approach Self-organization robot in aggregating swarms: A DW-KNNtopological BioSystems

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ContentslistsavailableatScienceDirect

BioSystems

j ourna l h o m e pa g e :w w w . e l s e v i e r . c o m / l o c a t e / b i o s y s t e m s

Self-organization

in

aggregating

robot

swarms:

A

DW-KNN

topological

approach

Belkacem

Khaldi

a

_,

_Fouzi

_Harrou

b,∗

_,

_Foudil

_Cherif

a

_,

_Ying

_Sun

b

a_LESIA_Laboratory,_Department_of_Computer_Science,_University_of_Mohamed_Khider,_R.P.₀₇₀₀₀_Biskra,_Algeria

b_King_Abdullah_University_of_Science_and_Technology_(KAUST)_Computer,_Electrical_and_Mathematical_Sciences_and_Engineering_(CEMSE)_Division,_Thuwal

23955-6900,SaudiArabia

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received21July2017

Receivedinrevisedform1December2017

Accepted22January2018

Availableonline2February2018

Keywords:

Swarmrobotics

Self-organizedaggregation

SmoothedParticlesHydrodynamics(SPH)

Virtualviscoelasticmodel

Distance-WeightedKNN

a

b

s

t

r

a

c

t

Incertainswarmapplications,wheretheinter-agentdistanceisnottheonlyfactorinthecollective behavioursoftheswarm,additionalpropertiessuchasdensitycouldhaveacrucialeffect.Inthispaper, weproposeapplyingaDistance-WeightedK-NearestNeighbouring(DW-KNN)topologytothebehaviour ofrobotswarmsperformingself-organizedaggregation,incombinationwithavirtualphysicsapproach tokeeptherobotstogether.Adistance-weightedfunctionbasedonaSmoothedParticleHydrodynamic (SPH)interpolationapproach,whichisusedtoevaluatetherobotdensityintheswarm,isappliedasthe keyfactorforidentifyingtheK-nearestneighbourstakenintoaccountwhenaggregatingtherobots.The intravirtualphysicalconnectivityamongtheseneighboursisachievedusingavirtualviscoelastic-based proximitymodel.WiththeARGoSbased-simulator,wemodelandevaluatetheproposedapproach, show-ingvariousself-organizedaggregationsperformedbyaswarmofNfoot-botrobots.Also,wecompared theaggregationqualityofDW-KNNaggregationapproachtothatoftheconventionalKNNapproachand foundbetterperformance.

1. Introduction

Aggregation is a fundamental behaviourthat is observed in manybiologicalorganisms,suchassocialinsectsandgroup-living animals(Camazineetal.,2002).Itisanimportantrequirementfor animalsocietiestoaccomplishcomplexswarmingbehaviours col-lectively.Thiscanbeachievedusingexternalgradients,suchas humidityforwoodlice(Brolyetal.,2012,2013)ortemperature forhoneybees(Kernbachetal.,2009);thiskindofaggregationis mostlyknownascue-basedaggregation(Arvin etal.,2014b).It canbealsoachievedinself-organizedmanner,wherethe aggrega-tionprocessisenabledwithoutanyexternalcues(Camazineetal., 2002).Fascinatingexamplesof suchbehaviourcanbeobserved in bird flocks, fish schools, and mammal herds (Hemelrijk and Hildenbrandt,2012).

Onthebasisofthesebiologicalstudies,aggregationisalsoa mat-terofinterestinvariousswarmroboticsstudies.Itisconsidered asafundamentaltaskthatallowsrobotswarmstoperform com-plextasks,suchascollectivemovement,self-assembly,andpattern formation,ortoexchangeinformation.

∗ Correspondingauthor.

E-mailaddress:[email protected](F.Harrou).

This study takes inspiration from the topological distance approachrevealedinstudiesofbirdsflockingandfishschooling. Ballerinietal.(2008a,b)showedempiricallythattheunpredictable, amazinglycomplexpatternsformedbybirdsemergefroma topo-logicaldistanceapproachratherthanametricdistanceapproach, meaningthatthebirdsonlyinteractwiththeirnearestsixorseven neighboursratherthanalloftheneighboursintheirfieldofvision. Computersimulationspredictthatasignificantlyhigher cohe-sionoftheaggregationisachievedusingatopologicalinteraction ratherthanthestandard metricone.Similarly,empiricalresults showthatfishandelephants,forexample,interactwithonlythree orfourneighbours(Niizatoetal.,2014).

Inswarmrobotics,fewmodelshaveappliedsuch topological approachestothestudyofcollectivebehaviours.Someresearchers proposedselectingstrategiesbeforeinteractingwithnearbyrobots so that only a fixed number of neighbours are used (Lee and Chong, 2008; Ercan et al., 2010). For example, Lee and Chong (2008)proposedabio-inspiredflockingcontrolbasedonselecting twoneighborsforteammaintenanceandlocalinteractions.Ercan etal.(2010)introducedaregulartetrahedronformationstrategy for selectingthe three neighbours that forms thebest tetrahe-drontoensureformation.Inourpreviouswork(KhaldiandCherif, 2016a),westudiedself-organizedpatternsthatemergefroman approachbasedontheintra-virtualphysicalconnectivityamong https://doi.org/10.1016/j.biosystems.2018.01.005

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thedensityoftherobotscouldplaytheprimaryrolewhilethe inter-robotdistancewouldplayonlyasecondaryroleasametricuseful forproximity controlorcollisionavoidance.WiththeDW-KNN topology,virtualphysicalconnectionsaredynamicallycreatedand destroyedamongtheweightedKNNratherthantheunweighted KNN.To achievethis topology,we introducea Smoothed Parti-cle Hydrodynamic(SPH)density-based approach(Section4.2.1) toevaluatethedensityoftherobotsintheswarmandtoweight thedistancesamongneighbours.SPHisamesh-freeLagrangian methodthatiswidelyusedformanydiverseapplications,including astrophysics,geophysics,engineering,andthefilmandcomputer gamesindustries.Themethodisbasedonaninterpolation tech-niquethatapproximatesphysicalquantitiesthataremovingwith particles(VioleauandRogers,2016);hereweapproximatethe den-sityasa physicalquantitymovingwiththerobots.AsinKhaldi andCherif(2016a,b),weuseavirtualviscoelastic-basedproximity model(Section4.1)astheintra-virtualphysicsconnectivityamong theneighbouringrobots;thevirtualviscoelasticconnectionsare Voigt links, andeach linkcan berepresentedby apurely elas-ticspringandapurelyviscousdamperconnectedinparallel.We alsoincorporatearepulsiveprimitivetoavoidobstaclesand colli-sionswiththerobotsthatarenotvirtuallylinkedtogether(Section 4.4).Theoverallcontrolmodel(Section5)withaswarmof foot-botrobotsisdesignedandevaluatedusingtheARGoSsimulator (Pincirolietal.,2012).Theresultsobtainedin(Section6)by sim-ulatingupto150foot-botrobotsinbothabsenceandpresenceof obstaclesshowthevariationthatemergedintheself-organized aggregationfromtheexecutionoftheoverallcontrolmodel.The performanceofthemodelisstudiedusingthreedifferentmetrics (Section7.1):thedistance-weighteddistributionquality(Fw),the

aggregationquality(Fag),andtheAveragedMeanDistanceError

(AMDE)metric. Withthesemetrics,weanalyzetheaggregation performanceinnormalcircumstances(Section7.2),andwelater assesshowtheaggregationperformanceisaffectedwhenthe read-ingsoftherobots’rangeandbearingsensorsarecorruptedbynoise (Section7.3).Furthermore,inSection8,weprovidecomparisonsof theproposedapproachwiththeconventionalKNNapproachand showedthatweachievebetterresults.

2. Relatedworks

Aggregationisacurrentfocusofswarmrobotics,andseveral studiesonbothcue-basedaggregationandself-organized aggre-gationhavebeencompletedinthelasttwodecades.

2.1. Worksoncue-basedaggregation

In one study of cue-based aggregation by Kube and Zhang (1993),alightsourcewasusedtoaggregatearobotsystemaround anobjectandtotransporttheobjectinacollaborativemannerto anothergoal;byfollowingsimplebehaviours(e.g.,tofindlightand followlight),theaggregationtaskwascompletedwithnoexplicit

bees,whichprefergatheringwherethetemperatureis36◦_C._The

BEECLUSTmodelproposedbyKernbachetal.(2009)wasthefirst algorithm that mimickedthis behaviour;a gradual light source wasusedtogenerateclusteringbehaviorinaswarmrobotics sys-tem.Itwasproventoactrobustlyinmanyresearches(Bodietal., 2012,2015;Arvinetal.,2014a,2016).Further,differentvariations of themodelhave beensuggestedtoincrease theperformance oftheaggregationprocess.Forinstance,Arvinetal.(2011) pro-posedanewaggregationalgorithminwhichadynamicvelocity andacomparativewaiting timewereintroducedtotheoriginal BEECLUSTmodel,whichcontributedtoasignificantimprovement intheaggregationtime.Furthermore,acomparisonbetweenthe originalBEECLUSTalgorithmandtwomodifiedversions–calledthe vectoraveragingalgorithmandthenaivealgorithm–showedthat boththevectoraveragingandnaïvealgorithmsoutperformedthe originalBEECLUSTmodel,andrevealedthatnoisehaslessimpact inthevectoraveragingmethodthanthenaïveone(Arvinetal., 2014a).

Later, those authors introduced a fuzzy-based aggregation approach to enhance the performance of BEECLUST in both computer-basedsimulationsandrealrobotswarms(Arvinetal., 2014b).Wahbyetal.(2016),proposedanotheradaptive variant ofBEECLUST,wheretheoriginalalgorithmwasextendedtoadapt automaticallytoanylightconditions.Recently,Bodietal.(2012) proposedamodelcalledODOCLUST,whichincorporated odome-tryasanadditionalcapabilitytoBEECLUST.TheODOCLUSTvariant achievedafastandaccurateaggregationwithoutrequiring high-fidelityodometry.

2.2. Self-organizedaggregationworks

In thispaper,we arespecificallyinterested instudying self-organizedaggregation,inwhichrobotswarmsachieveaggregation usingsimplelocalinteractionrulesamongindividuals.Different approacheshave beenproposedtoaddresstheproblemof self-organized aggregationin swarm robotics. Garnier et al. (2008) adoptedaprobabilisticapproach,inspiredbythecockroachmodel ofJeansonetal.(2005),toachieveaggregationusingaswarmof 20physicalAlicerobotsinhomogeneousenvironments.A simi-larworkbyCorrellandMartinoli(2011)showedthatwhenusing probabilisticaggregationrules,aminimumcombinationof com-municationrangeandlocomotionspeedwasneededtoachievea singleaggregatecluster.SoysalandSahin(2005)suggesteda prob-abilisticaggregationmethodinwhichastate-finitemachinewas usedtocombineasetofsimplebehavioursthatincludedavoiding anobstacle,approaching,repelling,andwaiting.

Inotherstudies,self-organizedaggregationwasstudiedusing deterministicapproaches.Inthesemethods,therobotsbuilda con-nectedvisibilitygraph,andensurethatthegraphispermanently maintained.Andoetal.(1999)usedanalgorithminthissenseto studyaggregationinagroupofmobilerobotswithalimitedsensing range.Later,thealgorithmwasgeneralizedbyAndoetal.(1999)to achieveanaggregationinarbitrarilyhighdimensions.The

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forma-Fig.1. Thefoot-botrobot,CADmodel(Pinietal.,2009).

tionofthegraphinthesealgorithmswasbasedontheassumption thattherobotswereabletomeasureboththerangeandtheangle oftheirneighbours.However,Gordonetal.(2004)wereableto achievesuchanaggregationusingonlytheanglemeasurementof therobot’sneighbours.Theaggregationperformanceofthislast algorithmwaslaterimproved,byintroducinganadditional,crude range-sensingcapabilityfordifferentiatingwhetherneighbouring robotswerenearorfarinGordonetal.(2008).Inanotherstudy,De GennaroandJadbabaie(2006)usedtheLaplacianmatrixtoallow eachrobottobuilditsownproximitygraph.TheLaplacianmatrixis analgebraicrepresentationofagraph,throughwhichmanyuseful graphpropertiescanbefound(Merris,1994).Therelatedcontrol wasfullydecentralized,andsimulatedresultsdemonstratedthat themodelwaseffectiveandevenincreasedtheconnectivityofthe entireswarm.

Insomeworks,self-organizedaggregationmodelshavebeen approached using artificial evolution techniques. For instance, aggregation with simple robots, called s-bots, was studied by Triannietal.(2003).Inthisstudy,generalsolutionstothe aggre-gation problem were produced using an evolutionary robotics mechanism.Themethodwasabletoproduceclusteringbehaviours withboth staticand dynamic behavioural strategies.With that model,Dorigoetal.(2004)revealedthat effectiveevolved con-trollerscouldbeachievedforboth aggregatingandcoordinated motionbehavioursinaswarmofs-bots.Inasimilarsetup,Soysal etal.(2007)investigatedtheeffectsofanumberofparameters, suchastherobots’number,thesizeofarena,andtheruntime. Inanotherstudy,Gaucietal.(2014a)proposedtwoalgorithms– areactivecontrollerwithnomemoryandarecurrentcontroller withmemory–tostudyaggregationinaswarmofe-puckrobots. Thealgorithmswerebasedonaclassicalevolutionary program-mingtechnique,andusedasimplebinarysensorwithasensing rangethatprovedsufficienttoachieveanerror-freeaggregation. Resultsfromboththesimulation andexperimentsshowedthat aggregationtowardoneclusterwassuccessfullyachieved. How-ever,asufficientlylongrangeinthebinarysensorwasneededto achieveanaccurateaggregation.

Otheraggregationstudiesapproachedavisionbasedonvirtual physics methods. Gasparri et al. (2012a) adopted an attrac-tive/repulsivevirtualforcemodeltostudyaggregationinaswarm ofmulti-robotsystemsbasedonlocalinteraction.Thismodelwas laterextendedtocopewithactuatorsaturationbyGasparrietal. (2012b)andtointegrateobstacleavoidancebyLecceseetal.(2013).

Inmostoftheworkscitedabove,theinter-robotdistanceistheonly factortakenintoaccountwheninterconnectingtherobots. How-ever,incertainapplicationswheresupplementaryfactorssuchas thedensityoftherobotscouldemergeasimportant,arobot’s den-sityplaysamorecrucialrolethantheinter-robotdistance.Inthis context,therelatedmethodologiesmainlyusetheSPHmodelto simulatefluid-likebehaviourinarobotswarm.Thistechniquehas successfullybeenusedtodemonstratehowmulti-roboticagents, modelledasastreamofincompressibleorcompressiblefluidare capableofbeingappliedtotaskssuchasgroupmotionandshape control,groupsegregation(Zhaoetal.,2011),patterngeneration (Pimentaetal.,2008,2013),andunknownareacoverage.

RatherthanusingtheSPHmodelasavirtualfluidforcelawto designtheproximitycontrolofarobotswarm,hereweusethe SPHmodel toevaluatethedensity oftherobotsin theswarm, thenwedefineaDW-KNNneighbouringtopologytoaggregatethe robotsbasedonthisdensity.UsingtheDW-KNN-based neighbour-ingtopology,self-organizedaggregationswereachievedbytaking intoaccounttwokeyfactors,theinter-robotdistanceandtherobot’ density.

3. Method

3.1. Problemformulation

We consideranareasurroundedbyfourwalls,containing a swarmofN(R1,...,RN)differentialtwo-wheeledmobilerobots

thatareinitiallydistributedinrandompositionsandheading arbi-trarydirections.Weassumethat,withinaspecificrangeDr,each

robotRiissufficientlyequippedwithsensorstomeasuretherange

d_ijandthebearing!ijofitsjthneighbour(j ∈Ni(t),whereNi(t)is

thesetofneighboursoftherobotRiatatimet).Inaddition,we

con-siderthattherobotcancommunicatewithitsneighbourswithin thesamerange.

Theobjectiveistostudytheself-organizedaggregationsthat emergethrougha topologicalneighbourhoodapproachwithout usinganycues.

3.2. Simulationplatform

Thesimulationsinthis studywereperformedontheARGoS platform(Pincirolietal.,2012).ARGoSisanopensource multi-robot simulatorthat wasfirst developed withinthe EU-funded Swarmanoidproject1_to_study_tools_and_control_strategies_for

het-erogeneous swarmsof robots. Now,it is one of themost-used simulation platforms in many researches and projects that are dedicatedtosynthesisswarmingbehaviourcontrols.ARGoScan simulatelarge-scale,heterogenousswarmsofrobotsinrealtime. Moreover,itcomeswithallthetoolsneededforthedevelopment cycleofrobotcontrolcode,fromdesigntovalidationonrealrobots; therefore,thereisnodifferencebetweencodingforsimulationor reality(Pincirolietal.,2012).ARGoShasabuilt-inmodelsfor sev-eralwell-knownrobotsuchasthefoot-bot,e-puck,kilobot,and fly-bots.

Therobotsusedinourexperimentalsimulationsarefoot-bots (seeFig.1),two-wheeleddifferentialmobilerobotsdevelopedby Bonani etal. (2010) that come withthe actuators and sensors requiredby ourcontrolmodel.In ourexperiments,we usethe following.

• A range and bearing perception and communication device (calledRAB),thatallowstherobotcommunicatingwithits

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Fig.2. Avirtualviscoelasticinteractionmodelbetweentworobotsmodeledasa

Voigtconnection.

bours, as wellas identifying their relative range and bearing measures.

• Proximitysensorsthatallowdetectingobjectsaroundtherobot. Thefoot-botrobotisequippedby24sensors,whichareuniformly distributedinaringaroundtherobotbody.

• Two wheels actuators to independently control the forward speedoftheleftandtherightwheelsoftherobot.

Thebodyofthefoot-botisabout8.5cmwideand14.7cmtall. Theinter-wheeldistanceis0.14cm.Thevelocitiesoftheleftand rightwheelsalongthegroundcanbesetindependentlytoa maxi-mumspeedof[13,13]cm/s.

4. Robotmodel

Inthisstudy,therobotmodelwasbuiltfollowinganartificial physicsdesignapproach(Spearsetal.,2004).Inthismethod,the robotssenseandrespondtovirtualforcesthataredrivenbyregular physicslaws.Here,therobotwasbasedonourpreviouslyproposed model(Khaldietal.,2017;KhaldiandCherif,2016a,b),wherethe robotissubjectedtothevirtualforcevector

ˆai= ˆpi+ ˆri. (1)

Theproximitycontrolvector, ˆpi,isusedtoencodethe

attract-ingrules.Therepulsivecontrolvector, ˆri,isusedtoencode the

repulsingrules. 4.1. Proximitymodel

Theproximitycontrolisthecrucialmodelinourstudy;itwas usedtoaggregatetherobotstogetherwhilemaintainingacertain distancebetweenthem.Toachievethismodel,atopological neigh-bourhoodstrategyisappliedfirstbyeachrobotR_itodecidewhich neighbours amongthoseavailableare takenintoaccountwhen arrangingtherobots.Then,ameshofvirtualviscoelasticlinksis builtbetweentherobotsofthetopologicalneighbourhoodby mod-ellingthesensingcapabilitiesusingVoigtinteractions(seeFig.2) (KhaldiandCherif,2016a,b).TheproximitymodelofeachrobotRi

iscomputedas ˆpi=

!

j∈ Ti(t) (ks ij(dij−d0)ˆdij+kdij(

v

i−

v

j)), with ks ij= ks

"

_d ij , kd ij=kd

#

ks ij, and i=/j, (2)

whereTi(t) ∈Ni(t)isthesetoftopologicalneighboursattimet

(moredetailsabouthowtherobotidentifiesthissetisdiscussed in the next subsection),and ks

ij and kdij are the spring and the

dampingcoefficients,respectively.Thevaluesofthesecoefficients

Fig.3. AschematicrepresentationofSPHparticleinteractionswithintheinfluence

domaingovernedbythekernelfunction.

aredynamicallychangingwithregardstotheactualdistancedij

betweenarobotRiandarobotRj,andthetwogainconstants,ks

andk_d.Theequilibriumlengthofthespringisd0,and ˆdijistheunit

vectorindicatingthedirectionoftheforceofthevirtualspring. Finally,

v

iand

v

jaretheforwardvelocitiesoftherobotsRiandRj,

respectively.

4.2. Topologicalneighborhoodapproach

WeproposetheDW-KNNmethodasatopologicalapproachto identifyTi.Theweighteddistancesarecomputedasfollows:

• First,arobotRicomputesitsdensity"ibasedonanSPHdensity

estimationtechnique,whichshouldbeimmediately communi-catedtoitsneighbours.

• Second, upon receiving the densities ("j) fromneighbours, a

weighted-distancefunctionwijisappliedtoweightthedistances

totheneighbours.

• Finally,basedonwij,thesetTiisidentifiedbysortingthe

neigh-boursinorderofthenearestK-weighteddistances,whereKrefers tohowmanyneighboursaretakenintoaccount.

The SPH density estimation technique and the weighted-distance-basedfunctionweuseinourstudyarediscussedinthe followingsubsections.

4.2.1. SPHdensityestimationtechnique

SPHisamesh-freeLagrangianmethod,inwhichthestateofthe simulatedsystemisrepresentedbyemployingafinitesetof dis-orderdiscreteparticlesinawaythatmakesbothafixedorderto organizetheparticlesandageneratedmeshtorepresentthe con-nectivityoftheparticlesunnecessary(VioleauandRogers,2016). SPHhasbeenfirstlyintroducedincomputationalastrophysics stud-iesandisappliedinsimulatingcompressibleflowproblems(Price, 2012).OneofthemainfeaturesoftheSPHtechniqueisthatatany givenpointinthesimulationdomain#,apropertyofaparticleican beapproximatedrelyingonasummationofaninterpolationkernel functionWwithhasthesmoothinglength(Pimentaetal.,2013).A schematicrepresentationofthissystemispresentedbelowinFig.3 IntheSPHapproach,interpolationisusedtoapproximate phys-icalquantitiesthataremovingwithparticles;hereweapproximate thedensity asa physicalquantitymoving withtherobots. The robotdensity"iisevaluatedastheweightedsumofdistancesover

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itsproximaterobotswithinaparticularrangeDr=2h(Zhaoetal., 2011): "i=

!

j∈Ni W(dij,h), (3)

TheweightfunctionsusedinthisworkaretheM4cubicspline functionstruncatedat2h(Price,2012):

W(d_ij,h)=$

⎧

⎪

⎨

⎪

⎩

1 4(2−q) 3 −(1−q)3, 0≤q<1 1 4(2−q) 3_, ₁ ≤q<2 0, q≥2 (4)

whereq=dij/hand$=10/(7%h2)isanormalizedconstant.The

computeddensityiscommunicatedtotheneighboursoftherobots. 4.2.2. Distance-WeightedK-NearestNeighbours

DW-KNNisaverypopularacronyminmachinelearning.Itis aclassificationmethodthatisspecificallyusedtoassignalabelto anewquerybasedonweightingcloserneighboursmoreheavily, accordingtotheirdistancesfromthequery(Gouetal.,2012). Tak-inginspirationfromthistechnique,anduponreceivingtheSPH densitiesfromtheneighbours,arobotRiweightsthedistancesto

itsneighboursas

wij="jdij, (5)

where"jdenotesthedensityreceivedfromthejthneighbour.

Further,therobotbuildsaDW-KNNconnectivity,Ti(t),bysorting

theneighboursinorderofthenearestK-weighteddistances,where K∈{1,2,...,N−1}referstohowmanyneighboursaretakeninto account.Thecardinalityoftheneighbourhood,|Ti(t)|isK,andj∈

Ti(t):wij≤wim,

∀

m∈Ti(t).InthecasewhereK=N−1,themesh

istheall-to-allconnectednetwork.

ByusingthefunctioninEq.(5)whereboththedistanceand thedensityareappliedasequalkeyfactors,aneighbourrobotRj

locatedfarawayfromtherobotR_iwithaheavydensitycouldhave agreaterimpactthanonelocatednearR_ibutwithaweakdensity. 4.3. Obstacle/collisionavoidancemodel

Thismodel wasusedspecificallytoavoid obstacles encoun-teredin thearena.Sincetherobotswereplaced inaninbound environmentsurroundedbyfourwalls,wedenotethesetofthe fourwallsandobstaclespresentinthearenabyOi.Inaddition,

sincetherobotsonlyinteractedwiththeirDW-KNN robots(Ti),

themodelwasalsoactivatedtoavoidcollisionswiththeremaining neighbouringrobots(Mi=Ni\Ti).Toachievethismodel,a

repul-sivepotentialfieldwasgeneratedaroundeachrobot.Thefieldhad astronginfluencewhenarobotwasclosetothepotentialfieldand adecreasingeffectastherobotmovedfurtheraway.Wecompute therepulsivecontrolvectorasfollows(KhaldiandCherif,2016a,b): ˆri=

⎧

⎨

⎩

kr(_L1 j− 1 L0)( 1 L2_j), Lj≤L0 0, elsewhere (6) wherekrisascalingconstant,Lj(j∈Ai=Oi∪Mi)isthe

dis-tancetothenearestobstacleorthenearestrobot,andL0 isthe

obstacleinfluencethreshold. 4.4. MotionControl

Ateachcontrolstep,eachrobotR_itransformsthetotalvector, ˆai,intosignalsthatupdatetheforwardspeeds(

v

li and

v

ri)ofits

Fig.4. Theoverallself-organizedaggregationcontrolmodel.

leftandrightwheels,respectively,asfollows(KhaldiandCherif, 2016a,b):

(

v

li

v

ri

)

=

⎡

⎢

⎣

1 b 2 1 −b 2

⎤

⎥

⎦

(

v

i ωi

)

, (7) ω_i=kω

0

₁₈₀ ∠ˆai %

1

,

v

i=

"

v

max |ωi|+1 , (8)

where

v

iandωi,respectively,aretheforwardandtheangular

speedsoftherobot,bistherobot’sinter-wheeldistance,∠ˆaiisthe

angleformedbythexandycomponentsofthevector ˆai,kωisa

gainconstant,and

v

maxisthemaximumforwardspeedthatthe

robotcanreach.Theangularspeedshouldbealsowithintherange [−ωmax,ωmax].

5. Implementationoftheoverallself-organized aggregationcontrolmodel

Aschemeoftheoverallcontrolmodelimplementedinthe foot-botrobotisshowninFig.4.Thedetailsoftheentirealgorithmare illustratedinA.

Toachieve theSPHDensityEstimatorControl(SPHDEC),the robotusestheRABdevicetomeasurethedistances,dj,toits

neigh-bourswithintheperceptionrange,Dr.Usingtheequationsfromthe

SPHDECmodel,thefoot-botisabletocomputeitsdensity,whichis immediatelysenttoitsneighboursviatheRAB.Sincethefoot-bot canexchangeonly10bytesofdata,thedensityisscaleddownby 103_before_being_sent_to_the_neighbours.

ToachievetheDW-KNNControl(DWKNNC),therobotreceives thedensities oftheneighboursand computesitscorresponding weighteddistances;then,basedonthevalueofK,therobotcan determinetheDW-KNN.

Torecognizetheproximitycontrol(PC),therobotusesonce moretheRABdevicetocommunicateitsforwardspeedtoits neigh-bouringrobots,aswellastomeasuretherangeandbearing(dj

and!j)towardstheseneighbouringrobots.Foraccomplishingthe

RepulsiveControl(RC),weuseproximitysensorstomeasurethe distance(Lj)and the angle(ϕj)ofthe closet detectedobstacle.

Finally,basedontheresultingvectorai,theMotionControl(MC)

actuatesthelinearspeedoftherobotwheels. 6. Simulationresults

Asmentionedabove,allofthesimulationsinthisstudywere performedusingARGoSplatform(Pincirolietal.,2012).Inthe fol-lowingsubsections,weillustratetheconfigurationsetupusedin thesimulationandsomeofthesimulationresults,explainthe met-ricsusedtoevaluatetheproposedmethod,andpresentsomeofthe resultsinthecontextofthemetrics.

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Fig.5.Self-organizedaggregationofaswarmofNfoot-botsrunningtheDW-KNNtopology.

Table1

Parametersandconstantsusedinthesimulation.

Parameter Description Value

b Inter-wheelsdistance 0.14m

vmax Maximumforwardspeed 0.10m/s

ωmax Maximumangularspeed 180◦s−1

d0 Equilibriumlengthofthespring 0.3m

L0 Obstacleinfluencethreshold 0.1m

kr Obstaclescalingconstant 1.75forceunit

kd Dampergainconstant 1.25forceunit

kω Angularspeedgain 1.5◦s−1

ks Springgainconstant 1.9forceunit

Dr Maximumperceptionrange 1.5m

h Thesmoothinglength 0.5m

6.1. Experimentalsetup

WeuseaswarmofNfoot-botsheadedinarbitrarydirections thatwerearbitrarilyplacedina(10×6)m2_arena._Noise_was_added totherangeandbearingmeasurementsoftheRABdevice.Noise wasmodelledasaGaussiandistributionwithzeromeanand0.01 deviation. The loss of packets during communication wasalso takenintoconsiderationbysettingthelossprobabilityto3%.We conductedthreeseparateexperimentswithdifferentnumbersof foot-botrobots(N={50,100,150})foradurationof2000timesteps (ts)each(1ts=0.1s).Startingfromtheinitialdistributionofrobots, asetofconstantsandparameters(seeTable1)wereinitiatedin eachexperiment.

WedemonstrateinFig.5theself-organizedaggregationsthat developed from the execution of theoverall control model by

a swarm of N={50, 100, 150} foot-bots robot at diverse time steps(t=400,t=1200,andt=2000)whenstartingfromtheinitial positions(t=0).Startingfromthesameinitialposition,therobot swarmachieveddifferentself-organizedaggregationsasonlythe neighbourhoodtopologywasvaried.Fig.5ahighlightstheresults obtainedfromaDW-2NN topology,Fig.5bpresentstheresults achieved froma DW-3NN topology, whereas resultswith DW-4NNandDW-5NNtopologiesarementionedinFig.5candFig.5d, respectively.

Also, we investigated theDW-KNN approach in presence of obstacleswhilekeepingthesameparameterssetupofthe previ-ousscenario.Todoso,threeobstaclesarerandomlyplacedinthe arena.Fig.6showstheevolutionoftheswarmtoachieveaccurate aggregationswhilesmoothlyavoidingobstacles.

Itcanbeseenthatinbothabsenceandpresenceofobstacles, therobotsswarmisabletoachieveself-organizedaggregationsvia theproposedapproach.Moreover,theapproachcanbeusefulin scenariossuchasdrivingalargescaleofrobotsfromoneareato another,whilemaintainingaconnectednessbetweentherobots andavoidingcollisions.Here,theconnectivitybetweenthe neigh-bours aremodelledusing virtualviscoelasticlinks betweenthe DW-KNN,andthedistancestowardthoseneighboursareweighted usinganSPHdensityestimationtechnicwhereM4cubicspline functionsareapplied.Thisfactcouldsmoothlydrive theswarm toemergecubicbasedself-organizedaggregationsasillustratedin snapshotsofFigs.5and6.

Itcanalsobenoticedthatinpresenceofobstacles,more clus-terscouldemergecompared toasituationwithoutobstaclesin thearena.See,forexample,snapshotsinFig.6a,c,anddforthe

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Fig.6.Self-organizedaggregationofaswarmofNfoot-botsrunningtheDW-KNNtopologywithexistingobstacles.

casewhenN={50,100},andsnapshotsinFig.6bforthecaseof N=50. Thisismainlyduetothefactthattherobot’field vision willbemuchinfluencedbytheexistenceofmoreobstacles,and thereforeitsrangeandbearingsensingcapabilitiescanbe signifi-cantlyaffected.Thismeansthatanobstaclelocatedinarobot’RAB rangewillblindthevisionofarobottosenseandcommunicate withneighbours.Asconsequenceofanincreaseofboththesizeof robotsintheswarmandthevalueofKintheDW-KNNapproach resultsinadecreaseofthetotalnumberoftheclustersthatcould emerge.

7. Performanceanalysis

Inthissectionweareinterestedinstudyingtheperformance ofourproposedmodel.Specifically,weevaluatehowtherobots, relyingonlyontheDW-KNNtopology,achievedaself-organized aggregationwhilemaintaininga certaindistance betweeneach robot.Wealsoanalyzetheeffectofnoiseonthemodel.Forthat, weusethefollowingmetricstoascertaintheattainmentofthis objective.

7.1. Performancemetrics

7.1.1. Distance-weighteddistributionquality

Wedefineanewmetrictomeasurethequalityoftheevolution oftheoverallweighteddistancesoftheentireswarm.First,the

weighteddistancesaveragedoverthedifferentrobotsand neigh-boursAWD(t)arecalculatedasfollows:

AWD(t)=_N.K1 N

!

i=1

⎛

⎝

K

!

j=1 w_ij(t)

⎞

⎠

. (9)

Thenthedistance-weightedqualitymetric,Fw(t),isgottenby

thefollowingequation:

Fw(t)=1−

"

1

AWD(t)+1. (10)

7.1.2. Aggregationquality

Theaggregationquality(Dorigoetal.,2004),Fag(t),isrelated

totheaveragedistanceoftherobotsfromtheircenterofmass.To measurethismetric,firstthedistanceci(t)ofeachrobotRifromthe

centerofmassofthegroupatsimulationcycletiscomputedas

c_i(t)=

6

X_i(t)−_N1 N

!

j=1 X_j(t)

6

, (11)

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Fig.7.Performancemetricsresultsinabsenceofobstacles:(a)Fw(t),(b)Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN={50,100,150}robots

whentakingintoconsiderationdifferentDW-KNNtopologies.Inallplots,thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtained

from25runswithdifferentinitialconfigurationsofrobots.

Fig.8. Performancemetricsresultsinpresenceofobstacles:(a)Fw(t),(b)Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN={50,100,150}

robotswhentakingintoconsiderationdifferentDW-KNNtopologies.Inallplots,thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvalues

obtainedfrom25runswithdifferentinitialconfigurationsofrobots.

whereXi(t)isthepositionvectoroftheithrobotattimestept.

ThisvalueisusedtocomputetheaggregationqualityAgi(t)ofthe

ithrobotasfollows:

Ag_i(t)=

⎧

⎪

⎨

⎪

⎩

0, c_i(t)< ˜r(n) R(n)−c_i(t) R(n)− ˜r(n), ˜r(n)≤ci(t)≤R(n) 1, ci(t)>R(n) (12) where R(n)= ˜r(n)+100, ˜r(n)=rs.(√n−1) is defined as an

approximationoftheradiusofthesmallestcirclethathasnrobots positionedonitsperimeter,andrsistheradiusofarobot(Dorigo

etal.,2004).

Thentheaverageaggregationquality,Fag(t),isdefinedas

fol-lows: Fag(t)=_N.K1 N

!

i=1

⎛

⎝

K

!

j=1 Ag_i(t)

⎞

⎠

. (13)

7.1.3. AveragedMeanDistanceError

TheAveragedMeanDistanceErrormetric,AMDE(t),isdefinedas theinter-robotsdistanceerroraveragedoverthedifferentrobots andneighbours.Itisusedmainlytomeasurehowwelltheswarm maintainsacertaindesireddistancebetweentheindividualrobots astheymovetogether.Themetriciscalculatedasfollows:

AMDE(t)=_N.K1 N

!

i=1

⎛

⎝

K

!

j=1 (dij(t)−d0)

⎞

⎠

. (14)

7.2. Analysisinnormalcircumstances

Toevaluateourproposedapproachmoreaccurately,we per-formed25runsofeachexperimentandtheexperimentduration of each run was setagain to 2000time steps.We plotted the resultsobtainedfromthesimulationofN={50,100,150}foot-bot robotswithintheDW-2NN,DW-3NN,DW-4NNandtheDW-5NN topologiesinabsenceandpresenceofobstacles.Figs.7and8show respectively thetimeevolution ofFw(t),Fag(t), andAMDE(t)for

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Fig.9.Performancemetricsresults:(a)Fw(t),(b)Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedon100robotswhentakingintoconsideration

differentDW-KNNtopologieswithdifferent$.Inallplots,thex-axisrepresentsthestandarddeviationofnoise$.Eachboxrepresentsvaluesobtainedfrom25runswith

differentinitialconfigurationsofrobots.Theredsquaresshowthemedianvalues,andthedashedredlinesshowsalinearleastsquaresregressionfittothesesquares;the

olivebandshowsalinearleastsquaresregressionfittothemeanofthe5redsquaresforeachtopology.Thelinearleastsquaresregressionfunctionsgeneratedtofitallof

thesepointsareoftheformf(x)=c1x+c0;thecorrespondingcoefficientscanbefoundinthetablestotherightofeachfigure.(Forinterpretationofthereferencestocolor

inthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

thefourDW-KNNcase studiesin bothabsenceandpresenceof obstacles.Inalloftheplots,thecurvesshowthemedianvalues ofthe25runs.Resultsshowthattheswarmrobotsystem suc-cessfullyconvergedintoastablevalueofFw(t),Fag(t)andAMDE(t)

forallNrobotsandforalltopologies. Thismeansthatthefinal self-organizedaggregationsareachievedwherealltherobotshad nearlythesamedensitywiththedesiredinter-robotdistance.We alsonotethattheperformanceoftheproposedstrategyimproved whenthesizeoftheswarmincrease.However,theoveralltime convergenceofthemetricsinabsenceofobstaclesismuchbetter whencomparedtotheobstaclecasestudies.

7.3. Theeffectofsensorynoise

Inthissection,weinvestigatehowtheaggregationperformance wasaffectedwhenthereadingsoftherobotsrangeandbearing

sen-sorswerecorruptedbynoise.Wemodellednoiseasinthenormal experimentalsimulation,i.e.,asaGaussiandistributionwithzero meanandastandarddeviation$={0.1,0.2,0.3,0.4,0.5},butthis timeweconsidereddifferentvaluesof$.Theotherexperimental settings(giveninTable1)remainedfixed.Wesetanexperiment foreachvalueof$andperformed25runsofeachexperiment.Inall ofthesimulations,wefixedN=100robots,andweusedthe follow-ingDW-KNNtopologiesineachrun:DW-2NN,DW-3NN,DW-4NN, andDW-5NN.

Fig.9plotsFw(t)(Fig.9a),Fag(t)(Fig.9b),andAMDE(t)(Fig.9c)

withrespecttothedifferentvaluesof$.Inalloftheplots,eachbox representsthemetricvaluesobtainedfrom25simulationswith dif-ferentinitialconfigurationsof100robots.Theredsquaresshowthe meanvalues,andthedashedredlineshowsaleastsquares regres-sionfittothefivepointsofeachtopology.Theolive-colouredband

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Fig.10.PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeMeanShiftnoisemodel(caseb=3):(a)Fdisp(t),(b)Fag(t)and

(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNNtopologies.Inallplots,

thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurationsofrobots.

Fig.11.PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeMeanShiftnoisemodel(caseb=6):(a)Fdisp(t),(b)Fag(t)and

(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNNtopologies.Inallplots,

thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurationsofrobots.

indicatestheevolutionofthemetricwithregardstotheDW-KNN topology;itrepresentsanotherlinearleastsquaresregressionfitto thepointsofthemeanvaluesofeachfiveredsquares.Alloftheleast squaresregressionfittingfunctionshavetheformf(x)=c1x+c0.The

coefficientsgenerated forthesefunctionsarehighlightedin the tablesbesideeachplotinFig.9.

WecanseeinFig.9a that,foragivenDW-KNNtopology,an increasein$decreasesFw(t).Table9aillustratesthecorresponding

leastsquaresregressionfunctionsgeneratedforeachtopology.It suggeststhatforagiventopology,Fw(t)decreasesslightlyand

sub-linearlyinfunctionof$.Moreover,theolivebandinFig.9ashows thatFw(t)issublinearlyincreasingwithregardtothenumberKin

theDW-KNNtopology.

InFig.9b,theaggregationqualityFag(t)inagiventopologyis

sublinearlyaffectedbyanincreasein$.Wecanseethatan increas-ing$yieldsadecreasingFag(t).Therelatedleastsquaresregression

functionsare depictedin Table9b. However,the olivebandin Fig.9bindicates thatFag(t)showsa small,decreasingdeviation

withregardtothenumberKintheDW-KNNtopology.Therefore, itremainsalmoststable.

ThemetricAMDE(t)wasalsoaffectedbynoise.Fig.9cshows thatasmorenoisewasintroducedintotheRAB,AMDE(t)deviated furtherfromzero.Ineachtopology,thereddashedlinesthat rep-resentlinearleastsquareregressionfittingfunctionsindicatesthat AMDE(t)sublinearlyincreaseswithregardstothevalueof$.On theotherhand,ananalysisoftheolivebandinFig.9cshowsthat, whateverthevalueofKis,AMDE(t)hasaconstantslight,sublinear increasethatseemsstable.

Theseobservationsareasexpected.MorenoiseintheRAByields moremismeasurementsofthedistances andbearingsto neigh-bouringrobots,which immediatelyaffects thequalityof allthe metrics.WhileattractingmorerobotsbyincreasingthenumberK intheDW-KNNtopologydoesnotseemstohaveagreatimpacton thequalityoftheaggregationortheaveragemeandistanceerror, itdoesincreasethequalityofFw(t).

8. ComparisontotheKNNaggregationapproach

Inourpreviousstudy(KhaldiandCherif,2016a),weproposed aKNNapproachforself-organizedaggregationinswarmrobotics

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Fig.12.PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeMeanShiftnoisemodel(caseb=6andvariance0.1):(a)

Fdisp(t),(b)Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNN

topologies.Inallplots,thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurations

ofrobots.

Fig.13. PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeAuto-Correlatednoisemodelwherea=0.5:(a)Fdisp(t),(b)

Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNNtopologies.

Inallplots,thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurationsofrobots.

systems.Inthatapproach,theinteractionbetweentheneighbours wasgovernedbytheK-NearestNeighbours.Inthissection,we com-paretheproposedDW-KNNaggregationapproachwiththeKNN one.Tobettercomparetheperformanceofthetwoapproaches, weusedanothermetriccalledadispersionmetric.Thismetrichas beenusedbyGrahamandSloane(1990)toaddresstheproblem ofpennypackinginatwo-dimensionalplane.Also,thismetricwas adoptedbyGaucietal.(2014b)tostudyaggregationtaskinaswarm ofe-puckrobotswiththeassumptionofusingminimalresources. Unliketheaggregationtaskthatseekstogathertherobotstogether inanareaofanenvironment,thedispersionmetricisgenerally usedtomeasurethedispersionofaswarmrobots-systeminthe environment.HenceasstatedinthestudyofGrahamandSloane (1990),Gaucietal.(2014b),lowerboundsofthemetricare ana-lyzedasabaddispersionquality,meaningthatthepennies,aswell astherobots,aremostclosetotheircentroid,andthereforethiscan beconsideredasagoodaggregationsign.Here,wefollowthesame ideaandweadoptedparticularlythedispersionmetricproposed byGaucietal.(2014b)tomeasurethedispersionofourrobots.We

callthemetricF_disp(t),andweusedittomeasurethequalityof dispersionoftheentireswarm.

Todefinethismetric,firstthedispersionqualityDisp_i(t)ofa givenrobotisaveragedoveritsdifferentKneighboursasfollows (Gaucietal.,2014b): Disp_i(t)=_4.r1 s2 K

!

i=1 ci(t)2. (15)

Thenthedispersionquality,Fdisp(t),oftheswarmisaveraged

overthenumberoftherobots, Fdisp(t)=_N1

N

!

i=1

Dispi(t). (16)

Wherersrepresentstheradiusoftherobotandci(t)represent

thedistanceoftherobotfromthecenterofmassofthegroup,and itiscomputedasinEq.(11).The4rs2inthedenominatorserves

tonormalizeF_disp(t)such thatit becomes independentofrs for

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Fig.14. PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeAuto-Correlatednoisemodelwherea=0.75:(a)Fdisp(t),(b)

Fag(t)and(c)AMDE(t)obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNNtopologies.

Inallplots,thex-axisrepresentstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurationsofrobots.

Fig.15.PerformancemetricsresultsfortheKNNandtheDW-KNNtopologicalaggregationapproacheswiththeuniformnoisemodel:(a)Fdisp(t),(b)Fag(t)and(c)AMDE(t)

obtainedfromtheoverallcontrollerimplementedonN=100foot-botrobotswhentakingintoconsiderationdifferentKNNandDW-KNNtopologies.Inallplots,thex-axis

representstheevolutionoftimestept.Eachcurverepresentsthemedianvaluesobtainedfrom5runswithdifferentinitialconfigurationsofrobots.

Now,weevaluatetheperformanceoftheproposedapproach as well as the KNN algorithm under the presence of complex noisemodels.Specifically,theperformancesofconventionalKNN approach and its extension, DW-KNN aggregationmethod, are comparedunderdifferentnoisemodels.

8.1. EffectofGaussiannoisewithmeanshiftontheperformance oftheproposedapproach

Inthissubsection,weinvestigatetheimpactofGaussiannoise withaconstantbiasontheperformanceoftheproposedapproach. Towardsthisend,weintroducedaGaussiannoisewithameanshift toboththerangeandbearingmeasurementsas:

Z′

i(t)=Zi(t)+εi(t), (17)

whereZ′

i(t)isthenoisymeasurement,Zi(t)isthetrue

measure-ment,andεi(t)isGaussiannoisewithconstantmeanbandunit

variance.Usingthisnoisemodel,weevaluatedtheperformanceof boththeDW-KNNandtheKNNaggregationapproachesunderthe presenceofmeasurementnoisewithtwodifferentlevels(i.e.,b=3,

6).Fig.10showstheresultsofthequalitymetricswhenaconstant biasofamplitudeequalb=3,whereasFig.11plotstheresultswhen b=6.FromFig.10,itcanbeseenthatbothapproachesconvergeto thesameFag(t)andAMDE(t)values(seeFig.10bandcfortheb=3

case,andFig.11bandcfortheb=6case).However,theDW-KNN hasaslightlylowerdispersioncomparedtothatoftheKNNmethod (seeFig.10aandFig.11a).Lowerdispersionmeansthattherobots areveryclosetotheirgroupcenterandthereforetheoverall aggre-gationperformanceoftheDW-KNNismuchbetterthantheKNN aggregationmethod.

Inanothersimulationsetup,weconductedsimulationwitha meanshiftconstantb=0.05andavariance0.1.Resultsareshown inFig.12.Similartotheabovenoisemodels,theKNNandthe DW-KNNaggregationapproachesconvergealmosttothesamequality ofaggregation,Fag(t)andtothesamemeandistanceerrorAMDE(t)

(seeFig.12bandc).However,theoverallaggregationperformance oftheDW-KNNapproachisbetterthantheKNNduetothefactthat itgiveslowerdispersionqualitythantheKNNapproachinthecase ofK={3,4,5}(seeFig.12a).Noticealsothatforbothapproaches intheabovecasesstudies,F_disp(t)convergesalmosttothesame

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valueincasek=2andthatitslightlyincreasesincasek={3,4,5}. Thisisasexpected,aggregatingrobotsbyformingameshofmore thantworobots(k>2),willattractmoreweightedrobotstothe centeroftheirgroupwhenusingaDW-KNNapproachthantheKNN approach.Itcanalsobeseenthatwhennoiselevelincreasesthe aggregationqualityintermofthedispersiondegradebyincreasing thedispersionmetric.Theresultsalsoshowthatmoderatenoise levelhasnosignificantimpactontheAMDE(t)andFag(t).

8.2. Effectofautocorrelatednoiseontheperformanceproposed approach

Inthissubsection,theperformanceoftheDW-KNNand KNN-basedneighborhoodtopologywillbeinvestigatedinthepresence of autocorrelated noise. Towards this end, the performance of DW-KNNandKNNapproacheshavebeenstudiedwhenthe mea-surement noise is generated from a first-order autoregressive process,orAR(1).Specifically,themeasurementnoiseoftherange andbearingsensorsaregeneratedusinganAR(1)modelasfollow:

7

Z′

i(t)=Zi(t)+)i(t)

)_i(t)=a)_i(t−1)+ε_i(t) , (18) whereZ′

i(t)isthenoisymeasure,Zi(t)isthetruemeasure,ais

theautocorrelationcoefficientwithlag1,and

*

i(t)isaGaussian

distributionof0meansandastandarddeviation1.

Withthisnoisemodel,weinvestigatedtheperformanceofthe twoapproachesbyconductingsimulationsfordifferentvaluesof theAR-parametera={0.5,0.75}.Figs.13and14plotsuccessively theresultsofthetwobasedtopologicalapproachesforthe differ-entvaluesofa.Inbothvaluesofa,theanalysisresultsareclose totheanalysisdepictedinthepreviousnoisemodelswherethe twoapproachesconvergenearlytothesameFag(t)andthesame

AMDE(t)inahand.Inanotherhand,theDW-KNNapproachgives moredispersionqualitythantheKNNapproach(alowervalueis better).Inaddition,thetwoapproaches convergealmosttothe samedispersionqualityincasek=2andthattheyslightlydiverge linearlyto different values in case k={3, 4, 5}. This is also as expectedwhengroupingtherobotsbyformingameshofmore thantworobots(k>2),usingaDW-KNNapproachwillattractmore weightedrobotstothecenteroftheirgroupthantheKNNapproach. Alikethepreviousnoisemodel,itcanbealsoseenthatwhenAR coefficientincreases(highlycorrelatednoise)theaggregation qual-ityintermofthedispersiondegradebyincreasingthedispersion metric,andtheresultsalsoshowthatnoisewithmoderate autocor-relationhasnosignificantimpactontheAMDE(t)andFag(t).Notice

alsothattheoverallaggregationmetricstakemuchtimestepstobe convergedwhilecomparedtothepreviousnoisemodels.Ascanbe seeninalltherelatedfigures,thetimeconvergenceofthemetrics isproportionaltothevalueoftheautoregressivecoefficient(fora greatervalueofathetimeconvergenceofthemetricsislong). 8.3. Effectofuniformnoiseontheperformanceproposed approach

Inthiscasestudy,rangeandbearingsensorsnoisearegenerated froma uniformdistributionof[−3,3cm]fora range measure-mentandof[−5◦_,₅◦_]_for_a_bearing_measure._Fig.₁₅_illustrates_the

resultsofKNNandDW-KNNbasedtopologicalapproaches.Like theothernoisemodels,boththeKNNandtheDW-KNN aggrega-tionapproachesconvergealmosttothesamequalityofaggregation andtothesamemeandistanceerror(seeFig.15bandc).Butthe DW-KNNapproachgiveslowerdispersionqualitythantheKNN approachin thecaseof K={3,4, 5}(seeFig.15c),which mean thattherobotsarebetterclosetotheircenterofmasse,andhence itis agoodsignofaggregationperformance.However,thetwo

approachesperformslightlyinasimilarwaywhenK=2.Thiscan beexpectedasinthepreviousnoisemodels,whengroupingrobots bymorethentworobots(k>2),theDW-KNNapproachwillattract moreweightedrobotstothecenteroftheirgroupthantheKNN one.Alsoasanalyzedinthepreviousnoisemodels,theaggregation qualityintermofthedispersiondegradebyincreasingthe disper-sionmetric,andtheresultsalsoshowthatuniformnoisemodelhas nosignificantimpactontheAMDE(t)andFag(t).

9. Conclusionsandfurtherwork

Whenstudyingthecollectivebehavioursofalargenumberof individualrobots,theinter-robotdistancecanbeofkey impor-tance,butadditionalpropertiessuchasthedensityofrobotscould havea greaterimpactonthecollectivebehaviourof thewhole swarm.Here wereliedonboththedistanceandthedensityas equalkeyfactors,andweproposedaDW-KNN-based neighbour-hoodtopologytostudyself-organizationinanaggregatingrobot swarm.Withthistopology,weusedthevirtualviscoelastic-based proximalmodeldevelopedinourpreviouswork(KhaldiandCherif, 2016a)tokeepandarrangetheneighbourstogether.DW-KNNwas achievedbydefining a distance-weightedfunction basedonan SPHinterpolationapproachtoestimatethedensityofrobotsinthe swarm.

Also, the proposed algorithm exhibited high efficiency to smoothly drive the robotsswarm to achieve cubic based self-organizedaggregationsinthepresenceandabsenceofobstacles inthearena.Theproposedapproachcouldbehelpfulinasituation whenattractingalargescaleofrobotsfromoneareatoanother whilemaintainingaconnectednessbetweentherobotsand avoid-ingcollisions.

Various self-organized aggregations are achieved using the ARGoSsimulatorinbothabsenceandpresenceofobstacles,and performanceanalysiswithinthreemetricsshowstheefficacyof theproposedapproach.Theeffectofnoiseintherobotrangeand bearingsensingcapabilitiesisalsoaddressedinthisstudyshowing howtheproposedmodelisbehavinginsuchcircumstance.

Furthermore,theproposed methodshowedsuperior perfor-mancecompared totheKNN approach inpresence ofdifferent noisemodels(i.e.,uniform,Gaussiannoisewithmean-shift,and Auto-Correlatednoisemodels).

Moreover,inthepresentapproachthechoiceofthedensity esti-mationmethodtobeusedtoweightthedistancescouldplay a crucialroleintheemergenceoftheself-organizedaggregations. Thisislettothescenarioinwhichtheapproachwillbeapplied.For example,webelievethatwiththeproposedaggregationapproach, various B-spline based self-organized aggregations couldeasily emergebyadoptingotherkernelfunctionsintheSPHdensity esti-mationmethodssuchasthosebasedontheSchoenbergB-spline functions(Schoenberg,1988)(i.e.,theM5quarticfunctionsand theM6quanticfunctions(Price,2012)).Moreover,different den-sityestimationmethodsthatdifferfromtheSPHapproach(i.e., Gaussiankerneldensityestimationmethod)couldbeusedasan alternativetoachievedifferentself-organizedaggregations,orto drivetheswarmofrobotstoadesireddensitydistribution. There-fore,weopenadoorforfurtherstudiesonhowtoselectthedensity estimationmethodforagivenscenario.

Inthelightoftheabovepointsandasfuturework,weplan tofiltermeasurementsnoisetoenhancetheperformanceofthe model.Weplanalsotostudytheimpactoftherobotdensityon self-organizedaggregationbyincorporatingdifferentSPH density-estimationmethodsandusingotherdensityestimationtechnics thatdiffersfromtheSPHone.Moreattentionshouldalsobegiven tohowarobotswarmcollectivelydecidestoadaptthevaluesofK todynamicallyswitchitsneighbouringrelationships.

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Algorithm2. PartII

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