Impulsive Ankle Push-Off Powers Leg Swing in Human Walking Susanne Lipfert, Michael Günther, Daniel Renjewski and Andre Seyfarth

CORRECTION

Impulsive ankle push-off powers leg swing in human walking Susanne Lipfert, Michael Günther, Daniel Renjewski and Andre Seyfarth

There was an error in J. Exp. Biol. 217, 1218-1228. A reference was missing from the reference list. The reference is listed below.

Zelik, K. E., Huang, T. W., Adamczyk, P. G. and Kuo, A. D. (2014). The role of series ankle elasticity in bipedal walking. J. Theor. Biol. 346, 75-85.

We apologise to the authors and readers for this omission.

1831 Allmandring 28, D-70569 Stuttgart, Germany.Allmandring 28,D-70569Stuttgart, Friedrich-Schiller-Universität, Seidelstraße20,D-07749Jena, Germany. USA. 97333, USA. Robotics Laboratory, OregonState University, 021Covell OR Hall,Corvallis, 1218 Received 18September 2013;Accepted19November 2013 1 Lichtwark andWilson, 2006). force withlittleornoshorteningvelocity(Robertsetal.,1997; muscle fibersactivelymaintaintensiononthespring,developing performed passivelybythestretchandrecoiloflegtendons, while and Cheng,1990).Mostofthisspring-likefunctioncan be 1977; Heglundetal.,1982;Hof,1998;Blickhan,1989;McMahon and extend(Cavagnaetal.,1964;AlexanderBennet-Clark, springs, storingandrecoveringmechanicalenergy asthelimbsflex minimizing mechanicalwork.Muscle–tendonunitscanoperate like may bemetmosteconomicallybymusclesthatproduceforcewhile and workmustbedonetoliftpropelthebody. Thesedemands But ofcourse,forcemustbeproducedtosupportthebodyweight the averagemechanicalenergy ofthebodyisconstantovertime. Steady-speed walkingoverlevelgroundisacyclicmotionwhere power, Impulse,Jerk KEY WORDS:Push-off,Poweramplification,Catapult,Jointforce 2.5 dynamic dataofhumanwalkingcollectedatspeedsbetween0.5and output oftheankleextensors.Inourstudy, weusekinematicand biomechanical processleadingtothiswidelyobservedhighpower walking. However, thereisalackofknowledgeaboutthe suggested topoweracatapult-likeactioninlatestanceofhuman kinetic energyfortheinitiationofswing.Also,muscleworkis research suggeststhatworkdonebytheankleextensorsprovides body centerofmassjustbeforetouch-downtheleadingleg.Other based studieshypothesizethatthispush-off causesredirectionofthe been widelyobservedduringpush-off inhumanwalking.Model- Rapid unloadingandapeakinpoweroutputoftheanklejointhave Susanne W. Lipfert Impulsive anklepush-off powers legswinginhumanwalking RESEARCH ARTICLE © 2014.PublishedbyTheCompanyofBiologistsLtd|JournalExperimentalBiology(2014)217,1218-1228doi:10.1242/jeb.097345 catapult withoutescapement. initiation profitsfromanimpulsiveanklepush-off resultingfroma power fromtheankletoremainingbody. Itappearsthatswing launching phase.Here,thebucklingkneejointinhibitstransferof trailing legthanfortheremainingbodyisobservedduring in theanklejointduringalleviationphase.A largerimpulseforthe Our resultsshowareleaseofjustsmallparttheenergystored a launchingphase,wherestoredenergyintheanklejointisreleased. trailing legisalleviatedfromsupportingthebodymass,andsecond, with positiveanklepoweroutput,analleviationphase,wherethe identify twodistinctphases,whichdividethepush-off: first,starting *Author ([email protected]) for correspondence Germany.Magdalenenstrasse 27,D-64289Darmstadt, INTRODUCTION ABSTRACT Human MotionEngineering, 5021SWPhilomath Boulevard, OR97333, Corvallis, m 2 Institut für Sport- undBewegungswissenschaft, Universität Stuttgart, Institut fürSport-

s − 1 for acomprehensiveanalysisofpush-off mechanics.We 5 Institut für Sportwissenschaft, TechnischeInstitut fürSportwissenschaft, Universität Darmstadt, 1, *, MichaelGünther 3 Institut fürSportwissenschaft, 2,3 , DanielRenjewski 4 Dynamic force (GRF)vector forthestancephaseofwalking. Anextending showsthedynamics ofthelowerlimband the groundreaction 1 Fig. launching phase,wherestored energy intheanklejointisreleased. is alleviatedfromsupporting thebodymass,andsecond,a push-off. Theseare,first,analleviationphase,wherethetrailing leg upper bodyaswelltheirimpulsesthroughouttwophasesof the calculating thelinearpowertransferbetweentrailinglegand the is thenacceleratedintoswing.We supportoursuggestionby mass fromthetrailingleg.Themuchsmallerof leg energy storedintheankleextensors isreleasedbyalleviatingbody walking. We proposea catapultwithoutescapement,whereelastic remarkable powerpeakobservedduringanklepush-off inhuman Meinders etal.,1998). kinetic energy forinitiation oftheswingphase(Bajdetal.,1997; Therefore, workdonebytheankleextensorswassuggestedtoprovide through thehip(Winter andRobertson, 1978; Hofetal.,1992). during push-off ispropagatedthroughthekneejointandevenless other researchimpliesthatonlyasmallpartoftheenergy generated 2002; Collinsetal.,2005;DeanandKuo,2009).Ontheotherhand, leg (McGeer, 1990;Donelanetal.,2002a;2002b;Kuo, impulsively pushingoff thegroundbeforeheelstrikeofleading discussed asonemethodofactuationtorestorethelostenergy by power levelwalking,positiveworkdonebythetrailingleghasbeen mechanical collisionbetweentheleadinglegandground.To negative workisperformedonthecenterofmass(CoM)ina that mechanicalenergy isdissipatedatthebeginningofeachstep,as (Meinders etal.,1998)shows.Ontheonehand,ithasbeenargued function oftheankleextensors,asresearchbyMeindersetal. et al.,2009).Butthereiscontroversyaboutthebiomechanical Hof etal.,1983;Ishikawa2005;Donelan2002b;Sawicki human walkinghasbeendescribedinalarge numberofstudies(e.g. mechanical descriptionofhowthisactuallyhappensismissing. muscle fibersoftheankleextensors,seeAppendix1).However, a output thanwhatmusclefiberscouldproduce(forpowerof 2007). Inhumanwalking,suchfunctionallowshigheranklepower Nishikawa, 1999;Burrows,2003;Wilson etal.,2003;Patek prevents themovementuntilalatertime(Gronenberg, 1996; catch ofsomesort(e.g.alatchorantagonisticmuscleactivity) force, elasticpotentialenergy isstoredinelasticelementswhilea Clark, 1975;Alexander, 1988).Asmusclesprovidethenecessary relatively slowmusclecontractionsprecederapidmovement(Bennet- Power-amplifying mechanismshavebeendepictedascatapults,where muscle–tendon unittooperatewithhighpoweroutputandefficiency. between musclefibersandtheattachedtendonallowsoverall stance, whilethefibersoperatenear-isometrically. Thisinteraction single supportphaseofwalkingandthenrecoilrapidlyduringlate the tendonsofhumanankleextensorsstretchslowlyduring Ishikawa etal.,2005;Lichtwark2007;Cronin2013)that RESULTS In ourstudy, weaimtodescribethemechanismbehind High poweractionoftheankleextensorsduringlatestancein It hasbeendemonstratedintheliterature(Fukunagaetal.,2001; 4 and AndreSeyfarth 5

The Journal of Experimental Biology akn pes(i.4A,Table walking speeds(Fig. remaining body(RB)isredirectedduringthelaunchingphase atall accelerate towardstheendofstance. TL’s impulsevector flexes andtheanklejointextends(Fig. Fig. the trailinglegallowskneebucklingalsobeforeTDc(Fig. down oftheleadingleg(TDc).Zerojointtorqueatknee of Positive anklepoweroutput the majorpartofthisstoredenergy 2C).Duringlaunching, reduced duringthealleviationphase(Fig. motion oftheanklejointisaccompaniedbyextendingtorque( joint, theGRFmovesfurtherinfrontofanklejoint.At51%stance,kneetorquebecomeszero( φ̇ ankle torqueτ s m (preferred transitionspeedbetweenwalkingandrunning;1.5 Fig. the flexinganklemotion( RESEARCH ARTICLE stance, theleadinglegtouchesdown.Theflexingmotionofkneejointiskeptundercontrolbyasmallextendingtorque( starts flexing( φ̇ resisted byextendingtorquesinbothjoints( and anklejoint,butflexionis pressure. (A)Thefootisflatonthegroundwithbeginningofsinglesupport(at20%stance).GRFflexesbothknee segments, respectively. Blackarrowsshowtheamountofangularvelocityaroundjoints.GrayGRF. ofmass;CoP, CoM,center centerof larger atthe highest walkingspeed.Fig. that oftheRBforallwalking speeds, andmorethanseventimes trailing legtakesoff theground,fastflexingmotionatkneejointandextending theanklejointareobserved. At 100%ofstance,i.e.whenthe at 84%ofstance,withincreasingflexingvelocitythekneejointandextendinganklejoint.(H) resisted byalargeextendingtorque( during thisphaseatallwalking speeds.FortheRB, flexing motionintheanklejointisfurtherdeceleratedbyanincreasingextendingtorque( during launching.TheTL’s relativeimpulse| at allwalkingspeeds,indicating horizontaldecelerationoftheRB alleviation phaseandincreasesconstantly. Itspeakof~150 ankle jerk observed (Fig. time, apeakoftheextendingankleangularacceleration 3A).Atthe same reached untilwellintothelaunchingphase(Fig. Kne Kne The verticalmomentum

<0 =0 1. Dynamics of the lower limb and ground reaction force (GRF) for the stance phase of one representative subject walking at 75% PTS Dynamicsofthelowerlimbandground reactionforce(GRF)forthestancephaseofonerepresentativesubjectwalkingat75% 1. 1E). Atthebeginningoflaunchingphase,kneejoint ). Atthesametime,anklejointisjustabouttostartextending( ). TheGRFhasapproachedthekneejointandmovedfurtherinfrontofanklejoint.extendingtorque( … φ τ Ank Kne Ank 3B). Thisaccelerationstartsfromzero.Theangular <0, φ̇ (Fig. builds upduringsinglesupportandisonlyslightly CoP 20% stance CoM Kne Δ A φ̇ 3B) showsamaximumshortlyaftertouch- =0 p Ank x ). The GRF has moved in front of the knee joint and further in front of the ankle joint. The flexing motion in the ankle joint is now ). TheGRFhasmovedinfrontofthekneejointandfurtheranklejoint.flexingmotion >0). (C)Towards midstance,theextendingkneetorque decreaseswiththeGRFfurtherapproachingkneejoint.Atankle indicates forwardacceleration oftheTL GRF p E 78% stance of boththetrailingleg(TL)and P τ 1). A positivex Ank Ank >0, φ̇ marks thebeginningof τ Δ Kne E Ank

Ank 4B showsthevectors of and τ <0 Δρ 26% stance

). (E)Withzerotorqueandflexingangularvelocity, thekneejointbucklesat78%ofstance( 1G). Thesemotions srlae Tbe1). is released(Table BCD r Ank appearslarger than | -component ofthe >0, φ̇ Δ p Kne x FGH 82% stance is negative and φ̇ τ Ank

W isnot φ >0, φ̇ − ¨ 1 Ank Ank ).

2B, τ Knee andanklejointtorquesaredisplayedaspinkarrowsaroundtheshankfoot Ank >0). (B)At26%ofstance,thekneestopsflexingandstartsextending( Ank is >0, φ̇ >0 51% stance ). (G)Withamaximuminangularanklejerk impulses indicateforwardaccelerationoftheTL ( impulse oftheTL (Table push-off inferthattheankle joint’s poweroutputmostlychanges the alleviation ( speeds (Δ launching, theTL isacceleratedupward,thoughalittlelessathigh downwards, otherwiseitisdecelerated duringalleviation( RB. positive power actsontheHAT segmentineitherdegree of vertically duringalleviation.During thelaunchingphase,almostno segment forwardandnegative power deceleratestheHAT segment 5E,F),positivepoweracceleratesthehead–arms–trunk(HAT) (Fig. velocity change While forwardvelocity running)]. Bothvectorsindicateaverticalredirectionofmomentum. ( decelerated verylittleornotatallduringalleviation( as muchduringlaunching.Intheverticaldirection, TL is during alleviation,forwardaccelerationisonlysmalloratmost half During launching,theRBisaccelerated upward( 1.5 Ank Δ The impulses Observing thelinearjointforce poweratthehip p =0; m x <0). OnlyathighspeedsistheRBverticallyaccelerated

s τ beginning ofthealleviationphase).(F)Shortlyafterthat,at82% Ank − 1 [75% PTS(preferredtransitionspeedbetweenwalkingand >0, φ̇ p 84% stance The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 y >0). TheRBisslightlyacceleratedforwardduring Δ Ank p x <0). (D)At64%ofstance,thekneestopsextendingand Δ >0) andclearlydeceleratedduringlaunching Δ p v τ of theTL andtheRBoverbothphasesof for TL andRBduringthelaunchingphaseat Kne 64% stance =0), yetthekneestillextends( v x

increases fortheTL,itdecreases 1). Inbothphases,positivehorizontal τ … φ Kne 100% stance Ank τ >0, φ̇ Ank , thelaunchingphasebegins >0) increases,whichresists Kne <0 ). Theextending φ̇ Δ Δ Kne p p Δ P y x >0). >0), however, >0 p τ x ,Trc Kne y τ =0). During Kne ). The =0, and >0, Δ p 1219 P y <0). y ,Trc

The Journal of Experimental Biology launching isnegligible,confirmingtheobservedimpulses. given in%. ankle jointisreleased;red).Thegaitcycle phase. Vertical linesrepresentthebeginningofalleviationphase indicate doublesupportphases,andnon-shadedareastheswing subject. Darkgrayareasindicatethesinglesupportphase,light 1220 RESEARCH ARTICLE ankle ( important tonote thatthegroundactsasablock fortheflatfoot. into elasticpotentialenergy astheankleextensorsareloaded. Itis body weight.Thekineticenergy ofthemovingbodyisconverted single supportphasewhilethe rotating stancelegcarriestheentire body andtheirimpulsesthroughout push-off. calculating powertransferbetween thetrailinglegandupper same time,weidentifytherecipientofpush-off powerby mechanistically elucidateacatapultwithoutescapement.At the extensors towardtheendofwalkingstance.Here, we biomechanical processleadingtohighpoweroutputofthe ankle swing initiation?Literatureislackingknowledgeabout the restore energy lostincollision ordotheyprovidekineticenergy for ankle extensorsduringlatestanceinhumanwalking.Do they This workismotivatedbythecontroversyaboutroleof the | leading leg,forverticaltranslationoftheHAT segmentas freedom. Theremustbeanotherenergy source,possiblyfromthe phase from supportingthebodymass;pink)andbeginningoflaunching (beginning ofpositiveanklepoweroutput,wherethetrailinglegisalleviated Fig. DISCUSSION Δ E In humanwalking,thefootis flat onthegroundformostof

y .Legjointtorques. 2. ,Trc τ t L Ank |>|Δ (instant ofmaximumjerkinankleangle,wherestoredenergythe ; C)forwalkingat75%PTS(1.5 E C B A

Ank τAnk (Nm) τKne (Nm) τHip (Nm) –20 –40 –80 40 80 40 80 | (Table 0 0 0 0 Joint torquesforthehip( ) rnfre oe Fg 6)during 1). Transferred power(Fig. t (% t 50 A

t isnormalizedtocycletimeand m t L cyc

s − ) 1 ) ofonerepresentative τ Hip ; A),knee(τ 100 Kne t ; B)and A ( balance betweengravitationalforceduetothebodymass during theslowerloading.Thescenariostartswithastaticforce mass, whichisaccordinglylowerduringthefasterreleasethan released elasticallymustcompulsivelyberelatedtoanaccelerated power amplificationwiththeankleextensorMTCloadedand increased poweroutputascomparedwithinput.Thisobserved faster thantheelasticpotentialenergy hasbeenstoredimpliesan in walking,conversionofelasticpotentialenergy intokineticenergy muscle–tendon complex(MTC)duringsinglestanceandpush-off the ankleextensorfibersaddingnearlynowork(energy) tothe gravitational force( Then, legalleviationisinitiated.Now, becauseofthereduced to angularpower generatedbytheanklejoint that, theratiooftransferredhorizontal powerthroughthehipjoint used forhorizontalHAT translationviathehipjoint.Inadditionto Then, onlyasmallfractionof this workdoneattheanklejointis themselves. not providedbyactivelengtheningandshorteningofthemusclefibers largely fromtheelastictendoninserieswithmusclefibersandis work totheAchillestendonorskeleton.Thus,powercomes 2007; Croninetal.,2013),whichmeanstheycannotaddsignificant stance (Fukunagaetal.,2001;Ishikawa2005;Lichtwark extensor musclefibersareoperatingnear-isometrically duringlate 2014). Forhumanwalking,itisimportanttonotethattheankle instantaneous changeofforce(DeanandKuo,2009;Zeliketal., proposed thatpositivepush-off powercanbegeneratedbyan by positivepoweroutputintheanklejoint.Inmodelstudiesitwas torque (Fig. throughout singlesupport,leadingtoapeakinextendingankle these observationsindicatethatloadingoftheanklejointincreases (Fig. of pressure(CoP)increasingthemomentarmforexternalforces Our experimentaldataclearlyshowforwardtravelingofthecenter extensors the alleviationphase,only10–20% oftheenergy storedinthe ankle increase inaccelerationmustbe relatedtoasmallermass.During good indicatorofcompletealleviationthetrailingleg,asasudden angular acceleration(jerk).We foundthemaximumjerktobea single supportandendswiththemaximumrateofchangein ankle The alleviationphasebeginswithpositiveanklepoweroutputin late ( becomes dynamicwithacorrespondinginertialcontribution (approximately one–sixthofthebodymass),forcebalance Alleviation phase extensor MTCrecoilat~40%ofthegaitcycle[comparefig. initiating kneeflexion,closelycoincideswiththebeginningofankle and thegroundblockingheel(Fig. is rapidlyreleasedtolaunchthetrailinglegintoaction. phase, wherethemajorityofstoredelasticenergy intheanklejoint from supportingthemassofremainingbody, and(2)alaunching (1) analleviationphase,duringwhichthetrailinglegisalleviated i.e. ajerk,oftheextendinganklejoint(Fig. extension. Thisshowsinasuddenchangeaccelerationfromzero, the legsegmentsisstarted,whichrapidlyacceleratesankleinto conversion ofstoredelasticpotentialenergy intokineticenergy of Cronin etal.(Croninal.,2013)andFig. F F = Power istherateatwhichenergy isconverted( The push-off phaseattheendofawalkingstepisusuallydefined A catchforthiscatapultisprovided bytheextendingkneejoint In viewoftheseconditions,thepush-off phasecanbedividedinto G = m m 1). Additionally, theGRFincreases aftermidstance.Bothof leg ·g · a ) andtheforceproducedbyankleextensors( leg Δ E The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 ). Thus,thelegisacceleratedduringrelease. 2C) justbeforeTDc. Ank is released(fortypicalwalking speeds, seeTable F G = m leg ·g ) ofthesmallerlegmass

1). Releasingthiscatch,i.e.

3).

1D]. Afterthat,the P x ,Trc P / P = Δ Ank E / decreases Δ t ). With F MTC

2 in m

1). leg m ).

The Journal of Experimental Biology Launching phase Launching phase alleviation phase butdoesnotacceleratethe HAT segmentinto downward movementoftheupper body(HAT segment)duringthe through thehipjoint.Thisindicates thatthetrailinglegbrakes forward accelerationatthehip joint. for thelegand,withmomentum oflegretraction,wouldcause Fig. retraction (seepositivehiptorquebeforeandaftertouch-down, forward accelerationbytheleadinglegcouldresultfromactive leg such astheinvertedpendulumorspring-massmodel. The opposing actionsofthetwolegspredictedbyconceptualmodels PTS, preferredtransitionspeedbetweenwalkingandrunning. trailing leg( alleviation phaseandlaunchingaredenotedby the impulsevectorsandtheircomponentsofTL andRBarenormalizedtothenormofrespectivemomentumvector during thealleviationphase(Fig. body inthisphase. that theanklejointpush-off contributes toforwardpropulsionofthe from 1earlyinthisphasetoalmost0(Fig. ( Two phasesofpush-off aredistinguished.Thealleviationphasebeginswithpositiveanklepoweroutputattheinstant Alleviation phase Phase RESEARCH ARTICLE Alleviation phase jerk inankleangleat Table Δ Δρ │Δ p Vertically, theunloading anklepowertransfersasnegative It isinterestingtonotethatbothlegsworktogetherhorizontally x r and Δ │ 1A). Theheelpiledintotheground wouldbetherotationpoint , Δρ 1. Impulsesandpowerintegrals x p and Δρ y m . Alldataaregivenasgrandmeans±s.d.of21subjectsforthefivemeasuredwalkingspeeds.Additionally, thenormsof forthelaunchingphase, TL =11.4±1.8 y ). Powerintegrals t L . Thelaunchingphasefollowsdirectlyafterthealleviationandendswithfoottakingoff theground(TO). ofthe Theduration Δ Δ Δρ │Δ │Δ t Δ Δ Δρ │Δ Quantity Δ Δ Δρ t Δ │ Δ L A kg), theremainingbody( p p E t t p p E p (% t (% t 1 p p r (s) (s) y x y x y x │ (J) (J) │ │ │ (Ns) (Ns) (Ns) (Ns) (%│ (%│ (s B3.±. 661. 0.±901262. 165.7±29.5 132.6±24.4 100.4±19.0 66.6±12.2 33.5±7.3 RB (Ns) (s B73261.±. 89592.±. 18.4±5.3 21.7±6.5 18.9±5.9 19.9±10.3 11.4±3.5 15.9±10.3 7.3±2.6 7.2±6.1 RB 3.8±1.9 (Ns) 2.7±1.0 RB (Ns) (%│ cyc cyc p p ) ) 1 1 p │ │ 1 B381. 29541.±. 44475.6±5.9 14.4±4.7 16.4±3.7 12.9±5.4 3.8±10.4 RB ) )RB │ Δ

B2.±. 69411.±. 634211.0±2.2 16.3±4.2 18.5±3.6 16.9±4.1 21.6±5.1 RB ) 5E). Thisisincontrasttothestrict E for bothphasesaregiventherotationofanklejoint(Ank)andtranslationhip( n .424 00±.61.349 83±.216.98±7.82 18.38±5.92 14.03±4.96 10.09±3.46 5.34±2.45 Ank y L431879161.±. 533217.0±3.7 15.3±3.2 11.0±2.2 7.9±1.6 4.3±1.8 CoM TL x n .107 .815 .434 .766 15.90±8.56 9.67±6.65 3.84±3.45 1.28±1.58 0.01±0.74 Ank CoM L6.±647.±097.±188.±7178.8±16.3 86.2±17.1 70.7±11.8 73.9±10.9 67.9±26.4 TL L73221.±. 58301.±. 22.6±5.1 18.6±4.0 15.8±3.0 11.0±2.0 7.3±2.2 TL ecitr0.52 Descriptor RB RB RB RB TL y L531686171.±. 553317.1±3.7 15.4±6.6 15.5±3.3 22.5±7.7 11.7±2.3 21.2±6.4 10.2±3.4 23.1±10.2 12.7±4.3 8.6±1.7 7.6±2.8 19.8±10.5 6.5±2.8 5.3±1.6 10.2±7.3 4.8±2.2 CoM TL 5.2±2.2 2.2±0.9 3.0±1.2 1.6±0.5 CoM TL L8.±607.±127.±188.±7279.5±16.2 87.3±17.2 75.0±11.8 79.8±11.2 81.7±26.0 TL TL x L3.±792.±. 43501.±. 1.1±8.5 11.3±6.5 24.3±5.0 28.7±6.6 39.2±17.9 TL TL CoM CoM Tc08±.00.49±1.56 0.88±1.30 ,Trc ,Trc Tc0.09±0.45 ,Trc ,Trc

6A). Sothereisnotmuch m RB Δ =59.5±9.8 t A and Δ − − 0.2±2.0 .5±.3 .1±.1 .9±.0 .9±.1 0.077±0.009 0.091±0.011 0.094±0.009 0.115±0.012 0.158±0.032 .0±.3 .3±.2 .7±.4 .1±.5 0.119±0.019 0.117±0.050 0.076±0.044 0.038±0.026 0.006±0.030 1.3±3.2 − 0.3±1.7 0.4±0.6 − 55.3±1.2 5 T 0 T 5 T 0%PS125%PTS 100% PTS 75%PTS 50%PTS 25% PTS 54.9±2.3 2.6±1.3 − − 0.0±0.8 3.9±4.3 0.4±0.8 t .±. .±. .±. .±. 4.7±3.8 6.0±3.5 3.0±3.0 2.3±3.0 1.5±3.3 0.28±1.38 5.8±3.0 0.1±0.4 .603 .607 .221 2.14±2.76 2.92±2.13 0.76±0.75 0.06±0.33 16.7±6.5 L kg) andtheentirebody( , respectively. Normsoftheimpulsevectors

s − 1 which ismostlyusedtodecelerate thefallingHAT segment. ankle jointduringalleviation is transferredthroughthehipjoint, delaying thebuild-upoflegforce. power isgenerated. However, mostofthepowergenerated atthe ends withthetrailinglegtaking off theground.Here,peak ankle The launchingphasefollowsdirectly afterthealleviationphaseand is duetothekneeforcedintoflexionafterTDc(Fig. result inashortperiodoftheHAT segmentfallingevenfaster. This of deceleratingtheHAT segment’s downwardmovement,whichcan this additionalenergy. However, atTDc, thereisabriefinterruption (Table translation ishigherthanenergy producedattheanklejoint moving upward(Fig. Launching phase − 2.5±2.4 − 0.2±0.4 − 52.9±1.0 49.6±2.6 8.8±3.8 2.3±2.2 1.1±0.8 − 1.04 3.1±0.9 1.6±0.9 11.9±4.3 2.6±1.6 To summarize,onlyasmall partofthepowergeneratedat 1.32±1.44 5.6±3.2 1.72±1.54 8.3±4.3 m 1). Itseemsmostlikelythattheleadinglegissourceof m

s CoM − 1 =70.9±11.7 The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 − − − − 50.9±0.8 43.2±4.3 − − − 16.8±5.7 0.1±8.2 1.9±1.3 1.55 3.8±1.1 20.6±6.5 4.4±2.1 6.2±3.1 1.43±1.77 2.52±1.53 8.0±2.9 8.24±5.01 0.5±9.2 0.6±1.1 7.9±2.3

kg). Alsopresentedaretheimpulsecomponents 5F). Energy usedforthis verticalhip m │Δ

s p − │ 1 forbothphasesarepresentedthe t A │ and endswiththeinstantofmaximum p 1 │ atthebeginningofthisphase − − 48.4±1.1 34.9±5.5 − − 19.1±6.7 − − − 2.07 1.5±1.2 2.1±1.2 21.2±7.5 − 6.9±3.3 8.3±3.5 5.06±2.89 5.73±3.73 9.3±3.2 11.66±5.67 14.2±13.4 1.2±0.9 7.0±1.9 13.1±12.8 m

s − 1 x ,Trc andy

2B), therefore − − 47.0±1.4 31.3±2.2 − − 9.0±10.7 − − − 9.4±12.1 2.59 − 0.3±2.6 0.4±1.9 − 9.1±4.0 9.4±2.9 4.02±1.97 18.03±7.90 12.3±4.3 7.63±6.83 19.8±10.9 0.9±1.5 7.4±2.2 19.0±10.5 2.28±6.25 ,Trc). m

1221 s − t 1 L

The Journal of Experimental Biology for walkingat75%PTS(1.5 1222 RESEARCH ARTICLE loss attouch-down oftheleadingleg(Kuo, 2002).Also,the the trailingleg’s push-off along the legaxisreducescollision Usherwood etal.,2012;Zelik al.,2014).Itwashypothesizedthat 2002b; Kuo,2002;Collinset al.,2005;DeanandKuo,2009; transition (McGeer, 1990;Donelanetal.,2002a; been discussedtocauseredirection oftheCoMatstep-to-step power output(Fig. of theanklejointduringlaunchingphasealongwithapeak in In accordancewithpreviousfindings,ourdatashowrapidunloading power ratioP as withincreasinganklepowerthereissteadydecreasingofthe ankle jointisnotlikelytobeusedforpropellingthebodyforward, angular jointpower Fig. lines representthebeginningofalleviationphase support phases,andnon-shadedareasindicatetheswingphase.Vertical areas indicatethesinglesupportphase,lightgraydouble This alsoshowsintherelativeimpulse| of theworkdoneatanklejointremainswithinleg(Table integrals calculatedforthelaunchingphaseindicatethatamajor part power reachesitsmaximum(seeFig. the launchingphase in %. the trailinglegintoswing(Fig. joint totheremainingbody. With that,itenablesrapidpropulsionof of thelaunchingphaseinhibitstransferpowerfrom ankle times higherinthetrailinglegthanremainingbody(Table Collision losses Our resultsindicatethatthebucklingkneejointatbeginning

.Angularjointpower, accelerationandjerkattheanklejoint. 3. C B A … –3 –2 P φAnk (deg s ) φ¨ Ank (deg s ) Ank (W) –6000 x 6000 ,Trc –3.5 –50 100 200 3.5 P / 0 0 0 t P Ank L . Thegaitcycleisnormalizedtotimeandgiven 2C, Fig. Ank (A), angularacceleration 0 50 ×10 , whichcrosseszeroevenbeforetheankle

5 s − 3A). Work donebytheanklejointhas 1 ) ofonerepresentativesubject.Darkgray

4B). t (% t A t

L Δρ cyc 3A, Fig. r ) φ ¨ |, whichisfourtoeight Ank t A (B) andjerk and thebeginningof 6A). Thepower

100 … φ Ank Ankle (C)

1). 1). by atriggersignal providedbythetreadmillcomputer. The remainingtime Kinematic anddynamicdatawere recordedsimultaneously, synchronized recorded atafrequencyof1000 height- andweight-specificregression curves(NASA,1978).GRFswere phase forwalkingat75%PTS(1.5 actual collisionlosses. behaviour, regardlessofitslocalmechanicalorigin,reducesthe phase inwalking(Lipfertetal.,2012).Thisglobalelasticleg elasticity ofthehumanlegcanbeassumedfordoublesupport remaining body. Inapreviousstudy, itwasobservedthatglobal take careofverticallyredirectingandhorizontallydeceleratingthe load transferfromonelegtotheotherduringdoublesupportcould CoM byitslocalizedaction,acceleratingthetrailingleg.Anelastic reduce thecollisionlossexperiencedbyHAT, butaffects the space. Thus,thepush-off inhumanwalkingisnotprimarilythereto along thelegaxiswithanimmediateeffect onHAT translationin small fractionoftheenergy storedintheanklejointistransferred Collins etal.,2005).However, ourfindingsindicatethatonlya crucial forthereductionofcollisionlosses(Donelanetal.,2002a; appearance ofapush-off atorbeforetouch-downwasfoundtobe are plottedatthemidconfigurationoflaunchingphase. body duringlaunching. Fig. center ofmasstheHAT segment(CoM 7).The over anatomicallandmarksofboth of thesubject’s lowerlimbs(Fig. camera recordingsofthesagittalpositions ofeightreflectivemarkersplaced recording atasamplingfrequencyof 240 mounted high-speedinfraredcameras(Qualisys,Gothenburg, Sweden) Boutheon, France).Motionanalysiswasperformedusingeight wall- instrumented treadmill(typeADAL-WR,HEFTecmachine, Andrezieux 75, 100and125%oftheirPTSbetweenwalkingrunning) on an from 21subjects(11 females,10males)walkingatdifferent speeds(25,50, three-dimensional (3D)lowerlimbkinematicsanddynamicswerecollected We usedexperimentaldatafromapreviousstudy(Lipfert,2010),where into swingbyefficiently utilizing elasticenergy storage. the smallermassoftrailinglegexhibitsapowerfulacceleration observed inhumanrunning).With that,launchingisenabled,where of thetrailingleg(contrastingin-phasemotionbothjoints is discontinuedbytheopposingmotionsofkneeandanklejoints and alaunchingphase.Duringalleviation,supportofthebodymass human walking.Thepush-off phaseconsistsofanalleviation scenario fortheobservedpoweramplificationduringpush-off in Our studyprovidesanexperimentallysupportedmechanical (B) Changeofvelocityforthetrailingleg vectors originateintheirrespectivecentersofmass(CoM remaining body Data collection MATERIALS ANDMETHODS Conclusions

.Momentumandvelocityofthetrailinglegremaining 4. AB p → The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 1,TL p RB are shownatthebeginningandendoflaunching p → end,TL (A) Momentumofthetrailingleg p → 1,RB p → end,RB y

Hz andweredown-sampledto240 m

s − x 1 ) ofonerepresentativesubject.The Δ

v Hz. Forthepresentstudy, weused TL HAT and theremainingbody ) wasderivedfromgender-, Δv → TL RB p TL and CoM and the Δv RB → Δ TL ). v RB

Hz.

The Journal of Experimental Biology to thenegative segments. Absolutesegmentanglesweremeasuredclockwisewithrespect collected markertrajectorieswereusedtodefinefoot,shank,thighand HAT (1.5 acceleration ( velocities oftheHAT segment received fromGRFdata(fordetails,seeLipfert,2010). 72 perspeedandsubject). combined). Intotal,weanalyzed5188walkinggaitcycles(between21 and averaged foreachsubjecttogiveindividualmeans(leftandright side down ofthesameleg)werelinearlyinterpolatedto100pointsand then cycles (startingattouch-downofonelegandendingwiththenexttouch- R2007b, TheMathWorks, Inc.,Natick,MA,USA).Signalsofdetectedgait All datawereprocessedandanalyzedusingcustomsoftware(MATLAB delay (2.5×10 Fig. RESEARCH ARTICLE identified andcorrectedafterthemeasurements(Lipfertetal.,2009). were definedtoincreasewith joint extension.Angularvelocity are addedasdashedlines.Take-off ofthetrailinglegismarkedby normalized tocycletimeandgivenin%.Shownontherightofeachdouble-panelishorizontalzoompush-off Curvesoftheleadingleg phase. assumptions (e.g. rigidsegments)andmeasuredkinematics (e.g.fluctuating dynamics algorithms.Inconsistencies betweeninversedynamicsmodel 2004). second-order Butterworthfilterwith acut-off frequencyof40 approximation. Allkinematicdata werelow-passfilteredusingazero-lag areas indicatetheswingphase.Vertical linesrepresentthebeginningofalleviationphase Data processing φ Ank Definitions ofsagittalplanekinematicsareillustratedinFig. CoM movementsweredeterminedbytwiceintegratingtheaccelerations Leg jointtorquesandforcescan becalculatedimplementinginverse

m .Hipjointforces,head–arms–trunk(HAT) segmentvelocityandlinear jointforcepoweratthehipjoint. 5. ) weremeasuredbetweenthecorrespondingtwoadjacentsegmentsand

s –1 Px (W) vx (m s ) Fx (N) − ,Trc ,HAT ,Trc –100 1 ) of one representative subject. Dark gray areas indicate the single support phase, light gray areas indicate double support phases, andnon-shaded ) ofonerepresentativesubject.Darkgrayareasindicatethesinglesupportphase,lightdoublephases, –50 100 –60 –30 1.4 1.5 1.6 1.7 50 30 60 0 0 –3 φ andjerkφ ¨ A E C x s) andtimedrift(2.0×10 0 -axis. Jointanglesatthehip( … t (% v x were derivedusingacentraldifference t ,HAT A 50 t L cyc and v ) –5 y Push-off phase ,HAT s

s − (C,D), andtheresultantlinearhipjointforcepower 1 ) betweenbothsystemswere 100 φ Hip ), knee(φ t A

t L

Kne

Hz (Winter, ) andankle t TO .

7. The t TO φ . , flexing jointtorquestobenegative.Anklepower frequency of15 by multiplyingankletorque joint forcesF Appendix torque andpowercontributionsinalinkedchainofsegments, see with thevelocityofadjoiningsegment’s CoM.Fordetailsonforce, We alsocalculatedlinearjointforcepowerbymultiplyingthe sagittal planetakingsofttissuedynamicsintoaccount(Güntheretal., 2003). Appendix algorithm basedonthesagittalcoordinatesoffourmarkersperleg (see sagittal ankle,kneeandhipjointtorquesforeachlegbyasequential guaranteed beforecalculatinginversedynamics.Essentially, wedetermined that constantsegmentlengthsthroughoutmeasuredsequenceswere corrected. Inouranalysis,rawskinmarkertrajectorieswereprocessedsuch segment lengthsduetoskinmarkermovement)canbeidentifiedand (Fig. of maximumjerkinankleangleand endswiththefoottakingoff theground maximum jerkinankleangle.The launchingphasebeginswiththeinstant phase beginswithpositiveanklepower outputandendswiththeinstantof contributions technical limitations (spatialresolutioninparticular), therepeatedderivation timing ofthemaximumanklejerk forindividualgaitcycles.Becauseof After theinversedynamicsprocedure,resultingjointtorques We dividedthepush-off phase intotwofunctionalphases.Thealleviation We noticedautomaticdetectionfailingtoreliablyreturncorresponding –1 3). Py,Trc (W) vy,HAT (m s ) Fy,Trc (N) –0.15 –200 –140 –0.3 0.15 200 400 600 –70 140 0.3 70 2. 2 forfurtherdetails).Equationsofmotionweresolvedthe 0 0 0 t A The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 and thebeginningoflaunchingphase x P B F D P 0 and x Hz. We definedextendingjointtorquestobepositive,and x ,Trc and and P F y P , aswelltheresultantlinearjointforcepower y , werefurtherlow-passfilteredwithacut-off y ,Trc t (% τ Ank (E,F) arepresentedforwalkingat75%PTS 50 by ankleangularvelocity cyc Hip jointforces ) 100 F x ,Trc and F t L P . Thegaitcycleis Ank φ y . was calculated ,Trc Ank . (A,B), CoM 1223 τ and

The Journal of Experimental Biology (angular jointwork of thealleviationphase the doublesupportphase(TDctoTO). Vertical linesrepresentthebeginning et al.,2003),and theremainingbody(RB)asentire bodywithoutthe (foot, shankandthigh)twowobbling masses(shankandthigh)(Günther respectively). 2, P 1224 RESEARCH ARTICLE where and (linear jointwork power transfer. Forbothphases,work contribute topropulsionandsupportofthebody, weonlyconsideredlinear and oneangular. Asangularpowertransferthroughthehipjointdoesnot be transferredthroughthehipjointinthreedegreesoffreedom,twolinear raggedness, whichoccurredpredominantlyattheslowestwalkingspeed. the jerkfurtherbyeyewherefilteringwasuselessduetoextreme cycles tocorrectmisdetectionwherenecessary. Specifically, wesmoothed Therefore, weoptedtomanuallycheckeachoftheanalyzed5188gait series, whichdidnotalwaysallowclearidentificationoftherightindex. of kinematicdatacollectedatonly240 contribution Fig. and P is normalizedtocycletimeandgivenin%. PTS (1.5 power t respectively. Thedarkgrayareaindicatesthelastpartofsinglesupportfrom and shanksegments,respectively) and P and 4forthethighHAT segments,respectively).Notetheterms As furtherdetailedinAppendix S x to TDc(touch-downoftheleadingleg),andlightgrayareaindicates ,Trc We definedthetrailingleg(TL)asconsistingofthreebonysegments In ourstudy, theangularpush-off powergeneratedattheanklejointcan

Ank .Powerratio. 6. / P P F y = P ,lin,34 Ank t ,Trc 1 P Ank m and τ = ,12 and P =

F is presentedforbothphasesofpush-off duringwalkingat75% s P P B A (indices 1and2forthefootshanksegments,respectively) − y x 1 x ,34 t ,Trc ,Trc ) ofonerepresentativesubject.AsfurtherdetailedinAppendix P P P P 2 y,Trc/ Ank [ ] x,Trc/ Ank [ ] y ·V specify thebeginningandendofeach phase,respectively. –0.5 –0.5 ,Trc + (A), verticalcontribution 0.5 0.5 P –1 –1 y ,4 / y Δ P 0 1 0 1 The ratiooflinearhipjointforcepower[horizontal ,Trc Δ Ank E (indices 3and4forthethigh HAT segments, E , whereP x t Ank ,Trc A are equivalenttod 48 t and thebeginningoflaunchingphase A , Eqn

and Δ Δ Δ EPt EPt EPt 50 yy xx n Ank Ank Tc,Trc ,Trc ,Trc ,Trc Δ x 3) byintegratingpowerovertime: ,Trc 2, E y = d (1)=d, d (3)=d, (2)=d,

,Trc P 52 t F L Ank ∫ ∫ ∫ x

t t t , Eqns ,34 1 1 1 t t t 2 2 2 P t (% Δ = ·

F,lin,34 P V E Hz hadledtoratherraggedtime P E y x τ ,Trc ,4 was calculatedforthehipjoint x ,12 ,Trc cyc and P 1 and2)theanklejoint = (B)] andangularanklejoint (indices 1and2forthefoot /dE P ) x ,Trc Ank y ,Trc , whereP and d = F y ,34 E · t 61 V TO y ,Trc y x ,4 ,Trc /dE (indices 3 = t F L Ank . Time x ,34 , · V x ,4 denoted bym Fig. and CoM trailing leg’s masswasdeterminedby: where bonysegmentsandwobblingmassesareenumeratedfrom1to5.The were calculatedusing: TL. Thepositionandmomentumofthetrailingleg’s centerofmassCoM components, were additionallynormalizedtothe norm oftherespective of thephasep calculated fortheTL andRBbysubtractingthemomentumatbeginning extension. inner jointanglesbetweentwoadjacentsegmentsandincreasewith CoM segment betweenKneandTrc, andtheHAT segmentbetweenTrc and between Mt5andAnk,theshanksegmentAnkKne,thigh specific regressioncurves(NASA,1978).Thefootsegmentisdefined mass (CoM)oftheHAT segmentisderivedfromgender-,height-andweight- metatarsal joint,Mt5),andtheankle(lateralmalleolus,Ank).Thecenterof (greater trochanter, Trc), theknee(lateraljointgap,Kne),toe(fifth The positionandmomentumoftheremainingbody’s centerofmass For thelaunchingphase,norm | For boththealleviationand launching phase,impulses

y .Kinematicsetup. 7. HAT RB CoM . Ankleangle were calculatedbyEqns . y The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 CoM 1 from themomentumatendof the phase and thecenterofmasspositionentirebodyby y x x CoM,RB CoM,RB φ =.(10) . = CoM φ Ank Ank p  RB Trc , kneeangle Sagittal markerpositionsarerecordedatthehip =,(6) , = HAT y x φ =,(9) =,(8) CoM,TL CoM,TL Kne ⎝ ⎜ ⎜ ⎛ ymymy xmxmx pm  o o LCoM,TL TL CoM CoM Δ− o o LCoM,TL TL CoM CoM vp mv TL vp mv mm pp pp   o CM,TL ,CoM CoM o CM,TL ,CoM CoM Ank TL (11) . = (5) , = , =

⋅−⋅ 8–10. Themassoftheentirebodyis ⋅−⋅ ∑ ∑ ∑ .(7) =. i i i φ mm mm φ =1 =1 n 1 end =1 5 5 ⋅− 5 ⋅− Hip o TL CoM o TL CoM ∑ m Kne m i my mx Δ =1 5 yy xx TL TL ii ii i p and hipangle ⎝ ⎜ ⎜ ⎛ ⋅ ⋅ | ofΔ i − − v v x y i i p Kne ⎠ ⎟ ⎟ ⎞ HAT , aswellthex Thigh Foot ⎠ ⎟ ⎟ ⎞ φ Shank Mt5 Hip are definedas p end : Δ - andy p were x CoM (4) TL -

The Journal of Experimental Biology contractions bysolving: optimum musclefiberlength (Yamaguchi etal.,1990).Maximumshorteningvelocity (Maganaris, 2001;Maganaris,2003)andYamaguchi etal. jerk inankleangle). oes30 . .40.25 0.25 0.04 0.05 5.0 8.0 3500 1300 Gastrocnemius Soleus F (2.4l fast-twitch fibers.Maximumactiveisometricforce weigh parametervalueswhencombininginformationonslow-and from Yamaguchi etal.(Yamaguchi etal.,1990)andwereusedto summarized inT gastrocnemius (GAS)musclesavailablefromtheliteratureare (Hill, 1938). estimated frommuscleparametersandbysolvingHill’s equation (Herzog, 2007).However, force–velocityrelationshipsmaybe obtained inexperimentsunderextremelyrestrictedconditions and shorteningvelocitiesofcontractileelementscanonlybe capability isnottrivial,asmuscleforcescanbemeasured muscle andwithitthedeterminationoftheirmaximumpower The descriptionofforce–velocitypropertiesintacthumanskeletal momentum vector| RESEARCH ARTICLE at 0.06 dependence (Bennett,1984)intoaccount. Bottinelli andothers(Bottinellietal.,1996)thermal velocity optimum musclefiber length; gastrocnemius muscles Table (Herzog, 2007). Detailed deductionofEqns where and: or maysimplybecalculatedbysolving: force–velocity relationship of 117 6–16l muscle fibers of thesoleusandgastrocnemius muscle of fibers the velocitiesMaximum power of outputandshortening APPENDIX 1 2006; Herzog,2007).Forv good averagevaluesforskeletalmuscleofvertebrates(Alexander, iso Maximum poweroutputP Instantaneous powerP Muscle parametersforthehumansoleus(SOL)and The force–velocityrelationship Results forthemuscles’ , Maximumactiveisometricforce; opt A1. Muscleparametersforthe human soleusand opt W isgivenfortheankleextending musclemass. p s m − s v and 1 − P ) fortheGAS.With that,amaximumtotal poweroutput

s 1 ,max − 1 (depending onfibertype)andacurvatureofthe q (1.5l a era f h oe–eoiycre(i.A1) may bereadoff thepower–velocitycurve(Fig. are factors(here0.095and0.31)dependingon p beA1. able 1 opt | atthebeginningoflaunchingphase(maximum F iso vF Fv s − vqv (N) ()= 1 PpFv P ) fortheSOL and50 ( a s max iso max C mxmax ,max v P vF v Fv Pv , curvatureoftheforce–velocity relationship. ) wasdeterminedby: Percentages offibertypesweretaken max ) ).(A2) . () ()= C

A3 andA4canbetracedinHerzog ,( =, max (A4) , = max of 0.25aregenerallyassumedtobe iso l and correspondingv opt v v and thecorrespondingshortening ⋅⋅ ⋅ ⋅ max F max we alsotookdatareportedby vv were averagedfromMaganaris ( max , maximumshorteningvelocity; v vv ( max ) wascalculatedforconcentric l opt + ⋅ s − –1 ⋅ ) C 1 .(

W at0.12 l opt P ,max (m) were 67 F iso v in vivo max

m A A C and l

opt s 3 1 W C of − , ) ) 1 . data (Croninetal.,2013)ofonesubjectwalkingat1.3 comfortable motionsuchasveryslowwalking.Recentlypublished (Alexander, 2006),whichisnotlikelytobehappeningduringa velocities, increasedmetaboliccostwouldbetheconsequence provide alloftheanklepower. shorten anywhereneartheirmaximumshorteningvelocity or slower thanthewholeMTU,andthusthatmusclefibersdo not confirm thatmusclefibersoftheankleextensorsshorten much very littlepoweroutput output oftheanklepush-off work. Forslowwalkingspeeds,theobservedmaximumpower corresponding shorteningvelocity calculated toobtaintheMTU’s shortening velocity Bobbert (vanSoestandBobbert,1993).Thederivativewasthen (Lipfert, 2010)withmomentarmstakenfromvanSoestand SOL andGAS.MTUlengthwasestimatedfromourkinematicdata shortening velocitiesoftheMTUforallfivewalkingspeeds to beneededforthisobservedankleextension.We derived than themusclefibersofankleextensorsarecapableseems Silverman etal.,2008;Lipfert,2010).Thus,higherpoweroutput is ~180 When humanswalkatacomfortablespeed(1.3 vertebrate skeletalmusclefibers. Normalized force–velocityandpower–velocityrelationshipsfor A1. Fig. case, definition andcorrespondingequation ofmotion,seebelow).Inthat A2).ThesymbolP indices (Fig. Here, asegmentisrepresentedbyoneindexandjoint two A jointisalinkorconstraintbetweentwosegmentsbodies. above orclosetothemusclefibers’ transferred fromsegment Then, ‘jointforce’ and‘constraint force’ would beequivalentterms. one actingonsegment as high50%ofv muscle mass;however, theshorteningvelocityisstillapproximately power outputoftheextendinganklejointduringpush-off during walking theSOLandGAS theMTUsof velocities of shortening thehumananklejointand Maximum power outputof segments andpowerForce, torque contributionsinalinked chain of APPENDIX 2 force’ carrying structurescausingajoint torque equals the‘constraintforce’ inatechnical joint onlyifnoforce- For typicalwalkingspeeds,thehighestshorteningvelocitieslie τ F ij ij is assumedtobecausedbytwo abstracttorquegenerators, W (Donelanetal.,2002b;LewisandFerris,2008; that isexertedbysegment The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 F/P (normalized) 0 1 0 max i . Ifmusclefiberswereshorteningatthese and oneactingonsegment P P i v max to segmentj , MTU 0.31 v v (normalized) P P Maximum poweroutput , max if itwereonlymusclemassdoing max,Ank are indicatedbytheverticalline. i F v on segmentj Force , ij max may berealizedsolelyby means thepowerthatis due tothe(resultant)‘joint , whichwouldentailnoor Power τ ij are specified(for

. This‘jointforce’ m v

MTU s P j − , respectively. max 1 1 ), maximum TbeA2). (Table

and the m

P s − max,Ank 1 1225 also τ ij

The Journal of Experimental Biology joint torque atonmu 001 . .24 60.52 0.52 56 56 41 55 0.22 0.11 0.4 0.2 0.12 0.06 50 67 Gastrocnemius Soleus 1226 RESEARCH ARTICLE is apurelyangularterm,thelatterduetotorquethat is apurelylinearterm,andthesecondaddend: The firstaddend: from Exampleofalinkedchainsegments. A2. Fig. CoM positionR with regardtothepower–velocityrelationship(seeFig. velocities higherthan power outputobservedfromdynamicdatafortheanklejointduringfivedifferent fortheMTUatshortening walkingspeeds.Note:Nopoweroutputisgiven aswellmaximum Maximum poweroutputandcorrespondingshorteningvelocitiesofmusclefibersmuscle–tendonunits(MTUs)theankleextensors maximum shorteningvelocityofmusclefibers; on segmentj Table simply as‘jointforce’ inthefollowing. terms areusedinanequivalentsense,sowealsodenominate ‘resultant jointforces’ havebeendescribed(Niggetal.,2007).Both 1975; RobertsonandWinter, 1980;Meindersetal.,1998)and In theliterature,‘jointforces’ havebeenanalyzed(Quanburyetal., velocity Eqn the ‘linearjointforcepower’ product symbols,respectively. We maycallthefirstterminEqn velocity ofsegment velocity ofsegment The ‘jointforcepower’ R A5, i.e.the‘angularjointforcepower’ A2. Maximumpoweroutputandshorteningvelocities j to ω → j r of segment ij , CoMvelocity τ ij , CoMposition (see Eqn A11). Here,V ω → j to thepositionr j v j max P around itsCoM. j j , and‘·’ and‘×’ arethescalarandvector max , → PLF τ . P PP PP V ij Fij ji ij j L i jFij F ij F Fij j R ,ang, (W) R max ln ,ang, ,lin, , → ji of segment = j j PFV P of segment =.( , maximumpoweroutputofmusclefibers; r Fij ijj F ji ,lin, , – ij (),(A7) , ) =( P L → R Segment r → consists oftwoterms: ij ji F j ,lin,ij  ,(A6)=, is thevectorfromsegment’s ji v = j P  × , velocity + ,max r (Eqn j j ij , position F ⋅ → is thecentreofmass(CoM) ij of jointij, j ( m v ⋅ω MTU Segment

A6). Thesecondtermin s  P − v 1 v , shorteningvelocityoftheMTUobservedfromkinematicdata; → F Vectors arejointforce Joint ij ) ij ij ,ang,ij of jointij,andangular r ij

A1); P V of jointij,vector → j ω (Eqn i v j max max,Ank is theangular A7), canbe (m F , maximumpoweroutputoftheextendinganklejointobservedfromdynamicdata. s ij − 1 ) exerts L A ji

A5 F v 5 F P ij ) , ,max ij v .3—252.59 2.07 1.55 1.04 2.59 295 2.07 283 1.55 206 1.04 124 295 — 283 — 206 — 124 15 — 0.53 — 0.52 — 0.45 7 0.34 0.33 0.32 0.26 0.19 MTU , shorteningvelocityofmusclefibersattheirmaximumpoweroutput; joint forcepowerP the jointforce contributions bystructuresspanningthejoint independently superposesthetorqueduetojointforce constituting acauseoftheangularaccelerationsegment segment is transmittedfromsegment the ‘jointtorquepower’: independent variablesintheequationsofmotion(Eqns P distinctly linearandangularpowercontributions,thecontribution due toactio= where v product, whichreformulatesEqn rearranged bycircularlyshiftingtheconstituentsofscalartriple jj–1 toitsdistalneighbor rigid bodythatrepresentsthesegment Eqns actio= definition, segmented chainconnectedbyallthejoints overall angularmomentumofthemassdistribution Eqn (in general,three componentsinthree-dimensional space,butonly ‘¨’ meansthesecondderivative. the dot‘·’ meansthefirsttimederivative; accordingly, adoubledot dimensional movementdescription asinthisstudy. Inthefollowing, principal components,while it isascalarparameterintwo- three-dimensional rotations,which isthendeterminedbythree around thesegment’s CoMisatensorofsecondorderincase which segmenti Equations of motion Equations of mass proximal neighbor (m τ , The equationsofmotionsegment By definition,thejointtorque ij s solely changestheangularenergy ofsegment A11). ‘Internal’ meansthatsuchconstituentsdonotchangethe − 1 A5–A9 arederivedfromtheequationsofmotionfree m ) reactio j ij is ascalarparameter. Themomentofinertia = j V back onsegment τ The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 j + ji =–τ . Likethejointtorque P ω F reactio v j ij ,MTU . Thejointtorque × exerts thejointforce ij L pertains ina(force-analogous)torqueruleto j+1 withinachainofsegments.Thesegment’s F (W) ji F LFPFV , ij i jjjj ij ji j j ij Fij symbolizes thevelocityofjointpositionat , thereactionforce . IncontrasttoP , = ) =( , = =() Fv VL FV    P i jij ij jjjji j j ij τ P . j i , –1 andaproximaljoint ⋅ ⋅ ⋅ P ji j ij ij v to segment ,MTU  max,Ank ,(A9)=,

+ A5 to: τ⋅ω  , poweroutputcorrespondingto + τ ij ω ω   W Walking speed(m (W) is definedasaninternaltorque τ F ij F j j × × τ , ij consist ofalinearequation ij and thejointforce connected byadistaljoint ij , consistingofthesum   on segmentj j independently fromthe represents alltorque F ⋅ ji =–F ij ij. Thatis,bysuch that arenotdueto j ij . is exertedby θ = . Vice versa, j

for rotating A10, A11), jj+1 toits v j max , which F F ij , v (A8) ij

MTU s (see − are 1 )

The Journal of Experimental Biology joint forcesL where forces actingonsegment m and Trc, whilerecalculatingboth components werepossible:(1)relyingonthe Five combinationsofrelyingonandrecalculatingmarker constant lengthsfortheshankandthighsegmentswerepresumed. Now, theremainingtwocomponentscouldberecalculatedwhere sagittal componentsfromtheknee(Kne)andhip(Trc) markers. the anklemarker(Ank).We furtherreliedontwoofthefour et al.,2003),wereliedonthemeasured ground reactionforces,eveninheavierimpactsituations(Günther phase shiftbetweenanklemarkeraccelerationsandtherespective taking themedianacrosseachsequence.Asthereisbasicallyno reliable androbustdeterminationofconstantsegmentlengthby measured sequencebutneglectedhigh-valueoutliers.We gaineda constant segmentlength,whichwasnearitsmaximumwithina further enhancedtheprocedureasfollows. of anumberdiscriminativesolutionsatanygivensample,we et al.(Güntheral.,2003).Allowingforaweightedcombination this constantsegmentlengthproblemhasbeendescribedbyGünther body modelassumption(Fig. segments toincreaseconsistencyoftheinputdatasetwithrigid modified themarkercoordinatesdefiningshankandthigh assumption thathumanbonesarerigid.Therefore,weslightly dynamics, modelingofhumanlocomotionisbasedonthe Applying theformalismofrigidbodydynamicsforinverse and → where (measured withrespecttotheinertialsystem): two componentsinthisstudy)forthesegment’s CoMposition RESEARCH ARTICLE occurs inEqn and one componentinthisstudy)forthesegment’s angularorientation F Constant segment lengths Constant segment the component ofTrc, whilerecalculatingthe components, (3)relyingonthe y and thex of Kneandthe y iteration converged tomodifiedKneandTrc markerpositions with instant. Within approximatelyfivestepstheso-implemented then takenastheinitialcondition forarecalculationatthesame of bothsegmentlengths,wascalculated. Thistransientsolutionwas five solutions,momentarilyneglecting therequirednominalvalues of Kne. ϕ -components ofTrc, whilerecalculatingthe -components ofKneandTrc, whilerecalculatingboth j j ) torquesactingonsegment +1j A criticalprerequisitewastoset anominalvalueforthe At eachpointintimealinearly weightedcombinationofthese in theinertialsystem: x , andanangularequation(alsogenerallythreecomponents;only → ϕ . τ -component ofTrc, (4)relyingonthe j j = +1j Ʃ Ʃ ω k m . Thesymbol F j -component ofTrc, whilerecalculatingthe τ is anothernotationfortheangularvelocity, whichalready ext,kj ext,mj A7. jj–1 symbolizes thesumoverallother(external;index symbolizes thesumoverallother(external;index θ⋅φ y × -component ofTrc, and(5)relyingonthe j RFFF F F mR (A11) , F   jjj jj j j j –1j ⋅ jjjjjjjjj   = → ϕ ¨ and j FLF LF (A10) , =  = j τ   −− ω   in additiontothejointforces . jjj jj 111 11 L − j − ×+×+ 1ext,11 jj+1 1ext,11

represents theangularacceleration, A3). Thecorrespondingsolutionto j + ++ × in additiontothetorquesby τ x y F -component ofKneandthe -components, (2)relyingonthe + j +1j + , andthejointtorquesτ + ∑ ++ ∑ m y x k τ -component ofKneand - andy y x x -component ofKne -components ofKne mj - andy kj -components of x -components -component F j –1j x - and and j –1j R y x k ) - - j from Güntheret al.(Güntheretal.,2003). parameters ofthenonlinearspring-damper elementsweretaken the jointforcesandtorques samplebysample.Thecoupling bony segmentequationsofmotion weresimultaneouslysolvedfor approximately fiveiterationsofrecalculatingateachpointintime. are displayedindarkgray. A finalsolution(black)wasobtainedafter circles. Modifiedmarkerdataobtainedfromconstraintcombination(i)and(v) lengths ofshankandthigh.Originalmarkerdataaredenotedbytheopen modified relyingontheankle(Ank)markerpositionassumingconstant combinations. Example ofmarkermodificationutilizingtwoconstraint A3. Fig. coupling forcesincurasexternal forcesinEqns those ofitscorrespondingbone CoM.Inourstudy, these three 1994). Initialconditionsofawobbling masswereassumedtoequal kinematics, withasimpleRunge–Kuttaalgorithm(Presset al., masses alongatimescale,draggedbymeasuredbonysegment state variablesforintegratingsecond-orderdynamicsofwobbling segment andwobblingmasspositionsvelocities.Thelatter are were calculatedfromcouplingforcesknownasfunctionsof rigid elements wasusedtorepresentwobblingmasses.Theirkinematics coupled witharigidsegmentmassbythreenonlinearspring-damper (NASA, 1978)implementedinCcode(Hahn,1993).A pointmass derived fromgender-, height-andweight-specificregression curves kinematics ofthesegmentmasses.Segmentalanthropometry was application pointsofjointforcesaswelllinearandangular of thesegments.With that,markerkinematics determinedboththe taken ascentersofjointrotationwelldistalandproximalends from forceplatemeasurements.Thecorrectedmarkerpositionswere interaction. GRFandpointofforceapplicationweredetermined and torquesateachtimesample,startingwiththefoot–ground Eqns dynamics determinedtheequationsofmotionhumanleg(see been detailedbyGüntheretal.(Güntheral.,2003).Rigidbody The two-dimensionalinversedynamicsprocedureusedherehas precision wassettoliebetween10 both segmentlengthsattheirnominalvalues.Therequestedrelative each solutionatpointintimewasnotdetermined. constraint combinationswereincluded.Theexactcontributionof combinations (1and5).Inouranalysis,allfiveequallyweighted for onlytwoconstraint A3 reduced formisshowninFig. demonstrate thisprocedure,anexampleofmarkermodificationin smooth trajectoriesovertimeresultedforallmeasuredtrials.To samples withoutafinalsolution.Constantsegmentlengthsand Inverse dynamics A10, A11) andweresequentiallysolvedforthejointforces y The JournalofExperimentalBiology(2014)doi:10.1242/jeb.097345 Sagittal knee(Kne)andhip(Trc) markercoordinatesare x Ank Kne (i) Trc –6 and 10 Thigh

–8 A10 andA11. The (v) . Therewereno Shank Mt5 1227

The Journal of Experimental Biology ezg W. Herzog, il A.V. Hill, 1228 RESEARCH ARTICLE Forschungsgemeinschaft (DFG). This studywasfundedbygrantSE1042/1toA.S.providedtheDeutsche manuscript. funding andequipmentforthestudycontributedtowritingof design ofthestudy, theanalysesandwritingofmanuscript.A.S.shared and preparedAppendix2.S.L.,M.G.D.R.contributedtotheconception inverse dynamicsprocedures.M.G.implementedprocedures S.L. organizedandexecutedalldatacollectionprocessingexceptfor The authorsdeclarenocompetingfinancialinterests. of thispaper. The authorsthankStenGrimmerforhiscontinuoussupportduringthepreparation eln,N . ea,M . alr .R n aan,G.A. Cavagna, and C.R. Taylor, M.A., Fedak, N.C., Heglund, R.M. Alexander, ad . tfni,M,Matjacić M., Stefancic, T., Bajd, H.C. Bennet-Clark, and R.M. Alexander, R.M. Alexander, Author contributions Competing interests Acknowledgements References Funding an U. Hahn, entCak H.C. Bennet-Clark, lchn R. Blickhan, en .C n u,A.D. Kuo, and J.C. Dean, ent,A.F. 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