Verbs and Adverbs: Multidimensional Motion Interpolation Using

17 Pages·1998·1.64 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
motion capture, the results are still difficult to reuse. The additional work to modify the animation may be
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
1
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≥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 θ (T), j = 1...NumDOF, where T represents clock time from
j
[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 θ 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 basisassociatedwithM
i
RadialCoefficient r i,j,k
LinearBasis A l Adverbindex 1..NumAdverbs
LinearCoefficient a j,k,l
Distance d i Distancetop
i
Time Variable range
Clocktime τ
Keytimetime T 0..K
NumKeyTimes
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 isthereforedefinedbyasetofexamplemotionsM ,
i
M ={θ (T),p ,K :i=1...NumExamples,
i ij i m
j =1...NumDOF,m=0...NumKeyTimes}
4
wheretheθ (T)aretheDOFfunctionsfortheithexampleM ,p thelocationoftheexampleintheadverb
ij i i
space,andK thesetofkeytimeswhichdescribethephrasing(relativetimingofthestructuralelements)of
theexample. Thetimeparameter,T,representsclocktimebutwiththeclockstartingat0atthebeginningof
themotionexample.
3.1 ExampleMotions
EachexamplemotionM isdefinedbyofanumberofDOFfunctions,denotedbyθ (T). Eachθ (i.e.,the
i ij ij
jth DOFfortheith example)isrepresentedasauniformcubicB-splinecurvespecifiedbyNumCP control
points,
NumCP
θ (T)= b B (T)
ij ijk k
k(cid:88)=1
wheretheB (T)aretheB-splines,andtheb arethescalarB-splinecoefficientsorcontrolpointsforthe
k ijk
ithexampleM . Bartelsetal.[3]containsanextensivediscussionofB-splinesandtheirproperties.
i
Theinterpolationtechniquepresentedheredecouplestheinterpolationschemefromtherepresentationof
theexamples. Theexamples,therefore,couldbeencodedusingothermethods,suchasawaveletorFourier
decomposition.
3.2 TimeWarping
The keytimes are used to define a piecewise linear mapping from T ∈ {0...K } to a generic
NumKeyTimes
timet ∈ {0...1}. Thefirstkeytimeoftheverb,K ,isdefinedtobe0. Thelastkeytime,K }
1 NumKeyTimes
marksthedurationoftheverb. Inthecaseofcyclicverbs,suchaswalking,itmustalsomarkthesameevent
asthefirst. Moregenerally,givenaT between0andK
NumKeyTimes
T −K 1
t(T)= (m−1)+ m (1)
(cid:18) K −K (cid:19)NumKeyTimes −1
m+1 m
forthelargestmsuchthatT >K andkeepinginmindthatt(0)=0. ThismappingisillustratedinFigure1.
m
Inotherwords,ateachkeytime,thegenerictimet(K ) = m−1 . Atthethirdoffourkeytimes,
m NumKeyTimes−1
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
5
T
Key-times for (cid:13)
Example Motion B
Interpolated Key-times(cid:13)
K (p)
m
Key-times for (cid:13)
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
6
crossed. If,forexample,whenakeytimeiscrossedtriggeringafootconstraint,thefoot’shorizontallocation
isat(x,z)andtheheightoftheflooratthatpointisy = 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∆θ = ∆x (2)
whereJistheJacobianoftheDOFwithrespecttothemotionoftheendeffector,∆xisthevectorfromwhere
theendeffectorwasplacedbytheverbtotheconstraintpoint,and∆θ 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
M (t).
i
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 ×NumDOF +NumKeyTimes individualinterpolationproblems).
Thestandardproblemofmultivariableinterpolationisasfollows: givenN distinctpointsp inRn and
i
N values v in R, find a function f : Rn → R such that for all i, f(p ) = v , and such that f does not
i i i
oscillatebadlybetweenvalues. Thehighdimensionalityofthespacedefinedbytheadverbs, coupledwith
thedesiretorequirefewexamplemotions(perhapsonlytwotothreetimesthenumberofadverbs),presents
difficultiesformanyinterpolationmethods. Giventhesedifficulties,acombinationofradialbasisfunctions
andloworder(linear)polynomialswasselectedforthisproblem. Thepolynomialfunctionprovidesanoverall
approximationtothespacedefinedbytheexamplemotions. Italsoallowsforextrapolationoutsidetheconvex
hullofthelocationsoftheexamplemotions. Theradialbasesthenlocallyadjustthepolynomialtointerpolate
theexamplemotionsthemselves.
Radialbasisfunctionshavetheform:
R (d (p))
i i
whereR istheradialbasisassociatedwithM andd (p)isameasureofthedistancebetweenpandp ,most
i i i i
7
oftentheEuclideannorm(cid:107)p−p (cid:107). Becausesumsofradialbasescannotrepresentanaffineorpolynomial
i
function,radialbasissetsareoftenaugmentedbyaddingapolynomialoffixeddegree.
DetailsofthemathematicsforthistypeofinterpolationcanbefoundintheseminalworkofMicchelli[10]
andinthesurveyarticlebyPowell[13]. Radialbasisfunctionshavebeenusedincomputergraphicsforimage
warpingbyRuprechtandMu¨ller[15],andAradetal.[2].
ThevalueofeachinterpolatedDOFcurvecontrolpointinthisspace,b (p),isdefinedas
jk
NumExamples NumAdverbs
b (p)= r R (p)+ a A (p) (3)
jk ijk i jkl l
(cid:88)i=1 (cid:88)l=0
wherether andR aretheradialbasisfunctionweightsandradialbasisfunctionsthemselvesandthea
ijk i jkl
andA thelinearcoefficientsandlinearbasesasexplainedbelow. Interpolatedkeytimesaresimilarlydefined
l
as
NumExamples NumAdverbs
K (p)= r R (p)+ a A (p). (4)
m im i lm l
(cid:88)i=1 (cid:88)l=0
For each verb there are (NumCP ×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
functionsaresimplyA (p)=p ,thelthcomponentofp,andA (p)=1. Anordinaryleastsquaressolution
l l 0
determinestheNumAdverbs +1linearcoefficients,a ,thatminimizethesumofsquarederrorsbetween
jkl
NumAdverbs
˜b (p )= a A (p ),
ijk i jkl l i
(cid:88)l=0
andb ,theactualB-splinecontrolpoint(orkeytime)beinginterpolated,wherep istheadverbsettingfor
ijk i
theith examplemotion. Lettingb andb˜ denotevectorsofeachb (p )and˜b (p )forafixedj and
jk jk ijk j ijk j
k,thelinearapproximationleavestheresiduals
b¯ =b −b˜ .
jk jk jk
Itisthejoboftheradialbasistointerpolatetheseresiduals.
4.2 RadialBasisFunctions
Wedefineoneradialbasisfunctionforeachexamplemotion. Theradialbasisfunctionsaresolelyafunction
ofthedistance,d (p)=(cid:107)p−p (cid:107)betweenapointintheadverbspace,p,andthepointintheadverbspace
i i
corresponding to example motion i, p . The radial basis itself, R (p), has its maximum at p (i.e., where
i i i
8
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/α). Thedilationfactor,1/α,ischosenforeachexamplemotiontocreatea
supportradiusfortheB-splineequaltotwicetheEuclideandistancetothenearestotherexamplemotion. For
α = 1,thecubicB-splinehasaradiusof2.0,thusαissimplytheminimumseparationtothenearestother
exampleintheadverbspace. Giventhisdefinition,itisclearthattheexamplemotionsmustbewellseparated.
Thecoefficients,r ,cannowbefoundforeachDOFB-splinecontrolpointandkeytimebysolvingthe
ijk
linearsystem,
Dr =b¯
jk jk
wherer isavectorofther termsforafixedj andk,andDisasquarematrixwithtermsequaltothe
jk ijk
valueoftheradialbasisfunctioncenteredonmotioni atthelocationofmotioni . Thus
1 2
R (p )
D =B i1 i2 .
i1,i2 i1(cid:18) α (cid:19)
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
T(tA)−T(tA) + T(tB)−T(tB)
e s e s
(cid:0) (cid:1) 2 (cid:0) (cid:1)
9
q A
(t)
tA(cid:13)
j
e
q B
tA(cid:13) (t)
s
j
tB(cid:13)
tB(cid:13) s
s
(cid:13)
(cid:13)
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,α = 0.5cos(βπ)+0.5isusedinthiswork.
Overthetransitionduration,βmoveslinearlyfrom0to1representingthefractionofthewaythroughthe
transitionintervalsineachverb. ThetransitionalmotionisdeterminedbylinearlyinterpolatingtheverbsA
andBwithweightsαand1−αrespectively. TheinternalDOFsarefoundbyinterpolatingthejointpositions,
asshowninFigure2,whichshowstheDOFfunctionθ asacombinationofthetwoDOFfunctionsθA and
j j
θB duringthetransitionregion. ThepaththejointtakesisdefinedbyθA beforethetransition,θA(cid:59)B during
j j j
thetransition,andθB 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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