Journal of Robotics

Volume 2015 (2015), Article ID 321781, 10 pages

http://dx.doi.org/10.1155/2015/321781

## Self-Organized Fission Control for Flocking System

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China

Received 31 October 2014; Revised 7 January 2015; Accepted 8 January 2015

Academic Editor: Yuan F. Zheng

Copyright © 2015 Mingyong Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies the self-organized fission control problem for flocking system. Motivated by the fission behavior of biological flocks, information coupling degree (ICD) is firstly designed to represent the interaction intensity between individuals. Then, from the information transfer perspective, a “maximum-ICD” based pairwise interaction rule is proposed to realize the directional information propagation within the flock. Together with the “separation/alignment/cohesion” rules, a self-organized fission control algorithm is established that achieves the spontaneous splitting of flocking system under conflict external stimuli. Finally, numerical simulations are provided to demonstrate the effectiveness of the proposed algorithm.

#### 1. Introduction

The fission behavior of flocking system is a widely observed phenomenon in biology, society, and engineering applications [1]. A bird flock may sometimes split into multiple clusters for food foraging or predator escaping [2]. An unmanned ground vehicle (UGV) swarm often needs to segregate into small subgroups for the multisite surveillance mission [3]. In these cases, members in the flock are often identical ones with limited sensing and computing capabilities and only rely on the local interaction with their nearest neighbors [4]. Therefore, the fission phenomenon of flocking system is virtually an emergent behavior that arises in a self-organized fashion [5]. How a cohesive flock splits and forms clusters remains a fascinating issue of both theoretical and practical interests.

Currently, the research on flocking system mainly focuses on the consensus based problems such as aggregation and formation [6, 7], of which the central rules are separation, alignment, and cohesion [8]. These rules have been extensively used in the distributed sensing of mobile sensor networks, formation keeping of satellite clusters, cooperative control of unmanned ground/aerial/underwater vehicles, and so forth [9]. However, the “average consensus” property of these rules gives the flock a “collective mind” [10] and leads to a group-level ability of “consensus decision making” [11, 12], which may dispel the conflict information and encumber the process of group splitting [13].

At present, literatures that address the fission control problem seem diverse. By predefining the leaders/targets to different individuals, fission behavior emerged in the multiobjective tracking process [14–16]; in [17], Kumar et al. assigned different coupling strength to heterogeneous robot swarm that leads weak coupling robots to separate and the strong coupling robots form clusters. In addition, a long range attractive, short range repulsive interaction as well as an intermediate range Gauss-shaped interaction was employed for flock aggregation and splitting in [18].

In this paper, we tend to study the self-organized fission control problem for flocking system without predefining the leaders or identifying the differences between individuals. Motivated by the fact that interaction intensity plays a crucial role in the fission behavior of animal flocks [2, 11, 19], information coupling degree (ICD) is used as an index to denote the interaction intensity between individuals. Then, a “maximum-ICD” based pairwise interaction rule is proposed to achieve the effective information transfer within the flock. Together with the traditional “separation/alignment/cohesion” rules, a self-organized fission control algorithm is established, which realizes the spontaneous splitting of a cohesive flock under conflict external stimuli. Finally, numerical simulations are performed to illustrate the effectiveness of the proposed method.

The remainder of this paper is organized as follows: in Section 2, the fission control problem for flocking system is formulated; in Section 3, the biological principle for fission behavior is described and an information coupling degree based fission strategy is established; in Section 4, a self-organized fission control algorithm that includes a new pairwise interaction rule is proposed and the theoretical analysis is given; in Section 5, various numerical simulations and discussions are carried out to demonstrate the effectiveness of the fission control algorithm; Section 6 offers the concluding remarks.

#### 2. Problem Formulation

Consider a flocking system consisting of identical individuals moving in the -dimensional Euclidean space with the following dynamics: where is the position vector of individual , is its velocity vector, and is the acceleration vector (control input) acting on it. For notational convenience, we let

The neighboring set of individuals is defined as , , , where is the Euclidean norm and is the sensing range of each individual. Also, we define the neighboring graph [20] to be an undirected graph consisting of a set of nodes , a set of edges , and an adjacency matrix with if and , otherwise. The adjacency matrix of undirected graph is symmetric and the corresponding Laplacian matrix is , where is the in-degree matrix of graph and is the in-degree of node .

Essentially speaking, flocking behavior is a self-organized emergent phenomenon of large numbers of individuals by interacting with their neighbors and surrounding environment [21]. Correspondingly, the control input acting on an individual member can be written as where is the internal interaction force between individual and its neighbors and is the force acting on individuals from external environment. Usually, not all the members are directly influenced by the environment; here we utilize to denote whether individual can sense environment information, where means the environment information can be directly obtained by individual and , otherwise.

Fission behavior often occurs when a cohesive flock encounters obstacles/danger on their moving path or observes multiple targets for tracking [2, 15, 22, 23]. In such occasions, only a small portion of individuals (e.g., lie on the edge of the flock) are directly influenced by the environment information and often response with fast maneuvering like abrupt accelerating or turning [2, 22]; the motion of other members is only governed by their internal interaction force . Therefore, environment information can only be seen as the trigger event of fission behavior; whether the whole flock has the ability to split is essentially determined by its local interaction rules [5].

Therefore, the objective of this paper is to design the distributed local interaction rules and synthesize the control input for a flocking system such that when conflict environment information acts on part of members in the flock, it can segregate into clustered subgroups spontaneously.

To better describe the fission behavior in a quantitative way, we first give the mathematical definition of fission behavior as follows.

*Definition 1. *A flocking system is said to be segregated (or a fission behavior occurs) if and only if it satisfies the following conditions.(1)The distance between individuals in the same subgroup remains bounded; that is, , , where is a constant value and denotes the subgroup .(2)For individuals in the same subgroup, their velocities will asymptotically converge to the same value; that is, , .(3)For any distance , there exists a time after which the distance between individuals in different subgroups is at least ; that is, , , , which means that the subgroups will ultimately lose connection with each other and hence the fission behavior emerges.

*Remark 2. *It is worth mentioning that the fission behavior studied in this paper is a spontaneous response to external stimuli, during which only a small portion of members directly sense the external stimuli and the whole flock governed by the fission control algorithm is able to segregate autonomously in a self-organized fashion. Therefore, it is fundamentally different from the aforementioned fission control approaches like assignment, identification, or centralized control [14–17] and is more consistent with the fission behavior of real flocking system [2, 23].

#### 3. Information Coupling Degree Based Fission Rule

Fission behavior is the result of “collective decision making” in the presence of motion differences of individuals in the flock [2, 24]. Couzin et al. revealed that, for significant differences in the preferences of individuals, the decision dynamics may bifurcate away from consensus and lead to the emergence of fission behavior [11]. In addition, research from biologists also suggests that fission behavior, to a large extent, depends on the mutual interaction intensity between individuals [1, 25]. Individuals with larger interaction intensity tend to have tighter correlation and form clusters in the presence of significant differences in the preferences of individuals [2, 11].

Inspired by the above results, we construct a new index named information coupling degree (ICD) to denote the mutual interaction intensity between individuals. ICD is a motion dependent variable that is relevant to many factors; for example, individuals are usually more influenced by the close neighbors (distance) [26], they tend to be more sensitive to fast moving neighbors (velocity) [19, 27], and individual with more neighbors are usually more dominated (number of neighbors) [28]. In particular, we choose the two most dominating factors, the relative position and relative velocity between individuals, to design ICD in the following form: where is the position coupling term determined by the relative position between individuals. As the influence of neighbors is decreasing with the increase of their relative distance due to the sensing ability [29], we write the position coupling term as follows: where is the coefficient of position coupling term.

In addition, is the velocity coupling term that is relevant to the relative velocity between individuals. Generally, individuals are very sensitive to some specific behavior of their neighbors like abrupt accelerating or turning [19]; therefore we design as where is the average velocity of the neighbors of individual , is the number of its neighbors, and is the coefficient of velocity coupling term. Here, reflects the degree of the difference between and , with being the relative velocity between two individuals. The bigger is, the more different motion of individual is from the neighbors of individual , the more attention will be paid by individual to individual , and the tighter correlation they will tend to have, correspondingly.

From the information transfer perspective, fission behavior can be generalized as the conflict stimulus information propagation process among members within the flock [2, 23, 30]. Individuals which change their direction of travel in response to the direction taken by their nearest neighbors can quickly transfer information about the predator or food source [30]. Therefore, we propose a “maximum-ICD” based strategy to maximize the stimulus information transfer, where individuals tend to have closer relation with the neighbor that has maximum ICD with it [19, 31].

Based on the above description, we formulate the “maximum-ICD” strategy as where is the threshold value of the fission behavior. When , fission behavior occurs; otherwise, it is not disturbed by conflict environment stimuli and moves in stable formation. In this paper, we choose to be an appropriate value to prevent the unexpected fission behavior due to random fluctuation or other unknown factors.

Utilizing the “maximum-ICD” strategy, we propose a pairwise interaction rule to realize the directional information flow among individuals. The internal interaction mechanism of individuals during the fission process is illustrated in Figure 1. Assume that, at time , individual (locates at with velocity ) is propelled by external stimuli and changes its motion rapidly to a new position with velocity in a small time interval . According to (4), individual is more influenced by and tends to have a relatively tighter correlation with it. Therefore, at time the force of neighbors acting on individual can be written as where is the pairwise interaction force of the most correlated neighbor acting on individual and denotes the sum of the forces of other neighbors acting on it, with being the number of neighbors.