The double contact phase in walking children
Unité de Physiologie et Biomécanique de la Locomotion, Université Catholique de Louvain, 1 Place P. de Coubertin, B-1348 Louvain-la-Neuve, Belgium
* Author for correspondence (e-mail: benedicte.schepens{at}loco.ucl.ac.be)
Accepted 14 May 2003
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Summary |
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Key words: child, double contact, locomotion, mechanics, walking, work
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Introduction |
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The total muscular work is often divided into two parts: the external work
and the internal work. The external work (Wext) is
performed to raise and accelerate the centre of mass of the body
(COM) relative to the surroundings. The walking gait in humans and
other terrestrial animals involves a pendulum-like transfer between potential
and kinetic energy of the COM, which substantially reduces the amount
of work required of the muscles to move the COM at a constant average
speed on level terrain (Cavagna et al.,
1977).
The internal work is performed to accelerate the body segments relative to
the COM, to overcome internal friction or viscosity, to overcome
antagonistic co-contractions and to stretch the series elastic components
(Cavagna et al., 1964).
Although any work that is not done on the environment nor changes the energy
level of the COM is internal work, typically only the work done to
accelerate the body segments relative to the COM has been measured as
classical internal work (Wint,k), using cinematographic
analysis (Cavagna and Kaneko,
1977
).
During the double contact phase of walking (DC), when both feet are on the ground, the muscles perform more than just Wext and Wint,k; they also have to perform work due to the fact that one leg is pushing against the other.
Recently, Donelan et al.
(2002a) measured the work done
by one leg pushing against the other during DC in walking adults. This
mechanism had been discussed by Alexander and Jayes as early as 1978
(Alexander and Jayes, 1978
),
but the mechanical work had never before been measured. During DC, both legs
are on the ground simultaneously and exert horizontal forces in opposite
directions; the back leg is pushing forwards while the front leg is pushing
backwards. The work performed during DC by the muscles of the back leg can be
considered in two parts: the first is to accelerate and raise the
COM, and the second is to compensate for the work simultaneously
absorbed by the muscles of the front leg to redirect the trajectory of the
COM. The first part is measured as Wext but the
second part, the work done by one leg against the other,
Wint,dc, is not measured as Wext nor
as Wint,k.
During growth, body dimensions change significantly, almost 4-fold for the
body mass and almost 2-fold for the leg length, between the age of 3 and
adulthood (Schepens et al.,
1998). The work done by one leg against the other should depend on
the forces exerted by each leg and the displacement of the COM during
DC, both of which may change with age. The horizontal component of the force
should increase as the angle between the legs increases, and the forward
displacement of the COM during DC may be related to the length of the
foot (after the moment of front foot heel-strike, the back foot is `peeled
off' the floor as the COM continues to move forwards;
Cavagna et al., 1976
). The
effect of age and body dimensions on Wint,dc is unknown,
yet Wint,dc may be an important factor in explaining the
higher energy cost at a given speed of walking in children
(DeJaeger et al., 2001
). In
this study, we measure Wint,dc in children and adults
during walking at different speeds and compare it with
Wext, since, to date, no data exist for the total power
output of walking children.
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Materials and methods |
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Informed written consent of the subjects and/or their parents was obtained. The experiments involved no discomfort, were performed according to the Declaration of Helsinki and were approved by the local ethics committee. All of the subjects wore swimming suits and gym shoes. They were asked to walk across a force platform at different speeds.
The mean speed (f) was measured
by two photocells placed at the level of the neck and set 1.5-5.0 m apart
depending upon the speed. In each age group, the data were gathered into speed
classes of 0.14 m s-1 (0.5 km h-1). In most cases, two
trials per subject were recorded in each speed class. A total of 895 steps
were analysed.
Force platform measurements
The mechanical energy changes of the COM due to its motion in the
sagittal plane during a walking step were determined from the vertical and
horizontal components of the ground reaction forces
(Cavagna, 1975). The work
necessary to sustain the lateral movements of the COM in adults is
small (Tesio et al., 1998
) and
was neglected.
The ground reaction force was measured by means of a force platform (6 m
long and 0.4 m wide) mounted at floor level 25 m from the beginning of a path
40 m long. The force platform was made of 10 separate plates, similar to those
described by Heglund (1981).
The plates were sensitive to forces in the fore-aft and vertical directions
and had a natural frequency of 180 Hz and a linear response to within 1% of
the measured value for forces up to 3000 N. The difference in the electrical
signal to a given force applied at different points on the surface of the 10
plates was less than 1%. The crosstalk between the vertical and forward axis
was less than 1% of the applied force. The individual signals of the 10 plates
were digitised by a 12-bit analogue-to-digital converter every 5 ms and
processed by means of a desktop computer.
A complete step was selected for analysis only when the feet were on
different force plates and when the subject was walking at a relatively
constant average height and speed. Specifically, the sum of the increments in
both forward and vertical velocity could not differ by more than 25% from the
sum of the decrements (Cavagna et al.,
1977). According to these criteria, the difference in the forward
speed of the COM from the beginning to the end of the selected step
was less than 6% of
f (except in
four instances at very low speeds below 0.56 m s-1, where it was up
to 9%), and the mean vertical force was within 5% of the body weight.
The step length (Lstep) was calculated as
f times the step period measured
from the force tracings. The distance the COM moves forward during
the period of double contact (Ldc) was calculated as the
mean speed during DC multiplied by the DC period measured from the force
tracings. The leg length, Lleg, was measured as the
distance from the ground to the greater trochanter as the subjects stood
vertically. The limb angle (measured in radians) was calculated as:
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Calculation of the positive muscular work done by one leg against
the other during double contact, Wint,dc
The positive muscular work done by one leg against the other during DC is
calculated in a three-step process: (1) measure all the work done by each leg
on the COM (including passive work and external work); (2) subtract
any work that may have been done passively, i.e. work that did not have to be
done by muscular force; and (3) subtract the external work,
Wext. The remaining work is equal to the positive muscular
work done by one leg against the other during double contact,
Wint,dc.
Step 1. Measure all the work done by each leg on the COM
This requires measuring the individual limb ground reaction forces, having
each foot on a separate force plate. The work curves shown in
Fig. 1B are calculated
independently for the back and the front limb as:
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Step 2. Subtract any work that may have been done passively
Since the purpose of these measurements is to determine the muscular work
done, any part of the work that is done passively, without the need of
muscular intervention, should be excluded. For example, the external work
(discussed in the next section) during walking involves a well-known pendular
energy transfer between the kinetic (Ek,f) and potential
energy (Ep + Ek,v) of the
COM. It is generally accepted that the positive muscular work done
will be overestimated if the pendular energy transfer is not allowed. Part of
this pendular transfer takes place during DC. Logically, if one allows the
transfer of energy in the external work calculations (the pendulum mechanism),
one must allow similar energy transfer in the Wint,dc
calculations; the phenomenon is the same, only the method of measurement has
changed.
In order to allow pendular energy transfers in the Wint,dc calculations while disallowing any non-pendular transfers (i.e. the work done by one leg against the other), the work done on the COM in the vertical direction (Wv) must be allowed to exchange with the work done on the COM in the horizontal direction by each of the legs (Wf,back and Wf,front). To obtain Wv, Wv,back and Wv,front can be summed with no loss or transfer of work because both legs are simultaneously doing positive or negative work, depending upon the vertical displacement of the COM and the fact that Fv,back and Fv,front are always positive. In contrast to Wv,back and Wv,front, Wf,back and Wf,front cannot be summed without allowing non-pendular transfers, i.e. Wf,back and Wf,front have to be treated separately.
Work transfers between Wv and Wf,back and between Wv and Wf,front occur during DC. A close examination of Fig. 1 shows two opportunities where positive work may be done passively and incorrectly counted as positive muscular work unless it is subtracted.
The first opportunity is during the beginning of DC when the COM is `falling down' off the back leg due to gravity (this is seen as the decrease in the Wv curve at the beginning of DC shown in Fig. 1B), passively pivoting around the point of contact of the back leg as the compliant front leg starts to take up the load. The curvilinear trajectory of the falling COM involves both tangential and normal accelerations, which can be decomposed into vertical and horizontal components. The horizontal component of the acceleration of the COM must result from a horizontal component of the force acting at the pivot. Part of this horizontal component of the force is derived passively from the acceleration of gravity and results in a forward acceleration of the COM. In other words, the passive displacement of the COM accelerating downwards and accelerating forwards under the influence of gravity would result in a passive horizontal component of the ground reaction force applied on the back leg. This force multiplied by the forward displacement of the COM results in a passive increase in Wf,back (this is part of the increase in the Wf,back curve at the beginning of the DC shown in Fig. 1B).
Thus, the positive work done by the horizontal component of the ground
reaction force acting on the back leg during the first part of DC (the
increase in the Wf,back curve during DC shown in
Fig. 1B) should be reduced by
the amount of simultaneous negative work resulting from the vertical forces
(the decrease in the Wv curve during DC shown in
Fig. 1B), as indicated by the
upward arrow in the figure. The resulting curve, Wback, is
the net muscular work done by the back leg during DC:
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The second opportunity for pendular transfer occurs during the latter part of DC when the COM starts to `ride up' onto the front leg, passively pivoting around the point of contact of the front leg (this is seen as the increase in the Wv curve at the end of DC shown in Fig. 1B) as the kinetic energy of forward motion is converted into potential energy. As before, the curvilinear trajectory of the COM can be decomposed into a forward deceleration and vertical acceleration of the COM. Part of the vertical component of the ground reaction force is derived passively from the forward deceleration of the COM. This vertical force multiplied by the vertical displacement of the COM results in a passive increase in Wv (the increase in the Wv curve at the end of the DC shown in Fig. 1B).
The positive work done to raise the COM during the second part of
double contact (the increase in the Wv curve during DC
shown in Fig. 1B) should be
reduced by the amount of simultaneous negative work done by the horizontal
component of the ground reaction forces acting on the front leg (the decrease
in the Wf,front curve during DC shown in
Fig. 1B), as indicated by the
downward arrow in the figure. The resulting curve, Wfront,
is the net muscular work done by the front leg during DC:
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The net positive muscular work done by each leg is equal to the sum of the
increments in the Wback and Wfront
curves, respectively, +Wback and
+Wfront.
Step 3. Subtract the external work
The external work is subtracted from the net positive muscular work done by
each leg in order to count only the work that is not already measured by the
Wext.
The external work method uses the resultant of the ground reaction forces acting on both limbs (i.e. the vertical forces and, separately, the horizontal forces from all of the force plates were summed), resulting in the mechanical cancellation of simultaneous positive and negative work performed by the limbs during double support periods.
The principle of the method to measure Wext and the
procedures followed to compute the velocity in the forward and vertical
directions, the vertical displacement, the changes in gravitational potential
energy and the changes in kinetic energy of the COM from the platform
signals have been described in detail by Cavagna
(1975) and by Willems et al.
(1995
) and are only briefly
described here. Provided that air resistance is negligible, the acceleration
of the COM in the forward (af) and vertical
(av) directions can be calculated by:
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The kinetic energy of the COM due to its motion in the forward
direction was calculated as
Ek,f=(m·Vf2)/2.
The energy of the COM due to its vertical movement was calculated as
Ep+Ek,v=(m·g·Sv)+[(m·Vv2)/2],
and the total mechanical energy of the COM was calculated as
Eext=(Ep+Ek,v)+Ek,f.
The sum of the increments in Ek,f,
Ep+Ek,v and Eext
represents the positive work done to accelerate the COM forward
(+Ef), the positive work done against
gravity and to accelerate the COM upward
(
+Ev; a+b in
Fig. 1A) and the positive work
done to maintain the motion of the COM in the sagittal plane
(Wext; c+d in
Fig. 1A), respectively.
As noted above, the Wext method sums all the vertical forces and, separately, all the horizontal forces before performing the work calculations on the resultant forces (Fig. 1A), while the Wint,dc method measures the vertical and horizontal force of each foot separately and performs the work calculations on the individual foot forces (Fig. 1B). The two methods are closely related. In fact, the Ek,f curve of Fig. 1A is exactly equal to the sum of the Wf,front plus Wf,back curves of Fig. 1B. Similarly, the Ep+Ek,v curve of Fig. 1A is exactly equal to the Wv curve, which is the sum of the Wv,front plus Wv,back curves of Fig. 1B. And the Eext curve of Fig. 1A is exactly equal to the sum of the Wf,front+Wf,back+Wv,front+ Wv,back curves of Fig. 1B. The sum of the Wback and Wfront curves results in the Wcom curve, which is exactly equal to the classic external energy (Eext curve in Fig. 1A) during the period of double contact.
The positive muscular work realised by one leg against the other during
double contact (+Wint,dc) is therefore
equal to:
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Results |
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The timing of the peak Ff,back force (indicated by the arrows in Fig. 2) also changes with speed and age. At slow and intermediate speeds, the peak Ff,back occurs during the first part of DC in both children and adults. At high walking speeds in adults, the peak Ff,back occurs before the DC period starts, while it remains within the DC period for the youngest children.
At all ages, Lstep increases with increasing walking speed. At a given speed, children younger than 11 have a smaller Lstep than adults, in spite of a larger limb angle (Fig. 3A,B). The limb angle increases with speed in children and adults, from 0.35 rad at slow speed to 1.22 rad at high speed.
At slow and intermediate speeds, the forward displacement of the
COM during double contact (Ldc) is independent of
speed for all subjects. At the maximal speed, Ldc shows a
tendency to decrease in all age groups. The relative importance of DC
(Ldc/Lstep) decreases with speed for
each age group (from 0.4 to 0.1), although young children show a slight
tendency to have an overall smaller value compared with adults (see
Fig. 3C). Note that the duty
factor ß, the fraction of stride duration for which a particular foot is
on the ground (Alexander and Jayes,
1978), is related to the
Ldc/Lstep ratio by:
ß=[(Lstep+Ldc)/2Lstep]=0.5[1+
(Ldc/Lstep)].
The work done by one leg against the other during DC
Wint,dc represents the work done by one leg against the
other during DC. The Wint,dc normalised to body mass and
distance travelled (J kg-1 m-1) shows an inverted U
shape as a function of walking speed, with a maximum at intermediate speeds
for all ages (Fig. 4A). For
comparison, the Wext curve tends to a minimum at
intermediate speeds (Fig. 4A;
see also fig. 4 in
Cavagna et al., 1983). The
Wint,dc/Wext ratio shows an inverted U
shape with speed (Fig. 4B),
attaining a maximum value of around 0.4 for all ages (except the 3-4-year-old
children, who seem to have a higher maximum value).
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Discussion |
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Relation to previous studies
In 1980, Alexander showed that the classic method ('energy' method) of
calculating the net work performed during walking, the sum of the increases in
mechanical energy of the body, did not measure the work done by one leg
against the other during DC (Alexander,
1980) and therefore makes an error in deficit. On the other hand,
the `hybrid energy/work' method used by Alexander and Jayes
(1978
) measured work as the
product of a force multiplied by a displacement when the angle between them
was 90° and therefore makes an error in excess. In order to avoid these
errors, Alexander subsequently developed a many-legged model where the work
done by the muscles was calculated separately for each leg
(Alexander, 1980
).
The method used here to calculate the work done by one leg against the
other (Wint,dc) is in complete agreement with the latter
analysis presented by Alexander
(1980). However, we do not
agree that the `energy' method is at fault because it does not measure every
case of simultaneous positive and negative work; in particular, because it
does not measure the Wint,dc (although it does correctly
measure the pendulum-like transfer between potential and kinetic energy of the
COM, as pointed out by Alexander). The classical method of measuring
Wext, originally conceived by Fenn
(1930
) and developed into an
easy to use tool by Cavagna
(1975
), precisely measures only
the work done to accelerate and/or lift the COM and some of the work
done on the environment (e.g. sand;
Lejeune et al., 1998
). In
addition, the classical method of measuring Wint,k
(Cavagna and Kaneko, 1977
;
Willems et al., 1995
) measures
only the work done to accelerate the body segments relative to the
COM, although all muscular work other than Wext
or work done on the environment should be classified as
Wint,k (including Wint,dc). Neither of
the classic Wext nor the Wint,k
methods can be expected to measure Wint,dc. Similarly,
neither the energy nor the work methods, nor a combination of the two, are
able to measure all of the muscular work done during locomotion; for example,
the positive and negative work done by different muscles in the same leg
cannot be measured by these methods
(Alexander, 1980
).
The work done by one leg against the other has been estimated in adults
using a method that explicitly does not allow any transfer of work from one
leg to the other, although it implicitly does allow transfers within each leg:
the `individual limb method' of Donelan et al.
(2002a). These authors state
that work by one leg cannot be transferred to the other leg because there are
no muscles that cross from one leg to the other. This, however, would not seem
to be the case. In general, if two actuators are attached to the same mass,
not only can both actuators do work on the mass but they can also do work on
each other provided they are not maintained in an exactly 90° orientation
to each other. In this case, the actuators are the legs and the mass is the
COM; two specific examples of energy transfers that would be
incorrectly disallowed by the individual limb method are given in the
Materials and methods.
The basis for all these work measurements is the Wf,back, Wf,front, Wv,back and Wv,front curves, which are derived from the vertical and fore-aft components of the ground reaction forces exerted by each leg on the COM, as shown in Fig. 1. If the four curves are summed instant-by-instant and then the increments in the resulting curve are added up, we obtain the classical external work Wext. The instant-by-instant summation of the curves allows decrements in one curve to cancel simultaneous increments in another curve. In other words, Wext allows all energy transfers, in particular the well-known pendular transfers that characterise the walking gait.
Alternatively, if the Wf,back and the
Wv,back curves, and similarly the
Wf,front and the Wv,front curves, are
summed instant-by-instant and then the increments in the two resulting curves
are added up, we obtain the individual limb method work. This method only
allows energy transfers within a limb and excludes any energy transfers
between the limbs. Furthermore, this work contains both external work (the
work to lift and accelerate the COM) and internal work done by one
leg against the other during DC (unfortunately, this work is incorrectly, and
very confusingly, called `external work' by Donelan et al.), although it
calculates neither one exactly. Part of the passive pendular energy transfer
taking place during DC is excluded by disallowing transfers from one leg to
the other, resulting in an external work component that is too large.
Likewise, part of the energy that is transferred from one leg to the other
via the energy of the COM during DC, as is detailed in the
Materials and methods (see equations 3, 4), is excluded as well, resulting in
a work done by one leg on the other that is too large. When the individual
limb method is applied to our adult data, we obtain the same results over the
limited speed range (0.75-2.0 m s-1) studied by Donelan et al.
(2002a). However, since fewer
energy-saving work transfers are allowed, over the entire speed range studied
in adults (0.49-2.61 m s-1) the individual limb method results in a
mean of 0.031±0.018 J kg-1 m-1 (mean ±
S.D., N=231) greater work done during DC than the sum of
Wint,dc plus the Wext done during DC
(
10% greater at intermediate speeds, increasing to
20% at low or
high speeds). Subsequent studies that have used the individual limb method
have incurred the same errors (Donelan et
al., 2002b
; Griffin et al.,
2003
).
In contrast to the individual limb method, Wint,dc is
calculated by first summing the Wv,back and
Wv,front curves instant-by-instant to obtain the
Wv curve (Fig.
1). Next, the Wv curve is summed
instant-by-instant with either the Wf,back or the
Wf,front curves to obtain the Wback
and Wfront curves, as detailed in the Materials and
methods. This summation allows all pendular energy transfers between the
vertical work of the COM and the fraction of the forward work on the
COM that can be attributed to each of the legs while avoiding any
cancellation of the work done on one leg due to the horizontal push of the
other. Finally, the increments in the Wback and
Wfront curves are added up and the external work
+Wcom is subtracted to obtain
Wint,dc.
Wint,dc includes only work that is not measured as
Wext nor as the `classical' Wint,k.
Consequently, Wint,dc can be directly compared with the
last 70 years of measurements of Wext and
Wint,k in humans and, with caution, may be simply summed
with Wext and Wint,k to obtain a
measure of total work (acknowledging, of course, the caveats mentioned in the
Introduction). The lateral forces were not measured in this study, and the
work done by one leg against the other during DC in the lateral direction was
ignored. This lateral Wint,dc work is a fraction of the
lateral component of the work as measured by the individual limb method, and
the whole of the lateral component is described as `small' in adults by
Donelan et al. (2002a
). The
effect of age on the lateral Wint,dc has never been
measured and is unknown but can be presumed to be small since, by the age of 4
years, children have a lateral displacement of the COM not
significantly different from that of adults
(Lefebvre et al., 2002
).
Wint,dc as a function of speed and
age
The +Wint,dc can only occur when one
leg is doing positive work and the other is simultaneously doing negative
work, as shown by simultaneous increases and decreases in the
Wback and Wfront curves
(Fig. 1B). However, the shape
of these curves is highly dependent upon work transfers to/from the
Wv and upon the relative timing of the horizontal forces.
All positive work done by either leg during DC, after taking into account any
passive work done, results in either
+Wcom or
+Wint,dc (equation 6).
The effect of horizontal force timing
In adults and children at slow and intermediate walking speeds, the peak in
the Ff,back curve occurs at about 20-40% of the way through
DC (Fig. 2). With increasing
speed, this peak shifts towards the beginning of the DC period, until at the
highest walking speeds the peak often occurs during the single contact phase
preceding DC (negative phase angle values in
Fig. 5). The push performed by
the back foot before the DC period is an additional means of increasing the
step length independent of an increase in the limb angle as speed increases.
This progressive shift in phase angle is observed in all age groups except the
youngest children, who always have the Ff,back peak within
the DC period (see 3-4-year-old children in
Fig. 5). At high walking
speeds, by the time of mid-DC, when the horizontal velocity is maximal, the
Ff,back has already fallen to about half its peak value,
thereby reducing the work that one leg could perform against the other and
thus reducing Wint,dc. The negative phase angle of the
Ff,back peak occurs simultaneously with, and is related to,
the lifting of the heel of the back foot before the front leg contacts the
ground at high walking speeds; the resulting reduction in the
Ldc further reduces Wint,dc at these
speeds. This phase angle does not become negative in the youngest children,
but nevertheless their Wint,dc still goes down to zero,
suggesting that this shift in phase angle alone cannot explain the
Wint,dc decrease at the highest speeds.
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The change in the timing of the Ff,back peak during the step, especially whether it occurs during or before DC, affects the amount of work done by one leg against the other. The work done during the forward push of the back foot to accelerate the COM forward during the single contact phase is counted entirely as Wext and is needed to maintain the forward speed of the body. On the other hand, the same work done to accelerate the COM forward during DC is counted as either Wext or Wint,dc, the latter representing energy lost from the system.
The effect of passive work transfers
At low walking speeds, the mass-specific work done by one leg against the
other, +Wint,dc (J kg-1), is
minimised in both children and adults because there is almost no simultaneous
increase and decrease in Wback and
Wfront (Fig.
6). This is a consequence of the flattening of the
Wback and Wfront curves due to passive
work transfers between Wf,back or
Wf,front and Wv (see arrows on the low
speed traces of Fig. 6A).
Inparticular, the Wfront curve becomes almost completely
flat during the latter part of DC as the body rides up onto the front leg,
minimising any possibility of
+Wint,dc.
Nearly all the positive work done during DC results in
+Wcom (external work done during DC)
rather than in
+Wint,dc.
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At intermediate walking speeds,
+Wint,dc is at a maximum in both
children and adults. The amplitude of the Wf,back and
Wf,front curves is much greater than at low speeds, while
the amplitude of the Wv curve remains relatively
unchanged. Consequently, although the passive work transfers between
Wf,back or Wf,front and
Wv are similar (see arrows on the mid-speed curves of
Fig. 6B), the resulting changes
in the Wback and Wfront curves are
large and of opposite sign at the intermediate walking speeds. Consequently
the
+Wint,dc is large, approximately
equal to
+Wcom.
At high walking speeds, +Wint,dc
again becomes reduced in both children and adults. In this case, the
Wv decreases throughout the DC period (as the body is
falling down) while the Wf,back increases. Consequently,
the amplitude of Wback is greatly reduced due to the work
transfer from Wv to Wf,back (see arrow
on the high speed curves of Fig.
6C), decreasing the work available to perform
+Wint,dc. In addition, the work transfer
has the effect of limiting the increase in the Wback curve
to earlier in the DC period, reducing the time during which
Wback increases and Wfront
simultaneously decreases and reducing the opportunity to perform
+Wint,dc. The net result is that
+Wint,dc approaches zero at the highest
walking speeds in all age groups (Fig.
4).
Normalisation using the Froude number
Wint,dc normalised for body mass and distance travelled
shows an inverted U shape with speed and maximum value independent of age;
however, the speed range and the speed at which the maximum value is attained
clearly change with age (Fig.
4A). When comparing subjects of different size, it is often useful
to normalise the speed based on the assumption that the subjects move in a
dynamically similar manner, i.e. assuming all lengths, times and forces scale
by the same factors (Alexander,
1989). In a situation where inertia and gravity are of primary
importance, such as in walking, expressing the speed by the dimensionless
Froude number is appropriate:
![]() | (7) |
The peak Ff,back phase angle and the mass-specific
Wint,dc per unit distance are shown as a function of
Froude number in Fig. 7; it can
be seen that the differences observed between children and adults for the most
part disappear, with the exception of the youngest age group. Therefore,
despite the changes in body dimensions with age, the Froude number indicates
that people above 5 years of age walk in a dynamically similar way. Dynamic
similarity has been demonstrated in several previous studies involving various
other walking parameters as a function of size
(Cavagna et al., 1983;
DeJaeger et al., 2001
;
Minetti, 2001
).
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The phase angle of the Ff,back peak decreases as a
function of Froude number for all subjects
(Fig. 7A), attaining negative
values at speeds greater than Froude 0.5, except in the youngest subjects. A
Froude speed of 0.5 is approximately the speed at which people and animals
spontaneously change from a walk to a run or trot, as shown for differently
sized subjects, i.e. children (Cavagna et
al., 1983), for males and females (Herljac, 1995), for pygmies
(Minetti et al., 1994
), for
short stature growth-hormone-deficiency patients
(Minetti et al., 2000
) and for
different species (Alexander,
1989
). Although it is possible to walk at Froude speeds up to 1.0
(Fig. 7), higher speeds require
a different gait. Theoretically, this is because at a Froude speed of 1.0 the
centrifugal force as the subject rotates on a rigid leg over the ankle
(m·
f2/h)
becomes equal to the gravitational force (m·g)
holding the subject in contact with the ground, and the subject starts to have
an aerial phase.
The mass-specific Wint,dc done per unit distance also
appears to be independent of size when expressed as a function of Froude speed
(Fig. 7B). The `optimal'
walking speed, in terms of maximum % recovery (a measure of the relative
quantity of energy saved by the pendulum mechanism of walking), minimum
Wext and minimum energy consumption, occurs at a Froude
speed of about 0.2-0.3 in children and adults
(Cavagna et al., 1983;
DeJaeger et al., 2001
;
Willems et al., 1995
).
Notably, at this speed, the energy consumption is minimal despite the fact
that the Wint,dc is maximal
(Fig. 7B). This is likely
because Wint,dc is, at most, only 40% of
Wext, despite the fact that Wext is at
its minimum at this speed (Froude 0.2-0.3). Wint,dc
represents a futile work, energy lost from the system for no gain; presumably,
if there were no Wint,dc the energy consumption would be
even less. The mere fact that Wint,dc is a maximum at
about the same speed that the energy consumption is a minimum shows that
Wint,dc is not a major determinant of the energy cost of
walking in humans. Nevertheless, the previously reported peak efficiency,
measured in adults but not taking into account Wint,dc
(Willems et al., 1995
), was
underestimated by about 10%. The effect of Wint,dc on the
total work and efficiency of children remains to be seen.
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Acknowledgments |
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References |
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