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First published online October 5, 2006
Journal of Experimental Biology 209, 4051-4060 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02344
The landing–take-off asymmetry in human running
Istituto di Fisiologia Umana, Università degli Studi di Milano, Via Mangiagalli, 32, 20133 Milan, Italy
e-mail: giovanni.cavagna{at}unimi.it
Accepted 23 May 2006
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionList of symbolsReferences |
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Key words: locomotion, running, muscle stretch–shorten cycle
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Introduction |
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). A spring-mass model was proposed to simulate the bounce of
the body (Blickhan, 1989;
McMahon and Cheng, 1990;
Seyfarth et al., 2002). This
model is largely used in the interpretation of experiments aimed to determine
the changes in spring stiffness and step frequency with speed
(Cavagna et al., 1988;
McMahon and Cheng, 1990), with
grounds of different compliance (Ferris
and Farley, 1997; Ferris et
al., 1999; Kerdok et al.,
2002) and the effect of spring stiffness on energy expenditure
(McMahon et al., 1987;
Kerdok et al., 2002).
The spring-mass model assumes (i) the same characteristics of the `spring',
i.e. of the muscle–tendon units, when the spring is compressed after
landing (the muscle–tendon units are stretched) as when the spring
extends to take-off (the muscle–tendon units shorten), and (ii) the same
height and velocity of the centre of mass at landing and take-off
(Blickhan, 1989;
McMahon and Cheng, 1990). As
the authors proposing the model pointed out, this is a simplification because:
(i) `the leg has to be more stiff during landing than during
take-off', and (ii) `animals in general do not have a similar
take-off and landing velocity as assumed in the model. They take-off with
straightened legs and land with bent legs'
(Blickhan, 1989).
The purpose of the present study was to determine if and to what extent (i) the functional characteristics of the limb in running are affected by, and adapted to, an eventual asymmetry of the movement during landing and take-off, and (ii) a possible different response of the muscle–tendon units to stretching and shortening affects the timing of negative and positive work performance during their stretch–shorten cycle in the running step.
Before describing the movement of the centre of mass during its apparent
bounce at each running step it is worth noting that landing and take-off may
not occur at all during trotting and low-speed running, when the vertical
oscillation of the centre of mass of the body takes place entirely in contact
with the ground (Cavagna et al.,
1988). It is obvious that in this case the contact with the ground
cannot be taken as an indication of landing and take-off when describing the
bounce of the elastic system. However, in this case the vertical oscillation
also takes place below and above an equilibrium position where the vertical
force equals body weight. As the running speed increases and an aerial phase
takes place, the contact time progressively decreases towards, but not to, the
time interval during which the elastic system is loaded beyond its equilibrium
position. It is therefore more appropriate to consider `landing' as the
instant where the vertical force increases above body weight (rather than the
instant where the foot contacts the ground) and `take-off' as the instant
where the vertical force drops below body weight (rather than the instant
where the foot leaves the ground). In the present study the height of the
centre of mass was measured both when the foot actually contacts and leaves
the ground, and at its equilibrium positions during the downward and the
upward displacement.
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Materials and methods |
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The subjects ran back and forth across a 4 mx0.5 m force platform
sensitive to the force exerted by feet in the forward and vertical directions;
lateral forces were neglected. The platform was inserted in a 50 m long
corridor with its surface at the level of the floor. Two photocells fixed
1–3 m apart (according to the speed) at neck height along the side of
the platform were used to determine the average running speed,
f. The characteristics of
the platform have previously been described
(Cavagna, 1975).
A microcomputer was used to acquire data at a sampling rate of 500 Hz per channel. Three channels were employed to acquire (i) the platform signal proportional to the force exerted in forward direction (Ff), (ii) the platform signal proportional to the force exerted in vertical direction (Fv), and (iii) the output of the photocell circuit.
From platform records to mechanical energy changes
A custom LabVIEW (6.1) software program was used to analyze the ground
reaction force records. The procedure used to derive the mechanical energy
changes of the centre of mass of the body from the ground reaction forces has
previously been described in detail
(Cavagna, 1975). In short, only
motion in a sagittal plane was considered when calculating the mechanical
energy of the centre of mass. Rotational kinetic energy of the body and
lateral translational energy were ignored. The time-average of the
Fv record was measured after each run over an integer
number of steps, encompassing those subsequently used for calculation,
searching for a minimum difference with the subject body weight; the ratio
between this time average and the weight of the subject was 1.00±0.01
(N=359). The velocity changes of the centre of mass of the body in
the vertical direction and in the forward direction were determined, by
integration, from the Fv and Ff
platform signals in the time interval between photocells. An integer number of
running cycles was then selected between two peaks (or valleys) of the
vertical velocity changes record searching by eye for a minimum drift of both
the vertical and forward records. The procedure followed to determine the
instantaneous velocity in the vertical direction
Vv(t) and forward direction
Vf(t) from the corresponding velocity changes has
been described (Cavagna, 1975).
The instantaneous vertical velocity Vv(t) was
used to calculate the instantaneous kinetic energy of vertical motion
Ekv(t)=0.5MbVv(t)2
(where Mb is the mass of the body) and, by integration,
the vertical displacement of the centre of mass,
Sv(t), with the corresponding gravitational
potential energy change Ep(t)=
MbgSv(t) (where
g is the acceleration of gravity). 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).
Since, as mentioned above, selection was initially made between peaks (or
valleys) of the vertical velocity changes, the records were expanded to
include the previous valley (or peak) of Ep(t)
until a clear picture of the step(s) was obtained
(Fig. 1). The downward vertical
displacement Sv,down and the upward vertical displacement
Sv,up were calculated from the total decrement,
respectively increment, of the Ep(t) curve,
taking place between two or more Ep peaks (or valleys),
divided by the number of steps. Algorithms were made to calculate the work
done during the selected steps between Ep valleys (or
peaks): Wv, Wkf and
Wext were calculated from the amplitudes of valleys and
peaks, and the initial and final values in the
Ep(t), Ekf(t) and
Ecm(t) records. Positive values of the energy
changes gave positive work, negative values gave negative work. In a perfect
steady run on the level the ratio between the absolute values of positive and
negative work done in an integer number of steps should be equal to one.
Experimental values were as follows (N=359):
W+v/W–v=0.99±0.07,
W+v
kf/W–kf=1.03±0.18,
W+ext/W–ext=1.00±0.07.
All the selected steps were used for analysis.
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0 [the distance between
the two reference levels was the minimal distance, which included the noise of
the dEcm(t)/dt record]. Due to the noise
of the dEcm(t)/dt record, the aerial
time was in some cases overestimated (by 6.5±7.6% in one randomly
selected subject; mean ± s.d., N=28). The brake duration, tbrake, i.e. the time during which external negative work is done, was calculated as the time interval during which the dEcm(t)/dt record was below the reference level. The push duration, tpush, i.e. the time during which external positive work is done, was calculated as the time interval during which the dEcm(t)/dt record was above the reference level. Due to the noise of the dEcm(t)/dt record, tbrake and/or tpush were in some cases underestimated (5.3±6.6% for tbrake and 7.4±7.1 for tpush, N=28). Similarly, the downward and upward displacements of the centre of mass during contact, Sc,down and Sc,up, were measured from the descending, respectively ascending portions of the Ep(t) curve during the time interval where dEcm(t)/dt was lower, respectively greater than the two reference levels.
Vertical displacement below and above the equilibrium points
The vertical force, Fv, applied by the foot on the
ground is:
 | (1) |
).
Maximum vertical velocity and kinetic energy attained during the downward displacement and the lift
The maximal vertical velocity and the maximal kinetic energy
Ek=Ekv+Ekf
attained by the centre of mass during the downward displacement,
Vv,mx,down and Ek,mx,down, and during
the lift, Vv,mx,up and Ek,mx,up, were
determined by comparison of the peaks in Ekv and
Ek with the simultaneous Ep curve.
Since the maximal vertical velocity may increase with the amplitude of the
vertical displacement, which may be different during the lift and the downward
displacement (see, e.g. the Ep curves in
Fig. 1), the ratio between
maximal downward and upward velocities was normalized in
Fig. 3 as
(Vv,mx,down/Vv,mx,up)
(Sv,up/Sv,down).
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Ek transduction):
 | (2) |
The Ep–Ek transduction is
complete [r(t)=1] during the aerial phase (ballistic lift
and fall) when no external work is done by the muscular force. However the
Ep–Ek transduction also occurs
during contact, when the body is partially supported by the foot on the ground
in the upper part of the trajectory of the centre of mass
[0<r(t)<1]. The
Ep–Ek transduction is nil
[r(t)=0] in two phases of the step, 
and ß,
where Ek increases, decreases, respectively,
simultaneously with the gravitational potential energy Ep
(Cavagna et al., 2002). Note
that positive and negative external work is done both in the phases
and ß of the step [r(t)=0] and in the phases of the
step where a transduction occurs between Ek and
Ep [0<r(t)<1].
In this study, r(t) was calculated from the absolute
value of the time derivative of Ep, Ek
and Ecm in the time interval between two or more
Ep peaks (or valleys). The cumulative value of energy
recovery, Rint(t), resulting from the
instantaneous Ek–Ep
transduction, was measured from the area below the r(t)
record divided by the step period, 
:
Rint(t)=[
0r(u)du]/.
At the end of the step
Rint()=Rint
(Cavagna et al., 2002).
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Statistics
The data collected as a function of running speed were grouped into classes
of 1 km h–1 intervals as follows: 2 to <3 km
h–1, 3 to <4 km h–1,.., 20 to <21 km
h–1. The data points in
Fig. 2 represent the mean
± s.d. in each of the above speed intervals and the figures near the
symbols give the number of items in the mean. A comparison was made to assess
the difference between positive and negative external work durations as a
function of speed (Fig. 6).
Since each couple of means at a given running speed has the same number of
items, with each item measured in the same step for each subject, a
t-test paired two-sample for means tool (Excel 2004, version 11.2)
was used to determine when the means are significantly different
(Table 1).
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Results |
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The relative amount of the vertical displacement of the centre of mass
taking place during contact changes markedly with speed, decreasing from unity
at the lowest speed (when the aerial phase is nil) to 
0.6 during the lift
and 0.4 during the downward displacement at 20 km
h–1 (Fig. 2).
In contrast, the fraction of the vertical displacement
Sce/Sv, from the lowest point of the
trajectory to the point where the vertical force equals body weight, changes
less with speed and approaches one half of Sv
(Fig. 2).
Even at the equilibrium points, however, the maximal vertical velocity attained by the centre of mass during the downward displacement Vv,mx,down is greater than that attained during the lift Vv,mx,up (Fig. 3). This results in a maximum of kinetic energy, Ek=Ek,f+Ek,v, which is greater during the downward displacement than during the lift (compare the two peaks of the green line in Figs 1 and 7). However, the peak of Ek is higher during the downward displacement than during the lift not just because the maximal vertical velocity is greater during the downward displacement than during the lift. The contribution of Ek,v to the maximum of Ek is greater during the downward displacement than during the lift also because the increment of Ek,v adds onto the maximum of Ek,f during the downward displacement, whereas it precedes the maximum of Ek,f during the lift (Fig. 1; see Discussion). During the downward displacement, the time difference between the maximum of Ek and the maximum of Ek,v is nil or possibly slightly negative (see vertical dotted lines in Fig. 1). During the lift, by contrast, the time difference between the maximum of Ek and the maximum of Ek,v is positive (increasing with speed to 20–25 ms; data not shown).
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The ratio Ek,mx,down/Ek,mx,up decreases with speed, approaching asymptotically unity at high running speeds (Fig. 4) due to the large increase of Ek,f (brown in Fig. 1) relative to Ek,v (red in Fig. 1). The difference Ek,mx,down–Ek,mx,up remains positive even at high running speeds (inset in Fig. 4).
Mechanical energy changes during the downward displacement and the lift
The transduction of gravitational potential energy of the centre of mass
Ep into kinetic energy Ek during the
downward displacement is greater than the transduction of
Ek into Ep during the lift, as shown
in Fig. 5 by a ratio
Rint,down/Rint greater than 0.5. This
is also true at the lowest running speeds when the aerial phase is often nil,
indicating that even in the absence of a ballistic fall the vertical support
on the ground is less during the downward displacement than during the
lift.
The total duration of positive and negative external work, including the
and ß phase durations [r(t)=0] and the time
during which external work is done in presence of a transduction between
Ek and Ep
[0<r(t)<1; see Materials and methods], is given in
Fig. 6. It can be seen that at
low and intermediate running speeds the time during which external positive
work is done (increment of the Ecm curve in
Fig. 1) is greater than the
time during which external negative work is done (decrement of the
Ecm curve in Fig.
1). With increasing speed the difference between the two times
decreases together with their duration, and tends to become not significantly
different above 14 km h–1 when both time values fall below
0.1 s (Fig. 6 and
Table 1).
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Discussion |
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The apparent bounce of the body is therefore better described by considering the succession of its four phases above and below the equilibrium points during the downward displacement and during the lift: (i) downward acceleration, (ii) downward deceleration, (iii) upward acceleration and (iv) upward deceleration (Fig. 7).
The downward acceleration occurs during the Sae,down fraction of the step, between the maximum of Ep and the maximum of Ekv (downward displacement with a vertical force exerted on the ground less than body weight; blue section of the Ep curve in Fig. 7 with lighter blue indicating the aerial phase). The Ep into Ek=Ekv+Ekf transduction takes place entirely during this phase and is indicated by the increment of the Rint(t) curve after the maximum of Ep. Note that a large Ep into Ek transduction also occurs in the absence of an aerial phase (upper left panel in Figs 1 and 7). The kinetic energy of the centre of mass Ek attains its maximum value in the running step in close proximity to the end of this phase, due to the increment of Ek,v caused by gravity (Fig. 1).
The downward deceleration occurs during the Sce,down fraction of the step, between the maximum of Ekv and the minimum of Ep (downward displacement with a vertical force exerted on the ground greater than body weight, red section of the Ep curve in Fig. 7). In this phase, translational kinetic energy Ek and gravitational potential energy Ep are simultaneously absorbed by the muscle–tendon units. The Ep–Ek transduction is nil during this phase, as indicated by the upper horizontal tract of the Rint(t) curve, which corresponds to the ß fraction of the step. Note that the maximum of kinetic energy Ek in the running step, attained during the negative work phase, is consistent with a high velocity of stretching of the muscle–tendon units and, according to the force–velocity relation, with a high force exerted by the forcibly stretched muscle.
The upward acceleration occurs during the Sce,up
fraction of the step, comprised between the minimum of Ep
and the maximum of Ekv (lift with a vertical force exerted
on the ground greater than body weight). In this phase of the step
muscle–tendon units deliver both Ek and
Ep. The
Ep–Ek transduction is nil during
most of this phase, as shown by the lower horizontal tract of the
Rint(t) curve, corresponding to the initial part
of the fraction of the step, when Ek and
Ep simultaneously increase (except for a small time
interval during which Ep begins to increase while
Ekf is still decreasing).
The upward deceleration occurs during the Sae,up
fraction of the step, between the maximum of Ekv and the
maximum of Ep (lift with a vertical force exerted on the
ground less than body weight). During this phase Ekv
decreases, but Ekf continues to increase: the leg is still
exerting a propulsive force to accelerate the body forwards even though the
vertical force has dropped below body weight. The increment in
Ekf exceeds the decrement of Ekv with
the result that Ek increases, but to a peak lower than
during the downward displacement due to the opposite changes of
Ekf and Ekv. As a consequence of the
push of the foot that is about to leave the ground, Ek
increases simultaneously with Ep, i.e. the
fraction of the step [lower horizontal tract of the
Rint(t) curve] extends into this phase whereas
the transduction of Ek into Ep is
limited to the last part of it [increment of the
Rint(t) curve taking place before the maximum of
Ep is attained]. It follows that the transduction of
Ek into Ep during the lift is smaller
than the transduction of Ep into Ek
during the downward displacement: the increment of the
Rint(t) curve during the downward displacement is
larger than during the lift, i.e.
Rint,down/Rint>0.5
(Fig. 5). In the presence of an
aerial phase the ballistic fall is larger than the ballistic lift (dotted
Ep line in Fig.
7). The protracted execution of positive external work during the
lift results in a blunt attainment of a plateau by Ecm
(black line in Fig. 1), which
contrasts with its sharper decrease at the end of the plateau. These findings
are the mechanical counterpart of the observation that in running the lower
limb is more extended at take-off than at landing [see e.g. the photographs of
Muybridge (Muybridge,
1955)].
Muscle–tendon unit response during stretching and shortening
The finding that at low and intermediate running speeds the time required
for external positive work production is greater than the time required for
external negative work production (Fig.
6) suggests that the muscle–tendon units act differently
during shortening (positive work) and during stretching (negative work). Above
14 km h–1, however, the difference between the two time
values tends to be not significantly different
(Table 1) suggesting that the
muscle–tendon units act as a spring with similar characteristics during
compression and recoil, as assumed in the spring-mass model of running. A
possible explanation of this finding is given below.
During running at a constant step-average speed, the momentum (force x time) lost at each step in the forward direction during the brake equals the momentum gained during the push. The force responsible for the change in momentum is mainly the muscular force (neglecting the negative work done by frictional forces outside muscle–tendon units). It follows that the shorter duration of the momentum lost during the brake implies an average force exerted by the muscle–tendon units during stretching (the brake) greater than during shortening (the push). This is in qualitative agreement with what is expected from the force–velocity relation of muscle for an equal number of active fibres.
According to the force–velocity relation of muscle, however, the difference in force between stretching and shortening, and as a consequence the difference in time for the same change in momentum, should increase with the velocity of the length change. This contrasts with the present findings, which indicate that the difference between push and brake times decreases with running speed, i.e. with the velocity of the length change.
The force–velocity relation of muscle applies to ramp stretches and
releases, when the sliding velocity of the actin filament relative to the
myosin filament influences cross-bridge attachment and detachment
(Huxley, 1957). Is it
reasonable to infer, from the results described in
Fig. 6, that the amount of
lengthening and shortening of muscle fibres (sliding of filaments) decreases
progressively with running speed, and with it the impact of the
force–velocity relation on the timing of positive and negative external
work production?
Lengthening imposed to muscle–tendon units is shared between muscle
fibres and the tendon in series with them. It has been suggested that the
energy expenditure during running is due to the cost of generating force in
quasi isometric contracting fibres, not due to the work done by them
(Kram and Taylor, 1990;
Alexander and Ker, 1990).
Experiments on turkeys (Roberts et al.,
1997) and kangaroos (Biewener
et al., 1998) have shown that the length change of the
muscle–tendon units bending and extending the ankle, while the foot is
on the ground in a running step on the level, is mostly due to tendon
stretching and recoil, with a small change in the length of muscle fibres. On
the other hand a contribution of the contractile machinery to the mechanical
work done is suggested at low running speeds in humans by a greater energy
expenditure when the mechanical work is made greater by decreasing the step
frequency below the freely chosen step frequency
(Cavagna et al., 1997). The
present results suggest that both views, i.e. tendon vs muscle
contribution to mechanical work, may possibly apply to human running,
depending on the speed of the run.
The lengthening of the fibres relative to lengthening of the tendons
depends on the stiffness of muscle relative to that of tendon; the stiffness
of the muscle in turn depends on its activation. In the relaxed muscle, most
of the lengthening imposed on the muscle–tendon units is taken by the
muscle fibres. In fact only few cross-bridges resist filament sliding in a
relaxed muscle fibre (Hill,
1968), with the consequence that during a stretch filaments will
oppose a force much smaller than when the muscle is activated and will
transmit this smaller force to the tendon in series with them. As a
consequence the tendons will not lengthen appreciably during stretching a
relaxed muscle. At low running speeds muscle activation is likely to be
moderate with the result that some of the length change of the
muscle–tendon units is sustained by muscle fibres. In this case the
force exerted during stretching is expected to be greater than during
shortening according to the force–velocity relation of muscle, and as a
consequence the time of positive work is expected to be greater than the time
of negative work, as explained above. When muscle activation is progressively
increased with speed, muscle fibres oppose a progressively greater force to
stretching and a progressively greater fraction of the lengthening imposed to
the muscle–tendon units must be absorbed by tendons. At speeds greater
than 14 km h–1 muscle activation may be increased to
such an extent that muscle fibres are held almost isometric, so that the
length change is taken almost completely by tendons. In this case the
muscle–tendon units would operate as a simple spring with similar
characteristics during stretching and recoil, as assumed in the spring-mass
model. Obviously a work input by the contractile machinery must always be
added to account for the losses due to tendon hysteresis which, however, is
small (Ker et al., 1987;
McMahon, 1987;
Alexander, 2002). In other
words, the decrease with speed of the difference between positive and negative
external work durations (Fig.
6) suggest a decrease in muscle length change relative to tendon
length change within the muscle–tendon units.
This interpretation is consistent with (i) the increase in peak muscle
stress with speed towards the peak stress developed in the isometric
contraction found in the limb muscles of the horse
(Biewener, 1998) and (ii) the
in vivo measurements of muscle fibre length in running turkeys
[Roberts et al. (Roberts et al.,
1997), their fig. 3], showing that the contribution of muscle work
to the total work production by the muscle–tendon units is greater at
low speeds than at high speeds.
In conclusion, the landing–take-off asymmetry consists of the
vertical support of the body being greater during the lift than during the
downward displacement of the centre of mass. Gravity is exploited during the
downward displacement to increase the downward velocity of the centre of mass,
thus attaining the maximum of kinetic energy in the running step. This
provides a high velocity of stretching to the muscle–tendon units during
the negative work phase, followed by a more prolonged push during the lift. A
fast stretch, during which the maximal force is quickly attained, followed by
a slower shortening, was the strategy used in isolated muscle experiments to
get the maximal efficiency of positive work production in ramp
stretch–shorten cycles (Heglund and
Cavagna, 1987). This is similar to what naturally happens during
running at low and intermediate running speeds. With increasing speed, the
work done by the contractile component decreases progressively for two
reasons: (i) the relatively larger length change taken up by tendons during
the stretch (as described above), and (ii) the decrease in force with the
velocity of shortening (as described by the force–velocity relation). It
follows that the mechanical energy released during shortening at high speeds
[the `irreducible work' (Hill,
1970)] is in practice only that previously stored by the elastic
elements, due to their negligible damping. The role of the contractile
machinery at high speeds is to provide a force large enough to exploit a large
fraction of the force–length relation of the elastic elements for the
storage of mechanical energy.
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List of symbols |
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f
, ß) or decrease (ß)
simultaneously
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Acknowledgments |
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References |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionList of symbolsReferences |
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