Effect of an increase in gravity on the power output and the rebound of the body in human running
1 Istituto di Fisiologia Umana, Università degli Studi di Milano,
20133 Milan, Italy
2 Unité de Physiologie et Biomécanique de la Locomotion,
Université catholique de Louvain, 1348 Louvain-la-Neuve,
Belgium
* Author for correspondence (e-mail: giovanni.cavagna{at}unimi.it)
Accepted 25 April 2005
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: locomotion, running, gravity, human
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The waste of energy during legged terrestrial locomotion is reduced by two
basic mechanisms: the pendular mechanism of walking and the bouncing mechanism
of running. In walking, the kinetic energy of forward motion is stored in part
as gravitational potential energy when the point of contact with the ground is
in front of the centre of mass: the body is lifted while it decelerates
forwards. The process is reversed when the point of contact is behind the
centre of mass with a transformation of potential energy back into kinetic
energy of forward motion. Gravitational potential energy and kinetic energy of
forward motion change in opposition of phase during a walking step
(Cavagna et al., 1963).
The same energy conserving mechanism is not possible in running because the
centre of mass is lowered while decelerating forwards and lifted while
accelerating forwards. Gravitational potential energy and kinetic energy of
forward motion change in-phase during the running step
(Cavagna et al., 1964). With
each step the muscletendon units must absorb and restore both the
kinetic energy change of forward motion, due to the braking action of the
ground, and the gravitational potential energy change, associated with the
fall and the lift of the centre of mass. This results in a large amount of
negative and positive work and the chemical energy cost per unit distance is
twice that spent in walking at the optimal speed
(Margaria, 1938
).
The metabolic energy expenditure is reduced in running, however, by an
elastic storage and recovery of mechanical energy, as in a bouncing ball. This
was initially suggested by the finding that in human running the ratio between
mechanical power output and metabolic energy expenditure exceeded the maximum
efficiency of transformation of chemical energy into mechanical work
(Cavagna et al., 1964).
Evidence for an elastic storage and recovery was also found in the kangaroo by
Alexander and Vernon (1975
) and
in the horse by Biewener
(1998
). A springmass
model of the bounce of the body at each running step
(Blickhan, 1989
;
McMahon and Cheng, 1990
;
Seyfarth et al., 2002
) is now
widely used in studies on the effect of the spring stiffness on energy
expenditure (McMahon et al.,
1987
; Kerdok et al.,
2002
) as well as the changes in spring stiffness and step
frequency with speed (Cavagna et al.,
1988
; McMahon and Cheng,
1990
) and with surfaces of different stiffness
(Ferris and Farley, 1997
;
Ferris et al., 1998
;
1999
;
Kerdok et al., 2002
).
As expected, the pendular mechanism of walking is drastically affected by
gravity. The range of walking speeds, optimal walking speed and external work
done to move the centre of mass are all increased by gravity
(Cavagna et al., 2000).
Walking on the Moon is practically not possible
(Margaria and Cavagna, 1964
):
in fact, locomotion takes place by a succession of bounces with a mechanism
similar to that of running on Earth. The effect of gravity on the mechanism of
running still needs investigation. Suspending the body from springs results in
a reduction of the peak force attained at each step during running, with
little change of the stiffness of the leg
(He et al., 1991
). No data
exist on the effect of an increase in gravity on the mechanics of running.
One purpose of the present study was to analyze the effect of an increase
in gravity on the characteristics of the elastic bounce of the body. The
vertical oscillation of the centre of mass during each running step can be
divided into two parts: one taking place when the vertical force exerted on
the ground is greater than body weight (lower part of the oscillation) and
another when this force is smaller than body weight (upper part of the
oscillation; Fig. 1). According
to the springmass model, the duration of the lower part of the
oscillation represents half-period of the bouncing system and the vertical
displacement of the centre of mass during this period represents the amplitude
of the oscillation (Cavagna et al.,
1988; Blickhan,
1989
; McMahon and Cheng,
1990
). The upper part of the oscillation, by contrast, may be
compared with the second half-period only in the absence of an aerial phase,
as often happens during trotting and very slow running
(Cavagna et al., 1988
)
(Fig. 1). On Earth, the rebound
is symmetric, i.e. the duration and the amplitude of the lower part of the
oscillation are about equal to those of the upper part up to a speed of
11 km h1 in both adult humans and children
(Cavagna et al., 1988
;
Schepens et al., 1998
). It
follows that up to this critical speed the step frequency equals the frequency
of the bouncing system. Above the critical speed the rebound becomes
asymmetric, i.e. the duration and the amplitude of the upper part of the
oscillation become greater than those of the lower part, and the step
frequency is lower than the frequency of the system. The asymmetry arises
because the average vertical acceleration upwards during the lower part of the
oscillation becomes greater than 1 g, whereas during the upper
part of the oscillation the average vertical acceleration downwards cannot
exceed 1 g. It was therefore suggested that the critical speed
may depend on gravity and that at higher gravity values this speed would
probably be greater (Schepens et al.,
1998
). The aim of the present study was therefore to determine
experimentally if and to what extent this hypothesis is true. Analysis of the
rebound of the body may also help to determine how an increase in gravity
affects the stiffness of the bouncing system and the preferred combination
between stiffness and landing angle of attack
(Seyfarth et al., 2002
).
|
In order to answer the questions outlined above, the motion of the centre of mass of the body during running at different speeds was analyzed in this study on Earth and on an aeroplane undergoing flight profiles, resulting in a simulated gravity of 1.3 g. The effect of the increase in gravity on the stiffness of the bouncing system and on the work done to move the centre of mass in a sagittal plane was measured.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Subjects
The experiments were made on five adult subjects, four males: A, 33 years,
6872 kg, 1.74 m height, leg length (hip to floor) 0.92 m; B, 68 years,
81 kg, 1.79 m height, leg length 0.94 m; C, 53 years, 8791 kg, 1.93 m
height, leg length 1.03 m; D, 46 years, 87 kg, 1.79 m height, leg length 0.93
m; and one female E, 25 years, 52 kg, 1.66 m height, leg length 0.87 m.
Informed consent was obtained from each subject. The studies were performed
according to the Declaration of Helsinki. The European Space Agency Safety
Committee approved the procedures in the experiments made on the
aeroplane.
Experiments at 1.3 g
Experiments were performed during the 32nd European Space Agency parabolic
flight campaign. A simulated gravity of 1.3 g was attained
during turns of an A300 Airbus. The experiments were done over 3 days with a
total of 23 turns. An aircraft orthogonal frame of reference was defined as
follows: the X-axis is parallel to the foreaft axis of the
aeroplane, the Y-axis is parallel to the lateral axis of the
aeroplane and the Z-axis is perpendicular to the floor of the
aeroplane. Three accelerometers (DS Europe, Milan, Italy) measured
simultaneously the X, Y and Z components of the acceleration
vector in the aircraft reference frame. In order to reduce the noise induced
by the aircraft vibrations (Fig.
2), the accelerometers were both mechanically and electrically
damped by a low-pass filter with a cut-off frequency at 5 Hz. In this study,
results are given as means ± S.D. During the time intervals
encompassing the 175 steps analysed, the acceleration was 12.90±0.51 m
s2 in the Z-direction, 0.03±0.06 m
s2 in the X-direction and 0.13±0.16 m
s2 in the Y-direction. The X and
Y-axis accelerations of the frame of reference were neglected and the
acceleration along the Z-axis of the frame of reference was
considered to be equivalent to the vertical on Earth. Therefore in the
following text, forward (or foreaft), lateral, and vertical refer to
the X, Y and Z-axes, respectively.
|
During each turn, the subjects ran at different speeds back and forth
across a 6.1 mx0.4 m force platform fixed to the aeroplane floor along
the X-axis. The platform
(Heglund, 1981) was sensitive
to the force exerted by the feet in the forward (X) and vertical
(Z) directions; lateral (Y) forces were neglected. The
lowest frequency mode of vibration for the unloaded platform was greater than
180 Hz. With each turn of the aircraft, the simulated gravity was maintained
for 4060 s: during this period the subjects could run several times
back and forth on the platform, one after the other. Two handrails, fixed on
each side of the platform, proved to be useful in case the subject lost
balance. Two photocells fixed 5.87 m apart at neck height along the side of
the platform were used to determine the average running speed,
f. Two additional
photocells, 1.93 m from the first and last, were used to detect rough
variations in speed between the first and the last photocells. Two video
cameras were used to check for obvious loss in balance or touching the
handrails. Before stepping on the platform, the subjects had, in one
direction, 12 m to accelerate, the last 6 m of which were at the same level of
the platform; the corresponding figures in the other direction were 8 and 4.8
m.
Data were gathered as follows. The five subjects made 309 runs, of which 175 were used for analysis. The other runs were unusable because the subject was accelerating or decelerating forward (indicated by a continuous velocity change greater than 0.3 m s1 from start to end of the run), lost balance, made irregular steps, the platform signals were out of scale, records between photocells were incomplete or the Z, Y and X acceleration were not steady due to turbulence or other factor. In particular, of the 175 runs analysed: subject A had 50 runs with 61 steps analysed (speed range: 5.820.5 km h1); B had 20 runs with 31 steps analysed (6.013.6 km h1); C had 12 runs with 16 steps analysed (7.511.2 km h1), D had 47 runs with 61 steps analysed (5.515.5 km h1); and E had 46 runs with 52 steps analysed (5.215.3 km h1).
A microcomputer was used at a sampling rate of 500 Hz to acquire (i) the platform signals, proportional to the force exerted by the feet in forward direction (Ff, along the X-axis of the aeroplane) and in vertical direction (Fv, along the Z-axis of the aeroplane), (ii) the output of photocell circuit, and (iii) the output of the accelerometers (Fig. 2). No subject suffered motion sickness. Subjects experienced a normal coordination of movements since the first runs at 1.3 g, but at an evidently greater effort; any instability tended to quickly result in a loss of balance due to the greater vertical acceleration.
Experiments at 1 g
Data were gathered during running in the laboratory of Louvain-la-Neuve
with the same platform used on the aeroplane (photocell distance 34.5 m
according to the running speed) and in the laboratory of Milan with the
platform described by Cavagna
(1975; photocells distance 3
m). The two platforms gave consistent results. Data were collected for the
same range of speeds obtained by each subject on the aeroplane (520 km
h1). Subject A had 32 runs with 45 steps analysed (made
before the flight, sampling frequency 500 Hz); B had 20 runs with 23 steps
analysed (three before the flight, 250500 Hz, the others after the
flight, 500 Hz); C had 14 runs with 25 steps analysed (after the flight, 500
Hz), E had 25 runs with 36 steps analysed (after the flight, 200400
Hz); D had 29 runs with 47 steps analysed (23 before the flight and 6 after,
500 Hz). Experimental records obtained at 1 g and at 1.3
g are given in Fig.
2.
Analysis of force platform records
A custom LabVIEW (6.1) software program was used to analyze the ground
reaction force records. The average of the first 50 points of the unloaded
platform base lines was measured in the short time interval between runs and
subtracted from the entire Ff and Fv
arrays to get the net changes of Ff and
Fv during each run. In case of the 1.3 g
experiments, the Ff and Fv records
where also corrected at each instant for the changes in acceleration during
the aircraft manoeuvres. This was made by subtracting the product of the
changes in the acceleration measured by the accelerometers along the
Z and X-axis times the mass of the suspended part of the
platform.
|
An integer number of running steps, encompassing those subsequently used
for calculation, was selected between peaks of Fv. The
time-average of the Fv record over an integer number of
cycles should equal body weight. In reality the ratio between this
time-average and the weight of the subject (body mass x 1.0
g for the Earth experiments and 1.3 g for
the aeroplane experiments) was 1.00±0.02 (N=120) in the
experiments at 1 g and 1.05±0.05 (N=175) at
1.3 g.
The velocity of the centre of mass of the body in the vertical direction
Vv and in the forward direction Vf was
determined by integration of the Fv and
Ff tracings (Cavagna,
1975). Only translational motion in a sagittal plane was
considered when calculating the mechanical energy of the centre of mass.
Lateral movements were ignored.
The instantaneous vertical velocity Vv(t) was used to calculate the instantaneous kinetic energy of vertical motion Ekv(t)=0.5MbVv(t)2 and, by integration, the vertical displacement of the centre of mass, Sv(t), with the corresponding gravitational potential energy change Ep(t)=MbgSv(t). The kinetic energy of forward motion was calculated as Ekf(t)=0.5MbVf(t)2, the total translational kinetic energy of the centre of mass in the sagittal plane as Ek(t)=Ekf(t)+Ekv(t), and the translational mechanical energy of the centre of mass in the sagittal plane as Ecm(t)=Ekv(t)+Ekf(t)+Ep(t) (Fig. 3).
Since this study refers to steady running at a constant step-average speed,
an integer number of running cycles was selected between two
Vv peaks (or valleys), searching by eye for a minimum
difference between the beginning and the end of both the
Vv and Vf records. The time-average
vertical velocity in the selected integral number of Vv
cycles was calculated and considered to be zero on the assumption that the
upward vertical displacement was equal to the downward vertical displacement
(Cavagna, 1975). This would
only be true if successive steps were exactly equal to each other. An attempt
to quantify the consequences of this assumption is described below.
The work done at each step to move the centre of mass in the sagittal plane was measured in the interval included between two or more peaks (or valleys) of the gravitational potential energy, Ep. Since, as mentioned above, selection was initially made between peaks (or valleys) of the vertical velocity, the record was expanded to include the previous valley (or peak) of Ep(t) until a clear picture of the selected steps was obtained (Fig. 3). The work done during the selected steps, Wv, Wk, Wkf and Wext, was calculated from the amplitudes of peaks and valleys and the initial and final values in the Ep(t), Ek(t), Ekf(t) and Ecm(t) records. Positive values of the energy changes gave positive work, negative values gave negative work. In a perfectly steady run on the level the ratio between positive and negative work should be equal to one. In reality the ratios were as follows. In the 1 g experiments (N=120): W+v/Wv=1.00±0.04, W+k/Wk= 1.01±0.06, W+kf/Wkf=1.02±0.10, W+ext/Wext= 1.01±0.05. In the 1.3 g experiments (N=175): W+v/Wv=1.01±0.10, W+k/Wk= 1.01±0.10, W+kf/Wkf=1.02±0.16, W+ext/Wext=1.01±0.09.
The error involved by the assumption that lifts equal falls in an integer number of steps was estimated from the difference in amplitude between the first and last of the selected Vv(t) peaks (or valleys), as if this difference were due to a drift of the whole Vv(t) record. Accordingly, the absolute value of the difference between Vv(t) peaks (or valleys) was multiplied by the time interval between them and divided by 2 to get the vertical displacement due to the hypothetical drift. This was then expressed as a fraction of the sum of the upward vertical displacements calculated during the same time interval: 0.08±0.07 at 1 g (N=120) and 0.15±0.12 at 1.3 g (N=175). These figures correspond to a random error less than 1% in the measured values of Ep (as found by simulating the Vv drift over one step with a sine wave).
Aerial time and vertical displacement during contact
Since the mechanical energy of the centre of mass is constant when the body
is airborne (air resistance is neglected), the aerial time was calculated as
the time interval during which the derivative
dEcm(t)/dt=0. This time interval was
measured using two reference levels set by the user above and below the
section of the record where
dEcm(t)/dt0. When vibrations or a
peak of Ecm at the end of the aerial phase (at high
speeds) disturbed this measurement, the aerial time could also be determined
by eye on the basis of the duration of the plateau of the
Ecm(t) record
(Fig. 3, after appropriate
expansion of the tracing): this procedure, however, was followed in only eight
of the 295 runs analyzed. The upward and downward displacement of the centre
of mass during contact was measured from the section of the
Ep(t) curve during which
dEcm(t)/dt
0 and
dEp(t)/dt was positive and negative,
respectively.
`Effective' contact and aerial times, vertical and forward displacements
As described in the Introduction, the springmass model can be
applied to the vertical displacement of the centre of mass taking place at
each running step provided that the period and the amplitude of the
oscillation of the springmass system are correctly measured. In all
conditions the half period of the oscillation equals the time interval where
the centre of mass decelerates downwards and accelerates upwards, i.e. the
time interval during which the vertical force is greater than body weight.
This time interval is called effective contact time, tce,
and is shorter than the total time of contact
(Fig. 1). The time interval
where the centre of mass decelerates upwards and accelerates downwards, i.e.
when the vertical force is less than body weight, is called effective aerial
time, tae: it occurs both during contact and during the
aerial phase and does not necessarily correspond to the other half period of
the oscillation. The amplitude of the vertical oscillation, i.e. the
compression of the spring from its equilibrium position, equals in all
conditions the vertical displacement Sce attained during
tce. The displacement upward, Sae,
attained during tae corresponds to the amplitude of the
vertical oscillation only in the absence of an aerial phase
(Cavagna et al, 1988). In the
springmass model, but not in the actual running step (see below), both
Sce and Sae are assumed to be equal
during the lift and the fall of the centre of mass.
|
|
The total vertical displacement of the centre of mass at each step (Fig. 4) was calculated as the average between the total vertical displacement during the lift and that during the fall, i.e. Sv=(Sv,up+Sv,down)/2.
The forward displacements of the centre of mass during
tce and tae were labeled
Lce and Lae
(Fig. 4) and were calculated
from the time integral of the instantaneous forward velocity of the centre of
mass Vf(t) during the corresponding times. These
measures are more precise than those hitherto made using the average forward
speed f
(Cavagna et al., 1988
) because
they take into account the difference in forward speed between the lower and
the upper part of the vertical oscillation.
Vertical stiffness
The vertical stiffness (kvert in
Fig. 5) was calculated from the
effective contact time tce on the assumption that this
time represents the half-period of oscillation of the elastic system:
kvert=Mb(/tce)2,
where Mb is the body mass of each subject.
Correspondingly, the resonant frequency of the bouncing system was calculated
as fs=1/(2tce)
(Fig. 5). The mass-specific
vertical stiffness, kvert/Mb, was also
calculated from the slope of a graph obtained by plotting the vertical
acceleration of the centre of mass av as a function of the
simultaneous vertical displacement of the centre of mass
Sv during the effective contact time (positive values of
av), as described by Cavagna et al.
(1988
). This method, however,
was found to be drastically affected by the oscillations of
av after foot contact, particularly at high speeds and at
1.3 g. In fact, the mass-specific vertical stiffness,
calculated in some of the records from the maximum compression of the spring
as av,mx/Sce (see Discussion)
approaches that calculated from the effective contact time
tce and is higher, particular in the presence of large
oscillations, than that measured from the average slope of the
avSv plot.
Leg stiffness
McMahon and Cheng (1990)
pointed out that the actual compression of the hypothetical spring over which
the body bounces each step includes not only the lowering of the centre of
mass after foot contact, but also the amplitude of the arc made by the
extended spring in its rotation during contact. These authors calculated leg
stiffness as the ratio between maximal vertical force attained during contact
and length change of the leg taking place during the contact time
tc. This leg change extends beyond the amplitude of the
oscillation of the springmass system and includes the beginning and the
end of `spring' loading, when the loadextension curve is often not
linear. In this study, leg stiffness kleg is calculated
within the amplitude of the oscillation, i.e. during the effective contact
time tce, as the ratio between the increment of the
vertical force above body weight
(Mbav,max) and the total length change
of the hypothetical leg-spring during this time
(Sce+
Y). The amplitude of the arc made by
the extended spring in its rotation during tce was
calculated as
Y=l(1cos
), where
=
arcsine(Lce/2l) and l is the hip-ground
distance of each subject (0.871.03 m).
Statistics
The data collected as a function of running speed were grouped into 1 km
h1 intervals as follows: 5 to <6 km h1,
6 to <7 km h1,.., 19 to <20 km h1
and 2020.5 km h1 at 1 g; and 5 to
<6 km h1, 6 to <7 km h1,.., 17 to
<18 and 19 to <20.5 km h1 at 1.3 g (no
data in the 18 to <19 km h1 range at 1.3
g). The data points in the figures represent the mean ±
S.D. in each of the above speed intervals and the figures near the
symbols in Fig. 4A,D give the
number of items in the mean. Given the limitations imposed by the experimental
conditions, a different number of subjects and of steps analyzed contributed
to the mean values reported at each speed. Comparison between groups of data
at each speed (Tables 1 and
2) was made using a two-factors
analysis of variance (ANOVA) with contrast (SuperANOVA version 1.11). The
contrast analysis was not made for the four highest speed groups (>16 km
h1) because, both at 1 g and at 1.3
g, the number of items was too small (see
Fig. 4A,D). For clarity, in
some of the figures lines are drawn through all of the data using Kaleidagraph
3.6.4 linear or weighted fits, as indicated in the legend of each figure. The
only purpose of these lines is to be a guide for the eye; they do not describe
the underlying physical mechanism.
|
|
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Dynamics of the rebound
In what follows an average is made of the upward and downward
displacements, and corresponding time intervals, in the lower and upper half
of the vertical oscillation of the centre of mass. This procedure was followed
by Cavagna et al. (1988) and
is the basis of the springmass model
(Blickhan, 1989
;
McMahon and Cheng, 1990
),
which assumes that during the running step the landing and take-off conditions
are the same.
The step period (, Fig.
4A,D), the vertical displacement of the centre of gravity during
each step (Sv, Fig.
4B,E) and the step length (L,
Fig. 4C,F) are given as a
function of running speed
(
f) at 1 g
and 1.3 g. As proposed by Cavagna et al.
(1988
),
,
Sv and L are divided into two parts,
corresponding to the lower and upper parts of the vertical oscillation of the
centre of mass: a lower part taking place when the vertical force is greater
than the body weight (tce, Sce and
Lce; Fig.
4, red squares), and an upper part taking place when the vertical
force is smaller than body weight (tae,
Sae and Lae;
Fig. 4, blue triangles). For
comparison,
, Sv and L have also been
divided, according to tradition, into parts occurring during the ground
contact phase and the aerial phase (Fig.
4, dotted lines).
It can be seen that tae=tce, and
Lae=Lce in a range of lower speeds at
1 g (<10 km h1) and of higher speeds at
1.3 g (>11 km h1; see
Table 1). As in previous
studies (Schepens et al.,
1998), this trend is not so clear for Sae and
Sce due to the larger scatter of the data. In addition,
, Sv and L are on average smaller at 1.3
g than at 1 g, mainly due to their reduction
in the phase of the step where the vertical force is less than body weight,
i.e. tae, Sae and
Lae are on average smaller than tce,
Sce and Lce at 1.3 g
(Table 2).
The whole body vertical stiffness kvert, the leg
stiffness kleg, the natural frequency of the bouncing
system fs and the freely chosen step frequency f
are given in Fig. 5 as a
function of running speed. Fig.
5A,C show the vertical stiffness and the leg stiffness. The
vertical stiffness increases linearly with speed whereas the leg stiffness is
about constant independent of speed. This is in agreement with the
springmass model predictions and experimental results reported in the
literature (McMahon and Cheng,
1990; Farley et al,
1993
). It can be seen that gravity tends to increase stiffness,
but its effect is not significant at intermediate running speeds
(Fig. 5C and
Table 2).
In Fig. 5B,D a comparison is
made between the step frequency f and the natural frequency of the
bouncing system, fs. At 1 g the overlap
between f and fs occurs at speeds (<11 km
h1) lower than the speeds (>10 km h1)
where the overlap occurs at 1.3 g
(Table 1). As reported in
previous studies (Cavagna et al.,
1988; Schepens et al.,
1998
), the natural frequency of the bouncing system exceeds the
step frequency beyond 10 km h1 at 1 g
(fs>f, see
Fig. 5 and
Table 1). The maximum speed
where f=fs is greater at 1.3 g.
Data in Figs 4,
5 and
Table 1 also suggest an
asymmetry in the opposite direction at low speeds
(tce>tae and
f>fs), particularly at 1.3 g;
this will be discussed below.
If the frequency of the bouncing system is calculated from the total
contact time as fc=1/(2tc) or from the
leg stiffness as
fk,leg=(klego/Mb)0.5/(2)
(Fig. 5B,D, continuous lines),
a large discrepancy is found with f. This indicates that the vertical
stiffness only is related to step frequency.
Work
The step-average positive external power to move the centre of mass of the
body in the sagittal plane is plotted as a function of the running speed in
Fig. 6A (1 g)
and Fig. 6C (1.3
g). Fig. 6B,D give the corresponding positive work done per unit distance. The results
obtained at 1 g (Fig.
6A,B) are in good agreement with those obtained previously (e.g.
Cavagna et al., 1976;
Schepens et al., 1998
). The
external power,
ext, seems
to increase linearly with speed. Since the gravitational potential energy
curve and the kinetic energy curve of forward motion are nearly in-phase
during the running step, the external power is practically equal to the sum of
the power output due to the kinetic energy changes of forward motion
f and the vertical lift
against gravity
v, i.e.
ext
f+
v.
The power spent to sustain the forward velocity changes
f increases with speed,
whereas the power spent against gravity
v attains a plateau at
11 km h1. The external work done per unit distance,
xt/
f,
is about constant at speeds greater than 10 km h1, due to
mirror changes of
f/
f
and
v/
f,
whereas it increases as the speed is reduced below 10 km h1
due to the positive intercept of
ext=f(
f)
linear relation.
|
A 1.3x increase in gravity causes a 1.3x increase of the
external power
ext and its
components
f and
v. This is shown by the
similarity of the experimental data in Fig.
6C, with the crosses obtained by multiplying the mean experimental
values of Fig. 6A (1
g) x1.3. The linear relationship between external power
and speed is retained but power increases more steeply with speed from a
smaller intercept at 1.3 g. Given the small intercept, the
slope approaches the work done per unit distance
(Fig. 6D).
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
What is the physiological significance of attaining equivalent step
frequencies and resonant bouncing frequencies? Runners maintain spring-like
mechanics over a wide range of speeds independent of the symmetry of the
rebound (Blickhan, 1989;
McMahon and Cheng, 1990
;
Seyfarth et al., 2002
;
Farley and Gonzalez, 1996
).
Furthermore, a mismatch of the resonant and step frequencies does not appear
to result in greater work required to maintain the motion of the center of
gravity (Fig. 6). However,
energy expenditure, rather than mechanical work, may be considered in this
respect. It is possible that the energy expenditure required to maintain the
oscillation will be smaller when the rebound is symmetric than when it is
asymmetric. When the rebound is symmetric the system is activated to bounce at
a frequency equal to its resonant frequency. The utility to adopt a step
frequency equal to the resonant frequency of the bouncing system is suggested
by the finding that at running speeds less than about 13 km
h1, an increase in step frequency above the freely chosen
step frequency increases the energy expenditure, despite a decrease in
mechanical power (Cavagna et al.,
1997
).
At low running speeds the increase in gravity often causes an asymmetry in the opposite direction, with tce>tae and Sce>Sae. This condition is also observed at 1 g in some subjects, particularly at low speeds, but is enhanced by an increase in gravity in all subjects (Fig. 4 and Table 1). The reversed asymmetry is due to the fact that the vertical stiffness when the vertical force is greater than body weight is, in some conditions, smaller than the vertical stiffness when the vertical force is less than body weight and the foot is still in contact with the ground. This is shown in Fig. 7, where the vertical acceleration of the centre of mass is plotted as a function of its vertical displacement for different running speeds.
|
|
A vertical stiffness during
SaeSa, when the vertical force
is lower than body weight, greater than that during Sce,
when the force is higher than body weight, does not reflect the
characteristics of the elastic structures over which the body may possibly
bounce. In fact, it has been shown both in vitro and in situ
that the muscletendon stiffness increases with load
(Hill, 1950;
Cavagna, 1970
;
Ker et al., 1987
). The finding
that kvert,aea>kvert,ce is
simply due to the fact that body weight minus the upward push results in a
restoring force, per unit of deformation of the spring, directed downward
during taeta, which is greater
than that directed upward during tce. In these subjects,
gravity during tae is more effective than the upward push
of the elastic system during tce in absorbing and
restoring the kinetic energy of vertical motion. Since the vertical momentum
lost and gained during tce must equal the vertical
momentum lost and gained during tae, a stiffer `spring'
during taeta will contribute to
make tae<tce
(Fig. 4) and, as a consequence,
the step frequency,
f=1/(tce+tae), greater than
the natural frequency of the bouncing system,
fs=1/(2tce). This was in fact found
during some of the runs at low speeds, particularly at 1.3 g
(Fig. 5 and
Table 1).
At low running speeds (<10 km h1), subjects
characterized by tcetae and
Sce
Sae show a smaller angle swept
by the leg during tce, a smaller Sv
and a higher mass-specific vertical stiffness than subjects characterized by
tce>tae and
Sce>Sae. Seyfarth et al.
(2002
) showed that different
running strategies are compatible with stable running as predicted by a
springmass system: `either stiff legs with steep angles of attack or
more compliant legs with flatter angles' (a steep angle of attack corresponds
to a small angle swept by the leg during tce). The first
strategy implies a smaller work done against gravity at each step, but a
higher step frequency, with the result that the average vertical power is
similar to that spent during the `softer' running of the second strategy with
Sce>Sae (this was in fact measured
comparing subject A with subject B, data not shown).
Effect of gravity on work
As shown in Fig. 6, a
1.3x increase in gravity results in an 1.3x increase in the
work done per unit distance against gravity and to sustain the forward speed
changes. An increase in the work done against gravity is to be expected. The
finding, however, that this increase is proportional to gravity gives the
additional information that the average vertical displacement of the centre of
mass per unit distance is the same at 1 g as at 1.3
g. In other words, the sum of all the vertical lifts made in 1
km is the same and is independent of gravity. Since the step frequency is
increased by gravity (Fig. 8
and Table 2), the vertical lift
per step must be smaller at 1.3 g
(Fig. 4), but the average lift
per unit distance must be the same.
Less obvious is an increase with gravity of the work done per unit distance
to sustain the forward speed changes. In fact kinetic energy of forward motion
does not contain a gravity component. One possible explanation is given by a
simplified model worked out by Alexander (Modelling step by step;
http://plus.maths.org/issue13/features/walking/).
Considering that the average vertical force over the step period (contact
phase plus aerial phase) must equal body weight, Alexander derived an equation
where the work done per unit distance to sustain the forward speed changes of
the centre of mass is proportional to gravity and to the tangent of the angle
made by the leg with the vertical (leg is assumed as a rigid rod connecting
point of contact on the ground with the centre of mass). Alexander's approach
is based on the assumption that at each running step the direction of the
resultant force exerted on the ground equals that of the link between centre
of mass and point of contact on the ground. It follows that an increase in the
vertical component of the force due to an increase in gravity, as in the
present experiments, must imply an increase in the horizontal component as
well. This is in agreement with the experimental results of Chang et al.
(2000) showing that gravity
affects both vertical and horizontal forces generated against the ground
during running so that the orientation of the resultant vector remains aligned
with the leg. This strategy, minimizing muscle forces, was first described by
Biewener (1989
). The finding
that a 1.3x increase in gravity results in an
1.3x increase
in the work done to sustain the forward speed changes
(Fig. 6) is in agreement with
Alexander's equation, and indicates that at any given speed the average
direction of the push is independent of gravity.
All together the above information suggest that, in running, a similar
centre of mass motion tends to be maintained when gravity is increased as well
as when stiffness of the ground is changed
(Ferris and Farley, 1997;
Ferris et al., 1998
,
1999
;
Kerdok et al., 2002
). The
similarity here refers to the average vertical displacement per unit distance
and the direction of the push, which are maintained in spite of an increase in
step frequency.
The increase in external mechanical power induced by the increase in
gravity is due to an increase in step frequency more than to an increase of
the external work done at each step (Fig.
8, Table 2). The
increase in step frequency in turn is explained to a lesser extent by an
increase in the stiffness of the bouncing system
(Fig. 5 and
kvert in Table
2), and to a greater extent by a decrease of the effective aerial
time (Fig. 4 and
tae in Table
2). He et al.
(1991) found that the leg
stiffness adopted at 1 g was retained when a lower gravity was
simulated by suspension with springs.
The finding that the increase in step frequency
f=1/=1/(tce+tae), is not
explained by an increase in the natural frequency of the bouncing system
fs=1/(2tce), but is mainly due to a
shorter duration of the upper half of the oscillation tae,
implies that the ratio tce/
is greater at 1.3
g. Since the average vertical force over the step period must
equal body weight, a greater fraction tce/
has the
beneficial effect of reducing the fraction of the step during which the
vertical force exerted on the ground is greater than body weight. On the other
hand the greater step frequency at 1.3 g must increase the
internal power spent to accelerate the limbs relative to the centre of mass
(Cavagna et al., 1991
).
List of symbols
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Alexander, R. McN. and Vernon, A. (1975). Mechanics of hopping by kangaroos (Macropodidae). J. Zool. Lond. 177,265 -303.
Biewener, A. A. (1989). Scaling body support in mammals: limb posture and muscle mechanics. Science 245, 45-48.[Medline]
Biewener, A. A. (1998). Muscle-tendon stresses and elastic storage during locomotion in the horse. Comp. Biochem. Physiol. 120B,73 -87.
Blickhan, R. (1989). The spring mass model for running and hopping. J. Biomech. 22,1217 -1227.[CrossRef][Medline]
Cavagna, G. A. (1970). Elastic bounce of the
body. J. Appl. Physiol.
29,279
-282.
Cavagna, G. A. (1975). Force platforms as
ergometers. J. Appl. Physiol.
39,174
-179.
Cavagna, G. A., Saibene, F. P. and Margaria, R. (1963). External work in walking. J. Appl. Physiol. 18,1 -9.[Medline]
Cavagna, G. A., Saibene, F. P. and Margaria, R. (1964). Mechanical work in running. J. Appl. Physiol. 19,249 -256.[Medline]
Cavagna, G. A., Thys, H. and Zamboni, A. (1976). The sources of external work in level walking and running. J. Physiol. 262,639 -657.[Abstract]
Cavagna, G. A., Franzetti, P., Heglund, N. C. and Willems, P. A. (1988). The determinants of the step frequency in running, trotting and hopping in man and other vertebrates. J. Physiol. 399,81 -92.[Abstract]
Cavagna, G. A., Willems, P., Franzetti, P. and Detrembleur, C. (1991). The two power limits conditioning step frequency in human running. J. Physiol. 437,95 -108.[Abstract]
Cavagna, G. A., Mantovani, M., Willems, P. A. and Musch, G. (1997). The resonant step frequency in human running. Pflügers Arch. 434,678 -684.[CrossRef][Medline]
Cavagna, G. A., Willems, P. A. and Heglund, N. C.
(2000). The role of gravity in human walking: pendular energy
exchange, external work and optimal speed. J. Physiol.
528,657
-668.
Chang, Y.-H., Huang, H.-W., Hamerski, C. M. and Kram, R.
(2000). The independent effects of gravity and inertia on running
mechanics. J. Exp. Biol.
203,229
-238.
Farley, C. T. and González, O. (1996). Leg stiffness and stride frequency in human running. J. Biomech. 29,181 -186.[CrossRef][Medline]
Farley, C. T., Glasheen, J. and McMahon, T. A.
(1993). Running springs: speed and animal size. J.
Exp. Biol. 185,71
-86.
Ferris, D. P. and Farley, C. T. (1997).
Interaction of leg stiffness and surface stiffness during human hopping.
J. Appl. Physiol. 82,15
-22.
Ferris, D. P., Louie, M. and Farley, C. T. (1998). Running in the real word: adjusting leg stiffness for different surfaces. Proc. R. Soc. Lond. B 265,989 -994.[CrossRef][Medline]
Ferris, D. P., Liang, K. and Farley, C. T. (1999). Runners adjust leg stiffness for their first step on new running surface. J. Biomech. 32,787 -794.[CrossRef][Medline]
He, J., Kram, R. and McMahon, T. A. (1991). Mechanics of running under simulated low gravity. J. Appl. Physiol. 23,65 -78.
Heglund, N. C. (1981). A simple design for a force-plate to measure ground reaction forces. J. Exp. Biol. 93,333 -338.
Hill, A. V. (1950). The series elastic component of muscle. Proc. R. Soc. Lond. B 137,273 -280.[Medline]
Ker, R. F., Bennett, M. B., Bibby, S. R., Kester, R. C. and Alexander, R. McN. (1987). The spring in the arch of the human foot. Nature 325,147 -149.[CrossRef][Medline]
Kerdok, A. E., Biewener, A. A., McMahon, T. A., Weyand, P. G.
and Herr, H. M. (2002). Energetics and mechanics of human
running on surfaces of different stiffnesses. J. Appl.
Physiol. 92,469
-478.
Margaria, R. (1938). Sulla fisiologia e specialmente sul consumo energetico della marcia e della corsa a varie velocità ed inclinazioni del terreno. Atti Accad. nazionale Lincei Memorie 7,299 -368.
Margaria, R. and Cavagna, G. A. (1964). Human locomotion in subgravity. Aerospace Med. 35,1140 -1146.
McMahon, T. A. and Cheng, G. C. (1990). The mechanics of running: how does stiffness couple with speed? J. Biomech. 23,65 -78.[CrossRef][Medline]
McMahon, T. A., Valiant, G. and Frederick, E. C.
(1987). Groucho running. J. Appl.
Physiol. 62,2326
-2337.
Schepens, B., Willems, P. A. and Cavagna, G. A.
(1998). The mechanics of running in children. J.
Physiol. 509,927
-940.
Seyfarth, A., Geyer, H., Günther, M. and Blickhan, R. (2002). A movement criterion for running. J. Biomech. 35,649 -655.[CrossRef][Medline]