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Leg stiffness and stride frequency in human running
J. Biomechanics, Vol. 29. No. 2. pp. 181-186, 1996 Copyright 0 1995 Elscvicr Science Ltd Printed in Great Britain. All rights reserved 0021-929Q96 s15.00 + .I0
Pergamon 0021-9290(95)00029-l
LEG STIFFNESS AND IN HUMAN
STRIDE FREQUENCY RUNNING
Claire T. Farley and Octavia Gonzalez Harvard University, Museum of Comparative Zoology, Concord Field Station, Old Causeway Road, Bedford, MA 01730, U.S.A. Abstract-When humans and other mammals run, the body’s complex system of muscle, tendon and ligament springs behaves like a single linear spring (‘leg spring’). A simple spring-mass model, consisting of a single linear leg spring and a mass equivalent to the animal’s mass, has been shown to describe the mechanics of running remarkably well. Force platform measurements from running animals, including humans, have shown that the stiffness of the leg spring remains nearly the same at all speeds and that the spring-mass system is adjusted for higher speeds by increasing the angle swept by the leg spring. The goal of the present study is to determine the relative importance of changes to the leg spring stiffnessand the angle swept by the leg spring when humans alter their stride frequency at a given running speed. Human subjects ran on treadmill-mounted force platform at 2.5 m s- ’ while using a range of stride frequencies from 26% below to 36% above the preferred stride frequency. Force platform measurements revealed that the stiffness of the leg spring increased by 2.3-fold from 7.0 to 16.3 kNm-’ between the lowest and highest stride frequencies. The angle swept by the leg spring decreased at higher stride frequencies, partially offsetting the effect of the increased leg spring stiffness on the mechanical behavior of the spring-mass system.We conclude that the most important adjustment to the body’s spring system to accommodate higher stride frequencies is that leg spring becomes stiffer. Keywords: Locomotion; Spring-mass model, Elastic energy; Biomechanics; Motor control.
INTRODUCTION
When animalsrun, they bouncealong the ground using musculoskeletalspringsto alternately store and return elasticenergy(Cavagnaet al., 1964,1977).Muscles,tendons and ligamentscan all behave as springs,storing elasticenergy when they are stretchedand returning it whenthey recoil (Alexander, 1988).During running, this complex system of musculoskeletalsprings behaves much like a singlelinear spring(the ‘leg spring’).In fact, a simple spring-massmodel (Fig. l), consistingof a singlelinear leg spring and a massequivalent to the animal’smass,has beenshown to describeand predict the mechanicsof running remarkably well (Alexander, 1992; Alexander and Vernon, 1975; Blickhan, 1989; Blickhan and Full, 1993;Cavagna et al., 1988;Farley et al., 1991,1993;He et al., 1991;Ito et al., 1983;McGeer, 1990; McMahon and Cheng, 1990; Thompson and Raibert, 1989). How can a simplespring-masssystembe adjustedto operateat different running speeds? As animalsrun faster, the vertical excursionof the centerof massduring the stancephasedecreases, and the time that eachfoot is on the ground decreases.In the spring-massmodel, these changescould bethe result of an increasedstiffnessof the leg spring or an increasedanglesweptby the leg spring during the ground contact phase.By increasingthe angle Received in final form 23 February 1995. Address correspondence to: Claire T. Farley, Department of Human Biodynamics, 103 Harmon, University of California, Berkeley, CA 94720, U.S.A.
swept by the leg spring, the vertical excursion of the center of massand the ground contact time can be reducedwithout changingthe stiffnessof the leg spring. Biomechanicalstudieshave shownthat as animalsrun faster,the body’s springsystemis adjustedto bounceoff the ground in lesstime by increasingthe anglesweptby the leg spring during the ground contact phaserather than by increasingthe stiffnessof the leg spring(Farley et al., 1993;He et al., 1991).The stiffnessof the leg spring remainsnearly the sameat all speedsin a variety of animalsincluding running humans,hopping kangaroos and trotting horses(Farley et al., 1993;He et al., 1991). Although the stiffnessof the leg spring remainsthe sameat all speedsduring forward running, experimental evidenceshowsthat it is possiblefor humansto alter the stiffnessof their leg spring.When humanshop in place and vary their hopping frequency,the stiffnessof the leg springcan be changedby asmuch astwofold to accommodatedifferent hoppingfrequencies(Farley et al., 1991). Similarly, when humansbounce vertically on a compliant board, the stiffnessof the legspringcan changeby twofold in responseto changesin kneeangle(Greeneand McMahon, 1979).Finally, when humansrun with increasedkneeflexion (‘Groucho Running’),the stiffnessof the leg spring appearsto decrease(McMahon et al., 1987).Thesestudiesclearly demonstratethat it is possible to change the stiffnessof the leg spring during bouncingmovements. The goal of the present study is to determine the relative importanceof changesto the leg springstiffness and the angle swept by the leg spring in adjusting the behavior
181
of the spring-mass
system when humans alter
182
C. T. Farley and 0. Gonzalez
sent the percent differencein stride frequency from the preferredstridefrequency.We defineda strideasthe time from the footfall of onelegto the next footfall of the same leg. On average, the preferred stride frequency was 1.33& 0.09stride per second(mean+ SD.). In all trials, including the trials at the preferredfrequency, the subjects matchedtheir footfalls to a metronomewhile they ran on a motorized treadmill. Each subject required about two l-h practice sessions to learn to match the frequency of the metronome to within 1%. At eachfreFig. 1. Runningis modeledas simplespring-mass system bouncingalongthe ground(McMahonandCheng,1990).The quency, the propertiesof the musculoskeletalspringsysmodelccnsistsbfa mass anda singlelegspring(whichconnects temwereanalyzed usingforce platform measurements in thefootandthecenterof mass of theanimal). Thisfiguredeicts order to addressthe questionof whether the stiffnessof the modelat the beginningof the stancedhase(leftmostposition),at the middleof the stancephase(legspringis oriented the legspringcan beadjustedto accommodatea rangeof vertically)andat theendof the stancephase(rightmostposi- stride frequenciesat a given speed. tion).Thelegspringhasaninitial length,Lo,at thebeginning of The vertical ground reactionforce during running was thestancephase, anditsmaximalcompression isrepresented by measured using a treadmill-mounted force platform AL. The dashedspring-mass modelshowsthe lengthof the (Kram and Powell, 1989).A strain-gaugedforce platform uncompressed legspring.Thus,thedifference between thelength of the dashedleg springand the maximallycompressed leg (model OR6-5-1, Advanced Mechanical Technology, springrepresents the maximumcompression of the legspring, Newton, MA) wasmounted under the tread belt (Kram AL. Thedownwardverticaldisplacement of themass duringthe and Powell, 1989).The cross-talkfrom a horizontal force stancephaseis represented by Ay andis substantially smaller to the vertical waslessthan 1%. The natural frequencyof thanAL. Half of the anglesweptby the legspringduringthe vertical vibration of the force platform mounted in the groundcontacttimeis denotedby 8. treadmill was 160Hz. The force signal was sampledat 1 kHz. The spring-massmodel that wasusedto analyze the their stride frequencyat a given running speed.It is well mechanicsof running consistedof a massand a single known that changingstridefrequencyat a given running linear massless ‘leg spring’(Fig. 1; Farley et al., 1993; He speedhasstrongeffectson energeticcost(Hogberg,1952; et al., 1991;McMahon and Cheng,1990).The stiffnessof Cavanagh and Williams, 1982),impact forces on the the leg spring,kleg,wasdefinedasthe ratio of the force in musculoskeletal system(Clarke et al., 1983),and mechan- the spring,F, to the displacementof the spring,AL, at the ical power of the centerof massandlimbs(Cavagnaet al., instant when the leg spring wasmaximally compressed: 1991;Kaneko et al., 1987).Basedon previousstudies,we kleg = FJAL. (1) predict that when stride frequency is increased,the spring-masssystemwill be adjustedusingoneor both of The leg springwasmaximally compressedat the middle the following mechanisms: (1) increasingthe stiffnessof of the stance phasewhen the leg spring was oriented the leg spring;and (2) increasingthe anglesweptby the vertically, and thus, F correspondedto the peak vertical leg spring. ground reaction force. The peak displacementof the leg spring, AL, was METHODS calculatedfrom the maximum vertical displacementof the centerof mass,Ay, the initial length of the legspring, Four male subjects(averagemass73.4& 8.3kg and L, (measuredasthe vertical distancefrom the ground to average leg length of 0.97 f 0.03m, meank S.D.) be- the greater trochanter during standing),and half of the tweenthe agesof 21 and 29 yr volunteeredto participate anglesweptby the legspringwhile it wasin contact with in this study. They werein good health and wereexperi- the ground, 0 (Fig. 1; McMahon and Cheng,1990): encedtreadmill runners.Informed consentwasobtained AL=Ay+L,(l -cost)). (2) from the subjects,and the experimental protocol was approvedby the Human SubjectsCommitteeat Harvard The vertical displacementof the center of mass,Ay, was University. calculatedby integrating the vertical accelerationtwice At the beginningof the experiments,eachsubjectper- asdescribedin detail elsewhere (Blickhan and Full, 1992; formed 10min runs on two separatedays to determine Cavagna,1975).From geometricconsiderations(Fig. l), the preferred stride frequency for running at 2.5 m s- ‘. an equation for calculating0 from the time of foot conThe running speedof 2.5m s- ’ was chosenbecauseit tact with the ground, t,, the forward speed,u, and LO was was low enough that a large range of stride frequencies obtained: above and below the preferredfrequency could be sus0 = sin- ’ (ut,/2L,). (3) tained. In subsequentexperiments,each subject ran at The vertical motionsof the systemduring the ground nine stride frequencies:the preferred, four below the preferred( - 5, - 11,- 18and - 26%) and four above contact phasecan be describedin terms of the ‘vertical the preferred( + 17, + 25, + 30 and + 36%).Through- stiffness’,k,.,. The vertical stiffness,k,,,, doesnot cormsout this paper, ‘A stride frequency’and ‘ASF will repre- pond to any physicalspringin the model.Rather,k,,,, de:.-..j
Leg stiffness and stride frequency in human running
2.5, A. Lowest
scribesthe vertical motions of the center of massduring the ground contact time and is important in determining how long the spring-masssystemremainsin contact with the ground. The effective vertical stiffnesswascalculated from the ratio of the force, F, to the vertical displacement of the centerof mass,Ay, at the momentwhenthe center of massreachedits lowestpoint: k wrt -- F/Ay.
k “Cl, = F/[( F/kleg) - Lo (1 - cos 0)].
(5)
Equation (5) showsthat when the leg springis oriented vertically (@= 0”) during the entire time that it is in contact with the ground (e.g.during hopping or running in place),k,,, = klcg. As tl increasesabove zero (e.g.luring forward locomotion), k,,,, increasesrelative to klcg This can be observedin that the vertical displacementof the centerof mass,by, decreases relative to the displacement of the leg spring,AL, when the anglesweptby the leg springis greater.In addition, analysisof equation(5) showsthat k,,, will increaseif klsg increases.A detailed discussionof the concept of the vertical stiffnessis includedin McMahon and Cheng(1990).
RESULTS
The generalpattern of vertical force applicationto the ground and of vertical displacementof the centerof mass was similar at all stride frequenciessuggestingthat a ‘spring-like’ gait was used over the range of stride frequencies(Fig. 2). The vertical ground reaction force beganto rise as the foot hit the ground and reachedan initial peak within the first 0.035s of the ground contact phase.This initial peak was associatedwith the heel hitting the ground at the beginningof the ground contact phase.After the initial peak, the vertical force smoothly increasedto a midstepmaximum that occurred at the sametime asthe centerof massreachedits lowestpoint. Then, during the secondhalf of the ground contact phase, the center of massmoved upward, and the vertical force smoothly decreased,reaching zero as the foot left the ground. The vertical displacementof the centerof massand the ground contact time decreasedat higher stride frequencies,showingthat there were substantialadjustmentsto the mechanicalbehavior of the musculoskeletalspring systemwhen stride frequency wasaltered (Fig. 2). First, as stride frequency was increased,the vertical displacement of the center of massduring the ground contact phase,Ay, decreasedsubstantially(Figs 2 and 6B). Be-
S.F. (-26%)
,
r 0.00 2 i c. - -0.05
ii 4 s k
--0.10
f 2
(4)
The center of massreachedits lowestpoint at the same time as the leg spring wasmaximally compressed. In the spring-massmodel, the effective vertical stiffness,L, is determinedfrom a combination of the stiffnessof the leg spring,kleg, half the anglesweptby the leg spring,8, and the force in the spring,F. Equation (4) can be rewritten to showthe interrelationshipbetweenthese aspectsof the systemby definingAy in termsof F, kleg, Lo and 0 [equation (2)]:
183
0.04
--I-0.y
2.5, B. Preferred
S.F.
f Igo-
k.;
1.5
r 0.00 $e
f: E.
- -0.05 E d c
a3
- -0.10 8 3 d
W” *ea.5
0.0
(
z -0.15 s
I
Time (s) Fig. 2. Vertical ground reaction force (thick line)and vertical displacement (thin line) as a function of time during theground contact phase for: (A) the lowest stride frequency, (B) the preferred stride frequency, and (C) the highest stride frequency. Note that the general pattern of vertical ground reaction force and displacement was similar for all stride frequencies suggesting that a spring-like gait was used at all stride frequencies. However, the time of foot-ground contact and the magnitude of the vertical displacement both decreased at higher stride frequencies. These are typical data from a 67 kg subject with a leg length of 0.94 m.
tween the lowest and highestfrequencies,Ay decreased by 76% (0.107+ 0.011m at the lowest frequency to 0.025+ 0.001m at the highest frequency, mean of all subjects+ S.E.M.). In addition, as frequency was increased,the time that a foot was on the ground (the ground contact time, t,) decreasedby 32% (Figs 2 and 3; 0.365& 0.009s at the lowestfrequencyto 0.248f 0.008s at the highestfrequency,Fig. 3). Theseobservationssuggestthat there were substantialchangesto the vertical stiffnessto accommodatechangesin stride frequency. The slopeof the relationshipbetweenthe vertical force and the vertical displacementof the center of massincreasedat higher stride frequencies,suggestingthat the
C. T. Farley and 0. GonzLlez
184
5 Y" $'3 i; 2 3 -Q g,
0.40 0.35 0.30 0.25
8
Downward vertical 10 displacement .
A stride frequency (%) Fig. 3. The time of foot-ground contact (the ground contact time, tJ decreased at higher stride frequencies (Fs,,, = 29.7, P < 0.01, ANOVA), suggesting that there were substantial adjustments to the mechanical behavior of the spring-mass system to accommodate changes in stride frequency. In this graph and others, ‘A stride frequency’ represents the percent difference in stride frequency from the preferred stride frequency. Each point represents the mean for all subjects, and each error bar represents the standard error of the mean. The line is the least-squares linear regression.
Fig. 5. When a subject was instructed to hit the ground with his forefoot rather than his heel, the general shape of the force-displacement relationship did not change substantially except that the initial force peak nearly completely disappeared. The force-displacement curves are labeled with stride frequencies. The thicker lines represent the landing phase and the thinner lines represent the takeoff phase. The convention used here is that the vertical displacement increases as the center of mass moves downward. The data shown in this figure are from the same subject as the data shown in Figs 2 and 4.
understandthe adjustmentsto the spring-masssystem for different stride frequencies.At the preferred stride frequency,
the vertical
ground
reaction
force began to
rise as the foot hit the ground (thicker line) (Fig.4). It rapidly increasedto about 1.6 times body weight (the ‘initial force peak’) and then decreasedslightly. This initial
force peak appeared
to be associated
with heel
strike and subsequentdecelerationof the shank.Inspection of videotapes of the trials confirmed
that the subjects
choseto strike the ground first with their heels.After the Downward vertical n initial force peak, the vertical force was nearly linearly 0.04 m displacement relatedto the vertical displacement.The force reachedits maximum of about 2.3 timesbody weight at the same Fig 4. After the force peak there was a nearly linear relationship between the vertical force andthe downwardverticaldisplace- time as the center of massreached its lowest point. ment during the ground contact phase at all stride frequencies, During takeoff (thinner line), the center of massmoved and the slope of the forc+displacement curve increased at upward as the force smoothly decreasedto zero. This bigher stride frequencies. This observation suggests that the generalpattern wassimilar at all stride frequencies,and vertical stiffness of the spring-mass system increased at higher the slopeof the vertical force-displacementrelationship stride frequencies. The force-displacement curves are labeled with stride frequencies. The ‘preferred’ stride frequency is the one increasedat higher frequencies. that the subject normally chose to use at this speed, and the When a subject ran at each stride frequency while other frequencies are labeled with their % difference from striking the ground with his forefoot rather than his heel, the preferred stride frequency. The thicker lines represent the landing phase and the thinner lines represent the takeoff the initial force peak wasnearly completely absent(Fig. phase. The convention used here is that the vertical displace5). This observation supports the idea that the initial ment increases as the center of mass moves downward. The data force peak observedin Fig. 4 wasassociatedwith heel shown in this figure are from the same subject and trials as the strike and subsequentdecelerationof the shank. This data shown in Fig. 2. This subiect, as well as all of the other conclusionis alsosupportedby the observationthat the subjects, chose to use heel-toe r&n&g. The data were similar for all of the subjects. time course of the initial force peak observed in Fig. 4 wasshort (0.035s) at all stride frequencies(Fig. 2). The vertical stiffnessof the spring-masssystemincreasedat higher stride frequencies,k,,, (Fig. 6C). Betweenthe lowestand highestfrequencies,k,,,, increased vertical stiffness,kveti,increased at higher stride frequenby 3.5-fold from 15.1( + 0.713kNm-‘) to 52.4kNm-’ cies (Fig. 4). Becausethe vertical stiffness,k,,,, of the ( + 2.34kNm- ‘). Over the range of stride frequencies, spring-masssystem is important in determiqing its peakvertical force, F, only changedslightly but the peak mechanicalbehavior, it is instructive to examine the vertical displacement, Ay, decreased substantially. vertical force-displacementrelationship to begin to Betweenthe lowest and higheststride frequencies,the
Leg stiffness and stride frequency in human running
185
30r
0’
’
’
-30 -20 -10
c*
’
’
’
’
’
0 10 20 30 40 A stride frequency (%)
Fig 7. Half the angle swept by the leg spring (8, see Fig. 1) decreased at higher stride frequencies (FUJI = 49.3, P < 0.01, ANOVA). This decreased angle swept by the leg spring at higher stride frequencies causes there to be a lower vertical stiffnessfor a given leg spring stiffness.Thus, the decreased angle swept by the leg spring stiffnesspartially offset the effects of the increased leg spring stiffnessat higher stride frequencies. Each point represents the mean for all subjects, and each error bar represents the standard error of the mean. Note that the error bars are contained within the points in some instances. The line is a linear least-squares regression.
601
-30 -20 -10 0
10 20 30 40 A stride frequency (%)
Fig. 6. The leg spring stiffness,F/AL, and the vertical stiffness, F/Ay, increased at higher stride frequencies. (A) The peak force decreased at stride frequencies above the preferred stride frequency (F8,24 = 19.5, P -c 0.01, ANOVA). (B) Both the vertical displacement of the center of mass (Ay; F8,24 = 63.4, P < 0.01, ANOVA) and the displacement of the leg spring (AC F 8,24 = 106.5, P -z 0.01, ANOVA) decreased substantially at higher stride frequencies. (C) Both the vertical stiffness (k,,,,; - 80.4, P -c 0.01, ANOVA) and the leg spring stiffness increased at higher stride pka.=; 8 24 = 43.5, P < 0.01, ANOVA) f&e&es. In parts A-C, each point represents the mean for all subjects, and each error bar represents the standard error of the mean. Note that the error bars are contained within the points in some instances. The lines are the least-squares regressions.
Half the angle swept by the leg spring decreasedat higherstride frequencies,partially offsettingthe effect of the increasein kleg on k,,, (Fig. 7). Betweenthe lowest andthe higheststridefrequencies,0 decreased from 28.2 ( + 0.4”)to 18.7”( f 0.3”).For a given klcg, reducing6 will reduce k,,,,. Thus, there was a lower k,,, for a given kleg at high stridefrequencies than therewould have been if 6 had not decreasedat higher stride frequencies. DISCUSSION
The spring-massmodelusedto analyzerunningin this study is the simplestmodelthat candescribethe mechanicsof runninggaits.This modelrepresentsthe entiremass of the animal as a singlepoint massand representsthe entire musculoskeletalsystemas a singlelinear spring. The simplicity of this modelcontrastswith the complexity of the actualmusculoskeletal system.The actual musculoskeletalsystemconsistsof a jointed skeletonwith muscles,tendons and ligaments acting across nearly every joint. Yet, despite the simplicity
of the model, it
describesand predictsthe mechanicsof running in a variety of animals including peak vertical force decreased slightly ( - 19%), and the peak vertical displacement, Ay, decreased much more ( - 76%, Figs 4, 6A and B). The vertical stiffness of the spring-mass system increased at higher stride frequencies because the leg spring stiffness increased (Fig. 6C). The leg spring stiffness more than doubled between the lowest frequency (7.03 f 0.21 kN m- ‘) and the highest frequency (16.34 + 0.42 kNm-‘). The ratio of the peak force to the peak displacement of the leg spring, F/AL, increased at higher stride frequencies mainly because AL decreased. Betweenthe lowestand higheststride frequencies,F decreased slightly ( - 19%), and AL decreased substantially ( - 65%, Figs 6A and B).
humans, horses, kangaroos
and
cockroaches(Blickhan and Full, 1993;Farley et al., 1993; He et al., 1991).This observationshowsthat the actions of all of the elements of the musculoskeletal system are integrated in such a way that the overall system behaves like a single linear spring during running.
In the presentstudy, weinvestigatethe relative importanceof changesto the leg spring stiffnessand the angle sweptby the leg spring in adjustingthe behavior of the spring-mass system when humans alter their stride frequency at a given running speed. We find that when humans increase their stride frequency at a given running speed, the most important adjustment to the body’s
springsystemis that leg spring becomesstiffer. Between the lowest and highest possible stride frequencies,
the
C. T. Farley and 0. Gonzalez
186
stiffness of the leg spring more than doubles. As a result, the vertical stiffness of the spring-mass system increases, the vertical displacement during the ground contact phase decreases, and the system bounces off the ground in less time at higher stride frequencies. The angle swept by the leg spring decreases at higher stride frequencies, partially offsetting the effect of the higher leg spring stiffness on the mechanical behavior of the spring-mass system. These findings about the adjustability of the leg spring during forward running parallel those of an earlier study showing that the stiffness of the leg spring can be altered substantially when humans hop in place at different frequencies (Farley et al., 1991). When humans hop in place, the stiffness of the leg spring increases by about twofold when they increase their hopping frequency by 65%. Similarly, when humans run forward at a given speed, the stiffness of the leg spring increases by about twofold when they increase their stride frequency by 65%. Thus, the relationship between leg stiffness and stride frequency is similar for hopping in place and for forward running. Although our study did not address the issue of the mechanism of stiffness adjustment, earlier studies (Greene and McMahon, 1979; McMahon et al., 1987) suggest that alterations in limb posture may lead to changes in leg stiffness, In addition, it is possible that the stiffness of the leg may be altered by changing the activation of muscles acting about the joints of the leg. In conclusion, we find that although the leg spring stiffness does not change with speed in forward running, it is possible for leg stiffness to be changed by more than two-fold to accommodate different stride frequencies at a given speed. The adjustability of leg stiffness may be important in allowing the body’s spring system to operate on the variety of terrain encountered in the natural world. In addition, our findings suggest that a variable leg stiffnessmay be an important designparameterfor spring-basedprosthesesfor running and for legged robots. Acknowledgements-This work was supported by a National Institutes of Health Grant (#AR18140)-to C. Richard Taylor and a National Institutes of Health Postdoctoral Fellowshin (#ARO8189) to C.T.F. The authors thank R.J. Full for helpful discussions. REFERENCES
Alexander, R. McN. (1988) Elastic Mechanisms in Animal Movement. Cambridge University Press, Cambridge. Alexander, R. McN. (1992) A model of bipedal locomotion on compliant legs. Phil. Trans. Roy. Sot. B338, 189-198. Alexander R. McN. and Vernon, A. (1975) Mechanics of hopping by kangaroos (Macropodidae), .I. 2001. Land. 177, 2655303.
Blickhan, R. (1989) The spring-mass model for running and hopping. J. Biomechanics 22, 1217-1227. Blickhan, R. and Full, R. J. (1992) Mechanical work in terrestrial locomotion. In Biomechanics: Structures and Systems (Edited by Biewener, A. A.), pp. 75-96. Oxford University Press, New York. Blickhan, R. and Full, R. J. (1993) Similarity in multilegged locomotion: bouncing like a monopode. .I. Comp. Physiol. A 173,509517. Cavagna, G. A. (1975) Force platforms as ergometers. J. Appl. Physiol. 39, 174179.
Cavagna,G. A., Franzetti,P., Heglund,N. C. andWillems,P. (1988) The determinants of step frequency in running, trotting, and hopping in man and other vertebrates. .I. Physiol. Land. 399, 81-92. Cavagna, G. A., Heglund, N. C. and Taylor, C. R. (1977) Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure. Am. J. Physiol. 233, R243-R261. Cavagna, 0. A., Saibene, F. P. and Margaria, R. (1964) Mechanical work in running. J. ADDZ. Phvsiol. 19. 249-256. Cavagna, G. A., Willems, P. ‘A;.,Franzetti, P: and Detrembleur, C. (1991) The two power limits conditioning step frequency in human running. J. Physiol. Lond. 437, 95-108. Cavanagh, P. R. and Williams, K. R. (1982) The effect of stride length variation on oxygen uptake during distance running. Med. Sci. Sports Exert. 14, 30-35. Clarke, T. E., Cooper, L., Clark, D. and Hamill, C. (1983) The effect of varied stride rate and length upon shank deceleration during ground contact in running. Med. Sci. Sports Exert. 15, 170. Farley, C. T., Blickhan, R., Saito, J. and Taylor, C. R. (1991) Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits. J. Appl. Physiol. 716,2127-2132. Farley, C. T., Glasheen, J. and McMahon, T. A. (1993) Running springs: speed and animal size. .I. Exp. Biol. 185, 71-86. Greene, P. R. and McMahon, T. A. (1979) Reflex stiffness of
man’santi-gravitymuscles duringkneebends whilecarrying
extra weights. J. Biomechanics 12, 881-891. He, J., Kram, R. and McMahon, T. A. (1991) Mechanics of runningundersimulated reduced gravity.J. Appl. Physiol. 71, 863-870. Hogberg, P. (1952) How do stride length and stride frequency influence energy output during running? Arbeitsphysiol. 14, 437441.
Ito, A., Komi,P.V., Sjodin,B.,Bosco,C.andKarlsson, J.(1983) Mechanical efficiency of positive work in running at different speeds. Med. Sci. Sports Exert. 15. 299-308. Kaneko, M., Matsumoto, M., Ito, A: and Fuchimoto, T. (1987) Optimum step frequency in constant speed running. In Biomechanics X-B (Editedby Jonsson, B.),pp.803-807.Human Kinetics Publishers, Champaign. Kram, R. and Powell, A. J. (1989) A treadmill-mounted force platform. J. Appl. Physiol. 67, 1692-1698. McGeer, T. (1990)Passive bipedalrunning.Proc. R. Sot. Lond. B240, 107-134. McMahon, T. A. and Cheng, G. C. (1990) The mechanics of running: how does stiffness couple with speed? J. Biomechanics 23(suppl. l), 65-78. McMahon, T. A., Valiant, G. and Frederick, E. C. (1987) Groucho running. J. Appl. Physiol. 62, 2326-2337. Thompson, C. and Raibert, M. (1989) Passive dynamic running. In International Symposium of Experimental Robotics (Edited by Hayward, V. and Khatib, 0.). pp. 74-83. Spring, New York.
Pergamon 0021-9290(95)00029-l
LEG STIFFNESS AND IN HUMAN
STRIDE FREQUENCY RUNNING
Claire T. Farley and Octavia Gonzalez Harvard University, Museum of Comparative Zoology, Concord Field Station, Old Causeway Road, Bedford, MA 01730, U.S.A. Abstract-When humans and other mammals run, the body’s complex system of muscle, tendon and ligament springs behaves like a single linear spring (‘leg spring’). A simple spring-mass model, consisting of a single linear leg spring and a mass equivalent to the animal’s mass, has been shown to describe the mechanics of running remarkably well. Force platform measurements from running animals, including humans, have shown that the stiffness of the leg spring remains nearly the same at all speeds and that the spring-mass system is adjusted for higher speeds by increasing the angle swept by the leg spring. The goal of the present study is to determine the relative importance of changes to the leg spring stiffnessand the angle swept by the leg spring when humans alter their stride frequency at a given running speed. Human subjects ran on treadmill-mounted force platform at 2.5 m s- ’ while using a range of stride frequencies from 26% below to 36% above the preferred stride frequency. Force platform measurements revealed that the stiffness of the leg spring increased by 2.3-fold from 7.0 to 16.3 kNm-’ between the lowest and highest stride frequencies. The angle swept by the leg spring decreased at higher stride frequencies, partially offsetting the effect of the increased leg spring stiffness on the mechanical behavior of the spring-mass system.We conclude that the most important adjustment to the body’s spring system to accommodate higher stride frequencies is that leg spring becomes stiffer. Keywords: Locomotion; Spring-mass model, Elastic energy; Biomechanics; Motor control.
INTRODUCTION
When animalsrun, they bouncealong the ground using musculoskeletalspringsto alternately store and return elasticenergy(Cavagnaet al., 1964,1977).Muscles,tendons and ligamentscan all behave as springs,storing elasticenergy when they are stretchedand returning it whenthey recoil (Alexander, 1988).During running, this complex system of musculoskeletalsprings behaves much like a singlelinear spring(the ‘leg spring’).In fact, a simple spring-massmodel (Fig. l), consistingof a singlelinear leg spring and a massequivalent to the animal’smass,has beenshown to describeand predict the mechanicsof running remarkably well (Alexander, 1992; Alexander and Vernon, 1975; Blickhan, 1989; Blickhan and Full, 1993;Cavagna et al., 1988;Farley et al., 1991,1993;He et al., 1991;Ito et al., 1983;McGeer, 1990; McMahon and Cheng, 1990; Thompson and Raibert, 1989). How can a simplespring-masssystembe adjustedto operateat different running speeds? As animalsrun faster, the vertical excursionof the centerof massduring the stancephasedecreases, and the time that eachfoot is on the ground decreases.In the spring-massmodel, these changescould bethe result of an increasedstiffnessof the leg spring or an increasedanglesweptby the leg spring during the ground contact phase.By increasingthe angle Received in final form 23 February 1995. Address correspondence to: Claire T. Farley, Department of Human Biodynamics, 103 Harmon, University of California, Berkeley, CA 94720, U.S.A.
swept by the leg spring, the vertical excursion of the center of massand the ground contact time can be reducedwithout changingthe stiffnessof the leg spring. Biomechanicalstudieshave shownthat as animalsrun faster,the body’s springsystemis adjustedto bounceoff the ground in lesstime by increasingthe anglesweptby the leg spring during the ground contact phaserather than by increasingthe stiffnessof the leg spring(Farley et al., 1993;He et al., 1991).The stiffnessof the leg spring remainsnearly the sameat all speedsin a variety of animalsincluding running humans,hopping kangaroos and trotting horses(Farley et al., 1993;He et al., 1991). Although the stiffnessof the leg spring remainsthe sameat all speedsduring forward running, experimental evidenceshowsthat it is possiblefor humansto alter the stiffnessof their leg spring.When humanshop in place and vary their hopping frequency,the stiffnessof the leg springcan be changedby asmuch astwofold to accommodatedifferent hoppingfrequencies(Farley et al., 1991). Similarly, when humansbounce vertically on a compliant board, the stiffnessof the legspringcan changeby twofold in responseto changesin kneeangle(Greeneand McMahon, 1979).Finally, when humansrun with increasedkneeflexion (‘Groucho Running’),the stiffnessof the leg spring appearsto decrease(McMahon et al., 1987).Thesestudiesclearly demonstratethat it is possible to change the stiffnessof the leg spring during bouncingmovements. The goal of the present study is to determine the relative importanceof changesto the leg springstiffness and the angle swept by the leg spring in adjusting the behavior
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of the spring-mass
system when humans alter
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C. T. Farley and 0. Gonzalez
sent the percent differencein stride frequency from the preferredstridefrequency.We defineda strideasthe time from the footfall of onelegto the next footfall of the same leg. On average, the preferred stride frequency was 1.33& 0.09stride per second(mean+ SD.). In all trials, including the trials at the preferredfrequency, the subjects matchedtheir footfalls to a metronomewhile they ran on a motorized treadmill. Each subject required about two l-h practice sessions to learn to match the frequency of the metronome to within 1%. At eachfreFig. 1. Runningis modeledas simplespring-mass system bouncingalongthe ground(McMahonandCheng,1990).The quency, the propertiesof the musculoskeletalspringsysmodelccnsistsbfa mass anda singlelegspring(whichconnects temwereanalyzed usingforce platform measurements in thefootandthecenterof mass of theanimal). Thisfiguredeicts order to addressthe questionof whether the stiffnessof the modelat the beginningof the stancedhase(leftmostposition),at the middleof the stancephase(legspringis oriented the legspringcan beadjustedto accommodatea rangeof vertically)andat theendof the stancephase(rightmostposi- stride frequenciesat a given speed. tion).Thelegspringhasaninitial length,Lo,at thebeginning of The vertical ground reactionforce during running was thestancephase, anditsmaximalcompression isrepresented by measured using a treadmill-mounted force platform AL. The dashedspring-mass modelshowsthe lengthof the (Kram and Powell, 1989).A strain-gaugedforce platform uncompressed legspring.Thus,thedifference between thelength of the dashedleg springand the maximallycompressed leg (model OR6-5-1, Advanced Mechanical Technology, springrepresents the maximumcompression of the legspring, Newton, MA) wasmounted under the tread belt (Kram AL. Thedownwardverticaldisplacement of themass duringthe and Powell, 1989).The cross-talkfrom a horizontal force stancephaseis represented by Ay andis substantially smaller to the vertical waslessthan 1%. The natural frequencyof thanAL. Half of the anglesweptby the legspringduringthe vertical vibration of the force platform mounted in the groundcontacttimeis denotedby 8. treadmill was 160Hz. The force signal was sampledat 1 kHz. The spring-massmodel that wasusedto analyze the their stride frequencyat a given running speed.It is well mechanicsof running consistedof a massand a single known that changingstridefrequencyat a given running linear massless ‘leg spring’(Fig. 1; Farley et al., 1993; He speedhasstrongeffectson energeticcost(Hogberg,1952; et al., 1991;McMahon and Cheng,1990).The stiffnessof Cavanagh and Williams, 1982),impact forces on the the leg spring,kleg,wasdefinedasthe ratio of the force in musculoskeletal system(Clarke et al., 1983),and mechan- the spring,F, to the displacementof the spring,AL, at the ical power of the centerof massandlimbs(Cavagnaet al., instant when the leg spring wasmaximally compressed: 1991;Kaneko et al., 1987).Basedon previousstudies,we kleg = FJAL. (1) predict that when stride frequency is increased,the spring-masssystemwill be adjustedusingoneor both of The leg springwasmaximally compressedat the middle the following mechanisms: (1) increasingthe stiffnessof of the stance phasewhen the leg spring was oriented the leg spring;and (2) increasingthe anglesweptby the vertically, and thus, F correspondedto the peak vertical leg spring. ground reaction force. The peak displacementof the leg spring, AL, was METHODS calculatedfrom the maximum vertical displacementof the centerof mass,Ay, the initial length of the legspring, Four male subjects(averagemass73.4& 8.3kg and L, (measuredasthe vertical distancefrom the ground to average leg length of 0.97 f 0.03m, meank S.D.) be- the greater trochanter during standing),and half of the tweenthe agesof 21 and 29 yr volunteeredto participate anglesweptby the legspringwhile it wasin contact with in this study. They werein good health and wereexperi- the ground, 0 (Fig. 1; McMahon and Cheng,1990): encedtreadmill runners.Informed consentwasobtained AL=Ay+L,(l -cost)). (2) from the subjects,and the experimental protocol was approvedby the Human SubjectsCommitteeat Harvard The vertical displacementof the center of mass,Ay, was University. calculatedby integrating the vertical accelerationtwice At the beginningof the experiments,eachsubjectper- asdescribedin detail elsewhere (Blickhan and Full, 1992; formed 10min runs on two separatedays to determine Cavagna,1975).From geometricconsiderations(Fig. l), the preferred stride frequency for running at 2.5 m s- ‘. an equation for calculating0 from the time of foot conThe running speedof 2.5m s- ’ was chosenbecauseit tact with the ground, t,, the forward speed,u, and LO was was low enough that a large range of stride frequencies obtained: above and below the preferredfrequency could be sus0 = sin- ’ (ut,/2L,). (3) tained. In subsequentexperiments,each subject ran at The vertical motionsof the systemduring the ground nine stride frequencies:the preferred, four below the preferred( - 5, - 11,- 18and - 26%) and four above contact phasecan be describedin terms of the ‘vertical the preferred( + 17, + 25, + 30 and + 36%).Through- stiffness’,k,.,. The vertical stiffness,k,,,, doesnot cormsout this paper, ‘A stride frequency’and ‘ASF will repre- pond to any physicalspringin the model.Rather,k,,,, de:.-..j
Leg stiffness and stride frequency in human running
2.5, A. Lowest
scribesthe vertical motions of the center of massduring the ground contact time and is important in determining how long the spring-masssystemremainsin contact with the ground. The effective vertical stiffnesswascalculated from the ratio of the force, F, to the vertical displacement of the centerof mass,Ay, at the momentwhenthe center of massreachedits lowestpoint: k wrt -- F/Ay.
k “Cl, = F/[( F/kleg) - Lo (1 - cos 0)].
(5)
Equation (5) showsthat when the leg springis oriented vertically (@= 0”) during the entire time that it is in contact with the ground (e.g.during hopping or running in place),k,,, = klcg. As tl increasesabove zero (e.g.luring forward locomotion), k,,,, increasesrelative to klcg This can be observedin that the vertical displacementof the centerof mass,by, decreases relative to the displacement of the leg spring,AL, when the anglesweptby the leg springis greater.In addition, analysisof equation(5) showsthat k,,, will increaseif klsg increases.A detailed discussionof the concept of the vertical stiffnessis includedin McMahon and Cheng(1990).
RESULTS
The generalpattern of vertical force applicationto the ground and of vertical displacementof the centerof mass was similar at all stride frequenciessuggestingthat a ‘spring-like’ gait was used over the range of stride frequencies(Fig. 2). The vertical ground reaction force beganto rise as the foot hit the ground and reachedan initial peak within the first 0.035s of the ground contact phase.This initial peak was associatedwith the heel hitting the ground at the beginningof the ground contact phase.After the initial peak, the vertical force smoothly increasedto a midstepmaximum that occurred at the sametime asthe centerof massreachedits lowestpoint. Then, during the secondhalf of the ground contact phase, the center of massmoved upward, and the vertical force smoothly decreased,reaching zero as the foot left the ground. The vertical displacementof the centerof massand the ground contact time decreasedat higher stride frequencies,showingthat there were substantialadjustmentsto the mechanicalbehavior of the musculoskeletalspring systemwhen stride frequency wasaltered (Fig. 2). First, as stride frequency was increased,the vertical displacement of the center of massduring the ground contact phase,Ay, decreasedsubstantially(Figs 2 and 6B). Be-
S.F. (-26%)
,
r 0.00 2 i c. - -0.05
ii 4 s k
--0.10
f 2
(4)
The center of massreachedits lowestpoint at the same time as the leg spring wasmaximally compressed. In the spring-massmodel, the effective vertical stiffness,L, is determinedfrom a combination of the stiffnessof the leg spring,kleg, half the anglesweptby the leg spring,8, and the force in the spring,F. Equation (4) can be rewritten to showthe interrelationshipbetweenthese aspectsof the systemby definingAy in termsof F, kleg, Lo and 0 [equation (2)]:
183
0.04
--I-0.y
2.5, B. Preferred
S.F.
f Igo-
k.;
1.5
r 0.00 $e
f: E.
- -0.05 E d c
a3
- -0.10 8 3 d
W” *ea.5
0.0
(
z -0.15 s
I
Time (s) Fig. 2. Vertical ground reaction force (thick line)and vertical displacement (thin line) as a function of time during theground contact phase for: (A) the lowest stride frequency, (B) the preferred stride frequency, and (C) the highest stride frequency. Note that the general pattern of vertical ground reaction force and displacement was similar for all stride frequencies suggesting that a spring-like gait was used at all stride frequencies. However, the time of foot-ground contact and the magnitude of the vertical displacement both decreased at higher stride frequencies. These are typical data from a 67 kg subject with a leg length of 0.94 m.
tween the lowest and highestfrequencies,Ay decreased by 76% (0.107+ 0.011m at the lowest frequency to 0.025+ 0.001m at the highest frequency, mean of all subjects+ S.E.M.). In addition, as frequency was increased,the time that a foot was on the ground (the ground contact time, t,) decreasedby 32% (Figs 2 and 3; 0.365& 0.009s at the lowestfrequencyto 0.248f 0.008s at the highestfrequency,Fig. 3). Theseobservationssuggestthat there were substantialchangesto the vertical stiffnessto accommodatechangesin stride frequency. The slopeof the relationshipbetweenthe vertical force and the vertical displacementof the center of massincreasedat higher stride frequencies,suggestingthat the
C. T. Farley and 0. GonzLlez
184
5 Y" $'3 i; 2 3 -Q g,
0.40 0.35 0.30 0.25
8
Downward vertical 10 displacement .
A stride frequency (%) Fig. 3. The time of foot-ground contact (the ground contact time, tJ decreased at higher stride frequencies (Fs,,, = 29.7, P < 0.01, ANOVA), suggesting that there were substantial adjustments to the mechanical behavior of the spring-mass system to accommodate changes in stride frequency. In this graph and others, ‘A stride frequency’ represents the percent difference in stride frequency from the preferred stride frequency. Each point represents the mean for all subjects, and each error bar represents the standard error of the mean. The line is the least-squares linear regression.
Fig. 5. When a subject was instructed to hit the ground with his forefoot rather than his heel, the general shape of the force-displacement relationship did not change substantially except that the initial force peak nearly completely disappeared. The force-displacement curves are labeled with stride frequencies. The thicker lines represent the landing phase and the thinner lines represent the takeoff phase. The convention used here is that the vertical displacement increases as the center of mass moves downward. The data shown in this figure are from the same subject as the data shown in Figs 2 and 4.
understandthe adjustmentsto the spring-masssystem for different stride frequencies.At the preferred stride frequency,
the vertical
ground
reaction
force began to
rise as the foot hit the ground (thicker line) (Fig.4). It rapidly increasedto about 1.6 times body weight (the ‘initial force peak’) and then decreasedslightly. This initial
force peak appeared
to be associated
with heel
strike and subsequentdecelerationof the shank.Inspection of videotapes of the trials confirmed
that the subjects
choseto strike the ground first with their heels.After the Downward vertical n initial force peak, the vertical force was nearly linearly 0.04 m displacement relatedto the vertical displacement.The force reachedits maximum of about 2.3 timesbody weight at the same Fig 4. After the force peak there was a nearly linear relationship between the vertical force andthe downwardverticaldisplace- time as the center of massreached its lowest point. ment during the ground contact phase at all stride frequencies, During takeoff (thinner line), the center of massmoved and the slope of the forc+displacement curve increased at upward as the force smoothly decreasedto zero. This bigher stride frequencies. This observation suggests that the generalpattern wassimilar at all stride frequencies,and vertical stiffness of the spring-mass system increased at higher the slopeof the vertical force-displacementrelationship stride frequencies. The force-displacement curves are labeled with stride frequencies. The ‘preferred’ stride frequency is the one increasedat higher frequencies. that the subject normally chose to use at this speed, and the When a subject ran at each stride frequency while other frequencies are labeled with their % difference from striking the ground with his forefoot rather than his heel, the preferred stride frequency. The thicker lines represent the landing phase and the thinner lines represent the takeoff the initial force peak wasnearly completely absent(Fig. phase. The convention used here is that the vertical displace5). This observation supports the idea that the initial ment increases as the center of mass moves downward. The data force peak observedin Fig. 4 wasassociatedwith heel shown in this figure are from the same subject and trials as the strike and subsequentdecelerationof the shank. This data shown in Fig. 2. This subiect, as well as all of the other conclusionis alsosupportedby the observationthat the subjects, chose to use heel-toe r&n&g. The data were similar for all of the subjects. time course of the initial force peak observed in Fig. 4 wasshort (0.035s) at all stride frequencies(Fig. 2). The vertical stiffnessof the spring-masssystemincreasedat higher stride frequencies,k,,, (Fig. 6C). Betweenthe lowestand highestfrequencies,k,,,, increased vertical stiffness,kveti,increased at higher stride frequenby 3.5-fold from 15.1( + 0.713kNm-‘) to 52.4kNm-’ cies (Fig. 4). Becausethe vertical stiffness,k,,,, of the ( + 2.34kNm- ‘). Over the range of stride frequencies, spring-masssystem is important in determiqing its peakvertical force, F, only changedslightly but the peak mechanicalbehavior, it is instructive to examine the vertical displacement, Ay, decreased substantially. vertical force-displacementrelationship to begin to Betweenthe lowest and higheststride frequencies,the
Leg stiffness and stride frequency in human running
185
30r
0’
’
’
-30 -20 -10
c*
’
’
’
’
’
0 10 20 30 40 A stride frequency (%)
Fig 7. Half the angle swept by the leg spring (8, see Fig. 1) decreased at higher stride frequencies (FUJI = 49.3, P < 0.01, ANOVA). This decreased angle swept by the leg spring at higher stride frequencies causes there to be a lower vertical stiffnessfor a given leg spring stiffness.Thus, the decreased angle swept by the leg spring stiffnesspartially offset the effects of the increased leg spring stiffnessat higher stride frequencies. Each point represents the mean for all subjects, and each error bar represents the standard error of the mean. Note that the error bars are contained within the points in some instances. The line is a linear least-squares regression.
601
-30 -20 -10 0
10 20 30 40 A stride frequency (%)
Fig. 6. The leg spring stiffness,F/AL, and the vertical stiffness, F/Ay, increased at higher stride frequencies. (A) The peak force decreased at stride frequencies above the preferred stride frequency (F8,24 = 19.5, P -c 0.01, ANOVA). (B) Both the vertical displacement of the center of mass (Ay; F8,24 = 63.4, P < 0.01, ANOVA) and the displacement of the leg spring (AC F 8,24 = 106.5, P -z 0.01, ANOVA) decreased substantially at higher stride frequencies. (C) Both the vertical stiffness (k,,,,; - 80.4, P -c 0.01, ANOVA) and the leg spring stiffness increased at higher stride pka.=; 8 24 = 43.5, P < 0.01, ANOVA) f&e&es. In parts A-C, each point represents the mean for all subjects, and each error bar represents the standard error of the mean. Note that the error bars are contained within the points in some instances. The lines are the least-squares regressions.
Half the angle swept by the leg spring decreasedat higherstride frequencies,partially offsettingthe effect of the increasein kleg on k,,, (Fig. 7). Betweenthe lowest andthe higheststridefrequencies,0 decreased from 28.2 ( + 0.4”)to 18.7”( f 0.3”).For a given klcg, reducing6 will reduce k,,,,. Thus, there was a lower k,,, for a given kleg at high stridefrequencies than therewould have been if 6 had not decreasedat higher stride frequencies. DISCUSSION
The spring-massmodelusedto analyzerunningin this study is the simplestmodelthat candescribethe mechanicsof runninggaits.This modelrepresentsthe entiremass of the animal as a singlepoint massand representsthe entire musculoskeletalsystemas a singlelinear spring. The simplicity of this modelcontrastswith the complexity of the actualmusculoskeletal system.The actual musculoskeletalsystemconsistsof a jointed skeletonwith muscles,tendons and ligaments acting across nearly every joint. Yet, despite the simplicity
of the model, it
describesand predictsthe mechanicsof running in a variety of animals including peak vertical force decreased slightly ( - 19%), and the peak vertical displacement, Ay, decreased much more ( - 76%, Figs 4, 6A and B). The vertical stiffness of the spring-mass system increased at higher stride frequencies because the leg spring stiffness increased (Fig. 6C). The leg spring stiffness more than doubled between the lowest frequency (7.03 f 0.21 kN m- ‘) and the highest frequency (16.34 + 0.42 kNm-‘). The ratio of the peak force to the peak displacement of the leg spring, F/AL, increased at higher stride frequencies mainly because AL decreased. Betweenthe lowestand higheststride frequencies,F decreased slightly ( - 19%), and AL decreased substantially ( - 65%, Figs 6A and B).
humans, horses, kangaroos
and
cockroaches(Blickhan and Full, 1993;Farley et al., 1993; He et al., 1991).This observationshowsthat the actions of all of the elements of the musculoskeletal system are integrated in such a way that the overall system behaves like a single linear spring during running.
In the presentstudy, weinvestigatethe relative importanceof changesto the leg spring stiffnessand the angle sweptby the leg spring in adjustingthe behavior of the spring-mass system when humans alter their stride frequency at a given running speed. We find that when humans increase their stride frequency at a given running speed, the most important adjustment to the body’s
springsystemis that leg spring becomesstiffer. Between the lowest and highest possible stride frequencies,
the
C. T. Farley and 0. Gonzalez
186
stiffness of the leg spring more than doubles. As a result, the vertical stiffness of the spring-mass system increases, the vertical displacement during the ground contact phase decreases, and the system bounces off the ground in less time at higher stride frequencies. The angle swept by the leg spring decreases at higher stride frequencies, partially offsetting the effect of the higher leg spring stiffness on the mechanical behavior of the spring-mass system. These findings about the adjustability of the leg spring during forward running parallel those of an earlier study showing that the stiffness of the leg spring can be altered substantially when humans hop in place at different frequencies (Farley et al., 1991). When humans hop in place, the stiffness of the leg spring increases by about twofold when they increase their hopping frequency by 65%. Similarly, when humans run forward at a given speed, the stiffness of the leg spring increases by about twofold when they increase their stride frequency by 65%. Thus, the relationship between leg stiffness and stride frequency is similar for hopping in place and for forward running. Although our study did not address the issue of the mechanism of stiffness adjustment, earlier studies (Greene and McMahon, 1979; McMahon et al., 1987) suggest that alterations in limb posture may lead to changes in leg stiffness, In addition, it is possible that the stiffness of the leg may be altered by changing the activation of muscles acting about the joints of the leg. In conclusion, we find that although the leg spring stiffness does not change with speed in forward running, it is possible for leg stiffness to be changed by more than two-fold to accommodate different stride frequencies at a given speed. The adjustability of leg stiffness may be important in allowing the body’s spring system to operate on the variety of terrain encountered in the natural world. In addition, our findings suggest that a variable leg stiffnessmay be an important designparameterfor spring-basedprosthesesfor running and for legged robots. Acknowledgements-This work was supported by a National Institutes of Health Grant (#AR18140)-to C. Richard Taylor and a National Institutes of Health Postdoctoral Fellowshin (#ARO8189) to C.T.F. The authors thank R.J. Full for helpful discussions. REFERENCES
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