Mechanics

of running

under

simulated

low gravity

JIPING

HE, RODGER KRAM, AND THOMAS A. McMAHON of Applied Sciences, Department of Organismic and Evolutionary Biology, Harvard and Center for Biological Information Processing, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138 Division

University,

measured the leg spring properties across a range of forward running speeds. We hypothesized that leg stiffness iol. 71(3): 863-870,1991.-Using a linear mass-springmodel of would be constant across both gravity and speed. Our the body and leg (T. A. McMahon and G. C. Cheng.J. Biomech. results include both experimental observations of run23: 65-78, 1990), we present experimental observations of hu- ning under conditions simulating low gravity and the abilman running under simulated low gravity and an analysis of ity of a lumped-mass spring-leg model to account for the these experiments. The purpose of the study was to investigate observed changes. how the spring properties of the leg are adjusted to different HE, JIPING, RODGER KRAM, AND THOMAS A. MCMAHON. Mechanics of running under simuluted low gravity. J. Appl. Phys-

levels of gravity. We hypothesized that leg spring stiffness would not change under simulated low-gravity conditions. To simulate low gravity, a nearly constant vertical force was ap-

METHODS

10). The “springs” in a running animal are presumably in the muscles and tendons. There is substantial neurological and biomechanical evidence that the muscles and tendons maintain constant mechanical stiffness over the range of physiologically relevant forces (1, 5-7). When animals run faster, the peak vertical forces increase little but they bounce off the ground in less time. If the vertical motions are controlled by a spring, this could only be accomplished with a spring that has a stiffness that increases with speed. To reconcile this apparent contradiction, McMahon and Cheng (9) suggested that a range of whole body vertical stiffnesses can be achieved with a single leg spring stiffness if the landing and takeoff angle

Subjects and protocols. Four healthy male subjects volunteered to participate in the study. Ages ranged from 23 to 35 yr and body weights from 598 to 705 N (Table 1). The subjects were familiar with treadmill running. All subjects ran at 3.0 m/s under conditions simulating 1.0, 0.7, 0.6,0.5, 0.4, 0.3, and 0.2 G. (Here and throughout, G means “times normal gravity.“) For comparison with the reduced-gravity experiments, the subjects ran at normal gravity at a variety of speeds: 2.0, 3.0, 4.0, 5.0, and 6.0 m/s. The only instructions given were to “run normally.” Experimental apparatus. The apparatus for applying a nearly constant vertical force to the body is shown in Fig. 1. The vertical force was produced by stretching a series of four steel springs of the type used to balance the weight of garage doors (Door Depot, Irving, TX). The tension in the springs could be increased by reeling in a cable attached to one end of the springs by use of a small hand winch fixed to a wall. As a safety precaution, the springs were enclosed in a &cm&am polyvinylchloride plastic pipe. Another steel cable ran from the other end of the springs to the runner, passing over two lubricated ball-bearing pulleys. After many attempts to make a comfortable and nonrestrictive device for transmitting the vertical force of the springs to the runner, we settled on a harness based on a racing bicycle saddle. The saddle was fixed to a U-shaped carrier made of plastic polyvinylchloride pipe. Four steel cables fixed to the carrier transmitted the vertical force from the springs. The subjects ran on a motorized treadmill with a force platform under the treadmill belt. The force platform was 1.21 m long and 0.46 m wide (model OR6-5-1, Advanced Mechanical Technology, Newton, MA) and had a natural frequency of 160 Hz (measured by rapping the unloaded force platform). Because the vertical forces measured during the running* experiments did not include components above 20 Hz in substantial amplitudes, the frequency response

of the leg vary with speed. The purpose of this study was to determine how leg spring stiffness changes when gravity (the primary force that muscles must overcome) is varied. In addition, we

of the loaded force platform did not compromise our measurements. Extensive previous tests and calibrations had shown that the treadmill-mounted force platform provided an accurate and reliable measurement of the verti-

plied to human

subjects via a bicycle

seat. The force was ob-

tained by stretching long steel springs via a hand-operated winch. Subjects ran on a motorized treadmill that had been modified

to include

a force platform

under the tread. Four sub-

jects ran at one speed (3.0 m/s) under conditions of normal gravity and six simulated fractions of normal gravity from 0.2 to 0.7 G. For comparison, subjectsalsoran under normal gravity at five speedsfrom 2.0 to 6.0 m/s. Two basic principles emerged from all .comparisons:both the stiffness of the leg, consideredas a linear spring, and the vertical excursion of the center of massduring the flight phasedid not change with forward speedor gravity. With these results as inputs, the mathematical model is able to account correctly for many of the changes

in dynamic

parameters

that do take place, including

the increasing vertical stiffness with speedat normal gravity and the decreasingpeak force observedunder conditions simulating

low gravity.

locomotion; muscleand reflex stiffness; reduced gravity; motor control

RUNNING

ANIMALS

have springlike properties (1, 3, 9,

0161-7567/91

$1.50

Copyright

0

1991

the

American

Physiological

Society

863

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864 TABLE

RUNNING

UNDER

SIMULATED

1. Subject data

Subj

Weight,

JPH RK JM TR

658.3 705.3 701.4

598.4

N

LOW GRA ‘ITY A

Leg Length, 0.910 0.940 1.003 1.003

m

Age,

yr

35 28 35 23

cal ground reaction force during walking and running (8). I 0.6 1 0.2 -\ 0.4 This apparatus, minus the treadmill, is similar in prinI \ Time (s) / ciple to those used by Spady (11) and by Cavagna et al. \ / \/V . (4), both of whom studied motions of the body under // conditions simulating the gravity of the moon. Under ideal circumstances, the vertical force applied to the body should not change as the body moves up and down during walking and running. An account of various calibrations, including direct measurements of the small fluctuations in cable tension during running at reduced gravity, is given in APPENDIX A. The conclusions reached there are that 1) the steel spring is only l/lOOth as stiff as the effective vertical stiffness of the body and therefore does not add significantly to the body stiffness and 2) fluctuations in cable tension due to stretching of the steel spring and rubbing of the cable, pulleys, and spring do not con-‘\ Time (s) tribute errors to the measured velocities or displace-0.5 . \ /’ ments that would change any conclusions of the study. ‘\J J Data reduction and analysis. The vertical ground reac-1.0 L tion forces sampled at 1.0 kHz during each run were used FIG. 2. Typical records for vertical reaction force (f, measured by as the raw data to derive other variables. The force data force platform), vertical velocity (u), and vertical displacement (y) vs. were low-pass filtered (attenuation 12 dB/octave above time. Curves show an average of 10 stride cycles (20 steps) for 1 individ50 Hz) and then aligned at the beginning of each stance ual. A: normal gravity (1 G). B: conditions simulating 50% gravity (0.5 G). Arrow, moment of contact of the next foot so that aerial time can be phase to overlay all strikes to obtain one averaged vertiseen. cal ground reaction force profile [f(t)] for a standard stride cycle (Fig. 2). Equating the sum of the vertical where y(t) is the vertical displacement, g = 9.81 m/s2 is forces acting on the center of mass to the total body mass the gravitational constant, and f, is the (assumed con(m) times the vertical acceleration gives stant) spring force applied by the apparatus during the experiment. One integration of Eq. 1 gives the velocity md2y( t)ldf = f(t) - (mg - f,) (I) [u(t)]. From f(t), y(t), and u(t) we determined the duration of contact (t,), the peak vertical force (f,,), the peak downward displacement of the center of mass during foot contact (Ay), and touchdown velocity (v,). The magnitude of the angle of the leg with respect to the vertical at the moment of foot contact was calculated as 0o = sin-l(utJ21,) (2) where u is the average forward velocity during foot contact and I, is the measured length of the fully extended leg (Table 1). Equation 2 was obtained from the geometric construction shown in Fig. 3. The two assumptions behind Eq. 2 are that 1) the forward velocity of the center of mass is nearly steady during the contact time and equal to the measured speed of the treadmill belt and 2) the length and absolute value of the angle of the leg is the same on touchdown and landing. Both assumptions are approximations that are justified for this purpose: for I I II I I //////////////f//////////////////////////// example, the forward velocity of the center of mass flucFIG. 1. Schematic diagram of apparatus. Motorized treadmill intuates by ~0.1 m/s, or 3.0%, when a human runs at a cludes a force platform (F) under the tread. A bicycle saddle (S), with steady speed of 3.3 m/s (2).

nose angled down, applies an upward force to subject’s pelvis. A wide belt (B) around the waist prevents subject from slipping forward on saddle. Cable (C) bearing vertical force runs over 2 pulleys and ends on series of steel springs (Sp) contained in open-ended polyvinylchloride pipe for safety. Springs may be tensioned by cranking the winch (W).

RESULTS

Vertical force, velocity, and displacement. Curves show-

ing the vertical ground-reaction

force (f), the vertical ve-

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RUNNING

UNDER

SIMULATED

LOW

865

GRAVITY

locity of the center of mass (v), and the vertical displacement of the center of mass (y) for an average of 10 complete stride cycles (20 steps) of a single subject running at 3.0 m/s are shown in Fig. 2. In Fig. 2A the subject is running at normal gravity, and in Fig. 2.B the simulated gravity is 0.5 G. Comparison of the curves in Fig. 2, A and& for a single subject reveals several features that were found in all subjects at all levels of reduced gravity. First, the maximum value of the force at midstep was decreased by 22% (mean of 20 steps for this one subject at 0.5 G) compared with the normal-gravity run, although the amplitude of the early peak in force due to the arrest of the downward -

FIG. 3. Schematic diagram showing geometry assumed McMahon and Cheng (9). Initial angle of leg with respect to is -8,. Trajectory is symmetrical with respect to the vertical, 8, on takeoff. Downward deflection of center of mass at the midstep. 1,, Initial and final lengths of the leg; u, forward duration of foot contact.

by model of the vertical so that 8 = hip is Ay at velocity; t,,

2.0

gL

1 0.0

0.2

0.4 0.6 0.8 1.0 Gravity ratio, G

A

-G= 1.0 --G=O.S

.O

Vertical displacement (mm) -LOG - - 4.7G - . - .0.6G .. .. .._s_.0.5G --JMG

1.2

0.0

20.0

40.0

60.0

80.0

1 100.0

Vertical displacement (mm)

0.0

0.2

0.4 0.6 0.8 1.0 Gravity ratio, G

1.2

0.0

0.2

0.4 0.6 0.8 1.0 Gravity ratio, G

1.2

Vertical displacement (mm)

FIG. 4. Comparison of several parameters vs. gravity ratio (G). A: peak force at midstep (fmax). B: contact time (t,). C: stride frequency. One stride cycle includes 2 contact periods and 2 aerial periods. Solid curves, predictions of model of McMahon and Cheng (9) with input parameters specified as in Fig. 7. Values are means t SD.

FIG. 5. Vertical force vs. vertical displacement. Each curve shows average of 20 steps (10 complete strides) at 3.0 m/s. A: a full step cycle, including 1 contact period and 1 aerial period, at normal gravity (1 G) and at 0.5 G. Vertical mark on horizontal axis, maximum height reached during aerial phase. B: second half of contact period comparing 7 conditions of G in 1 subject. C: second half of contact period for 1 subject at normal gravity running at 5 speeds. Arrows, direction of time.

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866

RUNNING

UNDER

SIMULA

D LOW

GRAVITY

motion of the foot and shank was 6.3% higher. Second, the duration

of the aerial phase (time between toe-off

of

one foot and contact of the next) was 4.1 times longer under conditions simulating 0.5 G, and the contact time (time of contact of one foot) was 12% less. In Fig. 4, it is

clear that fmax and stride frequency fall as G declines; tc changes little across the fivefold range of G. Effective verticd

stiffness. To consider a running

per-

son as a spring, it is illustrative to plot vertical force against displacement. This has been done for one subject running

at 3.0 m/s in Fig. 5A. After

01234567 Forward

the foot touches

down, the force rises abruptly to the early peak associated with stopping of the shank, and then falls briefly before rising again to the midstep maximum.

The verti-

Thereafter

the vertical

displacement

(y) rises

as the force falls. The vertical displacement on toe-off is always greater than on landing. Curves showing only the second half of the step, corresponding to the rising motion of the body, are shown in Fig. 5B for a running speed of 3.0 m/s. These curves, giving the results for experiments simulating reduced gravity down to 0.2 G, show a nearly linear relationship between vertical force and vertical displacement at each value of G. The change in vertical force divided by the change in vertical displacement has been called the “effective vertical spring stiffness” (&.J and has been the topic of previous investigations (3,9,10). As illustrated in Fig. 5B, there is little change in the slope of the curves and, therefore, in /zVetifor running under reduced gravity down to ~0.5 G, but kVertincreases somewhat at the lowest values of G. By contrast, kVert increases sharply with speed at any one value of G, as may be seen in Fig. 5C, giving the results for the second half of the step of a single subject

running

at normal

gravity at five speeds from 2.0 to 6.0 m/s. Leg stiffness. The parameter kleg is the effective stiffness of the leg (as if it were a linear spring) and is defined bY

kleg

=

(3)

fmax/Al

where Al is the change of the leg spring length during the ground contact and is obtained by AZ = Ay + 1,[1 - cos&)]

0

Figure 3 shows the geometry assumed in the above definitions. The definitions are part of a mathematical model of running proposed by McMahon and Cheng (9). A brief e assumptions and results of this model is e in the given in APPENDIX B. Another important variab 11 model of McMahon and Cheng is V, a dimension 1ess vertical velocity defined as V = uo/(Gglo)1’2 (5) where -u, is the downward velocity on landing. In Fig. 6, changes in kleg, kVert, &,, and V are sklawn as speed is changed while gravity is kept fixed (normal gravity). In Fig. 7, the same parameters are plotted against G for running at a constant speed (3.0 m/s). The solid curves in Figs. 6 and 7 give theoretical results to be dis1

cussed later. In Fig. 6, it is apparent

that the parameters

klewand V

(m/s)

0.6 0.5 0.4 y 0.3 s 0.2

cal force reaches a local maximum at the same moment the vertical height (y) of the center of mass reaches a minimum.

speed

0.1 0.0

01234567 Forward speed (m/s) FIG. 6. Comparisons of the parameters of running for 4 subjects running at a range of speeds under normal gravity. Points, mean for 20 steps (IO complete strides); error bars, SD. Solid curves, predictions from model of McMahon and Cheng (9). A: leg-spring stiffness (kjeg, an input to the model) and vertical stiffness (kVert, an output from the model). 23: dimensionless landing velocity [ V = u/(gZ0)1’2,an input] and touchdown

angle

of the

leg (8,, an output).

are nearly independent of the tread speed, even over a threefold change in speed from 2.0 to 6.0 m/s, while the parameters B, and /zVertincrease with speed. Because G, g, and 1, are all constants, the fact that V does not change with running speed over most of the speed range means that the vertical landing velocity (-u,) is roughly independent of speed. Comparing the effects of speed and gravity. There are also simple conclusions available from Fig. 7. In Fig. 7, A and B, both & and V are substantially independent of G over the full range down to 0.2 G. Note that if Vis fixed, then the actual vertical landing velocity (u,) must be smaller at 0.2 than at 1 G, and this is, in fact, what the experiments showed. The other data points on Fig. 7 illustrate that kVetiincreases at low G and B0 decreases. DISCUSSION

Interpretations using a mathematical model. When the subjects ran under conditions simulating reduced gravity, many variables changed: peak force, contact time, stride frequency, absolute vertical landing velocity, and B0fell; leg stiffness was nearly constant; and the vertical stiffness increased slightly. It would be valuable to know whether there is a connection between these observations. A theory that correlated the observed changes could give useful insights about the physiological mechanisms controlling the muscles in running. In the model for running proposed by McMahon and Cheng (9), the body mass is lumped at the hip and the leg is represented as a linear spring of stiffness klag. At the moment of contact, the horizontal and vertical velocities of the mass are U, and -uO, the leg length is ZO,and the

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RUNNING

UNDER

SIMULATED

LOW GRAVITY

867

model of McMahon and Cheng was then used to obtain 8, for each value of U, and then KvERT was computed. Finally, dimensional values of kvert were computed assuming a leg length and body weight for the model equal to the mean leg length and body weight of the runners, 0.96 m and 0.656 kN, respectively. The theoretical results were plotted as solid curves on Figs. 6 and 7 along with the experimental data. The reason we use dimensional parameters in Fig. 6A and dimensionless parameters in 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 6B is explained in the discussion of Fig. 7. McMahon and Cheng (9) found good agreement with data taken Gravity ratio, G from the literature on dogs, ostriches, and humans when 0.6 B they assumed both KLEG and V were specified constants, T . independent of both body size and forward speed. The 051 theoretical curve lies below the experimental points for kveti in Fig. 6A. With few exceptions, however, the experimental points in Fig. 6 lie within one standard deviation of the theoretical curves. In Fig. 7, we retain the assumption that V is fixed at 0.25, and Fig. 7B shows that the assumption is well justified, because the solid line comes close to the solid circles o.o! - g - ’ - ’ - ’ - i ’ 1 0,O 0.2 0.4 0.6 0.8 1.0 1.2 showing the means. Again k,, was fixed, although this Gravity ratio, G means that K&G = k,,,l,lmGg will be different at the different G levels. We assumed that the speed of the tread0.6 c mill belt (u) provided a good estimate of the forward 0.5 speed (u& on heel strike. Calculating U = u,/(Gg1,)1’2 completed. the input specifications, and the model was used to obtain 6, and KvERT. Finally, kvert = KvERTmGg/Zo 80 0.3 Og4 Iut-tI--I was calculated and plotted in Fig. 7A. Now it should be 0.2 i obvious why dimensional values have been used for Figs. 6A and 7A. Because G enters the definition of KVERT and . 01-l K LEG, it is more reasonable to plot the dimensional valo.o! ’ 1 - ’ - ‘ . ’ ’ * 1 ues for these parameters as functions of G in order that 0.0 0.2 0.4 0.6 0.8 1.0 1.2 the definition of the dimensionless quantity not hide an Gravity ratio, (G) important trend in the dimensional parameter. FIG. 7. Parameters of running compared at 1 speed (3.0 m/s) as G Returning to Fig. 4, it is apparent that the model also ranges from 1.0 to 0.2. Points, mean for 20steps (10 complete strides); makes good predictions of the changes in peak force as G error bars, SD. Solid curves, predictions from model of McMahon and changes, but it overpredicts the observed change in conCheng (9) A: leg-spring stiffness (k,,, an input to the model) and vertical stiffness (hert, an output). B: V (an input). C: & (an output). tact time and underpredicts the change in stride frequency. A limitation of the model that may account for angle of the leg with respect to the vertical -&. The di- its relatively poorer performance in predicting contact mensionless parameters 8,, U = u,l(gZ0)1’2, V = uol(gZ0)1’2, time and stride frequency is the requirement in the and KLEG model that leg length on takeoff be equal to leg length on ~-- = k~,,l,/mg are used to characterize a given running condition. One way to think of the model is to realize touchdown. This is not true of human runners where the that it provides a connection between the four dimenleg is longer at takeoff than on landing (Fig. 5A), reflectsionless variables, any three of which may be thought of ing the fact that the knee is almost fully extended and the as inputs. Thus, given numerical values for U, V, and ankle is plantar flexed on takeoff. Limitations of the experiments. We have proposed K LEG, the model specifies a value for 8, that results in no these experiments as simulations of locomotion in reSlOWi.ng down, speeding up, or than .ges in peak height from step to step. Once values are fixed for th .e four duced gravity, but several aspects of the experimental above-mentioned dimensionless parameters, other im- design limit the realism of the simulation. The most important of these is the fact that the vertical force is apportant dimensionless groups may be calculated directly, such as the dimensionless peak force F,, = f,,lmg, the plied to only one segment of the body. Because vertical forces were not added to all body segments, motions of duty factor DF = t,/( tc + t,), and the dimensionless vertibody parts with respect to the center of mass are not cal stiffness KvERT = kv,tiZolmg. The solid curves in Fig. 6 show the predictions of the simulated correctly. For example, if the subject allows an model of McMahon and Cheng (9) when klephas been arm to swing passively while he is standing quietly, the fixed at 11.33 kN/m and V is fixed at 0.25. Note that no natural frequency of the swinging motions is the same units need be given for Vbecause it is dimensionless. The whether or not the bicycle-seat harness is applying a vervalues for klepin the theoretical curves of Figs. 6 and 7 tical force to his pelvis. The same may be said, with perwere chose&o be near the respective experimental haps greater consequences, about swinging motions of means for G = 1.0 and u = 3.0 m/s in each case. The the legs.

1,;.v*,

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868

RUNNING

UNDER

SIMULATED

If the body really were a point mass, as assumed in the mathematical model, then applying a vertical force to the pelvis would be exactly equivalent to applying vertical forces to all body segments. Thus agreement between theory and experiment in this paper is evidence not only for the appropriateness of the theory but for the correctness of the experimental assumption that the motions of the body segments with respect to the center of mass are of secondary importance in the dynamics of running. Physiological significance. One fundamental conclusion is that ilZlegworking as a linear spring changes very little, even though running speed may triple (Fig. 6) or gravity may decrease to only one-fifth of normal (Fig. 7). The conclusion applies only to moderate and high leg forces: Fig. 7A shows that the spring stiffness of the leg declines slightly when the average force on the leg falls below about one-third of its normal-gravity level. Unlike kleg, k vert increases substantially with speed at constant G (Fig. 6A) and increases only slightly, if at all, with reduced gravity at constant speed (Fig. 7A). Why should kleg stay fixed? Alexander (1) has suggested that when the length of a tendon is much longer than the length of the muscle fibers in series with the tendon, the combined stiffness is dominated by the stiffness of the tendon, which would not vary with running speed. In experiments using decerebrate cats, the stiffness of the soleus muscle including reflexive components was found to vary only slightly with force at moderate and high force levels, although the stiffness fell sharply with decreasing force at low force levels (7). When human subjects bore weights on their shoulders, it was found that the stiffness of the antigravity muscles of the legs increased by 40% as the weight on the shoulders was increased from zero to more than twice body weight (5). Recently reported measurements of the length (from tine films) and force (using implanted tendon buckle transducers) in the medial gastrocnemius muscles of freely hopping wallabies showed that the effective spring stiffness of the muscle-plus-tendon combination changed little with hopping speed (6). All these studies lend support to the idea that the mechanical properties of the leg, including effects due to tendons, muscles, and reflexes, may be represented by an equivalent linear spring. Here we report not only experimental results but the outcome of calculations investigating the fundamental physical coupling among speed, gravity, and stiffness. McMahon and Cheng (9) showed that when a model for running is based on the idea that the leg is a spring of constant stiffness, faster speeds are achieved by landing with a larger 8, (Fig. 6B), which compresses the leg spring more and gives rise to a larger peak force. Although the peak force is larger at higher speeds, the vertical deflection of the center of mass (Ay in Fig. 3) is, if changed at all, smaller. Hence the vertical stiffness at higher speeds, being the ratio of the vertical force to Ay, is larger. This is true despite the basic assumption that the leg-spring stiffness does not change with speed. Another general conclusion of possible physiological significance is that the dimensionless vertical landing velocity V = u,l( GgZo)1’2 has a constant value near 0.25, independent of u and G. Because the vertical rise of the

LOW

GRAVITY

center of mass during the flight phase is Ah = u2/2Gg = (0.25)2GgZo/2Gg = 0.03125 I,, we may conclude that the center of mass rises a distance equal to -3% of the leg length in each flight phase. Our most significant conclusion, that leg-spring stiffness is independent of speed and leg force in running, has broad physiological relevance. It even has a potential application. An artificial leg suitable for running need have only one stiffness, rather than a range of stiffnesses for different speeds. For this reason, it could be fabricated quite simply. Summary of the findings. Here we have examined the hypothesis that the springlike properties of muscular action observed in animals running at normal gravity are still present under conditions simulating reduced gravity, when muscle forces may be as little as one-fifth of normal. We have confirmed this hypothesis and found that an essential property, the spring stiffness of the leg, is almost invariant across a threefold change in speed and a fivefold change in gravity. APPENDIX

Validation

A of

Experimental

Methods

In applying Eq. 1 to calculate the vertical velocity and displacement of the center of mass from force-plate measurements, we assumed that f,, the vertical force applied to the subject’s body by the harness, was constant. In fact, the finite compliance of the steel springs and friction throughout the spring-and-cable system give rise to small fluctuations in f, during a running stride cycle. In a series of tests, we included a force transducer (Kistler model 9203, Amherst, NY) in series with the cable at point C in Fig. 1. When the tension in the cable was equal to half of body weight for subject RK, the amplitude of the fluctuations was +6.5% of the mean cable tension. Under conditions stimulating 0.7 and 0.2 G, the fluctuations were 9.5 and 2.6% of the mean cable tension, respectively. The following is a short analysis estimating the errors this fluctuation causes in our measurements of u0 and kveti. Vertical ground reaction force. A good approximation to the vertical ground reaction force measured during the contact period as a subject runs at constant speed across a force platform is given by (10) f(t) = mgG[l

- cos w,t + (q&gG)

sin w,t]

(Al)

When an average subject runs at a speed near 3.0 m/s under normal gravity, the Groucho number w,u,lg is near unity (10). In the remainder of the analysis, we assume w,u,lgG = 1.0. Equation AI describes a force that begins at zero, rises to a maximum of 2.414times body weight at w,t = 3d4, and falls to zero at o,t = 3~/2. An aerial phase of duration w,t, = 7r/2 (where ta is time in the air) follows before the next foot strike. Fluctuations in cable tension. We assume that the variations in cable tension can be described by f,lmg

= 1 - G + A sin o,t + B cos o,t

(fw

where A and B are two constants obtained from a running experiment conducted at a particular G. In general, A and B change as G changes. Putting Eqs. AI and A2 into Eq. 1 and solving for the vertical acceleration gives d2yldt2 = g(B - G) cos o,t + g(A + G) sin w,t

(A3)

Vertical velocity and displacement. Integrating once, letting o,t = 0, and applying the condition o,u,/gG = 1, we obtain an expression for the vertical velocity

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RUNNING V =

-Wq-J(B

-

sin o,t + (A + G) cos o,t]

(3

so that the velocity on foot contact

wovO/gG = 1 and y(0) = yO,

- G) cos w,t + (A + G) sin w,t]

= -W:W

Y-Y0

The peak displacement (y,,) and solving for ~~~~~~~~~~~

(A6)

occurs when v = 0. Using Eq. A4

[(A + G)I(B - G)]

= tan-’

u04naxforce

(A4)

(A5)

+ G)

Eq. A3 twice, applying

SIMULATED

is

v, = -WOW Integrating we have

UNDER

(A7)

869

LOW GRAVITY

errors in the calculated vertical landing velocity and peak vertical displacement. These errors are expected to be generally

Mechanics of running under simulated low gravity.

Using a linear mass-spring model of the body and leg (T. A. McMahon and G. C. Cheng. J. Biomech. 23: 65-78, 1990), we present experimental observation...
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