Research Article Potential Odor Intensity Grid Based UAV ...downloads.hindawi.com/journals/mpe/2016/7802798.pdfthat it is di cult to combine the motion constraints of UAV itself, which

Research ArticlePotential Odor Intensity Grid Based UAV Path PlanningAlgorithm with Particle Swarm Optimization Approach

Yang Liu12 Xuejun Zhang1 Xiangmin Guan3 and Daniel Delahaye4

1School of Electronic amp Information Engineering Beihang University Beijing China2School of Information Science amp Electric Engineering Shandong Jiaotong University Jinan China3School of General Aviation Civil Aviation Management Institute of China Beijing China4MAIAA Laboratory Ecole Nationale de lrsquoAviation Civile Toulouse France

Correspondence should be addressed to Xuejun Zhang zhxjbuaaeducn

Received 18 March 2016 Accepted 21 August 2016

Academic Editor Mauro Pontani

Copyright copy 2016 Yang Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a potential odor intensity grid based optimization approach for unmanned aerial vehicle (UAV) path planningwith particle swarm optimization (PSO) technique Odor intensity is created to color the area in the searching space with highestprobability where candidate particles may locate A potential grid construction operator is designed for standard PSO based ondifferent levels of odor intensity The potential grid construction operator generates two potential location grids with highest odorintensity Then the middle point will be seen as the final position in current particle dimension The global optimum solution willbe solved as the average In addition solution boundaries of searching space in each particle dimension are restricted based onproperties of threats in the flying field to avoid prematurity Objective function is redesigned by taking minimum direction angleto destination into account and a sampling method is introduced A paired samples 119905-test is made and an index called straight linerate (SLR) is used to evaluate the length of planned path Experiments are made with other three heuristic evolutionary algorithmsThe results demonstrate that the proposed method is capable of generating higher quality paths efficiently for UAV than any othertested optimization techniques

1 Introduction

Unmanned aerial vehicle (UAV) is a kind of aircraft with-out onboard pilots that can be remotely controlled or flyautonomously based on the preplanned flying routes increas-ingly suitable for a real-world environment [1ndash4] CurrentlyUAV has been widely used in civil and military fields suchas aerial photography search and rescue tasks geophysi-cal survey environmental and meteorological monitoringsurveillance reconnaissance high-risk target penetrationsuppressing enemy air defense deep target attacking anddominating the battle space [5ndash7] But how to fly safelyin all the fields is vital for mission effectiveness in whichpath planning is one of the most important technologies forautonomous flight of UAV

UAV path planning is a key aspect of the autonomouscontrol module whose mission is to provide an optimalflying path from the starting point to the desired destination

avoiding artificial threats and some natural terrain con-straints with least cost and shortest length of flying distanceUsually the flying path of UAV is automatically providedby a path planner based on an objective function [8 9]considering all the constraints Planning intuitive flyingroutes for UAV in large real-world scenarios and in thepresence of obstacles is more complicated due to the mostlyopen structure of the airspace Particularly in the near futureUAV is to be integrated into national airspace system (NAS)

Inmost situations UAV path planning is often formulatedas a global optimization problem in which the feasibility ofthe candidate path depends on the mission environmentand UAV physical constraints Although the applications ofUAV are so different the optimality of a feasible path for anyof them can be defined by different optimization planningcriteria (such as minimal flying time andor path length)and fulfillment of some mission constraints (such as flyingat a given altitude or visiting some points) [10] Besides the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7802798 16 pageshttpdxdoiorg10115520167802798

2 Mathematical Problems in Engineering

physical characteristics of the UAV and the environment alsorestrict the feasibility of any path and should be consideredby a realistic path planning problem

In the past few years several path planning algorithmshave been proposed which can be mainly divided into twocategories graph-based and population-based evolutionaryalgorithms The former includes Voronoi diagram searchmethod [11] mathematical programming method [12] 119860lowastsearching algorithm [13 14] 119863lowast lite algorithm [15] andbilevel programming method [16] In these algorithms Epp-steinrsquos 119896-best paths algorithm [17] is used to find an optimalpath for UAV The biggest deficiency of graph-based ones isthat it is difficult to combine the motion constraints of UAVitself which means it usually cannot be used in practicalsituations UAV self-performance is an absolutely necessaryfactor needed to be taken into account when designing apath planning algorithm Another important category is thepopulation-based optimization algorithmsThey could makeUAV flying routes generated by reducing the complexity anddimensions which is a NP-hard problem These algorithmsmainly include genetic algorithm (GA) [18] particle swarmoptimization (PSO) [10] ant colony optimization (ACO) [19]artificial bee colony (ABC) [20] differential evolution (DE)[21] gravitational search algorithm (GSA) [22] intelligentwater drops optimization (IWD) [23] memetic computingmethod [24] and firefly algorithm (FA) [25] Each of thesecategories has its own advantages over others in certainaspects However among them GA PSO and FA are themost three popular for their simplicity and effectivenesswhich are becoming the hottest research topics and are mostsuitable for solving the global optimization problems withlarge scales For example GA is famous for the ease ofimplementation for both continuous and discrete problemsThere are no extra requirements for the continuity in responsefunctions and could be used efficiently with large numbersof parallel processors The generated global or near globalsolutions are more robust And FA is recently developedto solve nonlinear design problems In FA all fireflies areunisexual and any individual firefly will be attracted toothers based on their higher brightness But their brightnessdecreases as their mutual distance increases By iterations ofbrightness oriented movement the global optimal solutionscould be found finally

Another important one is called PSO which is short forparticle swarm optimization It is an evolutionary compu-tation algorithm first proposed by Kennedy and Eberhart[26] which is designed based on the study of the socialbehavior of bird flocking and fish schooling Each particleadjusts its flying positions in the searching space in terms ofits own flying experience and the whole swarm flying expe-riences Successful applications of PSO in neural networktraining function optimization and fuzzy system controlhave demonstrated that PSO is a promising and efficientoptimization method So far many significant improvementsare proposed to improve the performance of standard PSOalgorithm An excellent overview of the basic concepts ofPSO and its variants can be found in [27] PSO has beenseen as an attractive optimization tool for the advantages ofsimple implementation procedure good performance and

Start point

Way pointDestination point

S D

L1 Lk Lk+1 LM

W1

Wk

Wk+1

WM

T1

T2

T3

T4

Figure 1 Typical UAV path planning model

fast convergence speed However it has been shown that thismethod is easily trapped into local optima when coping withcomplicated problems

In this paper a potential odor intensity grid based UAVpath planning algorithm is proposed combining standardPSO technique By identifying the different levels of odorintensity a potential grid construction operator is designedfor standard PSO which is implemented easily and avoidslocal optima and slow convergence The potential grid con-struction operator indicates two potential location grids withhighest odor intensity including centers of the two grids andthe preset side length Then middle point of the two gridcenters will be seen as the final global position in currentparticle dimension space of current iteration The globaloptimum solution will be solved as the average Experimentalresults demonstrate that the proposed method is capable ofgenerating higher quality paths efficiently for UAV than anyother tested optimization algorithms

The remainder of this paper is organized as followsPath planning problem description is provided in Section 2Concise standard PSO algorithm and explicit realizationof proposed method are given in Section 3 respectivelySection 4 presents the settings in application conditions ofthe proposed algorithm and experimental results of globalroute planning are listed Finally the paper is concluded inSection 5

2 Problem Formation

UAV path planning can be seen as a global optimizationproblem to generate a serial of way points from the startto the destination with least cost values Terrain and threatmodeling and the design of objective function become twokey problems in route generation for UAV

21 Terrain and Threat Modeling As shown in Figure 1 themission of path planning is to generate a feasible flying routefrom the start point 119878 to destination point 119863 without flyingout of the designated map or being taken down by threats aswell as least flying cost There are some threat areas locatingin the map such as radars missiles and artillery which areall denoted as red circles in Figure 1 Once the UAV is in thecoverage of any threat it will be vulnerable to the threat witha certain probability proportional to the distance away fromthe threat centers Oppositely there will not be any danger if

Mathematical Problems in Engineering 3

UAV is flying out of the threat covered region Consideringthe threat areas and other affecting factors how to get to thedesired destination safely is the key problem

211 Terrain Restriction In real situations all UAVs will haveto fly in a specified region for different tasks all of whichare limited in power supply flying distance and flying timeAny UAV flying out of the region will lead to high risk outof control and crashing for the use up of energy In this wayany way point generated by the path planning algorithmsshould be in the flying region If out it will be punishedThe following equation is given to record the distances to thecoverage boundary of all way points

119869out =

sum

119894

Dis (119894) if (119909119894 119910119894)119879

notin 119877

0 otherwise(1)

Here in (1) (119909119894 119910119894)119879 is the current coordinate of way

point 119894 119877 is the designated flying region and Dis(119894) is thedistance from current way point 119894 to the nearest boundary ofdesignated flying region 119877 In most situations the flying areawill be set as a rectangle in which start and destination locate

212 Threat Modeling There are two kinds of threats oneis random and the other is deterministic For randomsituations path planning algorithm could generate paths byrefreshing itself to avoid the pop-up threats if real timeinformation could be obtained In this paper deterministicthreat areas are taken into account all of which are knownin advance as shown in Figure 1

It is desirable that all the generated UAV way points arekept away from the threat as far as possible and the fartherthe better For this purpose a new model for the threateningspace definition is developed as follows First let the vectorP119894threat = [119909

119894

threat 119910119894

threat 119903119894

threat]119879 be assigned to the threat 119894with

its center coordinate (119909119894threat 119910119894

threat)119879 and destruction range

119903119894

threat An effective destruction gain 119866119894is considered for each

threat 119894 which is different for different threats For any UAVway point 119895 with current coordinate (119909

119895 119910119895)119879 we can define

the effects of all threats on the planned path as follows

119869expo =

sum

119895

sum

119894

119866119894[(119903119894

threat)2

minus (119903119895119894)2

]

(119903119894

threat)2

if 119903119895119894lt 119903119894

threat

0 otherwise

(2)

Here 119903119895119894is the distance from current position of UAVway

point 119895 to the center of threat 119894 which can be written as

119903119895119894= radic(119909

119895minus 119909119894

threat)2

minus (119910119895minus 119910119894

threat)2

(3)

Equation (2) illustrates that once UAV is flying in thecovered areas of any threat the probability to be detected byradar or brought down by missiles will be high

o

y

x

Start point


D

L1

Lk

Lk+1

LM

W1

Wk

Wk+1 WM

T1

T2T3

T4

120579

y998400

x998400

o998400S

Figure 2 Transformation of coordinate system

22 Objective Function of UAV Route Optimization

221 Problem Analysis The task of path planning is to finda feasible flying route from the start point to destinationpoint with least flying cost and avoiding entering threat areas[28] as the green line does in Figure 1 Here the straight line119878119863 is divided into (119872 + 1) segments by 119872 vertical lines119871119896(119896 = 1 2 119872) In this way the flying route can be

denoted by119872 way points namely119882119896(119896 = 1 2 119872) on

each vertical line 119871119896(119896 = 1 2 119872) The core difficulty is

how to determine these way points with least global flyingcost and also without entering covered areas by threats

In order to simplify the computation process and acceler-ate the searching speed of global solutions in the path plan-ning algorithm the transformation of coordinate system ismade [29] as shown in Figure 2 in which the new coordinatesystem 119909

1015840

1199001015840

1199101015840 is transformed from origin coordinate system

119909119900119910 by setting the straight line 997888rarr119878119863 as the new 119909 axis and thestart point 119878 ofUAVas the neworigin of coordinateThe anglebetween 11990010158401199091015840 and 119900119909 is 120579 which means the origin 119909-axis 119900119909will be rotated anticlockwise with angle 120579

Suppose a way point in the original coordinate systemwith coordinate (119909

119874 119910119874)119879 and correspondingly the trans-

formed coordinate in the new coordinate system is (119909119879 119910119879)119879

and the relationship between the coordinates in two coordi-nate systems of the same way point can be written using thefollowing equation

[

119909119879

119910119879

] = [

cos 120579 sin 120579minus sin 120579 cos 120579

] [

119909119874minus 119909119878

119910119874minus 119910119878

] (4)

In (4) (119909119878 119910119878)119879 is the coordinate of start point in original

coordinate system Once the transformation is completed thedesired UAV flying path can be denoted as a sequence of waypoint 119882

119896from the start point 119878 to the destination point 119863

namely 1198781198821 119882

119872 119863 Combining Figures 1 and 2 the

coordinates of way points 119878 and 119863 in 119909101584011990010158401199101015840 can be easily


known as (0 0)119879 and (|119878119863| 0)119879 respectively In this way otherabscissas of any way point119882

119896can be obtained using

119909119896=

|119878119863|

119872 + 1119896 (5)

So the path planning problem becomes the optimizationproblem of longitudinal coordinates for any UAV way point119882119896with least global minimum cost value How to design

objective function to compute longitudinal coordinate withglobal optimum will be given in the following

222 Objective Function Design The objective function is toevaluate a candidate pathwhich is generated by path planningalgorithms Many factors should be taken into account suchas cost of the path the performance of UAVs themselvesand the mission constraints In our proposed method thefollowing factors are considered

(A) Cost of the Generated Path As described before the waypoints out of threat areas should be chosen Otherwise theywill be penalized which lead to high cost of the whole pathHere how to evaluate the cost of the generated UAV path isgiven In our proposed algorithm a samplingmethod for eachpath segment is proposed to better describe the quality ofpossible UAV path

As described in the section before after the coordinatesystem is transformed each path of UAV consisted of manyway points 119882

119894(119894 = 1 2 119872) and (119872 + 1) segments

are formed Suppose any path segment 997888997888997888997888997888997888rarr119882119894119882119894+1

as shownin Figure 3 is formed by two way points 119882

119894and 119882

119894+1

which locate on vertical lines 119871119896and 119871

119896+1 respectively The

samplingmethod for any possible UAV path segment997888997888997888997888997888997888rarr119882119894119882119894+1

is as followsFirst any path segment 997888997888997888997888997888997888rarr119882

119894119882119894+1

is detected whether it fallsinto the areas covered by threats If so sample points are seton the segment 997888997888997888997888997888997888rarr119882

119894119882119894+1

with the total number of 119871 Then therules of these sample points can be built using

210038161003816100381610038161198821198941198751119871

1003816100381610038161003816=100381610038161003816100381611987511198711198752119871

1003816100381610038161003816= sdot sdot sdot =

1003816100381610038161003816119875(119871minus1)119871

119875119871119871

1003816100381610038161003816

= 21003816100381610038161003816119875119871119871119882119894+1

1003816100381610038161003816

(6)

After the 119871 sample points are determined the cost of thesegment 997888997888997888997888997888997888rarr119882

119894119882119894+1

will be calculated as follows

119869119879(119894)

=

100381610038161003816100381610038161003816

997888997888997888997888997888997888rarr119882119894119882119894+1

100381610038161003816100381610038161003816

119871

119873119879

sum

119895=1

119866119895(

1

1198891119871119894119895

+1

1198892119871119894119895

+ sdot sdot sdot +1

119889119871119871119894119895

)

(7)

In (7) |997888997888997888997888997888997888rarr119882119894119882119894+1| is the distance of segment 997888997888997888997888997888997888rarr119882

119894119882119894+1

119871 is thetotal number of sample points in this segment997888997888997888997888997888997888rarr119882

119894119882119894+1

and119866119895

is the destruction gain of threat 119895 119889119898119871119894119895

denotes the distanceof the 119898th sample point that is on segment 119894 to the centerof threat 119895 and 119898 = 1 2 119871 Obviously the number ofthreats that could affect this path segment997888997888997888997888997888997888rarr119882

119894119882119894+1

is set as119873119879

Way pointSample point

PLL

Threat(j)

Threat(k)

middot middot middot


Li Li+1

Wi

Wi+1

P1LP2L

P3L

P4L

P(Lminus1)L

Figure 3 Sample method for UAV path segment 997888997888997888997888997888rarr119882119894119882119894+1

S D

Start point


(x1 y1)T

(x2 y2)T

(xk yk)T

(xM+1 yM+1)T

(xM+2 yM+2)T

Figure 4 Definitions of straight line rate (SLR)

The total effects of threat areas on one planned flying path canbe calculated by adding all the costs of path segments on theflying path by using (6) and (7)

119869path = sum119894

119869119879(119894) (8)

(B) Total Flying Distance for UAV Flying distance is anotherimportant factor that needs to be taken into account Inrealistic operations the generated flying path should be asshort as possible from the start to the destination pointwith all limitations satisfied Short path length means highefficiency which saves energy and time for UAV and also alower chance of being detected by unknown threats In theproposed method an index called straight line rate (SLR) isdefined to indicate performance of the total length of plannedUAV flying route for simplicity Suppose the coordinatesof start point 119878 and destination point 119863 as (119909

1 1199101)119879 and

(119909119872+2

119910119872+2

)119879 showed in Figure 4

As shown in Figure 4119872 is the number of way points onthe planned UAV path with coordinate of way point 119882

119896as

(119909119896 119910119896)119879 Straight line rate (SLR) can be obtained as follows

119869SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(9)

In (9) |119878119863| = radic(119909119872+2

minus 1199091)2

+ (119910119872+2

minus 1199101)2 is the Euclid-

ean distance from the start to the destination


Li

Wi1

(xi1 yi1)

Wi2

(xi2 yi2)

Wik

(xik yik)

Wik+1

(xik+1 yik+1)

120579i1

120579i2

120579ik

120579ik+1

D

(xD yD)

Figure 5 Definition of direction angle

(C) Minimum Direction Angle As shown in Figure 5 therewill be many candidate way points (119882

11989411198821198942 ) in the

searching space which is denoted as 119871119894 In order to improve

the searching speed and avoid falling into local optimumdirection angle of different way points119882

119894119896on vertical line 119871

119894

in the searching space is defined as 120579119894119896 They are the angles

between current way point and destination point which isdenoted as Figure 5 shows Obviously the way point onvertical line 119871

119894with smaller direction angle is better thanwith

larger ones This parameter can evaluate the performances ofgenerated UAV path as part of objective function

Equation (10) combines effects from all the directionangles of possible way points on different vertical linesmeaning different searching space

119869DA = sum119894

min119896

120579119894119896 = sum

119894

min119896

arctan1003816100381610038161003816119910119863minus 119910119894119896

10038161003816100381610038161003816100381610038161003816119909119863minus 119909119894119896

1003816100381610038161003816

(10)

(119909119863 119910119863) is the coordinate of destination and (119909

119894119896 119910119894119896) is

the candidate way point position on vertical line 119871119894

(D)Optimized Searching Scope To furthermake the searchingprocess of optimumway point simplified the searching scopeof each particle in the solution space is optimized [30] whichis shown in Figure 6 The red circle is the threat in the UAVflying field with center119879

119894and radius 119903

119894The searching space is

limited by upper limit 119871max and lower limit 119871minThemarginsfor upper limit and lower limit areΔ119889

119880andΔ119889

119863 respectively

The searching is realized in this constrained area Any waypoints out of this scopewill be penalizedThe two limits couldbe calculated as follows

119871max = max119894

119910threat (119894) + 119903threat (119894) + Δ119889119880

119871min = min119894

119910threat (119894) minus 119903threat (119894) minus Δ119889119863(11)

In (11) 119910threat(119894) and 119903threat(119894) are the longitudinal coordi-nate and radius of threat 119894 respectively In this way the cost of

D

Upper limit Lmax

ΔdD

ΔdU

rjTj

Tiri

Lower limit Lmin

Start pointDestination point

y998400

x998400

o998400S

Figure 6 Optimized searching scope

any UAV way points out of the limited searching space couldbe given in the following

119869out =

119872

sum

119894=1

[119910119880(119894) minus 119871max] if 119910

119880(119894) gt 119871max

0 otherwise119872

sum

119894=1

[119871min minus 119910119880 (119894)] if 119910119880(119894) lt 119871min

(12)

In the equation above119872 is the number of total way pointson the generated UAV path And 119910

119880(119894) is the longitudinal

coordinate of the 119894th way pointBased on the analysis above the complete objective

function can be given as follows to realize the evaluation ofone candidate UAV flying route and UAV self-performance

119869obj = 120572119869path + 120573119869SLR + 120594119869DA + 120575119869out (13)

Here in the equation 120572 120573 120594 and 120575 are four weightingparameters to assess the different effects of different elementson the objective functionThe value of the four parameters isbetween [0 1]

3 Realization of Proposed Algorithm

In this section the proposed method for UAV path planningin two-dimensional space is given explicitly based on thestandard particle swarm optimization (PSO) The standardPSO is described first

31 Standard PSO Thestandard particle swarmoptimization(PSO) is a population-based nondeterministic optimizationmethod which was first proposed by Kennedy and Eberhart[26] It simulates the movement of a swarm of particles in amultidimensional searching space towards a global optimalsolution by repeated iterations The position of each particlein each iteration represents a candidate solution which isinitialized randomly In each iteration the current particlevelocity is renewed based on the previous velocity of the


Standard PSO AlgorithmSet the generation counter 119896 = 0lowastInitializationlowast Generate 119878 individuals 119909

119894of119863 dimensions with random initial location and speed in searching space

lowastMain looplowastwhile termination criteria is not satisfied do

generation counter 119896 = 119896 + 1for 119894 = 1 119878 dolowastComputationlowast Calculate cost value of particle 119909

119894using its current position in current generation

lowastUpdatelowast Update the location and speed of particle 119909119894of current generation based on formula (17)

lowastUpdatelowast Update local best position Pibest and global best position Gbest based on current cost valueend for

end while

Algorithm 1 Pseudocode of standard PSO

particle the best position ever occupied by the particle(personal influence) and the best position ever occupied byany particle of the swarm (social influence) Then positioncould be renewed using the new updated velocity Themathematical descriptions are listed in the following

Suppose the size of the swarm is 119878 and the dimension ofeach particle 119894 is119863There are two parameters for each particle119894 namely position 119909

119894and velocity V

119894 whose dimension is the

same as119863 obviously And119863 also stands for the dimension ofthe problem to be solved The two of each particle 119894 can bewritten as a particle vector in the following equation

(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)

In this way a particle swarm with swarm size 119878 can bewritten as a matrix made up of 119878 particle vectors

(x1k1) (x2k2) (xSkS) (15)

Then cost value of each particle in each iteration withcurrent position could be calculated using specified objectivefunction for further steps There are two kinds of cost valuesin standard PSO local best value Pibest of each particle 119894and one global best value Gbest of all particles which can bewritten as (16)The number of local best values is the same asswarm size 119878

Pibest = (1199011198941best 1199011198942best 119901119894119863best)

Gbest = (1198921best 1198922best 119892119863best) (16)

Once the two values are obtained the position andvelocity of each particle in each dimension are updated bykeeping track of the two best positions using the followingequations

V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)

In (17) 119908 is inertia weight which reflects the impactsof the particle velocity in previous iteration on its current

iteration 120585 and 120578 are random numbers between 0 and 11198881and 1198882are positive constants named self-cognition and

social knowledge which stand for the inheriting abilitiesfrom particle itself and the whole swarm respectively 119903is constant factor used to constrain the position updatingAnd 119873 is the total iterations that the algorithm has to runIn most situations it is also set as termination criteria ofPSO The standard PSO will not stop until the terminationcriteria are satisfied Pseudocode of standard PSO is given inAlgorithm 1

32 Proposed Algorithm In this section the potential gridconstruction operator based on odor intensity is presentedfirst and then the realization of the proposed method isdescribed

321 Potential Grid Construction Operator The standardparticle swarm optimization technique falls into local opti-mum easily and sometimes it is hard to converge withlow speed Even in extreme situations there are no optimalsolutions which makes it fail to solve the UAV path plan-ning problem In our proposed algorithm potential grid isconstructed based on odor intensity of particles to get overthe defects of standard PSO which is called potential gridconstruction operator here

As described before the core step of UAV path planningis to find way points on different particle dimensions withleast global cost values as well as avoid all the threatsFor each dimension of each particle it can be seen assolution space in which the global solutions may exist Inour model it is supposed that there are 119878 particles denotedas 119875119894(119894 = 1 2 119878) with the number of dimensions 119863

Each dimension of particle 119875119894can be written as 119875

119894119895with

119895 = 1 2 119863 Each particle will leave odor pheromonetrails on the position in each dimension where it stayed andall the odor pheromone trails will be accumulated to formdifferent odor intensities Potential grid construction is givenin Figure 7 in the solution space which is based on odorintensities

In Figure 7 each blank square with the same side length119889119894119895denotes a candidate position in an iteration in the 119895th

dimension space of particle 119894 As shown in Figure 7 there will


Odor Intensity 5 Odor Intensity 4Odor Intensity 3 Odor Intensity 2

dij

dij

Wij

C1

C2

G1

G2

G3

dij

Figure 7 Potential grid construction in the 119895th dimension ofparticle 119894

be odor pheromone trails filled in each square evenly Theodor trails from different particles will be overlay and accu-mulated Subsequently different levels of odor intensities areformed Potential grid construction based on odor intensitywill be implemented in each 119875

119894119895as shown in Figure 7 with the

following steps

Step 1 (particle position square generation) For the 119896thgeneration in the 119895th dimension of particle 119894 there will be apossible particle locationwritten as 119871

119894119895(119896) A particle position

square 119861119894119895(119896) is generated by setting 119871

119894119895(119896) as center and 119889

119894119895

as the side length which can be written as the followingequation Also odor pheromone trails will be filled in thisposition square 119861

119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)

Step 2 (odor accumulation)The position that a particle stayswill leave odor pheromone trails in the particle positionsquare covered area and will be accumulated After all the119873 iterations are completed there will be some areas withhigher odor intensity as the different colorful grids whichis shown in Figure 7 As given in Figure 7 there are four odorintensities namely Odor Intensity 5 Odor Intensity 4 OdorIntensity 3 and Odor Intensity 2 These potential grids 119866

119897

which stand for potential positions can be obtained using thefollowing equation

1198661 1198662 = 119861

119894119895(1) cap 119861

119894119895(2) cap sdot sdot sdot cap 119861

119894119895(119873) (19)

Different odor intensities are colored in the end

Step 3 (candidate way point generation) Once odor accu-mulation is done two of potential grids with the biggest

odor intensities are formed The centers of the two grids aredenoted as 119888

1and 1198882 respectively So the particle position119882

119894119895

for the 119895th dimension of particle 119894 can be calculated using

119882119894119895=(1198881+ 1198882)

2 (20)

Step 4 (final particle position resolution) After the three stepsabove the final particle position 119871

119895for each dimension 119895 can

be resolved for each dimension using

119871119895=

sum119878

119894=1119882119894119895

119878 (21)

So once the four steps are realized all the way points thata UAV can fly along with avoiding all threat areas and terrainobstacles can be written as 119871

119894(119894 = 1 2 119863)

322 Realization of Proposed Algorithm Based on the afore-mentioned potential grid construction operator designed forstandard PSO all the elements required to build a completepath planning module for UAV are discussed in detail in thispart

Step 1 (path planning field formation) Two-dimensionalflying field for UAV is formulated first including the sizeof the area starting point and destination point designatedin advance Then terrain restriction and threat modelingare finished using (1) and (2) respectively Threat modelingincludes the position coordinates of threat center and thedestruction scope they can affect individually Another cru-cial step is to make coordinate system transformation using(4) All the calculations mentioned in the following will befinished under new coordinate system for simplicity

Step 2 (particle swarm initialization) In the proposed algo-rithm there are 119878 particles in a swarm each of which hasthe same particle dimension 119863 The initial positions andvelocities of particles are randomly assigned using (14) and(15) Initial cost values of each particle are calculated usingobjective function as shown in (13) Initial local best ofpersonal particle Pibest and global best Gbset are representedas (16)

Step 3 (start algorithm iterations) Once the initialization isfinished algorithm iterations start The termination criteriaare set as the total iterations which is also called generationswritten as 119873 Positions and velocities in current generationare updated by (17) based on the initial cost values Thenbased on the updated positions and velocities new cost valuesfor each particle are calculated New local best of personalparticle Pibest and global best Gbset of the swarm could befound out by comparing the minimum different cost values

Step 4 (call potential grid construction operator for the firsttime) Equation (18) is used to call potential grid constructionoperator for the first timeThe potential grid construction fordifferent dimensions of different particles in current iterationis realized which is stored in the dimension space Possible


Start

Scene creation terrain and threat modeling andcoordinate system transformation using equation (4)

Update positions and velocities of each particle incurrent generation are updated using equation (17)

Search local and global optimum positions in currentgeneration are found based on new calculated cost values

Call potential grid construction operator is called forthe first time using equation (18)

Termination criteria

Call potential grid construction operator is called forthe second time using equations (19) (20) and (21)

Output the final global optimum results

End

Calculation new cost values are calculated based on updatedpositions and velocities of particles using equation (13)

Yes

No

Set generation counter k = 1

Set k = k + 1

are given randomly in searching space Initial cost values of each particle are computedInitialization generate S particles with each dimension D and their initial positions and velocities

Figure 8 Flowchart of the proposed algorithm

particle location is formed as a center of a rectangular with adesignated side length 119889

119894119895

Step 5 (call potential grid construction operator for thesecond time) After all the iterations are stopped by thetermination criteria potential grid construction operator iscalled for the second time Then (19) (20) and (21) are usedto form the final global optimum way points for UAV whichare the best solutions

After the steps above are finished global optimum waypoints for path planning will be output The flowchart of thewhole algorithm is given in Figure 8

4 Experiment Validation and Comparison

To validate the effectiveness of the proposed algorithm insolving the two-dimensional path planning problem experi-ments are conducted and also comparisons of our proposed

method are made with stand PSO GA and FA with the sameparameters settings

41 Parameter Settings The path planning field for UAVin the experiments is a square with side length 100 kmnamely the acreage of the whole covered area is 100 km times

100 km There are some areas covered by many threats inthe flying field such as radars missiles and artilleries Heretwo experiment scenarios are defined based on the degreeof complexity One is general scenario (Scenario 1) with fivethreats and the other is the complicated scenario (Scenario 2)with nine threats in the terrain respectively As mentionedbefore the destruction gains 119866

119895of threat 119895 are all set the

same as grade 1 namely 119866119895= 1 The explicit experimental

parameters can be found in Table 1Besides external environment parameter settings there

are some crucial parameters in the algorithm itself which are


Table 1 Information of environment installations

Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint

Threatcenterlocation

Threatradius(km)

GeneralscenarioScenario 1

400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12

ComplicatedscenarioScenario 2

400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12

size of the particle swarm dimensions of a particle number ofiterations the number (119871) of sample points on path segment997888997888997888997888997888997888rarr119882119894119882119894+1

the size of constrained particle dimension searchinglimit Δ119889 (Δ119889 = Δ119889

119880= Δ119889

119863) and the side length (119889

119894119895) of

potential grid in the 119895th dimension of particle 119894 Anotherfour weighting parameters 120572 120573 120594 and 120575 which appearedin (13) are set with the same value of 1 which is written as120572 = 120573 = 120594 = 120575 = 1 Part of these crucial parameters in thealgorithms are all set as in Table 2

Three main indexes are listed as the performance indi-cator The first is the planned UAV path in two-dimensionalfield Whether the planned UAV path can avoid every threatsuccessfully reflects the accuracy of the algorithm itself Thesecond is normalized cost values The lower the cost is thebetter the algorithm is The third is the path length Here inthis paper an index called straight line rate (SLR) is definedfrom (9) which could be rewritten as follows

SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)

And (119909119894 119910119894)119879 is the way point for UAV in which the

start and destination points are includedThe fourmentionedpath planning algorithms will be implemented in the samesituations with the same self-parameter settings The resultswill be compared in the next part

42 Result Comparisons In this part performance com-parisons under Scenario 1 and Scenario 2 are given firstwith different parameter settings Then a paired samples 119905-test is made to analyze in a probability perspective Self-performance comparisons are implemented under Scenario1 in the end

421 Performance Comparisons under Scenario 1 Compar-isons of four algorithms mentioned before are implemented

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA

Figure 9 Planned path for UAV with 119878 = 100 and 119863 = 10 inScenario 1

in Scenario 1 with the same parameter settings in this partTwo groups of experiment results are given with differentswarm sizes One group is set with swarm size 119878 = 100 andparticle dimension 119863 = 10 and the other group is under119878 = 400 and 119863 = 10 both of which are to explore the effectsof different swarm sizes on the algorithms

As shown in Figure 9 there are five independent threatareas in Scenario 1 Themission is to make UAV fly from startpoint to the destination point safely It is obvious that PSOGA and our proposed algorithm could offer perfect flyingroute for UAV without entering any of these threats whichmeans they can solve the path planning problem with wellsolutions However FA fails to avoid all the threats Parts ofits planned trajectory fall into the areas covered by threats Assaid before once a UAV is flying in the dangerous fields therewill be a high risk of being detected or taken down FA is theworst of all the four algorithms in solving this kind of routeoptimization problem in Scenario 1

Another two indexes normalized cost values and straightline rate (SLR) are given in Figures 10(a) and 10(b) InFigure 10(a) FA is the worst with the highest cost values fromthe first iteration to the end which is mainly caused by fallinginto threats But it is stable during the whole process since itfinds a solution PSO is the most stable and it could easilyfind the optimal flying path for UAV after the 120th iterationBut when all iterations are finished PSO is not the bestwhich means the optimal solution it finds is possibly localoptimum GA fluctuates fiercely from the beginning to the300th iteration and then solutions could be found Its solutionis better than that of PSO after all the iterations finish Ourproposed algorithm presents a downtrend from the first tothe last iteration which reflects the process of optimizing Itfluctuates less than GA and convergence rate is also betterIt can find out the global optimum instead of local optimumwhich overcomes the defects of standard PSO By calculations


Table 2 Parameters of algorithm itself

Particle swarm size (119878) Particle dimension (119863) Number of sample points Searching limit Δ119889 Side length of grid (119889119894119895)

Variable Variable 5 0 001

0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA

(b)Figure 10 Performance comparisons of four algorithms with 119878 = 100 and119863 = 10 in Scenario 1 (a) Normalized cost comparisons of differentalgorithms (b) Path length comparisons of different algorithms

Table 3 Comparisons with 119878 = 100 and119863 = 10 in Scenario 1

Category Algorithm Best Worst Mean Std

Cost value

Proposed 04772 06245 04978 04410PSO 05100 06256 05079 04659GA 04939 1 04990 1FA 1 07221 1 01617

Path length


our proposed method is 637 5228 and 338 bettercompared with PSO FA and GA in cost values respectively

Figure 10(b) gives SLR comparisons of the planned pathfrom four methodsThe proposed one is the best option withshortest flying distance for UAV Also it is the most stablefrom the beginning to the end PSO is the worst with thelongest flying route GA and FA is between them It is hard forthe two to find the global optimum solutions whichmakes FAand GA produce a lot of fluctuationsThe optimizing processis unstable at all Our proposedmethod is 1206 608 and995 shorter than PSO FA and GA respectively with theplanned flying path

Table 3 shows other indexes under different situationsThere is no doubt that our proposed one is the best of allunder all the indexes except the standard deviation of costvalue Oppositely FA is the best which is because it could not

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA


find global best solutions and falls into local optimum at thebeginning till the endwhich can be illustrated in Figure 10(a)

In order to further explore the effects of different swarmsizes on the four algorithms this parameter is increased to400 So the planned path for UAV with 119878 = 400 and 119863 = 10

in Scenario 1 is given in Figure 11 The biggest difference from


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)

Figure 12 Performance comparisons of four algorithms with 119878 = 400 and119863 = 10 in Scenario 1 (a) Normalized cost comparisons of differentalgorithms (b) Path length comparisons of different algorithms

all the curves in Figure 9 is that FA realizes path planningperfectly with 119878 = 400 avoiding all the threats Because byincreasing the number of particles in the swarm the searchingspace and searching times are enlarged which makes morecandidate solutions included in which the global solutionsare included The other three algorithms keep the same withgood performances Particularly GA still tries to fly bypassingall the threats

The comparisons of cost and path length are given inFigures 12(a) and 12(b) respectively There are no doubts allthe algorithms are getting smooth which are caused by largenumber of particles in the swarm FA is still the worst with thebiggest cost values PSO and GA are in the middle place Thetwo give downtrend from the beginning to the last iterationThe proposed one is the best with least cost The proposedmethod become the best before the 150th generation It is1161 2360 and 745 better comparedwith PSO FA andGA respectively in cost values

In Figure 12(b) FA jumps up and down obviously as manysteps PSO and FA are similar with big fluctuation around anapproximation GA is themost unstable and the curve alwayschanges which is hard to find an optimal value The curveof proposed method is the smoothest of all almost withoutany big jumping which is 824 1239 and 733 bettercompared with PSO FA and GA respectively

Table 4 lists other properties of the four methods It canconclude that the proposed path planning algorithm couldfulfill the mission perfectly and its performances are the bestcompared with other three classic algorithms

422 Performance Comparisons under Scenario 2 To studythe feasibilities of the algorithms under other situations wemake experiments under a more complicated scene namelyScenario 2 with much more threat areas As stated before



Cost value


Path length


there will be nine threat areas in the flying fields and also theyare overlapped with each other which is shown in Figure 13

Figure 13 gives theUAVpath planned by the four differentalgorithms with 119878 = 100 and 119863 = 10 in Scenario 2 Underthis complicated scenario FA and PSO all failed to get to thedestinationwithout flying out of the threats Particularlymostof the flying route given by FA is in the threats and there arealso many turnings on the path which is difficult for a UAVto follow Because the motility of UAV itself is very limitedThere is a small part of PSO path also in the dangerous areabecause of so many threats in the field GA shows the samecharacteristics of flying bypassing all the threats But ourproposed method is still perfect Not only it avoids threatsbut also it is easy for UAV to follow for least turnings on itwith big turning angles

In Scenario 2 shown in Figure 14(a) PSO easily falls intolocal optimum at the beginning and stays to the end FA is theworst of all and it needsmuch time to find a solution GA is inthe middle position At the beginning it fluctuates and thenstays stable till the end The proposed one is the best except


0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA


a singular point at the 64th generation The global optimalsolution is found around 200 iterations with least cost It is446 6845 and 1881 better than PSO FA and GArespectively in path cost

Figure 14(b) gives the path length comparisons From theFA curve we can find it not stable at all with a lot of bigjitters which proves it is hard to find a reliable solution in thisscenario Many jumping points are located on the curve PSOmoves steadily and almost in a straight line but it is muchlonger than that of GA and proposed method Also we canfind PSO easily falls into local optimalThere are a lot of smallfluctuations on GA compared with FA But it is shorter thanthat of FA Combining the GA curves in Figure 13 GA alwaysflies a long distance to avoid all the threat in the flying areasFinally there is no doubt our proposed method is the bestsince the 79th generation It moves stably and produces theshortest path length for UAV in two-dimensional field

423 Paired Samples 119905-Test In order to find out how differentthe proposed method is from the other three by probabilityanalysis a paired samples 119905-test is made with PSO FA andGA respectively under the experiment conditions of 119878 = 100and119863 = 10 in Scenario 1 The data of cost values and plannedpath lengths are used for the realization of 119905-test In this waythere are three pairs as follows proposedmethod versus PSOproposed method versus FA and proposed method versusGA The results are shown in Tables 5 and 6

In Table 5 the original hypothesis 1198670and alternative

hypothesis 1198671are listed The original hypothesis 119867

0means

that there are no essential differences between the twocompared algorithms Oppositely the alternative hypothesis1198671holds that big differences exist in the related two methods

in a group and the two are totally different methods Table 6shows the results of 119905-test which is indicated in a bold font

By the 119875 values in the 119905-test as shown in Table 6conclusions are given in Table 5 namely all the original

Table 5 Paired samples 119905-test conclusions

119905-test 119878 = 100119863 = 10

1198670

Proposed = PSO Proposed = FA Proposed = GA1198671

Proposed = PSO Proposed = FA Proposed = GA

119875 value Cost Length Cost Length Cost Lengthlt0001 lt0001 lt0001 lt0001 lt0001 lt0001

Result Reject Reject Reject

hypothesis1198670is rejected and alternative hypothesis119867

1is all

accepted In this way it means that our proposed algorithmis a feasible path planning technique for UAV which is alsoessentially different from other three classic ones Also itsperformances are the best of all compared with the otherclassic three

424 Self-Performance Comparisons under Scenario 1 Inorder to explore the effects of different swarm sizes andparticle dimensions on the cost values and planned pathlengths experiments of our proposed method are madeunder Scenario 1 All the results are all shown in the following

The swarm sizes are set as 119878 = 50 119878 = 200 119878 = 300and 119878 = 400 respectively The particle dimensions are set as119863 = 10119863 = 15119863 = 20 and119863 = 25

As Figure 15(a) shows the number of swarm sizes ischanged from 50 to 400 The flying paths for UAV are allplanned well under different swarm sizes which avoids allthe threat areas Also when 119878 = 200 and 119878 = 300 thetwo trajectories are in coincidence most of the time Thedifferences between four curves are not obvious In thisway the swarm size can be set between 50 and 200 whichnot only saves calculation time and power consumption butalso can obtain good performances Figure 15(b) gives theeffects of particle dimensions which is changed from 10 to25 The proposed method can realize path planning underthe four different conditions avoiding all the threat areasThedifferences are not so big Figure 15 proves the accuracy ofour proposedmethod under all kinds of environment settingwhich could be used in a wide range of real applications

Once the effects of the two parameters on the plannedUAV path are determined they can be set as the environmentchanges under different applicationsThe cost values and pathlengths are compared in Figures 16(a) and 16(b)

Figure 16 gives the swarm sizersquos effects on cost values andpath lengths respectively The curves of cost values presentthe same trend till the global optimal solutions are found Aslisted in Figure 16(a) the worst two of them are 119878 = 50 and 119878 =400 When 119878 = 200 it has the least cost value 119878 = 300 is in themiddle position So conclusions can bemade in the followingToo many or too few particles in a swarm will both lead tohigher cost values However the medium numbers of swarmsizes could obtain better cost values Figure 16(b) shows howthe path length changes as the swarm sizes The trend of thefour is more or less the same especially when 119878 = 200 and119878 = 400 Also 119878 = 50 is the worst and 119878 = 300 is the bestSo it can be set as different values based on the UAV flyingdistance


0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)


Table 6 Results of the paired samples 119905-test

(a) 119905-test of cost values

Paired differences119905 df Sig

(2-tailed)Mean Std deviation Std error mean 95 confidence interval of the difference

Lower Upper

Pair 1 proposed method PSO minus05496 04522 01011 minus07612 minus03379 minus5435 19 000Pair 2 proposed method FA minus42184 04838 01082 minus44448 minus39919 minus38996 19 000Pair 3 proposed method GA minus06850 02255 00504 minus07905 minus05794 minus13582 19 000

(b) 119905-test of planned path lengths


(2-tailed)Mean Std deviation Std error mean 95 confidence interval of the differenceLower Upper


From the analysis above we can find the swarm size canmake effects on the cost values and planned UAV lengths ina similar way Different values will produce different resultsBut the changing trends on the curves are similar based onif it can meet many different requirements In order to makeperfect planned UAV flying route under different applicationscenarios different parameter settings are needed

Effects from different particle dimensions on the twoindexes are given in Figure 17 As shown in Figure 17(a) thedifferences in cost values caused by particle dimensions arenot obvious Larger particle dimensions bring less cost thansmaller ones Figure 17(b) gives that the path lengths of 119863 =

10 and 119863 = 15 are very similar as well as the changing trend

from the first iteration to the last one It will be the shortestpath length when119863 = 25

From Figure 17 we can conclude that more particledimensions will bring less cost and shorter path but cor-respondingly the calculation will become more complicatedand more calculation time will be needed So how to set thisparameter depends on the applied environments

5 Conclusion

In this paper a potential odor intensity grid based UAVpath planning algorithm is proposed by combining standardPSO technique A potential grid construction operator is


0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)

Figure 15 Planned path in Scenario 1 with different experiment settings (a) Effects of different swarm sizes (b) Effects of different particledimensions

0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)

Figure 16 Effects of different swarm sizes on cost values and planned path length (a) Cost value comparisons of different swarm sizes (b)Path length comparisons of different swarm sizes

designed in our model to identify the different levels ofodor intensity which is implemented easily and avoids localoptima and slow convergence Two of areas in the searchingspace with highest probability where candidate particles maylocate will be colored depending on different odor intensitiesincluding centers of the two grids and the set side lengthThen middle point of the two grid centers will be used asthe final position in current particle dimension of currentiteration The global optimum solution will be solved as

the average Also upper and lower boundaries of solutionspace in each particle dimension are restricted based onproperties of threats themselves in the field to avoid prema-turity In addition objective function is redesigned by takingminimum direction angle to destination into account and asampling method is introduced to better evaluate the costof path segments into the threat areas and straight line rate(SLR) is used to evaluate the planned path length A pairedsamples 119905-test and experimental results both demonstrate


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)

Figure 17 Effects of different particle dimensions on cost values and planned path length (a) Cost value comparisons of different particledimensions (b) Path length comparisons of different particle dimensions

the proposed method is capable of generating higher qualitypaths efficiently for UAV than any other tested optimizationalgorithms standard PSO FA and GA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no U1433203 and Grant nosU1533119 and L142200032) and the Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (Grant no 61221061)

References

[1] W Zhan W Wang N Chen and C Wang ldquoEfficient UAVpath planning with multiconstraints in a 3D large battlefieldenvironmentrdquoMathematical Problems in Engineering vol 2014Article ID 597092 12 pages 2014

[2] Y Zhang L Wu and S Wang ldquoUCAV path planning by fit-ness-scaling adaptive chaotic particle swarm optimizationrdquoMathematical Problems in Engineering vol 2013 Article ID705238 9 pages 2013

[3] Q Zhang J Tao F Yu Y Li H Sun and W Xu ldquoCooperativesolution of multi-UAV rendezvous problem with networkrestrictionsrdquo Mathematical Problems in Engineering vol 2015Article ID 878536 14 pages 2015

[4] J M Peschel and R R Murphy ldquoOn the human-machine inter-action of unmanned aerial system mission specialistsrdquo IEEETransactions on Human-Machine Systems vol 43 no 1 pp 53ndash62 2013

[5] V Roberge M Tarbouchi and G Labonte ldquoComparison ofparallel genetic algorithm and particle swarm optimization forreal-time UAV path planningrdquo IEEE Transactions on IndustrialInformatics vol 9 no 1 pp 132ndash141 2013

[6] J KimD Lee K Cho J Kim andDHan ldquoTwo-stage trajectoryplanning for stable image acquisition of a fixedwingUAVrdquo IEEETransactions on Aerospace and Electronic Systems vol 50 no 3pp 2405ndash2415 2014

[7] A Babaie and J Karimi ldquoOptimal trajectory planning for aUAVin presence of terrain and threatsrdquo Aerospace and MechanicsJournal of ImamHossein University vol 7 no 2 pp 57ndash69 2011

[8] E Besada-Portas L De La Torre A Moreno and J L Risco-Martın ldquoOn the performance comparison of multi-objectiveevolutionaryUAVpath plannersrdquo Information Sciences vol 238pp 111ndash125 2013

[9] A Altmann M Niendorf M Bednar and R Reichel ldquoIm-proved 3D interpolation-based path planning for a fixed-wingunmanned aircraftrdquo Journal of Intelligent and Robotic SystemsTheory and Applications vol 76 no 1 pp 185ndash197 2013

[10] F Yangguang D Mingyue and Z Chengping ldquoRoute Plan-ning for Unmanned Aerial Vehicle (UAV) on the sea usinghybrid differential evolution and quantum-behaved particleswarm optimizationrdquo IEEE Transactions on Systems Man andCybernetics-Part A Systems andHumans vol 43 no 6 pp 1451ndash1465 2013

[11] Y V Pehlivanoglu ldquoA new vibrational genetic algorithm en-hanced with a Voronoi diagram for path planning of autono-mous UAVrdquo Aerospace Science and Technology vol 16 no 1 pp47ndash55 2012

[12] Y Wu and X Qu ldquoPath planning for taxi of carrier aircraftlaunchingrdquo Science China Technological Sciences vol 56 no 6pp 1561ndash1570 2013

[13] P Melchior B Orsoni O Lavialle A Poty and A OustaloupldquoConsideration of obstacle danger level in path planning usingAlowast and fast-marching optimisation comparative studyrdquo SignalProcessing vol 83 no 11 pp 2387ndash2396 2003


[14] R J Szczerba P Galkowski I S Glickstein and N TernulloldquoRobust algorithm for real-time route planningrdquo IEEE Trans-actions on Aerospace and Electronic Systems vol 36 no 3 pp869ndash878 2000

[15] S Koenig and M Likhachev ldquoFast replanning for navigation inunknown terrainrdquo IEEE Transactions on Robotics vol 21 no 3pp 354ndash363 2005

[16] W Liu Z Zheng and K-Y Cai ldquoBi-level programming basedreal-time path planning for unmanned aerial vehiclesrdquo Knowl-edge-Based Systems vol 44 pp 34ndash47 2013

[17] D Eppstein ldquoFinding the 119896 shortest pathsrdquo SIAM Journal onComputing vol 28 no 2 pp 652ndash673 1999

[18] Y G Fu M Y Ding and C P Zhou ldquoPhase angle-encodedand quantum-behaved particle swarm optimization applied tothree-dimensional route planning for UAVrdquo IEEE Transactionson Systems Man and Cybernetics Part A Systems and Humansvol 42 no 2 pp 511ndash526 2012

[19] H B Duan and F P Li Bio-Inspired Computation in UnmannedAerial Vehicles Springer Berlin Germany 2014

[20] A Ponsich and C A C Coello ldquoDifferential evolution perfor-mances for the solution of mixed-integer constrained processengineering problemsrdquoApplied Soft Computing vol 11 no 1 pp399ndash409 2011

[21] A N Brintaki and I K Nikolos ldquoCoordinated UAV path plan-ning using differential evolutionrdquo Operations Research vol 5no 3 pp 487ndash502 2005

[22] F Neri and E Mininno ldquoMemetic compact differential evolu-tion for cartesian robot controlrdquo IEEE Computational Intelli-gence Magazine vol 5 no 2 pp 54ndash65 2010

[23] S Das and P N Suganthan ldquoDifferential evolution a survey ofthe state-of-the-artrdquo IEEE Transactions on Evolutionary Com-putation vol 15 no 1 pp 4ndash31 2011

[24] G Iacca F Caraffini and F Neri ldquoMemory-saving memeticcomputing for path-following mobile robotsrdquo Applied SoftComputing vol 13 no 4 pp 2003ndash2016 2013

[25] X-S Yang ldquoFirefly algorithm stochastic test functions anddesign optimisationrdquo International Journal of Bio-Inspired Com-putation vol 2 no 2 pp 78ndash84 2010

[26] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia December1995

[27] Y del Valle G K Venayagamoorthy S Mohagheghi J-C Her-nandez and R G Harley ldquoParticle swarm optimization basicconcepts variants and applications in power systemsrdquo IEEETransactions on Evolutionary Computation vol 12 no 2 pp171ndash195 2008

[28] P Yao H Wang and Z Su ldquoReal-time path planning ofunmanned aerial vehicle for target tracking and obstacle avoid-ance in complex dynamic environmentrdquo Aerospace Science andTechnology vol 47 pp 269ndash279 2015

[29] J Karimi and S H Pourtakdoust ldquoOptimal maneuver-basedmotion planning over terrain and threats using a dynamichybrid PSO algorithmrdquo Aerospace Science and Technology vol26 no 1 pp 60ndash71 2013

[30] X Zhang and H Duan ldquoAn improved constrained differentialevolution algorithm for unmanned aerial vehicle global routeplanningrdquoApplied Soft Computing Journal vol 26 pp 270ndash2842015

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


physical characteristics of the UAV and the environment alsorestrict the feasibility of any path and should be consideredby a realistic path planning problem

In the past few years several path planning algorithmshave been proposed which can be mainly divided into twocategories graph-based and population-based evolutionaryalgorithms The former includes Voronoi diagram searchmethod [11] mathematical programming method [12] 119860lowastsearching algorithm [13 14] 119863lowast lite algorithm [15] andbilevel programming method [16] In these algorithms Epp-steinrsquos 119896-best paths algorithm [17] is used to find an optimalpath for UAV The biggest deficiency of graph-based ones isthat it is difficult to combine the motion constraints of UAVitself which means it usually cannot be used in practicalsituations UAV self-performance is an absolutely necessaryfactor needed to be taken into account when designing apath planning algorithm Another important category is thepopulation-based optimization algorithmsThey could makeUAV flying routes generated by reducing the complexity anddimensions which is a NP-hard problem These algorithmsmainly include genetic algorithm (GA) [18] particle swarmoptimization (PSO) [10] ant colony optimization (ACO) [19]artificial bee colony (ABC) [20] differential evolution (DE)[21] gravitational search algorithm (GSA) [22] intelligentwater drops optimization (IWD) [23] memetic computingmethod [24] and firefly algorithm (FA) [25] Each of thesecategories has its own advantages over others in certainaspects However among them GA PSO and FA are themost three popular for their simplicity and effectivenesswhich are becoming the hottest research topics and are mostsuitable for solving the global optimization problems withlarge scales For example GA is famous for the ease ofimplementation for both continuous and discrete problemsThere are no extra requirements for the continuity in responsefunctions and could be used efficiently with large numbersof parallel processors The generated global or near globalsolutions are more robust And FA is recently developedto solve nonlinear design problems In FA all fireflies areunisexual and any individual firefly will be attracted toothers based on their higher brightness But their brightnessdecreases as their mutual distance increases By iterations ofbrightness oriented movement the global optimal solutionscould be found finally

Another important one is called PSO which is short forparticle swarm optimization It is an evolutionary compu-tation algorithm first proposed by Kennedy and Eberhart[26] which is designed based on the study of the socialbehavior of bird flocking and fish schooling Each particleadjusts its flying positions in the searching space in terms ofits own flying experience and the whole swarm flying expe-riences Successful applications of PSO in neural networktraining function optimization and fuzzy system controlhave demonstrated that PSO is a promising and efficientoptimization method So far many significant improvementsare proposed to improve the performance of standard PSOalgorithm An excellent overview of the basic concepts ofPSO and its variants can be found in [27] PSO has beenseen as an attractive optimization tool for the advantages ofsimple implementation procedure good performance and

Start point


S D

L1 Lk Lk+1 LM

W1

Wk

Wk+1

WM

T1

T2

T3

T4

Figure 1 Typical UAV path planning model

fast convergence speed However it has been shown that thismethod is easily trapped into local optima when coping withcomplicated problems

In this paper a potential odor intensity grid based UAVpath planning algorithm is proposed combining standardPSO technique By identifying the different levels of odorintensity a potential grid construction operator is designedfor standard PSO which is implemented easily and avoidslocal optima and slow convergence The potential grid con-struction operator indicates two potential location grids withhighest odor intensity including centers of the two grids andthe preset side length Then middle point of the two gridcenters will be seen as the final global position in currentparticle dimension space of current iteration The globaloptimum solution will be solved as the average Experimentalresults demonstrate that the proposed method is capable ofgenerating higher quality paths efficiently for UAV than anyother tested optimization algorithms

The remainder of this paper is organized as followsPath planning problem description is provided in Section 2Concise standard PSO algorithm and explicit realizationof proposed method are given in Section 3 respectivelySection 4 presents the settings in application conditions ofthe proposed algorithm and experimental results of globalroute planning are listed Finally the paper is concluded inSection 5

2 Problem Formation

UAV path planning can be seen as a global optimizationproblem to generate a serial of way points from the startto the destination with least cost values Terrain and threatmodeling and the design of objective function become twokey problems in route generation for UAV

21 Terrain and Threat Modeling As shown in Figure 1 themission of path planning is to generate a feasible flying routefrom the start point 119878 to destination point 119863 without flyingout of the designated map or being taken down by threats aswell as least flying cost There are some threat areas locatingin the map such as radars missiles and artillery which areall denoted as red circles in Figure 1 Once the UAV is in thecoverage of any threat it will be vulnerable to the threat witha certain probability proportional to the distance away fromthe threat centers Oppositely there will not be any danger if




119869out =

sum

119894

Dis (119894) if (119909119894 119910119894)119879

notin 119877

0 otherwise(1)





119894

threat 119910119894

threat 119903119894




119903119894



119895 119910119895)119879 we can define


119869expo =

sum

119895

sum

119894

119866119894[(119903119894

threat)2

minus (119903119895119894)2

]

(119903119894

threat)2

if 119903119895119894lt 119903119894

threat

0 otherwise

(2)



119903119895119894= radic(119909

119895minus 119909119894

threat)2

minus (119910119895minus 119910119894

threat)2

(3)


o

y

x

Start point


D

L1

Lk

Lk+1

LM

W1

Wk

Wk+1 WM

T1

T2T3

T4

120579

y998400

x998400

o998400S








1015840

1199001015840







[

119909119879

119910119879

] = [


] [

119909119874minus 119909119878

119910119874minus 119910119878

] (4)




namely 1198781198821 119882






119909119896=

|119878119863|

119872 + 1119896 (5)






119894(119894 = 1 2 119872) and (119872 + 1) segments



119894and 119882

119894+1





119894119882119894+1


119894119882119894+1


210038161003816100381610038161198821198941198751119871

1003816100381610038161003816=100381610038161003816100381611987511198711198752119871

1003816100381610038161003816= sdot sdot sdot =

1003816100381610038161003816119875(119871minus1)119871

119875119871119871

1003816100381610038161003816

= 21003816100381610038161003816119875119871119871119882119894+1

1003816100381610038161003816

(6)


119894119882119894+1


119869119879(119894)

=

100381610038161003816100381610038161003816

997888997888997888997888997888997888rarr119882119894119882119894+1

100381610038161003816100381610038161003816

119871

119873119879

sum

119895=1

119866119895(

1

1198891119871119894119895

+1

1198892119871119894119895

+ sdot sdot sdot +1

119889119871119871119894119895

)

(7)


119894119882119894+1


119894119882119894+1

and119866119895



119894119882119894+1

is set as119873119879


PLL

Threat(j)

Threat(k)



Li Li+1

Wi

Wi+1

P1LP2L

P3L

P4L

P(Lminus1)L


S D

Start point


(x1 y1)T

(x2 y2)T

(xk yk)T

(xM+1 yM+1)T

(xM+2 yM+2)T



119869path = sum119894

119869119879(119894) (8)


1 1199101)119879 and

(119909119872+2

119910119872+2



119896as


119869SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(9)

In (9) |119878119863| = radic(119909119872+2

minus 1199091)2

+ (119910119872+2




Li

Wi1

(xi1 yi1)

Wi2

(xi2 yi2)

Wik

(xik yik)

Wik+1

(xik+1 yik+1)

120579i1

120579i2

120579ik

120579ik+1

D

(xD yD)



11989411198821198942 ) in the




119894






119869DA = sum119894

min119896

120579119894119896 = sum

119894

min119896

arctan1003816100381610038161003816119910119863minus 119910119894119896

10038161003816100381610038161003816100381610038161003816119909119863minus 119909119894119896

1003816100381610038161003816

(10)


119894119896 119910119894119896) is



119894and radius 119903



119880andΔ119889

119863 respectively


119871max = max119894

119910threat (119894) + 119903threat (119894) + Δ119889119880

119871min = min119894



D

Upper limit Lmax

ΔdD

ΔdU

rjTj

Tiri

Lower limit Lmin


y998400

x998400

o998400S



119869out =

119872

sum

119894=1

[119910119880(119894) minus 119871max] if 119910

119880(119894) gt 119871max

0 otherwise119872

sum

119894=1


(12)


















end while







(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)


(x1k1) (x2k2) (xSkS) (15)





V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)








119894119895with






dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119869out =

sum

119894

Dis (119894) if (119909119894 119910119894)119879

notin 119877

0 otherwise(1)





119894

threat 119910119894

threat 119903119894




119903119894



119895 119910119895)119879 we can define


119869expo =

sum

119895

sum

119894

119866119894[(119903119894

threat)2

minus (119903119895119894)2

]

(119903119894

threat)2

if 119903119895119894lt 119903119894

threat

0 otherwise

(2)



119903119895119894= radic(119909

119895minus 119909119894

threat)2

minus (119910119895minus 119910119894

threat)2

(3)


o

y

x

Start point


D

L1

Lk

Lk+1

LM

W1

Wk

Wk+1 WM

T1

T2T3

T4

120579

y998400

x998400

o998400S








1015840

1199001015840







[

119909119879

119910119879

] = [


] [

119909119874minus 119909119878

119910119874minus 119910119878

] (4)




namely 1198781198821 119882






119909119896=

|119878119863|

119872 + 1119896 (5)






119894(119894 = 1 2 119872) and (119872 + 1) segments



119894and 119882

119894+1





119894119882119894+1


119894119882119894+1


210038161003816100381610038161198821198941198751119871

1003816100381610038161003816=100381610038161003816100381611987511198711198752119871

1003816100381610038161003816= sdot sdot sdot =

1003816100381610038161003816119875(119871minus1)119871

119875119871119871

1003816100381610038161003816

= 21003816100381610038161003816119875119871119871119882119894+1

1003816100381610038161003816

(6)


119894119882119894+1


119869119879(119894)

=

100381610038161003816100381610038161003816

997888997888997888997888997888997888rarr119882119894119882119894+1

100381610038161003816100381610038161003816

119871

119873119879

sum

119895=1

119866119895(

1

1198891119871119894119895

+1

1198892119871119894119895

+ sdot sdot sdot +1

119889119871119871119894119895

)

(7)


119894119882119894+1


119894119882119894+1

and119866119895



119894119882119894+1

is set as119873119879


PLL

Threat(j)

Threat(k)



Li Li+1

Wi

Wi+1

P1LP2L

P3L

P4L

P(Lminus1)L


S D

Start point


(x1 y1)T

(x2 y2)T

(xk yk)T

(xM+1 yM+1)T

(xM+2 yM+2)T



119869path = sum119894

119869119879(119894) (8)


1 1199101)119879 and

(119909119872+2

119910119872+2



119896as


119869SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(9)

In (9) |119878119863| = radic(119909119872+2

minus 1199091)2

+ (119910119872+2




Li

Wi1

(xi1 yi1)

Wi2

(xi2 yi2)

Wik

(xik yik)

Wik+1

(xik+1 yik+1)

120579i1

120579i2

120579ik

120579ik+1

D

(xD yD)



11989411198821198942 ) in the




119894






119869DA = sum119894

min119896

120579119894119896 = sum

119894

min119896

arctan1003816100381610038161003816119910119863minus 119910119894119896

10038161003816100381610038161003816100381610038161003816119909119863minus 119909119894119896

1003816100381610038161003816

(10)


119894119896 119910119894119896) is



119894and radius 119903



119880andΔ119889

119863 respectively


119871max = max119894

119910threat (119894) + 119903threat (119894) + Δ119889119880

119871min = min119894



D

Upper limit Lmax

ΔdD

ΔdU

rjTj

Tiri

Lower limit Lmin


y998400

x998400

o998400S



119869out =

119872

sum

119894=1

[119910119880(119894) minus 119871max] if 119910

119880(119894) gt 119871max

0 otherwise119872

sum

119894=1


(12)


















end while







(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)


(x1k1) (x2k2) (xSkS) (15)





V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)








119894119895with






dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119909119896=

|119878119863|

119872 + 1119896 (5)






119894(119894 = 1 2 119872) and (119872 + 1) segments



119894and 119882

119894+1





119894119882119894+1


119894119882119894+1


210038161003816100381610038161198821198941198751119871

1003816100381610038161003816=100381610038161003816100381611987511198711198752119871

1003816100381610038161003816= sdot sdot sdot =

1003816100381610038161003816119875(119871minus1)119871

119875119871119871

1003816100381610038161003816

= 21003816100381610038161003816119875119871119871119882119894+1

1003816100381610038161003816

(6)


119894119882119894+1


119869119879(119894)

=

100381610038161003816100381610038161003816

997888997888997888997888997888997888rarr119882119894119882119894+1

100381610038161003816100381610038161003816

119871

119873119879

sum

119895=1

119866119895(

1

1198891119871119894119895

+1

1198892119871119894119895

+ sdot sdot sdot +1

119889119871119871119894119895

)

(7)


119894119882119894+1


119894119882119894+1

and119866119895



119894119882119894+1

is set as119873119879


PLL

Threat(j)

Threat(k)



Li Li+1

Wi

Wi+1

P1LP2L

P3L

P4L

P(Lminus1)L


S D

Start point


(x1 y1)T

(x2 y2)T

(xk yk)T

(xM+1 yM+1)T

(xM+2 yM+2)T



119869path = sum119894

119869119879(119894) (8)


1 1199101)119879 and

(119909119872+2

119910119872+2



119896as


119869SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(9)

In (9) |119878119863| = radic(119909119872+2

minus 1199091)2

+ (119910119872+2




Li

Wi1

(xi1 yi1)

Wi2

(xi2 yi2)

Wik

(xik yik)

Wik+1

(xik+1 yik+1)

120579i1

120579i2

120579ik

120579ik+1

D

(xD yD)



11989411198821198942 ) in the




119894






119869DA = sum119894

min119896

120579119894119896 = sum

119894

min119896

arctan1003816100381610038161003816119910119863minus 119910119894119896

10038161003816100381610038161003816100381610038161003816119909119863minus 119909119894119896

1003816100381610038161003816

(10)


119894119896 119910119894119896) is



119894and radius 119903



119880andΔ119889

119863 respectively


119871max = max119894

119910threat (119894) + 119903threat (119894) + Δ119889119880

119871min = min119894



D

Upper limit Lmax

ΔdD

ΔdU

rjTj

Tiri

Lower limit Lmin


y998400

x998400

o998400S



119869out =

119872

sum

119894=1

[119910119880(119894) minus 119871max] if 119910

119880(119894) gt 119871max

0 otherwise119872

sum

119894=1


(12)


















end while







(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)


(x1k1) (x2k2) (xSkS) (15)





V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)








119894119895with






dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Li

Wi1

(xi1 yi1)

Wi2

(xi2 yi2)

Wik

(xik yik)

Wik+1

(xik+1 yik+1)

120579i1

120579i2

120579ik

120579ik+1

D

(xD yD)



11989411198821198942 ) in the




119894






119869DA = sum119894

min119896

120579119894119896 = sum

119894

min119896

arctan1003816100381610038161003816119910119863minus 119910119894119896

10038161003816100381610038161003816100381610038161003816119909119863minus 119909119894119896

1003816100381610038161003816

(10)


119894119896 119910119894119896) is



119894and radius 119903



119880andΔ119889

119863 respectively


119871max = max119894

119910threat (119894) + 119903threat (119894) + Δ119889119880

119871min = min119894



D

Upper limit Lmax

ΔdD

ΔdU

rjTj

Tiri

Lower limit Lmin


y998400

x998400

o998400S



119869out =

119872

sum

119894=1

[119910119880(119894) minus 119871max] if 119910

119880(119894) gt 119871max

0 otherwise119872

sum

119894=1


(12)


















end while







(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)


(x1k1) (x2k2) (xSkS) (15)





V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)








119894119895with






dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















end while







(xiki) = ((1199091198941 1199091198942 119909119894119863) (V1198941 V1198942 V119894119863)) (14)


(x1k1) (x2k2) (xSkS) (15)





V119896+1119894119895

= 119908V119896119894119895+ 1198881120585 (119901119896

119894119895best minus 119909119896

119894119895) + 1198882120578 (119892119896

119895best minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ 119903V119896+1119894119895

119894 = 1 2 119878 119895 = 1 2 119863 119896 = 1 2 119873 minus 1

(17)








119894119895with






dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











dij

dij

Wij

C1

C2

G1

G2

G3

dij




following steps




119894119895(119896) as center and 119889

119894119895


119894119895(119896)

119861119894119895(119896)

Center 119871119894119895(119896)

Side length 119889119894119895

(18)


119897


1198661 1198662 = 119861

119894119895(1) cap 119861


119894119895(119873) (19)





119894119895


119882119894119895=(1198881+ 1198882)

2 (20)




119871119895=

sum119878

119894=1119882119894119895

119878 (21)


119894(119894 = 1 2 119863)







Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start








End


Yes

No


Set k = k + 1




119894119895














Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Scenarionumber

Numberof

iterations

Startpoint

Destinationpoint


Threatradius(km)


400 [1 1] [95 95]

[15 25] 10[45 25] 15[55 58] 10[70 82] 8[81 58] 12


400 [1 1] [95 95]

[13 15] 10[18 48] 15[42 15] 8[49 45] 11[50 80] 15[62 31] 12[75 75] 13[80 15] 10[87 50] 12



119880= Δ119889


119894119895) of



SLR =sum119872+1

119894=1

radic(119909119894+1minus 119909119894)2

+ (119910119894+1minus 119910119894)2

|119878119863|

(22)





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA









0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0 50 100 150 200 250 300 350 400

04

05

06

07

08

09

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

(a)

11

12

13

14

15

16

17

18

19

SLR

0 50 100 150 200 250 300 350 400Iteration

Proposed methodPSO

FAGA




Cost value


Path length





0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA






0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

Iteration

SLR

Proposed methodPSO

FAGA

11

12

13

14

15

16

17

18

(b)









Cost value


Path length






0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

20

40

60

80

100 Destination

Start

T1

T2

T3

T5

T6

T9

T7

T8

T4

y(k

m)

x (km)

Proposed methodPSO

FAGA







0means





119905-test 119878 = 100119863 = 10

1198670






1is all








0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 40002

03

1

Iteration

Nor

mal

ized

cost

Proposed methodPSO

FAGA

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 4001

12

14

16

18

2

22

24

26

28

Iteration

SLR

Proposed methodPSO

FAGA

(b)






Lower Upper











5 Conclusion



0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100Destination

Start

T1

T2

T3

T5

T4

y(k

m)

x (km)

S = 50

S = 200

S = 300

S = 400

(a)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

Start

Destination

T1

T2

T3

T5

T4

y(k

m)

x (km)

D = 10

D = 15

D = 20

D = 25

(b)


0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 40008

082

084

086

088

09

092

094

096

098

1

Iteration

Nor

mal

ized

cost

S = 50

S = 200

S = 300

S = 400

(b)





0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400

1

Iteration

Nor

mal

ized

cost

D = 10

D = 15

D = 20

D = 25

04

05

06

07

08

09

(a)

0 50 100 150 200 250 300 350 400065

07

075

08

085

09

095

1

Iteration

Nor

mal

ized

pat

h le

ngth

D = 10

D = 15

D = 20

D = 25

(b)



Competing Interests


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Potential Odor Intensity Grid Based UAV ...downloads.hindawi.com/journals/mpe/2016/7802798.pdfthat it is di cult to combine the motion constraints of UAV itself, which