as those given in =-=[33]-=-, which hold only for sufficiently large values of . Theorem 5 is a more or less obvious consequence of the following lemma. Lemma 7: Let Then (61) (62) Moreover, for each finite set of indices in for... |