Comparison of mechanical energy expenditure of joint moments and muscle forces during human locomotion

Pergamon Copyright

0 1996 Elsevier Science Ltd. All rights resewed Printed in Gnat Britain GOZl-9290196 %lS.oO + .W

COMPARISON OF MECHANICAL ENERGY EXPENDITURE OF JOINT MOMENTS AND MUSCLE FORCES DURING HUMAN LOCOMOTION Biomech~ics

Boris I. Prilutsky,* Ludmila N. Petrova and Leonid M. Raitsin Laboratory, Central Institute of Physical Culture, Sirenevij Boulevard 4, Moscow 105483, Russia

Abstract-The mechanical energy expenditures (MEEs) of two human lower extremity models with different sources of mechanical energy-(l) muscles and (2) joint moments--were compared theoretically. Sources of mechanical energy producing movement of Model 1 were eight muscles, three of which were two-joint muscles. Sources of mechanical energy producing movement of Model 2 were net moments at its joints. These sources of mechanical energy were substituted by 11 one-joint muscles, with the assumption that antagonistic muscles did not produce force. Because of this assumption, summed MEE of all joint moments and all one-joint muscles of Model 2 were the same. It was shown that during the same movement the model with two-joint muscles could spend less mechanical energy than the model without two-joint muscles. This economy of mechanical energy realized by two-joint muscles was possible if(i) signs of the muscle powers which were produced by the two-joint muscle at both joints were opposite, (ii) moments produced by that muscle at each of the two joints had the same direction as the net joint moments at these joints, and (iii) muscles crossing these two joints from the opposite side did not produce force. Realization of these three conditions during human locomotion was checked experimentally. Electrical activity of eight lower extremity muscles of ten subjects was measured during treadmill walking and running. Based on this information, the periods where the muscles produce force were estimated. Moments and their Power at joints of the lower extremity of two subjects performing walking and running were calculated using kinematics and ground reaction force measurements, and an inverse dynamics approach. It was shown that MEE of models with different sources of mechanical energy appeared to be different during certain periods of the swing phase. However, the magnitude of this difference was probably relatively small. Keywords: M~hani~l energy expenditure; Two-joint muscles; Human locomotion; Electrical muscle activity. INTRODUCTION

There are severai approaches for estimating the mechanical energy expenditure (MEE) during human movements (Cavagna et nl., 1964; Fenn, 1930; Gersten et al.,

at joints (Winter, 1983a,b), it is assumedthat movement is realizedby the momentsat the joints. A detailed analysisof different approachesfor determining

MEE

was made by Aleshinsky

(1986a-e).

He

showed,in particular, that if sourcesof mechanicalen1969; Norman et al., 1975; Pierrynowski et al., 1980; ergy producing human movementwere assumedto be van Ingen Schenauand Cavanagh,1990; Williams and net momentsand forcesat joints, the best estimateof Cavanagh,1983; Winter, 1978; Zatsiorsky et al., 1982). MEE would be the summed mechanical work of net The different approachesgive different results [differ- moments at joints for the period of time during which encescan be up to ninefold (Prilutsky and Zatsiorsky, movement occurs.However, a human is able to move 1992)],de~nding on the assumptions about the sources becauseof forcescreatedby muscles,someof which cross of mechanicalenergy(term definedby Aleshinsky,1986a) severaljoints. One can expect that mechanicalwork at which make movement possible.For example,during a joint may not coincide with the summed mechanical calculationsof ‘externalwork’ (Cavagnaet al., 1964),it is work of musclescrossingthis joint, even if one neglects implied that human movementresultsfrom the general the activity of muscles with moments about the joint vector of external forcesappliedat the generalcentre of oppositein direction to the net momentat thisjoint. For massof the body. In calculationsof ‘pseudomechanical example,let us considermovementduring which only work’ (Norman et al., 1975) and its modifications (Piertwo muscle groups are active: the one-joint hip extensor rynowski et al., 1980),it isimpliedthat humanmovement musclesand the rectus femorismuscle.The hip joint is is realized by forcesand momentsapplied at centresof extended as a result of positive work done by the hip massof eachsegmentof the body. In calculationsof work extensormuslces,andthe two-joint rectusfemorismuscle contractsisometrically;i.e. its length doesnot change.If there are no external constrictions,the knee joint will extendbecauseof the action of the rectusfemorisandthe one-joint hip extensor muscles.In this example,work Received in jinal form 12 June 1995. *Current address: Human Performance Laboratory, Faculty doneby the kneejoint momentis positive (this work is of Physical Education, The University of Calgary, Calgary, equalto the product of the momentat the kneejoint and Alberta, Canada T2N lN4. Address correspondence. to: B. I. Prilutsky, Human Perfor- the knee angular displacement).However, the work of muscles crossing the knee joint (defined as the product of muscle force and change in the length of the muscles) is

mance Laboratory, Faculty of Physical Education, The University of Calgary, Alberta, Canada T2N lN4. 405

B. I. Prilutskyet al.

406

zero, becauseonly the rectusfemorismuscleis active and the length of this muscleis constant. The aimsof this study were(i)to comparetheoretically MEE of two humanlower extremity modelshaving different sourcesof mechanicalenergy: momentsat joints, andmuscles includingtwo-joint muscles;and(ii) to check experimentally whether or not the differencein MEE betweenthe modelsduring humanlocomotion is significant enough to warrant consideration. Preliminary resultsof the theoretical analysishave been published previously (Prilutsky et al., 1992).

was assumedthat sourcesof mechanicalenergyfor this model, muscleforces, were not ‘intercompensated’or ‘recuperative’ (Aleshinsky, 1986a).Under theseconditions, MEE of the mechanicalenergy sourcesof this model, WI, for a period of time of T1 to T2 is equal to w1 =

l-2 IP?,(t)I +

HTI 11

+ C IC”(t)l i=7

IP?i4tt)I

+

IP?6(t)I

1

(1)

dt,

whereP?(t) is the power producedby the ith muscleof the model (this power is equal to the product of muscle force and the rate of musclelength change)and t is time. To comparethe mechanicalenergy expendedby the The absolutevaluesin equation(1) meanthat mechanical modelswith different sourcesof mechanicalenergydur- energyspentby the musclecannot be reducedby simuling the samemovement,the differencein MEE between taneousabsorptionof energy(i.e.production of negative thesemodelswasestimatedusingthe theoreticalanalysis power) by another muscleat the sametime, or by the describedbelow. samemuscleduring different periodsof time. Model 1: muscle sources of mechanical energy Model 2: joint moment sources of mechanical energy The modelof the humanlower extremity with muscle Model 2 had different sourcesof mechanicalenergy: sourcesof mechanicalenergyconsistedof four rigid links (foot, shank,thigh and pelvis)interconnectedby friction- momentsat joints. We assumedthat thesesourcesof lessjoints, and eight muscles,three of which were two- mechanicalenergywere likewisenot ‘intercompensated joint muscles(Fig. 1A).Musclesweremodelledasweight- or ‘recuperative’sources(Aleshinsky, 1986a).The masslessthreadsand straightlinesof action;mucsleforce was inertia and geometricalcharacteristicsof this modeland simulatedby a pull of the thread (Alexandrovich, 1981; the modelwith musclesourcesof mechanicalenergywere Korenev and Pridvorov, 1977).The modelcould movein consideredto be the same.MEE of sourcesof mechanical the sagittal plane by meansof muscularforces.Move- energy of Model 2, momentsat joints, is equal to ment and the mass-inertiaand geometricalcharacteristicsof the model were assumedto be known. Also, it (2) THEORETICAL

ANALYSIS

A

B

10

Fig. 1. Two models of a lower extremity.A: Model 1. Legend: 1;2 is gastrocnemius muscle,3;4 is hamstrings, 5;6 is rectus

femorismuscle, 7 is tibialisanteriormuscle, 8 is soleus muscle, 9 is vastii group of muscles, 10 is iliacus muscle and 11 is gluteus maximus muscle. B: Model 2 (for explanations, see the text).

whereP;(t) is the powerof the momentat thejth joint of extremity (this power is equal to the product of the joint momentand the angular velocity at the joint). For convenienceof comparisonof MEE betweenthe models,Model 2, without muscles,waschangedby ‘identical transformations’;that is, the mechanicalenergy sourcesof Model 2 (joint moments)were substitutedby muscleforces of Model 1 in such a way that the total MEE of the sourcesof Model 2 remainedunchanged. The transformedmodel is presentedin Fig. 1B. It differs from Model 1by the absenceof two-joint muscles,which have beensubstitutedby pairs of one-joint muscleshaving the sameaction at the correspondingjoints. Thus, the gastrocnemius muscleof Model 1 indicatedby index 1; 2 (Fig. 1A) was substitutedby the two one-joint muscles with indices1 and 2 of the secondmodel (Fig. 1B). Similarly, the rectus femoris muscle(index 5;6) and the hamstrings(index 3;4) of Model 1 were substitutedby the one-joint musclesof Model 2 with indices5 and 6, and 3 and4, respectively(Fig. 1).During the substitution of the sourcesof mechanicalenergy,the following conditions were satisfied:

(1) Fdt)

= Fi(t) = Fdt); Fdt)

Fs;s(t) = F,(t) = Fadt);

= F,(t) = Fdt);

Comparison of mechanicalenergy expenditure

407

jth joint (q) is A&(t)

= AL,(t) + AL,(t);

AL,;,(t) = AL&) (3) &2(t) d:;&)

= d,(t);

+ AL&);

d:;*(t) = d,(t); 4;m

wherePC is the power producedby the ith musclecrossing thejth joint. Sinceit was assumedthat the net joint momentsand muscleforceswerenot ‘intercompensated or ‘recuperative”sourcesof m~hanical energy, the relation betweenMEE of the net momentat thejth joint and one-joint musclescrossingthis joint is

= fw);

= ddt); dl;;&) = dstt); d1;;& = &tt).

The next condition follows from conditions 1 and 2: (4) p%(t) P&(t)

= PW)

+ WtX

Fgt)

= ev)

+ PW,

= PT(t) + evb

where the subscriptsdesignatethe numbers of corresponding muscles(see Fig. 1); the superscriptsa, k and h designateankle, knee and hip joints, respectively; F(t), AL(t), d(t) and p”(c) are force, change of length, moment arm and power of a muscle,respectively. After the conducted substit~~tionof the sourcesof mechanicalenergy of Model 2, Model 1 and Model 2 had the samesources,muscleforces,that allowed one to estimate the difference in MEE between the two modelsduring the samemovement.Before making this estimation we need to show that the conducted substitution of the sourcesof Model 2, joint moments,by the sourcesof Model 1, muscleforces,did not change MEE of Model 2 [I%‘,, see equation (2)]. In other words, we needto show that

The left and right sides of inequality (6) are equal if antagonisticmusclescrossingthe joint do not produce fore (power).Sincethis conclusionholds for eachjoint of the transformed Model 2 (Fig. IB), equation (3) holds as well. Therefore, for the case where antagonistic muscles do not produce force, equation (2kMEE of Model 2-can be rewritten in the following way: (7)

Taking into account condition (4) of the model transformation, equation(I)-MEE of Model l&can alsobe rewritten:

S[ TZ

WI

=

IPW)

+

fw)l

+

IP%)

+ PWI

TI

+ IPl;t(t) -t Pg(r)l

wherethe integralon the left representsMEE of thejoint moment sourcesof Model 2 before the model transformation and the integral on the right representsMEE of the new sourcesof Model 2 (muscleforces)after the model transformation (see Fig. 1B). Consider a joint which operatesin the sagittal plane by one-joint flexor and extensormusclesexclusively.By definition (Andrews and Hay, 1983),the net (result~t) joint momentis equal to the algebraicsum of momentsof individual muscles crossingthe joint: Mj=CF,d
(4)

whereMj is the net moment at the jth joint, F, is the force of the ith musclecrossingthejth joint and dij is the momentarm of the ith musclecrossingthejth joint. Note that, by definition, momentsproducedby one-joint antagonisticmusclesabout the joint are opposite(havethe opposite sign) to the correspondingnet joint moment and to moments produced by the agonist& muscles. Given equation (4) and that dij = dLij/d#j (where dLiJd+j is the derivative of length, L, of the ith muscle crossingthe ph joint with respectto angleCpat the jth joint; An et al., 1984),the power of the net momentat the

+ F IP?(r)l i=7

1

dt.

As shownabove [seeinequality (611,the comparisonof MEE betweenModels 1 and 2 can be madeonly for the case where antagonistic musclesof the transformed Model 2 do not produceforce.According to condition(1) of the model transformation, forces produced by the two-joint musclesof Model 1 (Fig. lA, 1;2,3;4 and 5;6) are equal to forcesproducedby the correspondingonejoint musclesof Model 2 (Fig. 1B; 1 and 2, 3 and 4, and 5 and 6).Theseequalitiesandthe condition that antagonistic musclesof the transformedModel 2 do not produce force meanthat the two-joint muscle1; 2 of Model 1 can produce force only if there is an extensionmoment at the ankle and a flexion moment at the knee; two-joint muscle3;4 of Model 1 can produce force only if there is a flexion momentat the kneeand an extensionmoment at the hip; and two-joint muscle 5;6 can produce force only if there is an extensionmoment at the knee and a flexion moment at the hip. Condition (1) of the modeltransfo~ation and the condition that antagonistic musclesof the transformedModel 2 do not produce force alsohold if, during the above phasesof movement, two-joint muscles1;2, 3;4 and 5; 6 of Model 1 do not produce force. Thus, the comparison of MEE betweenModels 1 and 2 can be made for the phases

B. I. Prilutsky et al

408

of movement where the moments produced by two-joint muscles 1;2, 3;4 and 5; 6 of Model 1 at each of both joints they cross are zero or have the same direction as the net moments at these joints, and muscles crossing these joints from the opposite side do not produce force. We will consider only these phases of movement.

at each of both joints have the samedirection as the net joint moments at thesejoints, and (iii) muscles crossing these two joints from the opposite side do not produce force. The latter two conditions follow from the fact that we consideredonly phasesof movement wheretheseconditionsweremet. Theseconditions ensurethe sameMEE of Model 2 before and after its transformation.

Comparison of MEE of two models

The differencein MEE betweenModel 1 (WI) and Model 2 (W,) wasobtained.From equations(7) and (8) and the additive property of a definite integral,it follows that T2 wz-WI=

ClP%)l

s Tl

+ Ifw)l

- IP%) + f’Wl1 dt T2

CIPX)l + Ifml

+

s

Tl

- IPW + FWI dt T2 +

sTl

Clfw)l + IEM

- IP;l(t) + Pz(t)l]dt.

EXPERIMENTAL OF CONDITIONS JOINT DURING

CHECKING FOR

OF REALIZATION

DIFFERENCES

MOMENTS HUMAN

AND

IN MEE

BETWEEN

MUSCLES

LOCOMOTION

The purposeof this experimentalstudy was to check whether or not there are phasesin human locomotion where(i) signsof the musclepowerswhich are produced by a two-joint muscleat both joints are opposite,(ii) momentsproducedby that muscleat eachof both joints have the samedirection asthe net joint momentsat these joints, and (iii) musclescrossingthesetwo joints from the oppositesidedo not produceforce. During thesephases, accordingto the theoreticalanalysis,MEE of Model 1 is lower than that of Model 2. Estimating

phases of locomotion where individual

muscles

(9) produce force There are three integralsin equation(9). Each of them is Ten young, healthy males(rangingfrom 60 to 82 kg in equalto the differencein the mechanicalenergyexpendi- body mass)took part in the experiments.They walked ture betweenthe two-joint muscleandthe corresponding and ran on a treadmill operatedat a speedof 1.82m s- r two one-joint muscles.From equation (9) somecorolla- (6.5km h- ‘). Electrical activity of eight musclesof the riesfollow: lower extremity (Fig. 3) was registeredusing surfacebipolar silver electrodes8 mm in diameter,and two fourCorollary 1. W2 - WI > 0; i.e. MEE of these models, channel electromyographic amplifiers MG-400 (MedWI and WI, is not equal, and WI is not higher than Wz. icor, Hungary). After the electrical musclesignalswere amplifiedand high-passfiltered with a cutoff frequencyof Corollary 2. Possible dtfirences in MEE between Models 1 and 2 are caused by the two-joint muscles of 20 Hz, they were fed into a computer (Robotron, Germany) at a samplingfrequencyof 2083Hz for 2.8 s. Model 1. The full-wave rectified EMG of the muscleswaslowCorollary 3. Wz = WI if: passfiltered to obtain the linear envelopes.For the filtra(a) two-joint muscles are not active during a period of tion we usedthe zero phaselag, fourth-order Buttertime CT,, TJ; worth digital low-passfilter with a cutoff frequency of (b) two-joint muscles produce movement only at one 6 Hz (the correspondingfilter time-constant is about joint; i.e. they act like one-joint muscles; 25 ms). The linear envelopeand a cutoff frequency of (c) powers produced by two-joint muscles as a result of 6 Hz have been reported to reasonably representfrethe angle change at both joints have the same sign. quency characteristicsof changein force produced by human musclesduring locomotion (Hof, 1984;Winter, Corollary 4. Inequality W2 - WI > 0 occurs if powers 1979).Values of the linear envelopeobtained for each produced by two-joint muscles as a result of the angle musclewere normalizedwith respectto the peak envelchange at both joints have diferent signs. ope value of the correspondingmusclein a given cycle of Thus, the evaluation of human MEE using joint locomotion. For eachpercent of a cycle of locomotion, momentsdoesnot coincide with the mechanicalwork envelopesof each musclewere averagedover all cycles done by muscles,even if one neglectsthe activity of and subjectsavailable.A total of more than 25 cycleswas muscleswith momentsabout joints oppositein direc- processedfor both walking and running. tion to the correspondingnet momentsat thesejoints. Becauseof the excitation*ontraction coupling and The difference in MEE betweenjoint moments and muscle-tendoncomplex dynamics,there is a time shift muscles (Wz - WI > 0) realized by a two-joint betweena timecourseof the linearenvelopeof EMG and muscleoccurs if (i) signsof the musclepowerswhich the muscleforce (electromechanical delay, EMD). Thereare produced by the two-joint muscleat both joints fore, to find phasesof movement where a muscleproare opposite, (ii) moments produced by that muscle duces force based on muscle electrical activity, it is

Comparison of mechanical energy expenditure

0

20

40 Normalized

60

80

100

cycle time, %

Fig. 2. A diagram illustrating the determination of phases where a muscle produces force. 1 is the linear envelope obtained by low-pass filtering the full-wave rectified EMG with the zero phase lag; 2 is curve 1 shifted in time by 40 ms; 3 is curve 1 shifted in time by 100 ms. A muscle was considered to begin producing force at the instant where curve 2 exceeded the 20% threshold. A muscle was considered to stop producing force at the instant where curve 3 became less than the 20% threshold. The shaded rectangles show the estimated phases where the muscle produces force.

necessary to account for EMD. Unfortunately, available

information from the literature about EMD of human musclesis highly inconsistent.The reported valuesvary from 25 ms(Viitasaloand Komi, 1981)to 106ms(Vos et al., 1991),dependingon the muscle,type of contraction and motor task. Therefore, we used the following approach to estimatephasesof locomotion wheremuscles produceforce.To estimatethe instant of the beginningof force production, the obtainedaveragelinear envelopeof eachmusclewasshiftedin time by 40 ms(Fig. 2, curve 2). This time shift was reported to be a reasonableaverage estimateof EMD for humanmuscles(Hull andHawkins, 1990).A musclewasconsideredto beginproducingforce at the instant wherecurve 2 exceededthe 20% threshold. To estimatethe instant of the endof force production, the obtainedaverageenvelopeof eachmusclewasshiftedin time by 100ms(Fig. 2, curve 3).A musclewasconsidered to stop producing force at the instant wherecurve 3 becamelessthan the 20% threshold.A muscleforce correspondingto the level of the shiftednormalizedenvelopeof O-20% (open rectanglesin Fig. 2) was consideredvery low, and neglected.The above approachfor determining phasesof muscleforce production accountsfor the inconsistencyin experimentaldata about EMD and the fact that EMD during the beginningof force production is shorterthan during relaxation within one contraction (see,for example,Hof, 1984;Vos et al., 1991). Registration

of movement and inverse dynamics analysis

Two subjects(64and 68kg in body mass)participated in the secondseriesof experiments.Fewer subjectswere usedin this seriesof experimentscomparedto the first seriesbecausewe expectedlessvariability betweensubjects in kinematicsand kineticsthan in electricalmuscle activity during walking and running (see,for example, Winter, 1991).The subjectswalked and ran at constant speedson a specialwooden rostrum 40 m in length. In

409

the middle of this rostrum, two force platforms(VISTI, former U.S.S.R.) were embedded.The force platforms were usedfor recording the three componentsof the resultantvector of the ground reactionforcesand coordinatesof its point of application.A bilateral stereophotogrammetricfilming wasusedfor registrationof kinematics (for details, seePrilutsky and Zatsiorsky, 1992, 1994; Zatsiorsky et al., 1982).Reflectivemarkers attached to the main joints of the subjects’body were flashedby stroboscopesoperatedat 100Hz. Tracesof the markers wereexposedonto photoplatesof four photogrammetric cameras(UMK-10, Carl ZeissJena,Germany). Coordinates of the body markers were digitized with a precision of 1 pm using a stereocomparator‘Stecometer’(Carl ZeissJena, Germany).Trials of walking and running a$ speeds of 1.82ms-’ (6.5kmh-‘) and 1.57ms-’ (5.6km h- ‘), respectively, were chosen for further analysis. A 16-link,three-dimensional modelof the humanmusculoskeletal system and the software HUMMOT (Prilutsky, 1991, 1993)were used for computation of momentsand powersat joints of the lower extremity, in the sag&al plane.The main assumptions madein developing this model coincide with the assumptionscited earlier(Aleshinskyand Zatsiorsky, 1978).The dynamics of the model was describedusing equationsbasedon D’Alembert’s principle of ‘equilibrium’ of bodies in motion and the linear and angularmomentumtheorems for a rigid body. The inputs requiredfor computingwere the body marker coordinates,the force platform data, and the subjects’mass-inertialcharacteristicsderived from the regressionequations(Zatsiorsky et al., 1990). Joint moments in the sag&al plane and their powers were the outputs of these calculations. Results of experimental

study

During walking and running there were periods of time when muscles crossing the ankle, knee and hip joints

from the oppositesideswerenot active (Figs 3 and 4) or did not produce force simultaneously(Figs 5 and 6). During the periodsof &20%, 26-64% and 66-100% of the normalizedcycletime in walking,the tibialis anterior, and the soleusand gastrocnemius medialismusclesdid not produceforce simultaneously.During the periodsof O-27%, 32-42% and 6684% of the normalizedwalking cycle time, the gastrocnemiusmedialisand the quadriceps femoris muscles did not produce

force simulta-

neously. During the period of 42-100% of the normalized walking cycle time, the rectus femoris, and the hamstringsand gluteusmaximusmusclesdid not produceforce simultaneously(Fig. 5). During

running,

the normalized

time of simultaneous

forceproduction by musclescrossingthe ankle,kneeand hip joints from the opposite sides was longer than that during walking (see Fig. 6: tibialis anterior vs soleus and gastrocnemius medialis; gastrocnemius medialis vs vastus and rectusfemoris;rectusfemorisvs hamstringsand gluteus maximus). However, during running as well as during walking, there was a phase of movement when the two-joint muscles produced force (and, hence, moments

410

B. I. Prilutsky et al.

Stance 100 80 60 40 20 0 EILI

0

lo

20

30

40

50

60

70

80

90100

Normalized cycle time, %

0

1020304050607080w1w

Normalized cycle time, %

Fig. 3. The full-wave rectified, low-pass filtered with the zero phase lag EMG of eight muscles as a function of the normalized cycle time during walking. Thick lines represent the average values of ten subjects; thin lines represent the standard deviation. The vertical lines separate the stance and swing phases. TA is tibialis anterior muscle; SO is soleus muscle; GM is gastrocnemius medialis muscle; VM and VL are vastus. medialis and vastus lateralis muscles, respectively; RF is rectus femoris muscle; HA is hamstrings (long head of biceps femoris muscle); GLM is gluteus maximus muscle. The nominal speed of walking was 1.82 ms-’ (6.5 km h-l).

about adjacent joints) and muscles crossing these joints from the other side did not produce force. This phase almost coincided with the swing phase. The rectusfemorismuscleproducedforcefrom the beginning of the swingphaseto the middleof the swingphase. The hamstringsproducedforce from the middle to the endof the swingphase,and the gluteusmaximusproducedforce from 65to 100%of the normalizedrunningcycle time. During walking, there were four phasesin which powers at adjacentjoints had opposite signs(Fig. 7). During the first phase(S-15% of the normalized cycle time),powerat the kneejoint, producedby kneeextensor muscles,had a negativesign,and power at the hip joint, produced by hip extensormuscles,had a positive sign. During the secondphase(approximately M-35% of the cycle time),power at the anklejoint, producedby ankle extensormuscles,had a negativesign,powerat the knee joint, producedby kneeextensormuscles,had a positive sign,and power at the hip joint, producedby hip flexor muscles,had a negative sign. During the third phase (3540% of the cycle time), power at the ankle joint, producedby ankleextensormuscles,had a positivesign, and power at the kneejoint, producedby kneeextensor

muscles,had a negative sign. During the fourth phase (6O-75% of the cycle time), power at the knee joint, producedby kneeextensormuscles,had a negativesign, and power at the hip joint, produced by hip flexor muscles,had a positive sign. During running, there werealsofour phasesin which powers at adjacent joints had opposite signs(Fig. 8). During the first phase(&20% of the normalizedcycle time),powerat the kneejoint, producedby kneeextensor muscles,had a negativesign,and power at the hip joint, produced by hip extensormuscles,had a positive sign. During the secondphase(3440% of the cycle time), power at the knee joint, produced by knee extensor muscles,had a positive sign,and power at the hip joint, produced by hip flexor muscles,had a negative sign. During the third phase(approximately 50-65% of the cycle time), power at the kneejoint, produced by knee extensor muscles,had a negative sign, and power at the hip joint, produced by hip flexor muscles,had a positive sign. During the fourth phase (87-100% of the cycle time), power

at the knee joint, produced

by

knee flexor muscles,had a negativesign, and power at the hip joint, produced by hip extensor muscles,had a positive sign.

Comparison of mechanical energy expenditure

411

Stance

Swing

s 100 g a0 wao jso 4 20 u 0 0

IO

m

30

40

0

0

lo

20

3i

4iJ

io

so

70

a-0

90

10

20

30

40

!io

60

70

80

90

100

0 io 2i 3il 40so so m

loo

Normalized cycle time, %

Ncrmaliz~

cycle time, X

Fig. 4. The full-wave rectified, low-pass filtered with the zero phase lag EMG of eight muscles as a function of the normalized cycle time during running. Thick lines represent the average values of ten subjects; thin lines represent the standard deviation. The vertical lines separate the stance and swing phases. TA is tibialis anterior muscle; SO is soleus muscle; GM is gastrocnemius medialis muscle; VM and VL are vastus medialis and vastus lateralis muscles, respectively; RF is rectus femoris muscle; HA is hamstrings (long head of biceps femoris muscle); GLM is gluteus maximus muscle. The nominal speed of running was 1.82 m s- r (6.5 km h - ‘).

Stance

Stenee

Swing

GLlll

GLM

HA

MA

m

RF

RF

VL

VL

w-

VM

GM

1 m -

I

I

I

G)s

I

so l-A

swing

so 0

! 10

20

So

40

Nonnaked

50 cycle

60

, 70

, a0

, 90

TA loo

time, %

Fig. 5. The estimated phases of the walking cycle where muscles produce force. The vertical line separates the stance and swing phases. TA is tibialis anterior muscle; SO is soleus muscle; GM is gastrocnemius medialis muscle; VM and VL are vastus medialis and vastus lateralis muscles, respectively; RF is rectus femoris muscle: HA is hamstrings (long head of biceps femoris muscle); GLM is gluteus maximus muscle.

,““I”“I.“‘I”“1.“‘I”“(““I”“1”“,““’ 0 10 20 30

40

Normalized

50

00

70

00

90

loo

cycle lime, %

Fig. 6. The estimated phases of the running cycle where muscles produce force. The vertical line separates the stance and swing phases. TA is tibialis anterior muscle; SO is soleus muscles; GM is gastrocnemius medialis muscle; VM and VL are vastus medialis and vastus lateralis muscles, respectively; RF is rectus femoris muscle; HA is hamstrings (long head of biceps femoris muscle); GLM is gluteus maximus muscle.

B. I. Prilutsky et nl

412

Stance

phase

Swing

DISCUSSION

phase

C

B

The purposeof this study wasto compareMEE of the joint momentand the muscle force sources of mechanical energy during human locomotion. The results of the theoretical analysisshowed(seeCorollary 4) that the muscleforce sourcescould spendlessmechani~l energy than the joint momentsourcesundercertain conditions. The resultsof the experimentalstudy suggestthat these conditions occur during certain phasesof the swing in walking and running. Realization

of Corollary

4 during human locomotion

According to Corollary 4, MEE of Model 1,which had the musclesourcesof mechanicalenergy,can belessthan that of Model 2, having the joint moment sourcesof mechanicalenergy,if(i) signsof the musclepowerswhich are producedby the two-joint muscleat both joints are opposite,(ii) momentsproducedby that muscleat eachof 100 both joints have the samedirection as the net joint Normalized cycle time, % momentsat thesejoints, and (iii) musclescrossingthese Fig. 7. Powers of-net moments at the ankle (A), knee(B) and hip two joints from the oppositesidedo not produceforce. (C)joints in the sagittal plane as a function of the normalized Figures5 and 6 showthat during mostof the swingphase cycle time during walking in one subject. Opened and dosed areas represent powers produced by extensor and flexor muscles of walking and running, two two-joint muscies-the at thejoints,respectively (themuscles producingthepowerwere rectusfemorisand the hamstrings-appearedto produce determined using net moments at the joints). The arrows and force during different periods of time without simultanumbers indicatephases wherethe powersat adjacentjoints neousforce production by musclescrossingthe kneeand havethe oppositesigns.The speedof walkingwas1.82ms-l hip joints from appositesides.This result and the fact (6.5 km h-l). that the kneeand hip powershad oppositesignsduring certainphasesof the swingin walking(Fig. 7, phase4; see alsoWinter, 1983a;Zatsiorsky et al., 1982)and running (Fig. 8, phases3 and4; seealsoWinter, 19836Zatsiorsky Stance phase 1 Swing phase 300 et al., 1982)suggestthat thesepowerswereproducedby A I 1 the rectusfemorisand hamstringswithout force production by musclescrossingthe kneeandhip joints from the oppositesides.During phase4 in walking and phase3 in running, the rectus femoris muscleproduced negative power at the knee joint, and the rectus femoris and, probably, iliacus (seeInman, 1953)produced positive power at the hip joint. During phase4 in running, the hamstringsproducednegative power at the kneejoint, and the hamstringsand gluteusmaximusmusclesproducedpositive power at the hip joint. Thus, during the swingin walking (phase4) and running (phases3 and 4), it appearsthat Corollary 4 occurs, which meansthat during the above phasesof the swingin humanlocomotion, MEE of Model 1, having the musclesourcesof m~hanical energy, may be lessthan that of Model 2, having the joint moment sourcesof mechanicalenergy. This fact must be taken into account if MEE is estimated basedon the power at joints. 100

-lWB

swe

Normalized

cycle

time,

%

Possible in$luence of the assumptions about sources of mechanical energy on MEE estimates (C)jointsin thesagittalplaneasa functionof the normalized of both the models cycle time during running in one subject. Opened and closed areas represent powers produced by extensor and flexor muscles From Figs3-6, and the results reported by other at the joints, respectively (the muscles producing the power were authors(Elliott and Blanksby, 1979;Falconerand Windetermined using net moments at the joints). The arrows and ter, 1985;Nilssonet al., 1985),it follows that during the numbers indicate phases where the powers at adjacent joints stance phase of walking and especially of running, there have the opposite signs. The speed of running was 1.57 ms-t are phasesin which musclescrossingthe joints from (5.6 km h-r). Fig. 8. Powers of net moments at the ankIe (A), knee(B) and hip

Comparison of mechanicalenergy expenditure opposite sides are active and produce force simultaneously. Taking into account assumptions that the sources of mechanical energy of Models 1 and 2 are neither ‘intercompensated’ nor ‘recuperative’, MEE of the model having muscle sources of mechanical energy should be higher than that of the model with joint moment sources during phases of simultaneous force production by muscles on the opposite sides of the joints [see inequality (6)]. However, it may be assumed that the difference in MEE between the models is not very high in this case. First, the comparison of the calculated Achilles tendon force from the net ankle moment, and the directly measured Achilles tendon force during different types of jumps (Fukashiro et al., 1993) and during cycling (Gregor et al., 1991) showed that the calculated forces were not less than the measured forces. However, it could be expected that the calculated force would be underestimated if ankle flexor muscles produced a substantial moment about the joint. Notice that during jumping, there is co-activation of ankle flexor and extensor muscles (Pandy and Zajac, 1991; Prilutsky et al., 1989). Second, during the stance phase in running (Fig. 6; see also Elliott and Blanksby, 1979) and vertical jumps (Bobbert and van Ingen Schenau, 1988; Pandy and Zajac, 1991; Prilutsky et al., 1989), co-activation of some twoand one-joint muscles located on adjacent segments and crossing the same joints from the opposite sides-rectus femoris vs gluteus maximus, and gastrocnemius vs quadriceps femoris-was observed. Because of this co-activation, two-joint muscles can transfer mechanical energy between joints (Bobbert and van Ingen Schenau, 1988; Prilutsky and Zatsiorsky, 1994; van Ingen Schenau et al., 1990). This energy transfer is responsible for the difference between the power produced by the net moment at a given joint, and the summed power of muscles serving this joint (Prilutsky and Zatsiorsky, 1994; van Ingen Schenau et al., 1990; see also the example in the Introduction section). However, as was shown by Prilutsky and Zatsiorsky (1994), the sum of these differences over all joints is equal to zero if one neglects energy dissipated or generated by two-joint muscles during the energy transfer. According to estimates by Prilutsky and Zatsiorsky (1994; Table 2), this dissipated or generated energy is approximately 8-20% of the summed work of the net joint moments. Thus, the total MEE of the joint moments might not differ substantially from that of the muscles during the stance phase. In order to compare MEE of two models, one with the muscle sources and the other with the joint moment sources, we assumed that the muscle and the joint moment sources of energy were not ‘recuperative’. The absence of recuperative sources in both the models means that mechanical energy loss (generation of positive power) by a source cannot be returned into the system by the same source absorbing mechanical energy (generating negative power) during different periods of time (Aleshinsky, 1986a). However, in reality, a certain recuperation probably takes place in walking (Asmussen and Bonde-Petersen, 1974) and running (Cavagna et al., 1964): part of the negative work done by muscles (or joint

413

moments) may be stored in muscles’ elastic structures during their lengthening as strain energy, which may be used by the same muscles (or joint moments) to do positive work during the subsequent shortening of the elastic structures. The assumption about the absence of recuperative sources in both the models may influence the difference in MEE between them. From equation (9) it follows that a possible difference in MEE between two models during a stretch-shortening cycle may be caused by differences in behaviour of the two-joint muscles of Model 1 and the one-joint muscles of Model 2 which substitute for the corresponding two-joint muscles (Fig. 1A, B). The greatest difference in MEE wili occur if IV2 - W, > 0 (see Corollary 4) and a two-joint muscle of Model 1 contracts isometrically-that is, its MEE is zeroduring a stretch-shortening cycle (simultaneous lIexion is followed by simultaneous extension of the equivalent magnitude at the adjacent joints). In this case, the two one-joint muscles of Model 2 which substitute for this two-joint muscle do an equivalent amount of positive and negative work, respectively, during both flexion and the subsequent extension at the joints. If recuperation is allowed, then MEE of the one-joint muscle which does negative work during the flexion at the joints will be zero for the entire stretch-shortening cycle, because mechanical energy lost by this muscle during the subsequent shortening will be compensated by the energy absorbed during the preliminary muscle stretch. Correspondingly, MEE of the one-joint muscle which does positive work during the flexion at the joints will be zero for the entire stretch~ho~ening cycle, because m~hani~ energy lost by this muscle during the initial shortening will be compensated by the energy absorbed during the subsequent muscle stretch. Thus, if recuperation of mechanical energy is allowed, the difference Wz - W 1 is zero. Note, however, that in reality, recuperation occurs only to a limited degree-a substantial part of the energy absorbed by a muscle is lost (Morgan et al., 1978).Therefore, it may be expected that allowing recuperation decreases the difference Wz - W 1;however, the difference should be positive if the conditions for Corollary 4 are met. The assumption that the muscle sources of mechanical energy are not intercompensated means that if two muscles (for example, m. vastus medialis and m. iliacus) simultaneously do an equivalent amount of positive and negative work, the total MEE of these two muscles will not be zero. As shown in this study, the gumption that sources of mechanical energy of Model 2 (joint mom~ts) are not intercompensated may cause overestimation of mechanical energy spent by humans because of the presence of two-joint muscles [see equation (9) and the conditions for which MEE of Model 2 is greater than that of Model 11. Because of the difference in MEE between the models with and without two-joint muscles (Fig. lA, B), one can speak about the economy of mechanical energy by two-joint muscles which can be estimated using equation (9). Economy of mechanical energy by two-joint muscles Elftman (1940) estimated the economy of mechanical energy by two-joint muscles during sprint running as

414

B. I. Prilutskyet al.

47% of the total mechanicalwork of the joint moments. Elftman implied that in the phasesof movementwhere two-joint musclescould saveenergy(i.e.wherepowersat adjacentjoints had oppositesigns),one-joint agonistic and antagonisticmuscleswere not active. This assumption meansthat power at adjacentjoints is producedby two-joint musclesexclusively. As can be seen, for example,in Figs 5 and 7, powersat the ankle and knee joints having oppositesignsin phase3 of the walking cycle are not producedby one two-joint muscle,but by three different muscles:the soleus,gastrocnemiusand rectusfemoris.Therefore,Elftman’sestimatesof the energy savedby two-joint musclesappear to be significantly higher than the actual values. Estimatesof the mechanicalenergyeconomyby twojoint musclesduring walking (11.6%)and vertical jumps (6.4%) obtainedby Wells (1988)seemto be better than those by Elftman (1940). Wells did not assumethat powers at adjacent joints are produced by two-joint musclesexclusively.He calculated the power produced by eachone-and two-joint muscleincludedin the model. The energyeconomyby two-joint muscleswasthen estimated using an approach similar to that used in this study for estimatingthe differencein MEE betweenthe modelswith and without two-joint muscles[equation (9)]. Sincethe purposeof Well’sstudy wasnot to estimate the differencein MEE betweenthe muscleforce sources and the joint moment sourcesof mechanicalenergy, MEE of thejoint momentsand the correspondingequivalent one-jointmusclesproviding the samemovementof the model were not explicitly required to be the same. However, the algorithm usedto estimateforcesof twojoint musclesensuredimplicitly that muscleswith momentsabout joints oppositein direction to the corresponding net momentsat thesejoints did not produce force-the condition usedin this study to compareMEE of the modelswith and without two-joint muscles.This force estimationalgorithm predictsno force production by two-joint muscleswhen the net momentsat both adjacentjoints (for example,hip and knee,or knee and ankle) tend to extend thesejoints simultaneously(see Fig. 1 in Wells,1988).As shownin this and other studies, during the stancephasein walking (Fig. 5; Fig. 7, phases 2 and 3, knee and ankle joints), running (Figs6 and 8) and vertical jumps (Bobbert and van Ingen Schenau, 1988;Pandy and Zajac, 1991;Prilutsky et al., 1989),the net momentsat the ankle, knee and hip joints tend to extend them; however, the two-joint musclescrossing thesejoints are active and shouldproduce force. Estimates of the economy of MEE by two-joint musclescan also be obtained using equation (9) and resultsfrom this study regarding phasesin which the musclesproduce force and power during walking and running (Figs 5-8). If one assumesthat negative and positive powersat the knee and hip joints during phase 4 in walking (Fig. 7) and phases3 and 4 in running (Fig. 8) are producedby the rectusfemorisor hamstring musclesexclusively, the economy of MEE that results from the action of thesemuscleswill be 7.8 and 12.7J during walking and running, respectively.Thesevalues

are lessthan 3% of MEE of the 16-link, three-dimensionalmodelof the humanbody estimatedfor the corresponding types of locomotion by Prilutsky and Zatsiorsky (1992)usingequation (2). Thus, results obtained in this study show that total MEE of the modelswith different sourcesof mechanical energy,the joint moments and muscle forces, appears to be different during the swingphaseof locomotion.However, this differenceis probably relatively small. would like to thank Dr V.M. Zatsiorskyfor his valuablecomments and J. Falck for her Acknowledgements-We

help in editing the manuscript. We thank th- staff of the Human Performance Laboratory of The University of Calgary where the final variant of the manuscript was prepared. The preparation of the final variant of the manuscript was supported in part by the Alberta Heritage Foundation for Medical Research. REFERENCES

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