Verbs and Adverbs: Multidimensional Motion - CiteSeerX

17 Pages·1998·1.6 MB·English

Verbs and Adverbs:
Multidimensional Motion Interpolation Using Radial Basis
Functions
CharlesRose BobbyBodenheimer MichaelF.Cohen
[email protected] [email protected] [email protected]
MicrosoftResearch
Abstract
Thispaperdescribesmethodsanddatastructuresusedtoleveragemotionsequencesofcomplexlinked
figures. We present a technique for interpolating between example motions derived from live motion
captureorproducedthroughtraditionalanimationtools. Thesemotionscanbecharacterizedbyemotional
expressivenessorcontrolbehaviorssuchasturningorgoinguphillordownhill. Wecallsuchparameterized
motions“verbs”andtheparametersthatcontrolthem“adverbs.” Verbscanbecombinedwithotherverbs
to form a “verb graph,” with smooth transitions between them, allowing an animated figure to exhibit a
substantialrepertoireofexpressivebehaviors.
Acombinationofradialbasisfunctionsandloworderpolynomialsisusedtocreatetheinterpolation
spacebetweenexamplemotions. Inversekinematicconstraintsareusedtoaugmenttheinterpolationsin
ordertoavoid,forexample,thefeetslippingonthefloorduringasupportphaseofawalkcycle.
Oncetheverbsandverbgraphhavebeenconstructed,adverbscanbemodifiedinreal-timeproviding
interactiveorprogrammaticcontroloverthecharacters’actions. Thisallowsthecreationofautonomous
charactersinavirtualenvironmentthatexhibitcomplexandsubtlebehavior.
1 Introduction
Creating believable animated human figures is a difficult task, even with the most sophisticated software
available. Once an acceptable segment of animation has been created, either by an animator or through
motioncapture, theresultsarestilldifficulttoreuse. Theadditionalworktomodifytheanimationmaybe
almost as time consuming as creating the original motion itself. Furthermore, the exact style or structure
neededforaparticularmotionmaynotbeknownuntilruntime. Areal-timecontrollableanimationsystemis
neededforinteractiveapplications,suchasa3Dvideogame,oravirtualenvironment.
Researchintocontrollablehumanfigureanimationcanbedividedintothreemajorgroupings: procedural,
simulated,andinterpolated. Proceduralanimationusescodefragmentstoderivethedegreeoffreedom(DOF)
values at a particular time. The procedures can be as sophisticated as needed in order to provide different
motionstylesortoreacttodifferentconditionsofthesimulatedenvironment. Dynamicallysimulatedfigure
animation uses controllers together with a simulated human to generate motion. The degree to which this
methodsucceedsisboundedbyhowaccuratelyhumanmotionisunderstoodandmodeled. Boththesemethods
havethedisadvantagethattheycanalienateclassicallytrainedanimatorsandthattheydonoteasilymakeuse
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ofmotioncapturetechnology. Thisisimportantsinceanimatorsandmotioncapturesystemseachproduce
compellingresults. Toleveragetheirqualities,asystemmustusewhattheseresourcesprovide.
Interpolatedanimationisthethirdmajorgrouping. Thismethodusessetsofexamplemotiontogetherwith
aninterpolationschemetoconstructnewmotions. Theprimaryproblemsofthisapproachareinprovidinga
setofmeaningful,highlevelcontrolknobstotheanimatororruntimesystem,maintainingtheaestheticofthe
sourcemotionsintheinterpolatedmotions,andmotionextrapolation. Anotherconsiderationisthedifficulty
ofacquiringtheexamples. Eachisprecious. Additionally,foraninterpolatedmotionschemetobeusedina
run-timeenvironmentratherthanearlierinaproductionpipeline,itmustbeefficient. Forthesereasons,an
approximationschemeusingradialbasisfunctionswaschosen.
Thispaperdescribesasystemforreal-timeanimationbyexamplethataddressessomeoftheseproblems.
Throughthecreationofparameterizedmotions,whichwecall“verbs”parameterizedby“adverbs”,asingle
authoredverbproducesacontinuousrangeofsubtlevariationsofagivenmotionatreal-timerates. Asaresult,
simulatedfiguresaltertheiractionsbasedontheirmomentarymoodorinresponsetochangesintheirgoals
orenvironmentalstimuli. Forexample,wedemonstratea“walk”verbthatisabletoshowemotionssuchas
happinessandsadness,anddemonstratesubtlevariationsduetowalkinguphillordownhillwhileturningto
theleftandright.
Thepaperalsodescribestheconstructionof“verbgraphs”thatactasthegluetoassembleverbsandtheir
adverbsintoaruntimedatastructure. Verbgraphsprovidethemeansforthesimulatedfigurestoseamlessly
transitionfromverbtoverbwithinaninteractiveruntimesystem. Finally,webrieflydiscussthediscreteevent
simulatorthathandlestheruntimemainloop.
2 Related Work
The idea of altering existing animation to produce different characteristics is not new. Unuma et al. [16]
useFouriertechniquestointerpolateandextrapolatemotiondata. Amayaetal.[1]alterexistinganimation
byextractingan“emotionaltransform”fromexamplemotionswhichisthenappliedtoothermotions. For
example,“anger”fromanangrywalkisappliedtoaruntogeneratean“angry”run. Ourworkdoesnotfollow
thisapproachinthatitdoesnotapplycharacteristicsofonemotiontoanother,butinsteadassumesthatthe
initiallibraryofmotionscontainstheseemotions. Unlikethesetwotechniques, ourmethodisnotbasedin
thefrequencydomainandthuscanhandlenon-periodicmotionswhichearliermethodsfailtocapture.
Bruderlin and Williams [6] use multitarget interpolation with dynamic timewarping to blend between
motions, and displacement mappings to alter motions such as grasps. Witkin and Popovic´ [19] present a
similarsystemforeditingmotioncaptureclips. Theformerworkisinthesamespiritasours,andaddresses
many of the same difficulties, specifically the necessity of selecting appropriate key times for interpolation
andtheconsequentneedfortimewarping. Onedifferencebetweenthetwoapproachesliesinthechoiceof
interpolationtechniques: BruderlinandWilliamsusemultiresolutionfilteringofjointanglesinthefrequency
domain,whereasourtechniquedecouplessolutionrepresentationfrominterpolationmechanism. Perlin[11]
has approached this problem in a very different way by using noise functions to simulate personality and
emotioninexistinganimation.
BothWileyandHahn[18]andGuoandRoberge´[8]producenewmotionsusinglinearinterpolationona
setofexamplemotions. BothtechniquesrequireO(2d)examples,wheredisthedimensionalityofthecontrol
space. OurtechniqueusingradialB-splinesrequiresO(n)examplestoestablishthebaselineapproximation
andO(n3)tocomputetheresultinganswer. Tocompare,aDelaunaytriangulationofthedatawouldrequire
O(nceil(d=2))tocompute,whend(cid:21)3.
2
TheworkofWileyandHahniscloselyrelatedtoourown,theadditionaldifferencebeingthattheirtime
scalingisuniform,whereasoursisnon-uniformandbasedonkeyevents. Whileuniformtime-scalingobviates
theneedforananimatortoselectstructurallysimilarposesduringmotions,itassumedthattherepertoireof
motionsbeinginterpolatedbetweenmustbeverysimilarintime. Whenthisassumptionisviolated,oddities
inthemotioncanresult. WileyandHahnalsoreparameterizeandsampletheirmotionsonamultidimensional
gridandthenperformsimpleinterpolationsatruntime. Thisrequirescomputationandstorageexponentialin
thenumberofparameters.
Additionally,neitherWileyandHahnnorGuoandRoberge´ discusstheblendingofsubsetsofexamples,
whichwouldarisewhennewexamplesareplaced“between”oldexamplesasadesignerrefinestheinterpolated
motion space. Our technique, on the other hand, is refined by more examples as required. As we use an
approximation method based upon radial B-splines with compact support to perform the interpolation, our
exampleshavelimitedeffectoverthespaceofanimations,thusensuringthatsubsetsoftheexamplesareused
atruntimeasappropriate.
HodginsandPollard[9]interpolateoverthespaceofcontrollawsasopposedtojointtrajectories. The
controlfunctionsdefinethegainsinthejointswithinaphysicaldynamicsframework. Byinterpolatingthe
control,foranactionsuchasrunning,theycanalterthephysicalcharacteristicsofthefigurefromachildto
anadultorfromamaletoafemalecharacter.
An important distinction in our work is that the interpolation is performed simultaneously in real-time
over multiple dimensions, such as emotional content and physical characteristics. Although we apply the
techniquestointerpolatingtrajectoriescharacterizedbycoefficientsofsplinecurves,themethodspresented
here are also applicable to coefficients in the Fourier and other domains. It may also be possible to apply
similarideastocontroltheparametersofaphysicallybasedmodel.
In the context of autonomous agents, our work presents a back-end for applications such as games, the
Improv[12]system,andtheworkproposedbyBlumbergandGalyean[4]. Thehighlevelcontrolstructures
foundinsuchapplicationsarecapableofselectingverbsandadverbswhiletheworkwepresenthereprovides
thelowlevelanimationitself. Thus,wecreatethemotioninreal-timefor“directable”creaturesasdiscussed
byBlumbergandGalyean.
3 Verbs, and Adverbs
Thefigureanimationsystempresentedhereconsistsoftwomainparts. Anofflineauthoringsystemprovides
the tools for a designer to construct controllable motions, verbs, from sets of examples. In addition, the
authoring system provides tools for constructing transitions between verbs and for putting the verbs and
transitionstogetherinaverb-graph. Thisstructurebecomesthecontrolledobjectfortheruntimeportionof
theVerbsandAdverbssystem. Theruntimesystemcontrolstheinvocationofverbsandtheevaluationofthe
figure’sposeateachframeoftheanimation. Webeginwithadiscussionoftheauthoringsystemandthen
moveontotheruntimesystem. RefertoTable1forthesymbolsusedinthetext.
The simulated figures, or “creatures,” discussed here are assumed to be well represented as a hierarchy
ofrigidlinksconnectedbyjoints. Eachjointmaycontainoneormorerotationaldegreesoffreedom(DOF).
The root of the hierarchy has six special DOFs which represent the position and orientation of the figure
in the global coordinate frame. In particular, we use a 46 DOF human figure. Each DOF’s motion is
represented as a function through time (cid:18)j(T); j = 1:::NumDOF, where T represents clock time from
[0::durationofthemotion]whichwillshortlybedefinedaskeytimetime. GiventhesemotionsoftheDOFs,
onecanrenderthecreatureatanypointintime.
Traditionally hand-crafted or motion captured animation segments have DOF functions of 1 variable,
3
Object Variable Subscript subscriptmeaning subscriptrange
Motionexample M i Motionexamplenumber 1::NumExamples
DOF (cid:18) i Motionexamplenumber 1::NumExamples
j DOFindex 1::NumDOF
B-spline B k B-splineindex 1::NumCP
B-SControlPoint b i;j;k
PointinAdverbSpace p i
Keytime K m Keytimeindex 1::NumKeyTimes
Radialbasis R i basisassociatedwithMi
RadialCoefficient r i;j;k
LinearBasis A l Adverbindex 1::NumAdverbs
LinearCoefficient a j;k;l
Distance d i Distancetopi
Time Variable range
Clocktime (cid:28)
Keytimetime T 0::KNumKeyTimes
Generictime t 0::1
Table1: Terminology
time. TheDOFfunctionsforaverbareneverexplicitlyrepresented,butratherevolveatruntimethroughthe
interpolationofexamplemotionsweightedbychanginginterpolationparametersoradverbs. Theseadverbs
mayrepresentemotionalaxessuchashappy-sad, knowledgeable-clueless, andphysicalparameterssuchas
whether,inthecaseofawalkverb,thefigureiswalkinguphill,ordownhillandwhetherthemotioniscurving
totheleftorright. Thesetofadverbaxesdefineamultidimensionaladverbspaceofallpossiblevariations
foraparticularverb.
Verbsareconstructedfromsetsofsimilar,butdistinctexamplemotions. Theseexamplescanbeobtained
keyframingorwithamotioncapturesystem. Ineithercase,certainrestrictionsonthesetofexamplesapply.
The primary restriction is that all examples for a verb be structurally similar. A set of example walks, for
instance,mustallstartoutonthesamefoot,takethesamenumberofsteps,havethesamearmswingphase,
andhavenospuriousmotionssuchasahead-scratch. Theotherprimaryrestrictionisconsistentuseofjoint
angles. AnatomicallyimplausiblesettingsoftheDOFvaluescanyieldthesameoveralleffectasaplausible
settingduetotheredundantnatureoftwoandthreeDOFjoints. Bodenheimeretal.[5]presentmethodsto
ensureconsistentDOFvaluesformotioncaptureddata.
Eachexamplemotionisannotatedbyhandwithasetofadverbvalues,placingtheexamplesomewhere
intheadverbspace. Additionally,asetof“keytimes”,instantswhenimportantstructuralelementssuchasa
foot-down occurs, must be specified. These keytimes are used to preprocess the examples into a canonical
timelineforinterpolationandforlatersynthesisofthephrasingfortheinterpolatedmotions. Finally,examples
are defined by a set of intermittent constraints (e.g., that the foot should not slip while in contact with the
floor).
AverbM isthereforedefinedbyasetofexamplemotionsMi,
M =f(cid:18) (T);p ;K :i=1:::NumExamples;
i ij i m
j =1:::NumDOF;m=0:::NumKeyTimesg
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wherethe(cid:18)ij(T)aretheDOFfunctionsfortheithexampleMi,pithelocationoftheexampleintheadverb
space,andK thesetofkeytimeswhichdescribethephrasing(relativetimingofthestructuralelements)of
theexample. Thetimeparameter,T,representsclocktimebutwiththeclockstartingat0atthebeginningof
themotionexample.
3.1 ExampleMotions
EachexamplemotionMiisdefinedbyofanumberofDOFfunctions,denotedby(cid:18)ij(T). Each(cid:18)ij (i.e.,the
jth DOFfortheith example)isrepresentedasauniformcubicB-splinecurvespecifiedbyNumCP control
points,
NuXmCP
(cid:18) (T)= b B (T)
ij ijk k
k=1
wheretheBk(T)aretheB-splines,andthebijk arethescalarB-splinecoefficientsorcontrolpointsforthe
ithexampleMi. Bartelsetal.[3]containsanextensivediscussionofB-splinesandtheirproperties.
Theinterpolationtechniquepresentedheredecouplestheinterpolationschemefromtherepresentationof
theexamples. Theexamples,therefore,couldbeencodedusingothermethods,suchasawaveletorFourier
decomposition.
3.2 TimeWarping
The keytimes are used to define a piecewise linear mapping fromT 2 f0:::KNumKeyTimesg to a generic
timet 2 f0:::1g. Thefirstkeytimeoftheverb,K1,isdefinedtobe0. Thelastkeytime,KNumKeyTimesg
marksthedurationoftheverb. Inthecaseofcyclicverbs,suchaswalking,itmustalsomarkthesameevent
asthefirst. Moregenerally,givenaT between0andKNumKeyTimes
(cid:18) (cid:19)
T −K 1
t(T)= (m−1)+ m (1)
K −K NumKeyTimes −1
m+1 m
forthelargestmsuchthatT >Kmandkeepinginmindthatt(0)=0. ThismappingisillustratedinFigure1.
Inotherwords,ateachkeytime,thegenerictimet(Km) = NumKemy−T1imes−1. Atthethirdoffourkeytimes,
t= 2 asthekeytimeswillbeat0, 1, 2,and1. Betweenkeytimes,tislinearlyinterpolated.
3 3 3
Once all the examples have had their time reparameterized to generic time t, they are said to be in a
canonicalform. Foragivent,allexampleswillbeatthesamestructuralpointofthemotion,regardlessof
phrasingoroverallduration. Examplesareinterpolatedwithinthisgenerictimeframe. Thekeytimeequations1
themselvesarealsointerpolatedbetweenexamples. Theinverseofthisinterpolatedtimingfunctionisthen
appliedtountimewarptheinterpolatedmotiontorecapturephrasingandduration.
3.3 InverseKinematicConstraints
In addition to being used to timewarp the motions, keytimes also specify periods during which kinematic
constraintsneedtobeenforced. Forexample,keytimesofawalkmightbeheel-strikeandtoe-off. Between
thesekeytimes,thefootmustremainstationary.
Thevaluesforspecificconstraintconditionssuchasthelocationtheendeffector(e.g., thefoot)should
maintainarenotsetwhentheyaredesigned,butaresetatplaybackwhenakeytimethattriggersaconstraintis
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T
Key-times for
Example Motion B
Interpolated Key-times
K (p)
m
Key-times for
Example Motion A
T
1
t
K K K K K
1 2 3 4 5
0.0 t( T ) 1.0
1
Figure1: MappingBetweenGenericTimeandKeytimes
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crossed. If,forexample,whenakeytimeiscrossedtriggeringafootconstraint,thefoot’shorizontallocation
isat(x;z)andtheheightoftheflooratthatpointis y = floor(x;z), thentheconstraintlocationissetto
(x;y;z). Thisconstraintpositionismaintaineduntilanotherkeytimereleasingtheconstraintiscrossed.
Constraintsareenforcedatruntimewithafastinversekinematicsoptimization. Atagivengenerictime
t,acreatureisfirstpositionedindependentofanyconstraintsbyevaluatingtheactiveverb(s)atthattime. In
general,thiswillplacetheendeffectorclosetotheconstraintpoint. Theymakethesubtlechangesneededto
exactlysatisfytheconstraint,wewillonlyconsidermodifyingDOFsbetweentherootandtheendeffector.
ThechangesintheseDOFsneededtosatisfytheconstraintarefoundateachframeatruntimebysolvingthe
linearsystem(typicallyofsize5or6)oftheform
J(cid:1)(cid:18) = (cid:1)x (2)
whereJistheJacobianoftheDOFwithrespecttothemotionoftheendeffector,(cid:1)xisthevectorfromwhere
theendeffectorwasplacedbytheverbtotheconstraintpoint,and(cid:1)(cid:18) istheamounttheDOFneedstobe
perturbedtoholdtheendeffectorinplace. SeeGirard[7]orWatt[17]fordetailsonsuchinversekinematic
problems.
4 Verb Construction
Oncetheexamplemotionshavebeenspecifiedforaverbwiththeirassociatedkeytimesandadverbsettings,
thenextstepistoconstructacontinuous“space”ofmotionsparameterizedbytheadverbs. Thedimension
ofthisspaceisthenumberofadverbs,NumAdverbs. Apointinthisspaceisdefinedbythespecificvalues
of the individual adverbs. The point may move from moment to moment during the interactive animation
if,forexample,thecharacter’smoodchangesorthecharacterbeginstowalkupordownhill. Thegoalisto
produceatanypointpintheadverbspaceanewmotionM(p;t)derivedthroughinterpolationoftheexample
motions. Whenpisequaltotheadverbsettingsforaparticularexamplemotioni,thenM(p;t)shouldequal
Mi(t).
Each example motion has one free variable for each B-spline control point for each DOF and one free
variable for each keytime. The time warping described above ensures that corresponding control points in
each example motion specify similar moments in each motion, even if the overall lengths of the example
motionsdiffer. Thisallowsustotreattheexamplemotioninterpolationasaseparateproblemforeachcontrol
pointandeachkeytime(i.e.,NumCP (cid:2)NumDOF +NumKeyTimes individualinterpolationproblems).
Thestandardproblemofmultivariableinterpolationisasfollows: givenN distinctpointspi inRn and
N values vi in R, find a function f : Rn ! R such that for all i, f(pi) = vi, and such that f does not
oscillatebadlybetweenvalues. Thehighdimensionalityofthespacedefinedbytheadverbs, coupledwith
thedesiretorequirefewexamplemotions(perhapsonlytwotothreetimesthenumberofadverbs),presents
difficultiesformanyinterpolationmethods. Giventhesedifficulties,acombinationofradialbasisfunctions
andloworder(linear)polynomialswasselectedforthisproblem. Thepolynomialfunctionprovidesanoverall
approximationtothespacedefinedbytheexamplemotions. Italsoallowsforextrapolationoutsidetheconvex
hullofthelocationsoftheexamplemotions. Theradialbasesthenlocallyadjustthepolynomialtointerpolate
theexamplemotionsthemselves.
Radialbasisfunctionshavetheform:
R (d (p))
i i
whereRiistheradialbasisassociatedwithMianddi(p)isameasureofthedistancebetweenpandpi,most
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oftentheEuclideannormkp−pik. Becausesumsofradialbasescannotrepresentanaffineorpolynomial
function,radialbasissetsareoftenaugmentedbyaddingapolynomialoffixeddegree.
DetailsofthemathematicsforthistypeofinterpolationcanbefoundintheseminalworkofMicchelli[10]
andinthesurveyarticlebyPowell[13]. Radialbasisfunctionshavebeenusedincomputergraphicsforimage
warpingbyRuprechtandMu¨ller[15],andAradetal.[2].
ThevalueofeachinterpolatedDOFcurvecontrolpointinthisspace,bjk(p),isdefinedas
NumEXxamples NumXAdverbs
bjk(p)= rijkRi(p)+ ajklAl(p) (3)
i=1 l=0
wheretherijkandRiaretheradialbasisfunctionweightsandradialbasisfunctionsthemselvesandtheajkl
andAlthelinearcoefficientsandlinearbasesasexplainedbelow. Interpolatedkeytimesaresimilarlydefined
as
NumEXxamples NumXAdverbs
Km(p)= rimRi(p)+ almAl(p): (4)
i=1 l=0
For each verb there are (NumCP (cid:2)NumDOF) control point interpolations (eq. 3) and NumKeyTimes
keytimeinterpolations(eq.4).
This leaves us with the problem of choosing the specific shape of the radial bases and determining the
linearandradialcoefficients. Wesolvetheseproblemsintwosteps,byfirstsolvingforthelinearcoefficients
andthenfortheradialbasiscoefficients.
4.1 LinearApproximation
Inthefirststep,wesolveforthelinearcoefficientsbyfindingthehyperplanethroughtheadverbspacethat
bestfitsthevariationacrosstheexamplemotionsoftheselectedcontrolpointorkeytime. Thelinearbasis
functionsaresimplyAl(p)=pl,thelthcomponentofp,andA0(p)=1. Anordinaryleastsquaressolution
determinestheNumAdverbs +1linearcoefficients,ajkl,thatminimizethesumofsquarederrorsbetween
NumXAdverbs
~b (p )= a A (p );
ijk i jkl l i
l=0
andbijk,theactualB-splinecontrolpoint(orkeytime)beinginterpolated,wherepi istheadverbsettingfor
theith examplemotion. Lettingbjk andb~jk denotevectorsofeachbijk(pj)and~bijk(pj)forafixedj and
k,thelinearapproximationleavestheresiduals
b(cid:22) =b −b~ :
jk jk jk
Itisthejoboftheradialbasistointerpolatetheseresiduals.
4.2 RadialBasisFunctions
Wedefineoneradialbasisfunctionforeachexamplemotion. Theradialbasisfunctionsaresolelyafunction
ofthedistance,di(p)=kp−pikbetweenapointintheadverbspace,p,andthepointintheadverbspace
corresponding to example motion i, pi. The radial basis itself, Ri(p), has its maximum at pi (i.e., where
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d = 0). We would also like the radial bases to have compact support (i.e., have value zero beyond some
distance)tolimiteachexamplemotion’sinfluencetoalocalregionofadverbspace,thusallowingforlocal
refinement of the verbs. There are a number of choices for the specific shape of the radial basis. For its
combinationofsimplicityandC2 continuity,inthisworkwechosetousearadialbasiswithacrosssection
ofadilatedcubicB-spline,B(d=(cid:11)). Thedilationfactor,1=(cid:11),ischosenforeachexamplemotiontocreatea
supportradiusfortheB-splineequaltotwicetheEuclideandistancetothenearestotherexamplemotion. For
(cid:11) = 1,thecubicB-splinehasaradiusof2.0,thus(cid:11)issimplytheminimumseparationtothenearestother
exampleintheadverbspace. Giventhisdefinition,itisclearthattheexamplemotionsmustbewellseparated.
Thecoefficients,rijk,cannowbefoundforeachDOFB-splinecontrolpointandkeytimebysolvingthe
linearsystem,
Dr =b(cid:22)
jk jk
whererjk isavectoroftherijk termsforafixedj andk,andDisasquarematrixwithtermsequaltothe
valueoftheradialbasisfunctioncenteredonmotioni1atthelocationofmotioni2. Thus
(cid:18) (cid:19)
R (p )
D =B i1 i2 :
i1;i2 i1 (cid:11)
i1
5 Verb Graphs
Inadditiontotheconstructionofindividualverbs,itisimportanttocombinethemsothatsmoothtransitions
occur between verbs. To do so, an author builds a directed graph of verbs, or “verb graph,” in which the
nodesrepresenttheverbsandthearcsrepresenttransitionsbetweenverbs. Iftherearemultipletransitionarcs
leavingaverb,eachtransitionmaybegivena“likelihood”oftransitioning,usedtostochasticallychooseat
runtimebetweenmultiplepossibletransitionsifnonehavebeenexplicitlyspecified.
The adverbs are shared across verbs although each verb may or may not respond to each adverb; for
example,anadverbforuphill/downhillmayhavenomeaningtoa“sit”verb. Theverbgraphisstatic,thatis,
fixedatauthoringtime.
5.1 Transitions
Givencorrespondingtimeintervals(assetbythedesigner)intwoverbs,transitionssmoothlymovecontrol
fromoneverbtothenext. Forexample,intransitioningfromwalkingtorunning,thetimeintervalsmayboth
besettospantherightfootplacement.
AtransitionisamappingofsimilarsegmentsbetweentwoverbsAandB. Thecorrespondenceregion
isdeterminedbythefourvariables,tA,tA,tB,andtB,thestartandstoptimesoftheregionineachofthe
s e s e
twoverbs. NotethatthesetimesareexpressedasgenerictimessincetheverbsAandB mayhavedifferent
durationsbasedupontheircurrentadverbsettings. Inotherwords,thetransitionsaredesignedbasedonthe
similarityofthestructureofthemotions(generictime)andsowillworkeventhoughtheverbsthemselves
arechangingatruntime.
Thedurationofthetransitioniscalculatedbytakingtheaverageofthelengthsofthetwoblendingregions
(cid:0) (cid:1) (cid:0) (cid:1)
T(tA)−T(tA) + T(tB)−T(tB)
e s e s
2
9
q A
(t)
tA
j
e
q B
tA (t)
s
j
tB
tB s
s
Figure2: TransitionBlendingDuringCorrespondenceRegion
where the T function maps generic time to real time (keytime time) for the verb’s current adverb settings.
Transitions,astheyaremadeupofverbs,areaffectedbytheverb’sadverbsettingsandthereforetakeonthe
mood,duration,andphrasingoftheirconstituentverbs.
Spacetime transitioning mechanisms were discussed by the authors in [14]. Generation of spacetime
transitionsbetweentwomotionsisstillanofflineprocess. Sincethespecificmotionstheverbsgenerateare
not known until runtime, this method cannot be used here. Thus, in this work we will rely on a simpler
formulationtoallowruntimecalculationofthetransitionsbetweenverbs.
VerbsAandBaresimplyblendedbyfadingoutonewhilefadingintheother. Amonotonicallydecreasing
blendingfunctionwitharangeanddomainof[0;1]determinestherelativecontributionofthetwoverbs. A
sigmoid-likefunction,(cid:11) = 0:5cos((cid:12)(cid:25))+0:5isusedinthiswork.
Overthetransitionduration,(cid:12)moveslinearlyfrom0to1representingthefractionofthewaythroughthe
transitionintervalsineachverb. ThetransitionalmotionisdeterminedbylinearlyinterpolatingtheverbsA
andBwithweights(cid:11)and1−(cid:11)respectively. TheinternalDOFsarefoundbyinterpolatingthejointpositions,
asshowninFigure2,whichshowstheDOFfunction(cid:18)j asacombinationofthetwoDOFfunctions(cid:18)jA and
(cid:18)B duringthetransitionregion. Thepaththejointtakesisdefinedby(cid:18)A beforethetransition,(cid:18)A;B during
j j j
thetransition,and(cid:18)B afterthetransition. SmoothmotionfortherootDOFsisachievedbyinterpolatingthe
j
velocities(ratherthanpositions)ofthetwoverbsandthenintegratingtheresultsthroughthetransition.
5.2 VerbGraphatRuntime
Issuingcommandstothecreaturecontrolsthemovementfromverbtoverbintheverbgraph. Whendirected
toperformaparticularverb, asearchisexecutedtodetermineasequenceoftransitionsandverbsthatwill
reach the desired action from the one currently playing. The shortest path, where shortness is defined as
the number of needed transitions, is chosen. This path through the verb graph is represented as a queue of
transition,verb,transition,verb,andsoon.
Some bookkeeping is required to keep the verb graph in a useful state. The motion of the root of the
creaturemustbestoredandmaintainedateachtimestepinordertoreorientandrepositionthebaseverbsfor
concatenationontothecurrentaction. Forexample,inawalktransitioningtoitself,thehorizontalposition
specifiedbytheverbindicatesanoffsetfromwherethecreaturewaswhentheverbwasinvoked. Thus,the
creaturecontinuestowalkforwardanddoesnotjumpbacktotheoriginoneachstride. Also,thesetofcurrent
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