## SUMMARY

Mechanical energy expenditure was investigated in children who are just
learning to walk and compared with adult mechanical energy expenditure during
walking. First, we determined whether the inverted pendulum (IP) mechanism of
energy exchange was present in toddlers. It seems that new walkers partially
make use of this energy saving mechanism, but it is less efficient than in
adults. The reduced recovery values (*R*=40% at optimal speeds in
toddlers compared to 70% in adults) can be explained by their low
self-selected walking speed in combination with their tossing gait (large
vertical oscillations of the body) and by the observation that during as much
as 25–50% of the gait cycle kinetic and potential energy are oscillating
in-phase.

The second step was to calculate positive external mechanical work
(*W*_{ext}). Since the IP mechanism is less efficient in
toddlers, more mass-specific positive work has to be performed to lift and
accelerate the centre of mass than in adults walking at the same speed, even
when differences in body size are taken into account.

The amount of positive internal work (*W*_{int,k}) necessary
to move the body segments relative to the centre of mass was the third
parameter we calculated. In toddlers *W*_{int,k} is largely
determined by the kinetic energy of the lower limb. Compared to adults,
toddlers have to perform less mass-specific work per unit distance to
accelerate the body segments since the upper body is kept relatively stiff
during walking and there is no arm swing.

Apart from work performed on the centre of mass and work performed to move the body segments relative to the centre of mass, when walking some work is also performed during double contact as both legs are pushing against each other. Two methods were used to calculate this amount of work, both leading to the same conclusions. Mass-specific work during double contact is small in toddlers compared to adults because of their low walking speed.

Finally the total amount of mechanical work performed in toddlers was
compared to the work production observed in adults. *W*_{ext}
seems to be the major determinant for total mechanical energy expenditure. At
intermediate froude numbers work production is comparable between adults and
toddlers, but at low and high froude numbers *W*_{tot}
increases due to the steep increases in *W*_{ext}. Despite the
fact that mechanical work requirements in toddler gait are underestimated if
work during double contact is not taken into account, it is not a major
determinant of the energy cost of walking.

## Introduction

During walking metabolic energy is consumed, even if the average walking speed is constant and there is no net change in height of the body. Energy is lost at each step and has to be put into the system again. The total amount of positive muscle–tendon work that has to be performed can be divided into two categories: external work and internal work.

External work (*W*_{ext}) is the amount of work performed to
lift and accelerate the centre of mass. To minimize *W*_{ext}
adults make use of an imperfect inverted pendulum (IP) mechanism of energy
exchange (Fig. 1), which was
first formulated by Cavagna and others (Cavagna et al.,
1963, 1966,
1976) over 40 years ago. The IP
is characterized by an out-of-phase oscillation of potential
(*E*_{p}) and kinetic energy (*E*_{k}) allowing
energy exchange to occur. At preferred walking speed, as much as 70% of the
required external mechanical energy can be recovered due to this energy saving
mechanism (Cavagna et al.,
1977). The other 30% of external mechanical energy is lost from
the system and must be supplied by the muscles.

Internal work comprises all the work performed by the muscles and tendons
that does not directly lead to a displacement of the centre of mass. In the
past only the work necessary to accelerate the body segments relative to the
centre of mass (classical internal work, *W*_{int,k}) was
measured. Recently both Donelan et al.
(2002; individual limbs
method) and Bastien et al.
(2003;
*W*_{int,dc}) developed a method to determine another component
of internal work, the work done during double contact when both legs are
working against each other. At this time the propulsive back leg has to
overcome the energy absorbed by the braking front leg to maintain a constant
walking speed.

Children aged between 3 and 11 years old consume more energy per unit body
mass to walk at a given speed than do adults
(Schepens et al., 2004). Both
*W*_{ext} and *W*_{int,k} are larger at speeds
above 0.5 m s^{-1}. *W*_{int,dc} reaches a maximum at
lower speeds in younger subjects. However, differences between adults and
children above the age of 3 years disappear when mechanical work is expressed
as a function of the dimensionless froude number that takes into account the
differences in body proportions (Schepens
et al., 2004). This suggests it is the small stature of children
that makes them consume more energy than adults when walking at a given speed.
In other words, children above the age of 3 years are dynamically similar to
adults, at least with respect to the energetics of walking.

Is this also the case in toddlers who are just learning to walk? They are even smaller and show markedly different body proportions compared to adults. Balance problems and immature control of movement are compromising factors that make their gait pattern different from mature walking. Previous studies on infant walking have revealed differences between adults and toddlers in the spatio–temporal gait parameters, joint kinematics and ground reaction force patterns (e.g. Statham and Murray, 1971; Burnett and Johnson, 1971; Endo and Kimura, 1972; Sutherland et al., 1980; Grimshaw et al., 1998). Considering the toddler as a mechanical system, the observed kinematic and kinetic differences could result in different energy and power requirements of the system.

The aim of this paper is to find out whether toddlers differ from adults in
mechanical energy production. The amount of external and internal work was
calculated and evaluated over a range of speeds. To find out whether toddlers
make use of the IP mechanisms of energy exchange the time profiles of
*E*_{p} and *E*_{k} were considered and
percentages of recovered energy were calculated. We were also particularly
interested in the work performed during double contact, when the front and
back limbs are working against each other. The double support phase accounts
for a substantial portion of the gait cycle. We wanted to find out whether
this prolonged phase of double support had an effect on the mechanical work
requirements.

## Materials and methods

### Study subjects

Nine healthy children aged between 12 and 18 months participated in this study. Their walking experience ranged from 2 weeks to 6 months. Walking experience was defined as the time period between the onset of independent walking (ability to perform 2–3 consecutive steps) and the time of testing. Detailed information on the study subjects can be found in Table 1.

The ethical review board of the University of Antwerp approved the study protocol. Prior to participation, parents gave their informed consent. All experiments were carried out according to the guidelines stated in the Declaration of Helsinki.

### Experimental set-up

Data were collected at the HIKE campus of the department of Health Care (Hoger Instituut voor Kinesitherapie en Ergotherapie, Hogeschool Antwerpen, Belgium). The experimental set-up consisted of an instrumented walkway (3 m×1.5 m) surrounded by six infrared cameras (Mcam 460, 250 Hz; Vicon Motion Systems, Oxford, UK). Two force platforms (AMTI, MA, USA; 0.5 m×0.4 m, 250 Hz) were built into the walkway to record ground reaction forces under the left and right foot separately.

The Helen-Hayes marker set-up was used for measuring full body kinematics (Fig. 2). The retro-reflective markers (14 mm) were sewn on to a tight-fitting suit to prevent problems with `marker plucking' in young children. Foot markers were attached to socks or soft leather shoes.

The children were encouraged to walk over the platform towards a parent or experimenter at self-selected speeds. We tried to obtain five successful trials for each individual. A successful trial was defined as a trial for which ground reaction force measurements of both feet were available during at least one complete stride and all markers were visible throughout the trial. Sometimes the children became tired and failed to cooperate further before a sufficient number of trials could be obtained. The number of successful trials per individual is given in Table 1. After performing the calculations, the results from the different trials of each individual were averaged to prevent pseudo-replication.

In energetic analysis it is generally required that the average walking
speed is fairly constant over the whole trial. Toddlers, however, are
constantly accelerating and decelerating, taking a few steps and stopping
again. The average net speed change over a trial amounts to 0.10 m
s^{-1}. Because of their low walking speed this accounts for an
average variation in speed of 25%.

### Data analysis The body centre of mass (COM)

The three-dimensional (3-D) velocities of the COM and the vertical oscillations were determined by integrating the resultant of the ground reaction forces underneath both feet. This technique was first formulated by Cavagna (1975) and is well described in literature (e.g. Cavagna et al., 1983; Willems et al., 1995). Therefore it is only briefly discussed here.

If air resistance is neglected, the 3-D accelerations of the COM can be
calculated using Equations 1,
2,
3. Body weight was determined by
dividing the time impulse of the resultant vertical ground reaction force
during an entire gait cycle (period between two contacts of the same foot,
expressed from 0 to 100% in Fig.
1) by stride time, thus:
1
2
and
3
where (*A*_{x}, *A*_{y}, *A*_{z})
= 3-D linear accelerations of the COM, *F*_{x} = lateral force
component, *F*_{y} = fore–aft force component,
*F*_{z} = vertical force component and *M*_{tot}
= body mass

The 3-D linear velocities of the COM (*V*_{x},
*V*_{y}, *V*_{z}) were determined by numerical
integration of the 3-D accelerations (*A*_{x},
*A*_{y}, *A*_{z}). Integration constants were
determined so the average *V*_{x}, *V*_{y} and
*V*_{z} equalled the average 3-D linear velocities measured by
the video system. The vertical displacement (*z*) of the COM was
calculated by numerical integration of *V*_{z}.

### The inverted pendulum mechanism

Potential (*E*_{p}) and kinetic (*E*_{k})
energy fluctuations were calculated according to Equations
4 and
5 and plotted as a function of
gait cycle duration:
4
and
5
To find out how well the IP mechanism is working in toddlers, recovery values
(*R*) were calculated (Equation
6) and plotted as a function of walking speed (*v*) and
froude number [*v*^{2}/(9.81×leg length)]. *R* is
a measure of the pendulum-like transfer between *E*_{k} and
*E*_{p} observed in (mature) walking
(Cavagna et al., 1976).
6
where Δ^{+}*E*_{p} = the sum of the positive
increments in *E*_{p} over an integral number of steps,Δ
^{+}*E*_{k} = the sum of the positive increments
in *E*_{k} over an integral number of steps andΔ
^{+}*E*_{tot} = the sum of the positive
increments in the total mechanical energy curve over an integral number of
steps (total mechanical energy is the sum of *E*_{p} and
*E*_{k}).

The possibility of energy transfer depends not only on the shape of the
*E*_{p} and *E*_{k} curves but also on their
relative magnitude and phase relationship. Energy exchange is optimal when
both curves show equal amplitudes. Therefore their relative amplitudes
(*RA*) were calculated (Equation
7) and plotted as a function of walking speed and froude number:
7

Concerning the phase relationship between *E*_{p} and
*E*_{k}, energy exchange is optimal when they are oscillating
exactly 180° out-of-phase. In walking, however, *E*_{p} and
*E*_{k} are never exactly in- or out-of-phase. Percentage
congruity (Ahn et al., 2004)
measures the proportion of the gait cycle during which *E*_{p}
and *E*_{k} change similarly in direction. In an ideal inverted
pendulum, % congruity would equal zero. % congruity was determined as the
proportion of the gait cycle during which the time derivatives of
*E*_{p} and *E*_{k} showed the same sign.

### External mechanical work

The amount of positive work performed on the COM (*W*_{ext})
was determined by adding the positive increments in *E*_{tot}
over an integral number of steps. *W*_{ext} was plotted as a
function of walking speed and froude number.

### Classical internal mechanical work

To determine the kinetic energies of the body segments, a 12-segment body
model (Table 2) was developed
based on anthropometrical data for toddlers from Sun and Jensen
(1994)
(Table 3). For each body
segment the (*x, y, z*) positions of its segmental centre of mass were
calculated based on the filtered (quintic spline) 3-D marker trajectories
(Table 2). The 3-D linear
velocities of the body segments (*v*_{xi},
*v*_{yi}, *v*_{zi}) were obtained from the time
derivatives of the (*x, y, z*) positions of the segmental centres of
mass.

To determine the rotational kinetic energy of the body segments, only
rotations in the saggital plane were considered. For each segment a line
vector was created between two points: *a* (*y*_{1},
*z*_{1}) and *b* (*y*_{2},
*z*_{2}). Definitions of origins (*a*) and endpoints
(*b*) for each segment are found in
Table 2. The angular position
of this line vector (α_{i}) for each instant in time was
calculated using Equation 8:
8
The angular velocity (ω_{i}) of each line segment was then
determined by differentiating α_{i} with respect to time.

To calculate the kinetic energy of the body segments translational and
rotational terms were summed (Equation
9):
9
where *m*_{i} = segmental mass and *I*_{i} = the
segmental moment of inertia in the saggital plane

By adding the positive increments in the internal energy curve over an
integral number of steps, positive internal work (*W*_{int,k})
was determined. *W*_{int,k} was plotted as a function of
walking speed and froude number.

### Work during double contact

Alexander and Jayes (1978)
recognized that work is performed during double support, due to the fact that
one leg is pushing against the other. Classical methods for calculating
*W*_{ext} and *W*_{int,k} do not account for
this. Recently two models have been proposed to determine the amount of work
that has to be produced by the propulsive back leg to overcome the energy
absorption by the front leg.

The first model (Fig. 3A)
was proposed by Donelan et al.
(2002). The model pictures the
legs as two rigid struts joined by the centre of mass. During single support,
the mass moves over the supporting limb as an inverted pendulum and energy
exchange is allowed between *E*_{p} and *E*_{k}.
Double support is seen as a transition state during which the COM velocity is
redirected from one pendular arc to the next. Instead of calculating the
amount of work performed on the COM from the resultant of the ground reaction
forces of both limbs, positive work performed by the front limb and back limb
during a step is calculated separately (Equations
10,
11,
12) and then summed:
10
11
12
This *W*^{+}_{ILM} is the sum of work actually
performed on the COM (and equal to the classical *W*_{ext}) and
work resulting from the opposite action of both legs on the COM during double
support (which does not lead to a change in velocity or height of the centre
of mass and therefore is actually internal work).

The second model (Fig. 3B)
was proposed by Bastien et al.
(2003). The body consists of a
COM and two sticks for the legs. The legs are seen as two oscillating
actuators performing work on the COM. Since both actuators are performing work
on the same structure, they can also perform work on each other. This feature
is what distinguishes Bastien's model from the ILM. Energy exchange is not
only possible between *E*_{k} and *E*_{p} during
single support but also during double support, where a transfer of energy is
allowed between the front and back limb. Allowing this energy transfer will
decrease the amount of work that has to be performed by the back limb to
overcome the braking action of the front limb.

Bastien et al. (2003) ignore work resulting from the lateral force component. However, Donelan et al. (2001) showed that the lateral component might be important in the case of a wide base of support. Therefore, in the current study, the instantaneous work curve resulting from the lateral force component was compared to the instantaneous work curves resulting from the fore–aft and vertical force components. Lateral work showed to be negligible, despite the wide base of support, and was ignored for further analysis.

Following Bastien et al. (2003), we first calculated the four components of work from the individual ground reaction force measurements (Equations 13, 14, 15, 16) during double support: 9 10 11 12

The work done by the vertical ground reaction force components of both limbs can be added because they will always simultaneously do positive and/or negative work: 13

Following foot contact the downward movement of the COM is slowed down and
energy is absorbed by the front limb (*W*_{v} is decreasing
during the first half of double support in
Fig. 3B). If energy transfer is
allowed between both limbs, the back limb can use this energy to aid in
forward propulsion. Thus the work performed by the back limb equals the
horizontal work of the back limb minus the amount of energy absorbed by the
front limb.

If *W*_{v} is decreasing:
16A
if *W*_{v} is increasing:
16B

The front limb also slows down the forward movement of the COM following
foot contact. If energy transfer is allowed, the absorbed energy can be used
by the back limb to lift the COM again (*W*_{v} is increasing
during the second half of double support in
Fig. 3B). Thus the work
performed by the front limb equals the horizontal work of the front limb plus
the amount of positive vertical work transferred to the back limb.

If *W*_{v} is decreasing:
17A
if *W*_{v} is increasing:
17B

Adding *W*_{front} and *W*_{back}
instant-by-instant results in the amount of work performed on the COM during
double contact (*W*_{com}). To obtain the work resulting from
the opposite action of both legs on the COM (*W*_{int,dc}), the
positive increments in *W*_{com} have to be subtracted from the
sum of the positive increments in *W*_{back} and
*W*_{front}.
18

To be able to compare the ILM and the method proposed by Bastien et al.
(2003),
*W*_{ILM} and the sum of *W*_{ext} and
*W*_{int,dc} were plotted as a function of froude number and
compared to *W*_{ext}. Also, work during double contact was
expressed as a fraction of external mechanical work performed on the COM.

### Total mechanical work

To determine total mechanical work, *W*_{ext},
*W*_{intk} and *W*_{int,dc} were added and
plotted as a function of walking speed and froude number. The obtained result
was compared to the sum of *W*_{ILM} and
*W*_{int,k}.

### The effect of walking experience

Since the participating children show a fairly large range of walking
experiences (from 2 weeks to 6 months), the effect of increasing experience in
walking on average walking speed, the inverted pendulum (IP) mechanism,
*W*_{ext}, *W*_{int,k} and
*W*_{int,dc} were also investigated.

## Results

### The inverted pendulum mechanism

The average mechanical energy fluctuations in toddlers show a sinusoidal
oscillation, with two maxima and two minima occurring during one gait cycle
(Fig. 4A). The oscillations in
*E*_{p} are largest in amplitude and almost completely
determine the total mechanical energy fluctuations. Oscillations in
*E*_{k} are dependent upon walking speed. With increasing
walking speed, the *E*_{k} oscillations increase (cf. the
decrease in *RA* with increasing walking speed, which will be discussed
below). The forward kinetic energy is the most important component, almost
completely determining the fluctuations in *E*_{k}
(Fig. 4B).

Recovery values are plotted as a function of walking speed in
Fig. 5A. *R* shows an
inverted U-shape relationship with speed (*r*^{2}=0.63). At
optimal speed almost 40% of the external mechanical energy required to lift
and accelerate the COM can be recovered. Compared to adults, optimal speed in
toddlers is much smaller (0.6 m s^{-1} in toddlers compared to 1.65 m
s^{-1} in adults, Fig.
5A). To correct for the differences in size between adults and
toddlers, *R* was also plotted as a function of froude number
(Fig. 5A). At equal froude
number, *R* is much smaller in toddlers compared to adults. In both age
groups, energy exchange seems to be optimal around froude number 0.4.

Relative amplitudes are plotted as a function of walking speed
(Fig. 5E) and froude number
(Fig. 5F). With increasing
walking speed (and also increasing froude number), *RA* decreases but
the oscillations in *E*_{p} remain larger than the oscillations
in *E*_{k} (*RA* never below 1).

The phase relationship between *E*_{p} and
*E*_{k} is expressed as % congruity
(Fig. 5C,D), which equals the
percentage of the gait cycle during which *E*_{p} and
*E*_{k} are moving in the same direction. During as much as
25–50% of the gait cycle *E*_{p} and
*E*_{k} are oscillating in phase. % Congruity shows a U-shaped
relationship with speed (Fig.
5C; *r*^{2}=0.46), reaching a minimum at
approximately 0.6 m s^{-1} (around froude number 0.4).

### External mechanical work

*W*_{ext} plotted as a function of walking speed shows a
U-shaped curve (Fig. 6A;
*r*^{2}=0.71) reaching a minimum at a speed of 0.6 m
s^{-1}, which coincides with the speed when *R* was maximal.
When compared to adults, toddlers show a tendency to produce a higher amount
of mass-specific work per unit distance over the entire speed range observed
(0.1–1.0 m s^{-1}). Contrary to what is seen in children above 3
years of age, the difference in external work production between adults and
toddlers does not disappear when *W*_{ext} is plotted as a
function of the dimensionless froude number
(Fig. 6B).
*W*_{ext} remains larger in toddlers than in adults.

### Classical internal mechanical work

*W*_{int,k} shows a positive linear relationship with
walking speed (Fig. 6C,
*r*^{2}=0.60). Compared with adults, toddlers seem to perform
more mass-specific internal work when walking at the same speed. To eliminate
differences due to different body proportions, *W*_{int,k} was
plotted as a function of the dimensionless froude number in
Fig. 6D. If body size is taken
into account, *W*_{int,k} is smaller in toddlers compared to
adults.

The kinetic energies of the body segments are plotted in Fig. 7. Kinetic energies of the head, the trunk and the arms are relatively small compared to the kinetic energies of the thigh and shank.

### Internal work during double contact

In Fig. 6E,F
*W*_{ILM} and the sum of *W*_{ext} and
*W*_{int,dc} are compared to *W*_{ext}
calculated by the classical combined limbs method. Also in toddlers,
*W*_{ILM} as well as
*W*_{int,dc}+*W*_{ext} are larger. For example,
at optimal speed *W*_{ext}=0.84 J kg^{-1}
m^{-1}, *W*_{ext}+*W*_{int,dc}=0.88 J
kg^{-1} m^{-1} and *W*_{ILM}=0.94 J
kg^{-1} m^{-1}.

On average, the work performed by the back limb to overcome the braking action of the front limbs accounts for 7% (if the method of Bastien et al., 2003 is used) to 16% (if the ILM is used) of the work performed on the COM.

### Total mechanical work

*W*_{tot} shows a U-shaped relationship with walking speed,
reaching a minimum around 0.6 m s^{-1}, in contrast to adults, who
show a positive linear relationship with speed
(Fig. 6G). If the ILM is used,
*W*_{tot} values are slightly higher than when summing
*W*_{ext}, *W*_{int} and
*W*_{int,k} (Fig.
6G: compare at optimal speed:
*W*_{ext}+*W*_{int,dc}+*W*_{int,k}=1.15
J kg^{-1} m^{-1} while
*W*_{ILM}+*W*_{int,k}=1.24 J kg^{-1}
m^{-1}). However, the choice of method has no influence on the
position of the optimal walking speed.

At optimal speed, toddlers seem to use more than twice the energy per unit body mass that adults need to walk at the same speed. At speeds slower and faster than the optimal speed the difference is even larger.

*W*_{tot} was also plotted as a function of froude number
(Fig. 6H) to eliminate
differences between adults and toddlers due to body size. Energy consumption
is minimal around froude number 0.4. At this froude number,
*W*_{tot} is comparable in adults and toddlers. But when
walking faster or slower, toddlers again consume a lot more energy than
adults.

### The effect of walking experience

The individual mean *E*_{p} and *E*_{k}
curves are plotted in Fig. 8.
In the earliest walkers the sinusoidal oscillations of the energy curves
appear to be irregular. After 3 months of independent walking the stereotype
pattern with two maxima and two minima begins to show. However, variation
remains large, especially in *E*_{p}.

The effects of walking experience on preferred walking speed, mechanical
work production and recovery values are shown in
Fig. 9. With increasing walking
experience, there is a slight increase in preferred walking speed. Total
mechanical work as well as *W*_{ext} show a U-shaped
relationship with walking experience. No relationship is found between the
*W*_{int,k}, *R* values or work during double contact
and walking experience (*r*^{2}<0.01).

## Discussion

### The inverted pendulum mechanism

Our results suggest that in order to minimize energy expenditure, toddlers
are (at least partially) able to use the IP mechanism of energy exchange. At
optimal speed, 40% of external mechanical energy can be recovered. However,
the IP mechanism seems to be imperfect in toddlers. The relative amplitudes
show that the potential energy fluctuations are larger than kinetic energy
fluctuations, a feature that is disadvantageous for energy exchange to occur.
The large decreases in *E*_{p} during the second half of the
swing phase can be used to increase *E*_{k} when the centre of
mass accelerates downwards. But the oscillations in *E*_{k} are
too small to perform the required amount of work to lift the COM against
gravity. The large difference in relative magnitude of the
*E*_{p} and *E*_{k} oscillations can be
explained by toddlers' tossing gait (large vertical oscillations of the centre
of mass) in combination with their low walking speed.

Also the phase relationship between *E*_{p} and
*E*_{k} is suboptimal, as the % congruity measurements show
that during as much as 25–50% of the gait cycle, *E*_{p}
and *E*_{k} are moving in the same direction.
Fig. 8 shows that in some of
the children (e.g. a child with 13 weeks of walking experience) after foot
contact the COM is still moving down (decrease in *E*_{p})
during the first half of single support. At this time *E*_{k}
is also decreasing as this downward movement of the COM is slowed down by
flexing the hip and knee (A.H., personal observations).

### Positive external work

Positive external work in toddlers is minimal at the speed when energy
exchange is optimal (at approximately 0.6 m s^{-1}). Despite the fact
that the IP mechanism of energy exchange is observed in toddlers, they perform
a greater amount of work per unit distance to walk at a given speed than
adults, even when their small stature is taken into account (by expressing
speed as the dimensionless froude number). As argued above, an explanation can
be found in the differences in gait pattern between adults and toddlers. Both
their slow walking speed and tossing gait are disadvantageous for the pendular
exchange of energy. Consequently less energy can be recovered and more work
has to be performed to lift and accelerate the COM.

### Classical internal work

Positive internal work performed to accelerate the body segments relative
to the COM is a second important component of total positive mechanical work
performed. It is almost completely dependent on the work performed to swing
the limb forward. Work performed on the arms and upper body is negligible
since the upper body is kept relatively stiff and arm swing has not yet
developed in these children. Exactly these features cause
*W*_{int,k} to be smaller in toddlers compared to adults.

### Mechanical internal work during double contact

Work performed due to the fact that one leg is pushing against the other in
toddlers is also a contributing factor to mechanical work production. Contrary
to what might be expected due to the prolonged phase of double support in
toddlers, this amount of work is small compared to adults. Using the ILM, it
accounts for 16% of *W*_{ext}, whereas Donelan et al.
(2002) reported values of
*W*_{int,dc} reaching up to 33% of *W*_{ext} in
adults. Bastien et al. (2003)
reported that *W*_{int,dc} reaches 40% of
*W*_{ext} in older children and adults, while in toddlers this
value is only 7%. Again an explanation can be found in the combination of a
slow walking speed with a tossing gait. As a consequence of these features of
immature gait, work resulting from the vertical ground reaction force
components is much larger than opposite work resulting from the fore–aft
force components (compare the instantaneous work traces in
Fig. 10). In other words,
because of toddlers' tossing gait the most important component of work during
double contact is the work that has to be performed to lift the COM against
gravity. Compared to this, the amount of work that has to be performed to
overcome the opposite action of the front and back limb in order to maintain a
(slow) constant walking speed is relatively small.

Due to different assumptions of the model, the ILM leads to higher work
values than the method proposed by Bastienet al.
(2003).
Fig. 11 shows the difference
between both methods. Allowing for energy exchange to occur between the front
and back limb flattens the instantaneous work traces
(*W*_{front} and *W*_{back}) and reduces the
amount of opposite work during double contact. While quantitative results
differ upon using one or the other method, the conclusions of both methods
remain the same.

### Total mechanical work

Work performed on the COM is the major contributing factor of total
mechanical work performed during walking in toddlers. The second most
important contributor to *W*_{tot} is internal work performed
to accelerate the body segments relative to the COM. If opposite work
performed during double contact is ignored, *W*_{tot} will be
underestimated. However, this amount of work is not a determining factor of
*W*_{tot} since it does not influence the position of the
optimal walking speed.

At froude number 0.4, *W*_{tot} is comparable between adults
and toddlers. Recoveries are optimal at this dimensionless speed (at froude
number 0.4, *R* reaches a maximum of 40%) and thus
*W*_{ext} is minimal. Nevertheless it is still larger than the
external work production in adults at comparable froude number. Despite this,
*W*_{tot} is still comparable between adults and toddlers since
*W*_{int,k} is much smaller in toddlers due to the stiff upper
body and the absence of arm swing.

### The effect of walking experience

The individual *E*_{p} and *E*_{k} traces
suggest the IP is not fully mastered at the onset of independent walking and
starts to mature after 3 months of walking experience. However, these
observations were not confirmed by the *R* values, which did not change
over the range of walking experiences. Possibly by averaging different trials
per individual, speed effects are masked. Also, a cross-sectional study is not
the best set-up for exploring maturational effects, since in each child the
speed at which gait matures will differ. While this study is very valuable for
giving insight into the overall mechanisms of energy transfer and mechanical
energy costs of bipedal gait in very young walkers, a longitudinal set-up
would be more appropriate for investigating subtle changes in mechanical
energy expenditure due to motor development and growth. A longitudinal
follow-up study of young walkers, is therefore our future goal.

The U-shaped relationship of *W*_{ext} and consequently also
*W*_{tot} with walking experience can be explained by the
increase in preferred walking speed. After approximately 3 months of
independent walking, toddlers walk at a speed at which energy exchange is
optimal for their length. However, when growing older they choose to walk at
speeds above the optimal speed and thus mechanical energy production rises
again.

## Conclusion

*W*_{ext} is the major contributing factor of total
mechanical work in toddlers. They perform more mass specific work per unit
distance than adults to lift and accelerate the COM. An explanation can be
found in the fact that, despite the fact that the IP mechanism is observed in
toddlers to a certain degree, it has not yet been completely mastered. Because
of their tossing gait and slow walking speed, energy exchange is imperfect.
Apart from external work performed on the COM, internal work is performed to
swing each limb forward during walking. *W*_{int,k} increases
linearly with speed. If the small stature of toddlers is taken into account,
toddlers perform less internal work compared to adults since arm swing is not
yet present. Another component of internal work is the work performed during
double contact, due to the fact that one leg is pushing against the other.
This component of work is rather small in toddlers and is not a major
contributing factor to total mechanical work production.

**List of Symbols**

*A*_{x}*, A*_{y}*, A*_{z}- 3-D linear accelerations of the COM
- a
- origin
- b
- endpoint
- COM
- centre of mass of the body
- com
_{i} - centre of mass of the body segments
*E*_{k}- kinetic energy of the centre of mass of the body
*E*_{k,int}- kinetic energy of the body segments, which is the sum of translational and rotational kinetic energy
*E*_{p}- potential energy of the centre of mass of the body
*E*_{tot}- total mechanical energy of the centre of mass of the body (which is the sum of kinetic and potential energy)
*F*_{x}- lateral force component
*F*_{y}- fore–aft force component
*F*_{z}- vertical force component
*F*_{y,back}- fore–aft force component of the back leg
*F*_{y,front}- fore–aft force component of the front leg
*F*_{z,back}- vertical force component of the back leg
*F*_{z,front}- vertical force component of the front leg
*I*_{i}- moment of inertia of the body segments in the sagittal plane
- ILM
- individual limbs method
- IP
- inverted pendulum mechanism
*m*_{i}- mass of the body segments
*M*_{tot}- total body mass
- R
- % of recovered energy
- RA
- relative amplitide
*V*_{x}- lateral velocity of the centre of mass of the body
*V*_{y}- forward velocity of the centre of mass of the body
*V*_{z}- vertical velocity of the centre of mass of the body
- v
- walking speed
*v*_{xi}- lateral velocity of the segmental centre of mass
*v*_{yi}- forward velocity of the segmental centre of mass
*v*_{zi}- vertical velocity of the segmental centre of mass
*W*^{+}_{back}- positive work performed by the back leg
*W*^{+}_{front}- positive work performed by the front leg
*W*_{back}- work performed by the back leg
*W*_{com}- work performed on the COM
*W*_{ext}- the amount of positive work performed during a gait cycle to lift and accelerate the centre of mass
*W*_{front}- work performed by the front leg
*W*_{ILM}- the amount of work calculated by the ILM, which is the sum of external mechanical work performed on the centre of mass and some internal work due to the opposite action of the front and back limb during double support
*W*_{int,dc}- the amount of work performed during double contact due to the fact that both legs are working against each other, calculated by the Bastien method
*W*_{int,k}- the amount of positive work performed during a gait cycle to accelerate the body segments relative to the centre of mass
*W*_{y,back}- work performed by the horizontal force component of the back leg
*W*_{y,front}- work performed by the horizontal force component of the front leg
*W*_{z,back}- work performed by the vertical force component of the back leg
*W*_{z,front}- work performed by the vertical force component of the front leg
*w*_{v}- work performed by both legs in the vertical direction
*w*_{y}- work performed by both legs in the horizontal direction
- z
- vertical displacement of COM
- α
- angular rotation of the body segments in the sagittal plane
- ω
- angular velocity of the body segments in the sagittal plane

## ACKNOWLEDGEMENTS

We wish to thank all participating children and their parents for their time and effort. Also we would like to thank the HIKE for their willing participation in our research project. This project was funded by a personal grant to A. Hallemans and a BOF project to P. Aerts.

- © The Company of Biologists Limited 2004