Multi-robot path planning in a dynamic environment using improved gravitational search algorithm

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Availableonlineatwww.sciencedirect.com

ScienceDirect

JournalofElectricalSystemsandInformationTechnology3(2016)295–313

Multi-robot

path

planning

in

a

dynamic

environment

using

improved

gravitational

search

algorithm

P.K.

Das

a,∗

,

H.S.

Behera

a

,

P.K.

Jena

b

,

B.K.

Panigrahi

c

a_Dept._of_Comp._Sc._and_Engineering_and_Information_Technology,_VSSUT,_Burla,_Odisha,_India b_Dept._of_Mechanical_Engineering,_VSSUT,_Burla,_Odisha,_India

c_Dept._of_Electrical_Engineering,_IIT,_Delhi,_India

Received17August2015;receivedinrevisedform17November2015;accepted20December2015 Availableonline2August2016

Abstract

Thispaperproposesanewmethodologytooptimizetrajectoryofthepathformulti-robotsusingimprovedgravitationalsearch algorithm(IGSA)inadynamicenvironment.GSAisimprovedbasedonmemoryinformation,social,cognitivefactorofPSO (particleswarmoptimization)andthen,populationfornextgenerationisdecidedbythegreedystrategy.Apathplanningscheme hasbeendevelopedusingIGSAtooptimallyobtainthesucceedingpositionsoftherobotsfromtheexistingposition.Finally,the analyticalandexperimentalresultsofthemulti-robotpathplanninghavebeencomparedwiththoseobtainedbyIGSA,GSAand PSOinasimilarenvironment.ThesimulationandtheKheperaenvironmentalresultsoutperformIGSAascomparedtoGSAand PSOwithrespecttoperformancematrix.

Keywords: Gravitationalsearchalgorithm;Multi-robotpathplanning;Averagetotaltrajectorypathdeviation;Averageuncoveredtrajectorytarget distance;Averagepathlength

1. Introduction

Gravitationalsearchalgorithm (GSA)iseffective andefficientusing analternative approachtothemulti-robot pathplanning.Althoughmanyalgorithms(TuncerandYildirim,2012;GuoandParker,2002)havebeenproposed and provento be feasible andefficient for robotmotion planning andcollision avoidance, classictechniques for pathplanningproblem(Konar,1999;Banerjeeetal.,2011)aregeneralmethodslikeRoadmap,CellDecomposition, PotentialFields,OpticalTweezersandMathematicalProgramming.Manyauthorshaveproposedmulti-robotandthe

∗_{Corresponding}_author.

E-mailaddresses:[email protected](P.K.Das),[email protected](H.S. Behera),[email protected] (P.K.Jena),

[email protected](B.K.Panigrahi).

PeerreviewundertheresponsibilityofElectronicsResearchInstitute(ERI).

http://dx.doi.org/10.1016/j.jesit.2015.12.003

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singlerobotpathplanningproblemsusingdifferentclassicaltechniques(KcymeulcnandDecuyper,1994;Lietal., 2009),NeuralNetwork(YuandKromov,2001),artificialimmunesystem(Dasetal.,2012;LuhandCheng,2002)and heuristicoptimizationalgorithms(Dasetal.,2010,2011;Geemetal.,2001;Yang,2009;RegeleandLevi,2006).High timecomplexityinlargeproblemspacesandtrappinginlocaloptimumaredrawbacksforclassictechniquesandin manymeta-heuristicalgorithms.Thesedrawbackscausetheclassicaltechniquesandinefficientinthevariousproblem spaces.To improve theefficiency of classicalmethods, probabilisticalgorithms like PRMandRRTare proposed forimprovingthe localoptimizationproblem,manyevolutionaryalgorithmslike GeneticAlgorithms (Tuncerand Yildirim,2012;GongandLincheng,2001),PSO(Zhangetal.,2013;MasehianandSedighizadeh,2010),beecolony optimization(Bhattacharjeeetal.,2011)anddifferentialevolutionalgorithm(Chakrabortyetal.,2009)areusedin multi-robotpathplanningproblem.

Thegravitationalsearchalgorithm(Vermaetal.,2013;EldosandQasim,2013;Chatterjeeetal.,2011)isarecent algorithmthathasbeeninspiredbytheNewtonian’slawofgravityandmotion.GSAhasundergonealotofchangesto thealgorithmitselfandhasbeenappliedinvariousapplications.Atpresent,therearevariousvariantsofGSA(Precup etal.,2012;Rashedietal.,2010,2009b;Purcaruetal.,2013)whichhavebeendevelopedtoenhanceandimprovethe originalversion.Thealgorithmhasalsobeenexploredinmanyareas(Sabrietal.,2013;EldosandQasim,2013).

ForrealizationmultirobotpathplanningproblemwithdifferentgoaloftherespectiverobotswithGSA(Precup etal.,2012;TuncerandYildirim,2012)bythecentralizedapproach,afitnessfunctionisconstructedtodeterminethe nextpositionoftherobotsthatlieonoptimaltrajectoriesleadingtowardtherespectivegoals.Thefitnessfunctionof theGSA(AlbaandDorronsoro,2005)hastwomaincomponents: firstoneistheobjectivefunctiondescribingthe selectionofnextpositiononanoptimaltrajectorybasedonvelocity,andthesecondoneistheconstraintonacceleration representingavoidanceofcollisionwithotherrobotsandwithstaticobstacles.Thepath-planningproblemconsidered hereisformulatedbyacentralizedapproach,whereaniterativealgorithmisinvokedtodeterminethenextpositionof alltherobotssatisfyingalltheconstraintsimposedonthemulti-objectivefunction.Thealgorithmisiterateduntilall therobotsreachtheirdestination(goalposition).

Theadvantagesof GSAare(1) easytoimplementwithhigher computationalefficiency; (2)few parametersto adjust,butthedisadvantagesofthisalgorithmareasfollow(1)ifprematureconvergence occurs,therewillnotbe anyrecoveryfor thisalgorithm; (2)the algorithmlosesitsability toexploreandthenbecomesinactiveonlyafter becomesconvergence.DuetoabovedifficultiesinGSA,furtherimprovementsarerequiredfortheoptimalsolution

tothe complex problem.Here,we consider theimprovement of GSA whichis basedon the communicationand

memorycharacteristicsofPSO(particleswarmoptimization).Therefore,wecalleditimprovedgravitationalsearch algorithm.

Themainobjectiveofthispaperissummarizedasfollows:(i)westudytheproblemofmulti-robotpathplanning inaclutterenvironmentandformulatedtheaboveproblemasmulti-objectiveoptimizationproblemwithconstraints; (ii)anovelmethodtothesolutionofanoptimaltrajectorypathgenerationformultirobotpathplanningproblemusing IGSAisproposedinthisarticle;(iii)theproposedalgorithmhasbeenappliedformultirobotpathplanninginaclutter anddynamicenvironmentandobtainedresultsarecomparedtootheroptimizationalgorithmslikeGSA,DE;(iv)the performanceoftheproposedIGAS,asanoptimizingtoolinsolvingmultirobotpathplanningproblem,isappliedin thesimulationaswellasKhepera-IIenvironmentandresultispresented;(v)theperformancematrixoftheproposed approachissuccessfullyvalidatedinsimulationandKhepera-II.

Inthispaper,theimplementationofthemodifiedgravitationalsearchtechniquehasbeenproposedtodetermine thetrajectorypathformultiplerobotsfrompredefinedinitialpositionstopredefinetargetpositionsintheenvironment withanobjective tominimize the path length of allthe robots. Theresult shows that the algorithmcan improve thesolutionqualityinareasonableamountoftimeandalsoimprovestheconvergencerate.Thispaperimprovesthe gravitationalsearchalgorithm(IGSA)forimprovingtheglobalpathplanningproblemofthemulti-robotsbyimproving theconvergencerate.Finally,theefficiencyoftheIGSAhasbeenprovedthroughthesimulationaswellasKhepera environmentandaresultobtainediscomparedwithotherevolutionarycomputingsuchanGSAandDE.

Therestofthepaperisoutlinedasfollows:Section3brieflydescribestheimprovedgravitationalsearchalgorithm. Formulationoftheproblemformulti-robotpathplanninghasbeenelaboratedinSection4.Multi-objectiveoptimization problemsolvingusingimprovedGSAisdescribedindetailsinSection5.Section6demonstratestheresultofpath planningformulti-robotthroughsimulation.InSection7,theexperimenthasbeenconductedinKheperaIIenvironment andfinally,theconclusionoftheworkispresentedinSection8.

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2. Gravitationalsearchalgorithm(GSA)

Recently,thescientificcommunityhasgainedtheinterestonGSA.Itisameta-heuristicoptimizationalgorithm inspiredbynaturewhichisbasedontheNewton’slawofgravityandthelawofmotion(Rashedietal.,2009a;Sabri etal.,2013).GSAisgroupedunderthepopulationbasedapproachandisreportedtobemorenatural.Thealgorithm hasbeenplannedtoimprovetheperformanceintheexplorationandmanipulationcapabilitiesofapopulationbased algorithm,basedongravityrules.

GSAisbasedonthetwoimportantformulasaboutNewtongravitylawsgivenbyEqs.(1)and(2).Eq.(1)isthe gravitationalforceequationbetweenthetwo particles,whichisdirectlyproportionaltotheir massesandinversely proportionaltothesquareofthedistancebetweenthem.ButinGSAinsteadofthesquareofthedistance,onlythe distanceisused.Eq.(2)istheequationofaccelerationofaparticlewhenaforceisappliedtoit(Rashedietal.,2009a; Sabrietal.,2013).

F =GM1M2

R2 (1)

a= F

M (2)

Gisgravitationalconstant,M1andM2aremassesandRisthedistancebetweenthem,Fisgravitationalforce,anda

isacceleration.Basedontheseformulas,theheavierobjectwithmoregravityforceattractstheotherobjectsasitis seeninFig.1.

InGSA,eachmass(agent)hasfourcharacteristics,namely:position,inertialmass,activegravitationalmass,and passivegravitationalmass.Thepositionofthemasscorrespondstoasolutionoftheproblem,andthefitnessfunction isusedtodeterminethegravitationalandinertialmasses(Vermaetal.,2013;Sabrietal.,2013).Themoreprecisely massesobeythefollowinglaws.

Lawofgravity:Eachparticleattractseveryotherparticleandthegravitationalforcebetweentwoparticlesisdirectly

proportionaltotheproductoftheirmassesandinverselyproportionaltothedistancebetweenthem,R.Weusehere

RinsteadofR2,becauseaccordingtoourexperimentalresults,RprovidesbetterresultsthanR2inallexperimental cases.

Lawofmotion:Thecurrentvelocityofanymassisequaltothesumofthefractionofitspreviousvelocityandthe

variationinthevelocity.Variationinthevelocityoraccelerationofanymassisequaltotheforceactedonthesystem dividedbymassofinertia.

2.1. Agentsinitialization

ConsiderasystemwithNmassesinwhichpositionoftheith_mass_is_defined_as_follows:

Xi=

x1_i,...xd_i,...,xn_i for i=1,2,...N (3)

wherexd_i isthepositionofithmassindthdimensionandnisthedimensionofthesearchspace.

M2 F1 M1 F2 F3 M3

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2.2. Fitnessandbestﬁtnesscomputation

worst(t)andbest(t)aredefinedasfollowsfortheminimizationcase:

worst(t)=maxi∈pfiti(t), p=1,2,...,N (4)

best(t)=mini∈pfiti(t), p=1,2,...,N (5)

2.3. Gravitationalconstant(G)computation

ThegravitationalconstantGiscomputedatiteration(Sabrietal.,2013).

G(t)=Goe(-αt/T) (6)

Here,Tisthemaximumiteration,tisthecurrentiterationandα0istheweightfactor,computedasfollows.

α=αmax−

αmax−αmin

T ×t (7)

2.4. Massesoftheagents’calculation

Eachagent’smassiscalculatedaftercomputingcurrentpopulation’sfitnessas:

mi(t)= fiti(t)−worst(t) best(t)−worst(t) (8) Mi(t)= mi(t) N j=1mj(t) (9) whereMi(t)andfiti(t)representthemassandthefitnessvalueoftheagentiatiterationt,respectively.

2.5. Velocityandpositionsofagents

Thevelocityandpositionoftheagentsareupdatedas:

V_id(t+1)=βV_id(t)+ad_i(t) (10)

xd_i(t+1)=x_id(t)+V_id(t+1) (11)

Here,βistherandomnumber,0≤β≤1andanaccelerationoftheithagentsatiteration‘t’iscomputedas,

ad_i(t)= F

d i (t)

Mi(t)

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F_id(t)isthetotalforceactingonithagentcalculatedas:

F_id(t)=

j∈kbest,j=/i

βF_ijd(t) (13)

KbestisthesetoffirstKagentswiththebestfitnessvalueandbiggestmass,whichisafunctionoftime,initializedto

k0atthebeginninganddecreasingwithtime.Herek0issettoN(totalnumberofagents)andisdecreasedlinearlyto1.

F_ijd(t)iscomputedusingthefollowingequation:

F_ijd(t)=G(t)× Mpi(t)× Maj(t) disij(t)+ε ×Xd_j(t)−Xd_i(t) (14) HereXi andXjarethepositionvectoroftheithandjthagentindth dimension,F_ijd(t)istheforceactingonagenti

fromagentjatdthdimensionandithiteration.disij(t)istheEuclidiandistancebetweentwoagentsiandjatiteration

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gravitationalmassoftheagentiattheinstancet.Maj(t)istheactivegravitationalmassoftheagentjattimet,these

massesbeingcalculatedaccordingtoPrecupetal.(2012),Rashedietal.(2010,2009b)andPurcaruetal.(2013).

3. ImprovementofthegravitationalsearchalgorithmbasedonPSOandgreedystrategy

Mostofthemeta-heuristicsearchingalgorithmfinditsbestsolutionduetogoodbalanceofexplorationand exploita-tion(AlbaandDorronsoro,2005;Liuetal.,2013).Theexplorationcapabilityofthealgorithmprovidestheconnectivity relationshipofthesearchspace,whichhelpstofindglobaloptimalsolution.Theexploitationhelpstofindthebetter optimalsolutionsinthevisiteddomain,whichreinforcetheconvergencecapabilityoflocalsearch.So,good meta-heuristicalgorithmshouldimprovetheexplorationabilityinthefirststepandthenexploitationabilitywithincreasing of iterationinsecond step.Therefore, thegravitational searchalgorithm hasbeen improvedtomaintain the good balancebetweenexplorationandexploitation.InGSA,themomentdirectionofeachagentisbasedonthetotalforce actbyotheragentsonitandlackingthecommunicationbetweentheagents.Therefore,improvementofthesearching abilityofGSAbasedonthememoryandsocialinformationofPSOandtoacceleratetheconvergencerate,weight valueisassignedtoinertiamassofeveryagentineachiteration(Sarafrazietal.,2011)andthen,optimizedsolution savingstrategyisusedwithreferencetoDE(Sarafrazietal.,2011).ThePSOupdatesthevelocityusingthecognitive andsocialfactor.ThevelocityandpositionupdateequationofPSOareasfollow:

V_id(t+1)PSO=wVid+C1×ϕ1×(xdpbesti−x

d

i(t))+C2×ϕ2×(xdgbest−xdi(t)) (15)

xd_i(t+1)PSO=xdi(t+1)+Vid(t+1) (16)

vd_i(t+1)GSA=βvi(t)+adi(t) (17)

whereEq.(17)istheGSAvelocityformulationobtainedfromEq.(10).Inthispaper,GSAisimprovedbyadopting thememory,socialandcognitiveinformationofPSO.ThevelocityupdatingequationinGSAcanbedefinedas

V_id(t+1)IGSA=βVid(t)GSA+aid(t)+C1×ϕ1×(xdpbesti−x

d

i(t))+C2×ϕ2×(xdgbest−x

d

i(t)) (18)

xd_i(t+1)IGSA =xid(t)+Vid(t+1)IGSA (19)

whereEq.(19)istheIGSAvelocityformulation,whichisformulatedandupdatedusingPSOvelocitybyconsidering thememory,socialandcognitivefactorandGSAacceleration.C1andC2balancetheeffectivenessof“lawofgravity

andmemoryandsocialinformation”.Theoptimizedsolutionsavingstrategyisusedfordecidingthememberfornext generationt+1withreferencetodifferentialevolution(DE)(Sarafrazietal.,2011).The“survivaloffittest”strategy isusedtodecidethememberfornextgeneration.Here,thegreedystrategyhasbeendevisedfordecidingbettertarget vector.Thepopulationfornextgenerationisdecidedbycomparingthetrialvectorxd_i(t+1) withthetargetvector

xd_i(t).Theselectionprocedurecanbeexpressedbythefollowingexpression.

xd_i(t+1)=

xd_i(t), iffit(xd_i(t))<fit(xd_i(t+1))

xd_i(t+1), otherwise (20)

4. Problemformulationformultirobotpathplanning

Theproblemformulationfor multi-robotpathplanningistodeterminethenextpositionoftherobotfromtheir existingpositionsinitsworkspacebyavoidingthecollisionwithotherrobotsandobstacles(whicharestaticinnature) initspathtoreachatthegoal.Multi-robotpathplanningproblemisformulatedbyconsideringthesetofprinciples usingthefollowingassumptionsbyauniformtreatment.

Assumptions

a Foreachrobotthecurrentposition(recentposition)andgoalposition(targetposition)isknowninagivenreference coordinatesystem.

b Therobotcanselectanyactioninagiventimefromafixedsetofactionsforitsmotions.

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) 1 (t+ yi X Y ) (t i x ) (t i y ) 1 (t+ xi )) ( ), ( (xit yit ) (t v_i i θ

Fig.2.Representationofnextpositionfromcurrentpositionofthei-throbot.

Thefollowingprincipleshavebeentakencareforsatisfyingthegivenassumptions.

1. Fordeterminingthenextpositionfromitscurrentposition,therobottriestoalignitsheadingdirectiontowardthe goal.

2. Thealignmentmaycauseacollisionwiththerobots/obstacles(whicharestaticinnature)intheenvironment,hence, therobothastoturnitsheadingdirectionleftorrightbyaprescribedangletodetermineitsnextposition. 3. Ifarobotcanalignitselfwithagoalwithoutcollision,then,itwillmovetothatdeterminetheposition.

4. Ifrotatingtheheadingdirectionleftorrightrequiresthesameangleofrotationoftherobotaboutthez-axis,ifitis tiedthen,brokenrandomly.

Considertheinitialpositionoftheithrobotatatimetis(xi(t),yi(t)),thenextpositionofthesamerobotatatime

(t+δt)is(xi(t+δt),yi(t+δt)),vi(t)isthevelocityoftherobotRiand(x

goal

i ,y

goal

i )isthetargetorgoalpositionof

therobotRi.

So,theexpressionforthenextposition(xi(t+δt),yi(t+δt))canbederivedfromFig.2asfollows

xi(t+δt)=xi(t)+vi(t)cosθiδt (21)

yi(t+δt)=yi(t)+vi(t)sinθiδt (22)

Whenδt=1,Eqs.(21)and(22)arereducedto

xi(t+1)=xi(t)+vi(t)cosθi (23)

yi(t+1)=yi(t)+vi(t)sinθi (24)

Considerinitially,therobotRi isplacedinthelocationat(xi(t),yi(t)).Wewanttofindthenextpositionofthe

robot(xi(t+1),yi(t+1)),suchthat thelinejoining between{(xi(t),yi(t)),(xi(t+1),yi(t+1))}and{(xi(t+1),

yi(t+1)),(xgoali ,y

goal

i )} shouldnottouchtheobstacleintheworldmapisrepresentedinFig.3andminimizesthe

totalpathlengthfromcurrentpositiontoagoalpositionwithouttouchingtheobstaclebyformingconstraint.Then

X Y )) ( ), ( (xi t yi t )) ( ), ( (xit+δt yit+δt Obstacle

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objectivefunctionfit1thatdeterminesthelengthofthetrajectoryfornnumberofrobots, fit1= n i=1 ((xi(t)−xi(t+1))2+(yi(t)−yi(t+1)2)+ ((xi(t+1)−xgoali ) 2 +(yi(t+1)−yigoal) 2 ) (25) Byputtingthevaluexinextandynexti fromEqs.(21)and(24)intoEq.(25),weobtain,

fit1= n i=1 vi(t)+ (xi(t)+vi(t)cosθi−x goal i ) 2 +(yi(t)+vi(t)sinθi−y goal i ) 2 (26) Themulti-robotpath-planningisnowrepresentedasanoptimizationproblembyminimizingtheobjectivefunction inEq.(26)withconsideringthepenaltyfunctionastheconstraintsintheobjectivefunction.Minimizingtheobjective functioninEq.(26)showsthattherobotwillfollowtheshortestdistancefromtheinitialpointtotargetpoint.The constraintsherearetwotypesofpenalty.Thefirstpenaltyistoavoidcollisionbetweenteammates(anytwomobile robots),whereasthesecondpenaltyistoavoidcollisionof arobotwithastaticobstacle.Bycombiningthesetwo penaltiesalinearfuzzyfunctionisdevelopedforevaluatingtheobstaclepresentinthepath.So,theobjectivefunction formedbasedonthefuzzyfunctionisdenotedasfitj.

fitj= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 d(Oj)≤d(Oj)_min exp −α d(Oj)−d(Oj)min d(Oj)_max−d(Oj)_min d(Oj)_min≺d(Oj)≺d(Oj)_max 0 d(Oj)≥d(Oj)_max (27)

whereαisapositiveconstant,d(Oj)be thedistancebetween mobilerobotandobstacles,d(Oj)_max ismaximum

distanceandd(Oj)_min istheminimumdistancewithrespecttotheobstacleOj.Thepathissafeandcollisionfree

path,whend(Oj)≥d(Oj)_maxandpathisunsafeif,d(Oj)≥d(Oj)_min.

Thus,mathematically,theoptimizationproblemforobstaclescanbeformulatedasfollows:

fit2=maxj=1,2,...N(fitj) (28)

Thus,theoptimizationproblemcanberepresentedasfollows: fit=fit1+

λ

fit2

(29) Here,λispositiveconstant.TheaboveoptimizationproblemistominimizetheEuclideandistancebetweenthecurrent positionandtheirgoalpositionwhichispresentedbytheobjectivefunctionfit1andthesecondobjectivefunctionisa

constrainttofindthesafepath.

5. Multi-objectiveoptimizationproblemsolvingusingIGSA

Inthissection,multi-robotpathplanningalgorithmhasbeenproposedusingIGSA.TheproposedIGSAalgorithm isusedtoevaluatethenextpositionsofeveryrobotbypresumingthecurrentpositionsofrobotsandspeedsasthe parameterforoptimizingthegivenmulti-objectivefunction.Itdeterminestheoptimizedpathfromeachstatetothe goalstateinbothdynamicandstaticenvironmentsandtherobotmeasuresitsdistancetoobstacleswiththehelpof equippingsensors.

TheagentsaremovinginthesearchspacewiththehelpofthegravityisconsideredintheproposedIGSAbased pathplanning.Theoutlineoftheproposedalgorithmisdiscussedbelow:

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ProcedureIGSA(xcurr i,ycurr i,pos-vector)

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6. Computersimulation

Themulti-robotpath-planningalgorithmisimplementedinasimulatedenvironment.Thesimulationisconducted inaCenvironmentonaPentiumprocessorandtheexperimentwasperformedwith14robotsofcircularshape.The radiusofeachrobotisconsideredas6pixels.Beforeinitiatingtheexperimentformulti-robotpathplanning,eachrobot startingandgoalpointsarepredefined.Theexperimentswereperformedwithsevendifferentlyshapedobstaclesand withequalvelocitiesforalltherobotsinagivenrunoftheprogram;however,thevelocitieswereadjustedoverdifferent runsofthesameprogram.Oneofourexperimentalworld-mapswithaninitialconfigurationoftheworld-mapwith7 obstaclesandthestartingandthegoalpositionsof12circularsoft-botsareshowninFig.4.Theintermediatestepsof movementoftherobotsareshowninFigs.5and6.Thefinalstageofworldmaps,wherealltherobotsreachedintheir predefinegoalrespectivelyisshowninFig.7.Thesimulationisalsoconductedintheenvironmentaspresentedin

Fig.4forsamenumberofrobotsbyGSAandDE.TheoptimaltrajectoryofthepathhasbeenpresentedinFigs.8and9

forGSAandDErespectively.

Theexperimenthasbeen conductedusingacentralversionof thealgorithmusing thefitnessFunction(29)for decidingthenextpositionoftherobot.Inourexperiment,parametershavebeendescribedinTable1forsimulation andKheperaIIenvironment.

6.1. Averagetotaltrajectorypathdistance(ATTPD)

ConsideratrajectorypathfromthepredefinestartingpointSktothegoalpointGkgeneratedbytheprogramfor

robotRkinthejthrunisTPkj.IfTPk1,TPk2,....,TPkjarethetrajectorypathsgeneratedoverjthruns,thentheaverage

totaltrajectorypathtraversed(ATTPT)byarobotRkisgivenby

j

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Fig.4.Initialworldmapwith7obstaclesand5robots.

Fig.5.IntermediatestateoftheworldmapduringexecutionusingIGSAfor5robotsand7obstaclesafter9steps.

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Fig.7.Fiverobotsreachedintheirrespectivepre-definedgoal.

Fig.8.Allrobotsreachedintheirrespectivepre-definedgoalin29stepsbyGSA.

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Table1

ParameterusedinthesimulationandKhepera.

Parameters Values G0 100 αmin 0.2 αmax 0.4 λ 100 C1 0.5 C2 0.5 T(Maxiter) 100 W 0.72 β 0.5 N 50

forthisrobotisevaluatedbymeasuringthedifferencebetweenATPTandtheidealshortestpathbetweenSktoGk.If

theidealtrajectorypathforrobotRkobtainedgeometricallyisTPk−real,thentheaveragepathdistanceisgivenby

TPk−real− j r=1 Pir j .

Therefore,fornrobotsintheworkspacetheaveragetotalpathdistance(ATPD)is

n i=1 TPk−real− j r=1 Pir j

6.2. Averageuncoveredtrajectorytargetdistance(AUTTD)

GivenagoalpositionGkandthecurrentpositionCkofarobotona2-dimensionalworkspace,whereGkofCkare

2-dimensionalvectors,theuncoveredtrajectorydistancefortherobotkisGk−Ck,where.denotesEuclidean

norm.Fornrobots,uncoveredtrajectorytargetdistance(UTTD)isUTTD=

n

i=1

Gk−Ck.Forkrunsoftheprogram,

weevaluatetheaverageofUTTDs,andcallittheaverageuncoveredtrajectorytargetdistance(AUTTD).Fig.16shows thatbydecreasingthevelocity,AUTTDtakeslongertimetoconvergeandgraduallyterminatedwithiteration.Again, itisnotedthatlargerthevelocityoftherobot,thefasterfalloffintheAUTTD.Fig.17showsthat,largerthenumber ofrobots,slowertheconvergencerate.SlowertheconvergencecausesthedelayinfalloffinAUTTD.

TheperformanceanalysiswasundertakeninthesimulationenvironmentandtheATPTwasplotedfornrobots, calledaveragetotaltrajectorypathtraveled(ATTPT)byvaryingno.ofrobots1–5presentedinFig.18andgenerate pathsusingthreealgorithms,includingreal-codedDE,GSAandIGSA.ItisnoteworthyfromFig.18thatIGSApossess leastATTPTincomparisontothealgorithmsirrespectiveofthenumberofrobots.

TheperformanceanalysishasbeenperformedintermsofAUTTDoverthenumberofstepsinFig.19.Itprovides graphsbetweenAUTTDvs.no.ofstagesrequiredduringtheplanningof pathusingthreealgorithmswithnumber ofobstacles=7 andno.ofrobots=5.ItisapparentfromFig.19that AUTTDreturns thesmallestvalueforIGSA irrespectiveofthenumberofplanningsteps.

Theperformanceoftheresulthasbeenanalyzedbyplottingtheaveragetotaltrajectorypathdistance(ATTPD) withthenumberof robotsasvariableinFig.20.Thispathisgeneratedbythreedifferentevolutionaryalgorithms suchasGSA,DE,IGSA.Fig.20showstheresultofATTPDcomputation,whenthenumberofrobotsvariesbetween 1–5.Here,weobservedthatIGSAperformsbetterthantheothertwoalgorithmsasATTPDissmallestforIGSAin comparisontoothertwoalgorithmsirrespectiveofthenumberofrobots.

Now, the performanceanalysis was undertakenby comparing the running timeover the maximumnumber of

iterationsusingthreealgorithms.Fig.21providesthetimerequiredforrobotstoreachintheirrespectivegoalposition bythreedifferentalgorithmsanditshowsthatIGSAtakeslesstimeforrobotstoreachindestination.

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Table2

DescriptionofobstaclespresentsinFig.4.

Robotnumber No.ofsteprequiredtogoalinIGSA No.ofsteprequiredtogoalinGSA No.ofsteprequiredtogoalinDE

1 17 19 23 2 21 25 29 3 15 23 27 4 26 29 30 5 12 14 17 Table3

Comparisonofnumberofstepstaken,ATTPTandATTPDofdifferentalgorithmsfordifferentno.ofrobots.

No.ofrobots Algorithms(stepstaken) ATTPT(ininch.) ATTPD(ininch.)

IGSA DE GSA IGSA DE GSA IGSA DE GSA

2 12 16 18 35.7 36.5 38.4 3.7 4.7 5.7

3 15 18 20 37.8 38.6 40.4 4.9 5.6 6.6

4 19 21 24 39.7 40.5 42.6 6.8 7.3 7.9

5 21 24 26 41.3 44.6 45.7 7.6 8.4 9.3

Finally,the performanceof thesimulationresultisanalyzedintheterms ofthenumberturn,bywhichwe can abletominimizetheenergyconsumption.Thenumberofturnrequiredforthreedifferentalgorithmsfornumberof robots=6isdemonstrateinFig.22.ItshowsthatIGSAtakeslessnumberofturnthanothertwoalgorithmsandenergy consumptiontoreachinthedesignationislessthantheothertwoalgorithms.Thesimulationisonlypresentedforfive numbersofrobotsbutnumberofturnislessforirrespectiveoftherobotintheplanningschemeofthealgorithm.

TheexperimentisconductedintheenvironmentshowninFig.4bythethreealgorithmsforsamefitnessfunction inEq.(29)withsameparameterfor30iteration,thebestfitnessvalueforthreealgorithmsispresentedinFig.23.The fitnessvalueoftherobotspresentedinFig.23indicatesthatthereisnoconflictinthenextpositioncalculationbythe robots,itshowsthatthebestfitnessvalueobtainforIGSAafter26iterationis3.638,butthatachievedbyGSAand DEafter29and30is4.105and4.711respectively.ThispresentsthatIGSAisbetterthanGSAandDEintheterms ofavoidingproblematlocaloptimaandfasterconvergencerate.

Numberofoptimalstepsreqiuredfor differentrobots,numberfrom1 to5 ofthesimulationresultfordifferent algorithmispresentedinTable3.Table3showsthatthenumberofoptimalstepsrequiredforIGSAislessthanthe otheralgorithmsuchasGSAandDE.ThetotalnumberofoptimalstepsrequiredforIGSA,GSAandDEis26,29 and30respectively.

The result of the experimentsperformed is summarizedin Table2 inthe terms of three performancemetrics, namely,(1)totalno.ofstepsrequiredtoreachinthegoal,(2)ATTPTand(3)ATTPDhavebeenusedheretodetermine therelativemerits ofIGSAovertheotheralgorithmsfordifferentrobots.Table1confirmsthat theremainingtwo algorithmsperformwellwithrespecttoallthreemetricsfordifferentrobots.

7. ExperimentonKheperaIIrobot

KheperaII(Fig.10)isaminiaturerobot(diameterof8cm)equippedwith8built-ininfraredrangeandlightsensors, and2relativelyaccurateencodersforthetwomotors.Therangesensorsarepositionedatfixedanglesandhavelimited rangedetectioncapabilities.Thesensorsarenumberedclockwisefromtheleftmostsensor0tosensor7anditsinternal structure(Fig.12).Sensorvaluesarenumericalrangingfrom0(fordistance>5cm)to1023(approximately2cm).

Theonboardmicroprocessorhasaflashmemorysizeof256KB,andtheCPUof8MHz.Kheperacanbeusedon

adesk,connectedtoaworkstationthrough awiredserial link.Thisconfigurationallowsanoptionalexperimental configurationwitheverythingathand:therobot,theenvironmentandthehostcomputer.TheKheperaIInetworkand itsaccessoriesarepresentedinFig.11fortheconductofexperiments.

Theinitialworldmapforconductingthe experimentintheKheperaIIispresentedinFig.13to8obstaclesof differentshapeandpredefineinitialstateandgoalismarkedonthemap, wheredifferentmetaheuristicalgorithm

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Fig.10.TheKheperaIIrobot.

Fig.11.Kheperanetworkanditsaccessories.

Fig.12.PositionofsensorsandinternalstructureofKheperaII.

isapplied.Fig.14showstheintermediatemomentof therobotinthetrajectorypathtowardthegoalbyrespective robotusingIGSA.IGSAisimplementedintheKhepera-IIrobotwithconsideringtworobotsandcomparedwitha differentevolutionarycomputingalgorithmisdemonstratedinFig.15.Itshowsbetterconvergenceincomparingto theothermeta-heuristicalgorithmpresentedinFig.15.Finally,differentmeta-heuristicalgorithmshavebeenapplied inKheperaenvironmentandresultsofthetrajectorypathhavebeenpresentedinFig.15.

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Fig.13.Kheperaenvironmentsetupformulti-robotpathplanning.

Fig.14.Snapshotofintermediatestageofthemulti-robotpathplanningusingIGSAinKheperaenvironment.

Fig.15.Optimalpathrepresentationofdifferentalgorithmformulti-robotpathplanninginKheperaenvironmentisrepresentedbydifferentcolor code.

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Fig.16.Averageuncoveredtrajectorydistancevs.numberofstageswithvariablevelocityforfixednumberofobstacles=7.

Fig.17.Averageuncoveredtrajectorydistancevs.numberofstageswithvariablenumberofrobotsforfixednumberofobstacles=7(constant).

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Fig.19.Averageuncoveredtrajectorytargetdistancevs.numberofstepsindifferentalgorithms.

Fig.20.Averagetotaltrajectorypathdeviationvs.no.ofrobotsalgorithmwithfixedno.ofobstacles=7.

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Fig.22.Numberofturnvs.numberofrobotsinthreedifferentalgorithms.

Fig.23.FitnessvalueofIGSA,GSAandDEforfitnessfunctioninEq.(29).

8. Conclusionandfutureworks

Animprovedgravitationalsearchalgorithmwasproposedfortrajectorypathplanningofmulti-robotsinordertofind collisionfreesmoothnessoptimalpathfrompredefinestartpositiontoendpositionforeachrobotintheenvironment. Theobtainedresultsfromtheexperimentalworkperformbettercomparedwiththeproposedalgorithm.Comparing theperformancesamongdifferenttechniqueshavebeencarriedout.FromthesimulationandKhepera-IIenvironment, itisobservedthattheIGSAtechniqueisbestoverothertechniquefornavigationofmulti-mobilerobot.However,in thispaper,boththeenvironmentandobstaclesarestaticrelativetotherobots;whereasotherrobotsaredynamicfor priorityrobots.Infuture,workwillbecarriedoutusingdynamicobstaclesotherthanrobotssuchasrunningvehicle, animalsandonboardcameraduringmulti-robotpathplanning.

References

Alba,E.,Dorronsoro,B.,2005.Theexploration/exploitationtradeoffindynamiccellulargeneticalgorithms.IEEETrans.Evol.Comput.9(2), 126–142(IEEE).

(19)

Banerjee,G.,Chowdhury,S.,Losert,W.,Gupta,S.K.,2011.Realtimepathplanningforcoordinatedtransportofmultipleparticlesusingoptical tweezers.IEEETrans.Autom.Sci.Eng.9(October),669–678(IEEE).

Bhattacharjee,P.,Rakshit,P.,Goswami,I.,Konar,A.,Nagar,A.K.,2011.Multi-robotpath-planningusingartificialbeecolonyoptimizationalgorithm. ProceedingsofIEEECongressonNatureandBiologicallyInspiredComputing,IEEE,219–224.

Chakraborty,J.,Konar,A.,Jain,L.C.,Chakraborty,U.K.,2009.Cooperativemulti-robotpathplanningusingdifferentialevolution.J.Intell.Fuzzy Syst.20(1–2),13–27(IOSPress,Amsterdam).

Chatterjee,Mahanti,G.K.,Mahapatra,P.R.S.,2011.Generationofphase-onlypencil-beampairfromconcentricringarrayantennausinggravitational searchalgorithm.ProceedingsofInternationalConferenceonCommunicationsandSignalProcessing,IEEE,384–388.

Das,P.K.,Konar,A.,Laishram,R.,2010.Pathplanningofmobilerobotinunknownenvironment.Int.J.Comput.Commun.Technol.1(2–4),26–31 (Inderscience).

Das,S.,Mukhopadhyay,A.,Roy,A.,Abraham,A.,Panigrahi,B.K.,2011.Exploratorypoweroftheharmonysearchalgorithmanalysisand improvementsforglobalnumericaloptimization.IEEETrans.Syst.ManCybern.B41(1),89–106(IEEE).

Das,P.K.,Pradhan,S.K.,Patro,S.N.,Balabantaray,B.K.,2012.Artificialimmunesystembasedpathplanningofmobilerobot.SoftComput.Tech. Vis.Sci.395,195–207(Springer).

Eldos,T.,Qasim,R.A.,2013.Ontheperformanceofthegravitationalsearchalgorithm.Int.J.Adv.Comput.Sci.Appl.4(8),74–78(TheScience andInformationOrganization).

Geem,Z.W.,Kim,J.H.,Loganathan,G.V.,2001.Anewheuristicoptimizationalgorithm:harmonysearch.Simulation76(2),60–68(Sage).

Gong,C.,Lincheng,S.,2001.Geneticpathplanningalgorithmundercomplexenvironment.Robot23(1),40–43(Hindawi).

Guo,Y.,Parker,L.E.,2002.Adistributedandoptimalmotionplanningapproachformultiplemobilerobots.ProceedingofIEEEInternational ConferenceonRoboticsandAutomation(ICRA’02),IEEEvol.3,2612–2619.

Kcymeulcn,D.,Decuyper,J.,1994.Thefluiddynamicsappliedtomobilerobotmotion:thestreamfieldmethod.ProceedingsofIEEEInternational ConferenceonRoboticsandAutomation,IEEE,378–385.

Konar,1999.ArtificialIntelligenceandSoftComputing:BehavioralandCognitiveModelingoftheHumanBrain,1stedition.CRCPress.

Li,C.,Bodkin,B.,Lancaster,J.,2009.ProgrammingKheperaIIrobotforautonomousnavigationandexplorationusingthehybridarchitecture. Proceedingsof47thAnnualSouthEastRegionalConference,ACM,p.31.

Liu,S.Hsi,Mernik,M.,Hrnclc,D.,Crepinsek,M.,2013.Aparametercontrolmethodofevolutionaryalgorithmsusingexplorationandexploitation measureswithapracticalapplicationforfittingSovova’smasstransfermodel.Appl.SoftComput.13(9),3792–3805(Elsevier).

Luh,G.C.,Cheng,W.-C.,2002.Behavior-basedintelligentmobilerobotusingimmunizedreinforcementadaptivelearningmechanism.Adv.Eng. Inform.16(2),85–98(Elsevier).

Masehian,E.,Sedighizadeh,D.,2010.Multi-objectivePSO-andNPSObasedalgorithmsforrobotpathplanning.Adv.Electr.Comput.Eng.10 (4),69–76(StefancelMareUniversityofSuceava,FacultyofElectricalEngineeringandComputerScience).

Precup,R.E.,David,R.C.,Petriu,E.M.,Preitl,S.,Radac,M.B.,2012.Noveladaptivegravitationalsearchalgorithmforfuzzycontrolledservo systems.IEEETrans.Ind.Inform.8,791–800(IEEE).

Purcaru,C.,Precup,R.-E.,Iercan,D.,Fedorovici,L.-O.,David,R.-C.,Dragan,F.,2013.Optimalrobotpathplanningusinggravitationalsearch algorithm.Int.J.Artif.Intell.10,1–20(CESER).

Rashedi,E.,Nezamabadi-pour,H.,Saryazdi,S.,2009a.GSA:agravitationalsearchalgorithm.Inf.Sci.179(13),2232–2248(Elsevier).

Rashedi,E.,Nezamabadi-pour,H.,Saryazdi,S.,2009b.GSA:agravitationalsearchalgorithm.Inf.Sci.179,2232–2248(Elsevier).

Rashedi,E.,Nezamabadi-pour,H.,Saryazdi,S.,2010.BGSA:binarygravitationalsearchalgorithm.Nat.Comput.9,727–745(Springer).

Regele,R.,Levi,P.,2006.Cooperativemulti-robotpathplanningbyheuristicpriorityadjustment.ProceedingofIEEE/RSJInternationalConference onRoboticsSystem,IEEE,5954–5959.

Sabri,N.M.,Puteh,M.,Mahmood,M.R.,2013.Areviewofgravitationalsearchalgorithm.Int.J.Adv.SoftComput.5(3),1–39(SCRG).

Sarafrazi,S.,Nezamabadi-pour,H.,Saryazdi,S.,2011.Disruption:anewoperatoringravitationalsearchalgorithm.Sci.Iran.18(3),539–548 (Elsevier).

Tuncer,A.,Yildirim,M.,2012.Dynamicpathplanningofmobilerobotswithimprovedgeneticalgorithm.Comput.Electr.Eng.38(6),1564–1572 (Elsevier).

Verma,O.P.,Sharma,R.,Kumar,M.,Agrawal,N.,2013.Anoptimaledgedetectionusinggravitationalsearchalgorithm.Lect.NotesSoftw.Eng. 1(2),148–152(Springer).

Yang,X.-S.,2009.Harmonysearchasametaheuristicalgorithm.In:inGeem,Z.W.(Ed.),Music-inspiredHarmonySearchAlgorithm:Theoryand Applications,StudiesinComputationalIntelligence,vol.191.Springer,Berlin,pp.1–14.

Yu,J.,Kromov,V.,2001.Arapidpathplanningalgorithmofneuralnetwork.Robot23(3),201–205(Hindawi).

Zhang,Y.,Gong,D.-W.,Zhang,J.-H.,2013.Robotpathplanninginuncertainenvironmentusingmulti-objectiveparticleswarmoptimization. Neurocomputing103,172–185(Elsevier).