research article cooperative solution of multi-uav

Research ArticleCooperative Solution of Multi-UAV Rendezvous Problem withNetwork Restrictions

Qingjie Zhang Jianwu Tao Fei Yu Yingzhen Li Hongchang Sun and Wei Xu

Aviation University of Air Force Changchun 130022 China

Correspondence should be addressed to Qingjie Zhang nudtzhanghotmailcom

Received 2 February 2015 Accepted 11 May 2015

Academic Editor Yang Tang

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

By considering the complex networks the cooperative game based optimal consensus (CGOC) algorithm is proposed to solvethe multi-UAV rendezvous problem in the mission area Firstly the mathematical description of the rendezvous problem isestablished and the solving framework is provided based on the coordination variables and coordination function It can decreasethe transmission of the redundant information and reduce the influence of the limited network on the task Secondly the CGOCalgorithm is presented for the UAVs in distributed cooperative manner which can minimize the overall cost of the multi-UAVsystem The CGOC control problem and the corresponding solving protocol are given by using the cooperative game theory andsensitivity parameter method Then the CGOC method of multi-UAV rendezvous problem is proposed which focuses on thetrajectory control of the platform rather than the path planning Simulation results are given to demonstrate the effectiveness of theproposed CGOC method under complex network conditions and the benefit on the overall optimality and dynamic response

1 Introduction

In order to execute the missions such as simultaneous strikecooperative reconnaissance or SEAD (suppression of enemyair defense) multiple UAVs need to arrive at a selectedregion from different directions Each UAV plans its pathdynamically considering the restriction including enemyradars missiles and its own performance For the sake of thesuccessful task achievement it requires that all UAVs arriveat the goal position simultaneously or sequentially which iscalled rendezvous problem

In respect to the time coordination problem the methodusing the coordination function and coordination variables isadopted by [1ndash4] The optimal flight control sequence of theUAVs can be obtained to meet certain performance index bythismethod It can not only simplify the complexity of solvingthe problem but also reduce the redundant communicationlinks among the UAVs But it is still a kind of centralizedcontrol method in the view of the command and controlmanner After the UAVs flying into the mission area thecommunication will be reduced or interrupted In orderto ensure the survivability and increase the probability ofthe successful task a distributed control method based on

multiagent average consensus algorithm is proposed by [5ndash9]It can deal with the case of battlefield environment changesDue to the adoption of the average consensus algorithmit is a ldquocompromiserdquo collaborative result Therefore theconsensus-based flight control sequence is not consideredin an optimal manner To improve the performance of con-sensus algorithm literature [10] proposes a noncooperativebased optimal consensus (NCOC) algorithm which can beemployed to solve this rendezvous problem However theoverall cost ofmulti-UAV system is usually not theminimumsince these UAVs only minimize their own cost functionrather than the overall one This is due to the fact that theseUAVs are selfish or noncooperative Literature [11 12] focuseson the dynamic response and optimal cost by introducing theoutdated state difference and the optimized weighted matrixto the consensus algorithm This method accelerates thespeed of the consensus convergence Similarly the consensusalgorithm with virtual leader and state predictor has beenadopted in [13] to increase the convergence speed of therendezvous problem

Different from the existing methods using average con-sensus algorithm [5ndash9] this paper focuses on the optimalsolution rather than the convergence speed [11ndash13] Further

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 878536 14 pageshttpdxdoiorg1011552015878536

2 Mathematical Problems in Engineering

the NCOC method [10] is developed and the cooperativemethod of solving the rendezvous problem is proposed Thesystem achieves the overall optimal cost in this process Thenovel method can deal with the complex network restrictionsuch as switching topologies communication delay

The remainder of this paper is organized as followsFirstly the mathematical description of rendezvous problemis given Based on the results of [5ndash9] the solving frameworkis proposed using the coordination function and coordi-nation variable A novel distributed control method usingthe cooperative game based optimal consensus (CGOC)algorithm is proposed for the ldquocooperativerdquo UAVs Finallynumerical experiments and simulation results are illustratedto show the effectiveness and benefit of the proposedmethod

2 Problem Description

21 Basic Assumptions and Physical Constraints This paperfocuses on the distributed control method of the rendezvousproblem rather than the flight control of the platformHencewe assume that

(1) all UAVs are small size(2) each UAV is equipped with the autopilot which can

track the waypoint automatically(3) mission control and flight control can be decoupled

respectively

Thus the flight control problem can be simplified In orderto show the physical characteristics of the platform the flightconstraints and related performance parameters are listed asfollows

(i) the maximum and minimum speed Vmax Vmin(ii) the minimum turning radius 119877min(iii) the maximum flight time 119905max

22 Incomplete Kinematic Model of UAV Similar to [1ndash7 9]the 2D kinematic model of the UAV is selected to study therendezvous problem (forall119894 = 1 2 119899)

119894119909 (119905) = V

119894 (119905) cos120593119894 (119905)

119894119910 (119905) = V

119894 (119905) sin120593119894 (119905)

119894 (119905) = 119908119894 (119905)

(1)

where z119894(119905) = [119911

119894119909(119905) 119911

119894119910(119905)]

119879 denotes the position vector ofthe 119894th UAV and 120593

119894(119905) is the heading angle V

119894(119905) is the velocity

and119908119894(119905) is the changing rate of the heading angular velocity

119908119894(119905) and V

119894(119905) satisfy

119908119894 (119905) isin [minus119908head 119908head] (2)

V119894 (119905) isin [Vmin Vmax] (3)

The parameters 119908head Vmin and Vmax are determined by thephysical performance

The autopilot of each UAV maintains the expectedheading angle and velocity Its mathematical model can bedescribed by two first-order differential equations

119894 (119905) = 119886120593 (120593

119888

119894(119905) minus 120593119894 (119905))

V119894 (119905) = 119886V (V

119888

119894(119905) minus V119894 (119905))

(4)

The variable or vector with the superscript 119888 signifies thereference instruction Parameters 119886

120593and 119886V are the constant

coefficients of the heading and velocity channel of theautopilot

23 CommunicationModel Graph theory is used to describethe communication model among multi-UAV system seeFigure 1 Let 119866 = (VEA) denote the relationship betweenmultiple UAVs with the set of nodesV = 1205921 120592119899 the setof edgesE sub VtimesV and adjacent matrixA = [119886

119894119895]The node

indices belong to a finite index setI = 1 2 119899 The edgecan be depicted by (120592

119894 120592

119895) and the value of 119886

119894119895corresponds

to the edge of the graph that is (120592119894 120592

119895) isin E hArr 119886

119894119895gt 0 The

neighbors set of node 120592119894is defined byN

119894= 120592

119895isin V (120592

119894 120592

119895) isin

ESince the phenomenon such as communication delay

interruption and limited bandwidth may appear in practicalapplication in this paper two restrictions will be considered

(i) communication delay 120591119894119895 it denotes the transfer delay

of the message from the 119895th to the 119894th UAV(ii) time varying interconnections G

120590(119905)isin G

119886G

119887

G119873ss 120590(119905) [0infin] rarr 119886 119887 119873ss denotes the

switching signal with successive times to describe thetopology switches

24 Mathematical Model of Rendezvous Problem The math-ematical description of the rendezvous problem will be givenas follows The initial and target position of the 119894th UAVare defined as z

1198940 and z119894119891 Then the flight trajectory can be

defined as

Z119894(119879

119894) ≜ z

119894 (119905) 0le 119905 le119879119894 (5)

If there exist input sequence 120593119888

119894(119905) and V119888

119894(119905) 0 le 119905 le 119879

119894

satisfying z119894(119879

119894) = z

119894119891 kinematic model (1) and restricts (2)

(3) thenZ119894(119879

119894) is the feasible flight path of the 119894th UAV

For each feasible path UAV should avoid threats andreturn to the base safely with adequate fuel In fact there isa strong coupling relationship among these restrictions Onone hand avoiding threats means longer path and higherflight velocity It demands less residence time of the aircraftin danger On the other hand fuel saving means shorter pathand lower velocity Thus the rendezvous problem is a timecoordination problem considering the exposure time in themission area and fuel consumption

Define the cost function of the pathZ119894(119879

119894) as

119869 (Z119894(119879

119894)) = 1198961119869threat (Z119894

(119879119894)) + 1198962119869fuel (Z119894

(119879119894))

=

119873119879

sum

119901=1int

119879119894

0

119896110038171003817100381710038171003817ℎ119901minus 119911

119894 (119905)10038171003817100381710038171003817

4 119889119905 +int119879119894

01198962V

2119894(119905) 119889119905

(6)

Mathematical Problems in Engineering 3

UAV 3 UAV 3 UAV 3

UAV 1 UAV 1 UAV 1

UAV 2 UAV 2 UAV 2

UAV 4 UAV 4 UAV 4

1205911212059123

12059134

12059143

119970a = (119985a ℰa119964a)

119985a = 1205921 1205922 1205923 1205924

ℰa = (2 1) (3 2) (3 4) (4 2) (2 4)

119970c = (119985c ℰc 119964c )

119985c = 1205921 1205922 1205923 1205924

ℰc = (1 2) (2 1) (3 2) (3 4) (4 3)

119970b = (119985b ℰb119964b)

119985b = 1205921 1205922 1205923 1205924

ℰb = (2 1) (2 4) (4 3)

119964a =

0

0

0

1

0

1

1

0

0

0

0

0

0

1

0

1119964b =

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

1119964c =

0

0

0

1

1

1

0

0

0

0

1

0

0

1

0

0

Figure 1 The interconnection of multi-UAV system with communication delay and topology varying

where 1198961 1198962 isin [0 1] and 1198961+1198962 = 1The threat cost 119869threat(sdot) isdetermined by the exposure time under the threat radar If thesignals emitted by the radar in all directions are the same it isproportional to the fourth power of the distance between theUAV and the threat In (6) ℎ

119901is the position of the119901th threat

and H = ℎ1 ℎ2 ℎ119873119879 is the threat set Because the fuelconsumption rate is determined by the air drag torque thefuel cost 119869fuel(sdot) is proportional to the square of the velocity1198882 isin R is the proportion factor

With (5) and (6) the rendezvous problem is equivalent tothe following global optimization equation

min119899

sum

119894=1119869 (Z

119894(119879

119894))

st Z119894(119879

119894) is a feasible path 1198791 = 1198792 = sdot sdot sdot = 119879119899

= 119879ETA

(7)

Figure 2 demonstrates the rendezvous problem of threeUAVs (the dashed line denotes the feasible path) For exam-ple when number 2 UAV detects a new unknown threatall UAVs need to negotiate with each other and determinea new ETA (estimated time of arrival) Then a new path isreplanned see the solid line in Figure 2 The new path willensure the UAVs arrive at their destination simultaneouslyIt is obvious that the costs of number 1 and number 3 UAVare not optimal for that path But the bigger ETA nego-tiated will accomplish the mission Therefore rendezvousproblem is a typical cooperative control problem of multi-UAV including two aspects One is path planning EachUAV plans its path considering some constraints includingradars missile threats and platform performance The otherone is trajectory control Each UAV arrives at its destinationsimultaneously through adjusting its velocity and headingangle

Obviously there are also some challenges for solving theglobal optimization problem Firstly the expressions (1) (2)and (3) are noncomplete constraints of the motion modelSo it is difficult to generate the feasible path Secondly the

Forb

idde

n ar

ea

Forbidden area Forbidden area

Forbidden area

SAM launcher base Radar border

Threaten border

Threaten border

Threaten border

Threa

ten border

Threaten border

UAV 2

UAV 2

UAV 3

UAV 3UAV 1

UAV 1 New threaten

Figure 2 Typical scenery of multi-UAV rendezvous problem

gradient optimization technique is very sensitive to the initialpath Finally time constraints mean that all UAVs shouldplan their path simultaneously Therefore the suboptimal orfeasible solution of problem (7) is discussed

3 Distributed Solution ofthe Rendezvous Problem

31 Distributed Solving Structure

311 Coordination Function and Coordination Variable Thismethod was proposed to reduce the communication costand the difficulty of the problem by [1ndash4 14] Coordina-tion variable is the minimum amount of information tocoordinate multi-UAV system Coordination function is theperformance of the system achieving effective coordinationThe basic idea is listed as follows [3]

For the 119894th UAV the state space description of thebattlefield is defined as120594

119894and the state as119909

119894isin 120594

119894The decision

of each UAV would affect the overall cost U119894(119909

119894) stands for

the feasible decision set and 119906119894isin U

119894(119909

119894) is the decision

variable


In order to achieve effective coordination there is aminimum amount of information namely the coordinationvariable 120579 (120579 isin R119888

) If the variable 120579 and the correspondingfunction are known to each UAV the system can achievecollaborative behavior

312 Coordination Variable Selection Distributed structurefor solving multi-UAV rendezvous problem includes

(i) Waypoint Planner (WP) obtain the waypointsequence considering minimizing the threat and fuelconsumption

(ii) Kinematic Trajectory Smoothing (TS) generate thefine trajectory in accordance with the kinematicmodel (1) and the UAV platform physical constraints(2) (3)

(iii) Distributed Coordinator (DC) receive the ETA fromits adjacent UAVs and adjust its ETA by consensusalgorithm The reference velocity instruction V119888

119894is

generated according to the ETA Get the referenceheading instruction 120593

119888

119894according to the trajectory

point got fromDSmodule Send V119888119894 120593119888

119894to APmodule

(iv) autopilot (AP) ensure the aircraft fly to the destina-tion position according to V119888

119894and 120593119888

119894

(v) kinematic model (DM) describe the kinematic char-acteristics of the platform Assume that it satisfiesformula (1)

Obviously how to obtain the reference velocity is thekey to DC module Let 119889ETA119894

(119905) denote the remaining pathlength to the prespecified target location of the 119894thUAVThen119889ETA119894

(119905) = minusV119894(119905) Given 119889ETA119894

(119905) and V119894(119905) the ETA can be

calculated as follows

119879ETA119894 (119905) =119889ETA119894 (119905)

V119894 (119905)

(8)

The total flight time of the 119894th UAV is

TETA119894 (119905) = 119905 +119879ETA119894 (119905) 119905 isin [1199050 119905119891] (9)

The first derivative of the TETA119894(119905) is calculated as

TETA119894 (119905) = 1+[V

119894 (119905)119889len119894 (119905) minus 119889len119894 (119905) V119894 (119905)]

V2119894(119905)

= minus119879ETA119894 (119905) 119886119881 [V

119888

119894(119905) minus V

119894 (119905)]

V119894 (119905)

(10)

Thus the reference velocity can be obtained

V119888119894(119905) = V

119894 (119905) +V119894 (119905) TETA119894 (119905)

119886119881119879ETA119894 (119905)

(11)

For this rendezvous problem there are several advan-tages when the ETA time is selected as the coordinationvariable (a) reducing the redundant information negotiatingwith each other (b) lowering the difficulty of solving theoptimization problem (c) increasing the dynamic responsecapacity ofmultipleUAVs and (d) cutting down the influenceof the restricted networks on the solving structure

4 Optimal Consensus Based onCooperative Game

41 Cooperative Game Theory Considering a team with 119899

players the quadratic function (12) is constructed to describethe cost of the player in the team

119869119894=12int

T

0(119909

119879

119894Q

119894119909119894+119906

119879

119894R

119894119906119894) 119889119905 (12)

where Q119894and R

119894are positive definite matrices with proper

dimension Each player satisfies the following dynamicalmodel

= A119909+B11199061 + sdot sdot sdot +B119899119906119899 119909 (0) = 1199090 (13)

in which A and B119894(119894 = 1 119899) are constant matrices with

proper dimensions and 119906119894is the decision of the 119894th player

The state variable 119909 can be reflected by the other playerrsquosdecision in the minimization process That means that theplayers may have conflicting interests If a player decides tominimize its cost in a noncooperative manner the decisionchosen by the 119894th player can increase the cost of the others dueto the coupling relationship However if the players decideto cooperate individual cost may be minimized and hencewe can get a smaller team cost This will result in the set ofPareto-efficient solutions For the set of inequalities 119869

119894(U) le

119869119894(Ulowast

) 119894 = 1 2 119899 if there is not at least one solution UUlowast

= [1199061 1199062 119906119899] is called Pareto-efficient solution and

the corresponding costs Jlowast = [1198691(Ulowast) 1198692(Ulowast

) 119869119899(Ulowast

)] arePareto solution

The solution of the minimization problem cannot bedominated by any other solution

Ulowast(120572) = argmin

UisinU

119899

sum

119894=1120572119894119869119894 (U) (14)

where 120572 = (1205721 120572119899) and sum119899

119894=1 120572119894= 1 It is a set of Pareto-

efficient solutions of the above problemThe solutions are thefunctions of the parameter 120572 The final solution should beselected according to an axiomatic approach as our decisionfor the cooperative problemTheNash-bargaining solution isselected in this paper

120572 = argmax120572

119899

prod

119894=1[119869

119894(120572Ulowast

) minus 119869119905119901119894] (15)

where 119869119905119901119894

(119894 = 1 119899) are the individual costs calculatedby using the noncooperation solution that is obtained byminimizing the cost (12)

The coefficient 120572 can be obtained (see Theorem 610 in[15]) by

120572lowast

119895=

prod119894 =119895(119869

lowast

119894(120572

lowastUlowast

) minus 119869119905119901119894)

sum119899

119894=1prod119896 =119894(119869

lowast

119896(120572

lowastUlowast) minus 119869119905119901119896)

(16)


42 CGOC Problem Description When the multiagent sys-tems achieve consensus all agents get the same values Thusthe individual cost is defined as

119869119888

119894=12

sdot int

T

0[

[

sum

119895isinN119894

(120579119894minus 120579

119895)119879

Q119894119895(120579

119894minus 120579

119895) + 119906

119879

119894R

119894119906119894]

]

119889119905

(17)

whereQ119894119895andR

119894are symmetric positive definite matrices

The overall cost of the multiagent systems is obtained byweighted individual cost

119869119888119892=

119899

sum

119894=1120572119894119869119888

119894(U) = 1

2int

infin

0(120579

119879Q

119888119892120579 +U119879

R119888119892U) 119889119905 (18)

where the coefficient matrices are

Q119888119892= [Q

119894119895]119899times119899

Q119894119894= sum

119895isinN119894

120572119895Q

119895119894+ sum

119896isinN119894

120572119894Q

119894119896

Q119894119895=

minus120572119894Q

119894119895minus 120572

119895Q

119895119894119895 isin N

119894

0 119895 notin N119894

R119888119892= diag 1205721R1 120572119899

R119899

(19)

According to cooperative game theory the smallest over-all cost 119869

119888119892can be obtained when the multiagent system

achieve consensus Taking the communication delay (1205911 =

sdot sdot sdot = 120591119898= 120591) into account the dynamic model of the agent

system can be described as follows

(119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) (20)

in which 120579 = [120579119879

1 120579119879

119899]119879 is the state vector A

119888119892= [A

119894119895]119899times119899

A

119894119894= 0 A

119894119895= minus(12)119897

119894119895Hminus1

V119894Q119894119895 and B

119888119892= I HV119894

is the solution of the Riccati equation (21) and 119897119894119895is the

element of the Laplacian matrix which is used to describethe interconnection of the communication

minusH119879

V119894Rminus1119894HV119894 + sum

119895isinN119894

Q119894119895= 0 (21)

The cooperative game based optimal consensus (CGOC)can be described as the following optimization problem

minUrarrU

119869119888119892

st (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)

(22)

According to cooperative game theory the Pareto-optimal solution set can be obtained by


UisinU

119899

sum

119894=1120572119894119869119888

119894(U) (23)

It is easy to see that Pareto-efficient solution is a functionof the parameter 120572 Further the unique Nash bargainingsolution is calculated from the above set Consider

120572 = argmax120572

119899

prod

119894=1[119869

119888

119894(120572Ulowast

) minus 119869119905119901119894] (24)

in which 119869119905119901119894

is the individual cost calculated using thenoncooperative strategy (see [10])

119906119905119901119894

=12H

minus1V119894 sum

119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) minusR

minus1119894HV119894 (119905) 120579119894 (25)

When the multiagent systems achieve consensus theoptimal parameter 120572lowast is determined In the following we willdiscuss how to get the optimal control strategy of problem(22)

43 CGOC Problem Solving Due to the existence of thecommunication delay in the dynamicmodel (20) of the agentit is difficult to get the exact solution or a numerical solutionof such a problem The solving method of the two-pointboundary problem with delay and ahead term is adopted inthis section [16]

Firstly introduce the Lagrangian operator 120582(119905) and definethe Hamilton function

H119888119892 (120579 120582U) ≜

12[120579

119879Q

119888119892120579+U119879

R119888119892U]

+ 120582119879(119905) [A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)]

(26)

By the optimal control theory we can get

(119905) = nabla120582H

119888119892 119905 isin [0T]

(119905) =

minusnabla120579H

119888119892minus nabla

120579120591H

119888119892119905 isin [0T minus 120591]

minusnabla120579H

119888119892119905 isin [T minus 120591T]

(27)

where nabla120582H

119888119892 nabla

120579H

119888119892 and nabla

120579120591H

119888119892are the gradients at 120582 120579

and 120579120591 By formula (26) formulas (27) can be rewritten as

120579 (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905 + 120591) 119905 isin [0T minus 120591]

minusQ119888119892120579 (119905) 119905 isin [T minus 120591T]

(28)

in which the initial value is 120579(119905) = 120599(119905) 119905 isin [minus120591 0] Thus theoptimal control strategy is

U (119905) = minusRminus1119888119892B119879

119888119892120582 (119905) 119905 isin [0T] (29)


Rewrite (119905) and (119905) as follows

(119905) = A119888119892120579 (119905) +B119888119892

U (119905) minusA119888119892 [120579 (119905) minus 120579 (119905 minus 120591)] 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879

119888119892[120582 (119905 + 120591) minus 120582 (119905)] 119905 isin [0T minus 120591]

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)

The sensitivity parameter 120576 is introduced we have

(119905 120576) = A119888119892120579 (119905 120576) +B119888119892

U (119905 120576) minus 120576A119888119892 [120579 (119905 120576) minus 120579 (119905 minus 120591 120576)] 119905 isin [0T]

(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879

119888119892[120582 (119905 + 120591 120576) minus 120582 (119905 120576)] 119905 isin [0T minus 120591]

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)

and here 120576 is a scalar and it satisfies 0 le 120576 le 1Assume that 120579(119905 120576) 120582(119905 120576) and U(119905 120576) are differentiable

at 120576 = 0 and their Maclaurin series converges at 120576 = 1

120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)

Thus the suboptimal control strategy of the above prob-lem (31) is equivalent to the sum of the solutions of the two-point boundary problem including the 0th-order and the 119894th-order problems

(i) Two-point boundary problem with the 0th-order

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)

(ii) Two-point boundary problem with the 119894th-order

(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582

(119894minus1)(119905)] 119905 isin [0T minus 120591]

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)

431The Solution of the 0th-Order Problem Assume that theLagrangian operator is

120582(0)(119905) ≜ X120579(0) (119905) (39)

where X is a matrix with proper dimension

Substituting formula (34) with (39) we get

U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)

Moreover the Lagrangian operator should satisfy

(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)

On the other hand formula (35) can be rewritten as

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)

Comparing (42) with (43) the Riccati matrices equationcan be obtained as

A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)

Therefore the optimal control strategy U(0) of the 0th-order problem can be obtained through calculating X and120579(0)

432 The Solution of the 119894th-Order Problem Similar to the0th-order problem the Lagrangian operator is selected as

120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)

where X is a matrix with proper dimension and 120588119894(119905) is an

adjoint variable to be determinedSubstitute the Lagrangian operator 120582(119894)

(119905) into (36) and(38) We have

120588119894 (119905) =

minus [A119888119892minus B

119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588

119894minus1 (119905 + 120591) minus 120588119894minus1 (119905)]

V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)

In addition the state equation with the 119894th-order problem is

(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)

in which the initial values are

120579(119894)(0) = 0 119894 = 1 2 (49)

Thus the control strategy of the 119894th-order problem is

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)

and the suboptimal control strategy is obtained by the former119872 items

U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)

In (51) there exists an optimal control strategy when119872 tends to infin there exists an optimal control strategy Inparticular the control strategy isU(119905) = minusRminus1

119888119892B119879

119888119892X120579(119905)when

119872 = 0In the following the optimal algorithm of the CGOC

problem (22) is given with proper Maclaurin items119872

Algorithm 1

Step 1 Solve the Riccati equation (44) to get X

Step 2 Get 120579(0)(119905) and 120582(0)(119905) by (41) and (39) Let 119894 = 1

Step 3 Calculate 120588119894(119905) by (46)

Step 4 If 119894 lt 119872 solve 120579(119894)(119905) by (48) Otherwise execute Step6

Step 5 Let 119894 = 119894 + 1 and then execute Step 3

Step 6 Get the optimal control strategy Ulowast(119905) by (51)

According to the cooperative game theory the optimalcontrol strategy is a function of the parameter 120572 Thus arecursive algorithm can be given to get the Nash bargainingsolution

Algorithm 2

Step 1 Set the initial value 1205720 = [1119899 1119899]

Step 2 Calculate the suboptimal control strategy 119880lowast(120572

0) By

Algorithm 1


Step 3 Check the equation 119869119888

119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If

it does not hold there exists an 119890 satisfying 119869119888119890(Ulowast

) gt 119869119905119901119890

Update the parameter 1205720 with the following formula andthen execute Step 2

1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)

Step 4 Compute 119895by (16)

Step 5 Update parameter 120572 by

1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572

0119895| lt 0001 119895 = 1 119899 hold then terminate the

algorithm and let 120572 = Otherwise execute Step 2

Next we will discuss the convergence of the CGOCalgorithm When 119872 = 0 the CGOC algorithm can bedescribed as

(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)

The items minusB119888119892Rminus1

119888119892B119879

119888119892X and A

119888119892can be seen as the coef-

ficients I119899otimes H and L

120590otimes Γ in Theorem 2 (119897 = 0) in [17]

Thus the CGOC algorithm can converge with the givencommunication delay When 119872 gt 0 the (119905) can also berearranged with the items 120579(119905) and 120579(119905 minus 120591

119902) Similarly the

convergence range can be obtainedThe distributed solving method of the rendezvous prob-

lem is given in Algorithm 3

Algorithm 3

Step 1 Get the waypoints sequence of each UAV using theroute planning method [18] Consider

119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)

Step 2 Calculate the path length in the WGS84 coordinatesystem and get the ETA ranges which satisfy the restrict(3) Define the set 119877ETA119894

= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =

1 2 119899 If 119877ETA1 cap 119877ETA2 cap sdot sdot sdot cap 119877ETA119873= 0 holds then

execute Step 3 Otherwise return to Step 1

Step 3 Exchange the estimated arriving time 120579119894(119905) with the

other UAVs and confirm the communication delay 120591119894119895(119905) 119895 isin

119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)

Number of nodes is 500 and K (number of edge attempts) is 10

7142 71444472447444764478

Zoom in

dETA1 = 6496kmdETA2 = 5540kmdETA3 = 6167km

T1

T2

T3

T4

T5

T6

Figure 3 The typical scenery of multi-UAVs rendezvous problem

119970a 119970c119970b

1 1 1

2 2 23 3 3

Figure 4 The time varying interconnection of the communicationnetwork (the interval time of signal 120590(119905) [0 inf] rarr 119886 119887 119888 is 1 s)

Step 4 GetUlowast(120572) and 120572lowast by Algorithms 1 and 2Then obtain

the optimal control strategy Ulowast= [119906

119897

1 119906119897

119899]

Step 5 Select the 119894th component 119906119897

119894of Ulowast and substitute it

into (11) The optimal reference velocity instruction V119888119894(119905) can

be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)

Step 6 Calculate the reference heading instruction 120593119888

119894(119905)

considering the waypoints information

Step 7 Send the instructions V119888119894(119905) and 120593119888

119894(119905) to the autopilot

system (4) If theUAVs arrive at target position terminate thealgorithm otherwise execute Step 3

5 Simulations and Results

51 Simulation Environment andExperimental Conditions Inorder to validate the effectiveness of the proposed methodtwo experiments are illustrated in this section A typicalsimulation environment of the multi-UAV rendezvous prob-lem is given in Figure 3 Six artillery threats are placed in


0 20 40 60 80 1000

005

01

015

02

Coordination time (s)

Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s

)(b) Coordination variable

0 20 40 60 80 100Coordination time (s)

UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)

(c) Velocity variable

Figure 5 The response cure of 120579ETA and 119881 under delay I

the square area (100 km times 100 km) Let the initial positionsof three UAV be (20 km 20 km) (40 km 10 km) and (60 km15 km) and let the target positions be (50 km 70 km) (60 km60 km) and (70 km 70 km)The results of the route planningare shown in the figure The initial velocities of the threevehicles are 27ms 32ms and 30ms respectivelyThe initialpath lengths of the three UAVs are 6496 km 5540 km and6167 km using the TS module in the distributed solvingstructureTheir performance parameters are listed as follows119877min = 20m 119886V = 02 119886

120593= 005 119908head = 12 rad Vmin =

10ms and Vmax = 40ms

Two network restrictions are considered in the experi-ments One is the time varying communication delay 120591(119905)the other one is the time varying interconnection of thecommunication network

(1) time varying delay 120591(119905)

(i) delay I Markov random process 0 0092 02with the probabilities 03 03 and 04

(ii) delay II normal distribution 120591(119905) sim

119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)


Figure 6 The response cure of 120579ETA and 119881 under delay II

(iii) delay III sine function 120591(119905) = 02|sin(0902119905)|

(2) jointly connected topologies G119886G

119887G

119888 see

Figure 4

52 Experiment Results In the simulation we show theresponse of variable 120579ETA and velocity 119881 Let119872 = 0 We canget the Nash bargaining solution 120572 = [0423 0301 0276] byAlgorithms 1 and 2 Further the variable 120579ETA and the velocity119881

119888 can be obtained by Algorithm 3 The results are given inFigures 5 6 and 7

It is easy to see that the coordination variable 120579ETAachieves consensus from the Figures 5 6 and 7The time usedto adjust the 120579ETA is less than 5 s However there is an obviousoscillation in that process For the case of communicationdelays I II and III the rendezvous times are 2103 s 2106 sand 2102 s respectively The velocity instruction 119881

119888 can becomputed by Algorithm 3 and their response cures are alsoshown in Figures 5(c) 6(c) and 7(c)

Furthermore the simulation process of the rendezvousproblem is shown in Figure 8 The result illustrates theeffectiveness of the CGOC method



0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500

0 50 100Coordination time (s)

120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)


Figure 7 The response cure of 120579ETA and 119881 under delay III

53The Comparison of the Optimality andDynamic ResponseThecomparison of the energy consumptions is given by usingthe proposed CGOC method and the NCOC method [10]Considering the randomness and uncertainty property ofthe delay 20 statistical experiments are given in Table 1 withcommunication delays I II and III

The following conclusions can be obtained from Table 1

(1) The energy consumptions of the proposed CGOCmethod and the NCOCmethod are close The resultsimply that the CGOC method has better robustnessadaptability than NCOC method

(2) The overall cost using the CGOC method is less thanthat using the NCOC method The case of number3 UAV is the biggest one about 417 The secondone is number 2 UAV about 72 Number 1 UAV isthe minimum one about 25 In other words thereduction of the overall cost is about 171 Thus theproposed CGOC method is better than NCOC interms of the optimization

Another important issue is the dynamic response whennew incidents occur For the case of the threat appearing (thelength decrease 15 km) at 40 s and disappearing (the length


S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00

20 40 60 80 100x coordinate (km)

20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s

Figure 8 The simulation of the rendezvous process based on CGOC method


0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3

(a) Coordination variable

0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3

(b) Velocity variable

Figure 9 The response cure of 120579ETA and 119881 using NCOC method


1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3


Figure 10 The response cure of 120579ETA and 119881 using CGOC method

Table 1 The comparisons of the energy consumptions using NCOC method and CGOC method

Communication restrictions NCOC method (times103) CGOC method (times103)UAV1 UAV2 UAV3 UAV1 UAV2 UAV3

Delay I and topologiesG119886G

119887G

11988816506 15650 4422 16208 14617 2587

Delay II and topologiesG119886G

119887G

11988816506 15654 4425 16117 14532 2578

Delay III and topologiesG119886G

119887G

11988816513 15664 4436 15970 14403 2570


increase 15 km) at 80 s the dynamic responses are given inFigures 9 and 10 It is easy to see that CGOC requires lessadjustment time and lower oscillation

6 Conclusions

The cooperative game based optimal consensus (CGOC)algorithm has been proposed to solve the multi-UAV ren-dezvous problemwith complex networks Mainly the follow-ing contributions have been concluded in this paper (1) themathematical model and distributed solving framework ofthe rendezvous problem have been established (2) CGOCalgorithm has been presented to minimize the overall costof the multi-UAV system The solving strategy of CGOChas been given theoretically using the cooperative gameand sensitivity parameter method Numerical examples andsimulation results have been given to demonstrate the effec-tiveness with different network conditions and the benefit onthe overall optimality and dynamic response

Conflict of Interests

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

Acknowledgment

This work was supported by NSFC (61203355) and STDP(20130522108JH) funded by China government

References

[1] T W McLain and R W Beard ldquoCooperative path planning fortiming-criticalmissionsrdquo inProceedings of the AmericanControlConference pp 296ndash301 Denver Colo USA June 2003

[2] D R Nelson T W McLain R S Christiansen et al ldquoInitialexperiments in cooperative control of unmanned air vehiclesrdquoin Proceedings of the 3rd AIAA Technical Conference Workshopand Exhibit lsquoUnmanned-UnlimitedrsquomdashCollection of TechnicalPapers pp 666ndash674 Chicago Ill USA 2004

[3] T W McLain and R W Beard ldquoCoordination variables coor-dination functions and cooperative-timing missionsrdquo Journalof Guidance Control and Dynamics vol 28 no 1 pp 150ndash1612005

[4] D R Nelson T W McLain and R W Beard ldquoExperimentsin cooperative timing for miniature air vehiclesrdquo Journal ofAerospace Computing Information and Communication vol 4no 8 pp 956ndash967 2007

[5] RWWei RenBeard and TWMcLain ldquoCoordination variablesand consensus building inmultiple vehicle systemsrdquo in Proceed-ings of the 2003 Block Island Workshop on Cooperative ControlLecture Notes in Control and Information Sciences Series pp171ndash188 Block Island RI USA 2005

[6] D BKingstonWRen andRWBeard ldquoConsensus algorithmsare input-to-state stablerdquo in Proceedings of the American ControlConference (ACC rsquo05) pp 1686ndash1690 Portland Ore USA June2005

[7] L Yuan Z Chen R Zhou and F Kong ldquoDecentralized controlfor simultaneous arrival of multiple UAVsrdquo Acta Aeronautica etAstronautica Sinica vol 31 no 4 pp 797ndash805 2010 (Chinese)

[8] S Zhao and R Zhou ldquoCooperative guidance for multimissilesalvo attackrdquo Chinese Journal of Aeronautics vol 21 no 6 pp533ndash539 2008

[9] R Ghabcheloo I Kaminer A P Aguiar and A PascoalldquoA general framework for multiple vehicle time-coordinatedpath following controlrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 3071ndash3076 St Louis Mo USA June2009

[10] Q J Zhang J S Wang Z Q Jin and X Q Shen ldquoNon-cooperative solving method for a multi-UAV rendezvous prob-lem in a complex networkrdquo Journal of Southeast University vol43 supplement 1 pp 32ndash37 2013

[11] F Jia P Yao J Chen and B Wang ldquoDistributed cooperativeand optimized control for gathering mission of multiple UAVsrdquoElectronics Optics amp Control vol 21 no 8 pp 24ndash32 2014(Chinese)

[12] B Wang P Yao F Jia and J Chen ldquoOptimal self organizedcooperative control for mission rendezvous of multiple UAVsrdquoElectronics Optics amp Control vol 21 no 11 pp 5ndash13 2014(Chinese)

[13] X Fu H Cui and X Gao ldquoDistributed solving method ofmultiple UAVs rendezvous problemrdquo Systems Engineering andElectronics In press (Chinese)

[14] R W Beard T W McLain D B Nelson D Kingston and DJohanson ldquoDecentralized cooperative aerial surveillance usingfixed-wing miniature UAVsrdquo Proceedings of the IEEE vol 94no 7 pp 1306ndash1323 2006

[15] J Engwerda LQDynamic Optimization and Differential GamesJohn Wiley amp Sons 2005

[16] G-Y Tang and Z-W Luo ldquoSuboptimal control of linear systemswith state time-delayrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics pp 104ndash109Tokyo Japan October 1999

[17] Q Zhang Y Niu L Wang L Shen and H Zhu ldquoAverageconsensus seeking of high-order continuous-time multi-agentsystems with multiple time-varying communication delaysrdquoInternational Journal of Control Automation and Systems vol9 no 6 pp 1209ndash1218 2011

[18] Y Chen F Su and L-C Shen ldquoImproved ant colony algorithmbased on PRM for UAV route planningrdquo Journal of SystemSimulation vol 21 no 6 pp 1658ndash1666 2009 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the NCOC method [10] is developed and the cooperativemethod of solving the rendezvous problem is proposed Thesystem achieves the overall optimal cost in this process Thenovel method can deal with the complex network restrictionsuch as switching topologies communication delay

The remainder of this paper is organized as followsFirstly the mathematical description of rendezvous problemis given Based on the results of [5ndash9] the solving frameworkis proposed using the coordination function and coordi-nation variable A novel distributed control method usingthe cooperative game based optimal consensus (CGOC)algorithm is proposed for the ldquocooperativerdquo UAVs Finallynumerical experiments and simulation results are illustratedto show the effectiveness and benefit of the proposedmethod

2 Problem Description

21 Basic Assumptions and Physical Constraints This paperfocuses on the distributed control method of the rendezvousproblem rather than the flight control of the platformHencewe assume that

(1) all UAVs are small size(2) each UAV is equipped with the autopilot which can

track the waypoint automatically(3) mission control and flight control can be decoupled

respectively

Thus the flight control problem can be simplified In orderto show the physical characteristics of the platform the flightconstraints and related performance parameters are listed asfollows

(i) the maximum and minimum speed Vmax Vmin(ii) the minimum turning radius 119877min(iii) the maximum flight time 119905max

22 Incomplete Kinematic Model of UAV Similar to [1ndash7 9]the 2D kinematic model of the UAV is selected to study therendezvous problem (forall119894 = 1 2 119899)

119894119909 (119905) = V

119894 (119905) cos120593119894 (119905)

119894119910 (119905) = V

119894 (119905) sin120593119894 (119905)

119894 (119905) = 119908119894 (119905)

(1)

where z119894(119905) = [119911

119894119909(119905) 119911

119894119910(119905)]

119879 denotes the position vector ofthe 119894th UAV and 120593

119894(119905) is the heading angle V

119894(119905) is the velocity

and119908119894(119905) is the changing rate of the heading angular velocity

119908119894(119905) and V

119894(119905) satisfy

119908119894 (119905) isin [minus119908head 119908head] (2)

V119894 (119905) isin [Vmin Vmax] (3)

The parameters 119908head Vmin and Vmax are determined by thephysical performance

The autopilot of each UAV maintains the expectedheading angle and velocity Its mathematical model can bedescribed by two first-order differential equations

119894 (119905) = 119886120593 (120593

119888

119894(119905) minus 120593119894 (119905))

V119894 (119905) = 119886V (V

119888

119894(119905) minus V119894 (119905))

(4)

The variable or vector with the superscript 119888 signifies thereference instruction Parameters 119886

120593and 119886V are the constant

coefficients of the heading and velocity channel of theautopilot

23 CommunicationModel Graph theory is used to describethe communication model among multi-UAV system seeFigure 1 Let 119866 = (VEA) denote the relationship betweenmultiple UAVs with the set of nodesV = 1205921 120592119899 the setof edgesE sub VtimesV and adjacent matrixA = [119886

119894119895]The node

indices belong to a finite index setI = 1 2 119899 The edgecan be depicted by (120592

119894 120592

119895) and the value of 119886

119894119895corresponds

to the edge of the graph that is (120592119894 120592

119895) isin E hArr 119886

119894119895gt 0 The

neighbors set of node 120592119894is defined byN

119894= 120592

119895isin V (120592

119894 120592

119895) isin

ESince the phenomenon such as communication delay

interruption and limited bandwidth may appear in practicalapplication in this paper two restrictions will be considered

(i) communication delay 120591119894119895 it denotes the transfer delay

of the message from the 119895th to the 119894th UAV(ii) time varying interconnections G

120590(119905)isin G

119886G

119887

G119873ss 120590(119905) [0infin] rarr 119886 119887 119873ss denotes the

switching signal with successive times to describe thetopology switches

24 Mathematical Model of Rendezvous Problem The math-ematical description of the rendezvous problem will be givenas follows The initial and target position of the 119894th UAVare defined as z

1198940 and z119894119891 Then the flight trajectory can be

defined as

Z119894(119879

119894) ≜ z

119894 (119905) 0le 119905 le119879119894 (5)

If there exist input sequence 120593119888

119894(119905) and V119888

119894(119905) 0 le 119905 le 119879

119894

satisfying z119894(119879

119894) = z

119894119891 kinematic model (1) and restricts (2)

(3) thenZ119894(119879

119894) is the feasible flight path of the 119894th UAV

For each feasible path UAV should avoid threats andreturn to the base safely with adequate fuel In fact there isa strong coupling relationship among these restrictions Onone hand avoiding threats means longer path and higherflight velocity It demands less residence time of the aircraftin danger On the other hand fuel saving means shorter pathand lower velocity Thus the rendezvous problem is a timecoordination problem considering the exposure time in themission area and fuel consumption

Define the cost function of the pathZ119894(119879

119894) as

119869 (Z119894(119879

119894)) = 1198961119869threat (Z119894

(119879119894)) + 1198962119869fuel (Z119894

(119879119894))

=

119873119879

sum

119901=1int

119879119894

0

119896110038171003817100381710038171003817ℎ119901minus 119911

119894 (119905)10038171003817100381710038171003817

4 119889119905 +int119879119894

01198962V

2119894(119905) 119889119905

(6)


UAV 3 UAV 3 UAV 3

UAV 1 UAV 1 UAV 1

UAV 2 UAV 2 UAV 2

UAV 4 UAV 4 UAV 4

1205911212059123

12059134

12059143

119970a = (119985a ℰa119964a)

119985a = 1205921 1205922 1205923 1205924

ℰa = (2 1) (3 2) (3 4) (4 2) (2 4)

119970c = (119985c ℰc 119964c )

119985c = 1205921 1205922 1205923 1205924

ℰc = (1 2) (2 1) (3 2) (3 4) (4 3)

119970b = (119985b ℰb119964b)

119985b = 1205921 1205922 1205923 1205924

ℰb = (2 1) (2 4) (4 3)

119964a =

0

0

0

1

0

1

1

0

0

0

0

0

0

1

0

1119964b =

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

1119964c =

0

0

0

1

1

1

0

0

0

0

1

0

0

1

0

0






min119899

sum

119894=1119869 (Z

119894(119879

119894))

st Z119894(119879


= 119879ETA

(7)



Forb

idde

n ar

ea


Forbidden area


Threaten border

Threaten border

Threaten border

Threa

ten border

Threaten border

UAV 2

UAV 2

UAV 3

UAV 3UAV 1

UAV 1 New threaten








119894isin 120594

119894The decision


119894) stands for


119894(119909


variable








119894is


119888



119894to APmodule


119894and 120593119888

119894




(119905) = minusV119894(119905) Given 119889ETA119894



119879ETA119894 (119905) =119889ETA119894 (119905)

V119894 (119905)

(8)


TETA119894 (119905) = 119905 +119879ETA119894 (119905) 119905 isin [1199050 119905119891] (9)


TETA119894 (119905) = 1+[V

119894 (119905)119889len119894 (119905) minus 119889len119894 (119905) V119894 (119905)]

V2119894(119905)

= minus119879ETA119894 (119905) 119886119881 [V

119888

119894(119905) minus V

119894 (119905)]

V119894 (119905)

(10)


V119888119894(119905) = V

119894 (119905) +V119894 (119905) TETA119894 (119905)

119886119881119879ETA119894 (119905)

(11)





119869119894=12int

T

0(119909

119879

119894Q

119894119909119894+119906

119879

119894R

119894119906119894) 119889119905 (12)

where Q119894and R







119894(U) le

119869119894(Ulowast




) 119869119899(Ulowast




UisinU

119899

sum

119894=1120572119894119869119894 (U) (14)

where 120572 = (1205721 120572119899) and sum119899



120572 = argmax120572

119899

prod

119894=1[119869

119894(120572Ulowast

) minus 119869119905119901119894] (15)

where 119869119905119901119894



120572lowast

119895=

prod119894 =119895(119869

lowast

119894(120572

lowastUlowast

) minus 119869119905119901119894)

sum119899

119894=1prod119896 =119894(119869

lowast

119896(120572


(16)



119869119888

119894=12

sdot int

T

0[

[

sum

119895isinN119894

(120579119894minus 120579

119895)119879

Q119894119895(120579

119894minus 120579

119895) + 119906

119879

119894R

119894119906119894]

]

119889119905

(17)




119869119888119892=

119899

sum

119894=1120572119894119869119888

119894(U) = 1

2int

infin

0(120579

119879Q

119888119892120579 +U119879

R119888119892U) 119889119905 (18)


Q119888119892= [Q

119894119895]119899times119899

Q119894119894= sum

119895isinN119894

120572119895Q

119895119894+ sum

119896isinN119894

120572119894Q

119894119896

Q119894119895=

minus120572119894Q

119894119895minus 120572

119895Q

119895119894119895 isin N

119894

0 119895 notin N119894

R119888119892= diag 1205721R1 120572119899

R119899

(19)






(119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) (20)

in which 120579 = [120579119879

1 120579119879


119888119892= [A

119894119895]119899times119899

A

119894119894= 0 A

119894119895= minus(12)119897

119894119895Hminus1

V119894Q119894119895 and B

119888119892= I HV119894



minusH119879

V119894Rminus1119894HV119894 + sum

119895isinN119894

Q119894119895= 0 (21)


minUrarrU

119869119888119892

st (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)

(22)



UisinU

119899

sum

119894=1120572119894119869119888

119894(U) (23)


120572 = argmax120572

119899

prod

119894=1[119869

119888

119894(120572Ulowast

) minus 119869119905119901119894] (24)

in which 119869119905119901119894


119906119905119901119894

=12H

minus1V119894 sum

119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) minusR

minus1119894HV119894 (119905) 120579119894 (25)




H119888119892 (120579 120582U) ≜

12[120579

119879Q

119888119892120579+U119879

R119888119892U]

+ 120582119879(119905) [A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)]

(26)


(119905) = nabla120582H

119888119892 119905 isin [0T]

(119905) =

minusnabla120579H


120579120591H

119888119892119905 isin [0T minus 120591]

minusnabla120579H

119888119892119905 isin [T minus 120591T]

(27)

where nabla120582H

119888119892 nabla

120579H

119888119892 and nabla

120579120591H



120579 (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905 + 120591) 119905 isin [0T minus 120591]


(28)


U (119905) = minusRminus1119888119892B119879

119888119892120582 (119905) 119905 isin [0T] (29)



(119905) = A119888119892120579 (119905) +B119888119892


(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879


minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)


(119905 120576) = A119888119892120579 (119905 120576) +B119888119892


(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879


minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)



120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)



(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)


(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582


minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)


120582(0)(119905) ≜ X120579(0) (119905) (39)



U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










UAV 3 UAV 3 UAV 3

UAV 1 UAV 1 UAV 1

UAV 2 UAV 2 UAV 2

UAV 4 UAV 4 UAV 4

1205911212059123

12059134

12059143

119970a = (119985a ℰa119964a)

119985a = 1205921 1205922 1205923 1205924

ℰa = (2 1) (3 2) (3 4) (4 2) (2 4)

119970c = (119985c ℰc 119964c )

119985c = 1205921 1205922 1205923 1205924

ℰc = (1 2) (2 1) (3 2) (3 4) (4 3)

119970b = (119985b ℰb119964b)

119985b = 1205921 1205922 1205923 1205924

ℰb = (2 1) (2 4) (4 3)

119964a =

0

0

0

1

0

1

1

0

0

0

0

0

0

1

0

1119964b =

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

1119964c =

0

0

0

1

1

1

0

0

0

0

1

0

0

1

0

0






min119899

sum

119894=1119869 (Z

119894(119879

119894))

st Z119894(119879


= 119879ETA

(7)



Forb

idde

n ar

ea


Forbidden area


Threaten border

Threaten border

Threaten border

Threa

ten border

Threaten border

UAV 2

UAV 2

UAV 3

UAV 3UAV 1

UAV 1 New threaten








119894isin 120594

119894The decision


119894) stands for


119894(119909


variable








119894is


119888



119894to APmodule


119894and 120593119888

119894




(119905) = minusV119894(119905) Given 119889ETA119894



119879ETA119894 (119905) =119889ETA119894 (119905)

V119894 (119905)

(8)


TETA119894 (119905) = 119905 +119879ETA119894 (119905) 119905 isin [1199050 119905119891] (9)


TETA119894 (119905) = 1+[V

119894 (119905)119889len119894 (119905) minus 119889len119894 (119905) V119894 (119905)]

V2119894(119905)

= minus119879ETA119894 (119905) 119886119881 [V

119888

119894(119905) minus V

119894 (119905)]

V119894 (119905)

(10)


V119888119894(119905) = V

119894 (119905) +V119894 (119905) TETA119894 (119905)

119886119881119879ETA119894 (119905)

(11)





119869119894=12int

T

0(119909

119879

119894Q

119894119909119894+119906

119879

119894R

119894119906119894) 119889119905 (12)

where Q119894and R







119894(U) le

119869119894(Ulowast




) 119869119899(Ulowast




UisinU

119899

sum

119894=1120572119894119869119894 (U) (14)

where 120572 = (1205721 120572119899) and sum119899



120572 = argmax120572

119899

prod

119894=1[119869

119894(120572Ulowast

) minus 119869119905119901119894] (15)

where 119869119905119901119894



120572lowast

119895=

prod119894 =119895(119869

lowast

119894(120572

lowastUlowast

) minus 119869119905119901119894)

sum119899

119894=1prod119896 =119894(119869

lowast

119896(120572


(16)



119869119888

119894=12

sdot int

T

0[

[

sum

119895isinN119894

(120579119894minus 120579

119895)119879

Q119894119895(120579

119894minus 120579

119895) + 119906

119879

119894R

119894119906119894]

]

119889119905

(17)




119869119888119892=

119899

sum

119894=1120572119894119869119888

119894(U) = 1

2int

infin

0(120579

119879Q

119888119892120579 +U119879

R119888119892U) 119889119905 (18)


Q119888119892= [Q

119894119895]119899times119899

Q119894119894= sum

119895isinN119894

120572119895Q

119895119894+ sum

119896isinN119894

120572119894Q

119894119896

Q119894119895=

minus120572119894Q

119894119895minus 120572

119895Q

119895119894119895 isin N

119894

0 119895 notin N119894

R119888119892= diag 1205721R1 120572119899

R119899

(19)






(119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) (20)

in which 120579 = [120579119879

1 120579119879


119888119892= [A

119894119895]119899times119899

A

119894119894= 0 A

119894119895= minus(12)119897

119894119895Hminus1

V119894Q119894119895 and B

119888119892= I HV119894



minusH119879

V119894Rminus1119894HV119894 + sum

119895isinN119894

Q119894119895= 0 (21)


minUrarrU

119869119888119892

st (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)

(22)



UisinU

119899

sum

119894=1120572119894119869119888

119894(U) (23)


120572 = argmax120572

119899

prod

119894=1[119869

119888

119894(120572Ulowast

) minus 119869119905119901119894] (24)

in which 119869119905119901119894


119906119905119901119894

=12H

minus1V119894 sum

119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) minusR

minus1119894HV119894 (119905) 120579119894 (25)




H119888119892 (120579 120582U) ≜

12[120579

119879Q

119888119892120579+U119879

R119888119892U]

+ 120582119879(119905) [A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)]

(26)


(119905) = nabla120582H

119888119892 119905 isin [0T]

(119905) =

minusnabla120579H


120579120591H

119888119892119905 isin [0T minus 120591]

minusnabla120579H

119888119892119905 isin [T minus 120591T]

(27)

where nabla120582H

119888119892 nabla

120579H

119888119892 and nabla

120579120591H



120579 (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905 + 120591) 119905 isin [0T minus 120591]


(28)


U (119905) = minusRminus1119888119892B119879

119888119892120582 (119905) 119905 isin [0T] (29)



(119905) = A119888119892120579 (119905) +B119888119892


(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879


minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)


(119905 120576) = A119888119892120579 (119905 120576) +B119888119892


(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879


minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)



120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)



(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)


(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582


minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)


120582(0)(119905) ≜ X120579(0) (119905) (39)



U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119894is


119888



119894to APmodule


119894and 120593119888

119894




(119905) = minusV119894(119905) Given 119889ETA119894



119879ETA119894 (119905) =119889ETA119894 (119905)

V119894 (119905)

(8)


TETA119894 (119905) = 119905 +119879ETA119894 (119905) 119905 isin [1199050 119905119891] (9)


TETA119894 (119905) = 1+[V

119894 (119905)119889len119894 (119905) minus 119889len119894 (119905) V119894 (119905)]

V2119894(119905)

= minus119879ETA119894 (119905) 119886119881 [V

119888

119894(119905) minus V

119894 (119905)]

V119894 (119905)

(10)


V119888119894(119905) = V

119894 (119905) +V119894 (119905) TETA119894 (119905)

119886119881119879ETA119894 (119905)

(11)





119869119894=12int

T

0(119909

119879

119894Q

119894119909119894+119906

119879

119894R

119894119906119894) 119889119905 (12)

where Q119894and R







119894(U) le

119869119894(Ulowast




) 119869119899(Ulowast




UisinU

119899

sum

119894=1120572119894119869119894 (U) (14)

where 120572 = (1205721 120572119899) and sum119899



120572 = argmax120572

119899

prod

119894=1[119869

119894(120572Ulowast

) minus 119869119905119901119894] (15)

where 119869119905119901119894



120572lowast

119895=

prod119894 =119895(119869

lowast

119894(120572

lowastUlowast

) minus 119869119905119901119894)

sum119899

119894=1prod119896 =119894(119869

lowast

119896(120572


(16)



119869119888

119894=12

sdot int

T

0[

[

sum

119895isinN119894

(120579119894minus 120579

119895)119879

Q119894119895(120579

119894minus 120579

119895) + 119906

119879

119894R

119894119906119894]

]

119889119905

(17)




119869119888119892=

119899

sum

119894=1120572119894119869119888

119894(U) = 1

2int

infin

0(120579

119879Q

119888119892120579 +U119879

R119888119892U) 119889119905 (18)


Q119888119892= [Q

119894119895]119899times119899

Q119894119894= sum

119895isinN119894

120572119895Q

119895119894+ sum

119896isinN119894

120572119894Q

119894119896

Q119894119895=

minus120572119894Q

119894119895minus 120572

119895Q

119895119894119895 isin N

119894

0 119895 notin N119894

R119888119892= diag 1205721R1 120572119899

R119899

(19)






(119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) (20)

in which 120579 = [120579119879

1 120579119879


119888119892= [A

119894119895]119899times119899

A

119894119894= 0 A

119894119895= minus(12)119897

119894119895Hminus1

V119894Q119894119895 and B

119888119892= I HV119894



minusH119879

V119894Rminus1119894HV119894 + sum

119895isinN119894

Q119894119895= 0 (21)


minUrarrU

119869119888119892

st (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)

(22)



UisinU

119899

sum

119894=1120572119894119869119888

119894(U) (23)


120572 = argmax120572

119899

prod

119894=1[119869

119888

119894(120572Ulowast

) minus 119869119905119901119894] (24)

in which 119869119905119901119894


119906119905119901119894

=12H

minus1V119894 sum

119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) minusR

minus1119894HV119894 (119905) 120579119894 (25)




H119888119892 (120579 120582U) ≜

12[120579

119879Q

119888119892120579+U119879

R119888119892U]

+ 120582119879(119905) [A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)]

(26)


(119905) = nabla120582H

119888119892 119905 isin [0T]

(119905) =

minusnabla120579H


120579120591H

119888119892119905 isin [0T minus 120591]

minusnabla120579H

119888119892119905 isin [T minus 120591T]

(27)

where nabla120582H

119888119892 nabla

120579H

119888119892 and nabla

120579120591H



120579 (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905 + 120591) 119905 isin [0T minus 120591]


(28)


U (119905) = minusRminus1119888119892B119879

119888119892120582 (119905) 119905 isin [0T] (29)



(119905) = A119888119892120579 (119905) +B119888119892


(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879


minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)


(119905 120576) = A119888119892120579 (119905 120576) +B119888119892


(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879


minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)



120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)



(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)


(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582


minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)


120582(0)(119905) ≜ X120579(0) (119905) (39)



U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869119888

119894=12

sdot int

T

0[

[

sum

119895isinN119894

(120579119894minus 120579

119895)119879

Q119894119895(120579

119894minus 120579

119895) + 119906

119879

119894R

119894119906119894]

]

119889119905

(17)




119869119888119892=

119899

sum

119894=1120572119894119869119888

119894(U) = 1

2int

infin

0(120579

119879Q

119888119892120579 +U119879

R119888119892U) 119889119905 (18)


Q119888119892= [Q

119894119895]119899times119899

Q119894119894= sum

119895isinN119894

120572119895Q

119895119894+ sum

119896isinN119894

120572119894Q

119894119896

Q119894119895=

minus120572119894Q

119894119895minus 120572

119895Q

119895119894119895 isin N

119894

0 119895 notin N119894

R119888119892= diag 1205721R1 120572119899

R119899

(19)






(119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) (20)

in which 120579 = [120579119879

1 120579119879


119888119892= [A

119894119895]119899times119899

A

119894119894= 0 A

119894119895= minus(12)119897

119894119895Hminus1

V119894Q119894119895 and B

119888119892= I HV119894



minusH119879

V119894Rminus1119894HV119894 + sum

119895isinN119894

Q119894119895= 0 (21)


minUrarrU

119869119888119892

st (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)

(22)



UisinU

119899

sum

119894=1120572119894119869119888

119894(U) (23)


120572 = argmax120572

119899

prod

119894=1[119869

119888

119894(120572Ulowast

) minus 119869119905119901119894] (24)

in which 119869119905119901119894


119906119905119901119894

=12H

minus1V119894 sum

119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) minusR

minus1119894HV119894 (119905) 120579119894 (25)




H119888119892 (120579 120582U) ≜

12[120579

119879Q

119888119892120579+U119879

R119888119892U]

+ 120582119879(119905) [A119888119892120579 (119905 minus 120591) +B119888119892

U (119905)]

(26)


(119905) = nabla120582H

119888119892 119905 isin [0T]

(119905) =

minusnabla120579H


120579120591H

119888119892119905 isin [0T minus 120591]

minusnabla120579H

119888119892119905 isin [T minus 120591T]

(27)

where nabla120582H

119888119892 nabla

120579H

119888119892 and nabla

120579120591H



120579 (119905) = A119888119892120579 (119905 minus 120591) +B119888119892

U (119905) 119905 isin [0T]

(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905 + 120591) 119905 isin [0T minus 120591]


(28)


U (119905) = minusRminus1119888119892B119879

119888119892120582 (119905) 119905 isin [0T] (29)



(119905) = A119888119892120579 (119905) +B119888119892


(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879


minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)


(119905 120576) = A119888119892120579 (119905 120576) +B119888119892


(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879


minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)



120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)



(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)


(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582


minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)


120582(0)(119905) ≜ X120579(0) (119905) (39)



U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) = A119888119892120579 (119905) +B119888119892


(119905) =

minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) minus A119879


minusQ119888119892120579 (119905) minus A119879

119888119892120582 (119905) 119905 isin [T minus 120591T]

(30)


(119905 120576) = A119888119892120579 (119905 120576) +B119888119892


(119905 120576) =

minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) minus 120576A119879


minusQ119888119892120579 (119905 120576) minus A119879

119888119892120582 (119905 120576) 119905 isin [T minus 120591T]

U (119905 120576) = minusRminus1119888119892B119879

119888119892120582 (119905 120576) 119896 isin [0T]

(31)



120579 (119905 120576) =

infin

sum

119894=0

1119894120579(119894)(119905)

120582 (119905 120576) =

infin

sum

119894=0

1119894120582(119894)(119905)

U (119905 120576) =

infin

sum

119894=0

1119894U(119894)

(119905)

(32)

where 120579(119894)(119905) = lim120576rarr 0(120597

119894120579(119905 120576)120597120576

119894) 120582

(119894)(119905) =

lim120576rarr 0(120597

119894120582(119905 120576)120597120576

119894) and U(119894)

(119905) = lim120576rarr 0(120597

119894U(119905 120576)120597120576119894)



(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905) (33)

U(0)(119905) = minusR

minus1119888119892B119879

119888119892120582(0)(119905) (34)

(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892120582(0)(119905) (35)


(119894)

(119905) = A119888119892120579(119894)(119905) +B119888119892

U(119894)(119905) + 119894A119888119892

[120579(119894minus1)

(119905) minus 120579(119894minus1)

(119905 minus 120591)] (36)

U(119894)(119905) = minusR

minus1119888119892B119879

119888119892120582(119894)(119905) (37)

(119894)(119905) =

minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) minus A119879

119888119892[120582

(119894minus1)(119905 + 120591) minus 120582


minusQ119888119892120579(119894)(119905) minus A119879

119888119892120582(119894)(119905) 119905 isin [T minus 120591T]

(38)


120582(0)(119905) ≜ X120579(0) (119905) (39)



U(0)(119905) = minusR

minus1119888119892B119879

119888119892X120579(0) (119905) 119905 isin [0T] (40)

(0)(119905) = A

119888119892120579(0)(119905) +B119888119892

U(0)(119905)

= A119888119892120579(0)(119905) minusB119888119892

Rminus1119888119892B119879

119888119892X120579(0) (119905)


= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= [A119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(41)


(0)(119905) = X(0) (119905) = X [A

119888119892minusB

119888119892R

minus1119888119892B119879

119888119892X] 120579(0) (119905)

119905 isin [0T]

(42)


(0)(119905) = minusQ119888119892

120579(0)(119905) minusA119879

119888119892X120579(0) (119905)

= [minusQ119888119892minusA119879

119888119892X] 120579(0) (119905)

(43)


A119879

119888119892X+XA

119888119892minusXB

119888119892R

minus1119888119892B119879

119888119892X+Q

119888119892= 0 (44)



120582(119894)(119905) ≜ X120579(119894) (119905) + 120588119894 (119905) 119894 = 1 2 (45)



(119905) into (36) and(38) We have

120588119894 (119905) =


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894119908119888119892119894 (119905) 119905 isin [0 119879 minus 120591]


119888119892Rminus1

119888119892B119879

119888119892X]

119879

120588119894 (119905) minus 119894XA119888119892

V119888119892119894 (119905) 119905 isin [119879 minus 120591 119879]

(46)

where

119908119888119892119894 (119905) = A119879

119888119892X119906

119888119892119905119894 (119905) +XA119888119892V119888119892119905119894 (119905)

minusA119879

119888119892[120588


V119888119892119894 (119905) = [120579

(119894minus1)(119905) minus 120579

(119894minus1)(119905 minus 120591)]

119906119888119892119905119894 (119905) = [120579

(119894minus1)(119905 + 120591) minus 120579

(119894minus1)(119905)]

(47)


(119894)

(119905) = [A119888119892minusB

119888119892R

119888119892B119879

119888119892X]

119879

120579(119894)(119905)

minusB119888119892R

minus1119888119892B119879

119888119892120588119894 (119905) minus 119894A119888119892

V119888119892119905119894 (119905)

(48)


120579(119894)(0) = 0 119894 = 1 2 (49)


U(119894)(119905) = minusR

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)] 119905 isin [0T] (50)


U (119905) = minus

119872

sum

119894=1

1119894R

minus1119888119892B119879

119888119892[X120579(119894) (119905) + 120588119894 (119905)]

= minusRminus1119888119892B119879

119888119892[X120579 (119905) +

119872

sum

119894=1

1119894120588119894 (119905)]

(51)


119888119892B119879

119888119892X120579(119905)when



Algorithm 1








Algorithm 2



0) By

Algorithm 1



119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894(Ulowast

) le 119869119905119901119894 forall119894 = 1 119899 If


) gt 119869119905119901119890


1205720119890= 120572

0119890+ 0001

1205720119894= 120572

0119894minus0001119899 minus 1

119894 = 1 119899 119894 = 119890

(52)



1205720119895= 081205720

119895+ 02

119895 (53)

If |119895minus 120572




(119905) = minusB119888119892R

minus1119888119892B119879

119888119892X120579 (119905) +A119888119892

120579 (119905 minus 120591119902) (54)


119888119892B119879

119888119892X and A








Algorithm 3


119882ETA119894= 119882

0ETA119894

1198821ETA119894

119882119891

ETA119894 119894 = 1 119899 (55)


= 119889ETA119894Vmax 119889ETA119894

Vmin 119894 =





119873119894 Let 120591 = max(120591

119894119895(119905))

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

SUAV1

G

UAV2S

G

SUAV3

G

x coordinate (km)

y co

ordi

nate

(km

)


7142 71444472447444764478

Zoom in


T1

T2

T3

T4

T5

T6


119970a 119970c119970b

1 1 1

2 2 23 3 3




119897

1 119906119897

119899]




be obtained

V119888119894(119905)

= V119894 (119905)

+

V119894 (119905) sdot [(12)H

minus1V119894 sum119895isin119873119894

Q119879

119894119895120579119895(119905 minus 120591

119894119895) + 119906

119897

119894]

119886119881sdot 120579

119894 (119905)

(56)


119894(119905)








0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

005

01

015

02


Del

ay (s

)

(a) Delay I

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


UAV1UAV2

UAV3

120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)




120593= 005 119908head = 12 rad Vmin =






119873(015 0012)



013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











013

014

015

016

017

018

Del

ay (s

)

(a) Delay II

UAV1UAV2

UAV3

0 50 1001700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

25

26

27

28

29

30

31

32

V (m

s)





119887G

119888 see

Figure 4








0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

005

01

015

02

Del

ay (s

)

(a) Delay III

UAV1UAV2

UAV3

1700

1800

1900

2000

2100

2200

2300

2400

2500


120579ET

A(s



UAV1UAV2

UAV3

26

27

28

29

30

31

32

V (m

s)









S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S

G

S

G

S

G

UAV1

UAV2 UAV3

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

T2

T1T3

T4

T5

T6

(a) 242 s

S

G

S

G

S

G

UAV1UAV2 UAV3

00


20

40

60

80

100

y co

ordi

nate

(km

)

T2

T3

T4

T5

T6

T1

(b) 576 s

0

20

40

60

80

100

S

G

S

G

S

G

0 20 40 60 80 100x coordinate (km)

y co

ordi

nate

(km

)

UAV1 UAV2 UAV3

T2

T3

T4

T5

T6

T1

(c) 913 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100y

coor

dina

te (k

m)

S

G

S

G

S

G

UAV1UAV2 UAV3

T2

T3

T4

T5

T6

T1

(d) 1264 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1UAV2

UAV3T2

T3

T4

T5

T6

T1

(e) 1790 s

0 20 40 60 80 100x coordinate (km)

0

20

40

60

80

100

y co

ordi

nate

(km

)

S

G

S

G

S

G

UAV1

UAV2

UAV3

T2

T3

T4

T5

T6

T1

(f) 2102 s



0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 1001600

1800

2000

2200

2400

2600


120579ET

A(s

)

UAV1UAV2

UAV3


0 20 40 60 80 10022

24

26

28

30

32

34


V (m

s)

UAV1UAV2

UAV3




1600

1800

2000

2200

2400

2800

2600

120579ET

A(s

)

UAV1UAV2

UAV3



22

24

26

28

30

32

34

V (m

s)

UAV1UAV2

UAV3






119887G

11988816506 15650 4422 16208 14617 2587


119887G

11988816506 15654 4425 16117 14532 2578


119887G

11988816513 15664 4436 15970 14403 2570



6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Conclusions




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article cooperative solution of multi-uav

Documents