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V-FormationasOptimalControl
AshishTiwari
SRIInternational,MenloPark,CA,USA
BDA,July25th,2016
JointworkwithJunxingYang,RaduGrosu,andScottA.Smolka
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
V-Formation
• FlocksofbirdsorganizethemselvesintoV-formations
Eurasian Cranes migrating in a V-formation (HamidHajihusseini, Wikipedia)
Reason:Savesenergyasbirdsbenefit fromupwashregion;providesclearvisualfieldwithvisibilityoflateralneighbors
ReachingaV-Formation
• Rule-basedApproach:ØCombinationsofdynamicalflightrules asdrivingforcesØNotcompletelysatisfying
• ViewasaDistributedControlProblem:ØFlockwantstogetintoanoptimalconfigurationthatprovidesbestview,energybenefit,andstability
• OurApproach:ØUsesModel-Predictive Control(MPC)ØWhichusesParticle-SwarmOptimization(PSO)
Reynolds’Rules
Reynolds (1987)presentedthreerulesforgeneratingV-formations:
Alignment Cohesion SeparationAlignment:steertowardstheaverageheadingoflocalflockmatesCohesion:steertomovetowardtheaveragepositionoflocalflockmatesSeparation:steertoavoidcrowdinglocalflockmates
ExtendedReynoldsModel
Reynolds’modelwasextendedbyadditionalrules:• Arulethatforcesabirdtomovelaterallyawayfromanybirdthatblocksitsview(Flake(1998)).
• Dragreductionrule:computingthe induceddraggradientandsteeringalongthisgradient(Dimock&Selig(2003)).
Nathan&Barbosa’smodel(2008):• Coalescing:seekproximityofnearestbird• Gap-seeking:seeknearestpositionaffordingclearview• Stationingrule:movetoupwashofaleadingbird
ARule-basedAttempt
DesignedrulesthatgenerateaV-formation• Drivebirdstowardstheoptimalupwashpositionw.r.t.thenearestbirdinfront;unsatisfactory solution
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
TheV-FormationProblem
Assumeageneric2-ddynamicmodelofaflockof
birds
xi(t+1)=xi(t)+vi(t+1)
vi(t+1)=vi(t)+ai(t)
Goal:findbestaccelerationsai(t) ateachtimestep
thatwillfinallyleadtoaV-formation.
Thisisadistributedcontrolproblem
WhatisaV-Formation?
Wewantaformationthatachievestheoptimumvaluesforthefollowingthreefitnessmetrics:1. VelocityMatching2. ClearView3. UpwashBenefit
VelocityMatching(VM)
s=stateofthen-birds=npositions,nvelocitiesVM(s)=normalizedsumofpairwisevelocitydifferenceVM(s)=0ifallbirdshavethesamevelocityVM(s)increasesasthevelocitiesgetmoremismatched
VMisminimizedwhenallbirdshaveequalvelocity.
Velocitynotmatched Velocitymatched
ClearView(CV)
• Accumulatethepercentageofthebird’sviewthatisblocked• CV(s)=0ifeverybirdhasa100%clearview• CV(s)increasesasmoreoftheviewofanybirdisblocked
(b) i’s view is completelyblocked by j and k.Clear view: 1
UpwashBenefit(UB)
• AGaussian-likemodelofupwash anddownwash• UB(s)=sumofupwash benefiteachbirdgetsfromeveryother• UB(s)=1ifn-1birdsgetsmaxpossibleUBbenefit• UB(s)increasesasbirdsgetlesserupwash benefit
FitnessFunction
Fitnessofastateisasum-of-squares
combinationofVM,CVandUB
F(s)=(VM(s)-VM(s*))2+(CV(s)-CV(s*))2+(UB(s)-UB(s*))2
• stateachieving optimal fitnessvalue (i.e.,aV-
formation)
TheV-FormationProblem
Assumeageneric2-ddynamicmodelofaflockof
birds
xi(t+1)=xi(t)+vi(t+1)
vi(t+1)=vi(t)+ai(t)
Goal:findbestaccelerationsai(t) ateachtimestep
thatwillfinallyleadtoastatewithminimumF(s)
Thisisadistributedcontrolproblem
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
ModelPredictiveControl(1)
Ateachtimet,considerhowthemodelwillbehaveinthe
nextTstepsunderdifferentchoicesforthecontrolinputs
• Useamodel thatrepresentsthebehavioroftheplant
Useanoptimizationsolvertofindthebestcontrolinputs
overthisfinitepredictionhorizon
Onlyapplythefirstoptimalcontrolaction
Repeatatt+1
ModelPredictiveControl(2)
• Attimet+1,updatemodelstatewithnewmeasurementsoftheplant.• Repeattheoptimizationwithnewstates.
AdiscreteMPCscheme(Wikipedia):horizon=p,currenttime=k
ModelPredictiveControlforV-Formation(1)
Birdi attimetsolvesthefollowingoptimizationproblem:
a*i(t),…,a*i(t+T-1)=argmin ai(t),…,ai(t+T-1)F(sNi(t+T-1))
• sNi(t) :stateattimet consistingofpositionsandvelocitiesofbird’sneighbors
• Centralized controlifNi includesallbirds• F:fitness function.• T:predictionhorizon.
ModelPredictiveControlforV-Formation(2)
• Subjecttoconstraints:• Modeldynamics:Stateupdatesofeachbirdare
governedbythemodeldynamics
• Boundedvelocitiesandaccelerations:Thevelocitiesareupper-bounded byaconstant,andtheaccelerationsare
upper-bounded byafactorofthevelocities
• Finally,birdi usestheoptimalaccelerationforbird
itfoundfortime.
ParticleSwarmOptimization(1)
TheoptimizationproblemissolvedusingPSO• Inspiredbysocialbehaviorofbirdflockingorfishschooling.
• Initializeapopulation(swarm)ofcandidatesolutions(particles)thatmovearoundinthesearch-space.
• Eachparticlekeepstrackofthebestsolutionithasachievedsofar(pbest)andthebestsolutionobtainedsofarbyanyparticleintheneighborsoftheparticle(gbest).
ParticleSwarmOptimization(2)
• Repeatedlyupdatetheparticle’svelocityandpositionby:vi(t+1)=wvi(t)+c1 r1 (pbesti– xi(t))+c2 r2 (gbesti – xi(t))
xi(t+1)=xi(t)+vi(t+1)
where
w:inertiaweightr1,r2 :randomnumbersin(0,1)sampledeveryiterationc1,c2 :constantlearningfactors
• Terminatewhenmaximumiterationsordesiredfitnesscriteriaisattained.
DistributedMPCProcedure
Ateverytimestep:• EachbirdlooksatitsneighborsØPlaysseveralscenariosinitsheadtofindthebestconfiguration thattheneighborhoodcanreachin3 steps
ØThebirdthenappliesthefirstmove ofthatsolutiontoupdateitsposition
Inthenexttimestep,eachbirdupdatesitsknowledgeoftheneighbors (positionsandvelocities),whichmaynotbethesameofwhatthatbirdpredictedforitsneighbors
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
ExperimentalResults(1)
ExperimentalResults(2)
Outline
• Introduction
• TheV-FormationProblem
• ModelPredictiveControlforV-Formation
• ExperimentalResults
• Conclusions&FutureWork
Conclusions
• Usedistributedcontrolinsteadofbehavioralrulesto
achieveV-formation.
• IntegrateMPCwithPSOtosolvetheoptimization
problem.
OngoingandFutureWork
• Deploytheapproachtoactualplants(drones).
• Collisionavoidance.
• ImprovesuccessrateofconvergingtoV-formation.
• UseSMCtoquantifytheprobabilityofsuccess.
• Energyconsumptionandleaderselection.
Thankyou!
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