bio mechanics of walking and running

8/2/2019 Bio Mechanics of Walking and Running

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10Biomechanics of Walking andRunning:CenterofMassMovements toMuscle

ActionCLAIRET. AR;LEY, PH.D.DANIEL P.FERRIS, MS.

Locomotion is a particularly richly studied but hstrat ing aspect ofbiol-ogy,according to Loeb [91].This is especially true when one attemptsto understand how the numerous subdivisionsofthe neuromuscular and

musculoskeletal systems interact to produce locomotion.At lower levels of

organization, the structure and function ofthe system components becomeprogressively more complex, making it difficult to discern general princi-ples. One way of approaching the study of locomotion is to sequentiallyprogress from a level ofwhole body dynamics toward a level. of muscle-

tendon dynamics.We believe this approach canbe particularly useful inthat information from the higher levels oforganization can guide the questto understand fundamental mechanisms of locomotion at increasingly com-

plex lower levels oforganization. Thus, we will first examine the patternof center of mass movementS during locomotion in humans and otherlegged animals. Next,we will consider howjoint mechanics during locomo-tion are affectedby both center ofmass dynamics and leg posture. After

concentrating on the joint level,we will move to the level ofmuscle-tendon

mechanics by examining the techniques that have been usedto

investigatemuscle-tendon function during locomotion and the conclusions that have

been reached to date.

Throughout this review, the locomotion ofhumans will be compared to

the locomotion ofother animals.Awealth ofinformation showsthat humanlocomotion is not unique. Indeed, at the level ofthe center ofmass, the

dynamicsofwalking and running are similar in all legged animals that have

been studied. The similarities in the dynamicsoflocomotion among diverse

animals, including humans, suggest that there may also be similarities in

the mechanisms by which locomotion is produced at o,therlevelsoforgani-zationwithin the neuromuscular system. Thus,by comparing diverse spe-

cies, we can uncover common rules governing locomotion, and we can

assess the applicabilityofdata from animal models to biomedical issues inhumans.

253


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254 I F a r b , FmirCENTER OF MASS MECHANICSOur discussion of center ofmass mechanicswill be divided into four parr^.First, we will discuss the pattern ofground reaction force that occursduringlocomotion.Ground reaction force is the force exerted by the ground on

the feet It reflects the acceleration of the bodys center ofmass duringlocomotion. Second, the movements and mechanical energy fluctuations

ofthe center ofmas that occur as a result of ground reaction force willbe discussed. Third, we will discuss the transition froma walking gait to arunning gait, an event that is marked by a sudden and distinct change inthe pattern ofmovementof the center ofmass.By understanding the rea-

sons for the transition from one gait to another, we can gain insight intothe key factors that shape locomotion. Fourth,wewill describe behavioralmodelsfor locomotion that give insight intohow the musculoskeletal system

produces thedistinctly differentground reaction force and center ofmassmovements in walking as compared to running.

CmundReactionForceThe distinct difference between walking and running gaits is apparent inthe ground reaction force patterns for the twogaits (Figures 10.1and 10.2)[26, 27, 29 , 3 4 , 3 5 ] . In humanwalking, there is alwaysat least one foot incontact with the ground, and there are short phases ofdouble support

whenboth feet are in contact with the ground (Figure 10.1). In contrast,running is a series ofbouncing impacts with the ground that areusuallyalternatedwith aerialphaseswhen neitherfoot is incontactwith the ground(Figure 10.2). This difierence leads to a substantidy higher magnitudevertical component ofthe ground reaction force for running as comparedto walking. Thepattern ofthe horizontal componentofthe ground reactionforce is similar, however, for bothwalkingand running (Figures10.1and10.2). In the firsthalfof the stance phase, the horizontal ground reactionforce is negative, indicating that it is pushing backwardson the person. Inthe second half of the stance phase, the horizontal ground reaction forceis positive, pushing forward on the person. The ground reaction force pat-tern forwalking and running gaitsis similar in humansandin a wide varietyofother animalswith a mnge ofbody shapes, body masses, and numbersoflegs [14, 22, 40,50, 641.M e c h a n i d EnergyRudua&msof&Gntcr of MassThe differencein the ground reaction force pattern between&king andrunning translatesinto dramatically different patterns ofmechanical energyfluctuations for the center ofmass during the two gaits [ 2 528 ] .Duringwalking, the body vaults over a relatively stiffstance limb and the center ofmass reaches its highest point at the middle of the stance phase.As a result,the gravitational potential energy ofthe center ofmass ismaximized at the


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Biomechanics of Waacingand Running I 255

FIGURE 10.1.

Reprcsozm*vcgroundnaction force asa funaim of timeforwalking (1.25m/s)in a human. The dashed line r@mcnts the stancephase ofthe rightfoot, a d thesolid line npracnts the stanuphase ofihc Ley7fmt.(A) vertical component. (B)hotiumbl component. In bothparts, the grvund nadiollfia s txpnsJcd us amultipleofbodywe&.

A 1.5 Walk (1.25 d s )T

B 0.3 T

1 *B It *',;-0.3 II II i

00.4

0.8 1.2Time (s)


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FIGURE 10.2.Reprcrentafivepnd readionfmce as afunction oft i m eforrunning (3.8m/s) inahuman. Thedashedlinen$waentsthe slancephase ofthe'rightfoot, andtheSolidline reesentr thestancephase ofthelejfoot. (A) verticalcomponent. (B)horirontal

component. In bothparts, theground reactionforce is expressed asamultipleof&yweight.

A 3TRun(3.8 m / s >

B

Time (s)a


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BiomechanicsofWalking and Running I 257FXCURE 10.3.An i n d dpendulummodel anda stickfigure represmlationofa sin& stance

phaseofhuman wking.l7umodelumsistsofamassandangid strut that connectsthepointoff oo t - pnd conlad andthe centerofmassof the human. ThisfigLn&pi& thestick i g w e andthe moddat the beginning of thestance p h e &-most

position),the mi& ofthc stancephase (centerposition),andthe md ofthe stancephase (right-martposition).

middle of the stance phase (Figure 10.3and 10.5). In contrast, the stance

limb is compliant in runningso that the joints undergo substantial flexion

during the f irs t halfof stance and extension during the second half ofstance. This compliance causes the vertical displacement and gravitational

potential energy ofthe center ofmass to reach their minimum values at

mid-stance in running (Figures 10.4and 10.6). The pattern ofmovementofthe center ofmass hasbeen proposed as the defining difference betweena walking gait and a running gait [ l o l l .

Unlike the gravitationalpotential energy fluctuations, the pattern ofki-netic energy fluctuations is similar forwalking and running. In both gaits,

the kinetic energy ofthe center ofmass reaches its minimum value at mid-stance (Figures 10.5and 10.6) since the horizontal ground reaction forcetends to decelerate the body during the first halfofthe stancephase (Fig-

ures 10.1and 10.2).During the second halfof stance, the kinetic energy ofthe center ofmass increases due to the accelerating effect ofthe horizontal


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FIGURE 10.4.A spring-massmodeland a stick &re r@esentation ofa s i ng l e stance plruse o jhuman runfling. Themodel consists of a Linear spnng rgrescnfin g the kg andapointmassequivalent tobody mass. Thisfigure a!epi& t h e model utf h c briningof t / ustance phase ( lefi -mst position),at themiddle of the stanuphase (leg*ngoriented uertically),and ut the endof f h e stancephase (right-most position).

ground reaction force.Although the pattern ofkinetic energy fluctuationsis similar, the magnitude is much larger for running than forwalking (Fig-ures 10.5 and 10.6).

The kinetic energy and gravitational potential energy ofthe center ofmass are approximately 180 out of phase in walking. At mid-stance inwalking, the gravitationalpotentialenergy is at itsmaximumand the kineticenergy isat its minimum (Figure 10.5):Because these energies are approxi-mately a halfq.de out of phase with each other and their fluctuations aresimilar in magnitude, substantial pendulum-like exchange occurs betweenthem [25, 261.During the first half of the stance phase ofwalklng, thecenter ofmass loses kinetic energy but gainsgravitational potential energy.In this phase, kinetic energy can be converted to gravitational potentialenergy.During the second halfofthe stance phase,the center of~ilasslosesgravitational potential energy but gains kinetic energy. Thus, during

this

phase, gravitational potential energy can be converted to bnetic energy. Asimilar energy transfer mechanism occurs as a pendulum swings or as anegg rolls across the ground. As a result, the energy transfer mechanism

used in walking is often referred to as the inverted pendulum mechanism
http://halfq.de/http://halfq.de/


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Biomechanics of Walking and Running I 259FIGURE 10.5.

During madcralc speed w a N n g (1.25 m/s), thekinetic mrngy fructuations of t?ucenter of mqss arc approximately 186outofphase withthe gravitationalpotnh'almangy jluctuations, aUowingsubstantialpmdulum3ikc cnngy achangz T kthickhorizontal line att k bottom ofthe p u p epreUnts the phases when bothfmtarc incontact with the ground ("double su~o t t " haes) , and the thinlrorizontal linerepresents the phases w h m on4 a sin& foot is in contact withthe ground ("singlesuppOdpphases).

Gravitationalpotential energy

0 0.2 0.4 0.6 0.8 1Time (s)

or the "rolling egg mechanism."The pattern ofmechanical energy fluctua-tions is similar in other walking animals as it is in humans [14, 22, 40.50,64, 1071.As a result, many other animals, includingmammals,birds, r e ptiles, and arthropods, also conserve substantial mechanical energy by the

inverted pendulum mechanism during walking.

In human walking, as much as 60-70% of the mechanical energy required to lift and accelerate the center ofmass is conserved by this energy

transfer mechanism [28]. Mechanical energysavingsaremaximizedatmod-erate walking speeds, and fall toward zero at very low and veryhigh,walkingspeeds [28].At the walking speed where energy conservation is maximized,the magnitudes ofthefluctuations in kineticenergyand gravitational poten-tial energy are similar. Nevertheless, the maximum energy recovery by the

inverted pendulum mechanism is approximately 70%, substantially lessthan the theorerical maximum of100%.Atthe speed where energy transfer


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FIGURE 10.6.

Duringrunning (3.8m/J), thekinetic energyjluctuations of the centerof mass areappvxitnately in phase with the gravitational potential energy fluctuationr. astancephasesfw each limb amnoted atthe h t b m ofthe graph.

Run (3.8d s )

0 0.4Time (s)

0.8

is maximized, the metabolic energy costper unit distance (i.e., the cost oftransport) is lower than at any other walking speed [23]. It has been sug-gested that metabolic energy cost is minimized because the muscles haveto do the least mechanical work at the speed where energy transfer is opti-mal [23].

The speed at which energy transfer ismaximized during walking dependson body size. The optimum speed for energy transfer is lower for a small

child than for an adult [21].While a two-year-ld child's optimum speedis about 0.6 m/s, an adult's optimum speed is about 1.6 m/s. A similardifference in optimum speed occurs among animal species due to theirdifferent bodysizes. For example, a small lizard maximizes energy recovery

at a much lower absolute speed than a sheep [22, 401. In spite ofthisdifference in optimumwalking speed, energy transferby theinverted pen-

dulum mechanism reduces the mechanical work required for lifting andaccelerating the center ofmassby a similar fraction in animals of all body

sizes. A 0.005 kg lizard or a 70 kg sheep reduce the mechanical work re-quired to lift and accelerate the center ofmass by about 50% through the


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Biomechanics of WaLhing andRunning 1 261inverted pendulum mechanism of energy exchange [22,40]. Similarly, themechanical work required to lift and accelerate the center ofmass is re-duced by about 60-7096 for a tweyear-old child or an adult human [21].Thus, body size has a profound effecton the optimum walking speed but has

little effect on energy conservation by the inverted pendulum mechanism.In running, there cannot be substantial pendulum-like exchange be-tween kinetic energy and gravitational potential energy since their fluctua-tions are nearly in phase with each other (Figure 10.6) 1271.Both kineticenergy and gravitational potential energy reach their minimum values at

approximately the middle of the ground contact phase.As a result, theexchange of kinetic energy and gravitational potential energy conservesless than 5% of the mechanical work required to lift and accelerate thecenter ofmassduring running. Substantial mechanical energy, however, isconserved through the storage and return ofenergy in elastic tissues (tobe discussed). Because the movements ofthe center of mass during runningare similar toabouncing ball [27], running is often refei-red to asa bounc-inggait-*Asimilar pattern ofmechanical energy fluctuations occurs duringfast gaitsused by other animals [14,22,40,51,52,64]. Forexample, trottingdogs, hopping kangaroos, running quail, and trotting cockroaches all havea similar pattern ofmechanical energy fluctuations as running humans.

Although the pattern of limb movements and gaits varies among these

animals, all show the characteristic pattern of kinetic energy and gravita-tional potential energy being nearly exactly in phase with each other. All

ofthese g a i t s arereferred to as bouncing gaits.

Gail TransitionsIt is clear that walking and running are distinctly different in terms of

their patterns ofground reaction force and patterns ofmechanical energy

fluctuations Indeed, simply watching a person gradually increase her for-ward speed and break from a walk to a run makes it obvious that there aredistinct differences between walking and running. What determines the

speed range where each ofthese distinctly different gaits isused? It is likelythat by understanding the triggers for the gait transition, we will reach a

better understanding ofhow the neuromuscular system and the physical

characteristics ofthe body shape locomotion.

The transition from walking to running is not a smooth and continuous

event. Rather, there is a distinct transition from one gaitto

the other thatcanbe observed in both the kinematic and kinetic patterns [75, 77, 1391.For example, the transition from walking to running involves sudden

changes in ground contact time, duty factor, ground reaction force, and

movements of the center ofmass. It is not yet clear exacrly what triggersthe transition fromwalking to running or vice-versa in humans or other

animals [9, 33, 75-78, 86,102. 1031.


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262 Far@, FmisIt was long believed that humans and other animals choose their gait

transition speed based on minimization nf the metabolic energy cost oflocomotion [731. In humans, the most economical gait at low speeds is awalk. As walking speed is increased, a speed is eventually reached whererunning requires less metabolic energy than walking [95, 961.A similarpattern exists for quadrupedal animals [73]. Walking is most economicalat low speeds, trotting ismost economical at moderate speeds,and gallopingis most economical at the highest speeds. However, the speed where an . .animal prefers to switch gaits is not the speed that would minimize meb-bolic energy cost. Horses switch from a trot to a gallop at a speed substan-tially below the optimal speed for minimizing metabolic energy cost. Thus,galloping actuallyrequires more metabolic energy than trotting at the speedwhere a horse chooses to switch from a .trot to a gallop (411. Similarly,humans switch from a walk to a run at a speed that is not energeticallyoptimal [76,1031.These findingssuggest that another factor, perhaps bio-mechanical, actually triggers gait transitions.

Based on the inverted pendulum mechanics ofwalking, it is reasonableto think that gravity isan important factor in determining the speed wherethe walk-run transition occurs. In walking, the gravitational force on thecenter ofmms must be at least equal to the centripetal force needed tokeep the center ofmassmoving in a circular arc as it vaults over the stance

limb (Figure 3). The required centripetal force is equal to m$/L, wherem = body mass,v = forward velocity, and L = leg length.The ratiobetweenthe centripetal force and the gravitational force (rn?/L)/(mg) is theFroude number ($/gL). Basedon the mechanics ofan inverted pendulumsystem, it has been predicted that humans and other animafs should beable to use a walking gait only at speeds where the Froude number is lessthan or equal to 1 [l. 21. This is so because the gravitational force is SUE-cient to keep the center ofmassmoving in a circular arc when the Froude

number is less than or equal to 1.Experimental evidence has shown thathumans and other bipeds (e.g., birds) with a large range ofleg lengthsprefer toswitch from a walk to,a run at a similar Froude number but atdifferent absolute speeds [ l ,4, 53,78,1391. Furthermore, when humanswalk at different levels ofsimulated reduced gravity, they switch from a walkto a run at a similar Froude number (approximately0.5) but at very differ-ent absolute speeds@SI. These obsenations suggest that the ratio ofcentri-petal force to gravitational force is important in determining the gait transi-tion speed. Nonetheless, it is puzzling that the gait transition occurs at asubstantially lower Froude number than the-theoreticallypredicted Froudenumber of 1.

Most gait transition studies to date have examined gait choice when hu-mans or other animals move steadily at speeds near the transition speed.However, neither humans nor other animals naturally choose to move inthis way in their everyday lives. Generally, humans and other animalspreferto use speeds near the middle ofeach gait and rarely will choose other


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Bwmcchanim of Wauing a dRunning I 263speeds for an extended period oftime 1'73,1181.Thus, they tend to makerapid transitions from one gait to another, which occur in concert withabrupt changes in speed [1033. In the extreme case of accelerating froma standstill, humans and other animals seem to immediately choose the

appropriate gait for the speed to which they are accelerating. For example,

a human sprinter runs, not walks, out ofthe blocks [24]. Similarly, a dogimmediatelygallops at the beginning ofa sprint regardless of its startingspeed. These observations suggest that in the future itwillbe important toexaminegaitchoices in more natural locomotor patterns. including acceler-

ation and deceleration.

BehavioralM&& forWalkingandRunningOne way to gain insight into the behavior of the overall musculoskeletal

system during locomotion is to employ simple mechanical models ofwalk-ing and running. These behavioral models simulate the movements ofthe

center ofmass during locomotionbymodeling the output ofthe integratedmusculoskeletal system using mechanical elements. These models can pro-

vide a guide for studiesof lower levels oforganization within the musculo-skeletal andneuromuscular systems. They are particularlyvaluable in delim-iting the potential strategies that the neuromuscular system could use toproduce walking and running.WALKINGBEHAVIORALMODELS. The simplest behavioral model for walk-ing is an inverted pendulum. This model consists ofa rigid s t r u t that repre-sents the leg and a point mass equal to body mass (Figure 10.3) [11. In thismodel, the mass vaults over a rigid leg during the stance phase, and thecenter ofmass reaches its highest point at mid-stance. In the inverted pen-dulum model, like in a standard pendulum, the gravitational potential en-

ergy of the mass is exactly 180' out ofphase with the kinetic energy. As aresult, at mid-stance, the gravitational potential energy is maximized andthe kinetic energy is minimized. This pattern ofmechanical energy fluctua-

tions is qualitatively similar to the pattern observed during walking in hu-mans (Figure 10.5)and other animals. In bipedal animals, including hu-

mans, it is easy to visualize that the rigid strut in the inverted pendulummodel corresponds to the stance limb. For animals with four or more legs(e.g., a dog or a ghost crab), all of the legs in contact with the groundcooperate to produce movements ofthe center of mass similar to those of

a mass vaulting over a single rigid limb.

In an dedued inverted pendulum model, 100%recovery ofmechanicalenergy occurs due to the exchange between gravitational potential energy

and kinetic energy. As previously discussed, a walking human has a maxi-

mum recovery of mechanical energy of about 60-70% [28]. Clearly, partofthe reason why human walkers do not achieve 100% recovery is that

Lheir legs do not behave exactly like rigid struts. The functional leg length(i.e., distance from point offootground contact to the center ofmass)


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264 1 Far@, Fin is

changes to somc cxtent during the stance phase [134]. This is differentfrom the behavior of the rigid strut that represents the leg in the idealizedinverted pendulum modcl. In fact, sensitivity analyses on mtxhanicdmodels suggest that leg compression is an important parameter in deter-

mining the patternof

ground reaction force and center ofm a smovementg

during walking [109-

1111. Although an inverted pendulum model with arigid leg does a good job of predicting the mechanical energy fluctuation$of the center of mass, it does not accuntely predict the ground reactionforce pattern [1091.Adding compliance to the leg model greatly improvesthe prediction ofthe ground reaction force pattern [5,109-111,134,144],

Leggeomenyat the beginning ofthe stance phase also plays an importantrole in determining the pattern ofground reaction force and the pendu-lum-like exchange of mechanical energies during walking. When humans

are asked towalkwhile using exaggerated leg joint flexion during stance,the peak ground reaction force decreases [l36] and the pendulum-likeexchange ofcenter ofmass energy decreases [ 8 8 ] .Chimpanzees, animalsthat naturallywalkwith flexed limbs, have similar patterns ofground reac-tion force and energy exchange as humans walking with exaggerated limbflexion 1881.The role of leg geometry in determining the dynamics ofwalking is h t h e r emphasized by the observation that the peakgroundreaction force and loading rate increase when humans walkwith stiffer and

saaighter limbs than usual 1311.These studies suggest that one role ofnormaljoint flexion during the stance phase is to reduce the ground reac-

tion force and the vertical movements of the center ofmass. Pelvic tilt andpelvic rotation also sene to reduce the vertical movements of the centerofmass during walking [79,132].

In spite ofthese deviations from the simple inverted pendulum modelfor walking, anthropomorphic passive walking machines with rigid stance

legs demonstrate walking mechanics very similar to that of humans 198,991.These machines take advantage ofpendulum-like energy exchangebythe center ofmass ofthe body and also by the swinging leg. The idea of

having the swing limb move passively via pendulum-like energy exchangeis based on mathematicalmodels and observations of humans walking atmoderate speeds [53,97-99,104, 1051.Electromyographic measurementsshow that nearly no muscle activity is present in the swing limb at somewalkingspeeds [8]. It is thought that the limb swings forward passively afterthe muscles start thelimb into motion during the period of double support.

Becauseofthe energy exchanged by the swinglimb and the center ofmass,anthropomorphic passive walking machines only need the added energyinput ofmoving down a slight hill to counteract the small energy lossesthat occur with each stride. It is interesting to note that although passivewalking machines do not have any control systems, they are capable ofwalking in a stable and predictable pattern [98. 991.Their dynamics arcdetermined by the physical structure of the walker, demonstrating that


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Biomechanicsof Walking and Running I 265inherent mechanical properties ofthe body can greatly simplify the control

of locomotion.RUNNINGBEHAVIORALMODELS. Because the movements ofthe center ofmassduring running are similar to those ofabouncing ball, it is not surpris-

ing that running models rely upon springs. Running is often modeled asa simple spring-mass model that consists ofa single linear legspring and

a point mass that is equivalent to body mass (Figure 4) [3,IS,1001. Theleg spring stiffness represents the overall stiffnessofthe integrated muscubskeletal system. In bouncing gaits the leg spring compresses during the

first half of the ground contact phase and lengthens during the second

halfofthe ground contact phase. These changes in leg length result fromflexion and extension of leg joints. In spite ofits apparent simplicity, this

spring-mass model describes and predicts the dynamicsofrunning gaits inhumans and numerous other species remarkably well [15, 37-39, 42. 43,631.

Leg stiffness plays an important role in determining the dynamicsoftheinteraction between the stance leg and the ground. Many aspectsofrunning

depend on a runners leg stiffness, including the time offoot-ground con-tact, the vertical excursion ofthe bodys center ofmass during the ground

contact phase, and the ground reaction force [39, 001. Leg stiffness isdefined as the ratio of the ground reaction forceto the compression of theleg spring (AL) at the instant at midstance when the leg is maximally

compressed (Figure 4). In a running human, the leg stiffness represents

the average stiffness of the stance limb. In animals with more than onelimb simultaneously in contact with the ground, the leg stiffness in the

spring-mass model represents the average combined stiffness ofall of the

limbs in contact with the ground [15,381.Leg stifness remains the same at all forward speeds in running humans

(Figure 10.7) [63].They are able to run at higher speeds, andwith shorterground contact times, by increasing the angle swept by the leg during thestance phase (Figure 10.8). A variety of hopping, trotting, and running

animals keep leg stiffness the same at all speeds (Figure 7) and alter the

angle swept by the leg toadjust for different speeds [38].Although thestiffness of the leg remains the same at all speeds ofrunning, humans arecapable of altering their leg stiffness during bouncing gaits. Humans

change their leg stiffness in order to alter stride frequency during hopping

in place or fonvard running 137. 391. In addition, recent findings showthat humans adjust the stiffness oftheir legs to offset changes fn surfacestiffness [42,43].Ifleg stiffness were not adjusted toaccommodate surfacestiffness, then many aspects of the dynamicsofrunning would vary depend-ing on surface stiffness. By adjusting leg stiffness, humans are able to have

the same the peak ground reaction force, ground contact time, and vertical

displacement of the center of mass regardless of surface stiffness [42,43].


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FIGURE 10.7.

Leg sti#i.ness versu~speedfw mnning humans and fsotting dogs. For thc dogs, thrkg Stzjh?s.fepesa l s rhetotal stiffness ofthe two limbs on the ground during eachground confactphase.16--n

Human running$G-;4 - -cl

0 1 2 3 4 5 6 7I I

Dogtrotting

3 8 - -0Y

0 - I II I II II II IVelocity (m s-1)

FIGURE 10.8.The spn*ng--massmodelreprsenting low speedrunning and high speedrunning.The legstif~~ussandleg compreJsionare thesame inboth model. Theonly d1iJirm.cisthat f h e a @ swept bythe legspring (0) isgreater in the modelr+enting hi&

speed running. Because ofthe greaterangle Swept bythe legat thc h i g h s p 4 thev d i c a l d i sp t u m m l ofthemassduring theground conladphase issmaller atthehigher speed.

Low speed-- High speed


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Biomechanics ofWaudngandRunning I 267Just as human legs behave like springs during running, the fastest robots

also have spring-based legs [126-1291. We can gain insight into the controlSvategieSthat are possible in animal bouncing gaitsby exploring the rangeofworkable strategies in these robots. These robots use either compressed

air or metal springs in their legs to store and return elastic energy witheach step as they run, hop, or uot. There are many similarities betweenbouncing gaits in these robots and in animals. Both these robots and ani-mals run at different speedsby altering the angle sweptby their legs whilekeeping their leg stiffness the same [38,6S, 291. In addition,altering therobots leg stiffness leads to changes in stride frequency at the same speed[129],as is obsexved in humans [37,591.The control of these robots isgreatly simplified by relying on the passive dynamics of spring-mass system

of the robots body. The robots movements are largely determinedbyphysical parameters including the stiffness ofthe leg spring, the angle atwhich the leg spring is set down,and the mass of the robot [126]. Thecontrol algorithms workin concert with the physical properties of the ro-

bots body to produce stable locomotion. It seems logical to suggest thatanimals rely on the spring-mass dynamics of their bodies in a similar man-ner, thus simplifying the neural control oflocomotion.

JOINT MOMENTS, WORKAND POWERSo far, we have discussed the mechanics of locomotion at thewholebodylevel, including the ground reaction force, mechanical energy ofthe center

ofmass, and the behavior ofthe overall leg. During locomotion, musclesgenerate moments and perform mechanicalwork at thejoints, producingthe ground reaction force and the movements of the body. The next section

will concentrate on the current understanding ofjoint dynamics duringwalking and running. The focus of this section will be the moments and

mechanical work produced by muscles at eachjoint.Understanding loco-motion at the level of muscle action at each joint provides a bridge forunderstanding the link between the movements ofthe center ofmass and

the actions of individual muscle-tendon units. Little published information

is available about muscle moments and mechanical work at each joint dur-

ing locomotion in diverse animal species [44,94. 1361.Thus, it is diffkultto assess the similarities and differences between humans and other animals

at this level oforganization.

Net Muscle M m t sThe muscles of the body operate by exerting moments about joints. Tobegin to understand how the neuromuscular system produces walking and

running, researchers often examine the net muscle moment or general-ized muscle moment about a joint [149. 1631. A net muscle momentincludes h e moments produced by all of the muscles, tendons, ligaments,and contact forces at the joint. The moment produced by muscle-tendon


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forces is thought to be much higher than the moment produced by theligaments or other joint forces over the range of joint angles that occur

during locomotion [149, 1541.As a result, the net muscle moment gives areasonable approximation of the net moment produced by all the muscles

at a joint [12,461.An inverse dynamics approach can be used to determinethe net muscle moments during locomotion. This involves using force plat-

form, kinematic. and anthropomorphic measurements in concert with arigid linked segment model. TheNewtonian equations ofangular and translational motion are applied toeach segment starting distally and moving

proximally [34, 149, 1631.This type ofapproach has revealed that the net muscle moment during

the ground contact phase ofwalking is the largest at the ankle and is sub-stantially lower at the knee and hip 118, 19.32,34,115,135, 143, 146,148,1621.At the ankle, the net muscle moment tends to extend (equivalent toplantarflex) the joint throughout the ground contact phase. Electromyo-graphic (EMG) measurements have rewaled that both extensor muscles(e.g., gastrocnemius) and flexormuscles (e.g., tibialis anterior) are activeduring the ground contact phase, occasionally simultaneously [32].Theobservation that the net muscle moment tends to extend the ankle showsthat the ankle extensor muscles are creating a larger moment than theankle flexor muscles. The net muscle moment about the ankle is very smallduring the swingphase.

The net muscle moments at the knee and hip duringthe ground contact

phase of walking are much smaller and more variable than at the ankle

[l8, 19,34, 115, 116,135,143, 146,148, 1621.The ground reaction forcevector is closely aligned with the knee and hip.As a result, small net muscle

moments at the knee and hip are required in order to exert a given forceon the ground [135,1431.The exact pattern of net muscle moment at theknee and hip varies between subjects and is matched by variation in the

EMG patterns of the major limb muscles [1151.This observation has ledto the proposal that different individuals use different motor strategies forwalking [116,117].It has been suggested that these individual patterns areconsistent with a strategy that minimizes the total muscle effort for each

individual [1171.Walking kinematics are far less variable than the net muscle moments

or the muscle activation patterns at the knee and hip [148,1511.A compari-son of strides that have dramatically different net muscle moment patterns

at the knee and hip shows that the limb kinematics 5re remarkably similar.This obsemtion led Winter [146] to propose the idea ofasupport mclment equal to the sum of the net muscle moments at the ankle, knee,

and hip. Data on walkinghumans show that the support moment issubstan-

tially less variable than the net muscle moment at each individual joint.Thus, it seems that changes in muscle activation and net muscle momentat onejoint are offsetby changes at anotherjoint. This conclusion is further


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Biomechanics of Waking andRunning I 269

supported by the observation that humans with various injuries can stillwalk in a kinematically normal manner by changing the pattern ofmuscleactivation [150].As

one would expect, the peak magnitude ofthe net muscle moment ateach joint is higher during running than during walking [32.92,93, 1471.The leg is compliant during running, and the major leg joints undergosubstantial flexion and extension during the ground contact phase. In con-

trast,duringwalking, the limb behaves more like a stiffstrut, and thejointsundergo smaller angular displacements, remaining relatively extended

throughout the ground contact phase.As a result ofthepostural dserence,the muscles must generate largerjointmoments in order to exert a given

force on the ground during running than during walking. The net musclemoment tends to extend the joint at the ankle, knee, and hip during run-ning (Figure 10.9) [32,92,93,147]. The peak net muscle moment is largerat the knee than at the other joints. Indeed, this is a major difference

between running and walking. The magnitude ofthe peak net muscle rn6ment at the knee is much larger during running than walking. The knee

is substantially more flexed at the middle ofthe ground contact phase ofrunning. and as a result, a higher net muscle moment is required in order

to exert a given ground force during running compared to walking.In c o n m t to walking, running involves little variability in the pattern

and magnitude ofthe ground reaction force or the net muscle moments[1471. It has been speculated that the net muscle moments are less variablein running than in walking because the muscles are operating closer to

their force limits [1471.Joid P o w andWorkThe net power output at a joint can be calculated from the product ofthe

net muscle moment and the joint angular velocity. When the net muscle

moment and the angular velocity are both in the same direction, there is

net power production at thejoint. Conversely, when the net muscle moment

and thejoint angular velocity are in opposite directions, there is net power

absorption at the joint. The net muscle mechanical work can be calculatedfrom the integral of the power with respect to time.

I t is important to realize that the net power output measured at a joint

is not necessarily produced by muscles that cross that particular joint. This

is because there are many muscles in the body that cross more than one

joint. Thesemusclescan transportpower produced by muscles acting acrossone joint and allow them to con&ibute to the power output at another

joint [1401.The extent to which this transfer occurs during human walkingor running is not clear [80,122]. but there is evidence that energy transferby biarticular muscles is substantial during cat locomotion [1231.

The net power and net work output are substantially lower at all the

joints duringwalking than during running Duringwalking both


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FIGURE 10.9.

(A)Net musdemoment at the ankle, knee, and hip during running at2.5 m/s. Apositive net m w c k moment indicates ihat itten& to extend the joint. (B)Net musclep o woutput at the ankle,knee,andh 9 during running. Positive values indica&thatpower isproduced, andnegative values indicate that power isabsorbed.

A

B

-1200 Stance I Aerial0 0.1

I I0.2 0.3 0.4

Time (s)


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Biomechanics ofWalkingandRunning I 2'71

the net muscle moments and the joint angular velocities are lower than inrunning. T h e low net power and work outputs at the major limb jointsobserved during walking would beexpected for a limb that is behaving like

a stiffstrut.The net power output and work output are much higher during running

than walking.The net power outputs for slow jogging have been describedextensively [1471.At the ankle and knee, an extensor net muscle momentthroughout the ground contact phase exists (Figure 10.9). Meanwhile, theankle and knee both flex and absorb mechanical energy during the firstpart of the ground contact phase. Later in the ground contact phase, theankleand kneeboth extend and produce mechanical power. The net power

output is small and unpredictable at the hip. During low speed jogging,the ankle produces more mechanical energy than it absorbs. In contrast,

the knee absorbs more mechanical energy than it produces.

MUSCLE-TENDON MECHANICS

Examining the net muscle action at joints provides a link between whole

body dynamics andmuscle-tendon.dynamicsduring locomotion. This sec-tion will discuss muscletendon action during locomotion. Unfortunately,

it has been difficult to quantify muscle-tendon forces and length changes

during locomotion, although recent technological advances have yieldednew and exciting findings. In addition, the incredible complexity and a pparent redundancyofthe musculoskeletal system has made discerning gen-

eral principles about muscle-tendon action during locomotion extremely

challenging. In this section, we will examine information about muscle-tendon forces and muscle-tendon length changes during locomotion. Wewill then discuss how this information can be incorporated into forward

dynamicsand inverse dynamics approaches in order to uncover fundamen-

t a lrules about how the neuromuscular system produces locomotion.

Mucc.!e-TendonForce DuringLocomotionWhile center ofmass movementsand net muscle moments atjoints can becalculated relatively easily from force platform and video data using an

inverse dynamics approach, muscle-tendon forces during locomotion are

much more difficult to determine. Each leg joint has multiple muscle-ten-don units that span it , and each muscle-tendon has its own uniq.ue force-generating capabilities. Thus, the contribution ofeach muscle-tendon unitacting about a joint to the net muscle moment is not easily determined. Itmay seem reasonable touse simplifying assumptions to partition the netmuscle moment among different muscles that have a given action (e.g.. allsynergists experience equal stresses), but direct measurements in vivohave

shown that the distribution ofmuscle force is not so simple. In fact, the


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272 I Farby, Feniscontribution of each synergist changes for dBerent locomotion speeds,different gaits, and even during the course ofa single s t a ~ ~ c ephase (10,44,68, 70, 1451.

Oneway researchers have attempted to solve this problem isby employing

inverse optimization techniques. Inverse optimization (sometimes calledstatic aptimization [1611) uses a model ofthe musculoskeletal system andrequires it to produce specified movement dynamics while optimizing agiven cost function (e.& minimization of the sum ofmuscle forces) 11521.Although numerous optimization criteria have been suggested For use inthe cost function (e.g., minimal muscle force, minimal muscle stress,minkmal energy expenditure, minimal ligament force, minimal intra-articularcontact force, minimal instantaneous muscle power, and minimal m u &fatigue), no single best parameter hasbeen found. In fact, when seven1ofthemost commonlyused criteria were compared, they predicted remark-

ably similar patterns ofmuscle activation, but none demonsuated a closematch to the actual EMG patterns over a complete stride cycle [30].Atpresent, we do not yet sufilciently understand the disuibution offorcesamong synergists to identify any general rules [ 6 5 ] .

Muscle-tendon force calculations from an inverse dynamics approach are

also complicated by the possibility ofcoactintion ofantagonistic muscle

groups. When antagonistic muscle groups are simultaneoudy active, ahigher agonist force is required to exert a given net muscle moment. For

example, during the ground contact phase ofrunning,there isan extensornet muscle moment at the knee. EMG studies have shown that knee exten- .sor (e.g., vasti muscles) and knee flexor muscles (e.g., gastrocnemius)areactive simultaneously.As a result ofthis coactivation, a higher force is re- .quired from the knee extensors than if there were no coacuvation. Thecoactivation makes it impossible to determine the force in either muscle

group from an inverse dynamics approach, since there are an infrnitenum-ber ofcombinations of extensor and flexor forces that could produce theSame net muscle moment. It is interesting to note, however, that there islittle coactivationofextensor and flexormuscles at the ankleduringbounc-ing gaits.Thus,an inverse dynamics approach to calculating ankle extensorforce yields reasonably similard u e s asa direct measurementofthe muscle-tendon force [1 ,12,46] .There is substantial antagonist coactivation at theknee and the hip during locomotion, and as a result, an inverse dynamics

approach is less likely to yield accurate muscle force values at thosejoints.

Forces in muscle-tendon units axe measured in vivothrough the use offorce transducers on tendons (recently reviewed by Gregor and Abelew[571). In a fewcases, abuckle transducerwasplaced on the Achilles tendonofhumans [46,47,58,59,82,84] . The results from these studies show thatthe peak Achilles tendon force slightly decreases or remains about the

same (-2.6kN or 3.6 bodyweights) ashumans increasewalking speed from


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B i m c h a n i c s of WalkingandRunning I 273FIGURE 10.10.Horizontalground reactionfir&,verticalpnd reactionfmce,andAchilleJ tmdonfwcc (mcacured with a tendonbuckle) fm ahuman walkingutdiffoentspeed.Rgrinicd with pmission from Konu' etal. [84].

K . K F A

ab = 1.21.3 mm xx 5-1- 1

Tendon force

1.2 to 1.8m/s (Figure 10.10) [84].As they increase speed further and beginrunning, the peak tendon force increases to a maximum (-9 kN or 12.5bodyweights) at a speed of approximately 6m/s and then changes verylittle at higher running speeds [84]. Interestingly,both walking andmm-ning involve greater peak Achilles' tendon forces than maximal height 2squat jumps or countermovement jumps [84]. While these studies haveprovided m e in vivomuscle-tendon data for human locomotion, the inva-siveness of the technique limits the possibilities for human studies.

Alternatively, the use of force transducers on the tendons of animals has

enabled researchers to investigate numerous different research questions[lo, 57, 69, 121, 122,125, 1301. With animals, tendons can be surgicallyseparated so that forcedata can be collected from individual muscle-tendonunits. Unfortunately, current buckle-type transducers may affect themus-cle's force generation since they sometimes damage the tendon, causing it

to h y and break [57]. However, new types offorce transducers that areactually inserted within a tendon or ligament are being developed thatshould correct this problem 155. 66, 67. 71, 72,83,851.MuscloTendon h g t h ChangesDuring L a m t i o nPerhaps surprisingly, the calculation ofmuscle and tendon length changesduring locomotion is even more complicated than the determination of


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274 Fur@, Fmis

muscle-tendon forces. The most common method for calculating muscle.tendon length change has been to use a combination ofkinematic andanatomicaldata.In this approach, the instantaneous muscle-tendon lengthis estimated from the approximate origin and insertion sites for a givenmuscle-tendon unit, and fromjoint kinematic data [6,45,56,60, 106,1201.However, this technique does not partition the total displacement ofthemuscle-tendon unit into the length changes due to muscle fiber displace-

ment, muscle fiber pennation angle change, or tendon strain. Eachofthesefactors can substantially affect the total muscle-tendon length. and eachhas different implications for muscle-tendon Function [48,49,62.108,301,

The relevance ofpartitioning muscle-tendon displacements into respec.tive components becomes evident during"isometric"contractions inwhichisolated muscle-tendon units are held at a constant total length. m e nelectrically stimulated, the fibers of muscles with long compliant tendonscan shorten considerablyas the tendon is stretched [36,62].Even thoughthe whole muscle-tendon length remains unchanged, each component ofthe unit changes length substantially.Thus, although it is possible to es&mate the length changesof the overall muscle-tendon unit during locorn-tion, this information tells us little about the relative length changesof thedifferent components ofthe muscle-tendon unit

A recent technological innovation is the use ofsonomicrometry to mea-sure muscle fiber displacements and velocities in vivo [20, 611. By suturingpiezoelectric crystals into a muscle fiber bundle, the time required for ultra-sound pulses to travel from one crystal to another can be measured. The

transit time canthen be used to calculate the instantaneous muscle fiber

length. Studies on walking cats and running turkeys have shown that themuscle fiber does not always follow the same displacement pattern as the

whole muscle-tendon unit [62,1SO]. In walking cats, the medial gastrocne-mius muscle fibers shorten at the beginning ofthe ground contact phase,even though the overall muscletendon unit lengthens during this phase[62]. As a result, the tendon is stretched more and stores more elasticenergy thanwould be predicted based on the overall muscletendon unitlength change. Similarly,when turkeysrun on level ground, the gastrocne-

mius muscle fibers remain nearly isometric during the stance phase while

the tendon undergoes substantial length change (Figure 10.11) [130].Thetendon performs the majority ofthe combined muscle-tendon work whilethe muscle doesvery littlework. Thus,during level locomotion inboth cats

and turkeys, the work doneby some muscles is greatly reduced by tfiestorage and return of elastic energy in tendons.

Elastic energy storage in tendons is particularly important for bouncing

gaits [3]. The ankle extensor tendons have been studiedmost often, andthe results have shown that they play a key role in the storage ofelastic .

energy. In a running human, these tendons can store and return up to35% ofthe mechanical energy needed to lift and accelerate the center of


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Biomchanics ofWalking andRunning I 275FIGURE 10.11.

-a1 -ius rnusclefibcl.Imgth, EMG,andf i e in a tu% duringm n -ningat3m/s on kvelp d . usckjitUrlength wasmeasured usingsonomicrome8 ,andmuscle/me was measured un'ng astrain gmp attaciud to a caIcifidportion of the tmdon.Rep;ntcd with ~ h z f r o mo b n l s , T.J., RL Marsh,P.G. Wyand , andC.RTaylor.Muscular o r u inmnning turitcys: the emonry ominimizing worfir. Samu 275:1113-1115, 1997.Copyright 1997, AmericanAssociation for the Advancement ofScience.

nEE

53Y

cnt

1

hvZQ)

L02

00

50

00 0.2 0.4

Time (s)


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276 I Fa+, Fenismass during a stride [ B l l . Tendon buckle studies on hopping wallabieshave revealed that the ankle extensor tendons store and return c ~ ~ o u g helastic energy to reduce the metabolic cost of locomotion by about 50%

[lo]. It is important to realize that tendon canbe divided into two comp6nents-the external free tendon and the aponeurosis [1591.A number ofrecent studieshave examined the relative compliance ofthese twoportionsof the tendon and have come to different conclusions [36, 89,124, 125,133, 137, 641.As a result, it is not clear which part of the tendon is mostimportant in elastic energy storage during locomotion.

Role OfFOrwardQnamics Simulations in UnderstandingMusclcTendonFunction During LourmotionA promising alternative to invasive in viuo muscle-tendon measurements inhuman locomotion is forward dynamics computer simulations of humanlocomotion (1601. Instead of calculating net muscle momenb from theground reaction force and kinematicdata (i.e.. inverse dynamics),forwarddynamics simulations rely on musculoskeletal models and computer soft-

ware topredict the muscle activation patterns and muscle-tendon dynamicsduring normal locomotion. The difficulties with this approach, however,are that the results from detailed musculoskeletal models can be extremely

sensitive to the specificsoFtht model 187,1551, and experimental validationofthe muscle-tendon mechanics is difficult to obtain. Nonetheless, thisapproach has been used to simulatehuman walkingwith some success [119,157, 1581.

Forward dynamics simulations have also been used in conjunction withsensitivity a n a l y w s to determine which aspects of musculoskeletal designare most important in dictating the mechanics of locomotion [54]. In thefuture, this combination offorward dynamics simulations and sensitivityanalyses shouldprove valuable in providing insight into how muscle-tendonproperties and activation patterns can affect joint dynamics and center of

mass movements during locomotion.When forward dynamics simulations are employed in conjunction with

optimization techniques, they allow the researcher to probe the linkbe-tween muscle-tendon properties andmuscle activation patterns based onpossiblegoalsofthe central nervous system (called dynamic optimizationby Zajac [160] and forward dynamic optimization by Winters [152]. Inthis approach, a parameter or function is assumed to be optimizedby the

neuromuscular system, and the pattern of muscle activation that optimizes

thatpaxxmeter or function is identified. This approach is usedmost oftento study movemenb in which an optimality criterion is obvious,such as asinglemaximumheight vertical jump [7, 16, 17, 112-114, 14@142]. Forlocomotion, it isdifficult to determine which parameter or function should

be optimized. One possibility is that the minimization ofmetabolic energy

cost is the most important factor in determiningwhich neuromuscular strat-


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Biomchunics of W&ing andRunning I 277egy is used in sustained locomotion [4,198]. However, controlling the levelof musculoskeletal forces and stresses can alsoplay an important role [41,1181: Indeed, it seems most likely that multiple factors work in concert toshape locomotion, suggesting that multiple optimization criteria shouldbe

used [1531.Data fromexperimentson equine locomotion support the idea that mul-

tiple factorswork in concert to shape locomotion, and thus,that multipleoptimization criteria should be used. Experiments have revealed that there

are at leasttwo factors involvedin the choice of locomotor speed and gaitin horses-metabolic energy cost minimization andmusculoskeletal forceminimization. During unrestrained overground locomotion, horses only

use a smallm g e ofspeeds within each gait, and within this range of speeds,the metabolic energy cost is lower than at any other speed within the gait[74], This observation suggests that metabolic energy cost is an impo&tfactor that influences the choice ofspeed during locomotion [73]. Never-theless, when the mechanics and energetics of the transition from trottingto galloping are examined more closely, it is clear that horses donot choose

to switch fromone gait to another at the speed that would minimize themetabolic energetic cost oflocomotion [41]. Rather, the choice ofgaittrarsition speed seems to be most influenced by the levelof musculoskeletalforce [41]. Thus, at least two factors are important in shaping locomotionin horses, emphasizing the need to consider multiple optimization criteriain forward dynamics simulations of locomotion.

Rnlc o janInverse Dynamics AppToach in UnderstandingMusclPTendonFundionDuringLocomotionThe complexityofthe neuromuscular system has hindered progress in gain- ,inga fundamental understanding ofhow locomotionisproduced. Thiscom- -plexity has made it difficult to reach a synthesized understanding oflocometion.As a result, it may be helpful to pursue an alternative to a reductionistapproach that begins with detailed descriptions ofindividual muscles andtheir neural control. This possibility may have been best expressed by Loeb

who asked whether it is useful to collect yet more inexplicable data, and

suggested that it may be useful to consider the performance goals ofthewhole behavior [go]. Given the tremendous complexityofthe nervous sys-tem and the musculoskeletal system, we can imagine a seemingly infinite

number of possible neuromuscular strategies that could produce the loco-motion ofhumans and other animals. Indeed, numerous different patternsofjoint moments and muscle activation can produce normal walking [115,116,1481.By reaching an understanding ofoverriding performance goals(e.g., theneed fora given support moment duringwalking, [146],we canfocus in on a more limited range of potential strategiesthatwill produce nor-mal locomotion.An inverse dynamics approach thatbeginswith defined per-formancegoals at t h e whole organism level allowsus to do this.


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Simple behavioral models play an important role in an inverse dynamicsapproach tounderstanding 1ocomotion.Theyprovide a mechanicaldexription oftheoverallbehavior (orperfomancegoals) of themusculoskeleu\systemduringlocomotion.Behavioral models represent thebehaviorbutnotthe structureofthe musculoskeletalsystemduring locomotion.For xample,as described prcviously, a simple spring-mass system provides a reasonablyaccurate model of running. This model works because the human leg b

e-haves much likeaspring during running. However, we are allwell aware*at

the humanlegisnot actually a spring. Rather, it ismade up ofmultiplemu+cles, tendons, and ligaments that span severaljoints.Thus,after identifying

an appropriate behavioral model, we must begin to consider how the r edmusculoskeletal systemproduces the observed behavior.

In the example case of running, the observation that the overall leg be-haveslikeaspringsuggeststhattheproductionofspring-likebehaviormaybean important organizing principle for the actionsofmultiple muscle-tendon

units.Thegoalofproducingspring-likebehavior limits the numberofpoten-tial solutionsfor the behaviorofindividualjoints,interactions between multi-plejoints, andactionsofindividual muscle-tendon units that span eachjoint.It also makes it clear thatan important next step is to examine the spring-likeproperties ofindividualjoints andmuscle-tendon units during running.Atthispoint in theprogression,acombination of information frombehavioralmodels and realistic muxuloskeletalmodels islikely to bemost powerful indissecting the roles ofindividualjointsandmuscle-tendon unitsindetermin-

ing leg stiffness, and thus, the mechanicsofrunning.

CONCLUSIONS

Humans use the same basic mechanisms to walk and run asother leggedanimals. Walking gaits rely on a transfer ofkinetic and gravitational poten-tial energies with each step similar to an inverted pendulum. Runninggaits,on the other hand, can be characterized as bouncing gaits and modeled

with asimplespringmass system. These two.behavioral models, the invertedpendulum and the spring-mass system, provide researchers with simplede-scriptionsofoverall limb behavior and center ofmass movements duringwalking and running. This information can provide guidance in attemptsto discern generalrulesby which the neuromuscular system produces loco-motion at the increasingly complex lower levels of organization.

ACKNOWLEDGMENTSThis workwas supported in part by theNational Institutes ofHealth (mAR44008) and the National Aeronautics and Space Administration (NGT-51416).


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