Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 368369 9 pageshttpdxdoiorg1011552013368369

Research ArticleModeling and Flocking Consensus Analysis forLarge-Scale UAV Swarms

Li Bing Li Jie and Huang KeWei

School of Mechatronical Engineering Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Huang KeWei binlin lee163com

Received 31 July 2013 Accepted 7 October 2013

Academic Editor J A Tenreiro Machado

Copyright copy 2013 Li Bing et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently distributed coordination control of the unmanned aerial vehicle (UAV) swarms has been a particularly active topic inintelligent system field In this paper through understanding the emergent mechanism of the complex system further researchon the flocking and the dynamic characteristic of UAV swarms will be given Firstly this paper analyzes the current researchesand existent problems of UAV swarms Afterwards by the theory of stochastic process and supplemented variables a differential-integral model is established converting the system model into Volterra integral equation The existence and uniqueness of thesolution of the system are discussed Then the flocking control law is given based on artificial potential with system consensus Atlast we analyze the stability of the proposed flocking control algorithm based on the Lyapunov approach and prove that the systemin a limited time can converge to the consensus direction of the velocity Simulation results are provided to verify the conclusion

1 Introduction

UAV is an advanced system with high autonomy for intelli-gent combat [1] In the future UAVs will be used for complextasks such as surveillance reconnaissance and precisionstrike missions Many organizations have foreseen that in thenear future swarms of UAVswill replace single ones formorecomplicated missions in more uncertain and possibly hostileenvironments [2] Therefore many researchers are studyinggroups of cooperative UAVs

(A) Related Work on the UAV Swarms Problem The newchallenges imposed by UAV swarms have attracted manyresearchers New control mechanisms application domainssimulationmodels and simulation tools have been developedto tackle issues in different aspects of the swarm Currentlya completely new topic is opening up in the area of UAVswarms performing different missions cooperatively [3] themethod of evolutionary pinning control is applied to UAVswarms successfully Path planning and routing are investi-gated in [4 5] using multiobjective evolutionary algorithmThe path planning problem in three-dimensional environ-ment without any obstacles is addressed in [6 7] and withonly static obstacles in [8] [9] cooperative searching problem

is discussed for the purpose of detecting moving and evadingtargets in a hazardous environment A similar cooperativesearching problem is also discussed in [10 11] Reference [12]investigates the automatic target recognition (ATR) problemin UAV control and proposes a distributed strategy for UAVswarms Task allocation problem is discussed in [13ndash18] usingdifferent methodologies Some applications of using a UAVswarm to search and destroy targets could be found in [19ndash22] [23] a swarm simulator for target searching is imple-mented with Java Garcia introduces a multi-UAVs simulatorimplemented with X-Planemdasha commercial flight simulator[24] Russell et al present a parallel swarm simulationenvironment which utilizes an existing parallel emulationand simulation tool called SPEEDS [25] MASON [26] is ageneral purposemultiagents simulation library utilized in ourprevious work along with MATLAB based UAV simulator[27]

(B) Related Work on the Consensus Flocking Problem Mostresearch on formation of agent swarms uses distributed tech-niques by Reynoldsrsquo seminal work on the mobility of flocks[28] which prescribes three fundamental operations for eachrobot to realize distributed flockingmdashseparation alignmentand cohesion [29 30] One of the earliest attempts to realize

2 Mathematical Problems in Engineering

flocking through a set of basic behaviors including safewandering aggregation dispersion and homing to imple-ment flocking is by Mataric [31] Kelley and Keating realizeflocking with robots based on leader-following behavior[32] Hayes and Dormiani-Tabatabaei [33] propose a flockingalgorithm based on two behaviors collision avoidance andvelocity-matching flock centering Holland et al [34] pro-pose a flocking algorithm for UAV similar to Reynoldsrsquo Ahost is used as an intermediate station for receiving eachUAVrsquos range bearing and velocity and sending them toother UAVs to simulate the sensing process of one UAVfor perceiving range bearing and heading of its neigh-bors Ferrante et al [35] introduce a new communicationstrategy called the information aware communication foralignment behavior Recently Stranieri et al [36] performflocking with a swarm of behaviorally heterogeneous mobilerobots

In this paper we consider models for flocking swarmsFirstly a mathematical model of cooperative system isestablished by using Markov stochastic process and calculusanalysis Then the control law for UAV swarm is establishedbased on artificial potential field At last we analyze thestability of the proposed flocking control algorithm basedon the Lyapunov approach and prove the conclusion thatthe system in a limited time can converge to the consensusdirection of the velocity Simulation results are provided toverify the conclusion

2 The Model of the UAVs Swarms

21 Differential Integral Model Let 119862(119905) denote the state ofthe UAV swarms at time 119905 119862(119905) = 0 identifies the statethat UAV swarms are stable at time 119905 The state of UAV 119894 attime 119905 is denoted by 119888

119894(119905) = (119901

119894(119905) 119900119894(119905)) in which the first

element 119901119894(119905) = (119909

119894(119905) 119910119894(119905) 119911119894(119905)) is the UAVrsquos position in

the environment at time 119905 and the second element 119900119894(119905) is

the UAVrsquos orientation The UAVrsquos dynamics is subject to itsphysical curvature radius constraints leading to the fact thatit can only change its orientation by at most one step whichis described as go straight go up go down turn left turnupper left turn lower left turn right turn upper right andturn lower right

In order to obtain Markov random process the new stateof process is derived by supplement of variable [37 38] whichis described as follows

119875119894 (119905) = 119875 (119862 (119905) = 119894) 119894 = 0 1 2 119873

119875119895119896 (119909 119905) = 119875 (119862 (119905) = 119895 119909 lt 119910

119895119896 (119905) lt 119909 + 119889119909)

119895 = 119873 + 1 119872 119896 = 0 1 2 119873

(1)

where 119910119894(119905) is the dwell time after state 119894 So it is easy to verify

that 119880(119905) 0 le 119905 le 1198790 = (119862(119905) 119910(119905)) is a broad Markov

random process

The probability of state transition afterΔ119905 can be obtainedusing total probability theorem

1198750 (119905 + Δ119905) = (

119872

sum

119894=0

119875 (119862 (119905) = 119894 119862 (119905 + Δ119905) = 119894))

= 1198750 (119905) (1 minus

119873

sum

119894=1

1205820119894Δ119905 + 119900 (Δ119905)) +

119873

sum

119894=1

1205821198940119875119896 (119905) Δ119905

+

119872

sum

119895=119873+1

int

119879

0

119875119895 (119909 119905) 119903119895 (119909) Δ119905 119889119909 + 119900 (Δ119905)

(2)

According to (2) we can get the all probability

119875119896 (119905 + Δ119905) = 119875

119896 (119905) (1 minus

119873

sum

119894=1119894 = 119896

120582119896119894Δ119905 + 119900 (Δ119905))

+

119873

sum

119894=1119894 = 119896

120582119894119896119875119896 (119905) Δ119905 + 119900 (Δ119905)

119896 = 1 2 119873

119875119895 (119909 119905) = 119875 (119862 (119905) = 119895

the time of self-organised 119862system is 119909

119862 (119905 + Δ119905) = 119895)

= 119875119895 (119909 119905) (1 minus 119903

119895 (119909) Δ119905 + 119900 (Δ119905))

119895 = 119873 + 1 119872

(3)

where 120582119894119895is the average sustained rate of each state and 119903

119895(119909)

is the average repair rate at state 119895 Similarly the expression ofstate transition rate for 119875

119895(119909 + Δ119909 119905 + Δ119905) can be derivated

Differentiate the expression for state transition probabil-ity to derive its limit Then the mathematical model can bedescribed using integral-differential equations as follows

1198891198750 (119905)

119889119905+

119873

sum

119894=0119894 = 119896

12058201198961198750 (119905)

=

119873

sum

119894=0119894 = 119896

1205821198940119875119894 (119905)

+

119872

sum

119895=119873+1

int

119879

0

119903119895 (119909) 119901119895 (119909 119905) 119889119909

119889119875119896 (119905)

119889119905+

119873

sum

119894=0119894 = 119896

120582119896119894119875119896 (119905)

=

119873

sum

119894=0119894 = 119896

120582119894119896119875119894 (119905)

119896 = 1 2 119873

Mathematical Problems in Engineering 3

120597119901119895 (119909 119905)

120597119909+120597119901119895 (119909 119905)

120597119905+ 119903119895 (119909) 119901119895 (119909 119905) = 0

119895 = 119873 + 1 119872

(4)

The boundary and initial conditions are

1198750 (0) = 1 119875

1 (0) = 0 119875119894 (0) = 0 119875

119873 (0) = 0

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895119875119894 (119905) 119895 = 119873 + 1 119872

(5)

Theorem 1 The reliability of coordination system has unique-ness and nonnegative solution on 119862[0 119879]

Proof According to the initial conditions we can get theanalytic solution of the partial differential equation [39 40]

Set

1198860=

119873

sum

119894=1

1205820119894 119886

119895=

119873

sum

119894=1

120582119895119894 119886

119873=

119873

sum

119894=1

120582119873119894

119875119895 (119909 119905) = 119875

119894 (0 119905 minus 119909) exp(minusint119909

0

119903119894(120583) 119889120583)

119894 = 119873 + 1 119872

1198891198750 (119905)

119889119905= minus 119886

01198750 (119905) +

119873

sum

119894=0119894 = 119896

1205821198940119875119894 (119905)

+

119872

sum

119895=119873+1

int

119879

0

119875119895 (0 119905 minus 119909) exp(minusint

119909

0

119903119894(120583) 119889120583)

times 119903119894 (119909) 119889119909

1198750 (119905) = exp (minus119886

0119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) 119889119904

+ int

1198790

0

[

119872

sum

119894=119873+1

int

119904

0

119875119894 (0 120591) exp(minusint

119904minus120591

0

119903119894(120583) 119889120583)

times 119903119894 (119904 minus 120591) 119889120591]

times exp (minus1198860 (119905 minus 119904)) 119889119904

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) times 119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591) 119889120591

times int

1198790minus120591

0

exp(minus1198860(1198790minus 120591) + 119886

0V

minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) 1198960119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591)119870119894 (119905 minus 120591) 119889120591

(6)

where

1198960= exp (minus119886

0 (119879 minus 119904))

119896119894 (119879 minus 120591) = int

119879minus120591

0

1198960lowast exp(119886

0V minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

119894 = 119873 + 1 119872

119875119895 (119905) =

119873

sum

119894=0119894 = 119895

int

119879

0

exp (minus119886119895 (119879 minus 119904)) 120582119894119895119875119894 (119904) 119889119904

=

119873

sum

119894=0119894 = 119895

int

119879

0

119896119895120582119894119895119875119894 (119904) 119889119904 119895 = 1 2 119873

(7)

where 119896119895= exp(minus119886

119895(119879 minus 119904)) 119895 = 1 2 119873

So we can get the following equation

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895

119873

sum

119897=0119897 = 119895

int

119879

0

119896119895120582119897119895119875119897 (119904) 119889119904

= int

119879

0

119896119895(

119873

sum

119894=0

120582119894119895)(

119873

sum

119897=0119897 = 119895

120582119897119895119875119897 (119904))119889119904

119895 = 119873 + 1 119872

(8)

Assuming

119875 (119905) = (1198750 (119905) 1198751 (119905) 119875119873 (119905) 119875119873+1 (0 119905) 119875119872 (0 119905))

119891 (119905) = (1198910 (119905) 1198911 (119905) 119891119873 (119905) 119891119873+1 (0 119905) 119891119872 (0 119905))

= (exp (minus1198860119905) 0 0 0)

(9)

4 Mathematical Problems in Engineering

then the solution of the system can be converted into vectorsformat as follows

119875 (119905) = 119891 (119905) + int

119879

0

119896 (119879 minus 119904) 119875 (119904) 119889119904

119896 (119879 minus 120591) =

(((((((((((((((

(

0 120582101198960

sdot sdot sdot 12058211987301198960

119896119873+1 (119879 minus 119904) sdot sdot sdot 119896

119872 (119879 minus 119904)

120582011198961

0 sdot sdot sdot 12058211987311198961

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

1205820119873119896119873

1205822119873119896119873

sdot sdot sdot 0 0 sdot sdot sdot 0

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119873+1

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119873+1

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119873+1

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119872

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119872

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119872

0 sdot sdot sdot 0

)))))))))))))))

)

(10)

Any component of119865(119905) and119866(119905minusℎ) vector is nonnegativeThe functions119865

119895(119905) and119866

119895(119905minusℎ) 119895 = 1 2 119872 are limitary

on the domain 0 lt 119879 lt +infinThe solution of integral equationis unique and nonnegative on 119862[0 119879] So the reliability ofcoordination system has unique and nonnegative solution on119862[0 119879]

22 Probabilistic Analysis Based on State TransformationThe behavior evolution of the UAV swarm system is alimitedMarkov decision process Suppose that the probabilitydistribution of the system state is 119875(119873 119905) at time 119905 Then attime 119905+120591 the probability distribution is119875(119873 119905+120591) Accordingto the relationship of the probability density at different timethe marginal probability density 119875(119873 119905 + 120591) is 119875(119873 119905 + 120591) =

int119875(120578 119905)119875(119873 119905 + 120591 | 120578 119905)119889120578And the time derivative of the 119875(119873 119905 + 120591) is

119889119875 (119873 119905)

119889119905= lim120591rarr0

119875 (119873 119905 + 120591) minus 119875 (119873 119905)

120591 (11)

Define 119882119905(120578119873) as the transition probability density from

state 120578 to state119873 in unit time during time interval [119905 119905+120591] Sothe transition probability from state 120578 to state119873 during timeinterval [119905 119905 + 120591] is 120591119882

119905(120578119873) Then the probability by which

the transition does not happen is

1 minus 120591int119882119905(120578119873) 120575 (119873 minus 120578) 119889119899 (12)

where 120575(119873 minus 120578) = 119875(119873 119905 + 120591 | 120578 119905) when 120591 = 0 Thus

119889119875 (119873 119905)

119889119905= int 119882(120578 119899) 119875 (120578 119905) minus 119882 (119899 120578) 119875 (119899 119905) 119889120578

(13)

Equation (13) describes the evolution of the system statesover time which is the primary equation model of the UAVswarms behavior

3 Flocking Control of UAV Swarms

31 Flocking Control Law In this section first we design adistributed flocking control law Assuming that each UAVsenses its own position and velocity and is able to obtainits neighborsrsquo position and velocity the UAV swarms formflocking behaviour model structure control law as follows

119880119894= Uniformity (V)

119873

sum

119895=1119895 = 119894

119891 (119901119894119895) (119901119894minus 119901119895)

+ 120573 (119901119894minus 119901goal) +

119873

sum

119895=1119895 = 119894

119886119894119895(119901119894119895) (V119895minus V119894)

(14)

where Uniformity (V) = (120572sum119894 = 119895

V119894minus V1198952) Note that align-

ment at a common velocity is equivalent to Uniformity (V) =0 119875119894119895

is the distance between the individual 119894 and 119895 119880119894119895is

potential function and satisfies the condition [29 30]

(i) 119880119894119877(119901119894minus 119901119895) rarr infin 119901

119894minus 119901119895 lt 119877min

(ii) 119877min le 119901119894minus 119901119895 le 119877max existmin119880

119894119877

32 Stability Analysis Consider the following positive sem-idefinite function

119864 =1

2(

119873

sum

119894=1

(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal) + V119879

119894V119894) (15)

Mathematical Problems in Engineering 5

In order to facilitate writing we simplify the certificationprocess variable substitution as follows

119880119894= (

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

=1

2

119873

sum

119894=1

119880119894minus

119873

sum

119894=1

V119879119894119880119894

=1

2

119873

sum

119894=1

119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895) minus 119870

119894nabla119901119894119880119894119860(119901119894119895)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895)

minus 119870119894nabla119901119894119880119894goal (119901119894goal)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895minus 119870119894nabla119901119894119880119894goal

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minusV119879 (119871119862otimes 1198683) V

(16)

where 119871119888is UAV swarms system satisfying the Laplacian

matrix of the communication conditions Therefore thequadratic form is explicitly described as follows

= minusV119879119909119871119888V119909minus V119879119910119871119888V119910minus V119879119911119871119888V119911le 0 (17)

Consider the following collections V119894 119875119894119895

| 119864 le 119862 isa closed set The following is to verify that it is a compactset and there is a clear conclusion that 119875

119894119895le 119862 Similarly

V119879119894V119894le 119862 V

119894 le 119862

12 and according to the definition of thepotential field we obtain 119875

119894119895 le 119880

minus1

119894119895(119862(119873minus1)) According to

the LaSalle invariance principle the system will converge tothe largest invariant set in the area andmeet = 0 Accordingto = 0 when the system enters the steady state the speed ofeach individual is equal and all individualsmove to the targetposition119875goal making the overall potential energyminimum

Theorem 2 Consider the UAV swarms consisting of119873 UAVsThe position of individual 119894 is 119901

119894 All individuals in the swarms

will eventually build up to the spherical region

1003817100381710038171003817119901119894 minus 119901119888

1003817100381710038171003817 le2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(18)

Proof Consider

119875119888= 119881119888

119888=

1

119873

119873

sum

119894=1

V119894=

1

119873

119873

sum

119894=1

(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

minus 120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817) minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895) = 0

(19)

where 119875119888(119905) = (1)119873sum

119873

119894=1119901119894(119905) 119881119888= (1)119873sum

119873

119894=1120572119894119895V119894

By making the variable replacement 120576119894119901

= 119875119894minus 119875119888 120576119894119901

=

V119894minus V119888 we get

120576119894119901= 120576119894V

120576119894119901= (minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

minus120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817)

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minus120573119894120576119894119901minus 119873(

119873

sum

119895=1119895 = 119894

120572119894119895)120576119894V

minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

(20)

Then

120577119894= (

120576119894119901

120576119894V) = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))120577119894

minus(

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

119860 = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))

119861 = (

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

(21)

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

flocking through a set of basic behaviors including safewandering aggregation dispersion and homing to imple-ment flocking is by Mataric [31] Kelley and Keating realizeflocking with robots based on leader-following behavior[32] Hayes and Dormiani-Tabatabaei [33] propose a flockingalgorithm based on two behaviors collision avoidance andvelocity-matching flock centering Holland et al [34] pro-pose a flocking algorithm for UAV similar to Reynoldsrsquo Ahost is used as an intermediate station for receiving eachUAVrsquos range bearing and velocity and sending them toother UAVs to simulate the sensing process of one UAVfor perceiving range bearing and heading of its neigh-bors Ferrante et al [35] introduce a new communicationstrategy called the information aware communication foralignment behavior Recently Stranieri et al [36] performflocking with a swarm of behaviorally heterogeneous mobilerobots

In this paper we consider models for flocking swarmsFirstly a mathematical model of cooperative system isestablished by using Markov stochastic process and calculusanalysis Then the control law for UAV swarm is establishedbased on artificial potential field At last we analyze thestability of the proposed flocking control algorithm basedon the Lyapunov approach and prove the conclusion thatthe system in a limited time can converge to the consensusdirection of the velocity Simulation results are provided toverify the conclusion

2 The Model of the UAVs Swarms

21 Differential Integral Model Let 119862(119905) denote the state ofthe UAV swarms at time 119905 119862(119905) = 0 identifies the statethat UAV swarms are stable at time 119905 The state of UAV 119894 attime 119905 is denoted by 119888

119894(119905) = (119901

119894(119905) 119900119894(119905)) in which the first

element 119901119894(119905) = (119909

119894(119905) 119910119894(119905) 119911119894(119905)) is the UAVrsquos position in

the environment at time 119905 and the second element 119900119894(119905) is

the UAVrsquos orientation The UAVrsquos dynamics is subject to itsphysical curvature radius constraints leading to the fact thatit can only change its orientation by at most one step whichis described as go straight go up go down turn left turnupper left turn lower left turn right turn upper right andturn lower right

In order to obtain Markov random process the new stateof process is derived by supplement of variable [37 38] whichis described as follows

119875119894 (119905) = 119875 (119862 (119905) = 119894) 119894 = 0 1 2 119873

119875119895119896 (119909 119905) = 119875 (119862 (119905) = 119895 119909 lt 119910

119895119896 (119905) lt 119909 + 119889119909)

119895 = 119873 + 1 119872 119896 = 0 1 2 119873

(1)

where 119910119894(119905) is the dwell time after state 119894 So it is easy to verify

that 119880(119905) 0 le 119905 le 1198790 = (119862(119905) 119910(119905)) is a broad Markov

random process

The probability of state transition afterΔ119905 can be obtainedusing total probability theorem

1198750 (119905 + Δ119905) = (

119872

sum

119894=0

119875 (119862 (119905) = 119894 119862 (119905 + Δ119905) = 119894))

= 1198750 (119905) (1 minus

119873

sum

119894=1

1205820119894Δ119905 + 119900 (Δ119905)) +

119873

sum

119894=1

1205821198940119875119896 (119905) Δ119905

+

119872

sum

119895=119873+1

int

119879

0

119875119895 (119909 119905) 119903119895 (119909) Δ119905 119889119909 + 119900 (Δ119905)

(2)

According to (2) we can get the all probability

119875119896 (119905 + Δ119905) = 119875

119896 (119905) (1 minus

119873

sum

119894=1119894 = 119896

120582119896119894Δ119905 + 119900 (Δ119905))

+

119873

sum

119894=1119894 = 119896

120582119894119896119875119896 (119905) Δ119905 + 119900 (Δ119905)

119896 = 1 2 119873

119875119895 (119909 119905) = 119875 (119862 (119905) = 119895

the time of self-organised 119862system is 119909

119862 (119905 + Δ119905) = 119895)

= 119875119895 (119909 119905) (1 minus 119903

119895 (119909) Δ119905 + 119900 (Δ119905))

119895 = 119873 + 1 119872

(3)

where 120582119894119895is the average sustained rate of each state and 119903

119895(119909)

is the average repair rate at state 119895 Similarly the expression ofstate transition rate for 119875

119895(119909 + Δ119909 119905 + Δ119905) can be derivated

Differentiate the expression for state transition probabil-ity to derive its limit Then the mathematical model can bedescribed using integral-differential equations as follows

1198891198750 (119905)

119889119905+

119873

sum

119894=0119894 = 119896

12058201198961198750 (119905)

=

119873

sum

119894=0119894 = 119896

1205821198940119875119894 (119905)

+

119872

sum

119895=119873+1

int

119879

0

119903119895 (119909) 119901119895 (119909 119905) 119889119909

119889119875119896 (119905)

119889119905+

119873

sum

119894=0119894 = 119896

120582119896119894119875119896 (119905)

=

119873

sum

119894=0119894 = 119896

120582119894119896119875119894 (119905)

119896 = 1 2 119873

Mathematical Problems in Engineering 3

120597119901119895 (119909 119905)

120597119909+120597119901119895 (119909 119905)

120597119905+ 119903119895 (119909) 119901119895 (119909 119905) = 0

119895 = 119873 + 1 119872

(4)

The boundary and initial conditions are

1198750 (0) = 1 119875

1 (0) = 0 119875119894 (0) = 0 119875

119873 (0) = 0

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895119875119894 (119905) 119895 = 119873 + 1 119872

(5)

Theorem 1 The reliability of coordination system has unique-ness and nonnegative solution on 119862[0 119879]

Proof According to the initial conditions we can get theanalytic solution of the partial differential equation [39 40]

Set

1198860=

119873

sum

119894=1

1205820119894 119886

119895=

119873

sum

119894=1

120582119895119894 119886

119873=

119873

sum

119894=1

120582119873119894

119875119895 (119909 119905) = 119875

119894 (0 119905 minus 119909) exp(minusint119909

0

119903119894(120583) 119889120583)

119894 = 119873 + 1 119872

1198891198750 (119905)

119889119905= minus 119886

01198750 (119905) +

119873

sum

119894=0119894 = 119896

1205821198940119875119894 (119905)

+

119872

sum

119895=119873+1

int

119879

0

119875119895 (0 119905 minus 119909) exp(minusint

119909

0

119903119894(120583) 119889120583)

times 119903119894 (119909) 119889119909

1198750 (119905) = exp (minus119886

0119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) 119889119904

+ int

1198790

0

[

119872

sum

119894=119873+1

int

119904

0

119875119894 (0 120591) exp(minusint

119904minus120591

0

119903119894(120583) 119889120583)

times 119903119894 (119904 minus 120591) 119889120591]

times exp (minus1198860 (119905 minus 119904)) 119889119904

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) times 119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591) 119889120591

times int

1198790minus120591

0

exp(minus1198860(1198790minus 120591) + 119886

0V

minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) 1198960119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591)119870119894 (119905 minus 120591) 119889120591

(6)

where

1198960= exp (minus119886

0 (119879 minus 119904))

119896119894 (119879 minus 120591) = int

119879minus120591

0

1198960lowast exp(119886

0V minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

119894 = 119873 + 1 119872

119875119895 (119905) =

119873

sum

119894=0119894 = 119895

int

119879

0

exp (minus119886119895 (119879 minus 119904)) 120582119894119895119875119894 (119904) 119889119904

=

119873

sum

119894=0119894 = 119895

int

119879

0

119896119895120582119894119895119875119894 (119904) 119889119904 119895 = 1 2 119873

(7)

where 119896119895= exp(minus119886

119895(119879 minus 119904)) 119895 = 1 2 119873

So we can get the following equation

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895

119873

sum

119897=0119897 = 119895

int

119879

0

119896119895120582119897119895119875119897 (119904) 119889119904

= int

119879

0

119896119895(

119873

sum

119894=0

120582119894119895)(

119873

sum

119897=0119897 = 119895

120582119897119895119875119897 (119904))119889119904

119895 = 119873 + 1 119872

(8)

Assuming

119875 (119905) = (1198750 (119905) 1198751 (119905) 119875119873 (119905) 119875119873+1 (0 119905) 119875119872 (0 119905))

119891 (119905) = (1198910 (119905) 1198911 (119905) 119891119873 (119905) 119891119873+1 (0 119905) 119891119872 (0 119905))

= (exp (minus1198860119905) 0 0 0)

(9)

4 Mathematical Problems in Engineering

then the solution of the system can be converted into vectorsformat as follows

119875 (119905) = 119891 (119905) + int

119879

0

119896 (119879 minus 119904) 119875 (119904) 119889119904

119896 (119879 minus 120591) =

(((((((((((((((

(

0 120582101198960

sdot sdot sdot 12058211987301198960

119896119873+1 (119879 minus 119904) sdot sdot sdot 119896

119872 (119879 minus 119904)

120582011198961

0 sdot sdot sdot 12058211987311198961

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

1205820119873119896119873

1205822119873119896119873

sdot sdot sdot 0 0 sdot sdot sdot 0

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119873+1

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119873+1

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119873+1

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119872

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119872

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119872

0 sdot sdot sdot 0

)))))))))))))))

)

(10)

Any component of119865(119905) and119866(119905minusℎ) vector is nonnegativeThe functions119865

119895(119905) and119866

119895(119905minusℎ) 119895 = 1 2 119872 are limitary

on the domain 0 lt 119879 lt +infinThe solution of integral equationis unique and nonnegative on 119862[0 119879] So the reliability ofcoordination system has unique and nonnegative solution on119862[0 119879]

22 Probabilistic Analysis Based on State TransformationThe behavior evolution of the UAV swarm system is alimitedMarkov decision process Suppose that the probabilitydistribution of the system state is 119875(119873 119905) at time 119905 Then attime 119905+120591 the probability distribution is119875(119873 119905+120591) Accordingto the relationship of the probability density at different timethe marginal probability density 119875(119873 119905 + 120591) is 119875(119873 119905 + 120591) =

int119875(120578 119905)119875(119873 119905 + 120591 | 120578 119905)119889120578And the time derivative of the 119875(119873 119905 + 120591) is

119889119875 (119873 119905)

119889119905= lim120591rarr0

119875 (119873 119905 + 120591) minus 119875 (119873 119905)

120591 (11)

Define 119882119905(120578119873) as the transition probability density from

state 120578 to state119873 in unit time during time interval [119905 119905+120591] Sothe transition probability from state 120578 to state119873 during timeinterval [119905 119905 + 120591] is 120591119882

119905(120578119873) Then the probability by which

the transition does not happen is

1 minus 120591int119882119905(120578119873) 120575 (119873 minus 120578) 119889119899 (12)

where 120575(119873 minus 120578) = 119875(119873 119905 + 120591 | 120578 119905) when 120591 = 0 Thus

119889119875 (119873 119905)

119889119905= int 119882(120578 119899) 119875 (120578 119905) minus 119882 (119899 120578) 119875 (119899 119905) 119889120578

(13)

Equation (13) describes the evolution of the system statesover time which is the primary equation model of the UAVswarms behavior

3 Flocking Control of UAV Swarms

31 Flocking Control Law In this section first we design adistributed flocking control law Assuming that each UAVsenses its own position and velocity and is able to obtainits neighborsrsquo position and velocity the UAV swarms formflocking behaviour model structure control law as follows

119880119894= Uniformity (V)

119873

sum

119895=1119895 = 119894

119891 (119901119894119895) (119901119894minus 119901119895)

+ 120573 (119901119894minus 119901goal) +

119873

sum

119895=1119895 = 119894

119886119894119895(119901119894119895) (V119895minus V119894)

(14)

where Uniformity (V) = (120572sum119894 = 119895

V119894minus V1198952) Note that align-

ment at a common velocity is equivalent to Uniformity (V) =0 119875119894119895

is the distance between the individual 119894 and 119895 119880119894119895is

potential function and satisfies the condition [29 30]

(i) 119880119894119877(119901119894minus 119901119895) rarr infin 119901

119894minus 119901119895 lt 119877min

(ii) 119877min le 119901119894minus 119901119895 le 119877max existmin119880

119894119877

32 Stability Analysis Consider the following positive sem-idefinite function

119864 =1

2(

119873

sum

119894=1

(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal) + V119879

119894V119894) (15)

Mathematical Problems in Engineering 5

In order to facilitate writing we simplify the certificationprocess variable substitution as follows

119880119894= (

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

=1

2

119873

sum

119894=1

119880119894minus

119873

sum

119894=1

V119879119894119880119894

=1

2

119873

sum

119894=1

119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895) minus 119870

119894nabla119901119894119880119894119860(119901119894119895)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895)

minus 119870119894nabla119901119894119880119894goal (119901119894goal)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895minus 119870119894nabla119901119894119880119894goal

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minusV119879 (119871119862otimes 1198683) V

(16)

where 119871119888is UAV swarms system satisfying the Laplacian

matrix of the communication conditions Therefore thequadratic form is explicitly described as follows

= minusV119879119909119871119888V119909minus V119879119910119871119888V119910minus V119879119911119871119888V119911le 0 (17)

Consider the following collections V119894 119875119894119895

| 119864 le 119862 isa closed set The following is to verify that it is a compactset and there is a clear conclusion that 119875

119894119895le 119862 Similarly

V119879119894V119894le 119862 V

119894 le 119862

12 and according to the definition of thepotential field we obtain 119875

119894119895 le 119880

minus1

119894119895(119862(119873minus1)) According to

the LaSalle invariance principle the system will converge tothe largest invariant set in the area andmeet = 0 Accordingto = 0 when the system enters the steady state the speed ofeach individual is equal and all individualsmove to the targetposition119875goal making the overall potential energyminimum

Theorem 2 Consider the UAV swarms consisting of119873 UAVsThe position of individual 119894 is 119901

119894 All individuals in the swarms

will eventually build up to the spherical region

1003817100381710038171003817119901119894 minus 119901119888

1003817100381710038171003817 le2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(18)

Proof Consider

119875119888= 119881119888

119888=

1

119873

119873

sum

119894=1

V119894=

1

119873

119873

sum

119894=1

(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

minus 120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817) minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895) = 0

(19)

where 119875119888(119905) = (1)119873sum

119873

119894=1119901119894(119905) 119881119888= (1)119873sum

119873

119894=1120572119894119895V119894

By making the variable replacement 120576119894119901

= 119875119894minus 119875119888 120576119894119901

=

V119894minus V119888 we get

120576119894119901= 120576119894V

120576119894119901= (minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

minus120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817)

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minus120573119894120576119894119901minus 119873(

119873

sum

119895=1119895 = 119894

120572119894119895)120576119894V

minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

(20)

Then

120577119894= (

120576119894119901

120576119894V) = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))120577119894

minus(

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

119860 = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))

119861 = (

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

(21)

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120597119901119895 (119909 119905)

120597119909+120597119901119895 (119909 119905)

120597119905+ 119903119895 (119909) 119901119895 (119909 119905) = 0

119895 = 119873 + 1 119872

(4)

The boundary and initial conditions are

1198750 (0) = 1 119875

1 (0) = 0 119875119894 (0) = 0 119875

119873 (0) = 0

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895119875119894 (119905) 119895 = 119873 + 1 119872

(5)

Theorem 1 The reliability of coordination system has unique-ness and nonnegative solution on 119862[0 119879]

Proof According to the initial conditions we can get theanalytic solution of the partial differential equation [39 40]

Set

1198860=

119873

sum

119894=1

1205820119894 119886

119895=

119873

sum

119894=1

120582119895119894 119886

119873=

119873

sum

119894=1

120582119873119894

119875119895 (119909 119905) = 119875

119894 (0 119905 minus 119909) exp(minusint119909

0

119903119894(120583) 119889120583)

119894 = 119873 + 1 119872

1198891198750 (119905)

119889119905= minus 119886

01198750 (119905) +

119873

sum

119894=0119894 = 119896

1205821198940119875119894 (119905)

+

119872

sum

119895=119873+1

int

119879

0

119875119895 (0 119905 minus 119909) exp(minusint

119909

0

119903119894(120583) 119889120583)

times 119903119894 (119909) 119889119909

1198750 (119905) = exp (minus119886

0119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) 119889119904

+ int

1198790

0

[

119872

sum

119894=119873+1

int

119904

0

119875119894 (0 120591) exp(minusint

119904minus120591

0

119903119894(120583) 119889120583)

times 119903119894 (119904 minus 120591) 119889120591]

times exp (minus1198860 (119905 minus 119904)) 119889119904

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) exp (minus1198860 (119905 minus 119904)) times 119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591) 119889120591

times int

1198790minus120591

0

exp(minus1198860(1198790minus 120591) + 119886

0V

minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

= exp (minus1198860119905) +

119873

sum

119894=1

1205821198940int

1198790

0

119875119894 (119904) 1198960119889119904

+

119872

sum

119894=119873+1

int

1198790

0

119875119894 (0 120591)119870119894 (119905 minus 120591) 119889120591

(6)

where

1198960= exp (minus119886

0 (119879 minus 119904))

119896119894 (119879 minus 120591) = int

119879minus120591

0

1198960lowast exp(119886

0V minus int

V

0

119903119894(120583) 119889120583) 119903

119894 (V) 119889V

119894 = 119873 + 1 119872

119875119895 (119905) =

119873

sum

119894=0119894 = 119895

int

119879

0

exp (minus119886119895 (119879 minus 119904)) 120582119894119895119875119894 (119904) 119889119904

=

119873

sum

119894=0119894 = 119895

int

119879

0

119896119895120582119894119895119875119894 (119904) 119889119904 119895 = 1 2 119873

(7)

where 119896119895= exp(minus119886

119895(119879 minus 119904)) 119895 = 1 2 119873

So we can get the following equation

119875119895 (0 119905) =

119873

sum

119894=0

120582119894119895

119873

sum

119897=0119897 = 119895

int

119879

0

119896119895120582119897119895119875119897 (119904) 119889119904

= int

119879

0

119896119895(

119873

sum

119894=0

120582119894119895)(

119873

sum

119897=0119897 = 119895

120582119897119895119875119897 (119904))119889119904

119895 = 119873 + 1 119872

(8)

Assuming

119875 (119905) = (1198750 (119905) 1198751 (119905) 119875119873 (119905) 119875119873+1 (0 119905) 119875119872 (0 119905))

119891 (119905) = (1198910 (119905) 1198911 (119905) 119891119873 (119905) 119891119873+1 (0 119905) 119891119872 (0 119905))

= (exp (minus1198860119905) 0 0 0)

(9)

4 Mathematical Problems in Engineering

then the solution of the system can be converted into vectorsformat as follows

119875 (119905) = 119891 (119905) + int

119879

0

119896 (119879 minus 119904) 119875 (119904) 119889119904

119896 (119879 minus 120591) =

(((((((((((((((

(

0 120582101198960

sdot sdot sdot 12058211987301198960

119896119873+1 (119879 minus 119904) sdot sdot sdot 119896

119872 (119879 minus 119904)

120582011198961

0 sdot sdot sdot 12058211987311198961

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

1205820119873119896119873

1205822119873119896119873

sdot sdot sdot 0 0 sdot sdot sdot 0

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119873+1

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119873+1

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119873+1

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119872

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119872

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119872

0 sdot sdot sdot 0

)))))))))))))))

)

(10)

Any component of119865(119905) and119866(119905minusℎ) vector is nonnegativeThe functions119865

119895(119905) and119866

119895(119905minusℎ) 119895 = 1 2 119872 are limitary

on the domain 0 lt 119879 lt +infinThe solution of integral equationis unique and nonnegative on 119862[0 119879] So the reliability ofcoordination system has unique and nonnegative solution on119862[0 119879]

22 Probabilistic Analysis Based on State TransformationThe behavior evolution of the UAV swarm system is alimitedMarkov decision process Suppose that the probabilitydistribution of the system state is 119875(119873 119905) at time 119905 Then attime 119905+120591 the probability distribution is119875(119873 119905+120591) Accordingto the relationship of the probability density at different timethe marginal probability density 119875(119873 119905 + 120591) is 119875(119873 119905 + 120591) =

int119875(120578 119905)119875(119873 119905 + 120591 | 120578 119905)119889120578And the time derivative of the 119875(119873 119905 + 120591) is

119889119875 (119873 119905)

119889119905= lim120591rarr0

119875 (119873 119905 + 120591) minus 119875 (119873 119905)

120591 (11)

Define 119882119905(120578119873) as the transition probability density from

state 120578 to state119873 in unit time during time interval [119905 119905+120591] Sothe transition probability from state 120578 to state119873 during timeinterval [119905 119905 + 120591] is 120591119882

119905(120578119873) Then the probability by which

the transition does not happen is

1 minus 120591int119882119905(120578119873) 120575 (119873 minus 120578) 119889119899 (12)

where 120575(119873 minus 120578) = 119875(119873 119905 + 120591 | 120578 119905) when 120591 = 0 Thus

119889119875 (119873 119905)

119889119905= int 119882(120578 119899) 119875 (120578 119905) minus 119882 (119899 120578) 119875 (119899 119905) 119889120578

(13)

Equation (13) describes the evolution of the system statesover time which is the primary equation model of the UAVswarms behavior

3 Flocking Control of UAV Swarms

31 Flocking Control Law In this section first we design adistributed flocking control law Assuming that each UAVsenses its own position and velocity and is able to obtainits neighborsrsquo position and velocity the UAV swarms formflocking behaviour model structure control law as follows

119880119894= Uniformity (V)

119873

sum

119895=1119895 = 119894

119891 (119901119894119895) (119901119894minus 119901119895)

+ 120573 (119901119894minus 119901goal) +

119873

sum

119895=1119895 = 119894

119886119894119895(119901119894119895) (V119895minus V119894)

(14)

where Uniformity (V) = (120572sum119894 = 119895

V119894minus V1198952) Note that align-

ment at a common velocity is equivalent to Uniformity (V) =0 119875119894119895

is the distance between the individual 119894 and 119895 119880119894119895is

potential function and satisfies the condition [29 30]

(i) 119880119894119877(119901119894minus 119901119895) rarr infin 119901

119894minus 119901119895 lt 119877min

(ii) 119877min le 119901119894minus 119901119895 le 119877max existmin119880

119894119877

32 Stability Analysis Consider the following positive sem-idefinite function

119864 =1

2(

119873

sum

119894=1

(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal) + V119879

119894V119894) (15)

Mathematical Problems in Engineering 5

In order to facilitate writing we simplify the certificationprocess variable substitution as follows

119880119894= (

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

=1

2

119873

sum

119894=1

119880119894minus

119873

sum

119894=1

V119879119894119880119894

=1

2

119873

sum

119894=1

119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895) minus 119870

119894nabla119901119894119880119894119860(119901119894119895)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895)

minus 119870119894nabla119901119894119880119894goal (119901119894goal)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895minus 119870119894nabla119901119894119880119894goal

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minusV119879 (119871119862otimes 1198683) V

(16)

where 119871119888is UAV swarms system satisfying the Laplacian

matrix of the communication conditions Therefore thequadratic form is explicitly described as follows

= minusV119879119909119871119888V119909minus V119879119910119871119888V119910minus V119879119911119871119888V119911le 0 (17)

Consider the following collections V119894 119875119894119895

| 119864 le 119862 isa closed set The following is to verify that it is a compactset and there is a clear conclusion that 119875

119894119895le 119862 Similarly

V119879119894V119894le 119862 V

119894 le 119862

12 and according to the definition of thepotential field we obtain 119875

119894119895 le 119880

minus1

119894119895(119862(119873minus1)) According to

the LaSalle invariance principle the system will converge tothe largest invariant set in the area andmeet = 0 Accordingto = 0 when the system enters the steady state the speed ofeach individual is equal and all individualsmove to the targetposition119875goal making the overall potential energyminimum

Theorem 2 Consider the UAV swarms consisting of119873 UAVsThe position of individual 119894 is 119901

119894 All individuals in the swarms

will eventually build up to the spherical region

1003817100381710038171003817119901119894 minus 119901119888

1003817100381710038171003817 le2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(18)

Proof Consider

119875119888= 119881119888

119888=

1

119873

119873

sum

119894=1

V119894=

1

119873

119873

sum

119894=1

(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

minus 120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817) minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895) = 0

(19)

where 119875119888(119905) = (1)119873sum

119873

119894=1119901119894(119905) 119881119888= (1)119873sum

119873

119894=1120572119894119895V119894

By making the variable replacement 120576119894119901

= 119875119894minus 119875119888 120576119894119901

=

V119894minus V119888 we get

120576119894119901= 120576119894V

120576119894119901= (minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

minus120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817)

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minus120573119894120576119894119901minus 119873(

119873

sum

119895=1119895 = 119894

120572119894119895)120576119894V

minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

(20)

Then

120577119894= (

120576119894119901

120576119894V) = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))120577119894

minus(

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

119860 = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))

119861 = (

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

(21)

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

then the solution of the system can be converted into vectorsformat as follows

119875 (119905) = 119891 (119905) + int

119879

0

119896 (119879 minus 119904) 119875 (119904) 119889119904

119896 (119879 minus 120591) =

(((((((((((((((

(

0 120582101198960

sdot sdot sdot 12058211987301198960

119896119873+1 (119879 minus 119904) sdot sdot sdot 119896

119872 (119879 minus 119904)

120582011198961

0 sdot sdot sdot 12058211987311198961

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

1205820119873119896119873

1205822119873119896119873

sdot sdot sdot 0 0 sdot sdot sdot 0

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119873+1

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119873+1

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119873+1

0 sdot sdot sdot 0

sdot sdot sdot

sdot sdot sdot

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205820119872

(

119873

sum

119897=0119897 = 119895

120582119897119895)1205821119872

sdot sdot sdot (

119873

sum

119897=0119897 = 119895

120582119897119895)120582119873119872

0 sdot sdot sdot 0

)))))))))))))))

)

(10)

Any component of119865(119905) and119866(119905minusℎ) vector is nonnegativeThe functions119865

119895(119905) and119866

119895(119905minusℎ) 119895 = 1 2 119872 are limitary

on the domain 0 lt 119879 lt +infinThe solution of integral equationis unique and nonnegative on 119862[0 119879] So the reliability ofcoordination system has unique and nonnegative solution on119862[0 119879]

22 Probabilistic Analysis Based on State TransformationThe behavior evolution of the UAV swarm system is alimitedMarkov decision process Suppose that the probabilitydistribution of the system state is 119875(119873 119905) at time 119905 Then attime 119905+120591 the probability distribution is119875(119873 119905+120591) Accordingto the relationship of the probability density at different timethe marginal probability density 119875(119873 119905 + 120591) is 119875(119873 119905 + 120591) =

int119875(120578 119905)119875(119873 119905 + 120591 | 120578 119905)119889120578And the time derivative of the 119875(119873 119905 + 120591) is

119889119875 (119873 119905)

119889119905= lim120591rarr0

119875 (119873 119905 + 120591) minus 119875 (119873 119905)

120591 (11)

Define 119882119905(120578119873) as the transition probability density from

state 120578 to state119873 in unit time during time interval [119905 119905+120591] Sothe transition probability from state 120578 to state119873 during timeinterval [119905 119905 + 120591] is 120591119882

119905(120578119873) Then the probability by which

the transition does not happen is

1 minus 120591int119882119905(120578119873) 120575 (119873 minus 120578) 119889119899 (12)

where 120575(119873 minus 120578) = 119875(119873 119905 + 120591 | 120578 119905) when 120591 = 0 Thus

119889119875 (119873 119905)

119889119905= int 119882(120578 119899) 119875 (120578 119905) minus 119882 (119899 120578) 119875 (119899 119905) 119889120578

(13)

Equation (13) describes the evolution of the system statesover time which is the primary equation model of the UAVswarms behavior

3 Flocking Control of UAV Swarms

31 Flocking Control Law In this section first we design adistributed flocking control law Assuming that each UAVsenses its own position and velocity and is able to obtainits neighborsrsquo position and velocity the UAV swarms formflocking behaviour model structure control law as follows

119880119894= Uniformity (V)

119873

sum

119895=1119895 = 119894

119891 (119901119894119895) (119901119894minus 119901119895)

+ 120573 (119901119894minus 119901goal) +

119873

sum

119895=1119895 = 119894

119886119894119895(119901119894119895) (V119895minus V119894)

(14)

where Uniformity (V) = (120572sum119894 = 119895

V119894minus V1198952) Note that align-

ment at a common velocity is equivalent to Uniformity (V) =0 119875119894119895

is the distance between the individual 119894 and 119895 119880119894119895is

potential function and satisfies the condition [29 30]

(i) 119880119894119877(119901119894minus 119901119895) rarr infin 119901

119894minus 119901119895 lt 119877min

(ii) 119877min le 119901119894minus 119901119895 le 119877max existmin119880

119894119877

32 Stability Analysis Consider the following positive sem-idefinite function

119864 =1

2(

119873

sum

119894=1

(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal) + V119879

119894V119894) (15)

Mathematical Problems in Engineering 5

In order to facilitate writing we simplify the certificationprocess variable substitution as follows

119880119894= (

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

=1

2

119873

sum

119894=1

119880119894minus

119873

sum

119894=1

V119879119894119880119894

=1

2

119873

sum

119894=1

119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895) minus 119870

119894nabla119901119894119880119894119860(119901119894119895)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895)

minus 119870119894nabla119901119894119880119894goal (119901119894goal)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895minus 119870119894nabla119901119894119880119894goal

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minusV119879 (119871119862otimes 1198683) V

(16)

where 119871119888is UAV swarms system satisfying the Laplacian

matrix of the communication conditions Therefore thequadratic form is explicitly described as follows

= minusV119879119909119871119888V119909minus V119879119910119871119888V119910minus V119879119911119871119888V119911le 0 (17)

Consider the following collections V119894 119875119894119895

| 119864 le 119862 isa closed set The following is to verify that it is a compactset and there is a clear conclusion that 119875

119894119895le 119862 Similarly

V119879119894V119894le 119862 V

119894 le 119862

12 and according to the definition of thepotential field we obtain 119875

119894119895 le 119880

minus1

119894119895(119862(119873minus1)) According to

the LaSalle invariance principle the system will converge tothe largest invariant set in the area andmeet = 0 Accordingto = 0 when the system enters the steady state the speed ofeach individual is equal and all individualsmove to the targetposition119875goal making the overall potential energyminimum

Theorem 2 Consider the UAV swarms consisting of119873 UAVsThe position of individual 119894 is 119901

119894 All individuals in the swarms

will eventually build up to the spherical region

1003817100381710038171003817119901119894 minus 119901119888

1003817100381710038171003817 le2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(18)

Proof Consider

119875119888= 119881119888

119888=

1

119873

119873

sum

119894=1

V119894=

1

119873

119873

sum

119894=1

(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

minus 120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817) minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895) = 0

(19)

where 119875119888(119905) = (1)119873sum

119873

119894=1119901119894(119905) 119881119888= (1)119873sum

119873

119894=1120572119894119895V119894

By making the variable replacement 120576119894119901

= 119875119894minus 119875119888 120576119894119901

=

V119894minus V119888 we get

120576119894119901= 120576119894V

120576119894119901= (minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

minus120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817)

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minus120573119894120576119894119901minus 119873(

119873

sum

119895=1119895 = 119894

120572119894119895)120576119894V

minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

(20)

Then

120577119894= (

120576119894119901

120576119894V) = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))120577119894

minus(

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

119860 = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))

119861 = (

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

(21)

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

In order to facilitate writing we simplify the certificationprocess variable substitution as follows

119880119894= (

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

=1

2

119873

sum

119894=1

119880119894minus

119873

sum

119894=1

V119879119894119880119894

=1

2

119873

sum

119894=1

119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895) minus 119870

119894nabla119901119894119880119894119860(119901119894119895)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894119880119894+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(119901119894119895)

minus 119870119894nabla119901119894119880119894goal (119901119894goal)

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894nabla119901119894(

119873

sum

119895=1119895 = 119894

119880119894119895+ 119870119894119880119894goal)

+

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895minus 119870119894nabla119901119894119880119894goal

minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

=

119873

sum

119894=1

V119879119894(minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minusV119879 (119871119862otimes 1198683) V

(16)

where 119871119888is UAV swarms system satisfying the Laplacian

matrix of the communication conditions Therefore thequadratic form is explicitly described as follows

= minusV119879119909119871119888V119909minus V119879119910119871119888V119910minus V119879119911119871119888V119911le 0 (17)

Consider the following collections V119894 119875119894119895

| 119864 le 119862 isa closed set The following is to verify that it is a compactset and there is a clear conclusion that 119875

119894119895le 119862 Similarly

V119879119894V119894le 119862 V

119894 le 119862

12 and according to the definition of thepotential field we obtain 119875

119894119895 le 119880

minus1

119894119895(119862(119873minus1)) According to

the LaSalle invariance principle the system will converge tothe largest invariant set in the area andmeet = 0 Accordingto = 0 when the system enters the steady state the speed ofeach individual is equal and all individualsmove to the targetposition119875goal making the overall potential energyminimum

Theorem 2 Consider the UAV swarms consisting of119873 UAVsThe position of individual 119894 is 119901

119894 All individuals in the swarms

will eventually build up to the spherical region

1003817100381710038171003817119901119894 minus 119901119888

1003817100381710038171003817 le2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(18)

Proof Consider

119875119888= 119881119888

119888=

1

119873

119873

sum

119894=1

V119894=

1

119873

119873

sum

119894=1

(minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

minus 120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817) minus

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895) = 0

(19)

where 119875119888(119905) = (1)119873sum

119873

119894=1119901119894(119905) 119881119888= (1)119873sum

119873

119894=1120572119894119895V119894

By making the variable replacement 120576119894119901

= 119875119894minus 119875119888 120576119894119901

=

V119894minus V119888 we get

120576119894119901= 120576119894V

120576119894119901= (minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

minus120573119894nabla119901119894119880119894119860(1003817100381710038171003817119901119894 minus 119901

119888

1003817100381710038171003817)

119873

sum

119895=1119895 = 119894

120572119894119895(V119894minus V119895))

= minus120573119894120576119894119901minus 119873(

119873

sum

119895=1119895 = 119894

120572119894119895)120576119894V

minus

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)

(20)

Then

120577119894= (

120576119894119901

120576119894V) = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))120577119894

minus(

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

119860 = (

0 1

minus120573119894minus119873(

119873

sum

119895=1119895 = 119894

120572119894119895))

119861 = (

0

119873

sum

119895=1119895 = 119894

nabla119901119894119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817))

(21)

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Since119870119894 119873 gt 0

120582 (119860) =

minus119873(sum119873

119895=1119895 = 119894120572119894119895)

2

plusmn

radic(119873(sum119873

119895=1119895 = 119894120572119894119895))2

minus 4119896119894

2

Re (120582 (119860)) lt 0

(22)

The largest and the smallest eigenvalues of symmetric pos-itive definite matrix 119878 are 120582max(119878) and 120582min(119878) respectivelyThe symmetric positive definite matrix 119878 with appropriatedimensions satisfies the following conclusion [41 42]

120582min (119878)1003817100381710038171003817120577119894

1003817100381710038171003817

2le 120577119879

119894119878120577119894le 120582max (119878)

10038171003817100381710038171205771198941003817100381710038171003817

2 (23)

Finally select Lyapunov function

119864119894= 120577119879

119894119866120577119894 119866119879= 119866 120582 (119866) gt 0 (24)

Time derivative can be obtained

119864119894=

120577119879

119894119866120577119894+ 120577119879

119894119866 120577119894= (119860120577

119894minus 119861)119879119866120577119894+ 120577119879

119894119866 (119860120577

119894minus 119861)

= minus120577119879

119894(119860119879119866 + 119866119860) 120577

119894minus 2120577119879

119894119866119861

119880119894119895(10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817)10038171003817100381710038171003817119901119894minus 119901119895

10038171003817100381710038171003817le 119897119862

(25)

Therefore according to the above formula we obtain

119864119894le minus120582min (119860

119879119866 + 119866119860)

10038171003817100381710038171205771198941003817100381710038171003817

2+ 2120582max (119866)

10038171003817100381710038171205771198941003817100381710038171003817 (119873 minus 1) 119897119862

(26)

When

10038171003817100381710038171205771198941003817100381710038171003817 gt

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(27)

119894lt 0The system continues tomove closer to the population

centre Therefore eventually the system stabilizes at a knownsystem of

10038171003817100381710038171205771198941003817100381710038171003817 le

2120582max (119866) (119873 minus 1) 119897119862

120582min (119860119879119866 + 119866119860)

(28)

4 Simulation of System FlockingFormation Behavior

According to the UAVrsquos physical characteristics this paperwill discretize the time with high frequency Thus a UAV 119894

makes its path decision119875119894(119905+1) at time-step 119905 andwill execute

an action as the following equation

119875119894 (119905 + 1) = 119875

119894 (119905) + VΔ119905 (29)

The movement of the individual is not only controlledby itself but also affected by the state of other individuals

0 20 40 60 80 100 120 140 160 180 200t (s)

0

1

2

3

4

5

6

7

(rad)

Figure 1 Velocities with respect to time

minus2000

020004000

0 02000 2000

4000 40006000 6000

80008000

Y

X

Figure 2 Trajectories with respect to time

0 500 1000 1500 2000 2500 3000Time

minus13

minus12

minus11

minus1

minus09

minus08

minus07

minus06

minus04

minus05

minus03

Pitch

Figure 3 Angle of the Pitch with respect to time

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 500 1000 1500 2000 2500 3000Time

095

1

105

11

115

12

125

13

135

14

Roll

Figure 4 Angle of the Roll with respect to time

0 500 1000 1500 2000 2500 3000Time

0

002

004

006

008

01

012

014

016

018

AOA

Figure 5 Angle of the Attack with respect to time

0 500 1000 1500 2000 2500 3000Time

minus2

0

2

4

6

8

10times10

minus3

AOS

Figure 6 Angle of the Sideslip with respect to time

Therefore the individual direction of movement at a certaintime is not only relative to its direction one moment beforebut also relative to the directions of its surrounding individ-ualsrsquo movements The influence of all the individuals to theindividual 119894 can be described as the following equation

1

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

(30)

Then the speed direction of the UAV 119894 at time (119905 + 1) canbe modified as the following equation

120579119894 (119905 + 1) = 120572120579

119894 (119905) + 1205731

sum 1119901119894119895

119899

sum

119895=1

120579119895 (119905)

119901119894119895

+ 120574 arctan1199100minus 1199101

1199090minus 1199091

(31)

We consider the swarms of 100 UAVs with six degreesof freedom The weights of the cost function are set to 120572 =

03 120573 = 05 120574 = 02 119881 = [200 0 0]ms and 119898 = 25 kgDirection is the rand variable from minus2 lowast pi to 2 lowast pi Theposition of the UAVs is the rand variable The factors of theinfluence on the flight are wind and airstreamThe results forthe case of 100 UAVs are shown in Figures 1 and 2

From Figure 1 at 119905 = 19 s the velocities of the swarmsachieve consensus at 120579 = 32

Figure 2 describes the trajectories with respect to timeThe UAV swarms system will eventually be able to form astable distance between each individual and the same velocityvectors Collision between individuals is thus avoided

Figures 3 4 and 5 show the Pitch Roll and Attack withrespect to time From the simulation results we can concludethat the UAVs based on the method successfully fly after theadjustment at the initial stage

Figure 6 shows the Sideslip with respect to timeThroughthe analysis of the Sideslip Angle we can find that the Angleof the Sideslip is less than 05 degrees and tends to zero toensure the turning flight control

5 Conclusion

This paper analyzed current researches and existent problemsof UAV swarms Afterwards by the theory of stochasticprocess and supplemented variables a differential-integralmodel was established The existence and uniqueness ofthe solution of the system were discussed The flockingcontrol law is given based on artificial potential with systemconsensus At last we analyzed the stability of the proposedflocking control algorithm based on the Lyapunov approachand proved the conclusion that the system in 28 s canconverge to the consensus direction of the velocity And weperformed simulation tests to verify the conclusion

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no B222011XXXX)

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

References

[1] H B Duan Q N Luo and G J Ma ldquoHybrid particle swarmoptimization and genetic algorithm for multi-UAV formationreconfigurationrdquo IEEE Computational Intelligence Magazinevol 8 pp 16ndash27 2013

[2] W Yi M B Blake and R G Madey ldquoAn operation-time sim-ulation framework for UAV swarm configuration and missionplanningrdquo Procedia Computer Science vol 18 pp 1949ndash19582013

[3] Y Tang H J Gao J Kurths and J-A Fang ldquoEvolutionarypinning control and its application in UAV coordinationrdquo IEEETransactions on Industrial Informatics vol 8 no 4 pp 828ndash8382012

[4] G B Lamont J N Slear and K Melendez ldquoUAV swarmmission planning and routing using multi-objective evolu-tionary algorithmsrdquo in Proceedings of the 1st IEEE Symposiumof Computational Intelligence in Multicriteria Decision Making(MCDM rsquo07) pp 10ndash20 April 2007

[5] E Besada-Portas L de la Torre J M de la Cruz and Bde Andres-Toro ldquoEvolutionary trajectory planner for multipleUAVs in realistic scenariosrdquo IEEE Transactions on Robotics vol26 no 4 pp 619ndash634 2010

[6] S Kanchanavally R Ordonez and C J Schumacher ldquoPathplanning in three dimensional environment using feedback lin-earizationrdquo in Proceedings of the American Control Conference(ACC rsquo06) pp 3545ndash3550 Mineapolis Minn USA June 2006

[7] M Shanmugavel A Tsourdos R Zbikowski and B A Whiteldquo3D path planning for multiple UAVs using pythagorean hodo-graph curvesrdquo in Proceedings of the AIAAGuidance NavigationandControl Conference andExhibit pp 1576ndash1589HiltonHeadSC USA August 2007

[8] I Hasircioglu H R Topcuoglu and M Ermis ldquo3-D path plan-ning for the navigation of unmanned aerial vehicles by usingevolutionary algorithmsrdquo in Proceedings of the 10th AnnualGenetic and Evolutionary Computation Conference (GECCOrsquo08) pp 1499ndash1506 July 2008

[9] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMSymposium on Applied Computing pp 79ndash86 2004

[10] G Varela P Caamamno F Orjales A Deibe F Lopez-PenaandR J Duro ldquoSwarm intelligence based approach for real timeUAV team coordination in search operationsrdquo in Proceedingsof the 3rd World Congress on Nature and Biologically InspiredComputing (NaBIC rsquo11) pp 365ndash370 October 2011

[11] Y l Yang Cooperative search by uninhabited air vehicles indynamic environment [PhD thesis] University of CincinnatiCincinnati Ohio USA 2005

[12] P Dasgupta ldquoA multiagent swarming system for distributedautomatic target recognition using unmanned aerial vehiclesrdquoIEEE Transactions on Systems Man and Cybernetics A vol 38no 3 pp 549ndash563 2008

[13] M Yunhong J Zhe and Z Deyun ldquoA faster pruning optimiza-tion algorithm for task assignmentrdquo Journal of NorthwesternPolytechnical University vol 31 pp 40ndash43 2013

[14] B Di R Zhou and Q-X Ding ldquoDistributed coordinatedheterogeneous task allocation for unmanned aerial vehiclesrdquoControl and Decision vol 28 pp 274ndash278 2013

[15] W You Sh Wang and J Tao ldquoMulti-UAV dynamic taskassignment by ISODATA restrained clusteringrdquo ElectronicsOptics amp Control vol 17 pp 22ndash26 2010

[16] D Dionne and C A Rabbath ldquoMulti-UAV decentralizedtask allocation with intermittent communications the DTCalgorithmrdquo in Proceedings of the American Control Conference(ACC rsquo07) pp 5406ndash5411 July 2007

[17] P Dasgupta and M Hoeing ldquoDynamic pricing algorithms fortask allocation inmulti-agent swarmsrdquo inMassivelyMulti-AgentTechnology N Jamali P Scerri and T Sugawara Eds vol 5043of Lecture Notes in Computer Science pp 64ndash79 2008

[18] P Gaudiano B Shargel and E Bonabeau Swarm IntelligenceA New C2 Paradigm with an Application to Control Swarms ofUAVs Icosystem Cambridge Mass USA 2003

[19] J Finke K M Passino S Ganapathy and A Sparks ldquoModelingand analysis of cooperative control systems for uninhabitedautonomous vehiclesrdquo in Cooperative Control V Kumar NLeonard and A S Morse Eds vol 309 of Lecture Notes inControl and Information Science pp 79ndash102 Springer NewYork NY USA 2005

[20] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 2002

[21] A Moitra R Szczerba V Didomizio L Hoebel R Mattheysesand B Yamrom ldquoA novel approach for the coordination ofmulti-vehicle teamsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 608ndash618 MontereyCalif USA 2001

[22] P Vincent and I Rubin ldquoA framework and analysis for coop-erative search using UAV swarmsrdquo in Proceedings of the ACMApplied Computing pp 79ndash86 Nicosia Cyprus 2004

[23] H Hexmoor B McLaughlan and M Baker ldquoSwarm control inunmanned aerial vehiclesrdquo in Proceedings of the InternationalConference on Artificial Intelligence (ICAI rsquo05) pp 911ndash917 June2005

[24] R Garcia and L Barnes ldquoMulti-UAV simulator utilizing x-planerdquo Journal of Intelligent and Robotic Systems vol 57 no 1ndash4pp 393ndash406 2010

[25] M A Russell G B Lamont and K Melendez ldquoOn usingSPEEDES as a platform for a parallel swarm simulationrdquo inProceedings of the Winter Simulation Conference pp 1129ndash1137December 2005

[26] S Luke C Cioffi-Revilla L Panait K Sullivan and G BalanldquoMASON a multiagent simulation environmentrdquo Simulationvol 81 no 7 pp 517ndash527 2005

[27] S J Rasmussen JWMitchell P R Chandler C J Schumacherand A L Smith ldquoIntroduction to the Multi-UAV2 simulationand its application to cooperative control researchrdquo in Proceed-ings of the American Control Conference (ACC rsquo05) pp 4490ndash4501 June 2005

[28] T McLain R Beard and J Kelsey ldquoExperimental demon-stration of multiple robot cooperative target interceptrdquo inProceedings of the AIAA Guidance Navigation and ControlConference AIAA-2002-4678 Monterey Calif USA 2002

[29] H G Tanner A Jadbabaie and G J Pappas ldquoStable flockingof mobile agents part I fixed topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2010ndash2015December 2003

[30] H G Tanner A Jadbabaie and G J Pappas ldquoStable flocking ofmobile agents part II dynamic topologyrdquo in Proceedings of the42nd IEEE Conference on Decision and Control pp 2016ndash2021December 2003

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[31] M J Mataric Interaction and intelligent behavior [PhD thesis]Massachusetts Institute of Technology Cambridge Mass USA1994

[32] I Kelly and D Keating ldquoFlocking by the fusion of sonar andactive infrared sensors on physical autonomous robotsrdquo inProceedings of the Conference on Mechatronics and MachineVision in Practice pp 14ndash17 1996

[33] A T Hayes and P Dormiani-Tabatabaei ldquoSelf-organized flock-ing with agent failure off-line optimization and demonstrationwith real robotsrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation pp 3900ndash3905 May2002

[34] OHolland JWoods R deNardi andAClark ldquoBeyond swarmintelligence the ultraswarmrdquo in Proceedings of the IEEE SwarmIntelligence Symposium (SIS rsquo05) pp 217ndash224 June 2005

[35] E Ferrante A E Turgut N Mathews M Birattari and MDorigo ldquoFlocking in stationary and non-stationary environ-ments a novel communication strategy for heading alignmentrdquoin Parallel Problem Solving from NaturemdashPPSN XI R SchaeferC Cotta J Kolodziej and G Rudolph Eds vol 6239 of LectureNotes in Computer Science pp 331ndash340 2010

[36] A Stranieri E Ferrante A E Turgut et al ldquoSelf-organizedflocking with a heterogeneousmobile robot swarmrdquo Tech Rep2011

[37] B Li ldquoStochastic processmodel of themulti-UAVs collaborativesystem based on state transitionrdquo in Proceedings of Conferenceon Modeling Identification and Control pp 757ndash761 2012

[38] Z XChenPartial Differential Equations Science Press BeijingChina 2002

[39] L Guo H Xu C Gao and G Zhu ldquoStability analysis of a newkind series systemrdquo IMA Journal of Applied Mathematics vol75 no 3 pp 439ndash460 2010

[40] D p Gaver ldquoTime to failure and availability of paralleled systemwith repairrdquo IEEE Transactions on Reliability vol 12 pp 30ndash381963

[41] Q J Fan Key techniques research of cooperative formationbiomimetic flight control for multi-UAV [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2008

[42] C Yancai Research on distributed cooperative control for swarmUAVs [PhD thesis] Nanjing University of Aeronautics andAstronautics Nanjing China 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of