Dynamically stable control of articulated crowds

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Mukovskiy, Albert, Jean-Jacques E. Slotine, and Martin A. Giese.

“Dynamically Stable Control of Articulated Crowds.” Journal of

Computational Science 4, no. 4 (July 2013): 304–310.

As Published

http://dx.doi.org/10.1016/j.jocs.2012.08.019

Publisher

Elsevier

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Author's final manuscript

Citable link

http://hdl.handle.net/1721.1/103869

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Creative Commons Attribution-NonCommercial-NoDerivs License

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ContentslistsavailableatSciVerseScienceDirect

Journal

of

Computational

Science

jo u r n al hom ep a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o c s

Dynamically

stable

control

of

articulated

crowds

Albert

Mukovskiy

a,∗

_{, Jean-Jacques}

_E.

_Slotine

b

_,

_Martin

_A.

_Giese

a

a_Section_for_{Computational}_{Sensomotorics,}_Department_of_Cognitive_Neurology,_Hertie_Institute_for_Clinical_Brain_Research_&_Center_for_Integrative_{Neuroscience,}_University_Clinic

Tübingen,Germany

b_Nonlinear_Systems_Laboratory,_Department_of_Mechanical_Engineering,_MIT,_Cambridge,_MA,_USA

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received1August2011

Receivedinrevisedform12July2012 Accepted27August2012 Availableonlinexxx Keywords: Computeranimation Crowdanimation Multi-agentcoordination Distributedcontrol Dynamicstability Self-organization

a

b

s

t

r

a

c

t

Thesynthesisofrealisticcomplexbodymovementsinreal-timeisadifﬁcultproblemincomputer graph-icsandinrobotics.Highrealismrequirestheaccuratemodelingofthedetailsofthetrajectoriesfora largenumberofdegreesoffreedom.Atthesametime,real-timeanimationnecessitatesﬂexible sys-temsthatcanadaptandreactinanonlinefashiontochangingexternalconstraints.Suchbehaviors canbemodeledwithnonlineardynamicalsystemsincombinationwithappropriatelearningmethods. Theresultingmathematicalmodelshavemanageablemathematicalcomplexity,allowingtostudyand designthedynamicsofmulti-agentsystems.WeintroduceContractionTheoryasatooltotreatthe sta-bilitypropertiesofsuchhighlynonlinearsystems.Foranumberofscenarioswederiveconditionsthat guaranteetheglobalstabilityandminimalconvergenceratesfortheformationofcoordinated behav-iorsofcrowds.Inthiswaywesuggestanewapproachfortheanalysisanddesignofstablecollective behaviorsincrowdsimulation.

1. Introduction

The generationof realistic interactivehumanmovements is a difﬁcult taskwith highrelevance for computergraphics and robotics.Applicationssuchascomputergamesrequireonline syn-thesisofsuchmovements,atthesametimeprovidinghighdegrees ofrealismevenforcomplexbodymovements.Whilefortheoff-line synthesisofhumanmovements,themovementscanberecorded off-lineandretargetedtotherelevantkinematicmodel,this pro-cedure is not possible for online synthesis. Approaches based onphysicalor dynamicalmodels (e.g.[1])havefocused onthe simulationofsceneswithmanyinteractingagentsthatnavigate autonomouslyandshowinterestingcollectivebehaviors.Dueto thecomplexityoftheunderlyingmathematicalmodels,such sys-temsaretypicallydesignedinaheuristicmanner.Opposedtoother applicationsinengineering,thesystemdynamicsofsuchcomputer animationsystemsisusuallynotanalyzed,sothatrobustnessor stabilityguaranteesforthesystemdynamicscannotbegiven.

Inthis paperwe presentﬁrststepstowardthedevelopment of more systematic method for the design of the dynamics of

∗ Correspondingauthor.

E-mailaddresses:albert.mukovskiy@medizin.uni-tuebingen.de(A.Mukovskiy),

jjs@mit.edu(J.-J.E.Slotine),martin.giese@uni-tuebingen.de(M.A.Giese).

URLs:http://www.compsens.uni-tuebingen.de(A.Mukovskiy),

http://web.mit.edu/nsl/ (J.-J.E. Slotine), http://www.compsens.uni-tuebingen.de

(M.A.Giese).

interactive crowds. For this purpose, we approximate human movementsbyrelativelysimplemathematicalmodels,combining dynamicalmodelswithappropriatelearningmethods.Inaddition, weintroduceContractionTheory[2]asanewtoolforthestability designofcomplexnonlineardynamicalsystems.Wedemonstrate howthisapproachcanbeappliedforthestabilityanalysisofgroups ofautonomouslynavigatingcharacterswithfullbodyarticulation. Thecurrentresearchextendsourpreviouswork[3]bythe devel-opmentofstabilityanalysisformorecomplexscenarios,including thecontrolofheadingdirectionandoftheconsensusbehaviorof crowds.

Thepaperisstructuredasfollows:Thelearning-based dynam-icalmodelforcomplexhumanmovementsisbrieﬂysketchedin Section3.Section4describestherelevantcontroldynamicsthatis requiredtorealizecomplexinteractionsbetweendifferent char-acters in a crowd. In Section5 we review basicconcepts from ContractionTheory.Themajorresultsofourstabilityanalysisand somedemonstrationsoftheirapplicationtocrowdanimationare presentedinSection6,followedbysomeconclusions.

2. Relatedwork

Dynamicalsystemshavebeenfrequentlyappliedincrowd ani-mationforthesimulationofautonomouscollectivebehaviors(e.g. [4,5]).Someofthisworkwasinspiredbyobservationsinbiology showingthatcoordinatedbehavioroflargegroupsofagents,such asﬂocksofbirds,canbemodeledasemergentbehaviorsarising

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2 A.Mukovskiyetal./JournalofComputationalSciencexxx(2012)xxx–xxx

byself-organizationfromdynamicalcouplingsbetweeninteracting agents[6–8].

Simplerulesforsuchdynamicinteractionshavebeenderived fromtheobservationofthemotionofflocksofbirds[9],suchas: collisionavoidancebynearestneighbordistancecontrol,velocity matchingandflockcentering.Theobtainedmodelsforthe self-organization of behavior have beenanalyzed by application of methodsfromnonlineardynamics[10].Theyareinterestingfor computeranimationbecausetheypermittoreducethe compu-tationalcostsoftraditionalcomputeranimationtechniques,such asscripting orpathplanning[5].Inaddition, thegenerationof collectivebehavior by self-organization hasthe advantage that thesystemcanadaptautonomouslytoexternalperturbationsor changesinthesystemarchitecture,suchasthevariationofthe numberofcharacters[7].However,themathematicalanalysisof theunderlyingdynamicalsystemsistypicallyquitecomplicated. Thedynamicsdescribingindividualagentsistypicallyhighly non-linear[11],makingasystematictreatmentofstabilityproperties ofteninfeasible,evenforindividualcharacters.Inaddition,crowd animationrequiresthedynamicinteractionofmanysuchagents. Thecontrolofcollectivebehaviorsofgroupsofagentshasbeen treatedinmathematicalcontroltheory[12,13],howevertypically assuminghighlysimplifiedoftenevenlinearmodelsfortheagents. Groupcoordinationandcooperativecontrolhavealsobeen stud-iedinrobotics,e.g.inthecontextofthenavigationofgroupsof vehicles[14,15],orwiththegoaltogeneratecollectivebehaviorby self-organization.Examplesarethespontaneousadaptationto per-turbationsofinter-agentcommunicationorchangesinthenumber ofagents[16,17].Manyapproacheshaveonlyanalyzedasymptotic stabilityforconsensusscenarios,whileexponentialstabilitywhat alsoallowstoderiveboundsfortherateofconvergencehasrarely beentreated[15].Thesameresultcanbederivedbyapplicationof ContractionTheory[18],amethodforthederivationofconditions fortheuniformexponentialconvergenceofcomplexnonlinear sys-tems.Opposedtoclassicalstabilityanalysisfornonlinearsystems, ContractionTheorypermitstore-usestabilityresultsforsystem componentsinordertoderiveconditionsthatguaranteethe sta-bilityoftheoverallsystem.Itprovidesausefultoolspecificallyfor modularity-basedstability analysisanddesign.Contraction The-oryhasbeenalreadysuccessfullyimplementedforsynchronization ofDMPs(dynamicmovementprimitives)controllingUnmanned AerialVehicles[19].Butthe stability propertiesfor crowd ani-mationsystemsrealizinghumanbehaviorswithrealisticlevelsof complexityhavebasicallyneverbeentreatedbefore.

3. Systemarchitecture

The proposed new method for the design of the collective dynamics of interacting crowds is based on a learning-based approachforthemodelingofhumanmovementsusingdynamic movement primitives [20] (cf. Fig.1). For a relevant class of movements,likegaitswithdifferentstylesorstraightvs.curved locomotion,alow-dimensionalrepresentationintermsofasmall numberofbasiccomponentsislearnedusinganalgorithmfor ane-choicdemixing[21].Weshowedelsewherethatthismethodallows toapproximatetrajectoriesbyverysmallnumberofsourcesignals, outperformingotherdimensionreductionalgorithms,suchasICA orPCA[22,21].

Inordertogeneratethelearnedsourcesignalsonline,thestable solutionsofanonlineardynamicalsystem(dynamicprimitive)are mappedontotheformofthesourcefunctions.Forthesynthesis ofgaitpatternaprimitiveisnaturallymodeledbyoscillator[23]. Themappingfromthephasespaceofthedynamics,deﬁnedbythe statevariablesy=[y, ˙y]T,ontothevaluesofthesourcefunctions sjwaslearnedfromtrainingdatausingSupportVectorRegression

withgaussiankernel(seeRef.[20]fordetails).Fortheexamples presentedinthispapereachcharacterwasmodeledbyasingle Andronov–Hopfoscillator.Thegeneratedsourcesignalswerethen linearlycombinedwiththelinearweightswijandphasedelaysij inordertogeneratethejointangletrajectoriesi(t)accordingto thelearnedanechoicmixturemodelthatwasgivenbytheequation: i(t)=

j wijsj

t−ij

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Thecompletereconstructionofthetrajectoriesrequiresthe addi-tionoftheaveragejointanglesmi,whichalsowerelearnedfrom thetrainingdata.

Forthespecialcasewherethedynamicprimitivesaregivenby Hopfoscillators,whoselimitcycleforappropriate choiceofthe coordinatesystemisacirculartrajectoryinphasespace,thephase delayscanbeabsorbedinaninstantaneousorthogonalmapping (rotation)Mij inthephase planeoftheoscillator(Fig.1b).This allowsto derivedynamicsfor online synthesiswithout explicit delays,whichwouldgreatlycomplicatethesystemdynamics.(See Ref.[20]forfurtherdetails.)

Byblending of themixingweights wij and thephase delays ij,intermediategaitstylescanbegenerated.Thistechniquewas appliedtogeneratewalkingalongpathswithdifferentcurvatures, changesinsteplength,andemotionalgaitstyles.Interactive behav-iorofmultiplecharacterscanbemodeledbymakingthestatesof theoscillatorsandthemixingweightsdependentonthe behav-ioroftheothercharacters.Suchcouplings,whicharediscussedin moredetailinthenextsection,resultinahighlynonlinearoverall systemdynamics.Weshowedelsewherethatthesame architec-turecanbeappliedalsoforthegenerationofothertypesofbody motionthan locomotion[20],while suchexamplesarenot dis-cussedinthispaper.Thewalkingdirectionofthecharactersisalso changedbyinterpolationbetweenstraightwalkingandwalking alongcurvedpathstotheleftortotheright.Theparametersused forblending are describedin ourpreviouspublications [20,24]. Suchblendingisusedtosimulateacontroloftheheading direc-tionsinconsensusscenariosandforobstacleavoidanceduringthe autonomousreorderingofcrowds.Fortheimplementationof reac-tivelocalobstacleavoidanceweusedadynamicnavigationmodel thatoriginallywasdevelopedinrobotics[25].SeeRefs.[24,20]for detailsconcerningtheimplementation.

4. Controldynamics

Flexiblecontrolofthelocomotionofarticulatingagentsrequires thecontrolof multiplevariables, specifyinga controldynamics withmultiplecoupledlevels.Fortheexamplesdiscussedinthis paperoursystemincludedthecontrolofthefollowingvariables: (1)phasewithinthestepcycle,(2)steplength,(3)gaitfrequency, and(4)headingdirection.Thecontrolofstepphasewas accom-plishedby couplingof theAndronov–Hopf oscillators [26] that correspondto differentagents, resultingin phase synchroniza-tion.Theseoscillatorshaveastablelimitcyclethatcorrespondsto anoscillationwithconstantamplitudeandthe(time-dependent) phase(t).Inabsenceofexternal couplingsthephaseincreases linearly,i.e.(t)=ωt+(0),whereωisthestableeigenfrequencyof theoscillator.Controlofstepfrequencywasaccomplishedby vary-ingthisparameterinatime-dependentmannerindependenceof thebehaviorofthecharactersinthescene.Step-lengthand direc-tionwerecontrolledbymorphingbetweengaitswithdifferentstep lengthsorpathcurvatures,blendingtheparametersoftheanechoic mixingmodel(seeabove).Inthiscasethecontrolledvariablesare theblendingcoefﬁcientofthesemixtures.(SeeRef.[20]fordetails.) The formulation of the systemdynamics in terms of speed control is simpliﬁed by theintroduction of thepositions zi for

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Fig.1.(a)Architectureofthesystemforreal-timesynthesisofcomplexhumanmovements.Solutionsofdynamicalsystems(primitives)aremappedontosourcesignals, thathavebeenderivedbyanechoicdemixingfromtrainingdata.ThesolutionsofthedynamicalsystemsaremappedbySupportVectorRegressions(SVR)ontothesource signals.Thesesourcesignalsarethencombinedusingthelearnedanechoicmixingmodeltogeneratejointangletrajectoriesonline,whichspecifythekinematicsofthe animatedcharacters.(b)Whenthedynamicprimitivesaremodeledbynonlinearoscillatorsthetimeshiftsoftheanechoicmixingmodelcanbeabsorbedinaninstantaneous orthogonaltransformationMij,avoidingadynamicswithexplicitdelays.

eachindividualcharacteralong itspropagationpath(seeFig.2). Thisvariablefulﬁllsthedifferentialequation ˙zi(t)= ˙ig(i),where thepositivefunctiongdeterminestheinstantaneouspropagation speed ofthe characterdependingonthephase withinthe gait cycle.Thisnonlinearfunctionwasdeterminedempiricallyfroma kinematicmodelofacharacter.Byintegrationofthispropagation dynamicsoneobtainszi(t)=G(i(t)+0_i)+ci,withaninitialphase shift0

i and someconstant ci dependingontheinitialposition andphaseofavatari,andthemonotonouslyincreasingfunction G(i)=

i

0 g()d,whereweassumeG(0)=0.

Inthefollowing wewillanalyzefourdifferentcontrolrules, whosecombinationallowstogeneratequiteﬂexiblelocomotion behaviorofacrowdofcharacters:

(I)Controlofstepfrequency:asimpleformofspeedcontrol resultsifthefrequencyoftheoscillators ˙iismadedependenton thebehavioroftheothercharacters.Assumingthatω0bethe equi-libriumfrequencyoftheoscillatorswithoutinteraction,thiscanbe accomplishedbythecontroldynamics:

˙ i(t)=ω0−md N

j=1 Kij[zi(t)−zj(t)−dij] (2)

Theconstantsdijspecifythestablepairwiserelativedistancesinthe ﬁnalformedorderforeachpair(i,j)ofcharacters.Theelementsof thecouplinggraph’sadjacencymatrixKdeterminewhether char-actersiandjareinteractingandthusdynamicallycoupled.These parametersweresettoKij=1,ifthecharacterswerecoupled,and

Fig.2. Variablesexploitedforspeedandpositioncontrol.Everycharacteriis char-acterizedbyitspositionzi(t),thephasei(t)andtheinstantaneouseigenfrequency

ωi(t)= ˙i(t)ofthecorrespondingAndronov–Hopfoscillator,andastep-sizescaling

parameteri(t).

theyarezerootherwise(withKii=0).Forexample,wechooseKij=1,

∀

i /= jforall-to-allcoupling,andKij=1,

∀

mod(|i−j|,N)=1forring coupling.Theconstantmd>0determinesthecouplingstrength.

With theLaplacian matrix Ld _of _the_coupling _graph _(that_is assumedtobestronglyconnected[14,27,28]),deﬁnedbyLd

ij=−Kij fori /= jandLd

ii=

N

j=1Kij,andtheconstantsci=−

N

j=1Kijdij,the lastequationsystemcanbere-writteninvectorform:

˙ =ω01−md(LdG(+0)+c) (3)

(II)Controlofsteplength:steplengthwasvariedbymorphing betweengaitswithshortandlongsteps.Adetailedanalysisshowed thattheinﬂuenceofsteplengthonpropagationspeedcouldbe wellapproximatedbysimplelinearrescaling.Ifthepropagation velocityofcharacteriis

v

i(t)= ˙zi(t)= ˙i(t)g(i(t))=ωi(t)g(i(t)) forthenormalstepsize,thenthevelocityformodiﬁedstepsize couldbeapproximatedby

v

i(t)= ˙zi(t)=(1+i)ωi(t)g(i(t))with themorphingparameteri.Theempiricallymeasuredpropagation velocityasfunctionofgaitphaseisshowninFig.3(a)fordifferent valuesofthesteplengthparameteri.

In order torealize speed controlby steplengththe morph-ingparameteriwasmadedependentonthedifferencebetween

Fig.3.(a)Propagationvelocityfordifferentvaluesofthestep-lengthmorphing parameter(=[0...0.25])asfunctionofthegaitcyclephase.Empirical esti-matesarewellapproximatedbyalinearrescalingofthepropagationspeedfunction deﬁnedabove ˙z≈(1+)g(),forconstantω=1.(b)Theheadingdirectioncontrol dependsonthedifferencebetweentheactualheadingdirection heading_and_the

goaldirection goal_._Movement_along_parallel_lines_was_modeled_by_deﬁning_‘sliding

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actual and desired position differences dij between the agents i=−mz

N

j=1Kijz[zi(t)−zj(t)−dij],resultinginthecontrolrule: ˙zi(t)=ωi(t)g(i(t))(1−mz

N

j=1

K_ijz[zi(t)−zj(t)−dij])

withtheconstant couplingstrength mz>0. Here theadjacency matrixKz_of_the_coupling_graph_corresponds_to_the_Laplacian_matrix Lz_(according_to_the_equivalent_{relationships}_as_speciﬁed_above)._In vectornotationthedynamicsforthecontrolofspeedbysteplength canbewritten:

˙z=ωg(+0)(1−mz(Lzz+c)) (4)

(III)Controlofstepphase:bydeﬁningseparatecontrolsforstep lengthandstepfrequencythepositionandstepphaseofthe charac-terscanbevariedindependently.Thismakesitpossibletosimulate arbitraryspatialpatternsofcharacters,atthesametime synchro-nizingtheirstepphases.Theadditionalcontrolofstepphasecan beaccomplishedbysimpleadditionofalinearcouplingterminEq. (3):

˙=ω01−md(LdG(+0)+c)−kL (5) withk>0andtheLaplacianL_._(All_sums_or_differences_of_angular variableswerecomputedbymodulo2.)

(IV)Controlofheadingdirection:thecontroloftheheading directions iofthecharacterswasbasedondifferentialequations thatspecifyattractorsforgoaldirections _igoal,whichwere com-putedfrom‘slidinggoals’thatwereplacedalongstraightlinesat ﬁxeddistancesin front ofthecharacters (Fig.3b).Theheading dynamicswasgivenbyanonlineardifferentialequation, indepen-dentlyforeverycharacter[20]:

˙_i₌_ω_i_(t)(_−m _sin(_i₋ goal

i )+g (i(t)+ 0

i)) (6)

where goal_i =arctan(goal_i /zgoal_i ), withgoal_i specifyingthe distancetothegoallineorthogonaltothepropagationdirection and z_igoal being a constant (Fig.3b).The first termdescribes asimple directioncontrollerwhose gainis definedbythe con-stantm >0.Thesecondtermapproximatesoscillationsofheading direction,whereg isagainanempiricallydeterminedperiodic function.Controlisrealizedbymakingthemorphingcoefficients thatdeterminethecontributionsofleftvs.right-curvedwalking dependentonthechangerate ˙ioftheheadingdirection.

Themathematicalresultsderivedinthefollowingsectionsapply tosubsystemsderivedfromthecompletesystemdynamicsdeﬁned byEqs.(4),(5)and(6).Inaddition,simulationswillbepresented thatillustratetherangeofbehaviorsthatcanbemodeledbythe fullsystemdynamics.

5. ElementsfromContractionTheory

Thedynamicalsystemsforthemodelingofthebehaviorofthe autonomouscharactersderivedinthelastsectionareessentially nonlinear.In contrasttolineardynamicalsystems,amajor dif-ﬁcultyoftheanalysisofsuchnonlinearsystemsisthatstability propertiesofsystemspartsusuallydonottransfertocomposite systems.Contractiontheory(CT)[2]providesageneralmethodfor theanalysisofessentiallynonlinearsystemsthatpermitssucha transfer.Thismakesitsuitablefortheanalysisofcomplexsystems thatarecomposedfromcomponents.CTcharacterizesthesystem stabilitybythebehaviorofthedifferencesbetweensolutionswith differentinitialconditions.Ifthesedifferencesvanishexponentially overtimeindependentfromthechoseninitialstatesthesystemis calledcontracting.Inthiscasethesystemisgloballyasymptotically

stable,thatisallitssolutionsconvergetoasingletrajectory inde-pendentfromtheinitialstate.Forageneraldynamicalsystemof theform

˙x=f(x,t) (7)

weassumethatx(t)isonesolutionofthesystemand ˜x(t)=x(t)+ ıx(t)aneighboringonewithdifferentinitialcondition.The func-tionıx(t)isalsocalledvirtualdisplacement.WiththeJacobianof thesystemJ(x,t)=

∂

f(x,t)/

∂

xitcanbeshown[2]thatanyvirtual displacementdecaysexponentiallytozeroovertimeifthe symmet-ricpartoftheJacobianJs=(J+JT)/2isuniformlynegativedeﬁnite, denotedasJs<0,i.e.hasnegativeeigenvaluesforallrelevantstate vectorsx.Inthiscase,itcanbeshownthatthenormofthevirtual displacementdecaysatleastexponentiallytozerofort→∞.Ifthe virtualdisplacementissmallenough,then

d dtıx(t)=J(x,t)ıx(t) implies through _dtd||ıx(t)||2₌_2ıxT_(t)J s(x,t)ıx the inequality: ||ıx(t)||≤||ıx(0)||e

t

0max(Js(x,s))ds_._The_decay_of_the_virtual displace-ment occurs thus with a convergence rate (inverse timescale) that is bounded from below by the contraction rate c= −sup

x,t

max(Js(x,t)),wheremax(.)signiﬁesthelargesteigenvalue. Thishastheconsequencethatalltrajectoriesconvergetoasingle solutionexponentiallyintime[2].

Contractionanalysiscanbeappliedtohierarchicallycoupled sys-temsthataregivenbythedynamics

d dt

x1 x2

=

f1(x1) f2(x1,x2)

(8) where the ﬁrst subsystem is not inﬂuenced by the state of thesecond.ThecorrespondingJacobianF=

F11 0 F21 F22

implies for the dynamics of the virtual displacements: d

dt

ıx1 ıx2

=

F11 0 F21 F22

ıx1 ıx2

.IfF21isbounded,thentheexponential con-vergenceoftheﬁrstsubsystem,(followingfrom(F11)s<0),implies thusconvergenceofthewholesystem,ifinaddition(F22)s<0.This followsfromthefactthetermF21ıx1isjustanexponentially decay-ingdisturbanceforthesecondsubsystem.(SeeRef.[2]fordetails ofproof.)

Inpracticalapplicationsmanysystemsarenotcontractingwith respecttoalldimensionsofthestatespace,butrathershow con-vergenceonlywithrespecttoasubsetofdimensions.Thisbehavior canbemathematicallycharacterizedbypartialcontraction[18,28]. Theunderlyingideaistheconstructionofanauxiliarysystemthatis contractingwithrespecttoasubsetofdimensions(orsubmanifold) instatespace.Themajorresultisthefollowing[18]:

Theorem1. (Partialcontraction)Consideranonlinearsystemofthe form ˙x=f(x,x,t)andtheauxiliarysystem ˙y=f(y,x,t).Ifthe auxil-iarysystemiscontractingwithrespecttoyuniformlyforallrelevant xthentheoriginalsystemiscalledpartiallycontracting.Thisimplies thatifaparticularsolutionoftheauxiliarysystemveriﬁesaspeciﬁc smoothpropertythenalltrajectoriesoftheoriginalsystemalsoverify thispropertywithexponentialconvergence.

A ‘smoothproperty’ is a property ofthe solutionthat depends smoothlyonspaceandtime (assumingtherelevantderivatives orpartialderivativesexistand arecontinuous),suchas conver-genceagainstaparticularsolutionoraproperlydeﬁneddistance tosubmanifoldinphasespace([18]).

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Itthusissufficienttoshowthattheauxiliarysystemis contract-ingtoproveconvergencetoasubspace.Iftheoriginalsystemhas aflow-invariantlinearsubspaceM,whichisdefinedbythe prop-ertythattrajectoriesstartinginthisspacealwaysremaininit(

∀

t: f(M,t)⊂M),andassumingthatthematrixVisanorthonormal projectionontoM⊥_,_then_a_sufﬁcient_condition_for_global expo-nentialconvergencetoMisgivenby[27,28]:

V

∂

f

∂

x

s VT<0, (9)

whereA<0againindicatesthatthematrixAisnegativedeﬁnite. Finally,weintroduceherea theoremthat providessufﬁcient conditions for synchronization of a network that is composed fromNidenticaldynamicalsystemsthatcommunicatethrougha commonmediumorchannelwithstatevariable.Therelevant dynamicsisgivenby

˙x₌f(x,,t),

˙=g(, (x),t) , (10)

xcontainingthestatevariablesoftheindividualsystemsandall componentsoffhavingthesameformf.ExploitingthelastPartial contractiontheoremthefollowingresultcanbederived[29]: Theorem2. (Quorumsensing)Ifthereducedordervirtualsystem

˙y=f(y,,t)iscontractingforallrelevantthenallsolutionsofthe originalsystemconvergeexponentiallyagainstasingletrajectory,i.e. |xi(t)−xj(t)|→0ast→+∞.

6. Results:stabilityconditionsforcrowdcontrol

In the following we derive stability conditions for the for-mationofcoordinatedbehaviorofcrowds,providingcontraction boundsforfourscenarioscorrespondingtocontrolproblemswith increasinglevelsofcomplexity.Correspondingcrowdbehaviorsare illustratedbydemomoviesthatareprovidedassupplementsfor themanuscript.

(1)Controlofstepphasewithoutpositioncontrol:thissimple controlrulepermitstosimulatestepsynchronization,asinthecase ofagroupofsoldiers[28],[Demo1_]._The_dynamics_for_this_case_is

givenbyEq.(5)withmd=0(omittingthepositioncontrolterm).For Nidenticaldynamicalsystemswithsymmetricidenticalcoupling gainsKij=Kji=kthedynamicscanbewritten

˙xi=f(xi)+k

j∈Ni

xj−xi

,

∀

i=1,...,N (11)

whereNideﬁnestheindexsetspecifyingtheneighborhoodinthe couplinggraph,i.e.theothercharactersthataredirectly interac-tingwithcharacteri.Thesystemcanberewrittencompactly: ˙x= f(x,t)₋kLxwiththeconcatenatedphasevariablex₌[xT

1,...,xTN] T_. ThematrixL=LG

IpisderivedfromtheLaplacianmatrixofthe couplinggraphLG,wherepisthedimensionalityoftheindividual sub-systems(Ipistheidentitymatrixofdimensionp,and

signi-ﬁestheKroneckerproduct).TheJacobianofthissystemisgivenby J(x,t)=D(x,t)-kL,wheretheblock-diagonalmatrixD(x,t)contains theJacobiansoftheuncoupledcomponents_∂∂_xf(xi,t).

Thedynamicshasaﬂow-invariantlinearsubspaceMthat con-tainstheparticularsolutionx∗₁=···=x∗_n.Forthissolutionallstate variablesxiareidenticalandthusinsynchrony.Inthiscase,the couplingterminEq.(11)vanishes,sothattheformofthe solu-tionisidenticaltotheoneofanuncoupledsystem ˙xi=f(xi).IfV isa projectionmatrix ontotheinvariantsubspaceM⊥_,_then_by Eq.(9)thesufﬁcientconditionforconvergencetowardMisgiven

1_{http://www.uni-tuebingen.de/uni/knv/arl/avi/ct2012/video0.avi}

byV(D(x,t)− kL)sVT<0[28].Thisimpliesmin

V(kL)sVT

=k+_L > sup x,t

max(Ds),with+L beingthesmallestnon-zeroeigenvalueof

symmetricalpartoftheLaplacianLs.(Forstronglyconnected cou-plinggraphsallthenonzeroeigenvaluesof Ls arerealpositive, due to theGershgorin’s disctheorem [30].)The sufﬁcient con-dition forglobal stability ofthe overallsystemis given byk> sup x,t max

_∂_f ∂x(x,t)

/+_L. This implies the minimum convergence rate:c=−sup

x,t

max(V(D(x,t)−L)sVT).Differenttopologiesofthe couplinggraphsresultin differentLaplacians andthus stability conditions.Forexample+_L =2(1−cos(2/N))forsymmetric cou-plingoftheringtopology,and+_L =Nforall-to-allcoupling,where Nisthenumberofavatars.(SeeRefs.[18]and[28]formoredetails.) IntheparticularcaseofEq.(11)withmd=0,forthephasecoupling ofHopfoscillatorswithxi=i,wehavef(xi)=ω0=constandthe contractionconditionbecomesk+_L >0withtheuniform contrac-tionratec=k+_L fork>0.

(2)Speedcontrolbyvariationofstepfrequency:

thedynamicsofthisscenarioisgivenbyEqs.(3)and(4)for mz=0.Assumingarbitraryinitialdistancesandphaseoffsetsof dif-ferentpropagatingcharacters,implyingbyG(0

i)=cithatci /=cj, fori _/=j,weredeﬁnedijasdij−(ci−cj)inEq.(2),andaccordingly cinEq.(3).Furthermore,weassumeforthisanalysisascenario wherethecharactersfollowoneleadingcharacterwhosedynamics doesnotreceiveinputfromtheothers.Inthiscaseallphase trajec-toriesconvergetoasingleuniquetrajectoryonlyifci=cjforalli, j,asconsequenceofthestrictcorrespondencebetweengaitphase andpositionthatisgivenbyEq.(3).Inallothercasesthe trajec-toriesofthefollowersconvergetoone-dimensional,butdistinct attractorsthatareuniquelydeﬁnedbyci.Theseattractors corre-spondtoabehaviorwherethefollowers’positionsoscillatearound thepositionoftheleader.Thepartialcontractionofthedynamics withc=0guaranteesthattheresultingattractorareaisboundedin phasespace(cf.Ch.3.7.viiinRef.[2]).

Fortheanalysisofcontractionpropertiesweregardan auxil-iarysystemobtainedfromEq.(3)bykeepingonlythetermsthat dependon: ˙=−mdLdG(+0).AccordingtoTheorem 1the symmetrizedJacobianofthissystemprojectedontotheorthogonal complementoftheﬂow-invariantlinearsubspace∗₁+0

1=...= _N∗+0_Ndetermineswhetherthissystemispartiallycontracting. Byvirtueofalinearchangeofvariables,thestudyofthe contrac-tionpropertiesofthissystemisequivalenttostudythecontraction propertiesofthedynamicalsystem ˙=−mdLdG()ontrajectories convergingtowardtheﬂow-invariantmanifold∗₁=...=∗_N.

The sufficient conditions for (exponential) partial con-traction toward flow-invariant subspace are, (see Eq. (9)): VJs()VT=−mdVB()VT<0,introducingB()=LdDg+Dg(Ld)TandV signifyingtheprojectionmatrixontotheorthogonalcomplement oftheflow-invariantlinearsubspace.Fordiffusivecouplingwith symmetricLaplacianthelinearflow-invariantmanifold∗₁₌...₌ _N∗ isalsothenull-spaceoftheLaplacian.Inthiscase,the eigen-vectorsoftheLaplacianthatcorrespondtononzeroeigenvalues canbeusedtoconstructtheprojectionmatrixV.Forexample,in thecaseofNcharacterswithsymmetricalall-to-allcouplingwith Ld₌_NI₋₁₁T_≥₀ _we _obtain 1

2V(L d_D

g+Dg(Ld)T)VT=NVDgVT>0 for Dg>0. In this case the contraction rate is given by min= mdmin

(g()) +

Ld,with+_Ld=N.

Forgeneralsymmetriccouplingswithpositivelinkswithequal coupling strength md>0 a sufﬁcient contraction condition is: +_min(Ld)/+max(Ld)>max

(|g()−mean(g())|)/mean(g()), with mean(g())=1/T

₀Tg()d. This condition was derived from the fact that for symmetric (positive) matrices M1 and M2 for (M1−M2)>0 it is sufﬁcient tosatisfy min(M1)>max(M2). This

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Fig.4.Self-organizedreorderingofacrowdwith16characters.Controldynamicsaffectssimultaneouslydirection,distanceandgaitphase.See[Demo7_].

sufﬁcientconditionpermitstoconstraintheadmissiblecoupling topologiesdependentontheg().Alternatively,itispossibleto introducelow-passﬁlteringinthecontroldynamicstoincrease thesmoothnessofthefunctiong(),seeRef.[3].

Thesestabilityboundsareillustratedby[Demo2_]_that_shows

convergentbehaviorofthecharacterswhenthecontraction con-ditionmd>0,(Ld)s≥0issatisﬁedforall-to-allcoupling.[Demo3] showsthedivergentbehaviorofagroupwhenthisconditionis violatedwhenmd<0.Intheseandthenextdemonstrationsthe actualvaluesofinteractionparametersmd,mz,m >0(incases, when theyadditionallysatisfy thesufﬁcient contraction condi-tions)wereobtainedbymatchingthecorrespondingconvergence ratestothoseoftherealhumanbehaviorincrowds[31].

(3)Stepsizecontrolcombinedwithcontrolofstepphase: thedynamicsisgivenbyEqs.(4)and(5)withmd=0.This dynam-icsdefinesahierarchicallycouplednonlinearsystem(intypeof Eq. 8). While the dynamics would be difficult to analyzewith classicalmethods,thedynamicsforz(t)thatis givenbyEq. (4) ispartiallycontracting in thecaseof all-to-allcouplingforany boundedexternalinput(t)ifmz>0,Lz≥0,andω(t)>0.These suf-ficientcontractionconditionscanbederivedfromtherequirement ofthepositive-definitenessof thesymmetrizedJacobian apply-ingsimilartechniquesasabove.TheJacobianofthissubsystemis J(,ω)=−mzDzg(,ω)Lz,withthediagonalmatrix(Dzg(,ω))ii= ωig(i+0_i)thatispositivedefinitesinceg()>0andω>0.This subsystemis(exponentially)contractinganditsrelaxationrateis determinedbyz=mzmin

(g()) +

Lz (inthecaseofall-to-all

cou-pling)foranyinputfromthedynamicsof(t),cf.Eq.(5).Thelast dynamicsiscontractingwhen(L₎

s≥0anditsrelaxationrateis =k+_L,where+_L isthesmallestnon-zeroeigenvalueof(L

₎ s. Theeffectiverelaxationtimeoftheoveralldynamicsisthus deter-minedbytheminimumofthecontractionratesandz(Fig.4). Demonstrationsofthiscontroldynamicssatisfyingthe contrac-tionconditions are shown in [Demo4_], _without_control _of _step

phase,andin[Demo5_],_with_control_of_step_phase.

(4) Advanced scenarios:a simulation of a systemwith sta-bledynamicsincludingbothtypesofspeedcontrol(viastepsize andstepfrequency)andstepphasecontrolisshownin[Demo6_],

and a largercrowd with16 avatars simulated using the open-sourceanimationengineHorde3d[32],isshownin[Demo7_]. _In

thissimulationanadditionaldynamicsforobstacleavoidanceand thecontrolofheadingdirectionwasactivatedduringthe unsort-ingoftheformationofavatars.Thenthisnavigationdynamicswas deactivated,andspeedandpositioncontrolaccordingtothe dis-cussedprinciplesresultintheﬁnalcoordinatedbehaviorofthe crowd.(See[Demo8_].)_Finally,_[Demo9_]_shows_the_divergence_of

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thedynamicsformd<0,violatingthecontractionconditionforthe stepphasedynamics.Thetwosimulationsshownin[Demo10_]_and

[Demo11_]_illustrate_the_convergence_for_a_crowd_with₄₉_avatars_for

twodifferentvaluesofthestrengthofthedistance-to-stepsize cou-pling,theparametersofstepphasecouplingremainingconstant. Thedevelopmentofstabilityboundsandestimatesofrelaxation timesformorecomplicatedscenariosisthegoalofongoingwork. (5)Control ofheadingdirection:forthecontrolofheading directionsinpresenceofcouplingsthataffectthestepphases,the contractionconditionscanbederivedexploringtheresulton hier-archicallycoupledsystemsdiscussedinSection5.Fortheanalysis ofthestabilityofthedynamicsdeﬁnedbyEq.(6)itisthus sufﬁ-cienttoanalyzethecontractionpropertiesofthedynamicsforthe headingdirection ,treatingtheadditionaltermω(t)g ((t))asan externalinputtothe subsystem.

Assuming a constant goal direction, it was shown in Ref. [2] (Ch. 3.9) that the uncoupled dynamics for one character, givenby ˙ ₌_−ω(t)m sin( ₋ goal₎_is_contracting_in_the inter-vals] goal₋₊_2n, goal₊₊_2n[,_n_∈_Z_for_constant_m _>_0._(If (t)isasmoothstrictlyincreasingfunctionoftwiththe substitu-tion ((t))= ()(andω(t)=d/dt)thelastdifferentialequation canberewrittenthen:d /d=− m sin( ()− goal_)).

Anotherpossibilityistorealizedirectioncontrolistofeedback thecircularmeanaveragedirectionofallcharactersasjointcontrol parameter₌angle(1/N

_iexp[ _i√_−1]).Inthiscasethe dynam-icsisgivenby

˙_i₌_ω_i_(t)(sin(₋ _i₎₊_g _((t))),

_∀

_i_∈_[1_._._._N], ₍₁₂₎ whichis suitablefortheapplicationofTheorem2.Thisimplies that the overall dynamics is contracting if the dynamics ˙i= ωi(t)sin((t)− i)iscontractingforany(t).ThesameTheorem guarantees contraction,when the consensusvariable is esti-matedbyalow-passﬁlter(withtime-constant˛>0):˛˙=−+ angle(1/N

_iexp[ i

√

−1]). The simulation shown in [Demo12_]

illustratestheconsensusscenariodeﬁnedbyEq.(12),(withouta synchronizationofgaitcycles).

Conclusions

Theanalysisanddesignofthedynamicpropertiesofthe for-mationoforderedpatternsincrowdssofarhasbeenonlyrarely treatedincomputeranimation,andtreatmentsincontroltheory typicallyassumehighlysimplifiedagentmodels.Toourknowledge, thispaperpresentsthefirstsystematictreatmentofthedynamics oforderformationincrowdsusingmorecomplexagentmodels thatincludearticulationofthecharactersduringlocomotion. Com-biningasetofspecificapproximationsofthesystemdynamicswith ContractionTheoryasmathematicalframework forthe system-atictreatmentstabilitypropertiesofcomplexnonlinearsystems, wepresentedanumberofexamplesforthederivationofstability

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boundsfornontrivialscenariosofcoordinatedcrowdbehavior dur-inglocomotion.Wethinkthattheshownexamplesdemonstrate thefeasibilityoftheappliedapproachandmakeitplausiblethat itcanbeextendedforevenmorecomplexscenarios.Necessarily, thisﬁrstexploratorystudyishighlyincompleteandthespectrumof analyzedbehaviorsofcrowdsisstillverylimited.Futureworkwill havetoaddotherdynamicalprimitivestothemodelarchitecture, includingonessuitablefortherealizationofotherbehaviorsthan locomotion.Theintegrationof suchadditionalcomponentswill necessitatethedevelopmentofnewapproximationsand applica-tionsofadditionalmethodsfromnonlinearcontroltheoryinorder toderivetherelevantcontractionbounds.Thisdeﬁnestheresearch agendaforourfuturework.

Acknowledgement

SupportedbyDFGForschergruppe‘PerceptualGraphics’(GZ:GI 305/2-2),theECFP7projectsFP7-ICT-248311‘AMARSi’and FP7-ICT-249858‘TANGO’andtheHermannundLilly-Schilling-Stiftung. References

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AlbertMukovskiyisascientiﬁcresearcherinthe Sec-tionforComputationalSensomotorics,HertieInstitutfor ClinicalBrainResearch,whichispartoftheMedical Fac-ultyoftheUniversityofTübingen.HeobtainedhisM.Sc. inBiophysicsinMoscowStateUniversityin1992.His mainresearch interestsare intheareas ofbiological motioncontrol,robotics,multi-agentsystems, computa-tionalneuroscience.

Jean-JacquesSlotineisProfessorofMechanical Engineer-ing and InformationSciences, Professor of Brain and CognitiveSciences,andDirectoroftheNonlinearSystems Laboratory.HereceivedhisPh.D.fromtheMassachusetts InstituteofTechnologyin1983,atage23.After work-ingatBellLabsinthecomputerresearchdepartment,he joinedthefacultyatMITin1984.ProfessorSlotineteaches andconductsresearchintheareasofdynamicsystems, robotics,controltheory,computationalneuroscience,and systemsbiology.ResearchinProfessorSlotine’slaboratory focusesondevelopingrigorousbutpracticaltoolsfor non-linearsystemsanalysisandcontrol.ProfessorSlotineis theco-authoroftwopopulargraduatetextbooks,“Robot AnalysisandControl”(AsadaandSlotine,Wiley,1986),and“AppliedNonlinear Con-trol”(SlotineandLi,Prentice-Hall,1991)andisoneofthemostcitedresearcherin bothsystemsscienceandrobotics.HewasamemberoftheFrenchNationalScience Councilfrom1997to2002,andamemberofSingapore’sA*STARSigNAdvisory Boardfrom2007to2010.HeiscurrentlyamemberoftheScientiﬁcAdvisoryBoard oftheItalianInstituteofTechnology.HehasheldInvitedProfessorpositionsat Col-legedeFrance,EcolePolytechnique,EcoleNormaleSuperieure,UniversitadiRoma LaSapienza,andETHZurich.

MartinA.GiesehasreceivedB.Sc.degreesinPsychology andElectricalEngineeringandaM.Sc.andPh.D.degreein ElectricalEngineering.From1998to2001hewas postdoc-toralfellowattheCenterforBiologicalandComputational LearningatMIT,Cambridge,USA,andalsoheadofthe HondaResearchLaboratoryinBoston,USAfrom2000to 2001.From2001to2007heheadedtheLaboratoryfor ActionRepresentationandLearning(ARL)attheHertie InstituteforClinicalBrainResearch,Tübingen,Germany. From2007to2008hewasSeniorLecturerattheSchoolof PsychologyattheUniversityofBangor,UK.Since2008he isProfessorforComputationalSensomotoricsatthe Uni-versityofTübingenthatisjointlyembeddedintheHertie InstituteforClinicalBrainResearchandtheWernerReichardtCenterforIntegrative Neuroscience.Hisgroupworksonneuralmodelingofhigh-levelvision, computa-tionalmotorcontrol,andlearningtechniquesforthemodelingandanalysisofmotor behaviorandclinicalapplications.