appeared

Anewgaitparameterizationtechniquebymeansof

cyclogrammoments:

Applicationtohumanslopewalking

AmbarishGoswami

INRIARhˆone-Alpes

655avenuedel’Europe,ZIRST

38330MontbonnotSaintMartin,France

*****************************

Appearedin

Gait&Posture

August1998

Abstract

Anewparameterizationtechniqueforthesystematiccharacterizationofhumanwalkinggaitindiver

meterizationwemeanaquantitativeexpressionofcer-

taingaitdescriptorsasthefunctionofanexternalparameter,hematical

quantitiesderivedfromthegeometricfeaturesofthehip-kneecyclogramsarethemaingaitdescriptorscon-

nstratethatthedescriptors,expresdinageneralttingasthegeometric

momentsofthecyclogramcontours,canmeaningfullyreflecttheevolutionofthegaitkinematicsondifferent

ideanewinterpretationofthecyclogramperimeteranddiscovertwopotentialinvariantsof

mentalslope-walkingdataobtainedat

intervalwithintherangeofto

()onavariable-inclinationtreadmillwasudinthisstudy.

Theparameterizationprocedureprentedhereisgeneralinnatureandmaybeemployedwithoutrestric-

tiontoanyclodcurvesuchasthephadiagram,themoment-anglediagram,andthevelocity-velocity

hniquemaybeutilizedforthequantitativecharacterizationofnormalgait,

globalcomparisonoftwodifferentgaits,clinicalidentificationofpathologicalconditionsandforthetracking

ofprogressofpatientsunderrehabilitationprogram.

Keywords:gaitparameterization,slopewalking,cyclogram,geometricmoments,gaitkinematics,invariants

GoswamiGaitparameterization...

1Introduction

MotivationCharacterizationofhumangaitinaquantitativeandobjectivemannerhasmanypotentialben-

efitsinclinicaldiagnosisandrehabilitationaswellasintheenhancementofourbasicunderstandingofthe

eoftheimpressivesophisticationofourprentdaydatacollectionsys-

tems,communicabledescriptionsofcertaingaitconditionsremainsurprisinglydiffierdescribing

theprogressofapatient’lde-

scriptionofthejoint’skinem

isneededisacomprehensiveglobalpictureoranobjectivecaptureofinformation[40]whichmaybesubjected

toquantitativeanalysis.

Acondexample,whichweaddressinthisarticle,istheevolutionofthewalkinggaitinrespontoa

aitstudyliteraturethisisknownasslope-walking,grade-walkingand

edsurfacesarefrequentlyencounteredintheeverydaylifebuttheireffectonthegaitis

slopemodifiestheinfluenceofgravityonthehumanbody,whichisknown

lstudyofhumanlocomotion,ascanbe

eninthesimplifiedsketchofFig.1,revealsthatourgaitpatternschangeconsiderably,morethanisnecessary

teanidealsituationwhere

wehaveacompactmathematicaldescriptionwhichassignsagaitpatternnumber,say

,forthegaiton

roundslopechanges,thisnumberalsochangesrefl

powerfultoolwouldhaveveralpracticalussuchastheobjectivecharacterizationofnormalgait,global

comparisonoftwodifferentgaits,clinicalidentificationofpathologicalconditionsandthetrackingofprogress

ofpatientsunderrehabilitationprograms.

eginningofthelastdecade[44,51]udtheso-called

chain-encodingmethod[17],acomputerizedprocessingtechniqueoflinedrawings,tocorrelatetheshapes

metriccongruityofany

twoshapepatternswasconsistentlyreflectedbytheircross-correlationcoeffighthechain-code

reprentationofcontoursifficientforcomputerprocessingandufulfordeterminingthecross-correlation

coefficients,theyare,however,abstractnumbersanddonotgiveanyphysicalinsightintotheactualshapeof

thepatternsunderstudy.

ectiveofthispaperistointroduceageneralandphysi-

callyintuitivesystemoryto

thesimplisticsituationportrayedinFig.1,inreality,giventhecomplexityofthehumangait,wewouldneed

tidimensionalspacesuchquantitiescanbereprented

entgaitsarereprentedbydifferentpointsandtheevolutionof

agaitcanbecharacterizedbyalocusofpointsinthatspace.

Fromadynamicsystemspointofviewacompletetof“statevariables”uniquelydescribeasystem[36].

Forexample,thejointangleandthejointvelocityconstitutethestatevariablesofasimplependulumandcan

ghthelevelofourcurrentknowledgedoesnotpermitustoarch

forthecompletetofstatevariablesofthecomplexdynamicscalledthehumanlocomotion,weneverthe-

lesxtractsomemeasurable,

gaitdescriptorsmayreflectthechangesingaitpatterninrespontosomeexternallycontrollablefactoror

yspeaking,aparameterisanimpodconditionandadescriptoristheresultofthesystem’s

esofsomecommonlyudgaitdescriptorsarethesteplength,

stepfrequency,eter,ontheotherhand,couldbethe

Figure1:Asimplifiedsketchshowingthetypicalwalkingpatternsondifferentinclinationsandtheirquantification.

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GoswamiGaitparameterization...

weightofloadcarried,oraswestudyhere,thegroundslope1.

Byparameterizationwerefertothesystematicandobjectivedescriptionoftheevolutionofacertaingaitdescriptor

pleofthiswillbeacurveoramathematicalexpressionrelatingtheload

urrentstudy,thegroundslopehasbeenconsideredasthe

onlyparameterandthegaitdescriptorsareobtainedfromthegeometricfeaturesofthecyclograms[20],the

rationalebehindthechoicesbeinggiveninthefollowing.

ScopeofthisworkThegoalofthispaperistointroduceacoherent,meaningful,andefficienttechnique

ntedmethodisbaduponthegeometricmomentsof

ampleoftheapplicationofthistechniquewehaveconsiderednatural

dliketoemphasizethatthechoiceofthis

particularexampleismotivatedbyourownrearchinterests(e“Motivationfromrobotics”below)and

therelativefamiliarityofcyclogramsinthebiomechanicscommunityandhasnospecialconnectionwiththe

parameterizationmethod.

Infact,themoment-badparameterizationmethodcanbedirectlyappliednotonlytothecyclogramsof

otherrepetitiveactivitiesbutequallytootherreprentations(suchasthephadiagram[30],themoment-

anglediagram[18],andthevelocity-velocitycurves[51])ofrepetitiveactivities.

Whystudycyclograms?

simplypointoutthatalthoughmostofthemeasurablevariablesofthehumangaitrespondtoaparameter

change,andparameterizationmaybeperformed,inprinciple,withanyofthevariables,analysisbadon

nreasonforthisisthefactthatthe

clodtrajectoriesreprentformsorshapesthatprovideuswithimportantinsightsintothesystem[8,23]and

aredescribablebyappropriategeometricproperties[25,24].Wewilleinthefollowingthatastheground

slopegraduallychanges,thehip-kneecyclogram,obtainedbyplottingthehipangleversusthekneeangle

andbyomittingthetimevariablesfromthetwosignals,changesitsformgivingusaclearindicationofthe

modifier,cyclogramsreflectthegaitkinematicsduringthe

totalgaitcyclewhichisdifferentfromhavingotherdiscretemeasuressuchasthesteplength,orwalking

speed,whicharemorecommonintheliterature[35,53,52].Afurtherjustificationforchoosingcyclograms

overtime-angleplotsisthefactthatlocomotion,atightlycoordinatedmovementofverallimbgmentsis

morenaturallygraspedasthecoupledevolutionoftwoormorejointsratherthanfromthestudyofindividual

jointkinematics[8].

Whymoment-badshapecharacterization?Onecanimagineanumberofquantitiessuchasthe

perimeterandthearea,thatrefliceofmoment-badchar-

acterizationofthecyclogramfeaturesisjustifiedbythefactthatmoments(ofdifferentorders)canbeviewed

oment-badscheme,thecy-

clogramperimeteristhezeroth-ordermoment,thepositionofits“centerofmass”(CM)isacombinationof

thezeroth-orderandthefirlythehigherordermomentsreflectotherfeatures.

MotivationfromroboticsBeforeleavingthisctionweshouldmentionanothersourceofmotivation

thatguidemsfromour

challengeofformulatingacontrollawforabipedrobotwalkingondifferentinclinations[15].Acontrol

lawgeneheassumption

thatabiologicalsystemifficientoroptimalinsomen,weintendtoidentify“biologically”motivated

optimalitycriteriaoratleast,tomimichumanlocomotionwiththehopethattherobotwillbeendowedwith

ssiblethattheentiredynamicsofacomplicatedsystemcanbegenerated

fromasmalltofinflobrved

thatinsimplifiedpassivebipedrobotmodelsthegroundslopecompletelyspecifiesthedynamics[21].Our

long-termobjectiveistofindtheprinciplesbehindsuchhighlyorganizedmotionsandtoemploythemasthe

controllaws.

StructureofthispaperThestructureofthepaperisasfollows:Sections2and3providethebackgroundof

ribestheconstructionandinterpretation

ofcyclogramn3consists

prenttheexperimentalprotocolforthegait

1Somequantities,suchasstepfrequencyandwalkingspeedcanbeeitherdescriptorsorparameters(whenimpodbyametronome

andatreadmill,respectively)iteraturegaitdescriptorsandparametersaresometimecalledthe

dependentvariablesandtheindependentvariables,respectively.

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GoswamiGaitparameterization...

hip

t

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knee

ankle

s

h

a

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k

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i

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f

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Figure2:Astchalsoshowstypicalmarker

positionsonthelimbgmentswhicharerecordedwithacamerasystem.

dataudinthisworkandanoverallsnapshotoftheresults(Figs6and3.3).Section4formsthecoreofthis

ereportthetechniquesadoptedinthispaperforthecomputationofgeneralizedmomentsofthe

cyclograms(Section4.1)andthenapplythetechniquestocalculatevariousfeaturesofthegaitcyclogramsas

theyevolveasafunctionofthegroundslope(Section4.2).FinallySection5drawstheconclusionsandpoints

outsomeoftheopenquestions.

2Cyclogramsrevisited

Theconceptofcyclograms2,althoughknowntothebiomechanicscommunity,hasnotbeenenveryfre-

ewinthisctionhowtoconstructacyclogram,howto

interpretatypicalcyclogramoflevelwalkinggait,andprovidesomehighlightsofthehistoricaldevelopments

ofcyclograms.

2.1Whatarecyclogramsandhowtoconstructone?

Commonlyhumangaitdataconsistoftherecordedpositionsofretro-reflectivemarkerstapedontheskinat

theextremitiesofthelimbgments(thethigh,theshanketc.)lesbetweeneachtwo

gmentsaresu.2showsa

sketchoftheright

thehipangleandthekneeangle,respectively.

Fig.3(a)and3(b)showtwotime-angleplotscorrespondingtothekneeandthehipangleduringone

gramisformedbyignoringthetimeaxisofeachcurveanddirectlyplottingkneean-

gleVShipangleasshowninFig.3(c).Aformalwayofdescribingcyclogramsistoidentifythemasthe

so-called“parametriccurves”.Aparametriccurveisobtainedbydirectlyplottingtheassociatedvariables,

,whereeachvariableisafunctionofaparameter,.Intheprentcontextthejoint

antageofthisformaldefinitionisthatitcanbe

extendedtoincludeothercurvessuchasthephadiagrams[4,31,30]andthemomentanglediagrams[18].

Pleanotethatforthejointangleassignmentconventionadoptedinthispaper,theplanarcyclograms

sothatalthoughthepointsonthecurves3(a)and3(b)areequally

cingofpointsonacyclogramisdirectlypropor-

ejointsmoveslowly,thepointsareclospaced.

2Awebarchforthekeyword“cyclogram”pointstonumerousarticlesoncyclogramsrelatedtothefunctioningofinstrumentsud

inspaceflights!

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Figure3:Constructionofahnter-clockwi

rFig.(d)wasprovidedbytheGaitLab,Universityof

Waterloo,.

Steadyhumanlocomotionisalmostperiodicandcyclogramsobtainedfromjointanglesignalsofequal

timelengths,suchasFigs.3(a)and3(b),logramfeaturesthatweuin

thispaperforparameterizationareesntiallyvalidforsuchcontoursbutforalgorithmicsimplicitywewill

assumethatthecontoursareclod.

Finally,cyclogramsdonothavetobeplanaralthoughforvisualizationpurposweshouldlimitourlves

.3(d)showsa3-dimensionalcyclogramobtainedbysimultaneouslyplotting

thehip,[55]forexamplesofsometraditional3-dhip-knee-ankle

cyclogramsand[6]for3-dcyclogramsobtainedfromabsoluteelevationanglesofthigh-shank-foot.

CyclogramsandphadiagramsWehavementionedearlierthatourparameterizationmethodiqually

applicabletoothercyclicreprentationsoflocomotionsuchasthephadiagram[4,30,31],themoment-angle

diagram[18],andthevelocity-velocitycurves[51].Sincethephadiagramhasaformidablefollowingand

arephysicallymorefundamentalthanthecyclograms,itisimportanttodistinguishbetweenthetwo.

mostpopulardefinitionsofthe

phaspacedescribeitasthespaceconsistingofthegeneralizedcoordinate/generalizedmomentumvariables

andthegeneralizedcoordinate/generalizedvelocityvariables[3,26].Theconddefinition,accordingto

whichthephaspaceisidenticaltothestatespace,ofa

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GoswamiGaitparameterization...

systemisreprentedbyapointinitsphaspaceandtheevolutionofthesystemisgivenbyatrajectoryin

thephaspace,calledthephadiagram.

Inourcurrentcontext,phaspacewouldcontainthejointdisplacementsandthejointtrajectoriesofthe

movementunderstudyandthereforecanbeconsideredasasupertofcyclograms,whichcontainthejoint

r,sinceneithertheentirephadiagramnortheentirecyclogramofamulti-

degreeoffreedomsystemisgraphicallyvisualizable,wehavetobecontentwithlower-dimensional(most

frequently2-dimensional)lanarversionsofthediagramscarrysignif-

rangle-anglecyclogramprovidesinformationaboutthepostureofthe

legandrangle-velocitypha

diagram,ontheotherhand,reprentsthecompletedynamicsofasinglejointbutprovidesnoinformation

aboutthecoordinationoftwojoints.

Bothcyclogramsandphadiagramsareimportantsignaturesoflocomotionandeachhasitsownmer-

ghtraditionallycyclogramshavereceivedmoreexposureinthebiomechanicscommunity,pha

diagramshavestartedtobenoticedaswell[29,30,31].

2.2Interpretationofatypicalcyclogram

Itisinstructivetostudyatypicalcyclogramandrelateitsimportantfeatureswiththecharacteristicsofthe

rtoFig.2.2(andFig.2forthejointangleassignmentconvention)

completegaitcycleisdividedinto10equaltemporalgmentsandaremarkedby’*’n

importanteventsinagaitcyclearemarkedwithan‘o’onthecyclogramalongwithacorrespondingshort

description.

Letustravelalongthecyclogramfromtheinstantofheel-strike(markedhsinthefigure).Theperiodjust

aftertheheel-strikeisreprentedbyanalmostverticallinecharacterizingtherapidkneeflexionandlittlehip

ckcreatedbytheheelimpactwiththegroundisquicklyattenuatedduringthisperiod.

Afterfoot-flat(ff)thehipbeginstoextendalongwiththekneeshownbytheinclinedlineconnectingfoot-flat

andmid-support(msu).Thetimeperiodbetweenhsand(c)toiscalledtheloadingphawhichoccupies

about

tpha,theweight-bearingpha,ischaracterizedbyanextending

knee.

Theefer-extension

ofthehipreachesamaximumandgraduallyrevers,andthepreviouslyextendingkneesmoothlytranslates

tofl-offoccursbeforethekneeisfully

flexed.

Typicallytheswingphastartsat

thighextensionangleandakneeflexionofabout80%ofthemax-

-swingtheflexionofthethighiscompleteandtheknee,afterreachingitsmaximumflexion

salmostnothighmovementbetweenthe

mid-swingandtheheel-strikeandthephaiffectedbyasteadyreductionofthekneeflexion.

Weconcludethisctionbyprentingsomeofthehighlightsofthehistoricaldevelopmentoftheconcept

ofcyclogram.

2.3Historyofcyclograms

AliteraturearchofthecyclogramrevealsthenameofGrieveasthefirsttopropotheuofcyclograms

(theywerecalledtheangle-anglediagrams)[22,23].Grievearguedthatacyclicprocesssuchaswalkingis

betterunderstoodifstudiedwithacyclicplotsuchasacyclogramandpropodtheinclusionofauxiliary

informationsuchasthetimeinstantsofheel-strikeandtoe-offinthecyclogramtorenderthemmoreinforma-

ingthedeviationsofgaitcharacteristicsoncyclogramshesuggestedthatthedeviationsinthegait

characteristicscannotbeadequatelymodeledasameansquaredeviationsincethedeviationsarenotrandom

andthatboththedirectionandthemagnitudeofdeviationsincombinationwithothershavetobeconsidered

andthatonlycertaincombinationsofdeviationsaretoberegardedas“normal”.Grievealsorecognizedon

thecyclogramtheprominentshockabsorptionphaduringheel-strikeandthe“whiplasheffect”oftheleg

atfastergaits.

Sixdifferentcyclogramscorrespondingtosixdifferentwalkingspeedsforeachofhip-knee,hip-ankle,and

knee-anklecombinationswereprentedin[37].

Cyclogramsfromstandingbroadjump,stairclimbing,racewalkingaswellasnormalwalkingatdifferent

speedsareprentedin[8].Oneimportantcontributionofthispaperisthedemonstrationthatcyclograms,

insynergywithotherkinematicreprentationsofmulti-jointmovements,canbecomeapowerfulanalytical

tool.

WeencounterthesubquentworkofMilnerandhiscolleaguesduringtheventiesupuntil1980[40,

25,24].Uofcyclogramsasameansoftrackingtheprogressofpatientsundergoingtotalhipjointrecon-

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hs=heel-strike

msw=mid-swing

hr=heel-ri

msu=mid-support

to=toe-off

ff=footflat

(b)

Figure4:Atypicalhip-kneecyclogramforlevelwalk(adaptedfrom[9]).SeeSection2.2fordescription.

structionwaxploredin[40].Itwasdemonstratedthat,asonemightexpect,thecyclogramsofabnormal

gaitsaregeometricallyverydifferentandareeasytovisuallyidentifyfromthoobtainedfornormalgaits.

Thesameworkalsoreportedtheevolutionofthegeometricformofthecyclogramasafunctionofwalking

,noattempthasbeenmadetoanalyticallystudythequantitativegeometriccharacteristicsofthe

saddresdin[25,24]andsubquentpapers.

Theimportanceofaquantitativestudyofcyclogramshapesinordertoextractrelevantnumbers,tobe

udasgaitdescriptorsconcerns[25].Onlynormalhealthygaitsatdifferentspeedswereconsideredinthis

eometriccharacteristicsoftheclod-loopcyclograms,theperimeter

,thearea,andthe

hownthatalthoughtheperimeterandtheareaofcyclograms

areapproximatelylinearlyrelatedtotheaveragewalkingspeed,thequantitystaysroughlyconstant.

Thislatterqramareais

intgertherange,thelarger

ondconcurrentpaper[24]thesameauthorsstudiedcyclogramsobtainedfrom

ntityagainreflectedtheabnormalitiesinthe

gaits.

[10]comparesthegaitpatternsofhumananddogsbymeansofcyclograms(calledcyclographsandan-

gle/anglediagramsinthepaper).Itemphasizedtheutilityofcyclictracesofjointvariablesbypointingout

thatacoordinatedmotionofalegistobeperceivedasaninteractionbetweentwoormorelimbsratherthana

logrampatternisnotedtobeanextremelystable

mechanismtoidentifygaitbehavior.

Thegeometricsimilarityoftwocyclogramsortwovelocity-velocitycurvesoflocomotionwascomputed[44,

51]byemployingthediscretechain-encodingreprentation[17]kfollowsthesamephi-

losophy,thatofthequantificationofthesimilarityofdifferentmovementpatterns,whichisnicelyarticulated

in[51].

In[9]tifferentlevelsof

speedcorrespondingto0.5st/s(slow),0.9st/s(medium),and1.3st/s(fast)lograms

.2.2wehavefollowedthe

hologicalcaswereconsideredtoshowhowa“standard”cyclogramandsuperimpod

3Fivedifferentspeeds,from2.34km/hto6.91km/hwerestudied.

4Stature/snormalizedbytheleglength.

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GoswamiGaitparameterization...

standarddeviationdatacanbeudtodetectabnormalgaitconditions.

Inanunpublished(andpersonallycommunicated)article[49]levelanduphillanddownhillslopesof,

oundthatwalkingspeedatallgradeswasslightlyreducedcomparedtolevel

-kneecyclogramsudinthisarticledemonstratedthatdownhillwalkisassociatedwith

largerkneeflexioninstancephaandreducedhipflllwalkingboththe

kneeandthehipareflexedatfootcontactandintheswingpha.

Neuralnetworkhasbeenudtoperformautomateddiagnosisofgaitpatternsreprentedbycyclograms[2].

Oncetrained,thenetworkcanidentifywitha

successratethethreedifferentconditions–normalgait,

wthatkinematicanalysisisofgreat

helpinthediagnosisandrehabilitationoflocomotordisorderssuchascerebralpalsyandspasticdiplegiais

reinforced[2].

3-dcyclogramshavebeenrecentlyudinadifferentcontext,inordertoshowthemodalbifurcation

displayedinhumanlocomotion[55].Traditionally,cyclogramsaredrawnwiththeinter-gmentaljointvari-

kablerecentresultshowsthatifinstead,thecyclogramsareconstructedfromtheabsolute

elevationanglesofthelimbs(anglesmadebythigh,shankandfootwiththevertical),theresulting3-dcyclo-

gramofhumanlocomotionliesonaplane!Thismeansthatastrongunderlyingstrategyisinactionduring

locomotion.

Althoughitisaboutrunninggait(notwalking)gait,[7]reportsonethemostsystematicanalysonthe

clogramswereudtodemonstratethegradual

readrangeofthe

cushioningphakneeflexionwasnotedasaremarkablefeaturefordownhillrunningaswealsoobrvein

ourcyclogramsfordownhillwalking.

Thevalueofobjectivedescriptionofhumanlocomotioncanbeappreciatedfromtherecentwork[30,31]on

[30]theperiodicityofgaitwasdeterminedfromthePoincar´emap[3]ofthelocomotion

whereasin[31]thedifferenceintheareainsidethephadiagramsobtainedfromtheleftandtherightlegof

thesamepersonwassuccessfullyutilizedtoquantifythegaitasymmetryinpost-poliopatients.

Beforegettingintotheshapeanalysisofcyclogramswewillbrieflystudytheproblemofslopewalkingin

thenextction.

3Slopewalking

3.1Briefliteraturereview

Inordertofacilitatetheinterpretationandevaluationofourresultswebrieflyreviewtheexistingliterature

iewisnotmeanttobeexhaustiveandonlytheresultsthatarecomparabletoours

arementioned.

Oneofthefirstpapersinthisarea,[14]reportedthatthegroundslopewithintherangeoftodidnot

haveanysignificantinfluenceonthestridelengthandthesteprate.

[50]inastudyoflevelwalkandwalkonslopesofnoticedthataveragechonwalkingspeed

dkneeandhipflexionoftheforwardlimbduring

contactwasconsideredtobethemajorinfluenceofuphillwalkwhereasthofordownhillwalkwerean

increadkneeflexionofthesupportinglimbduringcontactanddecreadthighflexioninlateswing.

In[53]gaitdatafrom5healthymalesubjectswalkingonthelevelaswellason

andslopes.

Theauthorsfoundthattheonanuphillslopethesubjectstookshorterstepsatslowspeedsandlongersteps

atfastspeedscomparedtothointhelevelwalk.

Levelwalkandwalkonuphillanddownhillslopesof,,andwereconsideredin[35].The

authors’foundthatforhigherslopes(bothuphillanddownhill)thewalkingspeedsignificantlydecread.

Whereasonuphillslopesthespeedreductionwascaudbyareductioninthecadence,ondownhillslopes

findingspointtowardstheasymmetryinthewayhuman

beingsrespondtouphillanddownhillslopes.

Inthestudyreportedin[45]onlydownhillslopesof,,,5alongwiththelevelwalk

eportedthattheinfluenceofgroundslopeonthetemporalparametersofgaitwasnot

statisticallysignifi

firststrategy,perhapsunexpectedly,thesubjectsleanforwardandincreathesteplengthandinthecond

thesubjectsstayrelativelyerectbutdecreathesteplength.

[47]ledanalysisshows

thatforuphillwalkingthemeanwalkingspeed,cadence(steps/min),andsteplengthdecreassignificantly

nhillwalk,thespeedandcadencedidnotsignificantlyvarywithslopebutthe

5anslopecorrespondstoanangleof.

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SlopeAngle

HighSpeedCamera

DataAquisitiont-up

Treadmill

-Computer

-Software

-Video

-Stroboscope

Figure5:Theexperimentalt-up.

steplengthsignifintionedherethat[16]foundnosignificant

differencesbetweentheuphillanddownhillgaitsofurbanpedestriansonuptoslope.

[42]studiedtheinfluenceofdownhillslopesonthemechanismoffall,whichisariousissueespecially

horsalsocon

energyconsiderationsofslope-walkingwereconsideredin[46,5].

3.2Experimentalmethod

Thedataprentedinthepaperareobtainedfromtwohealthymalesubjects(28and23yearsofage,181cm

and182cmheight,respectively)withoutanyhistoryoflowerextremityinjury.11flatretro-reflectivemarkers

of1-2cmdikeronthe

shankwereplacedatthetibialepicondyleandtheexternalmalleolus,andthoonthethighwereatthe

eangleisdefinedastheanglebetweenthestraight

angle,ontheotherhand,istheangle

betweenthelinejoiningthetwothighmarkersandthelinejoiningthemarkeratthegreatertrochanterand

-drivenvariableinclinationtreadmillfromTechMachinewasudforall

jectschothe“mostcomfortable”V400video

eraaxiswasperpendiculartothelengthofthetreadmillthus

istereddatawereprocesdtaking

kerpositiondatawasfilteredwitha-orderButterworthfilter

.5forasketchoftheexperimentalt-up.

Theinclinationofthetreadmillwasvariedfrom

simplifythelogistics,twoparatessions,oneeachforuphillwalkanddownhillwalk,wereorganized.

However,inordertominimizethepossibilityofanticipationonthepartofthesubjects,thequenceinwhich

jectsworesoftshoes.

complete8minutesofwalkoneachslope,we

havelectedonecyclewhichisreprentativeoftheparticularslopeandwhichdoesnotshowanytransients.

Otherthanfilteringthetime-angledataasmentionedearlier,wedonotadoptanysophisticatedgmentation

technique(suchas[12])ethis,thetrendintheevolutionofthegaitdescriptors

asafunctionofthegroundslopeisclearlyidentifiablewhichillustratestheefficiencyandrobustnessofthe

ranexplanationiscalledforasthecommonpracticeistoaveragethegaitdata

oververaldifferentindividualsand/orveraltrailsfromthesameindividual.

Averaginggenerallyimprovestherobustnessofthedatabyreducingtheeffectsofthestatisticaloutliers.

Anunavoidableconquencei

6Werealizethatthedefinitionof“jointangle”aticdefinitionofjointanglesfrom

humangaitdataisatopicofourongoingstudy.

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GoswamiGaitparameterization...

indicatedintheliteraturethatgaitadaptationstrategiesinachangedenvironmentoftenvaryfromperson

toperson[45]andbydoingagrossaveragingwemayrisklosingsomeofthesubtlestrategiesadoptedby

nourlveswiththeview[41]thatitisimportanttostudytheresponsofindividual

subjects.

Thus,althoughthedataudhereisareprentativeofthenormalgait,wedonotimplythatthespecific

gaitdescriptorvaluextracte

importantpointisthattheparameterizationtechniqueiquallyvalidfortheaverageddataaswellasforthe

datafromanindividualsubject.

3.3Asnapshotoftheresult

.6showscyclograms

ondownhillslopeschangingfrom

toandFig.3.3showscyclogramsonuphillslopeschangingfrom

tainedforeachchangeinthegroundslopeprovidesuswitharichdatabaappropriate

fortheparameterizationtechniques.

Someoftheinterestingqualitativefeaturesofslope-walkingarevisiblydiscerniblefromFigs.6and3.3.

Fig6showstheprominentimpactcushioningphafordownhillslopesmarkedbyavirtualreversalofthe

tobecomparedwiththecyclogramsforuphillslopestonotetheutterlackof

impactinthelatter.

Therangeofhipmovementsteadilydiminishesfordownhillslopesasividentfromthehorizontally

“squashed”geofhipmovementhas,infact,alinearlyincreasing

trendaswegofrom

toslope,eplotFigs.8(b)and8(c).Thekneeanglebehavesinanopposite,

alrangeofkneeangleisalinearlydecreasingfunctionofslope(as

wegofromto)asshowninFigs.8(a)and8(d).HerewenotethesymmetricnatureoftheFig.8(a)

and8(b).Quantitativelywecansaythattherangeofkneejoint(hipjoint)decreas(increas)attherateof

perdegreeincreainthegroundslope.

Fig.9prentstheevolutionofthekneeandhipanglesatheel-strike(consideredtobelocatedatthepoint

of“folding”ofthecyclogram)alongwiththeirquadraticfitisinterestingtonotetheremarkably

symmetricnatureofevolutionofthehipandthekneeangles[13].

4Moment-badfeaturesofcyclograms

Theidentificationandclassificationofplaneclodcurves,asubjectofstudyoftencalled2Dshaperecog-

nition,isatopicofconsiderablerearchinterestinthefieldsofComputerVisionandPatternRecognition.

Theobjectiveoftherearch,simplystated,istoidentify,classify,anddescribe2Dobjectsorsceneswithun-

knownpositionandorientation,thePatternrecognitionapplications,

thereisaneedforlectingcertaingeometricpropertiesoftheobjectwhichareasinnsitiveaspossibleto

thevariationsinsize,displacement,ropertiesarecalledthe

fieldofcomputerized(handwritten)characterrecognitionalsohassimilar

requirements.

Thereareveraltechniquesforquantifyingplanarshapesandthereaderisdirectedto[33,32,19]for

concentrateontheuofmomentsforshapeidentification

andclassififirstsignificantworkontheuofmomentsinidentifying2Dshapesisby[28].The

ufulnessofthistechniquestimulatedalotofrearchandalgorithmsweredevelopedtorefineandextend

themethodandmakeitrobustagainstnoi[48].Somerelativelyrecentarticlesinthisdomainare[39,1].

Efficientalgorithmstocomputemomentswereprentedin[34,38,54].Inthispaperwehaveadaptedthe

methodin[34]tocomputeperimeter-badmoments.

Itcanbeshownthattheinfinitetofmomentsuniquelydetermineaplanarshapeandvice-versa[33].In

otherwords,themomentsaretheaxesinaninfinite-dimensionalspaceinwhichacontourisreprentedbya

ldbeaddedthatthehigher-ordermomentsarensitivetonoiandhardertointerpret

physically.

Area-badmomentsVSperimeter-badmomentsLetusrecognizecertaindistinctivecharacteris-

ticsoftheclosharacter-

fall,asshownintheintofFig.10,

thecyclogramsarenot,the

contourisratherunsmoothwhichisoftenafunctionoftheamountofnoiintheoveralldataregistration

,theplanecyclogramsfrequentlyconsistoflfinterctingloops.

10

GoswamiGaitparameterization...

0

0

(a)slope

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(n)slope

Figure6:o

Fig.2.2.

11

GoswamiGaitparameterization...

0

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(n)slope

Figure7:oFig.2.2.

12

GoswamiGaitparameterization...

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Gr

o

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n

d

sl

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p

e

(

d

e

g

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)

(d)

Figure8:Evolutionofthetotalrangeofmovementofa)kneeandb)ecurvesareshown

withstraightlinesofleast-squaresfit,c)andd)reprenttheactualjointrangesofthekneeandhip,respectivelyasfunctionsofslope.

Traditionally,thePatternRecognitioncommunityhasutilizedthearea-badmomentsofbinary2Dshapes.

Whereaswecoulduthesametechniqueshere,weprefertheperimeter-badmomentsofcyclogramcontours

,thecyclogramisnottheboundaryofanyrealobjectbutistheobjectwho

,theperimeter-badmomentsareequallyapplicabletohigher

dimensionalcyclograms(VSVS),eFig.3(d),wherethearea-badmomentslotheir

,the2Dcyclogramsoftenconsistoflf-interctingloopsandthearefreetoliepartially

resituationswheretheinterpretationofwhatareainsideacurve

ghwedonottreatthecasoflf-interctingatprenttheydooccur

erpretationofareaispecially

complicatedforaltherhand,

eraleffectofnoiinthe

ysignificantlyincreatheperimeterofthecyclogramwithout

tuitiveideaixploredmoreanalyticallyin[43].

Inordertocalculatetheperimeter-badmomentswemakeaphysicalanalogyofthecyclogramwitha

findtheexactmomentofeach

sideofthe

-cedureis

13

GoswamiGaitparameterization...

−15−10−5051015

−80

−60

−40

−20

0

20

40

60

80

Groundslope(degree)

K

n

e

e

a

n

d

hi

p

a

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gl

e

at

h

e

el

−

s

t

r

i

k

e

(

d

e

g

r

e

e

)

Figure9:Evolutionofthekneeandhipangleatheel-strikealongwiththeirquadraticfits.

analogoustothatofcalculatingarea-badmomentsbutthenumericalvaluesobtainedareobviouslydiffer-

ent.

4.1Computationofgeneralizedmoments

Inthisctionweformallyintroducetheideaofgeneralizedmoments,showtheirrelationshipstoveralbasic

featuresofthecontourandprentasimplerecursiveformula(following[34])tocalculatethegeneralized

momentsofhigherorders.

Thetwo-dimensionalmomentsoforderofacurveoflengthintheplaneare,

(1)

rtingpointonthecurve,correspondingtodoesnot

haveanyinfliceoforigin,doesnothaveasignificant

effectbutisfixedbyourangleconvention.

Theaboveequationcanbemodifiedforacontourcomprising

straightlinegmentsasfollows,

(2)

whereisthelengthofthegmentandixpresdas

(3)

Intheaboveequationtheinfinitesimallinegmenthasbeenreplaced,viaachangeofvariables,by

,whereistheslopeofthelinegment,needtomakesurethatthe

integrati

thisimpliedorderwewillhenceforthsimplyputtheintegrationfromto.

iableforthelinegmentcan

beexpresdas

(4)

emodifications,isrewrittenas

(5)

Followingtheprocedureprentedin[34]forthearea-momentsofa2Dshape,wearchforrecursive

goalwedothefollowing

14

GoswamiGaitparameterization...

(6)

(7)

Therecursiveequationsareoftheform

(8)

Anymomentoftheformsuchas,withwhichtherecursionmustbegin,canbecalculated

byasimpleintegration

(9)

.

Wehavetotakeintoaccountthesituationwheretheconsideredlinegmentisvertical;theslopeofthis

lineisnotdefica,insteadofre-parameterizingwiththevariableweratherdoitwith,as

.Thequantity,theinveroftheslope,

expressionforthegeneralizedmomentsforaverticallinegmentthereforereducesto,

(10)

ThefirstfewmomentsofastraightlinegmentareprentedintheTable1.

Table1:Momentcomputationoflinegments

momentnon-verticalgmentsverticalgments

Therecursionprocessmustfollowthequence:

4.2Parameterizationofcyclogramfeatures

Wenowconsidertheparameterizationofthemoment-baddescriptorsofthecyclogramsastheyevolveas

hca,wedefinethegaitdescriptor,plotitvolution

asafunctionofthegroundslopeanddiscussitstrend.

4.2.1Perimeter

.11(a)weprenttheevolutionofthe

cyclogramperimeternormalizedagainsttheperimeteratslope(whichiqualto).Theperimeter

isalinearlydecreasingfunctionofthegroundslopeascanbeinferredfromthestraightlineinthefigure

ghthe“jerkiness”ofthejointmotionwasgivenasapossible

reasonofanincreaintheperimeterofsimilarcurvesin[25],itisunlikelytobereasonhere.

15

GoswamiGaitparameterization...

Figure10:nimportrams

consistofatofunequalstraightlineachlineconnectingtwosuccessivedatapoints.

Inordertointerpretthecyclogramperimeter,weexpressthelengthofthestraightlinegmentcon-

nectingtwosuccessivedatapointsand,eFig.10as,

(11)

whereandaretheaverageangularvelocitiesofthehipandthekneejoints,respectivelyduringthe

interval,ingthat,theaboveequationscanbe

extendedtotheentirecyclogram.

AccordingtothefirstoftheEqns.11thecyclogramperimeteristhe“totaldistancetraveled”bythetwo

gerthejointexcursion,xcursion

isnottobeconfudwithjointrange(whichwediscusslater)–ajointcanhavealargeexcursionwithina

eexampleisFig12wherethecyclogramonthelefthasahigher

-jointrangebuta

.11(a)thusindicatesthatthetotaljoint

excursionlinearlydecreasastheslopechangesfromto.

ThecondofEqns.11,ontheotherhand,relatesthecyclogramperimetertotheaveragevelocityofthetwo

lesofequalduration,theperimeterisproportionaltotheaverage

ationofgaitcycleisalinearlyincreasingfunctionofslopeasshowninFig.11(b).The

plotoftheperimeter/cycledurationratiohasthesamenatureasthatofFig.11(a),plies

thatthenotethat

ahigheraveragejointvelocityduringacycledoesnotnecessarilyimplyahigherwalkingspeed.

Incidentally,alongercycledurationcorrespondstoasmallervalueofcadencewhichiswhatweobrve

foruphillslopesthusprecilycorroboratingtheobrvationsby[53,47].

4.2.2Area

Althoughtheareadoesnotdirectlyfallintoourschemeofperimeter-badmoments7,weincludeitforits

tion,cyclogramareaisrequiredtocalculateitscircularitycriterion(Sec-

tion4.2.3).

Thesignedareaofapolylinecontourcanbecomputedas

7Inthearea-badmomentscheme,theareaofaclodcontourisitszerothmoment

16

GoswamiGaitparameterization...

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0.4

0.5

0.6

0.7

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0.9

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1.1

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1.3

Groundslope(degree)

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Groundslope(degree)

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al

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ai

t

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y

cl

e

p

e

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i

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d

(b)

Figure11:Evolutionofnormalizedcyclogaight

linesshowlinearfitofthedata.

x

Y

RangeofX

RangeofX

Figure12:Thedifferencebetweenthejointexcursion(measuredbyperimeter)andthejointrange(relatedtocyclogramarea)is

demonstratedwithartifilogramonthelefthashigherjointrangebutalowerjointexcursioncomparedtothaton

theright.

(12)

whichassignsapositivevaluetotheareainsideacounterclockwicontourandanegativevaluetooneinside

troducerrorsinourcalculationsforcyclogramswithinterctingloops.

Fig.13(a)malizationis

donewithrespecttotheareaofacirclewhoperimeteriqualtothatofthecyclogramcorrespondingto

aboliclinereprentsaquadraticfitofthedata.

Notingthatthecyclogramareaisanindicationoftheconjointrangeofjointmovements[25]wecansay

thattherangeismaximumforlevelandshallowuphillslopes.

Althoughwearenotawareoftheuofcyclogramareainquantifyinggaitexceptthatby[25,24],thearea

insidethephadiagram,inadifferentcontext,wasudtoobtainameasureofgaitsymmetryinpost-polio

patients[31].

4.2.3Circularityorcompactnessorroundedness

Severalversionsofadimensionlesscriterioninvolvingtheperimeterandtheareaofaclodcontourand

characterizingitscircularity,[33]udtheexpres-

sionwhereasquantitiessuchas,[32],and[25,24]

17

GoswamiGaitparameterization...

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circle

le

square

Groundslope(degree)

C

y

cl

o

g

r

a

m

"

ci

r

c

ul

a

r

i

t

y

"

(b)

Figure13:Evolutioaboliclinesreprentquadratic

fi(b)thecircularityofwell-knowngeometricentities,thecircle,thesquareandtheequilateraltriangleareshownfor

comparison.

paper,weusimplybecauitisunityforacircle(whichisthemaximumpossiblevalue)andthata

parisonwenotethatthevalues

offorasquareandforanequilateraltriangleareand,llyoblongobjectshave

.13(b)prentstheevolutionofthecircularitycriterionasafunctionofthe

figureindicatesthatthecyclogramcircularityismaximumaroundslopeanddecreas

forlower(includingdownhill)qualitativelyverifi

circularityofthenegativeslopecyclogramsis,toalargeextent,duetothefactthatfollowingtheheel-strike

thecyclogramtrajectoryreversitsdirectionandretracesitlftherebyaddingsignificantlytotheperimeter

butverylittletothearea.

Itshouldhoweverbepointedoutthattwosignificantlydifferentshapesmayposssthesamecircular-

ity[56].Also,thecriterioncansignificantlyalterbetweenacontinuouscurveanditsdiscretecounterpart,

dependingonthefinenessofdiscretization.

4.2.4Location

Location,inthiscontext,meansthepositionofthecenterofmass(CM)ofawireofuniformmassintheshape

itionoftheCM()isgivenby

and(13)

whereandarethetwofirst-ordermomentsofthepolygonalcontour,and,asenbefore,isits

perimeter.

InFig.14weplotthedistanceofthecyclogramCMfromthecoordinateoriginasafunctionoftheground

eofquadraticfitisalso

.14(b)earfromtheplotsthat

theCMisthenearesttotheoriginfordownhillwalkonfeebleslopesandmovesawayforpositiveaswellas

negativeslopes.

Forthejointangleassignmentconventionudinthepaper(refertoFig.2)thecoordinateoriginofthe

cyclogramplanecorrespondstoastraightlegconfiedistanceofa

pointfromtheorigintothecyclogramreprentsthedeviationofthecurrentconfigurationfromthisconfig-

yclogramCMisviewedasthe“average”legconfigurationduringacompletewalkcycle,its

distancefromtheoriginwillbeaquantificationofthedeviationofthisaverageconfigurationfromthepassive

legconfigurationwhichisvertical8.

8Wenotethatthis“average”isapurelygeometricaverageandtimeisnotinvolvedinthisdefinition.

18

GoswamiGaitparameterization...

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Groundslope(degree)

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y

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M

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Hipangle(degree)

K

n

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(

d

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)

LocusofcyclogramCMwithgroundslope

(b)

Figure14:aticapproximationissuperpodonthe

econdfigureweethelocusoftheCMpositionwithslope.

4.2.5Orientation

Theangle(boundedbetween)betweenthepositiveabscissaandthelineofleastcond-ordermomentof

helineoftheleastcondmomentpass

throughtheCMofthecontour(),itquationwillbesimplerifwedisplacethecoordinateframes

suchthattheneworigincoincideswiththecenter[27].AftereffectingthisdisplacementasshowninFigs.15(a)

and15(b)thepointsonthecurvewithrespecttotheneworiginareexpresdas

.Weworkhenceforthwithpoints.

ThemomentscalculatedwithrespecttoacoordinateframesituatedattheCMofanobjectarecalled

tralmomentscanbecalculatedbythesameequations

prentedinTable1,ewcoordinates

.

Theorientationofthelineofminimumcond-ordermoment,isgivenby[32,33]

(14)

entationofthe

qns.14thesignscorrespondtothe

meshownthat

thecalculationoftheangleiquivalenttodeterminingtheeigenvectors(thedirectionsofthelines)andthe

eigenvalues(themagnitudesofthemoments)ofthematrixofcondmoments.

KeepingtheoriginfixedattheCM,ifwenowrotatethecoordinateframesothatitisalignedwiththemax-

imumandminimumcond-ordercentralmoments,themomentcalculationsaremuchsimplifi.15(c)

otatedcoordinateframeandthemoment

matrixwrittenaboveisdiagonal.

Themostcharacteristicfeatureintheevolutionoforientationwithgroundslope(Fig.16)isthetwocon-

stantoinspection

19

GoswamiGaitparameterization...

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(a)Cyclogram

−50050

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(c)Axesalignedwithprincipal

-ordermoments.

Figure15:Thedisplacementandrotationofthecoordinateframestosimplifymomentcalculations.

ofFig.3.3convinentation

analysisquantifiesthisgeometricfeature.

4.2.6Eccentricityandthebestfitellip

Thereareveraldefinitionsofeccentricityofaclodplanarcontourthesimplestofwhichistheratioof

themaximumandminimumcondmoments[27].Eccentricityisalsoanindicationoftheoblongnessof

thecontour,verpreferusingeccentricitywhichis

convenie.16(b)reprents

polynomialapproximationof

thedataisshownsuperpod.

Theabovedefiningtothis

definitiontheeccentricityofacircleisunitybutthatofalineisinfinity(astheminimumcondmomentfor

alineiszero).Abetterdefinitionisthus[32]

(15)

accordingtowhichthecurveofminimumeccentricityisacirclewhichhasandforastraightline.

Inyetanotherapproachtheeccentricityofacontourisdefinedastheratioofthelengthsofthemi-major

andthemi-minoraxesofthebest-fit-fitellipistheellipwhichhasthe

bethemi-majorandthemi-minor

axes,respectively,ofthebest-fistandthegreatestmomentsofinertiaoftheellipare

and(16)

Forthebest-fitellipwehave[33]

and(17)

Thenatureoftheplotsofthetwolatterdefinitionsofeccentricityareverysimilartothatoftheformer.

4.2.7Quantificationofcyclogramevolution

Inordertodemonstratehowwecanprovideaglobalpictureoftheevolutionofcyclogramswiththehelp

firstfigure(Fig.17)thenormalized

perimeter,theratioofmaximumandminimumcondmomentsandthedistanceofthecyclogramCMfrom

uddatafromthelinefi

thecondplotwehaveshownthenormalizedcyclogramarea,circularityandthenormalizedcycleduration.

20

GoswamiGaitparameterization...

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90

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Groundslope(degree)

A

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Groundslope(degree)

R

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mi

n

s

e

c

o

n

d

c

e

nt

r

al

m

o

m

e

nt

s

(b)

Figure16:Evolutionofa)cyclogramorientationandb)theratioofmaximumandminimumcondcentralmomentswithchangein

firstcurveisfieinthecondfigureshowscubicfitofthedata.

Thefiguresreprentsignaturesofnormalwalkingofthesubjectbycompactlycapturingamultitudeof

information.

4.2.8Higherordermoments–aninvariantofslopewalking?

Thehigherordercentralmomentsofthecontourarecalculatedwithrespecttothecoordinateaxesfixedto

gnmentisdonebymeansof

multiplyingeachdatapointbymeansofarotationmatrixas

(18)

whereandistheorientationangleofthecontour.

Ingeneral,unlessnormalized,thehigherordermomentshavehighnumericalvaluesaswecananticipate

ertheyareknowntobensitivetonoiwhichmeansasmallchangeinthecyclogramwill

lutionsofthefourthird-ordercyclogrammomentsareshownin

rifweconsidertheratios

ofandasshowninthebottomlineofFig.18wenoticethatinaremarkablemannerthequantities

tofisshownwithtwoconstant

datafitsasthereisadiscrete(butnotoflargemagnitude)lot

ofwehavesuppresdthevalue(=-21.537)correspondingtoslopewhichweconsideredtobean

therinterestingtonotethatinspiteofthelargemagnitudesofthemoments(ordersofto

)thoutaconcretephysicalinterpretationofthethird-order

cyclogrammoments,thequantitieshavethepotentialtoplaytheroleofinvariantsofslope-walking.

Anothercandrethe

combinationsofmomentsofacertainorderwhichareinnsitivetorotationandreflectionofthecontour

rd-orderinvariants[33]areand

.Theratio,turnsouttobeagoodcandidateofinvariantinslope-walkingas

confi,inthiscathehighvalueoftheratioforslopeisperhapsanoutlier.

21

GoswamiGaitparameterization...

5Discussion

Gaitparameterizationbymeansofcyclogrammomentscanbeofpotentialuinanumberoffieldssuchas

thequantitativecharacterizationofnormalgait,globalcomparisonoftwodifferentgaits,clinicalidentification

ofpathologicalconditionsandinthetrackingoftheprogressofpatientsunderrehabilitationprogram.

Whathasbeenachieveduptillnowistheestablishmentofastandardcyclogramcorrespondingtothe

normalwalkandtherecognitionofthefactthatthecyclogramscontinuouslydeforminapredictablemanner

tingtheconceptofperimeter-badmoments,eachcyclogramhasbeen

andardqualitativefeatures

ofthehip-kneecnthis

wecanexpectthatthegeneraltrendsoftheslopewalkinggaitdescriptorswillbesimilarforallindividuals.

Whatistobeeniswhethertheprecimoment-valuesreportedinthisworkarealsosubject-independent.

Inorderforthecyclogramstobecomeaneffectivetoolfortheclinicalidentificationofpathologicalcon-

ditionsweneedtheanalyticalskillofcorrelatingapathologywithacorrespondingfeatureofthecyclogram.

Thisnecessitatesathoroughacquaintancewithallthepertinentcyclogrampatternsavailablefromthehuman

r,ifwereprentacyclogramasapointinamultidimensionalspacecapturingallitsmoments,

apathologymightbesuspectedoridentifiedbynotingthepositionofthispointrelativetootherpointsrepre-

imaginejudiciouslyaddingotheraxesinthemultidimensionalspaceinorder

reworkinconcertwiththecliniciansisnecessary.

Asapatientunderrehabilitationcarereturnstoanormallyfunctioninggaitsoshouldhisgaitcyclograms.

Ascalarquantificationofthedifferencebetweenthepatient’sgaitfromanormalgaitisthemultidimensional

distancebetweenthepoints(similartothomentionedintheaboveparagraph)reprentingthetwogaits.

Thelocusofthispointoververalweeksisasignatureoftheprocessofrehabilitation.

Improvementsoftheprentedtechniqueshouldincludeapropertreatmentofthemultipleandlf-

includedloops,dnecessityisthatofacomprehensivensitivityanal-

sitivityofthecyclogrammomentswithrespecttodifferentdatasmoothing/filteringalgorithms

smorereasonabletoperformstatistical

analysonthegaitdescriptorsobtainedfromdifferentgaitcyclesthantoperformaveragingofthedatabe-

rkneedstobedonetoverify

this.

Gaitanalysisperformedonthebasisoftheentirecycleratherthanfromdiscretemeasuressuchasstep

itelikelythatameaningfulnormalizationofthe

cyclogramswouldsignifiimaginenormalizationsbadoncycleduration,

normalization–forexample,constrainingtheentirecyclogramtolieinsideaunitsquare–wouldcomplicate

rlinethefactthatthealgorithmsformomentcalculationsaredirectly

applicabletootherclodcurvesandthusgaitparameterizationcouldbeperformedaswellwithmoment-

anglediagrams[18],phadiagrams[4,31,30]orvelocity-velocitycurves[51].Itisalsopossibletouother

shapeparameterssuchasthebendingenergyofthecyclogramperimeter[56]oritlectricpotential[11]as

gaitdescriptors.

Acknowledgments

HelpfuldiscussionswithBernardEspiau,RearchDirectorofINRIARhˆone-Alpesaregratefullyacknowl-

elCordier,ateatINRIARhˆone-Alpescoordinatedtheexperimentsdescribed

-PierreBlanchiofthelaboratoryUFRSTAPSatJophFourierUniversity,Grenoble

elliofLaboratoryofExerciPhysiology,GIPExerci,

emadehislminary

versionoftheslope-walkingexperimentwasperformedonafixed

slopeintheUFRSTAPSMovement

elpissincerelyacknowl-

edged.

22

GoswamiGaitparameterization...

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24

GoswamiGaitparameterization...

(a)

0.8

1

1.2

0

5

10

15

0.9

1

1.1

1.2

1.3

1.4

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Normalizedperimeter

ratioofmaxandmin2ndmoments

C

y

cl

o

g

r

a

m

C

M

di

s

t

a

n

c

e

f

r

o

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o

r

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gi

n

−13

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+4

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Cyclogramarea

Cyclogramcircularity

C

y

cl

e

d

u

r

a

t

i

o

n

0

+4

−4

+8

−8

+13

−13

Figure17:Theevolutionofa)normalizedperimeter,theratioofmaximumandminimumcondmomentsandthedistanceofthe

cyclogramCMfromtheoriginandb)normalizedcyclogramarea,circularityandthenormalizedcycleduration,withrespecttoground

jectionofthecurveonthetwoverticalplanesareshownindashedlines.

25

GoswamiGaitparameterization...

0

00

(a)

0

0

0

(b)

0

0

(c)

0

0

(d)

−15−10−5051015

−2

−1.5

−1

−0.5

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Groundslope(degree)

R

at

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of

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(f)

Figure18:Evolutionoftheratioofe)andandf)rsmallercurvesa,b,c

anddshowtheevolutionoftheindividualcond-ordermoments,,and.

−15−10−5051015

0

0.5

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Groundslope(degree)

R

a

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v

a

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Figure19:Evolutionoftheratiooftwothird-ordermomentinvariantswithchangeingroundslope.

26