Research Article Sequence and Direction Planning ... - Hindawi

Research ArticleSequence and Direction Planning ofMultiobjective Attack in Virtual Navigation Based onVariable Granularity Optimization Method

Chengwei Ruan Lei Yu Zhongliang Zhou and Jinfu Wang

Aeronautics and Astronautics Engineering College Air Force Engineering University Xirsquoan 710038 China

Correspondence should be addressed to Chengwei Ruan rcwrff126com

Received 22 April 2015 Accepted 10 September 2015

Academic Editor Konstantinos Karamanos

Copyright copy 2015 Chengwei Ruan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In multiobjective air attacking tasks it is essential to find the optimal combat preplanning for the attacking flightThis paper solvesthe planning problem by decomposing it into two subproblems attacking sequence planning and attacking direction planningAccording to this decomposition we propose the VGHPSO (Variable Granularity Hybrid Particle Swarm Optimization) methodVGHPSO employs the Particle SwarmOptimization ametaheuristic global optimizationmethod to solve the planning problem inmultiple granularities including optimizing the high-level attacking sequence and optimizing the low-level attacking directions forengagements Furthermore VGHPSO utilizes infeasible individuals in the swarm in order to enhance the capability of searchingin the boundary of the feasible solution space Simulation results show that the proposed model is feasible in the combat and theVGHPSO method is efficient to complete the preplanning process

1 Introduction

The traditional multi-target-attack can be presented as aunique attack style that a single flight fighter ormultiple flightfighters launch and guideweapons towardsmultiple targets atthe same timeWith the development ofmany advanced tech-nologies the air combat scenario becomes more and morecomplex the needed condition to attack different targets atthe same time is hard to satisfy and then the sequential attackstyle is born to make the air combat more reasonable

The multi-target-attack in this paper is defined as anattack process that the multiple targets which are located indistributed spaces are attacked in sequence Here the attacksequence is of a great importance according to different rela-tive situations brought by different targets and the reasonableattack sequence may make the task easier and more effectiveto complete The other important point in this paper is aboutthe transform time during thementionedmulti-target-attackIt is not avoidable to research on how to enter the attack areaof next target after completing the last attack mission Theconcept of attack direction is given to describe the transformstyle duringmission process and a proper attack direction can

shorten the total time of executing the attack task and keepour flight fighter safer

Current researches on multi-target-attack problemmainly focused on the algorithms of calculating the targetthreat [1 2] methods of target assignment [3ndash5] tacticaldecision [6 7] and so on Only few references are concernedabout the transform time during the attack task and thecommon researches on sequential attack pay much attentionto the optimization algorithms [8] the systematic researchesfrom a broader tactical view still need to be focused on

In this study a new sequential attack method basedon the classic multi-target-attack problem is put forwardwhich can make the abstract route planning problem moreconcrete and the whole problem has been transformed into amultiobjective optimization problembased on virtual naviga-tion planning The classic weapon-target-assignment (WTA)problem [9] has been changed to a PSO (Particle SwarmOptimization) problem [10] in solution space The classiccomplicated route planning problem has been decomposedinto a hierarchical problem which pay more attentions to theoptimizing of attack sequence and attack direction In theprocess of optimization the attack sequence and the attack

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 807925 10 pageshttpdxdoiorg1011552015807925

2 Mathematical Problems in Engineering

direction are optimized with relative setting granularities anda surviving policy of noninferior solutions is given to makesure that the best solutions in the Pareto boundary [11ndash14] willnot be neglected

The paper is organized as follows Section 1 is introduc-tion Section 2 presents the basic sequential attack modeland its simplified method Section 3 describes the modifiedswarm optimization algorithm with a new surviving policySection 4 exhibits the simulation results and Section 5 isconclusion

2 Background-Basic Scenario

21 The Analysis of Target Threatening Area In moderncombat scenario the flight fighter may have many advancedsensor and weapon systems which can bring much morethreats [15] And in this paper we define the targetrsquos threatas the threat of targetrsquos comprehensive ability and the threatof targetrsquos destroy ability

Figure 1 describes the traditional 1 versus 1 air combatsituation 119860 is our flight fighter and 119879 is the target V

119886and

V119905are the velocities 119877 is the relative distance between 119860 and

119879 119902 is the targetrsquos approaching angle and 120593 is the targetrsquosazimuth angle In this paper we suppose that the attitudeangle exposed to the targetrsquos radar is equal to the targetrsquosazimuth angle on condition that 119860 and 119879 are in the samehorizontal plane

In air combat the greatest threat to our flight fightercomes from the targetrsquos detection threat and the targetrsquosweapon threat Thus we need to choose an appropriateapproaching method to avoid the threatening area in orderto increase our task success rate So it is necessary to analyzethe attack area and the exposure area

Attack Area We define the attack area in this content as aspecial air-to-air missile attack area around the target andonly in this area our flight fighter can get a probabilitywithin agiven threshold to destroy the target after launching a propermissile The destroy probability may decrease rapidly to zerowhen launching a missile outside this area [16]The real-timecalculating method to get the attack area is hard to satisfythe needs of the modern combat so we apply the data fittingmethod to acquire the approximate targetrsquos attack area [17]Suppose our member carries the middle-range-missile thenthe far and near boundary of the attack area around the targetcan be calculated by the following formulas

119863max = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050)

119863min = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050) (1)

where119863max119863min are far and near boundary of the attack areaaround the target119867 is attack height119881

0 is missile launching

velocity119881119879 is target velocity

119879 is target accelerated velocity

119881119903 is target angular velocity

119903 is target accelerated angular

velocity 119899119910 ismaneuver overload120596 ismaximummissile off-

boresight launch angle 119902119879 is target approaching angle and 119905

0

is missile working time

R

q

T

A

a

t

120593

Figure 1 The relative situation of friend and foe

Then we can get the whole attack area around the targetin polar coordinate

119885119860= (119902119879 119889 ℎ) | 119902

119879isin [minus120587 120587] ℎ

isin [119867min 119867max] 119889 (119902119879 ℎ)

isin [119863min (119902119879 ℎ) 119863max (119902119879 ℎ)]

(2)

where 119902119879is the targetrsquos approaching angle 119889 is the distance

of the target and ℎ is the relative height to the target where[119867min 119867max]means the attack range in height of missile

In this study we suppose that our member has the sameattack area with the targets and both of them have nearly thesame comprehensive combat capability If there was a greatgap between these two comprehensive combat capabilitiesthen we can describe the combat as an asymmetric combatand the model is not efficient According to current modelwe need to find a proper route planning project to get a betterattack situationwhich can be useful to avoid the targetrsquos attackand to make our member safer

Exposure Area It means an area we exposed to targetrsquos radarThe scanning range of airborne radar is about plusmn60∘ Then wecan also get the exposure area as follows

119885119864= (119902119879 119889 ℎ) | 119902

119879isin [minus120579119877 120579119877] ℎ

isin [119867min 119867max] 119889 (119902119879 ℎ) isin [0 119877119877 (119902119879 ℎ)] (3)

where 120579119877is the scanning angle of targetrsquos radar 119877

119877is the

maximum detection distance and [119867min 119867max] means thedetection range in height of radar

22The Basic Description of the Model Sequential attack [18]problem can be regarded as a multiobjective optimizationproblem and once the attack sequence and the attack direc-tion are assured the whole attack logic is born In order toanalyze the multi-target-attack process in detail we focuson the situation with single target for the first step Beforebringing in the model we should give some definitions aboutthe relative problems

Definition (i) Minimum safety distance this distance can beregarded as the maximum attack distance of the enemyrsquosWhen entering this area ourmembermay get the probabilityof being attacked by the enemies (ii) Attack preparing areathis area is defined between the minimum safety distanceboundary and the far attack area boundary where our flight

Mathematical Problems in Engineering 3

Far attack areaboundary

Minimum safetydistance

Attack preparing area

Threatening areaThreatening area

LAttack direction assuringTactical maneuvering

Retreat routing

2nd circle of tacticalnavigation points

1st circle of tacticalnavigation points

Target TTarget T

L

S1

S2 S3 TT 120579

120579

120578

l1

l2l3

l4

Figure 2 The simplified model of sequential attack

fighter should complete a series of actions to complete itsattack task

From the left side of Figure 2 we can see thewhole processof single-target-attack

119871 is the virtual navigation route across the attack prepar-ing area (an approximate area in this paper) and through thewhole cross process the flight fighter may deal with severalkey stages they are listed as follows

(a) Tactical Maneuvering Stage This stage may decide whichtargets to attack and which routes to choose to enter theminimum safety distance This stage may last till the missilelaunches

(b) Attack Direction Assuring StageThis stage is used to avoidthe threaten area of enemies and find a more effective way toincrease the success rate

(c) Retreat Routing Stage This stage is very important to offeran input for the next attack process and it is also useful to findan safe retreat route to get away from the last threaten area

The three stages can describe thewhole attack process andin order to build amathematical model of the different stageswe simplified the model as the right side of Figure 2

The attack preparing area is described by two circles oftactical navigation points The inner circle is the boundaryof far attack area and the outer circle is the boundaryof minimum safety distance The missile can be launchedtowards the target when the far attack area boundary isreached so we center the target and take sample points inthe far attack area boundary every 120578 angle then we canget the 1st circle tactical navigation points We define the1st circle tactical points of target 119905 as 119877119875119905

1= 119875119905

1 119875

119905

1198731

In order to enter the 1st circle tactical points we cannotavoid the minimum safety distance boundary which can bedecided by the threatening degree of the target as well as therelative combat situations So we also center the target andtake sample points in the minimum safety distance boundaryevery 120578 (120578 = 2120587119899 (119899 = 1 2 )) angle then we canget the 2nd circle tactical points We define the 2nd circletactical points of target 119905 as 119877119875119905

2= 119876119905

1 119876

119905

1198732

Then theproblem can be simplified as a route optimization problemwhich can be described as how to find a best route through

the two circles with shortest time and least threats Then thetransformation between two targets can be completed

If target 119894 would be attacked before target 119895 then thepartial order exists and we make it as 119876119894

1≻ 119876119895

1 The solution

space of virtual navigation can be described as

119875119878 = 1198751

1 119875

1

1198731

1198761

1 119876

1

1198732

119875119879

1 119875

119879

1198731

119876119879

1

119876119879

1198732

(4)

From the right side of Figure 2 we can see that 119871 meansthe attack route through two circles of navigation points and 120579is the angle between the velocity of target and the attack routeand it can describe the attack direction towards the target

From the analysis above we can conclude the problem intwo steps

Step 1 Choose four tactical points from the two circles anddecide the attack sequence according to every single targetin different space We describe it as 119876119905

119894rarr 119875

119905

119898rarr 119875

119905

119899rarr

119876119905

119900(119898 = 119899)

Step 2 Connect the four tactical points in sequence and anew virtual navigation route is born We can describe it as atuple

119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)

where 119873 means the number of navigation route fragmentswhich connect the two kinds of navigation points in differentcircles 119878 is the initial position of our member and it is alsothe first navigation point in the attack route 119879119892 is a set ofserial numbers of targets which records the attack sequence119875119903 contains all the chosen tactical points and their orders 119864ends the navigation

3 Modeling and Algorithm Optimizing

31 Analysis of Sequential Attack and the Attack DirectionThe single-target-attack model is given in Section 2 andthe multi-target-attack model is set up in this chapter FromFigure 2 we can get the general view of sequential attack inmultiple targetsrsquo situation


New target

Targets area

T1T2

T3

T4

Sequential attack routingManeuver direction routing

Figure 3 The sketch map of multiobjective sequential attack

Sequence attack connection graph

Attack directionconnection graph

Out-tactical point

Feasible solutionTargetTactical point

In-tactical point

Figure 4 Mission decomposition in variable granularities

In Figure 3 the blue line means the attack sequence andthe red linemeans the attack directionWe can easily find thatonce the tactical points are decided the whole attack planincluding attack sequence and direction is acquired Froma tactical view of the combat situation we can solve thisproblem in three different hierarchies

In Figure 4 the problem has been divided into threeparts which describe the feasible region sequential attack

and attack direction respectively The second layer canbe described as a directed graph 119892

119878= 119892(119879 119864

119904) 119879 =

1198791 1198792 1198793 119879

119899 where 119879 is a set of targets and 119864

119904is a set

of connection arcs 119890(119879119894 119879119895) isin 119864

119904means there is a partial

order between 119879119894and 119879

119895 and 119879

119894will be attacked first The

third layer can also be described as a connected directedgraph 119879

119894= 119892119894(119875 119864119901) where 119875 is a set of tactical navigation

points and 119864119901is a set of connection arcs Then the whole


problem can be rebuilt as a variable granularity directedgraph 119892

119872= 119892(119892

119894(119875 119864119901) 119864) 119864 means the connection arcs

between different granularities

32 Modeling

321 Targetsrsquo Reformation In current air combat scenariothe targetsrsquo flight fighters always appear in formation to exe-cute associated tactical task Thus we can divide these targetsinto different groups according to their unique characters inorder to simplify the complex combat situation And first thegroup should satisfy the following formula

119897avg =119897avg

119871avg (6)

where 119897avg is the mean distance between two targets in thesame group and 119871avg is the mean distance between twodifferent groups The minimum value of 119897avg can be regardedas the important index to choose the proper projects

We give the minimum distance 119871min between groups andmaximum distance 119897max between targets in the same group indifferent projects according to the combat needs When 119897 ge119897max then the target does not belong to the group When 119897 lt119871min we can decide that the target is in the same groupThuswe let 119871min = 119897max = 1198970 = const in order to use a commonmethod for completing the targetsrsquo reformation

The concrete method to reform the target is given asfollows

(a) Set a tactical targetrsquos group as

119866119905= 1198901 1198902 119890

119895 119890119895+1 (7)

1198901 1198902 119890

119895are members in the group and 119890

119895+1is the

119895 + 1member(b) Set the attribute of the member as

119890119894= 119890 (119909

119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)

119909119894 119910119894 119911119894is the space coordinate of the target in

geographic coordinate system V119894is the velocity in

geographic coordinate system 120595119894is the flight direc-

tion angle in geographic coordinate system 119896119894is the

targetrsquos types(c) Set the constraints to reform the target

st 10038161003816100381610038161003816119909119895+1(119905) minus 119909

119894(119905)10038161003816100381610038161003816lt Δ119909

10038161003816100381610038161003816119910119895+1(119905) minus 119910

119894(119905)10038161003816100381610038161003816lt Δ119910

10038161003816100381610038161003816119911119895+1(119905) minus 119911

119894(119905)10038161003816100381610038161003816lt Δ119911

10038161003816100381610038161003816V119895+1(119905) minus V

119894(119905)10038161003816100381610038161003816lt ΔV

10038161003816100381610038161003816120595119895+1(119905) minus 120595

119894(119905)10038161003816100381610038161003816lt Δ120595

119895 + 1 le 119872max 119894 isin (1 2 119895)

(9)

Δ119909 Δ119910 Δ119911 is the threshold to reform the targets intargetsrsquo location ΔV is the velocity threshold of targetΔ120595 is the flight direction angle threshold of target119872max is themaximumnumber of the group Differentconstraints may bring different consequence so it alldepends on the combat situation

322 Attack Sequence Optimization In air combat scenariothe advantages or disadvantages towards enemies can bejudged by four conditions such as aspect relative distancevelocity and height [19] We can get the attack sequence bythe formula below

119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]

119882 = 1199081 1199082 1199083 1199084

4

sum

119894=1

119908119894= 1

(10)

where 120583(120579) 120583(119877) 120583(119881) 120583(119867) are linguistic variables in whichgrade values are between 0 and 1 the closer the value ofmembership grade is to 1 the more advantages it is to attackthe target 119882 is the weight variable set which contains fourrelative weight variables in order to stress the importance ofthe linguistic variablesThen the fitness function can be givenas follows

119865 = max119899

sum

119894

119891119894 (11)

According to different targets we can get a sequence of119891119894(119894 = 1 2 119879) and then the total advantages can be

confirmed

323 Attack Direction Optimization In the model we men-tioned above we can easily find that the attack directioncan only be confirmed by connecting two tactical navigationpoints around the target So once the best tactical points arechosen the optimal attack direction is given automatically

33 The Evaluating Indexes of the Optimization The evalu-ation function is needed to evaluate the attack route andin this situation the distance between two different tacticalpoints is far enough with the support of the ability of long-middle-range missile Thus we set up four main indexesto evaluate the whole attack route which contains the totaldistance of attack route the exposure rate the constraints ofattack direction and the constraints when turning aroundFigure 5 describes the detailed definition of each variable

331 Total Distance of Attack Route Consider

119863119871=

119873

sum

119894=1

119897119894 (12)

where 119897119894is the distance of each navigation fragment


L1

L2

li

120579iT

ei

120583jj+1

Figure 5 Definition of each variable

Initialization Fitnesscalculating

Particleupgrading

Best individualcrossing crossing

Best groupvariation Complete EndYes

NoParticle

Figure 6 Calculating steps

332 Attack Route under Threat Consider

119864119871=

119873

sum

119894=1

119890119894 (13)

where 119890119894is the distance under threat (fire threat) of each

navigation fragment the distance under threat is in relation-ship with the targetrsquos attack area and it can be calculated byanalysis of the combat situation of enemies

333 Attack Direction Constraints The attack directionshould satisfy the maximum missile off-boresight launchangle so we give the constraint as

120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)

where 120579119894describes the attack direction towards target 119868 and

120579max is the maximummissile off-boresight launch angle

334 Constraints When Turning Around Consider

120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1

is the angle between route 119895 and route 119895 + 1119897119895means the distance of route 119895 119903min is the minimum turn

radius and 120583min is the minimum turn angleThese four indexes wementioned above canmake contri-

butions to building the fitness function as follows

119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)

where 119865(119871) is the fitness function and the smaller 119865(119871)gets the better the result will be If 119865(119871) rarr infin then thevalue would be meaningless 119863

119871is the total distance of the

attack route 119864119871is the distance under threat 119863

119872and 119864

119872are

normalized coefficients 1199081and 119908

2are weight coefficients

and 1198921(119871) and 119892

2(119871) are correction terms to correct the

conditions when the target approaching angle cannot satisfythe need of maximummissile off-boresight launch angle andthe turning angle is too small to support our action

34 Calculating Process According to the problem simpli-fication above we have changed the complex model intoa typical TSP model thus we use hybrid Particle SwarmOptimization method to solve this problem The detailedsteps can be acquired in Figure 6

341 Encoding Method From Figure 7 we can see that thelength of the gene code is 4119879 + 2 each target may get 4 codeswhich can be described by decimalism and each code rangesfrom 1 to the maximum number of tactical points For betterinitializing the values wemake the two codes in themiddle ofthe gene unequal and all the code will get a random but notrepeated number from 1 to the maximum number of tacticalpoints

342 Route Modifying The two-circle-encode style hasforced all the routes to cross through the tactical points so thesituations of route crossing and repeating exist According tothe rule in Figure 8 we can get the new attack route

Theproject needs to be accessed again aftermodifying thevirtual navigation If the accessing result was not better thanthe old one we would keep the old policy


4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O

middot middot middot

Figure 7 The encoding method

Deleted routfragment

Planning virtualnavigation

Modifying virtualnavigation

Figure 8 The display of route modifying method

343 Result Space Expanding Policy When calculating theresult may not get out of the local optimization so after eachiteration we get the global optimal value

bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2

119874sdot sdot sdot 119875

V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)

Then we can deal with the existing result as follows andthe description of the space expanding policy can be seen inFigure 9

(a) Rescue the abandoned route when the followingcondition is satisfied1003816100381610038161003816gbest minus fitness

119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)

where 119892119887119890119904119905 is the global optimal value in currenttime and fitness

119894is the abandoned value in the current

circulation when the given condition is satisfied werescue the relative 119877119900119906119905

119894(119894 = 1 2 3 119899) and this

route will take part in the next circulation

(b) Get 119899 random points around each best tactical navi-gation point within the distance 119903

119894= 1205791198771198942 (119894 = 1 2)

(119877119894defines the radius of each tactical area)

(c) Expand the result space with choosing points and geta new result space Ωnew

(d) Calculating 119891 = 119892119887119890119904119905119905 119905 is the simulation time If119891 lt 120575 then we save the current result

344 Calculating Steps See Algorithm 1

4 Simulation

41 Hardware Condition CPU is P4 28GHz and RAM is15 G Matlab 2010

42 The Tactical Scenario Our flight fighter is flying towardsthe enemiesrsquo formation and the initial locations of enemiesand our member are given in Table 1

The relative situation can be seen in Figure 10

43 Simulation Result We suppose the detecting area ofenemies is stable at the moment and we need to find the bestattack sequence and attack direction at this moment We set3000 loops to optimize the final result and the result can beseen in Figures 11 and 12

From Figure 11 our member choose the proper directionto avoid the threaten areas in order to complete the task ina relative short way The attack sequence can be describedas 1198791rarr 1198793rarr 1198792rarr 1198794 and the attack direction (the

direction of enemy velocity is the initial angle 0∘) can bedecided by the angles given in Table 2

Figure 12 shows the updating process when choosing thebest solution and after 600 more loops we get the stablevalue the best route we choose can be shown as follows

bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)

In Figure 12 three different methods are compared witheach other The blue line shows the result when using tradi-tional PSOmethod without dividing them in different granu-larities then it has costmore time to get a stable result becauseof much more formations of the available tactical points Thered line means the VGHPSO method we mentioned above

it has cost less time to get a stable value but it is easy to gettrapped in the local optimal value Thus we use VGHPSOlowastmethodwhichmeans the VGHPSOmethodwith result spaceexpanding policy We can see the result reflected in cyan line

In the former chapters we can know that the numbers oftactical points are depended on the sampling angle Thus we


Table 1 Initial locations

Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)

Table 2 Attack direction display

Attack sequence 1198791

1198793

1198792

1198794

Approach angle (∘) 10728 16076 7543 13306Attack angle (∘) 80 12793 2914 17293Retreat angle (∘) 16673 13673 2051 16051

PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout

Figure 9 Result space expanding policy

400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)

Figure 10 The initial locations sketch map

do some simulation with different sampling angles and theresults can be seen in Figure 13

From left side of Figure 13 the blue parts mean thetotal distances in different sampling angles and the red partsdescribe the distances under enemiesrsquo threats Then we canget the ratio as shown in Table 3

400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)

Figure 11 The optimal route planning project

0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast

Figure 12 Updating curves in different algorithms

FromTable 3 we can see thatwhen choosing the samplingangle as 15∘ we can get a minimum comprehensive indexwhich indicates that we may get a more proper resultefficiently with less threats and shorter times

The usage of the results in this paper can only beregarded as the equal description of attack sequence andattack direction and the tactical points in virtual navigationare not the real navigation points but of great importance tosimplify the engineering problems


Input (1) Targetsrsquo attributes including the targetsrsquo number119873 the coordinate 119879(119909119879 119910119879 119911119879) the velocity V

119879and the attack area

119882 (2) Our flight fightersrsquo attributes including the number of our fighters119872 the angle 120579 to decide the number of tacticalnavigation points the coordinate 119865(119909

119865 119910119865 119911119865) and the results space Ω (3) A result space expanding policy 120588lowast to optimize

the resultSuppose Condition (1) The targets stay flying in a stable velocity and an unchanged direction (2) The tactical pointsrsquoformation should obey the constraints 120585sequence directionOutput (1) The global optimal fitness result 119892119887119890119904119905 (2) Best tactical points formation 119887119890119904119905119877119900119906119905Initialize randomΩ set the initial stateBeginFor each iteration 119894 = 1 119899 119899 is the iteration number

Cross(119865) rarr Ωnew executing the cross process towards the setting codesIf 119865119894(119894 = 1 2 120587120579) (119865

119894isin Ωnew) do not obey the constraints 120585sequence direction

While (119865119894obey the constraints 120585sequence direction)

Random 119865119894 make sure that all the elements inΩnew obey the constraints

endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894

Variate(119865) rarr Ω1015840

new executing the variation process towards the setting codesCycling the process as Cross methodExecuting (120588lowast) the saving policy is used to expand the result space to optimize the final result

EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90

Running time (s)Sampling angle (∘)

Sam

plin

g an

gle(

∘ )

Total distanceThreatened distance

Figure 13 The comprehensive values in different sampling angles


Table 3 Relative calculating results display

Sampling angle (∘) 5 15 30 45 60 72 90Threats ratio 0013 0017 0044 0090 0129 0153 0257Running times (s) 78 45 35 26 24 19 13Comprehensive index 1014 0765 154 234 3096 2907 3341

5 Conclusion

The multiobjective sequence attack and attack directionplanning methods are given in the paper We put forwarda simplified model to create the tactical points in virtualnavigation and the complex air-to-air combat planningproblemhas been transformed into a variable granularity TSPproblem then theVGHPSOmethod is used to solve the givenproblem In the end the result of the simulation shows theefficiency of the method and it also indicates that the modelis available

The model and method we mentioned in the paper areavailable only in static combat planning which belongs toprecombat planning problem But in real combat scenariowe need a dynamic data management system to optimize thereal-time policiesThen we need to pay much attention to theprediction of the future situation according to the historicaldata [20] Thus we may get a more proper and more accurateresult

Conflict of Interests

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

References

[1] S Kang H Park S Noh et al ldquoAutonomously decidingcountermeasures against threats in electronic warfare settingsrdquoin Proceedings of the International Conference on ComplexIntelligent and Software Intensive Systems (CISIS rsquo09) pp 177ndash184 IEEE Fukuoka Japan March 2009

[2] H Guo Y Lv P Wang P Ren and L Zhu ldquoTarget threatassessment of air combat based on intervals and SVRrdquo FireControl amp Command Control vol 39 no 8 pp 17ndash21 2014

[3] A Salman I Ahmad and S Al-Madani ldquoParticle swarmoptimization for task assignment problemrdquoMicroprocessors andMicrosystems vol 26 no 8 pp 363ndash371 2002

[4] P Y R Ma E Y S Lee and M Tsuchiya ldquoA task allocationmodel for distributed computing systemsrdquo IEEE Transactionson Computers vol 31 no 1 pp 41ndash47 1982

[5] D Xinghua F Yangwang X Bingsong and et al ldquoTask alloca-tion in cooperative air combat based on multi-agent coalitionrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol40 no 9 pp 1268ndash1275 2014

[6] X-G Gao Y-H Wang and B Li ldquoDecision-making methodfor the optimal coordination of fire and electronic warfarebased on integrative effectivenessrdquo Systems Engineering andElectronics vol 29 no 12 pp 2081ndash2084 2007

[7] H Guo H Xu D Liu and D Wang ldquoAir combat decision-making for cooperative multiple target attack based on adaptive

hybrid particle swarm algorithmrdquo Journal of Air Force Engineer-ing University vol 11 no 2 pp 16ndash19 2010

[8] K Horie and B A Conway ldquoOptimal fighter pursuit-evasionmaneuvers found via two-sided optimizationrdquo Journal of Guid-ance Control and Dynamics vol 29 no 1 pp 105ndash112 2006

[9] S Matlin ldquoA review of the literature on missile-allocationproblemrdquo Operations Research vol 18 no 2 pp 334ndash373 1970

[10] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on Neural Net-works pp 1942ndash1948 Perth Australia November-December1995

[11] W A Crossley ldquoOptimization for aerospace conceptual designthrough the use of genetic algorithmsrdquo in Proceedings of the1st NASADoD Workshop on Evolvable Hardware pp 200ndash207Pasadena Calif USA

[12] M Camara J Ortega and F de Toro ldquoA single front geneticalgorithm for parallel multi-objective optimization in dynamicenvironmentsrdquo Neurocomputing vol 72 no 16ndash18 pp 3570ndash3579 2009

[13] C-K Goh and K C Tan Evolutionary Multi-objective Opti-mization in Uncertain Environments Issues and AlgorithmsSpringer Berlin Germany 2009

[14] Y Yu and Z-H Zhou ldquoOn the usefulness of infeasible solutionsin evolutionary search a theoretical studyrdquo in Proceedings of theIEEECongress on Evolutionary Computation (CEC rsquo08) pp 835ndash840 IEEE Hong Kong June 2008

[15] J S McGrew J P How B Williams and N Roy ldquoAir-combatstrategy using approximate dynamic programmingrdquo Journal ofGuidance Control and Dynamics vol 33 no 5 pp 1641ndash16542010

[16] Z ZhouGeneral Aviation Fire ControlTheory NationalDefenseIndustry Press Beijing China 2008

[17] L Xiao and J Huang ldquoStudy on calculation for air-to-airmissile attack area based on improved RBF neural networkrdquo inProceedings of the International Conference on Computer Scienceand Artificial Intelligence Advances in Artificial Intelligence pp631ndash637 Sydney Australia December 2012

[18] H Zhang and D Zhou ldquoRoute planning method in air-to-ground attackrdquo Electronics Optics amp Control vol 1 no 1 pp 37ndash42 1999

[19] T-Y Sun S-J Tsai Y-N Lee S-M Yang and S-H Ting ldquoThestudy on intelligent advanced fighter air combat decision sup-port systemrdquo inProceedings of the IEEE International Conferenceon Information Reuse and Integration (IRI rsquo06) pp 39ndash44 IEEEWaikoloa Village Hawaii USA September 2006

[20] X Hai and Y Lei ldquoA situation forecast method based onintuitionistic fuzzy reasoningrdquo Electronics Optics ampControl vol18 no 4 pp 33ndash37 2011

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


direction are optimized with relative setting granularities anda surviving policy of noninferior solutions is given to makesure that the best solutions in the Pareto boundary [11ndash14] willnot be neglected

The paper is organized as follows Section 1 is introduc-tion Section 2 presents the basic sequential attack modeland its simplified method Section 3 describes the modifiedswarm optimization algorithm with a new surviving policySection 4 exhibits the simulation results and Section 5 isconclusion

2 Background-Basic Scenario

21 The Analysis of Target Threatening Area In moderncombat scenario the flight fighter may have many advancedsensor and weapon systems which can bring much morethreats [15] And in this paper we define the targetrsquos threatas the threat of targetrsquos comprehensive ability and the threatof targetrsquos destroy ability

Figure 1 describes the traditional 1 versus 1 air combatsituation 119860 is our flight fighter and 119879 is the target V

119886and

V119905are the velocities 119877 is the relative distance between 119860 and

119879 119902 is the targetrsquos approaching angle and 120593 is the targetrsquosazimuth angle In this paper we suppose that the attitudeangle exposed to the targetrsquos radar is equal to the targetrsquosazimuth angle on condition that 119860 and 119879 are in the samehorizontal plane

In air combat the greatest threat to our flight fightercomes from the targetrsquos detection threat and the targetrsquosweapon threat Thus we need to choose an appropriateapproaching method to avoid the threatening area in orderto increase our task success rate So it is necessary to analyzethe attack area and the exposure area

Attack Area We define the attack area in this content as aspecial air-to-air missile attack area around the target andonly in this area our flight fighter can get a probabilitywithin agiven threshold to destroy the target after launching a propermissile The destroy probability may decrease rapidly to zerowhen launching a missile outside this area [16]The real-timecalculating method to get the attack area is hard to satisfythe needs of the modern combat so we apply the data fittingmethod to acquire the approximate targetrsquos attack area [17]Suppose our member carries the middle-range-missile thenthe far and near boundary of the attack area around the targetcan be calculated by the following formulas

119863max = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050)

119863min = 119891 (1198671198810 119881119879 119879 119881119903 119903 120578119910 120596 119902119879 1199050) (1)

where119863max119863min are far and near boundary of the attack areaaround the target119867 is attack height119881

0 is missile launching

velocity119881119879 is target velocity

119879 is target accelerated velocity

119881119903 is target angular velocity

119903 is target accelerated angular

velocity 119899119910 ismaneuver overload120596 ismaximummissile off-

boresight launch angle 119902119879 is target approaching angle and 119905

0

is missile working time

R

q

T

A

a

t

120593

Figure 1 The relative situation of friend and foe

Then we can get the whole attack area around the targetin polar coordinate

119885119860= (119902119879 119889 ℎ) | 119902

119879isin [minus120587 120587] ℎ

isin [119867min 119867max] 119889 (119902119879 ℎ)

isin [119863min (119902119879 ℎ) 119863max (119902119879 ℎ)]

(2)

where 119902119879is the targetrsquos approaching angle 119889 is the distance

of the target and ℎ is the relative height to the target where[119867min 119867max]means the attack range in height of missile

In this study we suppose that our member has the sameattack area with the targets and both of them have nearly thesame comprehensive combat capability If there was a greatgap between these two comprehensive combat capabilitiesthen we can describe the combat as an asymmetric combatand the model is not efficient According to current modelwe need to find a proper route planning project to get a betterattack situationwhich can be useful to avoid the targetrsquos attackand to make our member safer

Exposure Area It means an area we exposed to targetrsquos radarThe scanning range of airborne radar is about plusmn60∘ Then wecan also get the exposure area as follows

119885119864= (119902119879 119889 ℎ) | 119902

119879isin [minus120579119877 120579119877] ℎ

isin [119867min 119867max] 119889 (119902119879 ℎ) isin [0 119877119877 (119902119879 ℎ)] (3)

where 120579119877is the scanning angle of targetrsquos radar 119877

119877is the

maximum detection distance and [119867min 119867max] means thedetection range in height of radar

22The Basic Description of the Model Sequential attack [18]problem can be regarded as a multiobjective optimizationproblem and once the attack sequence and the attack direc-tion are assured the whole attack logic is born In order toanalyze the multi-target-attack process in detail we focuson the situation with single target for the first step Beforebringing in the model we should give some definitions aboutthe relative problems

Definition (i) Minimum safety distance this distance can beregarded as the maximum attack distance of the enemyrsquosWhen entering this area ourmembermay get the probabilityof being attacked by the enemies (ii) Attack preparing areathis area is defined between the minimum safety distanceboundary and the far attack area boundary where our flight







Retreat routing



Target TTarget T

L

S1

S2 S3 TT 120579

120579

120578

l1

l2l3

l4










1= 119875119905

1 119875

119905

1198731


2= 119876119905

1 119876

119905

1198732




1≻ 119876119895

1 The solution


119875119878 = 1198751

1 119875

1

1198731

1198761

1 119876

1

1198732

119875119879

1 119875

119879

1198731

119876119879

1

119876119879

1198732

(4)




119894rarr 119875

119905

119898rarr 119875

119905

119899rarr

119876119905

119900(119898 = 119899)


119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)





New target

Targets area

T1T2

T3

T4





Out-tactical point


In-tactical point





119878= 119892(119879 119864

119904) 119879 =

1198791 1198792 1198793 119879


119904is a set




119895 and 119879







119872= 119892(119892



32 Modeling


119897avg =119897avg

119871avg (6)





119866119905= 1198901 1198902 119890

119895 119890119895+1 (7)

1198901 1198902 119890


119895+1is the


119890119894= 119890 (119909

119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)






st 10038161003816100381610038161003816119909119895+1(119905) minus 119909

119894(119905)10038161003816100381610038161003816lt Δ119909

10038161003816100381610038161003816119910119895+1(119905) minus 119910

119894(119905)10038161003816100381610038161003816lt Δ119910

10038161003816100381610038161003816119911119895+1(119905) minus 119911

119894(119905)10038161003816100381610038161003816lt Δ119911

10038161003816100381610038161003816V119895+1(119905) minus V

119894(119905)10038161003816100381610038161003816lt ΔV

10038161003816100381610038161003816120595119895+1(119905) minus 120595

119894(119905)10038161003816100381610038161003816lt Δ120595

119895 + 1 le 119872max 119894 isin (1 2 119895)

(9)



119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]

119882 = 1199081 1199082 1199083 1199084

4

sum

119894=1

119908119894= 1

(10)


119865 = max119899

sum

119894

119891119894 (11)


confirmed




119863119871=

119873

sum

119894=1

119897119894 (12)



L1

L2

li

120579iT

ei

120583jj+1



Particleupgrading



NoParticle



119864119871=

119873

sum

119894=1

119890119894 (13)




120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)




120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1




119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)




119872and 119864

119872are



and 1198921(119871) and 119892








4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Retreat routing



Target TTarget T

L

S1

S2 S3 TT 120579

120579

120578

l1

l2l3

l4










1= 119875119905

1 119875

119905

1198731


2= 119876119905

1 119876

119905

1198732




1≻ 119876119895

1 The solution


119875119878 = 1198751

1 119875

1

1198731

1198761

1 119876

1

1198732

119875119879

1 119875

119879

1198731

119876119879

1

119876119879

1198732

(4)




119894rarr 119875

119905

119898rarr 119875

119905

119899rarr

119876119905

119900(119898 = 119899)


119871 = ⟨119873 119878 119879119892 119875119903 119864⟩ (5)





New target

Targets area

T1T2

T3

T4





Out-tactical point


In-tactical point





119878= 119892(119879 119864

119904) 119879 =

1198791 1198792 1198793 119879


119904is a set




119895 and 119879







119872= 119892(119892



32 Modeling


119897avg =119897avg

119871avg (6)





119866119905= 1198901 1198902 119890

119895 119890119895+1 (7)

1198901 1198902 119890


119895+1is the


119890119894= 119890 (119909

119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)






st 10038161003816100381610038161003816119909119895+1(119905) minus 119909

119894(119905)10038161003816100381610038161003816lt Δ119909

10038161003816100381610038161003816119910119895+1(119905) minus 119910

119894(119905)10038161003816100381610038161003816lt Δ119910

10038161003816100381610038161003816119911119895+1(119905) minus 119911

119894(119905)10038161003816100381610038161003816lt Δ119911

10038161003816100381610038161003816V119895+1(119905) minus V

119894(119905)10038161003816100381610038161003816lt ΔV

10038161003816100381610038161003816120595119895+1(119905) minus 120595

119894(119905)10038161003816100381610038161003816lt Δ120595

119895 + 1 le 119872max 119894 isin (1 2 119895)

(9)



119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]

119882 = 1199081 1199082 1199083 1199084

4

sum

119894=1

119908119894= 1

(10)


119865 = max119899

sum

119894

119891119894 (11)


confirmed




119863119871=

119873

sum

119894=1

119897119894 (12)



L1

L2

li

120579iT

ei

120583jj+1



Particleupgrading



NoParticle



119864119871=

119873

sum

119894=1

119890119894 (13)




120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)




120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1




119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)




119872and 119864

119872are



and 1198921(119871) and 119892








4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










New target

Targets area

T1T2

T3

T4





Out-tactical point


In-tactical point





119878= 119892(119879 119864

119904) 119879 =

1198791 1198792 1198793 119879


119904is a set




119895 and 119879







119872= 119892(119892



32 Modeling


119897avg =119897avg

119871avg (6)





119866119905= 1198901 1198902 119890

119895 119890119895+1 (7)

1198901 1198902 119890


119895+1is the


119890119894= 119890 (119909

119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)






st 10038161003816100381610038161003816119909119895+1(119905) minus 119909

119894(119905)10038161003816100381610038161003816lt Δ119909

10038161003816100381610038161003816119910119895+1(119905) minus 119910

119894(119905)10038161003816100381610038161003816lt Δ119910

10038161003816100381610038161003816119911119895+1(119905) minus 119911

119894(119905)10038161003816100381610038161003816lt Δ119911

10038161003816100381610038161003816V119895+1(119905) minus V

119894(119905)10038161003816100381610038161003816lt ΔV

10038161003816100381610038161003816120595119895+1(119905) minus 120595

119894(119905)10038161003816100381610038161003816lt Δ120595

119895 + 1 le 119872max 119894 isin (1 2 119895)

(9)



119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]

119882 = 1199081 1199082 1199083 1199084

4

sum

119894=1

119908119894= 1

(10)


119865 = max119899

sum

119894

119891119894 (11)


confirmed




119863119871=

119873

sum

119894=1

119897119894 (12)



L1

L2

li

120579iT

ei

120583jj+1



Particleupgrading



NoParticle



119864119871=

119873

sum

119894=1

119890119894 (13)




120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)




120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1




119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)




119872and 119864

119872are



and 1198921(119871) and 119892








4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872= 119892(119892



32 Modeling


119897avg =119897avg

119871avg (6)





119866119905= 1198901 1198902 119890

119895 119890119895+1 (7)

1198901 1198902 119890


119895+1is the


119890119894= 119890 (119909

119894 119910119894 119911119894 V119894 120595119894 119896119894) (8)






st 10038161003816100381610038161003816119909119895+1(119905) minus 119909

119894(119905)10038161003816100381610038161003816lt Δ119909

10038161003816100381610038161003816119910119895+1(119905) minus 119910

119894(119905)10038161003816100381610038161003816lt Δ119910

10038161003816100381610038161003816119911119895+1(119905) minus 119911

119894(119905)10038161003816100381610038161003816lt Δ119911

10038161003816100381610038161003816V119895+1(119905) minus V

119894(119905)10038161003816100381610038161003816lt ΔV

10038161003816100381610038161003816120595119895+1(119905) minus 120595

119894(119905)10038161003816100381610038161003816lt Δ120595

119895 + 1 le 119872max 119894 isin (1 2 119895)

(9)



119891 = 120593 [120583 (120579) 120583 (119877) 120583 (119881) 120583 (119867) 119882]

119882 = 1199081 1199082 1199083 1199084

4

sum

119894=1

119908119894= 1

(10)


119865 = max119899

sum

119894

119891119894 (11)


confirmed




119863119871=

119873

sum

119894=1

119897119894 (12)



L1

L2

li

120579iT

ei

120583jj+1



Particleupgrading



NoParticle



119864119871=

119873

sum

119894=1

119890119894 (13)




120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)




120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1




119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)




119872and 119864

119872are



and 1198921(119871) and 119892








4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L1

L2

li

120579iT

ei

120583jj+1



Particleupgrading



NoParticle



119864119871=

119873

sum

119894=1

119890119894 (13)




120579119894= 119891 (119875

119894

119868 119876119894

119868 119879119894) le 120579max (14)




120583119895119895+1

= infin if 119897119895+ 119897119895+1lt 119903min

120583119895119895+1

ge 120583min else

119895 = 1 2 119873 minus 1

(15)

where 120583119895119895+1




119865 (119871) = 1199081

119863119871

119863119872

+ 1199082

119864119871

119864119872

+ 1198921(119871) + 119892

2(119871) (16)




119872and 119864

119872are



and 1198921(119871) and 119892








4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4T

1 2 T

P1I Q

1I Q

1O P

1O P

2I Q

2I Q

2O P

2O P

119879I Q

119879I Q

119879O P

119879O








bestRout

= (119875V1

119868119876

V1

119868119876

V1

119874119875V1

119874119875V2

119868119876

V2

119868119876

V2

119874119875V2


V119895

119868119876

V119895

119868119876

V119895

119874119875V119895

119874)

(17)



119894

1003816100381610038161003816 lt 120576 (119894 = 1 2 3 119899) (18)




119894(119894 = 1 2 3 119899) and this



119894= 1205791198771198942 (119894 = 1 2)





4 Simulation








bestRout = (11987531198681198763

11986811987623

11987411987522

11987411987510

11986811987615

11986811987617

11987411987518

11987411987510

11986811987615

11986811987617

11987411987518

1198741198752

1198681198765

11986811987619

11987411987517

1198741198759

1198681198764

11986811987621

11987411987523

119874) (19)






Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Start point 1198791

1198792

1198793

1198794

End point(400 700) (700 1200) (1000 1000) (1100 1150) (1500 1350) (1700 1000)



1198793

1198792

1198794


PjO

QjO

r1

r2

n1 n2

n2

n3

bestRout


400 600 800 1000 1200 1400 1600

700

800

900

1000

1100

1200

1300

1400

1500

T1 T2

T3

T4

y(k

m)

x (km)




400 600 800 1000 1200 1400 1600600

700

800

900

1000

1100

1200

1300

1400

1500

1600

T1T2

T3

T4

y(k

m)

x (km)


0 500 1000 1500 2000 2500 30001800

1850

1900

1950

2000

2050

2100

2150

2200

2250

x (times)

VGHPSOPSO

y(k

m)

VGHPSOlowast














endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















endUpdate (Ωnew)

endCalculating (119891119887119890119904119905

119894)

If 119891119887119890119904119905119894lt min119891119887119890119904119905

1 119891119887119890119904119905

2 119891119887119890119904119905

119894minus1 Then

119892119887119890119904119905119894= 119891119887119890119904119905

119894



EndReturn (119892119887119890119904119905 119887119890119904119905119877119900119906119905 119887119890119904119905119894119899119889119890119909)

End

Algorithm 1

5 15 30 45 60 72 900

500

1000

1500

2000

2500

Dist

ance

(km

)

10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

90


Sam

plin

g an

gle(

∘ )






5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












5 Conclusion





References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Sequence and Direction Planning ... - Hindawi