Emergence of Consensus in a Multi-Robot Network: from Abstract Models to Empirical Validation

5 MB·

by  Vito Trianni

Emergence of Consensus in a Multi-Robot Network:
from Abstract Models to Empirical Validation
VitoTrianni1,DanieleDeSimone2,AndreagiovanniReina3 andAndreaBaronchelli4
1ISTC,NationalResearchCouncil,00185Rome,Italy
[email protected]
2DIAG,SapienzaUniversityofRome,00185Rome,Italy
[email protected]
3UniversityofSheffield,SheffieldS14DP,UK
[email protected]
4CityUniversityLondon,LondonEC1V0HB,UK
6
[email protected]
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0
2
n Abstract
a
Consensus dynamics in decentralised multiagent systems are subject to intense studies, and several different models have
J
beenproposedandanalysed. Amongthese,thenaminggamestandsoutforitssimplicityandapplicabilitytoawiderangeof
9
phenomenaandapplications,fromsemioticstoengineering. Despitethewiderangeofstudiesavailable,theimplementationof
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theoreticalmodelsinrealdistributedsystemsisnotalwaysstraightforward,asthephysicalplatformimposesseveralconstraints

thatmayhaveabearingontheconsensusdynamics.Inthispaper,weinvestigatetheeffectsofanimplementationofthenaming
]
A gameforthekilobotroboticplatform,inwhichweconsiderconcurrentexecutionofgamesandphysicalinterferences.Consensus
M dynamicsareanalysedinthelightofthecontinuouslyevolvingcommunicationnetworkcreatedbytherobots,highlightinghow
thedifferentregimescruciallydependontherobotdensityandontheirabilitytospreadwidelyintheexperimentalarena. We
. findthatphysicalinterferencesreducethebenefitsresultingfromrobotmobilityintermsofconsensustime, butalsoresultin
s
c lowercognitiveloadforindividualagents.
[

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1 Introduction
v
2
5 Collective decision-making is an essential capability of large-scale decentralised systems like robot swarms, and is often key
9 to achieve the desired goal. In swarm robotics, a large number of robots coordinate and cooperate to solve a problem, and
4
oftenconsensusamongtherobotsisnecessarytomaximisethesystemperformance[12,18,15]. Thedesignofcontrollersfor
0
consensus decision is often inspired by models of collective behaviour derived from studies in the ethology of social systems
.
1 [19, 14], as well as from studies about the emergence of social conventions and cultural traits [21, 6, 5]. Theoretical models
0
represent idealised instances of collective decentralised systems in which consensus can be somehow attained. Among the
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differentavailablemodels,aparticularlyinterestingcaseistheoneofthenaminggame(NG),whichrepresentstheemergence
1
: of conventions in social systems, such as linguistic, cultural, or economic conventions [23, 4, 7]. The appeal of this model
v
consistsintheabilitytodescribetheemergenceofconsensusoutofavirtuallyinfinitesetofequivalentalternatives,yetrequiring
i
X minimalcognitiveloadfromtheagentscomposingthesystem[4,2]. Moreover,theNGhasbeensuccessfullydemonstratedon
r anetworkofmobilepoint-sizeagents[3]. Suchacollectivedecision-makingbehaviourcanbeveryusefulinswarmroboticsin
a
caseconsensusisrequiredwithrespecttoapossiblylargenumberofalternatives(e.g.,thelocationandstructureforcooperative
construction[24,20],orthemostfunctionalshapeforself-assembly[11,17]).
When dealing with the implementation of physical systems starting from theoretical models, however, several constraints
mayarisewhichmayhaveabearingonthecollectivedynamics. Indeed,smallimplementationdetailsatthemicroscopicscale
may have a large impact at the macroscopic level. Hence, it is important to study the effects of such constraints in relation to
thedynamicspredictedbythetheoreticalmodels. Inthispaper,weproposeanimplementationoftheNGforthekilobotrobotic
platform[16]. Kilobotsarelow-costautonomousrobotsdesignedforexperimentationwithlargegroups[17]. Theycanmoveon
aflatsurfaceandinteractwithcloseneighboursbyexchangingshortmessagessentonaninfraredchannel. Thecollectivebe-
haviourofakilobotswarmresultssolelyfromtheindividualdecisionsandinter-individualinteractions,withoutanycentralunit
directingthegroupdynamics. Asaconsequence,theimplementationoftheNGforthekilobotsneedstobefullydecentralised
withgamesautonomouslytriggeredbyanyrobotatanytime. Additionally, withinadecentralisedsystem, theconcurrentexe-
cutionofgamesbyneighbouringrobotsbecomespossible,inoppositiontotherigorouslysequentialschemetypicallyadopted
intheoreticalstudies. Hence, theinteractionpatternamongrobotsmaybesignificantlyaltered, andthecorrespondingdynam-
icsneedtobecarefullycharacterised. Finally,theembodimentoftherobotsdeterminesphysicalinterferences(i.e.,collisions)
that strongly influence the overall mobility pattern. It follows that abstract models of agent mobility must be contrasted with
experimentationwithrobots,inwhichallthedetailsofthephysicalplatformcanbetakenintoaccount.
1
procedureNG(nm,ns) (cid:46)ImplementationoftheNG
nt←nt+1
ifnt modnm=0then (cid:46)Changemotiondirection
RandomTurn()
endif
MoveStraight()
W←ReceiveWords() (cid:46)Playthehearerrole
Randomise(W)
forw∈Wdo
UpdateInventory(w)
endfor
ifnt modns=0then (cid:46)Playthespeakerrole
w←SelectWord()
Broadcast(w)
endif
endprocedure
Figure1: TheNGalgorithmexploitedinmulti-agentsimulations.
Inthispaper,westudytheeffectsofthemotionandinteractionpatternsontheconsensusdynamics,andwepayparticular
attention to both concurrent executions of games and physical interferences. First, we provide an abstract model of mobile
agentsplayingtheNG,inwhichphysicalinterferencesareignored. Followingpreviousstudies[3],weanalysethismodelinthe
lightofthecommunicationnetworkestablishedbytheagents, weshowhowtheconsensusdynamicsaredeterminedbyagent
density,mobilityandinteractionfrequency,andwelinkourempiricalfindingswiththeoreticalstudies[2,10]. Then,wecontrast
abstractmodelswithlarge-scalesimulationsofthekilobots,aswellaswithreal-worldexperiments. Here,physicalinterferences
impactontheconsensusdynamicsbylimitingthefreediffusionofrobotsintheexperimentalarena,henceincreasingconsensus
times. Still, the cognitive load for the individual agents is reduced for physical implementations, due to the lower number of
alternativesthateachagentmustconsiderinaverage. OuradaptedimplementationoftheNGispresentedinSection2,whilethe
correspondingconsensusdynamicsarediscussedinSection3. ConclusionsandfuturedirectionsarepresentedinSection4.
2 Model and Implementations
Thenaminggameinitsbasicform[4]modelspairwiseinteractionsinwhichtwoplayers—thespeakerandthehearer—interact
by exchanging a single word chosen by the speaker, and updating their inventory on the basis of the game success. Previous
extensionsofthemodeltakeintoaccountdifferentinventoryupdatingandcommunicationschemes[1]andalsoconsidermobile
agents[3]. Inthiswork,weadoptabroadcastingschemeforthespeakeragent,whileinventoryupdatingisperformedonlyby
theheareragent,asdetailedinthefollowing(fordetails,see[1]).
WhenengaginginaNG,thespeakeragenta selectsawordweitherrandomlyfromitsinventory,orinventingitanewshould
s
theinventorybeempty(i.e.,thesetofpossiblechoicesforanewwordwisvirtuallyinfinite). Then,itbroadcastswtoallagents
initsneighbourhood. Uponreceptionofw, theheareragenta updatesitsinventorybyeitherstoringwifitwasnotfoundin
h
a ’sinventory,orbyremovingallwordsbutwifthelatterwasalreadyknowntoa . Byiteratingthegamemultipletimes,the
h h
entiresystemconvergestowardtheselectionofasinglewordsharedbyallagents[4,1].
2.1 Multiagentsimulations
WeimplementadecentralisedversionoftheNGbylettingeachagentaautonomouslytaketheroleofspeakereveryτ s. Given
s
that agents update their state at discrete steps of δ =0.1s, they communicate every n steps so that a word is broadcast every
t s
τ =n δ stoallneighbourswithintheranged =10cm. Inthisway, concurrentexecutionofgamesbecomespossible, hence
s s t i
introducing an important difference from previous theoretical studies in which at any time only one game is executed by a
randomlychosenagentandoneofitsneighbours[3,4,1,10]. Atthehearerside,multipleinteractionsarepossiblewithinany
timeintervalδ,dependingonthelocaldensityofagents. Hence,allwordsreceivedinasingleδ periodareusedsequentially
t t
to update the inventory. The list of received words is randomised before usage to account for the asynchronous reception of
messagesbythephysicalplatform(seeSection2.3). ThealgorithmforthemultiagentimplementationisshowninFig.1.
Atthebeginningofeachsimulationrun,NagentsaredeployeduniformlyrandomwithinasquaredboxofsideLwithperiodic
boundaryconditions(e.g.,atorus). Thisallowstofocusontheeffectsofagentmobilityanddensitywithoutconstraintsfroma
boundedspace[3]. Agentsaredimensionlessparticlesandthereforedonotcollidewitheachother. Theagentsneighbourhood
is determined by all the agents within the interaction range d. By moving in space, the agent neighbourhood varies so that a
i
dynamiccommunicationnetworkisformed. Agentmobilityfollowsanuncorrelatedrandomwalkscheme,withconstantspeed
v=1cm/s and fixed step length vτ , where τ represents the constant time interval between two consecutive uncorrelated
m m
changesofmotiondirection. ThisleadstoadiffusivemotionwithcoefficientD∼ v2τ [3]. Inpractice,agentschangedirection
m
everyn steps,sothatτ =n δ (seeFig.1).
m m m t
2
2.2 Robotsimulations
We have developed a custom plugin for simulating kilobots within the ARGoS framework [13], paying particular attention to
matchtherealrobotfeaturesintermsofbodysize,motionspeedandcommunicationrange. Communicationisimplementedby
allowingtheexchangeofmessagesbetweenneighbourswithintheranged. Nofailuresincommunicationhavebeensimulated,
i
assumingthatthechannelcansupportcommunicationevenwithhighdensities. Wewilldiscussthischoiceinthelightofthe
obtainedresultsinSection4.
Concerningthemotionpattern,kilobotsarelimitedtothreemodesofmotion:forwardmotionwhenbothleftandrightmotors
areactivated,andleftorrightturnswhenonlyonemotorisactivated. Turningisperformedwhilepivotingononeofthekilobot
legs. Wehavethereforeimplementedadifferentialdrivemotionschemecentredbetweenthetwobackwardlegsofthekilobot,
withspeedv=1cm/sforforwardmotionandangularspeedω=π/5s−1forturning. Amultiplicativegaussiannoiseappliedat
everysimulationcycle(standarddeviationσ =0.4)simulatestheimprecisemotionofkilobots. Withsuchanimplementation,
theindividualmotionisstilldiffusive,butwithalowercoefficientduetothedelayintroducedbyturning. Additionally,collision
avoidanceisnotpossiblewiththekilobotonboardsensors,androbotsareletfreetocrashintowallsandeachother. TheARGoS
framework provides a 2D dynamics physics engine that handles collisions between robots and with walls with an integration
stepsizeδ =0.1s,whichprovessufficientforourpurposes. Collisionsdetermineafurtherreductioninthediffusionspeed,as
t
we will discuss in Section 3.2. Robots are deployed randomly within a squared box of side L surrounded by walls. To avoid
overlappingofrobots,theinitialpositionsaredeterminedbydividingthearenaincellswideenoughtocontainasinglekilobot,
andrandomlyplacingkilobotsintofreecells.
2.3 Kilobotimplementation
The implementation of the NG for kilobots requires handling transmission and reception of messages, and implementing the
randomwalk.WeusethekilobotAPIfromKilobotics[22],whichprovidestwocallbackfunctionsfortransmissionandreception
of 10-byte messages, functions for distance estimation of the message source, and a counter that is updated approximately 32
times per second (i.e., δ (cid:39)1/32). Broadcast is allowed every τ seconds by opportunely activating the transmission callback.
t s
Communication interferences among robots are treated through the CSMA-CD protocol (carrier-sensing multiple access with
collisiondetection)withexponentialback-off,meaningthatupondetectionoftheoccupiedchannel,messagesendingisdelayed
withinanexponentiallyincreasingrangeoftimeslots.Thisintroducesanadditionallevelofasynchronythatmustbetoleratedby
thecollectivedecision-makingprocess,astheexacttimingofcommunicationcannotbecompletelycontrolled. Uponreception
ofanymessage,thecorrespondingcallbackfunctionisactivated,andtheNGisimmediatelyplayedexploitingthecontentofthe
receivedmessage. Giventhatthemaximumcommunicationdistancemayvaryacrossdifferentrobots,wecappedthemaximum
distancetod bysoftware,estimatingthesourcedistanceandignoringmessagesfromsourcesfartherthand.
i i
Themotionpatternimplementstherandomwalkexactlyasperformedinsimulation,exploitingtheinternalrandomnumber
generatorforuniformlydistributedturningangles. Forwardmotionvandangularspeedω ofeachkilobothavebeencalibrated
toobtainaroughlyconstantbehaviouracrossdifferentrobotsandtomatchtheparametervaluesusedinsimulation. Thecode
forthecontrolleriswritteninaC-likelanguage(AVR C)andfitsinabout200lines. Inexperimentalruns,kilobotsareinitially
positioned randomly following indications from the ARGoS simulator in equivalent conditions. This provides an unbiased
initialisationandsupportscomparisonwithsimulationsinSection3.3.
3 Consensus Dynamics
The most important quantity to evaluate the consensus dynamics following the NG process is the time of convergencet , i.e.,
c
the time required for the entire group to achieve consensus. Previous studies demonstrated that consensus is the only possible
outcome, even though in particular cases it can be reached only asymptotically [4]. Another relevant metric for the NG in
multiagent systems is the maximum memory M required for the agents, in average, until convergence: given that each agent
needs to store a possibly large number of words, it is important to study how the memory requirements scale with the system
size,especiallyintheperspectiveoftheimplementationforrealrobotsthatentaillimitedmemoryandminimalprocessingpower
tosearchlargeinventories.
Followingpreviousstudies,itisusefultolookatthe(static)interactionnetworkresultingbylinkingallagentsthatarewithin
interactionrange. GivenN agentsconfinedinaL×L spaceandinteractingoveraranged, theresultingnetworkhasaverage
i
degree(cid:104)k(cid:105)=πNd2/L2 [8,3]. Giventhatinourcaseallparametersareconstantbuttheagentdensity(asdeterminedbyN),two
i
valuesarecritical:
1. N =N isthegroupsizeatwhichtheaveragedegreeisaround1,meaningthateachagenthasinaverageoneother
1 (cid:104)k(cid:105)=1
agenttointeractwith. Belowthisvalue,interactionsaresporadicanddeterminedbytheagentmobility,whileabovethis
valueinteractionsarefrequentassmallclustersofagentsappear.
2. N =N correspondstothecriticalgroupsizeforapercolationtransition[8].AboveN ,thenetworkischaracterised
c (cid:104)k(cid:105)(cid:39)4.51 c
byagiantcomponentofsizeN.
GiventhebroadcastingruleemployedfortheNGinthispaper,itisclearthatthecharacteristicsoftheinteractionnetworkare
fundamental. Ifthereexistsagiantcomponent,informationcanspreadquickly. Ifotherwiserobotsaremostlyisolated,theywill
notbeabletointeractandconvergencewouldbeslower,asdiscussedinthefollowing.
3
τ =10s τ =50s
s s
N1 Nc τm N1 Nc τm
10 10
20 20
30 30
103 40 103 40
50 50
tc
5 5
4 4
M M
3 3
N N
2 2
101 102 103 101 102 103
102 102
101 102 103 101 102 103
τ =10s τ =50s
m m
N1 Nc τs N1 Nc τs
10 10
20 20
30 30
103 40 103 40
50 50
tc
5 5
4 4
M M
3 3
N N
2 2
101 102 103 101 102 103
102 102
101 102 103 101 102 103
N N
Figure2: Resultsfrommultiagentsimulations. Eachpanelshowsthedependenceoftheconvergencetimet onthesystemsize
c
N fordifferentparameterisation. Statisticalerrorbarsarenotvisibleonthescaleofthegraphs. Verticaldottedlinesindicatethe
thresholds for N =32 and N =143. The insets show the memory requirements M plotted against the system size N for the
1 c
sameparameterisations.
3.1 Influenceofdensity,mobilityandinteractionfrequency
Todetermine theconsensusdynamics andtheeffects ofthe differentparametersof thesystem, we runmultiagentsimulations
withsmallandlargegroupsinasquaredarenaofsizeL=1m.Inthiscondition,wehaveN (cid:39)32andN (cid:39)143.Figure2reports
1 c
theconsensustimet fordifferentparameterisationsvaryingN∈[10,500]andτ ,τ ∈[10,50]s.
c m s
Lookingattheresults,wenotethatt isadecreasingfunctionofN. Indeed,ahigherdensitycorrespondstoahighernumber
c
ofconcurrentlyexecutedgames,andresultsinafasterconvergence. ForN>N andsmallτ ,theconsensustimecollapsestothe
c s
samevalueforvaryingτ (seeforinstancethetop-leftpanelinFig.2). Abovethepercolationthreshold,agentmobilityplays
m
a minor role and the consensus dynamics can be related to the characteristics of the static network of interactions. Especially
forlowvaluesofτ ,thedynamicscloselycorrespondtothoseofstaticagentsinteractingonarandomgeometricnetwork[10].
m
Here,theagreementprocessproceedsthroughtheformationofclustersofagentswithlocalconsensusseparatedbyaninterface
of“undecided”agents,andconsensusdynamicsrecallthecoarseningonregularlattices[2]. Thisisconfirmedbytheleftpanel
in Fig. 3, which shows how the convergence timet scales with τ ∈[1,500]s for N =300. It is possible to appreciate a kind
c s
of power-law scalingt (cid:39)τγ, with γ (cid:39)0.5. This indicates that the convergence dynamics are mostly determined by τ , while
c s s
τ plays a relatively minor role, hence confirming the above mentioned resemblance with coarsening on lattices or random
m
geometricnetworks. Similarlytofully-connectednetworks[4],log-periodicoscillationsarevisibleinthepowerlawscaling,so
thatforsomevaluesofτ ,mobilityhappenstobemorerelevant,withlargeτ determiningalowerconvergencetime(seealso
s m
Fig.2top-right).
BelowthepercolationthresholdN ,agentsformtemporaryclustersthatdissolveduetotheagentmobility. Ifthedensityis
c
stillhighenoughtoensurefrequentinteractions(N>N ),thedynamicsaredeterminedmorebythemobilityofagentsthanby
1
thebroadcastingperiodτ . Thisisvisibleinthebottom-leftandbottom-rightpanelsofFig.2,whereconvergencetimestendto
s
coalescefordifferentvaluesofτ andN∈[N ,N ]. Instead,forverylowdensities(N<N ),agent-agentcontactsareinfrequent
s 1 c 1
andlastforshortperiodsoftime,sothatmanybroadcastsgounnoticed. Inthiscondition,highmobilityisimportantasmuchas
shortbroadcastingperiodstoensurefasterconvergence(seeFig.2).
Toevaluatetheeffectsofthebroadcastingperiodmorethoroughly,itisusefultolookattherescaledtimet /τ indicatingthe
c s
averagenumberofbroadcastseachagenttransmitted(seeFig.3right). Wenotelowervaluesoftherescaledagreementtimefor
largervaluesofτ ,meaningthatthenumberofbroadcastsrequiredforconvergencediminishesforlongerbroadcastingperiods,
s
recallingtheslower-is-fastereffectobservedinmanycomplexsystems[9].
AlookatthememoryrequirementsrevealsthatM isgenerallyconstrainedtolowvalues,whichmakestheNGimplemen-
tationaffordableforphysicalsystems(seetheinsetsinFig.2). ForN <N <N ,mobilityplaysasignificantrole,withlarger
1 c
4
N =300 102 τm=30s
τm N1 Nc τs
10 10
103 20 20
30 30
40 40
50 50
tc /τs
tc
5
M4
102 3
2 τs 101
100 101 102
100 101 102 101 102 103
τs N
Figure3: (Left)Scalingofconvergencetimeasafuncitonofτ . Theblacksolidlinet (cid:39)τ0.5 servesasaguidefortheeyeto
s c s
appreciatethepowerlawscalingoftheconvergencetimet . (Right)Rescalingconvergencetimebyτ ,representingtheaverage
c s
numberofbroadcastsbeforeconvergence. Statisticalerrorbarsarenotvisibleonthescaleofthegraphs.
valuesofτ correspondingtolargerM.Thetransientformationofsmallclustersenhancestherequirementsofmemorythemore
m
theagentsareabletotravelbetweenclustersthatagreeondifferentwords. Similarly,ifwelookatthebottompanels,wenotice
thathighervaluesofτ determinehighervaluesforM. Here,slowconvergenceleadstoagentsdiffusinginthearenaandbeing
s
exposedtomultipleoptions,henceincreasingthememoryrequirements. ForN>N ,instead,mobilityislessimportantandthe
c
memoryrequirementsareboundtotheinteractionperiod. ThescalinganalysispresentedintheleftpanelofFig.3showsthatthe
memoryrequirementsincreasedrasticallywithτ ,confirmingthatslowerconvergenceimpliesalsolargermemoryrequirements.
s
Ontheotherhand,frequentinteractionsleadtothequickformationoffewclusters,sothattheindividualmemoryrequirements
arelimitedtofewwords,especiallyforthoseagentsattheinterfacebetweenclusters. Asτ decreasesfurther,theeffectofcon-
s
currentexecutionsofgamesstartstobevisiblewithanincreaseinthememoryrequirementsasaresultofthehigherprobability
ofagentstosimultaneouslyexchangedifferentwords.
3.2 Influenceofphysicalinterferences
Theconsensusdynamicsdescribedaboverefertoanidealsystemthatneglectsthephysicalembodimentofrobots. Embodiment
leads to collisions with walls and among robots that constrain mobility. We study the influence of embodiment by comparing
multiagent with robotics simulations performed in similar conditions to what described above (see Fig. 4 for selected param-
eterisations). We first note that the convergence time t is in general higher for robotic simulations, as a consequence of the
c
slower diffusion in space resulting from the turning time, which introduces a stochastic delay in the random walk pattern, and
duetothephysicalboundariesthatpreventrobotstofreelymove. Forinstance,inthetop-leftpanelofFig.4weshowthecase
for τ =50s: here, collisions lead to an approximately constantt for N <N <N , no matter what is the broadcasting time
m c 1 c
τ . Indeed, the slower diffusion and the formation of small clusters determine the convergence time more than the interaction
s
frequency. Collisionswithwallsandwithotherrobotsleadtotheformationofstableclustersinwhichconsensuscanbequickly
achieved. Suchclustersdissolveataslowerpaceforlargervaluesofτ ,duetorobotsturningawaylessoften. Hence,theeffects
m
of mobility are diluted especially when it is supposed to play an important role, i.e., when N <N . Collisions influence the
c
convergencedynamicsalsoforN>N ,althoughtoalesserextent,ascanbeseeninthetop-rightpanelinFig.4: forlargeτ ,
c m
convergenceisslowerduetotheformationofclustersthatdonotinteractfrequently,ascollisionspreventrobotstomixasmuch
asintheidealmultiagentcase.
The low ability to mix due to collisions has an effect also on the required memory M, which is in general lower for robot
simulations (see bottom panels of Fig. 4). The slower diffusion of robots in space and the existence of boundaries limit the
spreadingofdifferentwordsintotherobotnetwork,henceresultinginlowermemoryrequirements.
3.3 Experimentswithrealrobots
To validate our results with respect to the real robotic platform, we performed comparative experiments in a smaller arena
(L=45cm). Inthiscondition, wehaveN (cid:39)6andN (cid:39)29, whichledustousesmallergroupsofrobots(N ∈{5,20,35})to
1 c
explorethesystembehaviourasthecharacteristicsofthestaticinteractionnetworkvary. Giventhesmallerdimensions,wealso
explored smaller values for the latencies τ and τ , and we decided to set both to the same value τ ∈{2.5,5,7.5}s. We have
s m a
performed20runswithrealrobotsforeachofthe9experimentalconditions(3groupsizes×3latencies),foratotalof180runs.
Figure5showsasequenceofframesfromonerunperformedwith20kilobots. Itispossibletonotethatinitiallymultiplesmall
clustersarepresentinwhichrobotshavethesameword(hererepresentedbythecoloroftheonboardLED).Astimegoesby,
clustersdisappearandeventuallyonesinglewordischosen.
For each experimental run, we have recorded the convergence time t , and the obtained results have been contrasted with
c
5
τ =50s τ =10s
m s
N N N N
1 c 1 c
103 103
tc
τ τ
s m
10 10
20 20
30 30
40 40
50 50
102 102
101 102 103 101 102 103
N
5.0 5.0
N N N N
1 c 1 c
4.5 4.5
4.0 4.0
M3.5 3.5
3.0 3.0
2.5 2.5
2.0 2.0
101 102 103 101 102 103
N N
Figure4: Comparisonbetweenmultiagentandrobotsimulations. Averagevaluesfrommultiagentsimulationsareshownwith
transparentcolours,andserveasreferencetoappreciatetheresultsfromroboticsimulations. Statisticalerrorbarsarenotvisible
onthescaleofthegraphs. (Top)Convergencetimet . (Bottom)RequiredmemoryM. (Left)Resultsforτ =50s,andvarying
c m
τ . (Right)Resultsforτ =10s,andvaryingτ .
s s m
simulationsincomparableconditions—samearenadimensionsLandsamelatenciesasdeterminedbyτ —withmultiagentand
a
roboticsimulations. Figure6showsthatthestatisticsarealignedbetweenkilobotsimulationsandrealkilobots,andbothpresent
aslightlylargerconvergencetimewithrespecttomultiagentsimulations. Wealsonotethatingeneral,theconvergencetimefor
τ =2.5s is lower in case of real robots than in simulation, while this is not always true for larger latencies. This is an effect
a
of interferences in communication due to simultaneous broadcasts, which leads to the loss of some communication messages.
When messages get lost, the convergence dynamics are actually faster because the exchanging of different words by robots
broadcasting at the same time gets reduced. This is in line with the observations made in Section 3 about the influence of the
broadcastingperiod,indicatingthatconvergenceisfasterwhentherearelessbroadcasts.Withkilobots,areductionofthenumber
ofbroadcastsduetointerferenceresultsfromsmallτ andlargeN (seeFig.6).
s
4 Conclusions
Thisstudyfindsitselfattheinterfacebetweentheoreticalinvestigationsandroboticsimplementation.Theresultsobservedinthe
multiagentsimulationscanbeofinterestforcomplexsystemsstudiesastheyhighlighttheeffectsofconcurrencyandofdifferent
latencies in the motion and interaction patterns, as determined by implementation constraints. Concurrency is customary in
t=10s t=23s t=45s t=60s t=80s
Figure5: Differentshotsofanexperimentwith20kilobots,withτ =τ =2.5s. RobotslightingtheirLEDhaveonlyoneword
s m
in their inventory, while no color signal indicates more than one word or an empty inventory. Different words correspond to
differentcolours.
6
N=5 N=20 N=35
tc110023 lllllllllll llllllll llllll llll l lllllll lllllllll l llllllllllll llllllllllllllll lll lllllllllll ll lllllllll llllll ll llllll llllllllll l lll llllll lllll lllllll l
10
t =5 t =7.5 t =2.5 t =5 t =7.5 t =2.5 t =5 t =7.5
1 t a=2.5 a a a a a a a a
multiagent kilobot sim. kilobot real
Figure6: Comparisonbetweenmultiagentsimulations,kilobotsimulationsandrealkilobots. Foreachcondition,200runswere
performedinsimulation,while20runswereperformedwithphysicalrobots. Boxesrepresenttheinterquartilerange,horizontal
linesmarkthemedian,whiskersextendto1.5timesthefirstquartiles,anddotsrepresentoutliers.
multi-robot systems and artificial decentralised systems in general. Hence, accounting for it into abstract models is important
to provide usable predictions. We have found here that concurrent executions of games are particularly important for aspects
likethemaximummemoryM,andfuturestudiesshouldbettercharacterisesucheffectsintermsoftheprobabilityofobserving
concurrentexecutionsatanytime.
Roboticssimulationsandexperimentswithkilobotsshowedhowembodimentinfluencestheconsensusdynamicsbylimiting
thediffusionofinformationintothesystem: ontheonehand,collisionsleadtotheformationofclustersthatdissolveslowerfor
largerτ , leadingtoslowerconvergencetimes. Ontheotherhand, thememoryrequirementsofrobotsisreducedasonlyfew
m
robotsattheinterfacebetweenclustersexperiencemorethantwowordsatthesametime. Futurestudiesshouldattemptamore
precisedescriptionofthediffusivemotionofagentsunderphysicalconstraints,inordertoobtainbetterpredictionsintermsof
theexpectedinteractionnetwork.Additionally,thecommunicationprotocolemployedbykilobotsandtheobservedinterferences
needtobebettercharacterised. Simulationsshouldaccountforuncertainreceptionofmessages,aswellasfortheexponential
back-offusedduringtransmissionwhenthechannelisbusy. Byincludingsuchfeatures,weexpecttodeliverpreciseestimations
ofthesystembehaviourevenforverylargegroupsizesandshortbroadcastingperiods.
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