## SUMMARY

The effect of age and body size on the total mechanical work done during
walking is studied in children of 3–12 years of age and in adults. The
total mechanical work per stride (*W*_{tot}) is measured as the
sum of the external work, *W*_{ext} (i.e. the work required to
move the centre of mass of the body relative to the surroundings), and the
internal work, *W*_{int} (i.e. the work required to move the
limbs relative to the centre of mass of the body, *W*_{int,k},
and the work done by one leg against the other during the double contact
period, *W*_{int,dc}). Above 0.5 m s^{–1}, both
*W*_{ext} and *W*_{int,k}, normalised to body
mass and per unit distance (J kg^{–1} m^{–1}), are
greater in children than in adults; these differences are greater the higher
the speed and the younger the subject. Both in children and in adults, the
normalised *W*_{int,dc} shows an inverted U-shape curve as a
function of speed, attaining a maximum value independent of age but occurring
at higher speeds in older subjects. A higher metabolic energy input (J
kg^{–1} m^{–1}) is also observed in children,
although in children younger than 6 years of age, the normalised mechanical
work increases relatively less than the normalised energy cost of locomotion.
This suggests that young children have a lower efficiency of positive muscular
work production than adults during walking. Differences in normalised
mechanical work, energy cost and efficiency between children and adults
disappear after the age of 10.

## Introduction

Children consume more energy per unit body mass to walk at a given speed
than do adults (DeJaeger et al.,
2001). The difference in the net mass-specific metabolic energy
cost per unit distance (i.e. the cost of transport, the energy required to
operate the locomotory machinery) between adults and children is greater the
higher the speed and the younger the subject. For example, at a speed of 1 m
s^{–1}, a 3–4-year-old has a net oxygen consumption 33%
greater than adults. This difference disappears by the age of 11–12
years.

In order to take into account the difference in size between children and
adults, the speed of progression can be normalised using the dimensionless
Froude number,
(*g**l*),
where *V̄*_{f} is mean walking
speed, ** g** is acceleration of gravity and

*l*is leg length (Alexander, 1989). In this case, the difference in the cost of transport between children and adults for the most part disappears. This indicates that, after the age of 3–4 years, the difference in the cost of transport may be explained mostly on the basis of body size (DeJaeger et al., 2001).

As previously observed in running
(Schepens et al., 2001), body
size can also affect the positive muscle–tendon work
(*W*_{tot}) performed during walking. *W*_{tot}
naturally falls into two categories: the external work
(*W*_{ext}), which is the work necessary to sustain the
displacement of the centre of mass of the body (*COM*) relative to the
surroundings, and the internal work (*W*_{int}), which is the
work that does not directly lead to a displacement of the *COM*. Only
some of *W*_{int} can be measured: (1) the internal work done
to accelerate the body segments relative to the *COM*
(*W*_{int,k}) and (2) the internal work done during the double
contact phase of walking by the back leg, which generates energy that will be
absorbed by the front leg (*W*_{int,dc}). On the contrary, the
internal mechanical work done for stretching the series elastic components of
the muscles during isometric contractions, to overcome antagonistic
co-contractions, to overcome viscosity and friction cannot be directly
measured (although this unmeasured internal work will affect the efficiency of
positive work production; Willems et al.,
1995).

Walking is characterised by a pendulum-like exchange between the kinetic
and potential energy of the *COM*. In children, the `optimal speed' at
which these pendulum-like transfers are maximal increases progressively with
age from 0.8 m s^{–1} in 2-year-olds up to 1.4 m
s^{–1} in 12-year-olds and adults
(Cavagna et al., 1983). At all
ages, the optimal speed is close to the speed at which the mass-specific work
to move the *COM* a given distance, *W*_{ext}, is at a
minimum. Above the optimal speed, the energy transfers decrease. This decrease
is greater the younger the subject. The decreased transfers result in a
greater power required to move the *COM*: at 1.25 m
s^{–1}, the mass-specific external power
(*Ẇ*_{ext}) is twice as great in
a 2-year-old child than in an adult. When normalising the speed with the
Froude number, *Ẇ*_{ext} is
similar in children and in adults.

The work done by one leg against the other (*W*_{int,dc})
was not counted in the `classic' measurements of the positive muscular work
done during walking, which was calculated as
*W*_{tot}=*W*_{ext}+*W*_{int,k}
(Cavagna and Kaneko, 1977;
Willems et al., 1995);
consequently, *W*_{int,k} has previously been referred to
simply as *W*_{int}. Using force platforms, Bastien et al.
(2003) studied the effect of
speed and age (size) on *W*_{int,dc} in 3–12-year-old
children and in adults. *W*_{int,dc} as a function of speed
shows an inverted **U**-shape curve, attaining a maximum value of
approximately 0.15–0.20 J kg^{–1} m^{–1},
which is independent of size but occurs at higher speeds in larger subjects.
The differences due to size disappear for the most part when
*W*_{int,dc} is normalised with the Froude number.

These observations indicate that, as for the energy expenditure, the
speed-dependent changes in *W*_{ext} and
*W*_{int,dc} are primarily a result of body size changes. To
our knowledge, the internal work due to the movement of the limbs relative to
the *COM* (*W*_{int,k}) has never been measured in
walking children. In the present study, we measure simultaneously
*W*_{ext}, *W*_{int,k} and
*W*_{int,dc} in children and in adults walking at different
speeds and calculate *W*_{tot} and efficiency.

*W*_{tot} is calculated from *W*_{ext},
*W*_{int,k} and *W*_{int,dc} over a complete
stride, taking into account any possible energy transfers that would reduce
the muscular work done. Transfers between *W*_{ext} and
*W*_{int,k} were analysed by Willems et al.
(1995). Energy transfers
between the back and the front legs in the computation of
*W*_{int,dc} were discussed by Bastien et al.
(2003). In the present study,
we analyse the possible transfers between *W*_{int,k} and
*W*_{int,dc}; we show that, during the double contact phase,
some positive work done by the back leg in pushing the body forwards can
result in an increase of the kinetic energy of the front leg moving backwards
relative to the *COM*.

The total mechanical work is compared with the energy expenditure to evaluate the efficiency of positive work production. It is shown that children younger than 6 years are less efficient than adults in producing positive work during walking.

## Materials and methods

Measurements of the mechanical work during walking were performed on the
same subjects and during the same sessions as the previously published
measurements of the mechanical work during running
(Schepens et al., 2001). The
details of the methods used to compute *W*_{ext} are given in
Cavagna (1975) and Willems et
al. (1995), those used to
compute *W*_{int,k} are given in Cavagna and Kaneko
(1977) and Willems et al.
(1995), and those used to
compute *W*_{int,dc} are given in Bastien et al.
(2003); these methods are only
summarised here.

### Subjects and experimental procedure

Twenty-four healthy children of 3–12 years of age and six healthy adults participated in the experiments. They were divided into six age groups: the 3–4-year-old group included subjects 3 years to <5 years old; the 5–6-year-old group included subjects 5 years to <7 years old, etc. Each group comprised 4–6 children; the averaged physical characteristics of these groups are given in table 1 of Schepens et al. (2001). Written informed 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.

Subjects were asked to walk across a 6 m-long force platform at different
speeds. The mean speed (*V̄*_{f})
was measured by two photocells placed at the level of the neck and set
0.7–5.5 m apart depending upon the speed. In each age group, the data
were gathered into speed classes of 0.13–0.14 m s^{–1}
(0.5 km h^{–1}).

### Measurement of positive mechanical work done per stride

The kinetic internal work (*W*_{int,k}), the external work
(*W*_{ext}) and the work done during the double contact phase
(*W*_{int,dc}) were measured simultaneously on 531 complete
strides according to the procedures described below. A stride was selected for
analysis only when the subject was walking at a relatively constant average
height and speed. Specifically, the sum of the increments in both vertical and
forward velocity of the *COM* could not differ by more than 25% from
the sum of the decrements (Cavagna,
1975). According to these criteria, the average vertical force was
within 4% of the body weight, and the difference in the forward velocity of
the *COM*, from the beginning to the end of the selected stride, was
less than 5% of *V̄*_{f} in 95%
of the trials (at very low speeds, it was less than 10% of
*V̄*_{f}).

### Measurement of positive internal work due to the segment movements per stride

*W*_{int,k} was computed from the segment movements and
anthropometric parameters. The body was divided into 11 rigid segments
(Willems et al., 1995): one
head/neck/trunk segment and two thigh, two shank, two foot, two upper arm and
two lower-arm/hand segments. The head/neck/trunk segment and the right limb
segments were delimited by infrared emitters placed at their extremities (see
table 2 in Schepens et al.,
2001). The coordinates of the infrared emitters in the forward and
vertical directions were measured every 5 ms by means of a Selspot
II® system (Selcom®, Göteborg, Sweden). The
coordinates were smoothed with a cubic spline function
(Dohrmann et al., 1988).

A `stick man' of the position of each segment relative to the
head/neck/trunk segment was constructed every frame
(Fig. 1). The movements of the
head/neck/trunk segment relative to the *COM* were ignored because
their contribution to *W*_{int,k} is negligible
(Willems et al., 1995). The
left side of the subject, opposite to the camera, was reconstructed from the
right-side data on the assumption that the movements of the segments of one
side were equal and 180° out of phase with the other side. The angular
velocity of each segment and the translational velocity of its centre of mass
relative to the head/neck/trunk segment were calculated from the derivative of
their position *versus* time relationship. The position of the centre
of mass and the moment of inertia of the body segments were calculated using
the anthropometric parameters of table 2 in Schepens et al.
(2001).

The kinetic energy of each segment due to its displacement relative to the
head/neck/trunk segment and due to its rotation was then calculated as the sum
of its translational and rotational energy. The kinetic energy *versus*
time curves of the segments in each limb were summed. The internal work due to
the movements of the upper limbs,
, was then calculated by
adding the increments in their kinetic energy–time curves
(Fig. 1). In order to minimise
errors due to noise, the increments in kinetic energy were considered to
represent positive work actually done only if the time between two successive
maxima was greater than 20–110 ms, according to the speed of
progression. The same procedure was used with the kinetic energy–time
curves of the lower limbs to compute the internal work due to their movements,
(Fig. 1).
*W*_{int,k} was then computed as the sum of
and
. This procedure allowed
energy transfers between segments of the same limb but disallowed any energy
transfers between different limbs (Willems
et al., 1995).

### Measurement of positive external work per stride

*W*_{ext} was calculated from the vertical and forward
components of the force exerted on a 6 m×0.4 m force platform mounted 25
m from the beginning of a 40 m walkway. The platform was made of 10 different
plates, similar to those described by Heglund
(1981). The plates measured
the fore–aft and vertical components of the forces exerted by the feet
on the ground. The responses were linear within 1% of the measured value for
forces up to 3000 N. The natural frequency of the plates was 180 Hz.

The signals from the platform were digitised synchronously with the camera
system. The integration of the vertical and forward components of the ratio
force/mass yielded the velocity changes of the *COM*, from which the
kinetic energy (*E*_{k}) was calculated after evaluation of the
integration constants (Cavagna,
1975; Willems et al.,
1995). The kinetic energy of the *COM* is equal to
*E*_{k}=½*m*(**V _{f}^{2}+V_{v}^{2}**),
where

*m*is body mass and

**V**

_{f}and

**V**

_{v}are the forward and vertical components, respectively, of the velocity of the

*COM*. A second integration of the vertical velocity yielded the vertical displacement of the

*COM*, from which the gravitational potential energy (

*E*

_{p}) was calculated. Potential energy of the

*COM*is equal to

*E*

_{p}=

*m*

*g*S_{v}, where

**is the gravitational constant and**

*g***S**

_{v}is the vertical displacement of the

*COM*.

The mechanical energy of the *COM* (*E*_{ext}) was
the sum of the *E*_{k} and *E*_{p} curves over a
complete stride. *W*_{ext} was the sum of the increments in the
*E*_{ext} curve (Fig.
1). Similarly, *W*_{k}, the positive work done to
sustain the velocity changes of the *COM*, was the sum of the
increments of the *E*_{k} curve, and *W*_{p},
the positive work done against gravity, was calculated from the increments in
the *E*_{p} curve (Fig.
1). The increments in mechanical energy were considered to
represent positive work actually done only if the time between two successive
maxima was greater than 20 ms.

Walking can be compared to a pendular mechanism where potential energy is
transformed into kinetic energy and *vice versa*
(Cavagna et al., 1976). The
recovery (*R*) of energy due to this pendular mechanism was estimated
by:
1
This equation differs slightly from the `classical' equation of recovery (e.g.
Cavagna et al., 1976;
Willems et al., 1995), which
is given by:
2
where *W*_{f} is the sum of the increments of the
*E*_{kf} curve
(),
and *W*_{v} is the sum of the increments of the
*E*_{p}+*E*_{kv} curve
().
Equation 2 evaluates the amount
of energy recovered through the transfer between energy due to the forward
motion of the *COM* (*E*_{kf}) into energy due to its
vertical motion (*E*_{p}+*E*_{kv}). *R*
differs slightly from *R*_{c} because *E*_{kv}
is included in the *W*_{k} term of
equation 1 and is included in the
*W*_{v} term of equation
2.

### Measurement of positive internal work made by one leg against the other during double contact

In walking, during the double contact phase, positive work is done by the
back leg pushing forwards while negative work is done by the front leg pushing
backwards. The forces exerted by each lower limb on the ground were measured
separately. The powers generated against the external forces by the front and
back legs were calculated from the dot product of the vertical and horizontal
components of the ground reaction forces acting under each leg multiplied,
respectively, by the vertical and horizontal velocity of the *COM*
(Donelan et al., 2002;
Bastien et al., 2003). The
positive work done by the ground reaction forces was calculated independently
for the back (*W*_{back}) and the front
(*W*_{front}) limb from the time-integral of the power curves,
taking into account any energy transfers
(Bastien et al., 2003). Part of
the positive work done by the limbs results in an acceleration and/or an
elevation of the *COM*. In order not to count the same work twice, the
positive muscular work realised by one leg against the other during double
contact *W*_{int,dc} was evaluated by:
3

Since *W*_{int,dc} was computed from the individual limb
ground reaction forces, it was necessary that the two feet were on different
plates during the double contact phase. This requirement could not often be
fulfilled over consecutive double contact phases. For this reason,
*W*_{int,dc} was measured on a single double contact phase of
the stride and the result was doubled to obtain the
*W*_{int,dc} for the whole stride. In 10% of the trials, two
successive measurements of *W*_{int,dc} were possible within a
stride; the two measurements were not statistically different
(*t*=–0.995, *P*<0.32, *N*=53).

### Evaluation of total positive muscular work done each stride

In order to compute the total positive muscular work done
(*W*_{tot}), it is necessary to account for the possible energy
transfers between *W*_{ext}, *W*_{int,k} and
*W*_{int,dc}. Willems et al.
(1995) showed that
*W*_{tot} was best evaluated when no transfers of energy were
allowed between *W*_{ext} and *W*_{int,k}.
Bastien et al. (2003) carefully
analysed which part of the positive mechanical work done by the legs can be
attributed to *W*_{ext} and to *W*_{int,dc}. In
the following paragraphs, we analyse the possible energy transfers between
*W*_{int,k} and *W*_{int,dc}.

During the double contact phase, the push of the back leg that increases
the forward speed of the *COM* relative to the surroundings also
increases the backward speed of the front limb relative to the *COM*.
This is shown, for example, in the first period of double contact in
Fig. 1. The back leg (in this
case, the left leg) does positive muscular work to lift and to accelerate the
*COM* (increment *a* in *E*_{ext};
Fig. 1). The back leg also does
positive muscular work on the front leg (increment *b* in
*W*_{int,dc}; Fig.
1). The work done on the front leg can appear as an increase in
the rotational and translational kinetic energy of the front leg relative to
the *COM* (increment *c* in
;
Fig. 1) rather than just being
absorbed and dissipated as negative work in the muscles of the front leg
(decrement *d* in *W*_{int,dc};
Fig. 1). In order to allow this
transfer, the and
*W*_{int,dc} curves of each leg are added instant-by-instant.
Due to this transfer during the double contact phase, the increment *e*
in
(Fig. 1) is smaller than
increment *c* in
. The sum of the
increments of the resulting curve,
, is the internal work
done on a lower limb
().

The total positive work done by the muscles during walking, after allowing
reasonable energy-saving transfers, is:
4
The total positive mechanical work done, not allowing energy transfers between
*W*_{ext}, *W*_{int,k} and
*W*_{int,dc}, would be:
5

### Normalisation of the mechanical work done during a stride

In order to compare subjects of different body size, the work done per
stride was divided by the subject's body mass. This mass-specific work can
then be either divided by the stride length to obtain the work done per unit
distance or divided by the stride period to obtain the mean mechanical power
expended during walking. In the sections that follow, the work symbols
(*W*) usually refer to the mass-specific work done per unit distance (J
kg^{–1} m^{–1}), and the symbols with a dot
(*Ẇ*) refer to the mass-specific power (W
kg^{-1}).

### Efficiency of positive work production

Efficiency of positive work production is calculated as the ratio of the
total positive muscular mechanical power to the net steady-state energy
consumption rate (*Ṁ*_{net}). The
*Ṁ*_{net} is the energetic
equivalent of the total oxygen consumption rate minus the standing oxygen
consumption rate. The total oxygen consumption rate for children was taken
from the data of DeJaeger et al.
(2001).

## Results

*Internal power due to the movements of the limb segments relative
to the* COM

As walking speed goes up, the velocity of the head/neck/trunk segment
relative to the surroundings increases. As a consequence, the backward
velocity of the supporting limb relative to the *COM* increases.
Furthermore, the stride length becomes longer and the time to reset the limbs
becomes shorter. As a result, the forward velocity of the swing leg increases
relative to the *COM*. For these reasons, the mass-specific
*W*_{int,k} done per stride increases with speed, both in
children and in adults (top row, Fig.
2). Note that, at a given speed, children and adults do the same
amount of *W*_{int,k} per unit body mass each stride.

Since the stride frequency (*f*) at a given speed is higher in
children than in adults (middle row, Fig.
2), the mass-specific internal power
(*Ẇ*_{int,k} = work per stride
multiplied by stride frequency) is greater in the children (bottom row,
Fig. 2). The difference is
greater at high speeds and in the young subjects and becomes negligible at
speeds less than 1 m s^{–1} and in subjects older than 10
years.

### External, internal and total mechanical work

At all ages, the recovery (*R*;
equation 1) of mechanical energy
*via* the pendulum-like transfer between *E*_{p} and
*E*_{k} attains a maximum of ∼65% at intermediate walking
speeds (circles in upper row of Fig.
3), although the speed of the maximum *R* increases with
age during growth. In the same panels of
Fig. 3, the crosses show the
recovery *R*_{c} (equation
2). *R* and *R*_{c} are very similar because
the variations of *E*_{kv} are small compared with the
variations of *E*_{p} and *E*_{kf}. Since the
vertical velocity of the *COM* is nil when it reaches its highest and
lowest point, the maximum and minimum of the *E*_{p} curve are
likely to be the same as those of the
*E*_{p}+*E*_{kv} curve.

The mass-specific external work per unit distance
(*W*_{ext}; second row, Fig.
3) reaches a minimum at a speed slightly lower than where
*R* is maximal. Above this `optimal' speed, *W*_{ext}
increases more in children than in adults since the decrease of *R*
with speed is greater in children than in adults (the effect of speed and
growth on the link between *R* and *W*_{ext} was
discussed in detail by Cavagna et al.,
1983).

Both in children and in adults, the mass-specific
*W*_{int,dc} per unit distance shows an inverted **U**-shape
curve as a function of speed (third row of panels in
Fig. 3).
*W*_{int,dc} attains a maximum value of ∼0.15 J
kg^{–1} m^{–1}, independent of age, although the
speed at which this maximum occurs increases from ∼1.1 m
s^{–1} at the age of 3 years to ∼1.6 m s^{–1}
above the age of 10 years. As a consequence, the maximum mass-specific power
developed by one leg against the other (which is the product of mass-specific
*W*_{int,dc} per unit distance multiplied by speed) increases
with age, from 0.15 W kg^{–1} at the age of three to 0.25 W
kg^{–1} above the age of 10.

The mass-specific *W*_{int,k} per unit distance (fourth row,
Fig. 3) represents the work
done to move the limbs relative to the centre of mass; it is the sum of
.
The difference between *W*_{int,k} in children and in adults is
greater the younger the subject and the higher the speed and becomes
negligible after the age of 10.

The mass-specific internal work per unit distance, *W*_{int}
(fifth row, Fig. 3), is not
equal to the sum of *W*_{int,dc} and
*W*_{int,k}. Indeed, due to the energy transfer between the
*W*_{int,dc} and
curves
(Fig. 1),
*W*_{int} is very similar to *W*_{int,k}.
Compared with adults, *W*_{int} is noticeably greater in
subjects younger than 11 years and at speeds higher than 1.5 m
s^{–1}.

The total mass-specific muscular work per unit distance,
*W*_{tot} (bottom row, Fig.
3), is calculated as the sum of the mass-specific external work
per unit distance, *W*_{ext,} plus the mass-specific internal
work per unit distance, *W*_{int}. The arrows in the bottom
panels indicate the speed at which the net energy cost is minimal (see
fig. 3 of
DeJaeger et al., 2001).
Contrary to the net energy cost of walking, the curve of
*W*_{tot} as a function of speed does not have a well-defined
minimum, although this could occur at speeds lower than the ones explored.
Above 0.5 m s^{–1}, *W*_{tot} increases with
walking speed more steeply in children than in adults. A two-way
repeated-measures analysis of variance with contrasts (SuperANOVA, 1.11) was
made to determine the speed at which *W*_{tot} differs between
children and adults. Specifically, the effect of speed was analysed within
each age group, and the speed at which *W*_{tot} in children
became significantly different from that in adults was determined
(Table 1). In children younger
than 11 years, *W*_{tot} is always greater at high walking
speeds. The contrast analysis also shows a statistical difference in
*W*_{tot} between the 5–6 years group and the adults,
even at very low speeds. This difference is of the order of 0.04 J
kg^{–1} m^{–1}, which represents ∼5% of the
adult values. Even if this difference is statistically significant it is
unlikely to be biologically significant.

## Discussion

In the present study, the internal work done to accelerate the limb segment
relative to the *COM* (*W*_{int,k}) is measured for the
first time in children walking at different speeds. *W*_{int,k}
is measured simultaneously with the external work (*W*_{ext})
and the work done by one leg against the other (*W*_{int,dc})
per stride. The *W*_{int,k}, *W*_{ext} and
*W*_{int,dc} are combined, taking into account possible energy
transfers, to get the total mechanical work (*W*_{tot}) per
stride. The results obtained for the mass-specific *W*_{ext}
and *W*_{int,dc} per unit distance are in agreement with those
of Cavagna et al. (1983) and
Bastien et al. (2003),
respectively.

### Efficiency of positive work production during walking

The efficiency of positive work production by the muscles and tendons
during walking is calculated as the ratio of the total mechanical power
(*Ẇ*_{tot}; i.e. the work per
stride multiplied by the stride frequency) to the net energy consumption rate
(*Ṁ*_{net}; i.e. the gross energy
consumption rate minus the standing energy expenditure rate):
6

The total mechanical power is shown as a function of walking speed in the
top row of Fig. 4.
*Ṁ*_{net}, the cost of operating
the locomotory machinery, is presented in the middle row of
Fig. 4 (data from
DeJaeger et al., 2001). At all
ages, both *Ẇ*_{tot} and
*Ṁ*_{net} increase with walking
speed, although the increase is greater the younger the subject. The
differences in *Ẇ*_{tot} and in
*Ṁ*_{net} between adults and
children disappear after the age of 10.

The efficiency of positive work production is presented in the bottom row
of Fig. 4. Assuming the
precision of the mechanical power measurement is as good as 0.1 W
kg^{–1}, then at low speeds, where the total mechanical power is
small, a change in *Ẇ*_{tot} of
0.1 W kg^{–1} would result in a change in the efficiency of>
0.05. For this reason, the values of efficiency are considered to be
robust only at speeds above ∼0.75 m s^{–1}.

In adults, the efficiency reaches a maximum of 0.30–0.35 at ∼1.25
m s^{–1}; at lower and higher speeds the efficiency decreases.
These values are in good agreement with those of Cavagna and Kaneko
(1977) and Willems et al.
(1995). At speeds greater than∼
1 m s^{–1}, the efficiency of positive work production is
greater than the maximal efficiency of the conversion of chemical energy into
positive work by muscles (≤0.25;
Dickinson, 1929), suggesting
that elastic energy is stored during the phase of negative work to be
recovered during the following phase of positive work
(Willems et al., 1995).

Before the age of seven, the increase in
*Ṁ*_{net} cannot be explained
only by an increase in *Ẇ*_{tot}.
Part of the extra cost of walking in young children appears to be due to a
reduction in the efficiency of positive work production. For example, in
3–4-year-old children, the efficiency is 0.15–0.25, while after
the age of six the efficiency is similar in children and adults. The lower
efficiency in young children could be explained, at least in part, by the
immature muscular pattern observed during walking before the age of five
(Sutherland et al., 1988),
which may require more isometric and/or antagonistic contractions to stabilise
the body segments. These contractions would result in an increased energy
expenditure without any increase in the mechanical work
(Griffin et al., 2003).

### Contribution of external and internal power to the total mechanical power during walking

At speeds below 1 m s^{–1}, the internal power is smaller
than the external power in all age groups
(Fig. 5). This is due to the
fact that the two components of the internal power tend towards zero as speed
approaches zero. On the contrary, the external power, specifically the power
necessary to sustain the vertical movements of the *COM*, does not tend
towards zero as speed approaches zero
(Cavagna et al., 1983). As
speed is increased above 1 m s^{–1}, *W*_{int}
increases faster than *W*_{ext}, and at high walking speed it
is 20–40% greater than *W*_{ext} (except in the
3–4-year olds).

In the present study, the two components of the internal power, and the
energy transfers between them, are taken into account for the first time in
the computation of the total muscular power of walking.
*Ẇ*_{tot}, based upon
equation 4 (i.e. allowing all
reasonable energy transfers, as explained in the Materials and methods), is
shown by the solid line in Fig.
5. If, on the other hand, the total power is based upon
equation 5 (i.e. assuming no
energy transfers), then the resulting total power is shown by the upper broken
line in Fig. 5. At the other
limit, if the total power is calculated simply as the sum of
*Ẇ*_{ext}+*Ẇ*_{int,k},
ignoring *Ẇ*_{int,dc} as has been
done in the past, the result is the lower broken line in
Fig. 5. It can be seen that at
all ages, *Ẇ*_{int,dc} represents
a small fraction of the total muscular power spent during walking:
*Ẇ*_{int,dc} represents ∼10%
of the total power at intermediate speeds, decreasing to zero at low speeds
and <5% at high speeds.

### Normalisation for body size

At a given speed, *W*_{int,k} per unit body mass and per
stride is the same in all age groups (top row,
Fig. 2) in spite of large
differences in stride frequency, movement amplitude and limb dimensions. In
other words, this normalisation of *W*_{int,k} makes it
independent of the amplitude/duration of the oscillation and takes into
account the different dimensions of the children and adults.

Different size subjects can be compared at equivalent, size-independent speeds if the mean velocity is normalised using the Froude number (Alexander, 1989). This assumes that children and adults move in a dynamically similar manner, i.e. all lengths, times and forces scale by the same factors. In a situation such as walking, where inertia and gravity are of primary importance, size-dependent speed differences should disappear if the assumption of dynamic similarity is justified.

The upper panel of Fig. 6
shows *Ẇ*_{int} as a function of
the Froude speed. For the most part, the differences between children and
adults disappear, although at the same Froude speed the data of the smaller
subjects tend to be lower than those of the larger subjects, indicating that
not all differences can be explained simply on the basis of size. The same can
be seen for *Ẇ*_{tot}
(Fig. 6, lower panel).
Likewise, when *Ẇ*_{ext},
*Ṁ*_{net} and
*Ẇ*_{int,dc} are expressed as a
function of the Froude number, the differences between children and adults
also tend to disappear (Cavagna et al.,
1983; DeJaeger et al.,
2001; Bastien et al.,
2003). These observations indicate that, after the age of three,
the differences observed in the mechanics and energetics of walking during
growth may be explained, for the most part, on the basis of dynamic
similarity. The fact that efficiency is lower in very young children compared
with in adults suggests that factors other than size scaling, such as
developmental changes in the neuromuscular system, may play a role before the
age of six.

## List of symbols

- COM
- centre of mass of the body
*E*_{ext}- mechanical energy of the
*COM* - internal energy of the lower limb
- kinetic energy change of the lower limbs due to their velocity relative
to the
*COM* - kinetic energy change of the upper limbs due to their velocity relative
to the
*COM* *E*_{k}- kinetic energy of the
*COM* *E*_{kf}- kinetic energy due to the forward motion of the
*COM* *E*_{kv}- kinetic energy due to the vertical motion of the
*COM* *E*_{p}- gravitational potential energy of the
*COM* - f
- stride frequency
- g
- gravitational acceleration
- l
- leg length
- m
- body mass
*Ṁ*_{net}- net steady-state energy consumption rate
- R
- recovery of mechanical energy through a pendular mechanism
**S**_{v}- vertical displacement of the
*COM* *V̄*_{f}- mean walking speed
**V**_{f}- forward component of the velocity of the
*COM* **V**_{v}- vertical component of the velocity of the
*COM* *W*_{back}- positive muscular work done by the back leg during double contact
*W*_{ext}- external work
*Ẇ*_{ext}- mass-specific external power
*W*_{f}- work done to sustain the forward motion of the
*COM* *W*_{front}- positive muscular work done by the front leg during double contact
*W*_{int}- internal work
*Ẇ*_{int}- mass-specific internal power
*W*_{int,dc}- positive muscular work realised by one leg against the other during double contact
*Ẇ*_{int,dc}- mass-specific power expended by one leg against the other during the double contact phase
*W*_{int,k}- work required to move the limbs relative to the
*COM* *Ẇ*_{int,k}- mass-specific internal power expended to move the limbs relative to the
*COM* - internal work done on a lower limb
- internal work due to the movements of the lower limbs
- internal work due to the movements of the upper limbs
*W*_{k}- work done to accelerate the
*COM* *W*_{p}- potential gravitational energy changes of the
*COM* *W*_{tot}- total mechanical work
*Ẇ*_{tot}- total mass-specific mechanical power
*W*_{v}- work done to sustain the vertical motion of the
*COM*

## ACKNOWLEDGEMENTS

This study was supported by the Fonds National de la Recherche Scientifique of Belgium.

- © The Company of Biologists Limited 2004