Research ArticleSelf-Organized Fission Control for Flocking System

Mingyong Liu Panpan Yang Xiaokang Lei and Yang Li

School of Marine Science and Technology Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Panpan Yang yangpanpan1985126com

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

Academic Editor Yuan F Zheng

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

This paper studies the self-organized fission control problem for flocking system Motivated by the fission behavior of biologicalflocks information coupling degree (ICD) is firstly designed to represent the interaction intensity between individuals Then fromthe information transfer perspective a ldquomaximum-ICDrdquo based pairwise interaction rule is proposed to realize the directionalinformation propagation within the flock Together with the ldquoseparationalignmentcohesionrdquo rules a self-organized fission controlalgorithm is established that achieves the spontaneous splitting of flocking systemunder conflict external stimuli Finally numericalsimulations are provided to demonstrate the effectiveness of the proposed algorithm

1 Introduction

The fission behavior of flocking system is a widely observedphenomenon in biology society and engineering applica-tions [1] A bird flock may sometimes split into multiple clus-ters for food foraging or predator escaping [2] An unmannedground vehicle (UGV) swarm often needs to segregate intosmall subgroups for the multisite surveillance mission [3]In these cases members in the flock are often identical oneswith limited sensing and computing capabilities and onlyrely on the local interaction with their nearest neighbors[4] Therefore the fission phenomenon of flocking system isvirtually an emergent behavior that arises in a self-organizedfashion [5] How a cohesive flock splits and forms clustersremains a fascinating issue of both theoretical and practicalinterests

Currently the research on flocking systemmainly focuseson the consensus based problems such as aggregation andformation [6 7] of which the central rules are separa-tion alignment and cohesion [8] These rules have beenextensively used in the distributed sensing of mobile sensornetworks formation keeping of satellite clusters cooperativecontrol of unmanned groundaerialunderwater vehicles andso forth [9] However the ldquoaverage consensusrdquo property ofthese rules gives the flock a ldquocollectivemindrdquo [10] and leads toa group-level ability of ldquoconsensus decision makingrdquo [11 12]

which may dispel the conflict information and encumber theprocess of group splitting [13]

At present literatures that address the fission controlproblem seem diverse By predefining the leaderstargetsto different individuals fission behavior emerged in themultiobjective tracking process [14ndash16] in [17] Kumar et alassigned different coupling strength to heterogeneous robotswarm that leads weak coupling robots to separate and thestrong coupling robots form clusters In addition a longrange attractive short range repulsive interaction aswell as anintermediate range Gauss-shaped interaction was employedfor flock aggregation and splitting in [18]

In this paper we tend to study the self-organized fissioncontrol problem for flocking system without predefining theleaders or identifying the differences between individualsMotivated by the fact that interaction intensity plays acrucial role in the fission behavior of animal flocks [211 19] information coupling degree (ICD) is used as anindex to denote the interaction intensity between individualsThen a ldquomaximum-ICDrdquo based pairwise interaction ruleis proposed to achieve the effective information transferwithin the flock Together with the traditional ldquosepara-tionalignmentcohesionrdquo rules a self-organized fission con-trol algorithm is established which realizes the spontaneoussplitting of a cohesive flock under conflict external stimuli

Hindawi Publishing CorporationJournal of RoboticsVolume 2015 Article ID 321781 10 pageshttpdxdoiorg1011552015321781

2 Journal of Robotics

Finally numerical simulations are performed to illustrate theeffectiveness of the proposed method

The remainder of this paper is organized as follows inSection 2 the fission control problem for flocking system isformulated in Section 3 the biological principle for fissionbehavior is described and an information coupling degreebased fission strategy is established in Section 4 a self-organized fission control algorithm that includes a new pair-wise interaction rule is proposed and the theoretical analysisis given in Section 5 various numerical simulations anddiscussions are carried out to demonstrate the effectiveness ofthe fission control algorithm Section 6 offers the concludingremarks

2 Problem Formulation

Consider a flocking system consisting of119873 identical individ-uals moving in the 119899-dimensional Euclidean space with thefollowing dynamics

119894= 119902119894

119902119894= 119906119894 119894 = 1 2 119873

(1)

where 119901119894isin R119899 is the position vector of individual 119894 119902

119894isin R119899

is its velocity vector and 119906119894isin R119899 is the acceleration vector

(control input) acting on it For notational convenience welet

119901 =

[[[[

[

1199011

1199012

119901119873

]]]]

]

119902 =

[[[[

[

1199021

1199022

119902119873

]]]]

]

isin R119873119899 (2)

The neighboring set of individuals 119894 is defined asN119894(119905) =

119895 119901119894minus 119901119895 le 119877 119895 = 1 119873 119895 = 119894 where

sdot is the Euclidean norm and 119877 is the sensing range ofeach individual Also we define the neighboring graph G =

(VEA) [20] to be an undirected graph consisting of a setof nodes V = V

1 V2 V

119873 a set of edges E = (V

119894 V119895) isin

V times V V119894sim V119895 and an adjacency matrix A = [119886

119894119895]

with 119886119894119895gt 0 if (V

119894 V119895) isin E and 119886

119894119895= 0 otherwise The

adjacency matrixA of undirected graphG is symmetric andthe corresponding Laplacian matrix is L = D minus A whereD = diag119889

1 1198892 119889

119873 isin R119873times119873 is the in-degree matrix of

graphG and 119889119894= sum119873

119895=1119886119894119895is the in-degree of node V

119894

Essentially speaking flocking behavior is a self-organizedemergent phenomenon of large numbers of individuals byinteracting with their neighbors and surrounding environ-ment [21] Correspondingly the control input acting on anindividual member can be written as

119906119894= 119906

in119894+ 119892119894119906out119894 (3)

where 119906in119894is the internal interaction force between individual

119894 and its neighbors and 119906out119894

is the force acting on individualsfrom external environment Usually not all the membersare directly influenced by the environment here we utilize119892119894to denote whether individual 119894 can sense environment

information where 119892119894= 1 means the environment infor-

mation can be directly obtained by individual 119894 and 119892119894= 0

otherwiseFission behavior often occurs when a cohesive flock

encounters obstaclesdanger on their moving path orobserves multiple targets for tracking [2 15 22 23] In suchoccasions only a small portion of individuals (eg lie on theedge of the flock) are directly influenced by the environmentinformation and often response with fast maneuvering likeabrupt accelerating or turning [2 22] the motion of othermembers is only governed by their internal interaction force119906in119894 Therefore environment information can only be seen as

the trigger event of fission behavior whether the whole flockhas the ability to split is essentially determined by its localinteraction rules [5]

Therefore the objective of this paper is to design thedistributed local interaction rules and synthesize the controlinput 119906

119894for a flocking system such that when conflict

environment information acts on part of members in theflock it can segregate into clustered subgroups spontaneously

To better describe the fission behavior in a quantitativeway we first give the mathematical definition of fissionbehavior as follows

Definition 1 A flocking system is said to be segregated (or afission behavior occurs) if and only if it satisfies the followingconditions

(1) The distance between individuals in the same sub-group remains bounded that is 119901

119894(119905) minus 119901

119895(119905) le 120575

119894 119895 isin 119866119896 where 120575 is a constant value and 119866

119896denotes

the subgroup 119896

(2) For individuals in the same subgroup their velocitieswill asymptotically converge to the same value that is119902119894(119905) minus 119902

119895(119905) rarr 0 119894 119895 isin 119866

119896

(3) For any distance 119863 gt 119877 there exists a time afterwhich the distance between individuals in differentsubgroups is at least119863 that is min 119901

119894(119905)minus119901

119895(119905) ge 119863

119894 isin 119866119896 119895 isin 119866

119897 which means that the subgroups will

ultimately lose connection with each other and hencethe fission behavior emerges

Remark 2 It is worth mentioning that the fission behaviorstudied in this paper is a spontaneous response to externalstimuli during which only a small portion of membersdirectly sense the external stimuli and the whole flockgoverned by the fission control algorithm is able to segregateautonomously in a self-organized fashion Therefore it isfundamentally different from the aforementioned fissioncontrol approaches like assignment identification or central-ized control [14ndash17] and is more consistent with the fissionbehavior of real flocking system [2 23]

3 Information Coupling DegreeBased Fission Rule

Fission behavior is the result of ldquocollective decision makingrdquoin the presence of motion differences of individuals in the

Journal of Robotics 3

flock [2 24] Couzin et al revealed that for significantdifferences in the preferences of individuals the decisiondynamics may bifurcate away from consensus and lead to theemergence of fission behavior [11] In addition research frombiologists also suggests that fission behavior to a large extentdepends on themutual interaction intensity between individ-uals [1 25] Individuals with larger interaction intensity tendto have tighter correlation and form clusters in the presence ofsignificant differences in the preferences of individuals [2 11]

Inspired by the above results we construct a new indexnamed information coupling degree (ICD) to denote themutual interaction intensity between individuals ICD is amotion dependent variable that is relevant to many factorsfor example individuals are usually more influenced by theclose neighbors (distance) [26] they tend to bemore sensitiveto fast moving neighbors (velocity) [19 27] and individualwith more neighbors are usually more dominated (numberof neighbors) [28] In particular we choose the two mostdominating factors the relative position and relative velocitybetween individuals to design ICD in the following form

119888119894119895= 120585119894119895sdot 120596119894119895 (4)

where 120585119894119895is the position coupling term determined by the

relative position between individuals As the influence ofneighbors is decreasing with the increase of their relativedistance due to the sensing ability [29] we write the positioncoupling term as follows

120585119894119895=

119903119894119895

1 +10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817

2 (5)

where 119903119894119895gt 0 is the coefficient of position coupling term

In addition 120596119894119895is the velocity coupling term that is

relevant to the relative velocity between individuals Gener-ally individuals are very sensitive to some specific behaviorof their neighbors like abrupt accelerating or turning [19]therefore we design 120596

119894119895as

120596119894119895= 120583119894119895

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

sum119895isinN119894(119905)

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

(6)

where 119902119894= (1119873

119894) sum119895isinN119894(119905)

119902119895is the average velocity of the

neighbors of individual 119894 119873119894is the number of its neighbors

and 120583119894119895gt 0 is the coefficient of velocity coupling term Here

(119902119894minus 119902119895) times 119902119894 reflects the degree of the difference between

(119902119894minus 119902119895) and 119902

119894 with (119902

119894minus 119902119895) being the relative velocity

between two individuals The bigger (119902119894minus 119902119895) times 119902119894 is the

more different motion of individual 119895 is from the neighborsof individual 119894 the more attention will be paid by individual119894 to individual 119895 and the tighter correlation they will tend tohave correspondingly

From the information transfer perspective fission behav-ior can be generalized as the conflict stimulus informationpropagation process among members within the flock [223 30] Individuals which change their direction of travel inresponse to the direction taken by their nearest neighborscan quickly transfer information about the predator or

utminusΔtf119894

uti

sum utminusΔti

R

i

(ptf119894

qtf119894)

(ptminusΔtf119894

qtminusΔtf119894)

Figure 1 Internal interaction mechanisms of individuals based onldquomaximum-ICDrdquo strategy

food source [30] Therefore we propose a ldquomaximum-ICDrdquobased strategy tomaximize the stimulus information transferwhere individuals tend to have closer relation with theneighbor that has maximum ICD with it [19 31]

Based on the above description we formulate the ldquomaxi-mum-ICDrdquo strategy as

119891119894= 119895 | max 119888

119894119895 119888119894119895gt 119888lowast 119895 isinN

119894 (119905) (7)

where 119888lowast is the threshold value of the fission behaviorWhen 119888

119894119895gt 119888lowast fission behavior occurs otherwise it is

not disturbed by conflict environment stimuli and movesin stable formation In this paper we choose 119888lowast to be anappropriate value to prevent the unexpected fission behaviordue to random fluctuation or other unknown factors

Utilizing the ldquomaximum-ICDrdquo strategy we propose apairwise interaction rule to realize the directional informa-tion flow among individualsThe internal interactionmecha-nism of individuals during the fission process is illustrated inFigure 1 Assume that at time 119905 minus Δ119905 individual 119891

119894(locates at

119901119905minusΔ119905

119891119894with velocity 119902119905minusΔ119905

119891119894) is propelled by external stimuli and

changes its motion rapidly to a new position 119901119905119891119894with velocity

119902119905

119891119894in a small time interval Δ119905 According to (4) individual 119894

is more influenced by 119891119894and tends to have a relatively tighter

correlation with it Therefore at time 119905 the force of neighborsacting on individual 119894 can be written as

119906119894 (119905) = 119906

119905minusΔ119905

119891119894+

119873119894minus1

sum

119894=1

119906119905minusΔ119905

119894 (8)

where 119906119905minusΔ119905119891119894

is the pairwise interaction force of the mostcorrelated neighbor 119891

119894acting on individual 119894 and sum119873119894minus1

119894=1119906119905minusΔ119905

119894

denotes the sum of the forces of other neighbors acting on itwith119873

119894being the number of neighbors

4 Journal of Robotics

Remark 3 It can be seen from (4)sim(6) that informationcoupling degree combines both the position and velocityinformation between individuals and their neighbors mean-while they also tend to have closer relation with the fastmaneuvering neighbors that are near to them which isconsistent with the property of real flocking system [19]

4 Self-Organized Fission Control Algorithm

Based on the above fission control principle we take themost correlated neighbor to form a pairwise interactionwith individual 119894 By integrating the motion informationof the most correlated neighbor into the coordinated lawtogether with the ldquoseparationalignmentrepulsionrdquo rules thedistributed fission control algorithm is formulated as

119906119894= 119891119901

119894+ 119891

V119894+ 119891119891

119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906in119894

+ 119892119894119891119890

119894⏟⏟⏟⏟⏟⏟⏟

119906out119894

(9)

where the environment force 119906out119894

is represented by the exter-nal stimulus term 119891

119890

119894 it causes the rapid movement change

of individuals and evokes the fission behavior and 119892119894denotes

whether individual 119894 can sense external stimuli Usually 119891119890119894

has diverse forms according to different environment stimulisuch as the direction to a known source or a segment of amigration route [11 32] Here for the convenience of analysiswe design the external stimulus term as the following simpleposition feedback form

119891119890

119894= minus (119901

119894minus 119901119890

119894) (10)

where119901119890119894is the desired position driven by external stimuli and

it is supposed to be a constant value at every sampling timeinterval

Obviously the internal interaction force 119906in119894consists of

three components(1) 119891119901119894is used to regulate the position between individual

119894 and its neighbors This term is responsible for collisionavoidance and cohesion in the flock that is when individualsare too far away from each other the attraction force willmake them move together when individuals are too closethe repulsion force will propel them away to avoid collisionIt is derived from the field produced by a collective functionwhich depends on the relative distance between individual 119894and its neighbors and is defined as

119891119901

119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590) (11)

where nabla is the gradient operator the 120590-norm sdot 120590of a vector

is a map R119899 rarr R+ defined by 119911120590= (1120598)[radic1 + 1205981199112 minus 1]

with a parameter 120598 gt 0 and 119911 = [1199111 1199112 119911

119899]Tisin R119899

Note that the map 119911120590is differentiable everywhere but

119911 = radic1199112

1+ 1199112

2+ sdot sdot sdot + 1199112

119899is not differentiable at 119911 = 0 This

property of 120590-norm is used for the construction of smoothcollective potential functions for individuals To construct asmooth pairwise potential with finite cutoff we follow the

0 5 10 15 20 250

10

20

30

40

50

60

120595120572

z

Figure 2 The pairwise attractiverepulsive potential function

work of [7] and integrate an action function 120601120572(119911) that varies

for all 119911 ge 119903120572 Define this action function as

120601120572 (119911) = 120588ℎ (

119911

119903120572

)120601 (119911 minus 119889120572)

120601 (119911) =1

2[(119886 + 119887) 1205901 (119911 + 119888) + (119886 minus 119887)]

(12)

where 1205901(119911) = 119911radic1 + 1199112 and 120601(119911) is an uneven sigmoidal

function with parameters that satisfy 0 lt 119886 le 119887119888 = |119886 minus 119887|radic4119886119887 to guarantee 120601(0) = 0 The pairwiseattractiverepulsive potential function 120595

120572(119911) is then defined

as

120595120572 (119911) = int

119911

119889120572

120601120572 (119904) 119889119904 (13)

and is depicted in Figure 2Additionally 119901

119894119895= 119901119894minus 119901119895is the relative position vector

between individuals 119894 and 119895 119901119891119894119895= 119901119891119894minus 119901119891119895is the relative

position vector between the most correlated neighbors ofindividuals 119894 and 119895

(2) 119891V119894is the velocity coordination term that regulates the

velocity of individuals

119891V119894= minus sum

119895isinN119894(119905)

119886119894119895(119902119894119895minus 119902119891119894119895) (14)

where 119902119894119895= 119902119894minus 119902119895is the relative velocity vector between

individuals 119894 and 119895 and 119902119891119894119895= 119902119891119894minus 119902119891119895is the relative velocity

vector between the most correlated neighbors of individuals 119894and 119895MoreoverA(119905) = [119886

119894119895(119905)] is the adjacencymatrixwhich

is defined as

119886119894119895 (119905) =

0 if 119895 = 119894

120588ℎ(

10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590

119903120590

) if 119895 = 119894(15)

Journal of Robotics 5

with the bump function 120588ℎ(119911) ℎ isin (0 1) being

120588ℎ (119911) =

1 119911 isin [0 ℎ)

1

2[1 + cos(120587119911 minus ℎ

1 minus ℎ)] 119911 isin [ℎ 1]

0 otherwise

(16)

(3) 119891119891119894is the pairwise interaction term that produces a

closer relation between individual 119894 and its most correlatedneighbor 119891

119894 which is used to guide the motion preference of

individual 119894 Specifically we employ an attractive force andvelocity feedback approach to generate 119891119891

119894as

119891119891

119894= minus1198961nabla119901119891119894119880119891119894minus 1198962(119902119894minus 119902119891119894) (17)

where 1198961 1198962gt 0 are the constant feedback gains and 119880

119891119894=

(12)(119901119894minus 119901119891119894)2 is the potential energy of its most correlated

neighbor This term guarantees the velocity convergence ofindividual 119894 to its most correlated neighbor 119891

119894and ultimately

induces the fission behavior

Remark 4 The specifically designed pairwise attractivere-pulsive artificial potential 120595

120572and adjacent matrix 119886

119894119895here are

to guarantee the smoothness of the potential function andLaplacian matrix which will facilitate the theoretical anal-ysis with the traditional Lyapunov-based stability analysismethod

Remark 5 Note that in our fission control algorithm (9) themost correlated neighbor of individual 119894 is chosen accordingto (7) and it is dynamically updating during the fissionprocess of the flock which realizes the implicit stimulusinformation transfer among members in the flock Thusthere is an essential difference between our work and theresearches of [14ndash16] as the latter assume the leader or targetof each member is predefined and fixed during the fissionprocess

Remark 6 From (7) we can also see that if 119888119894119895lt 119888lowast the most

correlated neighbor of individual 119894 does not exist In this casethe fission control algorithm (9) degrades into

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) + 119892119894119891119890

119894

(18)

which is equivalent to the first flocking control law proposedin [7] This situation occurs when the differences of motionpreference between individuals are relatively small and hencethe fission behavior does not occurTherefore algorithm (18)in [7] can be seen as a special case of oursHowever algorithm(18) is not able to achieve the spontaneous fission behaviordue to its average consensus property A detailed comparativesimulation and analysis is demonstrated in part 5

Here we define the sum of the total artificial potentialenergy and the total relative kinetic energy between allindividuals and the external stimuli as follows

119876 (119901 119902) =1

2

119873

sum

119894=1

(119880119894(119901) + (119902

119894minus 119902119891119894)T(119902119894minus 119902119891119894)) (19)

where

119880119894(119901) = 119881

119894(119901) + 119896

1(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

= sum

119895isinN119894(119905)

120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590)

+ 1198961(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

+ 119892119894(119901119894minus 119901119890

119894)T(119901119894minus 119901119890

119894)

(20)

Obviously 119876 is a positive semidefinite function We have thefollowing result

Theorem 7 Consider a flocking system consisting of 119873individuals with dynamics (1) Supposing the initial energy1198760(1198760= 119876(119901(0) 119902(0))) of the flock is finite and the initial

graph G is connected before the fission process when conflictexternal stimuli cause the rapid maneuvering of individualsthat lie on the edge of the flock under the fission control law(9) a cohesive flocking system will segregate into clusteredsubgroups due to the differences of information coupling degreebetween individuals Then the following statements hold

(i) The distance between individuals in the same subgroupis not larger than 120575 where 120575 = (119873

119894minus 1)radic2119876

01198961and

119873119894is the number of individuals in the subgroup

(ii) The velocities of individuals in the same subgroup willasymptotically converge to the same value

(iii) The distance between different subgroups is larger than119877 as time goes by

Proof We take a cluster of 119873119894individuals in the neighbor-

hood of individual 119894 into consideration and assume that graphG119873119894

is connected in the small sampling time interval from 119905

to 1199051(i) Let 119901119890

119894= 119901119894minus 119901119890

119894be the position difference vector

between individual 119894 and the external stimuli and let119901119894= 119901119894minus

119901119891119894 119902119894= 119902119894minus119902119891119894be the position difference vector and velocity

difference vector between individual 119894 and its most correlatedneighbor 119891

119894 respectively Then the following equations hold

119901119894119895minus 119901119891119894119895= 119901119894minus 119901119895= 119901119894119895

119902119894119895minus 119902119891119894119895= 119902119894minus 119902119895= 119902119894119895

(21)

The control input 119906119894can then be rewritten as

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895)

minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894

(22)

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

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DistributedSensor Networks

International Journal of

2 Journal of Robotics

Finally numerical simulations are performed to illustrate theeffectiveness of the proposed method

The remainder of this paper is organized as follows inSection 2 the fission control problem for flocking system isformulated in Section 3 the biological principle for fissionbehavior is described and an information coupling degreebased fission strategy is established in Section 4 a self-organized fission control algorithm that includes a new pair-wise interaction rule is proposed and the theoretical analysisis given in Section 5 various numerical simulations anddiscussions are carried out to demonstrate the effectiveness ofthe fission control algorithm Section 6 offers the concludingremarks

2 Problem Formulation

Consider a flocking system consisting of119873 identical individ-uals moving in the 119899-dimensional Euclidean space with thefollowing dynamics

119894= 119902119894

119902119894= 119906119894 119894 = 1 2 119873

(1)

where 119901119894isin R119899 is the position vector of individual 119894 119902

119894isin R119899

is its velocity vector and 119906119894isin R119899 is the acceleration vector

(control input) acting on it For notational convenience welet

119901 =

[[[[

[

1199011

1199012

119901119873

]]]]

]

119902 =

[[[[

[

1199021

1199022

119902119873

]]]]

]

isin R119873119899 (2)

The neighboring set of individuals 119894 is defined asN119894(119905) =

119895 119901119894minus 119901119895 le 119877 119895 = 1 119873 119895 = 119894 where

sdot is the Euclidean norm and 119877 is the sensing range ofeach individual Also we define the neighboring graph G =

(VEA) [20] to be an undirected graph consisting of a setof nodes V = V

1 V2 V

119873 a set of edges E = (V

119894 V119895) isin

V times V V119894sim V119895 and an adjacency matrix A = [119886

119894119895]

with 119886119894119895gt 0 if (V

119894 V119895) isin E and 119886

119894119895= 0 otherwise The

adjacency matrixA of undirected graphG is symmetric andthe corresponding Laplacian matrix is L = D minus A whereD = diag119889

1 1198892 119889

119873 isin R119873times119873 is the in-degree matrix of

graphG and 119889119894= sum119873

119895=1119886119894119895is the in-degree of node V

119894

Essentially speaking flocking behavior is a self-organizedemergent phenomenon of large numbers of individuals byinteracting with their neighbors and surrounding environ-ment [21] Correspondingly the control input acting on anindividual member can be written as

119906119894= 119906

in119894+ 119892119894119906out119894 (3)

where 119906in119894is the internal interaction force between individual

119894 and its neighbors and 119906out119894

is the force acting on individualsfrom external environment Usually not all the membersare directly influenced by the environment here we utilize119892119894to denote whether individual 119894 can sense environment

information where 119892119894= 1 means the environment infor-

mation can be directly obtained by individual 119894 and 119892119894= 0

otherwiseFission behavior often occurs when a cohesive flock

encounters obstaclesdanger on their moving path orobserves multiple targets for tracking [2 15 22 23] In suchoccasions only a small portion of individuals (eg lie on theedge of the flock) are directly influenced by the environmentinformation and often response with fast maneuvering likeabrupt accelerating or turning [2 22] the motion of othermembers is only governed by their internal interaction force119906in119894 Therefore environment information can only be seen as

the trigger event of fission behavior whether the whole flockhas the ability to split is essentially determined by its localinteraction rules [5]

Therefore the objective of this paper is to design thedistributed local interaction rules and synthesize the controlinput 119906

119894for a flocking system such that when conflict

environment information acts on part of members in theflock it can segregate into clustered subgroups spontaneously

To better describe the fission behavior in a quantitativeway we first give the mathematical definition of fissionbehavior as follows

Definition 1 A flocking system is said to be segregated (or afission behavior occurs) if and only if it satisfies the followingconditions

(1) The distance between individuals in the same sub-group remains bounded that is 119901

119894(119905) minus 119901

119895(119905) le 120575

119894 119895 isin 119866119896 where 120575 is a constant value and 119866

119896denotes

the subgroup 119896

(2) For individuals in the same subgroup their velocitieswill asymptotically converge to the same value that is119902119894(119905) minus 119902

119895(119905) rarr 0 119894 119895 isin 119866

119896

(3) For any distance 119863 gt 119877 there exists a time afterwhich the distance between individuals in differentsubgroups is at least119863 that is min 119901

119894(119905)minus119901

119895(119905) ge 119863

119894 isin 119866119896 119895 isin 119866

119897 which means that the subgroups will

ultimately lose connection with each other and hencethe fission behavior emerges

Remark 2 It is worth mentioning that the fission behaviorstudied in this paper is a spontaneous response to externalstimuli during which only a small portion of membersdirectly sense the external stimuli and the whole flockgoverned by the fission control algorithm is able to segregateautonomously in a self-organized fashion Therefore it isfundamentally different from the aforementioned fissioncontrol approaches like assignment identification or central-ized control [14ndash17] and is more consistent with the fissionbehavior of real flocking system [2 23]

3 Information Coupling DegreeBased Fission Rule

Fission behavior is the result of ldquocollective decision makingrdquoin the presence of motion differences of individuals in the

Journal of Robotics 3

flock [2 24] Couzin et al revealed that for significantdifferences in the preferences of individuals the decisiondynamics may bifurcate away from consensus and lead to theemergence of fission behavior [11] In addition research frombiologists also suggests that fission behavior to a large extentdepends on themutual interaction intensity between individ-uals [1 25] Individuals with larger interaction intensity tendto have tighter correlation and form clusters in the presence ofsignificant differences in the preferences of individuals [2 11]

Inspired by the above results we construct a new indexnamed information coupling degree (ICD) to denote themutual interaction intensity between individuals ICD is amotion dependent variable that is relevant to many factorsfor example individuals are usually more influenced by theclose neighbors (distance) [26] they tend to bemore sensitiveto fast moving neighbors (velocity) [19 27] and individualwith more neighbors are usually more dominated (numberof neighbors) [28] In particular we choose the two mostdominating factors the relative position and relative velocitybetween individuals to design ICD in the following form

119888119894119895= 120585119894119895sdot 120596119894119895 (4)

where 120585119894119895is the position coupling term determined by the

relative position between individuals As the influence ofneighbors is decreasing with the increase of their relativedistance due to the sensing ability [29] we write the positioncoupling term as follows

120585119894119895=

119903119894119895

1 +10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817

2 (5)

where 119903119894119895gt 0 is the coefficient of position coupling term

In addition 120596119894119895is the velocity coupling term that is

relevant to the relative velocity between individuals Gener-ally individuals are very sensitive to some specific behaviorof their neighbors like abrupt accelerating or turning [19]therefore we design 120596

119894119895as

120596119894119895= 120583119894119895

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

sum119895isinN119894(119905)

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

(6)

where 119902119894= (1119873

119894) sum119895isinN119894(119905)

119902119895is the average velocity of the

neighbors of individual 119894 119873119894is the number of its neighbors

and 120583119894119895gt 0 is the coefficient of velocity coupling term Here

(119902119894minus 119902119895) times 119902119894 reflects the degree of the difference between

(119902119894minus 119902119895) and 119902

119894 with (119902

119894minus 119902119895) being the relative velocity

between two individuals The bigger (119902119894minus 119902119895) times 119902119894 is the

more different motion of individual 119895 is from the neighborsof individual 119894 the more attention will be paid by individual119894 to individual 119895 and the tighter correlation they will tend tohave correspondingly

From the information transfer perspective fission behav-ior can be generalized as the conflict stimulus informationpropagation process among members within the flock [223 30] Individuals which change their direction of travel inresponse to the direction taken by their nearest neighborscan quickly transfer information about the predator or

utminusΔtf119894

uti

sum utminusΔti

R

i

(ptf119894

qtf119894)

(ptminusΔtf119894

qtminusΔtf119894)

Figure 1 Internal interaction mechanisms of individuals based onldquomaximum-ICDrdquo strategy

food source [30] Therefore we propose a ldquomaximum-ICDrdquobased strategy tomaximize the stimulus information transferwhere individuals tend to have closer relation with theneighbor that has maximum ICD with it [19 31]

Based on the above description we formulate the ldquomaxi-mum-ICDrdquo strategy as

119891119894= 119895 | max 119888

119894119895 119888119894119895gt 119888lowast 119895 isinN

119894 (119905) (7)

where 119888lowast is the threshold value of the fission behaviorWhen 119888

119894119895gt 119888lowast fission behavior occurs otherwise it is

not disturbed by conflict environment stimuli and movesin stable formation In this paper we choose 119888lowast to be anappropriate value to prevent the unexpected fission behaviordue to random fluctuation or other unknown factors

Utilizing the ldquomaximum-ICDrdquo strategy we propose apairwise interaction rule to realize the directional informa-tion flow among individualsThe internal interactionmecha-nism of individuals during the fission process is illustrated inFigure 1 Assume that at time 119905 minus Δ119905 individual 119891

119894(locates at

119901119905minusΔ119905

119891119894with velocity 119902119905minusΔ119905

119891119894) is propelled by external stimuli and

changes its motion rapidly to a new position 119901119905119891119894with velocity

119902119905

119891119894in a small time interval Δ119905 According to (4) individual 119894

is more influenced by 119891119894and tends to have a relatively tighter

correlation with it Therefore at time 119905 the force of neighborsacting on individual 119894 can be written as

119906119894 (119905) = 119906

119905minusΔ119905

119891119894+

119873119894minus1

sum

119894=1

119906119905minusΔ119905

119894 (8)

where 119906119905minusΔ119905119891119894

is the pairwise interaction force of the mostcorrelated neighbor 119891

119894acting on individual 119894 and sum119873119894minus1

119894=1119906119905minusΔ119905

119894

denotes the sum of the forces of other neighbors acting on itwith119873

119894being the number of neighbors

4 Journal of Robotics

Remark 3 It can be seen from (4)sim(6) that informationcoupling degree combines both the position and velocityinformation between individuals and their neighbors mean-while they also tend to have closer relation with the fastmaneuvering neighbors that are near to them which isconsistent with the property of real flocking system [19]

4 Self-Organized Fission Control Algorithm

Based on the above fission control principle we take themost correlated neighbor to form a pairwise interactionwith individual 119894 By integrating the motion informationof the most correlated neighbor into the coordinated lawtogether with the ldquoseparationalignmentrepulsionrdquo rules thedistributed fission control algorithm is formulated as

119906119894= 119891119901

119894+ 119891

V119894+ 119891119891

119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906in119894

+ 119892119894119891119890

119894⏟⏟⏟⏟⏟⏟⏟

119906out119894

(9)

where the environment force 119906out119894

is represented by the exter-nal stimulus term 119891

119890

119894 it causes the rapid movement change

of individuals and evokes the fission behavior and 119892119894denotes

whether individual 119894 can sense external stimuli Usually 119891119890119894

has diverse forms according to different environment stimulisuch as the direction to a known source or a segment of amigration route [11 32] Here for the convenience of analysiswe design the external stimulus term as the following simpleposition feedback form

119891119890

119894= minus (119901

119894minus 119901119890

119894) (10)

where119901119890119894is the desired position driven by external stimuli and

it is supposed to be a constant value at every sampling timeinterval

Obviously the internal interaction force 119906in119894consists of

three components(1) 119891119901119894is used to regulate the position between individual

119894 and its neighbors This term is responsible for collisionavoidance and cohesion in the flock that is when individualsare too far away from each other the attraction force willmake them move together when individuals are too closethe repulsion force will propel them away to avoid collisionIt is derived from the field produced by a collective functionwhich depends on the relative distance between individual 119894and its neighbors and is defined as

119891119901

119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590) (11)

where nabla is the gradient operator the 120590-norm sdot 120590of a vector

is a map R119899 rarr R+ defined by 119911120590= (1120598)[radic1 + 1205981199112 minus 1]

with a parameter 120598 gt 0 and 119911 = [1199111 1199112 119911

119899]Tisin R119899

Note that the map 119911120590is differentiable everywhere but

119911 = radic1199112

1+ 1199112

2+ sdot sdot sdot + 1199112

119899is not differentiable at 119911 = 0 This

property of 120590-norm is used for the construction of smoothcollective potential functions for individuals To construct asmooth pairwise potential with finite cutoff we follow the

0 5 10 15 20 250

10

20

30

40

50

60

120595120572

z

Figure 2 The pairwise attractiverepulsive potential function

work of [7] and integrate an action function 120601120572(119911) that varies

for all 119911 ge 119903120572 Define this action function as

120601120572 (119911) = 120588ℎ (

119911

119903120572

)120601 (119911 minus 119889120572)

120601 (119911) =1

2[(119886 + 119887) 1205901 (119911 + 119888) + (119886 minus 119887)]

(12)

where 1205901(119911) = 119911radic1 + 1199112 and 120601(119911) is an uneven sigmoidal

function with parameters that satisfy 0 lt 119886 le 119887119888 = |119886 minus 119887|radic4119886119887 to guarantee 120601(0) = 0 The pairwiseattractiverepulsive potential function 120595

120572(119911) is then defined

as

120595120572 (119911) = int

119911

119889120572

120601120572 (119904) 119889119904 (13)

and is depicted in Figure 2Additionally 119901

119894119895= 119901119894minus 119901119895is the relative position vector

between individuals 119894 and 119895 119901119891119894119895= 119901119891119894minus 119901119891119895is the relative

position vector between the most correlated neighbors ofindividuals 119894 and 119895

(2) 119891V119894is the velocity coordination term that regulates the

velocity of individuals

119891V119894= minus sum

119895isinN119894(119905)

119886119894119895(119902119894119895minus 119902119891119894119895) (14)

where 119902119894119895= 119902119894minus 119902119895is the relative velocity vector between

individuals 119894 and 119895 and 119902119891119894119895= 119902119891119894minus 119902119891119895is the relative velocity

vector between the most correlated neighbors of individuals 119894and 119895MoreoverA(119905) = [119886

119894119895(119905)] is the adjacencymatrixwhich

is defined as

119886119894119895 (119905) =

0 if 119895 = 119894

120588ℎ(

10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590

119903120590

) if 119895 = 119894(15)

Journal of Robotics 5

with the bump function 120588ℎ(119911) ℎ isin (0 1) being

120588ℎ (119911) =

1 119911 isin [0 ℎ)

1

2[1 + cos(120587119911 minus ℎ

1 minus ℎ)] 119911 isin [ℎ 1]

0 otherwise

(16)

(3) 119891119891119894is the pairwise interaction term that produces a

closer relation between individual 119894 and its most correlatedneighbor 119891

119894 which is used to guide the motion preference of

individual 119894 Specifically we employ an attractive force andvelocity feedback approach to generate 119891119891

119894as

119891119891

119894= minus1198961nabla119901119891119894119880119891119894minus 1198962(119902119894minus 119902119891119894) (17)

where 1198961 1198962gt 0 are the constant feedback gains and 119880

119891119894=

(12)(119901119894minus 119901119891119894)2 is the potential energy of its most correlated

neighbor This term guarantees the velocity convergence ofindividual 119894 to its most correlated neighbor 119891

119894and ultimately

induces the fission behavior

Remark 4 The specifically designed pairwise attractivere-pulsive artificial potential 120595

120572and adjacent matrix 119886

119894119895here are

to guarantee the smoothness of the potential function andLaplacian matrix which will facilitate the theoretical anal-ysis with the traditional Lyapunov-based stability analysismethod

Remark 5 Note that in our fission control algorithm (9) themost correlated neighbor of individual 119894 is chosen accordingto (7) and it is dynamically updating during the fissionprocess of the flock which realizes the implicit stimulusinformation transfer among members in the flock Thusthere is an essential difference between our work and theresearches of [14ndash16] as the latter assume the leader or targetof each member is predefined and fixed during the fissionprocess

Remark 6 From (7) we can also see that if 119888119894119895lt 119888lowast the most

correlated neighbor of individual 119894 does not exist In this casethe fission control algorithm (9) degrades into

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) + 119892119894119891119890

119894

(18)

which is equivalent to the first flocking control law proposedin [7] This situation occurs when the differences of motionpreference between individuals are relatively small and hencethe fission behavior does not occurTherefore algorithm (18)in [7] can be seen as a special case of oursHowever algorithm(18) is not able to achieve the spontaneous fission behaviordue to its average consensus property A detailed comparativesimulation and analysis is demonstrated in part 5

Here we define the sum of the total artificial potentialenergy and the total relative kinetic energy between allindividuals and the external stimuli as follows

119876 (119901 119902) =1

2

119873

sum

119894=1

(119880119894(119901) + (119902

119894minus 119902119891119894)T(119902119894minus 119902119891119894)) (19)

where

119880119894(119901) = 119881

119894(119901) + 119896

1(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

= sum

119895isinN119894(119905)

120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590)

+ 1198961(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

+ 119892119894(119901119894minus 119901119890

119894)T(119901119894minus 119901119890

119894)

(20)

Obviously 119876 is a positive semidefinite function We have thefollowing result

Theorem 7 Consider a flocking system consisting of 119873individuals with dynamics (1) Supposing the initial energy1198760(1198760= 119876(119901(0) 119902(0))) of the flock is finite and the initial

graph G is connected before the fission process when conflictexternal stimuli cause the rapid maneuvering of individualsthat lie on the edge of the flock under the fission control law(9) a cohesive flocking system will segregate into clusteredsubgroups due to the differences of information coupling degreebetween individuals Then the following statements hold

(i) The distance between individuals in the same subgroupis not larger than 120575 where 120575 = (119873

119894minus 1)radic2119876

01198961and

119873119894is the number of individuals in the subgroup

(ii) The velocities of individuals in the same subgroup willasymptotically converge to the same value

(iii) The distance between different subgroups is larger than119877 as time goes by

Proof We take a cluster of 119873119894individuals in the neighbor-

hood of individual 119894 into consideration and assume that graphG119873119894

is connected in the small sampling time interval from 119905

to 1199051(i) Let 119901119890

119894= 119901119894minus 119901119890

119894be the position difference vector

between individual 119894 and the external stimuli and let119901119894= 119901119894minus

119901119891119894 119902119894= 119902119894minus119902119891119894be the position difference vector and velocity

difference vector between individual 119894 and its most correlatedneighbor 119891

119894 respectively Then the following equations hold

119901119894119895minus 119901119891119894119895= 119901119894minus 119901119895= 119901119894119895

119902119894119895minus 119902119891119894119895= 119902119894minus 119902119895= 119902119894119895

(21)

The control input 119906119894can then be rewritten as

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895)

minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894

(22)

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

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DistributedSensor Networks

International Journal of

Journal of Robotics 3

flock [2 24] Couzin et al revealed that for significantdifferences in the preferences of individuals the decisiondynamics may bifurcate away from consensus and lead to theemergence of fission behavior [11] In addition research frombiologists also suggests that fission behavior to a large extentdepends on themutual interaction intensity between individ-uals [1 25] Individuals with larger interaction intensity tendto have tighter correlation and form clusters in the presence ofsignificant differences in the preferences of individuals [2 11]

Inspired by the above results we construct a new indexnamed information coupling degree (ICD) to denote themutual interaction intensity between individuals ICD is amotion dependent variable that is relevant to many factorsfor example individuals are usually more influenced by theclose neighbors (distance) [26] they tend to bemore sensitiveto fast moving neighbors (velocity) [19 27] and individualwith more neighbors are usually more dominated (numberof neighbors) [28] In particular we choose the two mostdominating factors the relative position and relative velocitybetween individuals to design ICD in the following form

119888119894119895= 120585119894119895sdot 120596119894119895 (4)

where 120585119894119895is the position coupling term determined by the

relative position between individuals As the influence ofneighbors is decreasing with the increase of their relativedistance due to the sensing ability [29] we write the positioncoupling term as follows

120585119894119895=

119903119894119895

1 +10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817

2 (5)

where 119903119894119895gt 0 is the coefficient of position coupling term

In addition 120596119894119895is the velocity coupling term that is

relevant to the relative velocity between individuals Gener-ally individuals are very sensitive to some specific behaviorof their neighbors like abrupt accelerating or turning [19]therefore we design 120596

119894119895as

120596119894119895= 120583119894119895

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

sum119895isinN119894(119905)

10038171003817100381710038171003817(119902119894minus 119902119895) times 119902119894

10038171003817100381710038171003817

(6)

where 119902119894= (1119873

119894) sum119895isinN119894(119905)

119902119895is the average velocity of the

neighbors of individual 119894 119873119894is the number of its neighbors

and 120583119894119895gt 0 is the coefficient of velocity coupling term Here

(119902119894minus 119902119895) times 119902119894 reflects the degree of the difference between

(119902119894minus 119902119895) and 119902

119894 with (119902

119894minus 119902119895) being the relative velocity

between two individuals The bigger (119902119894minus 119902119895) times 119902119894 is the

more different motion of individual 119895 is from the neighborsof individual 119894 the more attention will be paid by individual119894 to individual 119895 and the tighter correlation they will tend tohave correspondingly

From the information transfer perspective fission behav-ior can be generalized as the conflict stimulus informationpropagation process among members within the flock [223 30] Individuals which change their direction of travel inresponse to the direction taken by their nearest neighborscan quickly transfer information about the predator or

utminusΔtf119894

uti

sum utminusΔti

R

i

(ptf119894

qtf119894)

(ptminusΔtf119894

qtminusΔtf119894)

Figure 1 Internal interaction mechanisms of individuals based onldquomaximum-ICDrdquo strategy

food source [30] Therefore we propose a ldquomaximum-ICDrdquobased strategy tomaximize the stimulus information transferwhere individuals tend to have closer relation with theneighbor that has maximum ICD with it [19 31]

Based on the above description we formulate the ldquomaxi-mum-ICDrdquo strategy as

119891119894= 119895 | max 119888

119894119895 119888119894119895gt 119888lowast 119895 isinN

119894 (119905) (7)

where 119888lowast is the threshold value of the fission behaviorWhen 119888

119894119895gt 119888lowast fission behavior occurs otherwise it is

not disturbed by conflict environment stimuli and movesin stable formation In this paper we choose 119888lowast to be anappropriate value to prevent the unexpected fission behaviordue to random fluctuation or other unknown factors

Utilizing the ldquomaximum-ICDrdquo strategy we propose apairwise interaction rule to realize the directional informa-tion flow among individualsThe internal interactionmecha-nism of individuals during the fission process is illustrated inFigure 1 Assume that at time 119905 minus Δ119905 individual 119891

119894(locates at

119901119905minusΔ119905

119891119894with velocity 119902119905minusΔ119905

119891119894) is propelled by external stimuli and

changes its motion rapidly to a new position 119901119905119891119894with velocity

119902119905

119891119894in a small time interval Δ119905 According to (4) individual 119894

is more influenced by 119891119894and tends to have a relatively tighter

correlation with it Therefore at time 119905 the force of neighborsacting on individual 119894 can be written as

119906119894 (119905) = 119906

119905minusΔ119905

119891119894+

119873119894minus1

sum

119894=1

119906119905minusΔ119905

119894 (8)

where 119906119905minusΔ119905119891119894

is the pairwise interaction force of the mostcorrelated neighbor 119891

119894acting on individual 119894 and sum119873119894minus1

119894=1119906119905minusΔ119905

119894

denotes the sum of the forces of other neighbors acting on itwith119873

119894being the number of neighbors

4 Journal of Robotics

Remark 3 It can be seen from (4)sim(6) that informationcoupling degree combines both the position and velocityinformation between individuals and their neighbors mean-while they also tend to have closer relation with the fastmaneuvering neighbors that are near to them which isconsistent with the property of real flocking system [19]

4 Self-Organized Fission Control Algorithm

Based on the above fission control principle we take themost correlated neighbor to form a pairwise interactionwith individual 119894 By integrating the motion informationof the most correlated neighbor into the coordinated lawtogether with the ldquoseparationalignmentrepulsionrdquo rules thedistributed fission control algorithm is formulated as

119906119894= 119891119901

119894+ 119891

V119894+ 119891119891

119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906in119894

+ 119892119894119891119890

119894⏟⏟⏟⏟⏟⏟⏟

119906out119894

(9)

where the environment force 119906out119894

is represented by the exter-nal stimulus term 119891

119890

119894 it causes the rapid movement change

of individuals and evokes the fission behavior and 119892119894denotes

whether individual 119894 can sense external stimuli Usually 119891119890119894

has diverse forms according to different environment stimulisuch as the direction to a known source or a segment of amigration route [11 32] Here for the convenience of analysiswe design the external stimulus term as the following simpleposition feedback form

119891119890

119894= minus (119901

119894minus 119901119890

119894) (10)

where119901119890119894is the desired position driven by external stimuli and

it is supposed to be a constant value at every sampling timeinterval

Obviously the internal interaction force 119906in119894consists of

three components(1) 119891119901119894is used to regulate the position between individual

119894 and its neighbors This term is responsible for collisionavoidance and cohesion in the flock that is when individualsare too far away from each other the attraction force willmake them move together when individuals are too closethe repulsion force will propel them away to avoid collisionIt is derived from the field produced by a collective functionwhich depends on the relative distance between individual 119894and its neighbors and is defined as

119891119901

119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590) (11)

where nabla is the gradient operator the 120590-norm sdot 120590of a vector

is a map R119899 rarr R+ defined by 119911120590= (1120598)[radic1 + 1205981199112 minus 1]

with a parameter 120598 gt 0 and 119911 = [1199111 1199112 119911

119899]Tisin R119899

Note that the map 119911120590is differentiable everywhere but

119911 = radic1199112

1+ 1199112

2+ sdot sdot sdot + 1199112

119899is not differentiable at 119911 = 0 This

property of 120590-norm is used for the construction of smoothcollective potential functions for individuals To construct asmooth pairwise potential with finite cutoff we follow the

0 5 10 15 20 250

10

20

30

40

50

60

120595120572

z

Figure 2 The pairwise attractiverepulsive potential function

work of [7] and integrate an action function 120601120572(119911) that varies

for all 119911 ge 119903120572 Define this action function as

120601120572 (119911) = 120588ℎ (

119911

119903120572

)120601 (119911 minus 119889120572)

120601 (119911) =1

2[(119886 + 119887) 1205901 (119911 + 119888) + (119886 minus 119887)]

(12)

where 1205901(119911) = 119911radic1 + 1199112 and 120601(119911) is an uneven sigmoidal

function with parameters that satisfy 0 lt 119886 le 119887119888 = |119886 minus 119887|radic4119886119887 to guarantee 120601(0) = 0 The pairwiseattractiverepulsive potential function 120595

120572(119911) is then defined

as

120595120572 (119911) = int

119911

119889120572

120601120572 (119904) 119889119904 (13)

and is depicted in Figure 2Additionally 119901

119894119895= 119901119894minus 119901119895is the relative position vector

between individuals 119894 and 119895 119901119891119894119895= 119901119891119894minus 119901119891119895is the relative

position vector between the most correlated neighbors ofindividuals 119894 and 119895

(2) 119891V119894is the velocity coordination term that regulates the

velocity of individuals

119891V119894= minus sum

119895isinN119894(119905)

119886119894119895(119902119894119895minus 119902119891119894119895) (14)

where 119902119894119895= 119902119894minus 119902119895is the relative velocity vector between

individuals 119894 and 119895 and 119902119891119894119895= 119902119891119894minus 119902119891119895is the relative velocity

vector between the most correlated neighbors of individuals 119894and 119895MoreoverA(119905) = [119886

119894119895(119905)] is the adjacencymatrixwhich

is defined as

119886119894119895 (119905) =

0 if 119895 = 119894

120588ℎ(

10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590

119903120590

) if 119895 = 119894(15)

Journal of Robotics 5

with the bump function 120588ℎ(119911) ℎ isin (0 1) being

120588ℎ (119911) =

1 119911 isin [0 ℎ)

1

2[1 + cos(120587119911 minus ℎ

1 minus ℎ)] 119911 isin [ℎ 1]

0 otherwise

(16)

(3) 119891119891119894is the pairwise interaction term that produces a

closer relation between individual 119894 and its most correlatedneighbor 119891

119894 which is used to guide the motion preference of

individual 119894 Specifically we employ an attractive force andvelocity feedback approach to generate 119891119891

119894as

119891119891

119894= minus1198961nabla119901119891119894119880119891119894minus 1198962(119902119894minus 119902119891119894) (17)

where 1198961 1198962gt 0 are the constant feedback gains and 119880

119891119894=

(12)(119901119894minus 119901119891119894)2 is the potential energy of its most correlated

neighbor This term guarantees the velocity convergence ofindividual 119894 to its most correlated neighbor 119891

119894and ultimately

induces the fission behavior

Remark 4 The specifically designed pairwise attractivere-pulsive artificial potential 120595

120572and adjacent matrix 119886

119894119895here are

to guarantee the smoothness of the potential function andLaplacian matrix which will facilitate the theoretical anal-ysis with the traditional Lyapunov-based stability analysismethod

Remark 5 Note that in our fission control algorithm (9) themost correlated neighbor of individual 119894 is chosen accordingto (7) and it is dynamically updating during the fissionprocess of the flock which realizes the implicit stimulusinformation transfer among members in the flock Thusthere is an essential difference between our work and theresearches of [14ndash16] as the latter assume the leader or targetof each member is predefined and fixed during the fissionprocess

Remark 6 From (7) we can also see that if 119888119894119895lt 119888lowast the most

correlated neighbor of individual 119894 does not exist In this casethe fission control algorithm (9) degrades into

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) + 119892119894119891119890

119894

(18)

which is equivalent to the first flocking control law proposedin [7] This situation occurs when the differences of motionpreference between individuals are relatively small and hencethe fission behavior does not occurTherefore algorithm (18)in [7] can be seen as a special case of oursHowever algorithm(18) is not able to achieve the spontaneous fission behaviordue to its average consensus property A detailed comparativesimulation and analysis is demonstrated in part 5

Here we define the sum of the total artificial potentialenergy and the total relative kinetic energy between allindividuals and the external stimuli as follows

119876 (119901 119902) =1

2

119873

sum

119894=1

(119880119894(119901) + (119902

119894minus 119902119891119894)T(119902119894minus 119902119891119894)) (19)

where

119880119894(119901) = 119881

119894(119901) + 119896

1(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

= sum

119895isinN119894(119905)

120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590)

+ 1198961(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

+ 119892119894(119901119894minus 119901119890

119894)T(119901119894minus 119901119890

119894)

(20)

Obviously 119876 is a positive semidefinite function We have thefollowing result

Theorem 7 Consider a flocking system consisting of 119873individuals with dynamics (1) Supposing the initial energy1198760(1198760= 119876(119901(0) 119902(0))) of the flock is finite and the initial

graph G is connected before the fission process when conflictexternal stimuli cause the rapid maneuvering of individualsthat lie on the edge of the flock under the fission control law(9) a cohesive flocking system will segregate into clusteredsubgroups due to the differences of information coupling degreebetween individuals Then the following statements hold

(i) The distance between individuals in the same subgroupis not larger than 120575 where 120575 = (119873

119894minus 1)radic2119876

01198961and

119873119894is the number of individuals in the subgroup

(ii) The velocities of individuals in the same subgroup willasymptotically converge to the same value

(iii) The distance between different subgroups is larger than119877 as time goes by

Proof We take a cluster of 119873119894individuals in the neighbor-

hood of individual 119894 into consideration and assume that graphG119873119894

is connected in the small sampling time interval from 119905

to 1199051(i) Let 119901119890

119894= 119901119894minus 119901119890

119894be the position difference vector

between individual 119894 and the external stimuli and let119901119894= 119901119894minus

119901119891119894 119902119894= 119902119894minus119902119891119894be the position difference vector and velocity

difference vector between individual 119894 and its most correlatedneighbor 119891

119894 respectively Then the following equations hold

119901119894119895minus 119901119891119894119895= 119901119894minus 119901119895= 119901119894119895

119902119894119895minus 119902119891119894119895= 119902119894minus 119902119895= 119902119894119895

(21)

The control input 119906119894can then be rewritten as

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895)

minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894

(22)

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

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DistributedSensor Networks

International Journal of

4 Journal of Robotics

Remark 3 It can be seen from (4)sim(6) that informationcoupling degree combines both the position and velocityinformation between individuals and their neighbors mean-while they also tend to have closer relation with the fastmaneuvering neighbors that are near to them which isconsistent with the property of real flocking system [19]

4 Self-Organized Fission Control Algorithm

Based on the above fission control principle we take themost correlated neighbor to form a pairwise interactionwith individual 119894 By integrating the motion informationof the most correlated neighbor into the coordinated lawtogether with the ldquoseparationalignmentrepulsionrdquo rules thedistributed fission control algorithm is formulated as

119906119894= 119891119901

119894+ 119891

V119894+ 119891119891

119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906in119894

+ 119892119894119891119890

119894⏟⏟⏟⏟⏟⏟⏟

119906out119894

(9)

where the environment force 119906out119894

is represented by the exter-nal stimulus term 119891

119890

119894 it causes the rapid movement change

of individuals and evokes the fission behavior and 119892119894denotes

whether individual 119894 can sense external stimuli Usually 119891119890119894

has diverse forms according to different environment stimulisuch as the direction to a known source or a segment of amigration route [11 32] Here for the convenience of analysiswe design the external stimulus term as the following simpleposition feedback form

119891119890

119894= minus (119901

119894minus 119901119890

119894) (10)

where119901119890119894is the desired position driven by external stimuli and

it is supposed to be a constant value at every sampling timeinterval

Obviously the internal interaction force 119906in119894consists of

three components(1) 119891119901119894is used to regulate the position between individual

119894 and its neighbors This term is responsible for collisionavoidance and cohesion in the flock that is when individualsare too far away from each other the attraction force willmake them move together when individuals are too closethe repulsion force will propel them away to avoid collisionIt is derived from the field produced by a collective functionwhich depends on the relative distance between individual 119894and its neighbors and is defined as

119891119901

119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590) (11)

where nabla is the gradient operator the 120590-norm sdot 120590of a vector

is a map R119899 rarr R+ defined by 119911120590= (1120598)[radic1 + 1205981199112 minus 1]

with a parameter 120598 gt 0 and 119911 = [1199111 1199112 119911

119899]Tisin R119899

Note that the map 119911120590is differentiable everywhere but

119911 = radic1199112

1+ 1199112

2+ sdot sdot sdot + 1199112

119899is not differentiable at 119911 = 0 This

property of 120590-norm is used for the construction of smoothcollective potential functions for individuals To construct asmooth pairwise potential with finite cutoff we follow the

0 5 10 15 20 250

10

20

30

40

50

60

120595120572

z

Figure 2 The pairwise attractiverepulsive potential function

work of [7] and integrate an action function 120601120572(119911) that varies

for all 119911 ge 119903120572 Define this action function as

120601120572 (119911) = 120588ℎ (

119911

119903120572

)120601 (119911 minus 119889120572)

120601 (119911) =1

2[(119886 + 119887) 1205901 (119911 + 119888) + (119886 minus 119887)]

(12)

where 1205901(119911) = 119911radic1 + 1199112 and 120601(119911) is an uneven sigmoidal

function with parameters that satisfy 0 lt 119886 le 119887119888 = |119886 minus 119887|radic4119886119887 to guarantee 120601(0) = 0 The pairwiseattractiverepulsive potential function 120595

120572(119911) is then defined

as

120595120572 (119911) = int

119911

119889120572

120601120572 (119904) 119889119904 (13)

and is depicted in Figure 2Additionally 119901

119894119895= 119901119894minus 119901119895is the relative position vector

between individuals 119894 and 119895 119901119891119894119895= 119901119891119894minus 119901119891119895is the relative

position vector between the most correlated neighbors ofindividuals 119894 and 119895

(2) 119891V119894is the velocity coordination term that regulates the

velocity of individuals

119891V119894= minus sum

119895isinN119894(119905)

119886119894119895(119902119894119895minus 119902119891119894119895) (14)

where 119902119894119895= 119902119894minus 119902119895is the relative velocity vector between

individuals 119894 and 119895 and 119902119891119894119895= 119902119891119894minus 119902119891119895is the relative velocity

vector between the most correlated neighbors of individuals 119894and 119895MoreoverA(119905) = [119886

119894119895(119905)] is the adjacencymatrixwhich

is defined as

119886119894119895 (119905) =

0 if 119895 = 119894

120588ℎ(

10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590

119903120590

) if 119895 = 119894(15)

Journal of Robotics 5

with the bump function 120588ℎ(119911) ℎ isin (0 1) being

120588ℎ (119911) =

1 119911 isin [0 ℎ)

1

2[1 + cos(120587119911 minus ℎ

1 minus ℎ)] 119911 isin [ℎ 1]

0 otherwise

(16)

(3) 119891119891119894is the pairwise interaction term that produces a

closer relation between individual 119894 and its most correlatedneighbor 119891

119894 which is used to guide the motion preference of

individual 119894 Specifically we employ an attractive force andvelocity feedback approach to generate 119891119891

119894as

119891119891

119894= minus1198961nabla119901119891119894119880119891119894minus 1198962(119902119894minus 119902119891119894) (17)

where 1198961 1198962gt 0 are the constant feedback gains and 119880

119891119894=

(12)(119901119894minus 119901119891119894)2 is the potential energy of its most correlated

neighbor This term guarantees the velocity convergence ofindividual 119894 to its most correlated neighbor 119891

119894and ultimately

induces the fission behavior

Remark 4 The specifically designed pairwise attractivere-pulsive artificial potential 120595

120572and adjacent matrix 119886

119894119895here are

to guarantee the smoothness of the potential function andLaplacian matrix which will facilitate the theoretical anal-ysis with the traditional Lyapunov-based stability analysismethod

Remark 5 Note that in our fission control algorithm (9) themost correlated neighbor of individual 119894 is chosen accordingto (7) and it is dynamically updating during the fissionprocess of the flock which realizes the implicit stimulusinformation transfer among members in the flock Thusthere is an essential difference between our work and theresearches of [14ndash16] as the latter assume the leader or targetof each member is predefined and fixed during the fissionprocess

Remark 6 From (7) we can also see that if 119888119894119895lt 119888lowast the most

correlated neighbor of individual 119894 does not exist In this casethe fission control algorithm (9) degrades into

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) + 119892119894119891119890

119894

(18)

which is equivalent to the first flocking control law proposedin [7] This situation occurs when the differences of motionpreference between individuals are relatively small and hencethe fission behavior does not occurTherefore algorithm (18)in [7] can be seen as a special case of oursHowever algorithm(18) is not able to achieve the spontaneous fission behaviordue to its average consensus property A detailed comparativesimulation and analysis is demonstrated in part 5

Here we define the sum of the total artificial potentialenergy and the total relative kinetic energy between allindividuals and the external stimuli as follows

119876 (119901 119902) =1

2

119873

sum

119894=1

(119880119894(119901) + (119902

119894minus 119902119891119894)T(119902119894minus 119902119891119894)) (19)

where

119880119894(119901) = 119881

119894(119901) + 119896

1(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

= sum

119895isinN119894(119905)

120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590)

+ 1198961(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

+ 119892119894(119901119894minus 119901119890

119894)T(119901119894minus 119901119890

119894)

(20)

Obviously 119876 is a positive semidefinite function We have thefollowing result

Theorem 7 Consider a flocking system consisting of 119873individuals with dynamics (1) Supposing the initial energy1198760(1198760= 119876(119901(0) 119902(0))) of the flock is finite and the initial

graph G is connected before the fission process when conflictexternal stimuli cause the rapid maneuvering of individualsthat lie on the edge of the flock under the fission control law(9) a cohesive flocking system will segregate into clusteredsubgroups due to the differences of information coupling degreebetween individuals Then the following statements hold

(i) The distance between individuals in the same subgroupis not larger than 120575 where 120575 = (119873

119894minus 1)radic2119876

01198961and

119873119894is the number of individuals in the subgroup

(ii) The velocities of individuals in the same subgroup willasymptotically converge to the same value

(iii) The distance between different subgroups is larger than119877 as time goes by

Proof We take a cluster of 119873119894individuals in the neighbor-

hood of individual 119894 into consideration and assume that graphG119873119894

is connected in the small sampling time interval from 119905

to 1199051(i) Let 119901119890

119894= 119901119894minus 119901119890

119894be the position difference vector

between individual 119894 and the external stimuli and let119901119894= 119901119894minus

119901119891119894 119902119894= 119902119894minus119902119891119894be the position difference vector and velocity

difference vector between individual 119894 and its most correlatedneighbor 119891

119894 respectively Then the following equations hold

119901119894119895minus 119901119891119894119895= 119901119894minus 119901119895= 119901119894119895

119902119894119895minus 119902119891119894119895= 119902119894minus 119902119895= 119902119894119895

(21)

The control input 119906119894can then be rewritten as

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895)

minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894

(22)

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

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International Journal of

Journal of Robotics 5

with the bump function 120588ℎ(119911) ℎ isin (0 1) being

120588ℎ (119911) =

1 119911 isin [0 ℎ)

1

2[1 + cos(120587119911 minus ℎ

1 minus ℎ)] 119911 isin [ℎ 1]

0 otherwise

(16)

(3) 119891119891119894is the pairwise interaction term that produces a

closer relation between individual 119894 and its most correlatedneighbor 119891

119894 which is used to guide the motion preference of

individual 119894 Specifically we employ an attractive force andvelocity feedback approach to generate 119891119891

119894as

119891119891

119894= minus1198961nabla119901119891119894119880119891119894minus 1198962(119902119894minus 119902119891119894) (17)

where 1198961 1198962gt 0 are the constant feedback gains and 119880

119891119894=

(12)(119901119894minus 119901119891119894)2 is the potential energy of its most correlated

neighbor This term guarantees the velocity convergence ofindividual 119894 to its most correlated neighbor 119891

119894and ultimately

induces the fission behavior

Remark 4 The specifically designed pairwise attractivere-pulsive artificial potential 120595

120572and adjacent matrix 119886

119894119895here are

to guarantee the smoothness of the potential function andLaplacian matrix which will facilitate the theoretical anal-ysis with the traditional Lyapunov-based stability analysismethod

Remark 5 Note that in our fission control algorithm (9) themost correlated neighbor of individual 119894 is chosen accordingto (7) and it is dynamically updating during the fissionprocess of the flock which realizes the implicit stimulusinformation transfer among members in the flock Thusthere is an essential difference between our work and theresearches of [14ndash16] as the latter assume the leader or targetof each member is predefined and fixed during the fissionprocess

Remark 6 From (7) we can also see that if 119888119894119895lt 119888lowast the most

correlated neighbor of individual 119894 does not exist In this casethe fission control algorithm (9) degrades into

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) + 119892119894119891119890

119894

(18)

which is equivalent to the first flocking control law proposedin [7] This situation occurs when the differences of motionpreference between individuals are relatively small and hencethe fission behavior does not occurTherefore algorithm (18)in [7] can be seen as a special case of oursHowever algorithm(18) is not able to achieve the spontaneous fission behaviordue to its average consensus property A detailed comparativesimulation and analysis is demonstrated in part 5

Here we define the sum of the total artificial potentialenergy and the total relative kinetic energy between allindividuals and the external stimuli as follows

119876 (119901 119902) =1

2

119873

sum

119894=1

(119880119894(119901) + (119902

119894minus 119902119891119894)T(119902119894minus 119902119891119894)) (19)

where

119880119894(119901) = 119881

119894(119901) + 119896

1(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

= sum

119895isinN119894(119905)

120595119894119895(100381710038171003817100381710038171003817119901119894119895minus 119901119891119894119895

100381710038171003817100381710038171003817120590)

+ 1198961(119901119894minus 119901119891119894)T(119901119894minus 119901119891119894)

+ 119892119894(119901119894minus 119901119890

119894)T(119901119894minus 119901119890

119894)

(20)

Obviously 119876 is a positive semidefinite function We have thefollowing result

Theorem 7 Consider a flocking system consisting of 119873individuals with dynamics (1) Supposing the initial energy1198760(1198760= 119876(119901(0) 119902(0))) of the flock is finite and the initial

graph G is connected before the fission process when conflictexternal stimuli cause the rapid maneuvering of individualsthat lie on the edge of the flock under the fission control law(9) a cohesive flocking system will segregate into clusteredsubgroups due to the differences of information coupling degreebetween individuals Then the following statements hold

(i) The distance between individuals in the same subgroupis not larger than 120575 where 120575 = (119873

119894minus 1)radic2119876

01198961and

119873119894is the number of individuals in the subgroup

(ii) The velocities of individuals in the same subgroup willasymptotically converge to the same value

(iii) The distance between different subgroups is larger than119877 as time goes by

Proof We take a cluster of 119873119894individuals in the neighbor-

hood of individual 119894 into consideration and assume that graphG119873119894

is connected in the small sampling time interval from 119905

to 1199051(i) Let 119901119890

119894= 119901119894minus 119901119890

119894be the position difference vector

between individual 119894 and the external stimuli and let119901119894= 119901119894minus

119901119891119894 119902119894= 119902119894minus119902119891119894be the position difference vector and velocity

difference vector between individual 119894 and its most correlatedneighbor 119891

119894 respectively Then the following equations hold

119901119894119895minus 119901119891119894119895= 119901119894minus 119901119895= 119901119894119895

119902119894119895minus 119902119891119894119895= 119902119894minus 119902119895= 119902119894119895

(21)

The control input 119906119894can then be rewritten as

119906119894= minus sum

119895isinN119894(119905)

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895)

minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894

(22)

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Robotics

Substituting (21) into (19) yields

119876 =1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590) + 119902

T119894119902119894

+ 1198961119901T119894119901119894+ 119892119894(119901119890

119894)T119901119890

119894)

(23)

Due to the symmetry of the artificial potential function120595119894119895and adjacency matrixA we have

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894119895

=

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119894

= minus

120597120595119894119895(10038171003817100381710038171003817119901119894119895

10038171003817100381710038171003817120590)

120597119901119895

(24)

Taking the derivative of 119876 we can obtain

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

+ 21198961119902T119894119901119894+ 2119902

T119894119906119894+ 2119892119894(119901119890

119894)

T119901119890

119894)

=1

2

119873119894

sum

119894=1

( sum

119895isinN119894(119905)

119901T119894nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119901T119895nabla119901119895120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590) + 2119896

1119902T119894119901119894

+ 2119902T119894(minus sum

119895isinN119894

nabla119901119894120595119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817120590)

minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198961119901119894minus 1198962119902119894minus 119892119894119901119890

119894)

+ 2119892119894119902T119894119901119890

119894)

=

119873119894

sum

119894=1

119902T119894(minus sum

119895isinN119894(119905)

119886119894119895(119902119894minus 119902119895) minus 1198962119902119894)

= minus119902T[(L (119901) + 119896

2119868119873119894) otimes 119868119899] 119902

(25)

where L(119901) is the Laplacian matrix of graph G(119901) 119901 =

col(1199011 1199012 119901

119873119894) isin R119873119894times119899 119902 = col(119902

1 1199022 119902

119873119894) isin R119873119894times119899

and 119868119873119894

is the119873119894dimensional identity matrix

As L(119901) is positive semidefinite lt 0 hence 119876 isa nonincreasing function As the initial energy 119876

0of the

subgroup is finite for any sampling time interval 119905 sim 1199051 119876 lt

1198760 Form (23) we have 119896

1119901T119894119901119894le 21198760 Accordingly the dis-

tance between individual 119894 and its most correlated neighboris not greater than radic2119876

01198961 Due to the connectivity of the

subgroup there exists a joint connected path from individual

119894 to119873119894in the small time internal 119905 sim 119905

1 therefore the distance

between any two individuals is less than 120575 = (119873119894minus1)radic2119876

01198961

in the same subgroup(ii) From above 119876 lt 119876

0 the set 119901 119902 is closed and

bounded Therefore the set

Ω = [119901T 119902

T] isin R2119873119894times119899 | 119876 le 119876

0 (26)

is a compact invariant set According to LaSallersquos invariantset principle all the trajectories of individuals that start fromΩ will converge to the largest invariant set inside the regionΩ = [119901

T 119902

T] isin R2119873119894times119899 | = 0 Therefore the velocity

of individual 119894 will converge to that of its most correlatedneighbor 119891

119894 that is

119902119894= 119902119891119894 (27)

Without loss of generality we assume the119873119894th individual

in the subgroup is the only one that senses the externalstimulus and it responses with fast maneuvering Accordingto (4) 119873

119894is the most correlated neighbor with largest

information coupling degree In the small time internal 119905 sim 1199051

there exists a joint connected path from individual 119894 to119873119894 the

velocities of all the individuals in the subgroup will convergeto that of the119873

119894th individual asymptotically Hence we have

1199021= 1199022= sdot sdot sdot = 119902

119873119894minus1= 119902119873119894 (28)

where 119902119873119894

is the velocity of individual 119873119894that senses the

external stimuli(iii) For individuals 119894 and 119895 in different subgroups as has

been illustrated in (i) and (ii) their motion is substantiallydetermined by their most correlated neighbors Therefore ifthe most correlated neighbors of individuals 119894 and 119895 movein opposite directions under different external stimuli themotion of the two individuals will diverge as time goes byHence we will ultimately have

lim119905rarrinfin

119863 gt 119877 (29)

where 119863 = 119901119894minus 119901119895 is the relative distance between

individuals 119894 and 119895From the above proof we can conclude that under the

fission control algorithm (9) the subgroups are graduallymoving out of the sensing range of each other Meanwhilethe subgroup itself will keep a cohesive whole and move information separately

5 Simulation Study

To verify the effectiveness of the proposed fission controlalgorithm 40 individuals are chosen to perform the simula-tion in two-dimensional plane

51 Simulation Setup Suppose the 40 individuals lie ran-domly within the region of 20 times 20m2 and their initialdistribution is connected The initial velocities are set witharbitrary directions and magnitudes within the range of[0 10]ms The sensing range of each individual is con-strained to 119877 = 5m The other simulation parameters are

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 7

10 20 30 40 50 60

0

5

10

15

20

25

30

minus5

minus10

y(m

)

x (m)

(a) 119905 = 6 s

10 20 30 40 50 60 70

0

10

20

30

minus10

y(m

)

x (m)

(b) 119905 = 9 s

10 20 30 40 50 60 70 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(c) 119905 = 12 s

20 40 60 80

0

10

20

30

40

minus20

minus10

y(m

)

x (m)

(d) 119905 = 15 s

Figure 3 Snapshots of the trajectories of individuals during the fission process where ldquo∘rdquo represents the initial position of individuals andldquo∙rdquo denotes the final position of each individual Specifically ldquored dotrdquo are the two individuals that sense external stimuli with ldquored arrowrdquobeing their desired directions

chosen as 119903119894119895= 05 120583

119894119895= 5 119888lowast = 03 Δ119905 = 0005 s 120598 = 02

and 1198961= 1198962= 1

In particular we consider the case where a flock seg-regates into two clustered subgroups by introducing twoconflict external stimuli towards different directions Beforethe fission behavior takes place we first let individualsaggregate and move in formation with the same velocity of[5 0]

T ms Suppose that at 119905 = 7 s two individuals (eglie on the edge of the flock we let them be individual 1 andindividual 2) sense the two external stimuli separately othermembers are not able to sense the external stimuli and wehave 119892

1= 1198922= 1 119892

119894= 0 119894 = 1 2 Meanwhile the

desired positions of individual 1 and individual 2 are updatingaccording to (1) with speeds 119902119890

1= [5 3]

T ms and 1199021198902=

[5 minus 5]T ms respectively

52 Simulation Results Steered by the fission control algo-rithm (9) a self-organized fission behavior will occur and thesimulation results are shown in Figures 3sim6

Figure 3(a) is the snapshot of the flocking system at 119905 =6 s from which it can be seen that individuals aggregatefrom random distribution and form a cohesive formationFigure 3(b) shows the response of the flock when individual 1and individual 2 (denoted by red solid dots) sense the externalstimuli individuals in the flock tend to change their move-ments to different directions due to the pairwise interactionrule In Figure 3(c) the flock segregates into subgroups andtwo clusters emerge In Figure 3(d) the subgroups run out ofthe sensing range of each other and the segregated subgroupsmove in formation separately

Figures 4(a) and 4(b) give the plot of the velocities ofindividuals during the fission process in both 119909- and 119910-axis from which we can see that before two external stimuliappear (119905 lt 7 s) individuals move in formation and theirvelocities asymptotically converge to [5 0]T ms at 119905 = 7 sunder the fission control algorithm fission behavior occursand the velocities of the two subgroups tend to that of theindividuals who sense the external stimuli Consequently

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Robotics

0 5 10 150

10

20

30

Time (s)

x(m

middotsminus1)

(a) Velocity of 119909-axis

0 5 10 15minus20

minus10

0

10

20

Time (s)

y(m

middotsminus1)

(b) Velocity of 119910-axis

Figure 4 Velocity of individuals during the fission process

0 5 10 150

5

10

15

20

25

30

35

Average distance of all individualsAverage distance of individuals in sub-group 1 Average distance of individuals in sub-group 2

dav

g(m

)

65 7 75 8 85 9

12

10

8

6

4

2

t (s)

Figure 5 Average distance of the whole group and two separatedsubgroups

the velocities of individuals in subgroup 1 converge to[5 3]

T ms while the velocities of individuals in subgroup 2converge to [5 minus 5]

T msIn addition we utilize the average distance of individuals

to illustrate the fission behavior in amore intuitional wayTheaverage distance 119889avg is represented by

119889avg =1

119873119894

119873119894

sum

119894=1

119873119894

sum

119895=1

10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817 (30)

where119873119894is the number of individuals in the flock

Figure 5 shows the average distance between individualsduring the simulation where blue line denotes the averagedistance of all the individuals while red dash line and blackdash dot line represent the average distance of individualsin subgroup 1 and subgroup 2 respectively We can clearlysee that the three lines tend to converge to a fixed valuebefore segregation (119905 lt 7 s) When the fission behavior

0 5 10 150

01

02

03

04

05

06

07

08

ICD1

t (s)

Figure 6 Information coupling degree between individual 1 and itsneighbors

occurs the average distance of all the individuals increaseswith time denoting the divergence of the two subgroups Onthe contrary the average distances of the two subgroups tendtomaintain a constant value (about 3m)which demonstratesthat the two subgroups are moving in stable formation

Figure 6 gives the information coupling degree betweenindividual 1 and its neighbors during the fission processwhich clearly demonstrates that before the external stimuliappear (119905 lt 7 s) the ICD between individual 1 and itsneighbors converge to a constant value This is because inthe steady formation state the relative position and velocitybetween individual 1 and its neighbors remain stable Withthe introduction of the external stimuli the ICD betweenindividual 1 and its neighbors begins to vary due to their rapidmovement variation and the ICD between individual 1 andits neighbors in the same subgroup converge to a steady statewhile ICD of individual 1 and other individuals in the othersubgroup quickly decreases to 0 for the loss of connectionbetween each other

From the above simulation results (Figures 3sim6) wecan conclude that the proposed fission control algorithm iscapable of realizing the self-organized fission behavior whenthe introduction of the external stimuli causes the motion

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Robotics 9

0

10

20

30

40

50

60

70

80

90

0 02 04 06 08 1

S

Figure 7 Histogram statistics for the size ratio of the separatedsubgroups

preferences conflict in the flock Particularly our approachis neither negotiation nor appointment based but mimicsthe fission behavior of animal flocks which is also morefeasible robust and efficient in engineering application thanthe centralized approach

53 Further Discussion To better understand the size dis-tribution of subgroups we define the size ratio 119878 as aspecification to evaluate the fission behavior

119878 =119873min119873max

(31)

where 119873min and 119873max are the size of the smallest subgroupand biggest subgroup respectively Obviously 119878 isin [0 1]the bigger 119878 is the smaller size difference between the twosubgroups is Particularly when 119878 = 1 the numbers ofindividuals in the two subgroups is the same

Figure 7 is the histogram of the size ratio of 40 individualsfor 100 times of simulation fromwhich we can see that about80of the results show the size ratio equal to 1 and nearly 10are close to 1 and only a very small portion of the size ratio israndomly located from 0 to 1 Therefore it can be concludedthat from a probabilistic perspective the algorithmproposedin this paper can realize the equal-sized fission control

Finally we carry out a comparative simulation usingthe first algorithm (18) proposed in [7] based on ldquosepara-tionalignmentrepulsionrdquo rules At 119905 = 7 s we introduce twoconflict external stimuli on individual 1 and individual 2 andthe desired positions of individual 1 and individual 2 are alsoupdating according to (1) with speeds 119902119890

1= [5 3]

T ms and119902119890

2= [5 minus 5]

T ms respectively Then we have the followingthe algorithm

119906119894= sum

119895isinN119894

120595120572(10038171003817100381710038171003817119901119895minus 119901119894

10038171003817100381710038171003817120590)n119894119895+ sum

119895isinN119894

119886119894119895(119901) (119902

119895minus 119902119894)

+ 119892119894(119901119894minus 119901119890

119894) (119892

1= 1198922= 1)

(32)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

35

40 60 806

8

10

12

14

minus5

minus10

y(m

)

x (m)

Figure 8 Trajectories of individuals under the external stimuli withalgorithm (32)

With the same simulation parameters the moving trajec-tories of individuals are shown in Figure 8

From Figure 8 we can see that only two individuals thatsense the external stimuli can split from the group whilethe rest move in a cohesive formation along its previoustrajectory It can also be seen from the amplified figure thatwhen individual 1 and individual 2 split from the flock themotion of other individuals tends to fluctuate but they finallyreturn to the cohesive formation state The main reason forthe above result lies in the ldquoaverage consensusrdquo approachadopted in (32) which counteracts the external stimuli andleads the flock move towards the average direction of thestimulus signal

Therefore the traditional ldquoseparationalignmentcohe-sionrdquo based coordination rules decrease the flexibility andmaneuverability of flocking system especially in the case ofdangerobstacle avoidance or multiple objects tracking As amatter of fact under algorithm (32) fission behavior onlyoccurs when all the individuals can directly sense externalstimuli which is essentially different from our work

6 Conclusion

This paper addresses the fission control problem of flock-ing system To overcome the shortcomings of the ldquoaverageconsensusrdquo based interaction rules that encumber the fis-sion behavior a new pairwise interaction rule is proposedto implement the fission behavior Firstly we propose aninformation coupling degree based approach to describe theinternal interaction intensity between individuals Then bychoosing the neighbors with largest information couplingdegree as the most correlated neighbor we integrate themotion information of the most correlated neighbor intothe distributed fission control algorithm to form a strongcorrelated pairwise interaction which realizes the directionalinformation flow of external stimuli and induces the fissionbehavior of the flock in a self-organized fashion More-over theoretical analysis proves that a flock will segregate

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Robotics

into clustered subgroups when external stimuli cause themotion information conflict in it Finally simulation studiesdemonstrate the effectiveness of the proposed fission controlalgorithm

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China under Grants 51179156 and51379179 The authors are also grateful for the constructivesuggestions from anonymous reviewers that significantlyenhance the presentation of this paper

References

[1] I D Couzin and M E Laidre ldquoFission-fusion populationsrdquoCurrent Biology vol 19 no 15 pp R633ndashR635 2009

[2] I L Bajec and FHHeppner ldquoOrganized flight in birdsrdquoAnimalBehaviour vol 78 no 4 pp 777ndash789 2009

[3] H M La and W Sheng ldquoDynamic target tracking and observ-ing in a mobile sensor networkrdquo Robotics and AutonomousSystems vol 60 no 7 pp 996ndash1009 2012

[4] T Vicsek and A Zafeiris ldquoCollective motionrdquo Physics Reportsvol 517 no 3 pp 71ndash140 2012

[5] R Lukeman Y-X Li and L Edelstein-Keshet ldquoInferringindividual rules from collective behaviorrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 no 28 pp 12576ndash12580 2010

[6] V Gazi and K M Passino ldquoA class of attractionsrepulsionfunctions for stable swarm aggregationsrdquo International Journalof Control vol 77 no 18 pp 1567ndash1579 2004

[7] R Olfati-Saber ldquoFlocking for multi-agent dynamic systemsalgorithms and theoryrdquo IEEE Transactions on Automatic Con-trol vol 51 no 3 pp 401ndash420 2006

[8] C W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo ACM SIGGRAPH Computer Graphics vol21 no 4 pp 25ndash34 1987

[9] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[10] I Couzin ldquoCollective mindsrdquo Nature vol 445 no 7129 p 7152007

[11] I D Couzin J Krause N R Franks and S A Levin ldquoEffectiveleadership and decision-making in animal groups on themoverdquoNature vol 433 no 7025 pp 513ndash516 2005

[12] L Conradt and T J Roper ldquoConsensus decision making inanimalsrdquo Trends in Ecology and Evolution vol 20 no 8 pp449ndash456 2005

[13] X Lei M Liu W Li and P Yang ldquoDistributed motion controlalgorithm for fission behavior of flocksrdquo in Proceedings of theIEEE International Conference on Robotics and Biomimetics(ROBIO rsquo12) pp 1621ndash1626 IEEE December 2012

[14] H Su X Wang andW Yang ldquoFlocking in multi-agent systemswith multiple virtual leadersrdquo Asian Journal of Control vol 10no 2 pp 238ndash245 2008

[15] G Lee N Y Chong and H Christensen ldquoTracking multiplemoving targetswith swarms ofmobile robotsrdquo Intelligent ServiceRobotics vol 3 no 2 pp 61ndash72 2010

[16] X Luo S Li and X Guan ldquoFlocking algorithm with multi-target tracking for multi-agent systemsrdquo Pattern RecognitionLetters vol 31 no 9 pp 800ndash805 2010

[17] M Kumar D P Garg and V Kumar ldquoSegregation of heteroge-neous units in a swarm of robotic agentsrdquo IEEE Transactions onAutomatic Control vol 55 no 3 pp 743ndash748 2010

[18] Z Chen H Liao and T Chu ldquoAggregation and splitting inself-driven swarmsrdquo Physica A Statistical Mechanics and ItsApplications vol 391 no 15 pp 3988ndash3994 2012

[19] J Li and A H Sayed ldquoModeling bee swarming behaviorthrough diffusion adaptation with asymmetric informationsharingrdquo Eurasip Journal on Advances in Signal Processing vol2012 no 1 article 18 2012

[20] C D Godsil G Royle and C Godsil Algebraic Graph Theoryvol 207 Springer New York NY USA 2001

[21] D J T Sumpter ldquoThe principles of collective animal behaviourrdquoPhilosophical Transactions of the Royal Society B BiologicalSciences vol 361 no 1465 pp 5ndash22 2006

[22] S-H Lee ldquoPredatorrsquos attack-induced phase-like transition inprey flockrdquoPhysics Letters A vol 357 no 4-5 pp 270ndash274 2006

[23] B L Partridge ldquoThe structure and function of fish schoolsrdquoScientific American vol 246 no 6 pp 114ndash123 1982

[24] L Conradt and T J Roper ldquoConflicts of interest and theevolution of decision sharingrdquo Philosophical Transactions of theRoyal Society B Biological Sciences vol 364 no 1518 pp 807ndash819 2009

[25] L Conradt J Krause I D Couzin and T J Roper ldquolsquoLeadingaccording to needrsquo in self-organizing groupsrdquo American Natu-ralist vol 173 no 3 pp 304ndash312 2009

[26] H-X Yang T Zhou and L Huang ldquoPromoting collectivemotion of self-propelled agents by distance-based influencerdquoPhysical Review E vol 89 no 3 Article ID 032813 2014

[27] H-T Zhang Z Chen T Vicsek et al ldquoRoute-dependent switchbetween hierarchical and egalitarian strategies in pigeon flocksrdquoScientific Reports vol 4 article 5805 7 pages 2014

[28] J Gao Z Chen Y Cai and X Xu ldquoEnhancing the convergenceefficiency of a self-propelled agent system via a weightedmodelrdquo Physical Review E vol 81 no 4 Article ID 041918 2010

[29] M Nagy Z Akos D Biro and T Vicsek ldquoHierarchical groupdynamics in pigeon flocksrdquoNature vol 464 no 7290 pp 890ndash893 2010

[30] D Sumpter J Buhl D Biro and I Couzin ldquoInformationtransfer in moving animal groupsrdquo Theory in Biosciences vol127 no 2 pp 177ndash186 2008

[31] S-Y Tu and A H Sayed ldquoEffective information flow overmobile adaptive networksrdquo in Proceedings of the 3rd Interna-tional Workshop on Cognitive Information Processing (CIP 12)pp 1ndash6 Bayona Spain May 2012

[32] V Gazi and K M Passino ldquoStability analysis of social foragingswarmsrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 1 pp 539ndash557 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of