A qualitative dynamic analysis of reiterant speechproduction: Phase portraits, kinematics, and dynamic modeling J. A. S. Kelso

HaskinsLaboratories, 270 CrownStreet.NewHaven,Connecticut 06511andDepartments ofPsychology and Biobehavioral Sciences, The University of Connecticut, Storrs,Connecticut 06268 Eric Vatikiotis-Bateson

Haskt•sLaboratories, 270 CrownStreet,NewHaven,Connecticut 06511andDepartmentof Linguistics, Indiana University, Bloomington, Indiana 47405 Elliot L. Saltzman Haskins Laboratories, 270 Crown Street, New Haven, Connecticut 06511

Bruce Kay HaskinsLaboratories, 270 CrownStreet.NewHaven, Connecticut 06511andDepartmentof Psychology, The

University of Connecticut. $torrs,Connecticut 06268

(Received 4 May 1984;accepted for publication 7 September 1984I The departurepointof thepresentpaperis oureffortto characterize andunderstand the spatiotemporal structureof articulatorypatternsin speech. To do so,weremovedsegmental variationasmuchaspossiblewhileretainingthe spokenact'sstressand prosodicstructure. Subjects producedtwo sentences fromthe "rainbowpassage" usingreiterantspeechin which normalsyllableswerereplacedby foa/or/ma/. This taskwasperformedat two serf-selected rates,conversational andfast.InfraredLEDs wereplacedonthejaw andlipsandmonitoredusing a modifiedSELSPOTopticaltrackingsystem.As expected,whenpauses markingmajorsyntactic boundarieswereremoved,a high degreeof rhythmicitywithin ratewasobserved,characterized by well-defined periodicities andsmallcoefficients of variation.Whenarticulatorygestures were examined geometrically onthephaseplane,thetrajectories revealed a scalingrelationbetweena gcsture's peakvelocityanddisplacement. Furtherquantitative analysis of articulatormovement asa functionof stress andspeaking ratewasindicativeofa language-modulated dynamicalsystem with linearstiffness and equilibrium(or rest}positionaskey controlparameters,Preliminary modelingwasconsonantwith thisdynamicalperspective which,importantly,doesnot require that timeper sebe a controlledvariable. PACS numbers:43.70.Aj, 43.70.Bk INTRODUCTION

It hasoftenbeensupposed that temporalorganization in biologicalsystemsis ultimately governedby neural rhythmgenerators, biological clocks,metronomes, etc.Physiologists and psychologists, confrontedwith order in the time domain, have not hesitated to posit clocks whose "ticks"definewhenmuscles will activate(e.g.,Kozhevnikov and Chistovich, 1965; Rosenbaumand Patashnik, 1980}. Our approach,however,hasbeendirectedtowardsidentifying and understanding spatiotemporalpatternin articulatory eventsasa dynamicpropertyof naturalsystems rather thanasthe resultof theoperationof somespecialneuralor mentaltime-keeping device(of.Kelsoet al., 1981}.Onceelaborated,we believethisdynamicalperspective may afforda

substantial changes (induced by stressandrate)in thedurationof thevocaliccycle.Thesedata--alongwith otherevidence(reviewed by Fowler,1983}---suggest a vowel-to-vowel organization thatplaces constraints onspeech timing. Althoughspeechcertainlyinvolvesmanyof the same bodypartsaschewing, itsrhythmicbasisisnotclear,in spite

ofthefactthatlinguists andothers havelongclaimed speech

to berhythmic,andpeopleperceiveit to beso(e.g.,Lehiste, 1972;Lenneberg,1967;Lisker,1975;Pike, 1945).Yet experimentershave had enormousdifficultyidentifyingrhythmicityin eitherthearticulatoryor theacoustic domain.One possiblereason--aspointedout by Fowler (1983)with respearto acousticstudies---is that experimentalmeasurementstypicallyusedmaybeinappropriate for capturingthe principledaccountof the ubiquityof temporalconstraintsin natural,temporalstructureof spokensequences. Speaking is an inherentlymultidimensional process; duringspeechdifmovementin generaland in speechin particular.For examferent articulatorsare involvedto differentdegreesand the ple, the internalphasingrelationsamongmusclesand kinespatiotemporal overlapamongmovementsis considerable. matic componentsin rhythmic activitiessuch as locomotion, scratching,respiration,and masticationare preserved Confrontedwith so many simultaneousor nearly simultaneousevents,thereseemslittle chanceof our identifyingany acrossscalarchangesin force and rate (cf. Kelso, 1981; basic temporalregularity,eventhoughour perceptualimGrillnet,1982,forreviews). similarly,i n electromyographic leadusto suppose that oneexists. and kinematicwork on speech{Tulleret al., 1982c,1983; pressions Our approachin thepresentworkwasto stripaway,as Tuller andKelso,1984),timingof consonant productionrelmuch as possible,the influenceof segmentalvariationon ative to vowelproductionwasfoundto be invariantover 266

J. Acoust.Soc.Am.77 (1), January1985

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articulatory movement, byaskingsubjects to speak"reiterantly." That i•., speakerssubstitutedthe syllable/bad or /ma/for eachrealsyllable in theutterance, whilemimicking theutterance's normalprosodic structure.Thebenefitof the reiteranttechniqueis that, by minimizingsegmentalvariabilitywhilepreserving theprosodicpattern(Libermanand Streeter,1978;Nakatani, 1977),we are ableto measurethe movements of articulators (inthiscasethelipsandjaw) that are consistentlyinvolvedin the productionof/ha/and /ma/. In principle,thisprocedureaffordsananalysisof articulatorpatternsin a simpleandaccessible form. We recognize thattherelationship betweenrealspeech and reiterantspeechis not alwaystransparent. We should

stress, however, thatthemainthrustofthepresent workisto

mass-spring system,namely:equilibrium(or rest)position, whichis the positionat whichthe net forceon the massis zero;andlinearstiffness, whichisthereactiveforceperunit displacement.

I. METHODS AND PROCEDURES

Two adultspeakers [onemale(SK, the firstauthorand a nativespeakerof an Ulsterdialectof English},and one female(DW, a speakerof a New Jerseydialectof American English)]recitedthe firstandlastsentences of the "rainbow passage": (1)"Whenthesunlightstrikesraindropsin theair, theyactlikea prismandforma rainbow,"and(2)"Thereis, according to legend,a boilingpotof goldat oneend."After recitingeachsentence, speakers mimickedtheprosodic pattern2-4 times,substituting onlyfoa/or only/ma/for each syllable.So,for example,"Whenthe sunlightstrikesraindropsin theair" wouldbemimickedas"bababababababa haba ba" (whereitalicsindicates a hypothetical stress patternfor thesyllables}. Uponcompletion of thetaskat a normal,conversational rate,it wasthenrepeated at a fasterrate. One of the speakers(SK} repeatedthis procedureat a later date.In all, 392syllables at eachratewereanalyzed.We also obtainedmeasures of eachspeaker's preferredfrequency of

usereiterantspeech asa tool to examinearticulatormotions in a speechlike task.We donotclaimanynecessary generalizationto realspeech although onemightexist(seealsoLarkey,1983).Forinstance, LibermanandStreeter(1978)show the patternof acousticsyllabledurationsto be similar betweenrealandskilledreiterantspeechalthoughthe absolutedurationalvaluesare verydifferent.In termsof production, it seemsunlikelyto us that the controlof the lip-jaw systemfor theproductionofa reiterant/ba/is fundamentally differentwhenthesamesyllableisproduced duringnatural speech.Indeed,we shallquantitatively describecertain jaw•movement overanextended period oftime,byasking the kinematicrelationships (e.g.,betweenan articulator'speak subjectto "wag" thejaw at a comfortable amplitudeand velocityanddisplacement) thathavebeenobserved in many frequency "asif youweregoingto doit all day.""Wagging" othernonreiterant speech production studies. movements werethensampledovera 30-sinterval. In thepresentpaper,we outlinea geometric approach Forspeech andnonspeech tasks,verticaldisplacements forcharacterizing thedynan•ic properties underlying articu- of the lipsandjaw weretrackedusinga devicesimilarin !atory movements during reit•rant speech. Weusethephase principleto the commerciallyavailableSELSPOTsystem, portraitto facilitatetheanalysisof relevantarticulatoryvarwhichemploysinfraredLEDs that canbeplacedmidsagitiableswhenspeakers producethesesimplesequences of syltallyonthenose,lips,andpointof thechin.Modulatedlight lables.To our knowledge,phaseportrait techniques have from the diodesis capturedby a cameraequippedwith a rarely beenemployedin speechproductionstudies,even Schottkyplanardiodelocatedin itsfocalplane.Theoutput thoughtheirroleis to describe theformsof motionin com- ofthephotodiode isfedtoassociated electronics thatdecode plex, multidegree-of-freedom systems(cf. Abraham and thesignals andcompute pairsofx andy coordinates. Up to Shaw,1982).Wereoneto countthe neurons,muscles, and eight channelsof coordinatepotentialsmay be generated joints that cooperateto produceevena simpleutterance, simultaneously, eachwith a bandwidthof 0-500 Hz. These literallythousandsof suchelementswouldbe involved.Yet potentialsare thenfed to first-stage dc offsetpreamplifiers normalspeechis usuallycoherentand organized:A low diwhichcenterthesignalsaboutthe0-dclevel.Followingthe mensional patternemerges froma systemof highdimension- offsetadjustment,the coordinatevaluesare transmittedvia ality that canbe controlledwith relativelyfew dynamicpadc coupledamplifiers,checkedby meansof a monitoring rameters. 1Thusourapproach isonein whichweattemptto oscilloscope, andrecorded.Oncethesubjectwasseatedwith characterizeregularitiesof articulatorpatternin termsof a theLEDs in place,calibration wasachieved by raisingthe relativelyabstractfunctional organization(cf. Kelso and cameraa knowndistance(2 cm)andrecordingtheoutputof Tuller, 1984a).We do notattemptto modelperipheral bio- thelowerlipLED.Simultaneous aeonstie recordings were mechanicsor neurophysiological mechanisms. Rather we alsomade.The movementdatawererecordedon FM tape usethe phaseportraitas a way of uncoveringqualitatively and sampledat 200 Hz in later computeranalysis.This inthesystem's controlstructureandasa prefaceto a quantita- eludednumericalsmoothing (usinga 25-mstriangularwintive treatmentof articulatorytrajectories.In doing so we dow) and differentiation(usinga two-pointcentral differobserveboth invariantand systematically varyingfeatures encealgorithm;Jameset aL, 1977)for obtainingthe of motionwhenstressand speakingrate are changed.Perderivativesof motion(velocity,acceleration). hapsmost important,our results,analyzedgeometrically Figure I showsan exampleof the positionand velocity andinterpretedfrom a dynamicperspective, do not require of thelowerlip andjaw (i.e.,the LEDs attachedto lowerlip the assumptionthat time itself is a controlledvariable.Inandjaw) for the firstpart of sentence1, "When the sunlight stead,theformof articulatortrajectories overtimeisseenas strikesraindropsin the air," where/ha/is the reiterated a consequence of a controlstructurewhosedynamicparamsyllable.In the movementtraces,peaksand valleysdenote etersare functionallyequivalentto thoseof a mechanical thehighandlowverticalpositions achieved bytheindicated 267

J. Acoust.Sec. Am., Vol. 77, No. 1, January 1985

Kelsoetal.: Qualitativedynamicanalysis

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A. Global temporal regularity

JAW

First we showseparatelyfor the two ratesand two reitcrantsyllables themeandurationbetweensuccessive peaks and the associated standard deviations. The values shown in

POSITION

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TableI areaveragedacrosssubjects and sentences for both jaw andlowerlip motions(i.e.,motionsof thejaw andlower lip LEDs).In orderto studyarticulatorymotionsper se,we haveremovedintervalsthatspanmajorsyntactic breaksand the firstandlastsyllablesof thesentence, i.e.,wherestartup, pauses, andlengthening effectspredominate. The durationaldatashowquitelow variabilityregardlessof rate, with coefficientsof variation in the 10%-20% range.Thetwospeakers arealsoverysimilarin theirdurational behavioras revealedin the small between-subject

y,v7V V ¾¾YVI LOWER LIP

standarddeviation of the means.Mean eyele durationsfor

POSITION

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ix,/,,, v v v v-v vyv,I FIG. 1. Positionand velocityovertime of lowerlip andjaw LEDs for the

reiterant production of "Whenthesunlight strikes ra•.ndrops in theair." /ba/is the reiterantsyllable.

articulators.Thuspeaksoccurduringlip closurefor the bilabial stopand valleysoccurduringproductionof the low vowel/a/. In the velocitytraces,peaksand valleysare the maximumvelocitiesattainedgoinginto andout of a closure, respectively.The peaksand valleyswere determinedby a computer program which also calculatedmeans (M) and standarddeviations(s.d.)for peak-to-peakcycle duration and displacementof opening(peak-to-valley)and closing (valley-to-peak) gestures. II. RESULTS

thethreeexperimentalsessions were211 ms(approximately 5 Hz} for the normal rate and 167 ms (approximately6 Hz} for the fastrate. In this case,thejaw exhibitsa periodicity similarto that of the lowerlip. Not surprisingly, the data contrastwith thoseof Ohala's(1975)earlierstudyin which 10000 consecutive jaw openinggestures wereobtainedduring a 1.5-hreadingperiod.Ohala(1975}foundlargedura-

AND DISCUSSION

Each of the followingsectionsis.designedto be selfcontainedin that a discussion accompanies eachsetof empiricalfindings.First we presentdata pertainingto the global temporal regularity of articulator movementthat was observedin the experiments.Second,a qualitativedynamic

pausesandsegmental factors}accompanied by a dominant, but weakly defined,periodicityof about 250 ms (4 Hz}. Ohalaandothers{e.g.,Lindbiota,1983)havesuggested that thisperiodicitymay correspond to the "preferredfrequency of the mandible."However,the preferredwaggingfrequenciesofjaw movement forourtwospeakers (0.81and2.04Hz, s.d.s-- 0.06and0.21 Hz, respectively} aremuchslowerthan thefrequencies foundeitherby usfor rciterantspeechor by Ohalafor readspeech. It isclearthen,thatneitherthesharply definedperiodicityobserved by usin reiterantspeech nor theweaklydefinedcyclingfoundby Ohalain readspeechis the sameas the preferredfrequencyof the mandiblein our nonspeechtask (seealsoNelsonet al., 1984,for differences betweenpreferred frequenciesof mandible movementin speechlike andnonspeech tasks).We alsofoundthat the peTABLE I. Meansandstandard deviations of peak-to-peak durationin ms and frequency {f) in Hz for jaw (•) and lowerlip ILL} duringreiterant speech attworates.Between-subject standard deviations areinparentheses.

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portrait to describethe formsof motionthat are produced. Followingisa quantitativekinematicanalysisof motionand its derivativesthat detailseffectsof the local changesinducedby stressandspeakingratetransformations.We try to maintaincontinuityof presentation in thisquantitativesection by proceedingfrom lower-orderto higher-orderkinematic relations.Finally we presentsomeof our preliminary effortsto model the presentarticulatoryfindingsusingan approachbasedin dynamicalsystems theoryandsupported by recentresultsin the fieldof physiologicalmotor control.

f4.70(0.11} 4.72(0.10) 4.72.(0.06)4.73{0.06) 4.72(0.08) 4.73(0.14)

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d.Acoust. Sec.Am.,Vol.77, No.1,January1985

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fiodicity was unaffected bythesyllable thatwas used to mimicrealspeech• The largestmeandurationaldifference regardless of ratecondition between lea/and/ma/for any articulatorwas3 ms{seeTableI}. In short,whensegmental variationis minimized,it is possibleto identifyrelatively stablearticulatoryperiodicity.The periodicityis notperfectly isochronousbecausethere arc systematicvariationsconcomitantwith stressandrate (seeSec.II C).

SINE WAVE C5 Hz)

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B. A geometric (qualitativedynamic)analysis=

-315

In thefollowinggeometric analysis, phaseplanetrajectoriesare generated by continuously plottingthe relationshipbetween, in thiscase,articulatorposition x anditsderivative,velocity•. Asanexample, consider theidealized case shownin Fig. 2. The uppertraceis a computergenerated sinewaveof 5 Hz with a peak-to-valleydisplacement defined to be 20 min. The peakpositioncorresponds to the consonantclosure,andthevalleypositionto themaximumopeningfor the vowel.Pointsof maximumdownward(opening} and upward(closing)velocityfall at the midpointsof the positiontrace.To createa phaseplanetrajectoryshownon the lowerpart of Fig. 2, we plot successive positionpoints andtheircorresponding velocities ascoordinates on a plane whoseverticalaxisdenotespositionand whosehorizontal

CLOSED

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axisdenotes velocity?Thearrowheads onthecircledenote thedirectionof motionon the plane.Thusonecycleor orbit corresponds to theintervalbetweensuccessive closures,with

theopeninggesture onthelefthalfandtheclosinggesture on theright.•rotethattimeitselfisnotanexplicitvariablein this description.

Figure 3shows phase pldne trajectories forthejawand lowerlip LEDs of "Whenthe sunlightstrikesraindropsin theair," usingreiterantlea/spoken at a normalrate.Qualitatively,the shapesof the trajectoriesare quite similar acrossthe ten syllablesplotted.There is a strongtendency, for example,for displacementand peak velocityto covary directly (seeSec.II C). Normal and fast reiterant produc-

0

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VELOCITY (mm/s)

FIG. 2. Top:idealizedpositionandvelocityovertimeof articulatormovement.Bottom:corresponding phaseplanetrajectories. Abscissa isvelocity, ordinateis position(seetextfor details}.

tiens forsubjects SKandDWofthesecond partdfthefirst sentence,"they act like a prismand form a rainbow,"are shownin Figs.4 and 5. The mutualrelationshipbetweenthe kinematicvariablesof positionand velocityis accentuated by the rate manipulation,particularlyfor subjectSK. Once

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Kelsoeta/.: Qualitativedynamicanalysis

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NORMAl

RATE

smallerdisplacements and peakvelocitiesthan the stressed syllables,thus maintaininga global similarity of (elliptical) trajectory shapeacrossunstressedand stressedgestures.

CLOSED

Also observed,however, are subtie differencesbetweentra-

jectory shapesassociatedwith differentgesturaldisplacements.For example,the orbitsappearto be slightlymore compressed horizontallyfor largerdisplacement gestures relativeto shorterdisplacement gestures.In Sec.II C, we will quantifyboth the globalsimilaritiesand subtledifferencesamonggesturaltrajectoryshapes.

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C. Quantitative kinematic analysis

RATE

In Sec.I! C we proceedto quantifyspecificeffectsof

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speakingrate and stresson articulatory movementsin an effort to answerthe followingquestions.First, what kinematicvariab]esor relationsamongvariablesmightinformus aboutthe centre/of speechgestures? Second,what kind of

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FIG. 4. Phase planetrajectories oflowerlipmotions forthesecond partof sentence 1,"Theyactlikea prismandforma rainbow" produced atnormal

andfastspeaking rates with,/ha/as thereiterant syllable; subject isSK. again, even when there is a clear distinctionbetweenthe

trajectories corresponding to stressed andunstressed syllables, their orbital shapesare generallysimilar. The unstressed (sometimes reduced)syllablesare characterizedby NORMAl

RATE

CLOSED

regularity,if any,existsin themotionsof speecharticulaters acrosschangesin stressand speakingrate, and how might suchregularityberationalized? Althoughweappreciatethat there are many idiosyncraticdifferencesamongspeakers, dialects,andlanguages, our emphasishereis on identifying

whatiscommon across suchdiversity. In short, canwebegin to definea "deepstructure"for speechmotorcontrolthat can be recognizedin the faceof muchsurfacevariability, and,if so,on whatprinciple(s) is it based? We beginwithananalysis of thespace-time characteristicsof articulatormovementand its derivatives,with the emphasis nowon the gesture(opening andclosing)rather. than the cycle.Becauseof the enormousamountof kinematic datainvolved, werestrictourconcerns (unless otherwise

indicated) to(a)themotions ofthejawandlowerlipcomplex forthesyllable/ha/duringreiterantspeech, and(b)thesingleexperimental session for eachspeaker, i.e.,omittingthe repeated session. Thisamounts to 232gestures for speaker DW (116 openingand 116 closing)and 464 gestures for

speaker SK(232opening and232closing).

I - 290

340

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The generalstatisticalanalysisof the kinematicvariablestakesthe form of a gesture(opening,closing)X stress (stressed, unstressed) X rate (norma],fast)analysisof variancefor eachdependentvariab]e,followedby correlational analysisbetweenvariab]cs(e.g.,displacement versustime) whereappropriate.In order to facilitatecommunicationof theresultswereportthedegrees of freedomfor thestatistical maineffectsandinteractionsonlyonce.For subjectDW the numeratorand denominatordegreesof freedomare 1 and 224;for subjectSK theyare 1 and456.

RATE

CLOSED

1. Displacement, movement t/me, and the/r re/at/on 14.0

OPEN

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- 300

Tables II and III provide the mean displacementand mean movementtimesof the openingand closinggestures for the syllable/ba/, as a functionof speakingrate and stress.The meandataordersystematically for bothkinemat-

0

VELOCITY

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(mm/s)

S: DW

FIG. 5.Phase plane t•jectories oflowerlipmotions forthesecond partof sentence 1,"Theyactlikea prismandforma rainbow" produced atnormal

andfastspeaking rateswith/ba/asthereiterant syllable; subject isDW. 270

J. Acoust. Sec.Am.,Vol.77,No.1,January 1985

ic variablesin both subjects,althoughthe magnitudeof changeacrossrateandstressis idiosyncratic. Similarresults havebeenreportedby others(e.g.,KuehnandMoll, 1976; Tulleretal., 1982b).For displacement, sincethelipsalways 'returnto closure,the maineffectof gesturetype (opening versusclosing)wasnot significantin eithersubject'sdata; Kelsoeta/.: Qualitative dynamic analysis

270

TABLEII. Kinematic values of displacement, time,andpeakVelocity across rateandstress variations (opening gestures,/ba/). Stressed

Unstressed

For movementtime, openinggestures as a classtook longerthanclosing gestures. All themovement timevalues for similar conditionsreportedin Table II (opening)are

greaterthan thosereportedin Table III (closing),a finding substantiated by a significantgesturemain effectfor DW, F DW m 14.58 123.9 229.2 11,80 112.4 204.0 =171.43, p. 300 tsingle-dimensional movements in discretetargetingtasks canbegenerated by second-order, linearmodelswhosepaO, 240 ILl rameters includedamping,stiffness, andrestangle(cf.Bizzi, v 180 1980; Cooke, 1980; Fel'dman and Latash, 1982;Kelsoand o Holt, 1980for reviews).In short,thelimb exhibitsbehavior 120 qualitatively similarto a dampedmass-spring systemfor thesetasks(Fel'dman,1966).Suchsystems areintrinsically 6O self-equilibrating in the sensethat the "endpoints"or S:DW "movementtargets"areachieved regardless of initialconditions.In normaland deafferented animals,for example,it 5 10 15 20 25 hasbeenshownthatdesiredhead(Bizzietal., 1976)andlimb AMPLITUDE (mm} positions (PolitandBizzi,1978)areattainable withoutstarting position information even when the limb is perturbedon FIG. 9. Scatterplot of peakvelocityversus(valley-to-peak) amplitude(lower lip) of eachsubjoct's closinggestures associated with the vowel-conso- its path to the goal.Similarly,Kelso (1977)demonstrated nantportionof the syllable.Legendspecifies conditions. that fingerlocalizationabilityis not seriously impairedin functionallydeafferented humans,or individualswith the metacarpophalangeal joint capsulesurgicallyremoved,in ableacross subcategories. Howmighttheslopes of thekine- spiteofchanges in initialconditions or unexpected perturbamaticrelation between Vpandd beihterpreted withrespect tions(KelsoandHolt, 1980;seealsoKelsoandTuller, 1983, to the controlprocesses underlyingthe reiterantspeech and Tye etal., 1983for evidencein speech). Closed-loop task?First we addressthe significance of the overall Vp-d notionsthat rely on peripheralfeedbackbreakdownin the faceof suchevidence.Further, kinematic variablesneednot relation;then we considerthe specificeffectsof rate and stress. becontrolled explicitly.In a dynamicsystemlikea damped Recenttheoreticalconsiderations and empiricalfindmass-spring (orpointattractor,Abrahamand Shaw,1982), ingsin themotorcontrolfieldsupportanaccountof the Vpkinematicvariations in displacement, velocity,andaccelerad relationthat is basedon a movement's dynamics, not its tionoccurasa resultof thespecified parameter, values,and kinematics.Relationsamongkinematicvariablesare useful sensory"feedback"in its conventional formis not required. to describe the space-timebehaviorof articulaters,but it is Nor, importantly,is durationa controlled variable(seeSec. dynamics thatcause such motions. Thatis,it isimportant to [I B). For sustained,stable cyclic movementsof dissipative realizethat changes in displacement andits timederivatives (velocityand acceleration)are consequences of dynamical systemsthe appropriatedynamicregimeis a limit cycle(or 420

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periodicattractor,Abrahamand Shaw,1982).In suchsystems,the sameorbit is achievedregardlessof initial conditionsor temporaryperturbations. In the absence of imposed perturbations,such systemscan display near-sinusoidal steady-state motionsthatmaybetreatedasifthey weregeneratedby simplernondissipative mass-spring dynamics.As mentioned earlier, a constant slope in the relationship between eachgesture's peakvelocityanddisplacement for a givensetof gestures indicates thatthegestures areperfectly isochronous. With regardto anhypothesized underlyinglinear(harmonic) or nonlinear(anharmonic) mass-spring model, the Vp-dslopeisindicativeof thestiffness overtherange of gesturaldisplacements examined.Roughly speaking,a constantVp-d slopefor a givengesturalsubsetimpliesthat the averagemass-nommlized stiffness(K**v)of the spring functionsunderlyingthe gestures is the sameacrossthe observedrange. 4 Recently,Ostryet al. (1983)haveshownin a studyof tonguedotsummovement thattheslopeof the Vp-d relation variessystematicallywith stress,but lessso as a functionof rate.In their data,particularlyfor openinggestures,theslopeof therelationship wasgreaterfor unstressed than stressedgestures,suggesting to them that the tongue musclesystemwasactuallystifferin the unstressed environment (seealsoLaferriere, 1982,for evidenceleadingto the sameconclusion).More recently,observationsof tongue dorsumkinematicsasa functionof rate,vowel(/u/,/o/, and /a/}, andconsonant (/k/ and/g/} havebeeninterpretedas

indicative of anunderlying mass-spring controlregimeWith constantlinearstiffness for a givengesture{OstryandMunhall, 1984;seealsoMunhall and Ostry, 1984). Our dataalsosuggest that unstressed gestures are characterizedby greaterstiffness (K,*v)values(asrevealedin Table V by the slopesof the Vp-d relationsand the phaseportraits}thanstressed ones.This isapparentin threeout of four casesfor both.openinganddosinggestures (TableV). Interestingly,we showalsothat the Vp-d slopesfor closinggestures(againwith a singleexception)are greaterthan those for openinggestures,particularlyfor unstressed syllables. Like theOstryetaL (1983)tonguedata,therateeffectson the slopeof the gp-d relationship arelessclearcut. In onlyfive of eightpossible cases,slopeincreases asa functionof rate.

tude, unstressed gesturesthan largeramplitude,stressed gestures.An alternativenotion is that during reiterant speechthe jaw-lip systembehaveslike a soft, nonlinear mass-spring system where,for example, springforceequals

-- kx + exs, withk ande denoting linearandcubicstiffnesses, respectively (of.JordanandSmith,1977;Kelsoet aL, 1983}.For suchsprings,the net stiffness decreases nonlin-

earlywithdeviation fromtheequilibrium position. Hence, shorter amplitude gestures, involving relatively smalldevia-

tionsfromequilibrium, arecharacterized byMgheraverage stiffnesscs overthecourse of themovement thanlargeramplitudegestures {see alsofootnote 4).Thissecond hypothesis ispresaged on theassumption that all the motionswe have observed arisefrom a singleunderlying nonlinearspring function with constantlinear and cubic stiffnesscoefficients.

Sincea gesture's linearstiffness coefficient isindexedby the slopeof the aceelerat.ion-displacemcnt functionnear the gesture's midpoint{corresponding rougMytoitsequilibrium

position}, wecandistinguish between theseforegoing alternatives.

Theacceleration dataofthelowerlip-jawcombination wereobtainedby velocitydifferentiationand smoothedover

a 25-msinterval(seeSec.I). Linearinstantaneous, mass-nor-

realized stiffness, K*, wasestimated usinga computer routinethatfirstfoundthemidpoint ofa givenopening orclosing gestureand then obtainedthe position(x) and acceleration {X}coordinates of thedatasample toeachside of themidpoint. Thisprocedure allowedusto compute the slope around thehypothesized equilibrium position. IfK * is unchanged across conditions theslopes should bestatistically equal. Thusif thedatalieona single spring function {linear or nonlinear}K * shouldbe identicalcloseto the move-

ment'smidpoint.Differentslopesof the x, X function, however, wouldsuggest separate springfunctions withdistinct linearstiffness components? Figure10{inset}showshowK * wasestimatedandalso OPEN

Withonenotable exception, however, in whicha fourfo ld

:

•(1--X2

• i

SLOPE = Cx•X•

increasein slopeoccurs,slopechangesbetweenfastand norreallyproducedgestures are fairly small.

Although thedataingeneral suggest thatstiffness (K,*v) isa keysystem parameter, a comparison of the Vp-dslope data(whichindexes K•*v)andthedisplacement datashown in Tables II and III revealsthat stiffnessis not constantfor

movements of different displacements withina givenstress condition(seealsoOstryandMunhall, 1984).In fact,stiffnesschanges invariablywith displacement bothwithin and acrossstresscategories.

-

(x2 \

CLOSED

3700

'•

6.4•

11.8

3. Theacceleration-displacement function

Thereare at leasttwo possibilities that canaccountfor

theobserved changein stiffness (K •*•)asa functionof displacement. Oneis that a linearspringfunctionholds,for whichspring forceequals--/or andforwhichdifferent valuesof linearstiffness k areelected for,say,shorter amp]i-

276

J.Acoust. Sec.Am., VoL77,No.1,January 1985

DISPLACEMENT

(turn}

FIG. 10.Acceleration versus displacement fromrestposition fortheopeningandclosing gestures associated witha stressed andunstresscd syllable {see text}.Thesmaller displacements andsteeper slopes correspond to the unstressed gestures. Theopening gestures startatthebottom right;closing gestures startat top left.

Kelso oral.:Qualitative dynamic analysis

276

an example of the acceleration-displacement differences Thesedatacorrespond ratherwellto thepeakvelocitybetween theopening anddosing gestures ofa stressed versus displacement findingsdiscussed in Sec.II C 3. The present a• onstressed syllable,the fourthandfifthsyllables (itali- acceleration-displacement results,however,affordan addicized)of the reiterantversions of "Thereis ac-cordingto legend..." (normalrate,SK).Differences in slopeareapparent,withtheshorteramplitude, unstressed gestures displaying greaterK * valuesthan the longeramplitude,stressed gestures.

Statisticalanalysisbearsthispictureout.The meanestimatedK * andits standarddeviationare providedfor each subject andeachgesturetypeasa functionof conditions in TableVI. Stressed gestures asa classhavelowerK * values thanunstressed gestures, Fs = 9.38and192.13,ps < 0.01for DW and.SK,respectively. SubjectDW displaysa gesture typemaineffect,F = 19.16,p < 0.0001withK * significantly greaterfor closingthanopeninggestures. Additionally, for SK thereisa gesturetypeX stress interaction, F = 20.39, p 0.1.(2)The in-

flueneeof thedoubledifferentiation (i.e.,acceleration derivedfromposition)and concomitant smoothing procedures wastestedat a movement frequency (5 Hz) comparabic to thatusedby speakers in thepresent study (seeTableI), by comparing theoutputof an Entranaccelerometer (model EG-C-240-10D)to the secondderivativeof positionoutputsmoothedtwice

at 25 ms.Takingintoaccount thegainreduction induced bysmoothing (see above},wefoundtheaverage,absolute differneebetweentransducer (unsmoothed) andnumerical{smoothed twiceat 25 ms)acceleration to heless than 5% of the rangeof measuredmovements. The crosscorrelation between the raw, unsmoothedand the smoothed,derived signal was r ----0.98. Note:Not all the x,X functionsapproximatedstraightlinesas closelyas thoseillustratedin Fig. 10. SomewereS shaped("hooked"at

displacement extrema).However,our smoothing procedures did not re-

move the"hooks." Moreimpoflant, ourestimates ofK* wer•notaffected by thepresence of such"hooks."

namies,whichplotspositionon the horizontalaxisandvelocityon the verticalaxis.Sincethedisplacements measured herearevertical,nothoriAbbs,J.H. {1973)."Theinfluence ofthegammamotorsystem onjaw movezontal,wehavesimplyswitched theaxestoconformto thebehavior ofthe menCs duringspeech: A theoretical framework andsomepreliminary oblip-jawsystem andto facilitatevisualization of theprocess. servations," J. SpeechHear.Res.16, 175-200. 4Bothlinearandnonlinear maas-spring systems candisplaynear-sinusoidal Abraham, R. H., andShaw,C. D. {1952). Dynamics•TheGeometry ofBecyclicmotions whose observed peak-to-peak periodT-- 2•r//•½where havior{Aerial,SantaCruz,CA). denotes the observed angularfrequency for the cycle.For systems with

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T----D• 4-Do----2D----2D•,and/•c=//o =/2•. Moregenerally, incases

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