## SUMMARY

During walking, the centre of mass of the body moves like that of a `square
wheel': with each step cycle, some of its kinetic energy,
*E*_{k}, is converted into gravitational potential energy,
*E*_{p}, and then back into kinetic energy. To move the centre
of mass, the locomotory muscles must supply only the power required to
overcome the losses occurring during this energy transduction. African women
carry loads of up to 20% of their body weight on the head without increasing
their energy expenditure. This occurs as a result of an unexplained, more
effective energy transduction between *E*_{k} and
*E*_{p} than that of Europeans. In this study we measured the
value of the *E*_{k} to *E*_{p} transduction at
each instant in time during the step in African women and European subjects
during level walking at 3.5-5.5 km h^{-1}, both unloaded and carrying
loads spanning 20-30% of their body weight. A simulation of the changes in
*E*_{k} and *E*_{p} during the step by
sinusoidal curves was used for comparison. It was found that loading improves
the transduction of *E*_{p} to *E*_{k} during
the descent of the centre of mass. The improvement is not significant in
European subjects, whereas it is highly significant in African women.

## Introduction

During each step of walking, the gravitational potential energy
*E*_{p} and the kinetic energy *E*_{k} of the
centre of mass of the body oscillate between a maximum and a minimum value.
*A priori*, active movements of an organism are assumed to be powered
by muscles: positive muscle work to increase potential energy and kinetic
energy, and negative muscle work to absorb potential energy and kinetic
energy. Both positive and negative muscular work require the expenditure of
chemical energy. During walking, both the positive and the negative work
actually done by the muscles to sustain the mechanical energy changes of the
centre of mass (positive and negative external work) are reduced by the
pendular transduction of potential energy to kinetic energy and *vice
versa* (Cavagna et al.,
1963).

The fraction of mechanical energy recovered due to this transduction,
*R*_{step}, has been defined as:
1
where *W*^{+}_{v} represents the positive work
calculated from the sum, over one step, of the positive increments undergone
by the gravitational potential energy, *E*_{p}=*Mgh*
(where *M* is the mass of the body and *h* is the height of the
centre of mass), *W*^{+}_{f} is the positive work
calculated from the sum, over one step, of the positive increments undergone
by the kinetic energy of forward motion *E*_{kf}.
*E*_{kf}=0.5*MV*_{f}^{2} (where
*V*_{f} is the instantaneous forward velocity of the centre of
mass), and *W*^{+}_{ext} is the positive external work
calculated from the sum over one step of the positive increments undergone by
the total mechanical energy of the centre of mass,
*E*_{cg}=*E*_{p}+*E*_{kf}+*E*_{kv}
(Cavagna et al., 1976). The
kinetic energy of vertical motion,
*E*_{kv}=0.5*MV*_{v}^{2} (where
*V*_{v} is the instantaneous vertical velocity of the centre of
mass), has not been taken into account when obtaining
*R*_{step} from *W*^{+}_{v} and
*W*^{+}_{f}. *E*_{kv} has no effect on
*W*^{+}_{v} because the vertical velocity is zero at
the top/bottom endpoints of the *E*_{p} curve. As will be shown
below, using *W*^{+}_{k}, measured from the total
kinetic energy curve
*E*_{k}=*E*_{kv}+*E*_{kf},
instead of *W*^{+}_{f}, measured from the
*E*_{kf} curve, has a negligible effect on
*R*_{step} (see Fig.
8).

*R*_{step}, as defined in Equation 1, represents the
fraction of the maximum positive energy increments possibly undergone by the
centre of mass (measured assuming no energy transduction) that is recovered by
the pendular mechanism over the whole step cycle: it does not give information
about the time course of this transduction within the step. Factors that are
expected to affect *R*_{step} are: (i) the relative amplitude
of the potential and kinetic energy curves, (ii) their shape and (iii) their
relative phase. In a frictionless pendulum, energy recovery, *R*,
equals unity because the changes in potential energy mirror the changes in
kinetic energy. During walking, *R*_{step} attains a maximum at
an intermediate speed when the difference in amplitude of the potential and
kinetic energy curves approaches zero and the phase difference between the
potential and kinetic energy curves approaches 180° (Cavagna et al.,
1976,
1983;
Griffin and Kram, 2000).

More information about the pendular mechanism of walking may be obtained by analyzing how the pendular transduction of the mechanical energy occurs during the step cycle. The factors affecting the pendular transduction of mechanical energy within the step are not known. The aim of this study is to define these factors by following the transduction between potential and kinetic energy at each instant of time during the step.

We applied this new approach to the great skill of African women carrying
loads (Maloiy et al., 1986):
African women carry loads more economically than Europeans as a result of
their greater *R*_{step}
(Heglund et al., 1995);
however, it is not known how this greater *R*_{step} is
attained. The within-step analysis of the potential kinetic energy
transduction demonstrates the phases of the step in which the difference
between African women and European subjects is most apparent.

## Materials and methods

### Subjects and experimental procedure

In this study, we analyzed the changes in *E*_{k},
*E*_{p} and *E*_{k}+*E*_{p} of
the centre of mass of the body during one step of level walking at a constant
speed with and without a load being carried by the subject (loaded and
unloaded walking steps). Data were obtained for 11 Europeans (five male and
six female, 65.6±7.1 kg, mean ± S.D.) and four African women
(three Luo and one Kikuyu, 73.9±14.4 kg, mean ± S.D.) previously
described by Heglund et al.
(1995). The steps analyzed
were recorded during walking at 3.5-5.5 km h^{-1}, both for unloaded
subjects and for subjects carrying loads spanning 20-30% of their body weight.
The speed range corresponds to freely chosen walking speeds. The load range
was chosen because it results in the maximum difference between
*R*_{step} measured in the European subjects and
*R*_{step} measured in the African subjects (see
fig. 2 in
Heglund et al., 1995). Data
collected within these speed and load ranges were averaged (see
Table 1) neglecting any effect
of the speed and load change, because the scatter of the data was too large to
define a trend within such a narrow speed and load range. Loads were
head-supported by the African women and shoulder-supported by the Europeans.
The Kikuyu woman carried the loads on her back supported solely by a strap
over their head. Two of the Luo women carried the loads on top of her head,
and the other Luo woman carried the loads both ways. A step was considered to
be suitable for analysis when the sum of the increments of
*E*_{p} and *E*_{k} over the step cycle did not
differ by more than 10% from the sum of the decrements due to variability
between successive steps of the subject. All the usable steps (*N*=32)
of the African women during walking with loads were analyzed. An equal number
of steps was randomly chosen for analysis from a larger pool of data for
African women during unloaded walking and for European subjects during both
unloaded and loaded walking. The five European males walked both loaded (18
steps analyzed) and unloaded (15 steps analyzed) whereas, of the six European
females, four walked loaded and unloaded, one walked unloaded only and one
walked loaded only, giving a total of 14 loaded steps and 17 unloaded steps
analyzed.

The subjects walked across a force platform sensitive to the vertical and horizontal (fore—aft) components of the force exerted by the feet on the ground. The lateral component of the force was neglected (Cavagna et al., 1963). The force platform had a natural frequency greater than 180 Hz in both directions and was mounted at ground level in the middle of a walkway. The dimensions of the platform were 1.8 m×0.4 m in the case of the African women and 6.0 m×0.4 m in the case of the Europeans. The mean walking speed was measured by means of photocells placed 1.2 m apart (Africans) and 1.9-3.6 m apart (Europeans) alongside the platform.

The platform signals were collected by a microcomputer for analysis using a
sampling rate of 500 Hz for the African subjects and 100 Hz for the European
subjects. The changes in *E*_{k}, *E*_{p} and
*E*_{k}+*E*_{p} of the centre of mass of the
body were determined from the platform signals using the procedure described
in detail by Cavagna (1975). In
short, integration of the horizontal force and of the vertical force minus the
body weight, both divided by the body mass, yielded the velocity changes of
the centre of mass. The instantaneous velocity in the forward direction was
obtained using the mean walking speed, measured from the photocell signal, to
determine the integration constant. A first integration was made in the
vertical direction on the assumption that the initial and final velocities of
the step cycle were equal.

Contrary to our previous studies, in which the kinetic energy of forward
and vertical motion were calculated separately, the kinetic energy of both
forward and vertical motion, *E*_{k}, was calculated from the
velocity of the centre of mass in the sagittal plane. *W*_{k}
is the work necessary to sustain the kinetic energy changes (positive when
*E*_{k} increases, negative when *E*_{k}
decreases).

A second integration of the vertical velocity yields the vertical
displacement of the centre of mass. This integration assumes that the net
vertical displacement over the whole step cycle was zero. The oscillations of
the gravitational potential energy *E*_{p} were calculated from
the vertical displacement. *W*_{v} is the work necessary to
sustain the gravitational potential energy changes (positive when
*E*_{p} increases, negative when *E*_{p}
decreases). The total energy of the centre of mass, *E*_{cg},
due to its motion in the sagittal plane, is the algebraic sum at each instant
of *E*_{p} and *E*_{k}. *W*_{ext}
is the sum of the changes in *E*_{cg} during one step (the sum
of the positive increments corresponds to the external positive work done by
the muscular force, the sum of the negative increments corresponds to the
external negative work done by the muscular force).

During walking, the negative work done by external friction is small and was neglected: a maximum error of 10% due to this assumption was measured in sprint running (Cavagna et al., 1971). During level walking at a constant speed, the net changes in mechanical energy of the centre of mass of the body are zero over the whole step cycle. It follows that the external positive work done by the muscular force equals the external negative work (neglecting the negative work done by friction outside the muscles), i.e. during each step, the muscles and elastic structures deliver and absorb an equal amount of mechanical energy and the net work (positive + negative) is zero. As mentioned above, chemical energy is expended to perform positive work and also, to a lesser extent, to perform negative work. To reduce energy expenditure, the mechanical energy changes of the centre of mass (both positive and negative) should be reduced to a minimum.

### Within-step analysis of the potential—kinetic energy transduction

The mechanisms resulting in the measured value of *R*_{step}
(Equation 1) were analyzed in this study by measuring the fraction of
mechanical energy recovered due to the transduction between
*E*_{p} and *E*_{k} at each instant in time
during the step. The step period, τ, was divided into equal time intervals
(2 ms for the Africans and 10 ms for the Europeans), and the recovery,
*r*(*t*), was calculated from the absolute value of the changes,
both positive and negative increments, in *E*_{p},
*E*_{k} and *E*_{cg} during each time interval:
2
where *t* is time. The signal-to-noise ratio in *r*(*t*)
decreased when the changes in energy during a particular time interval
approached zero (see Fig.
7).

### Simulation of the energy transduction within the step

To examine the trend of *r*(*t*) within the step (Equation
2), the changes in *E*_{p} and *E*_{k} taking
place during a walking step were simulated by two sinusoidal curves. This
simulation is not meant to represent a model of the complex walking mechanics
but offers a useful background within which to interpret the experimental
recording of *r*(*t*) and to distinguish different phases within
the step. In addition, it allows us to define the relationship between the
mean pendular energy transduction derived from the present analysis and
*R*_{step}, previously used in the literature (see below).

Since, during walking, *E*_{p} and *E*_{k} of
the centre of mass change roughly out of phase, we assumed in the simulation
that *E*_{p}=-sin*x* and
*E*_{k}=sin(*x*-α), where the phase shiftα
=0° when the *E*_{p} and *E*_{k}
curves are exactly 180° out of phase. In previous studies
(Cavagna et al., 1983;
Griffin et al., 1999), this
phase shift was defined as α=360° Δ*t*/τ, whereΔ
*t* is the difference between the time at which
*E*_{kf} is at a maximum and the time at which
*E*_{p} was at a minimum and τ is the step period. At low
and intermediate walking speeds, such as those considered in the present
study, α >0°; at high walking speeds (>6 km h^{-1}),α<0° (Cavagna et al.,
1983). For most walking speeds,
45°>α>-45°.

The total mechanical energy of the centre of mass *E*_{cg}
was calculated as the algebraic sum of the curves for *E*_{p}
and *E*_{k}:
3
The effect of a change in the phase shift α is shown in
Fig. 1. *E*_{cg}
attains a maximum or a minimum when its derivative is zero, i.e. when
cos*x*=cos(*x*-α). In contrast,
cos*x*=cos(2π-*x*=cos(-*x*), so the angle for a
maximum of *E*_{cg} will be (π+α/2) and the angle for
a minimum of *E*_{cg} will be α/2
(Fig. 1). Substituting these
angles into Equation 3, one obtains the maximum and minimum values of the
total mechanical energy in the simulation, i.e.
*E*_{cg,max}=2sin(α/2) and
*E*_{cg,min}=-2sin(α/2). The changes in
*R*_{step} in the simulation with the phase shift α
(Fig. 2, dotted line) can then
be defined as:
4
The recovery of mechanical energy at each instant during one cycle,
*r*(*x*), was calculated in the simulation according to Equation
2 by substituting |Δ*E*_{p}(*t*)|,|Δ
*E*_{k}(*t*)| and|Δ
*E*_{cg}(*t*)| with the absolute
value of the derivative of the functions: -sin*x*,
sin(*x*-α) and -sin*x*+sin(*x*-α), for
*x*=0-360° in increments of 1°
(Fig. 1, upper panels, thick
lines):
5
The calculated value of *r*(*x*) for sinusoidal curves was equal
to that measured on the same sinusoidal curves with the procedure used to
determine *r*(*t*) on the experimental tracings.

If the two sinusoidal curves, representing *E*_{p} and
*E*_{k}, are exactly out of phase (α=0°) and have the
same amplitude, *E*_{cg} is constant,
*W*^{+}_{ext}(*x*) is zero and
*r*(*x*=1 over the whole cycle. If the two sinusoidal curves are
exactly out of phase (α=0°) but have different amplitudes,
*E*_{cg}(*x*) oscillates in phase with the curve of
larger amplitude, *W*_{ext}(*x*)≠0, and
*r*(*x*) decreases below unity, maintaining a constant value
over the cycle. For example, if the amplitude of *E*_{p} is
half that of *E*_{k}, *r*(*x*)=1-(1/3)=0.66
(Equation 2).

If the two energy curves have the same amplitude, but are not exactly out
of phase (α≠0°), *r*(*x*) changes as described in
Fig. 1 for α=10° and
20°. In each cycle, there are two periods when *r*(*x*=0:
the changes in potential energy and in kinetic energy have the same sign and,
as a consequence,|Δ
*E*_{cg}(*x*)|=|*E*_{p}(*x*)|+|Δ*E*_{k}(*x*)|
(see Equation 2). During one of these periods, henceforth called
*t*_{pk+}, lasting from the minimum of *E*_{p}
to the maximum of *E*_{k}, positive external work is done to
increase *E*_{p} and *E*_{k} simultaneously.
During the other, *t*_{pk-}, lasting from the maximum of
*E*_{p} to the minimum of *E*_{k}, negative
external work is done to absorb *E*_{p} and
*E*_{k} simultaneously. The phase shift between
*E*_{p} and *E*_{k} was calculated both asα
=360°*t*_{pk+}/τ and asβ
=360°*t*_{pk-}/τ
(Table 1).

The two periods when *r*(*x*) is zero are separated by two
time intervals when *r*(*x*) increases to unity and then
decreases to zero following a bell-shaped curve: one period after
*t*_{pk+} during most of the lift phase (increment of the
*E*_{p} curve in Fig.
1), the other after *t*_{pk-} during most of the
lowering phase (decrement of the *E*_{p} curve in
Fig. 1). According to equation
2, *r*(*x*)=1 is attained whenΔ
*E*_{cg}(*x*) is zero (i.e.
*E*_{cg} attains a maximum or a minimum and
*E*_{p}=*E*_{k}).

The mean value of *r* over the whole step cycle was calculated, both
for the simulation, *r*(*x*), and for the experimental tracings,
*r*(*t*), as the time integral divided by the period:
where *z*=360° for the simulation and *z*=τ for the
experimental tracings. In the case of two sinusoidal curves of different
amplitude and exactly out of phase (α=0°),
*R*_{int}=*R*_{step} (*R*_{step}
is defined as in Equation 1). In the case of two sinusoidal curves of the same
amplitude but with a phase shift α between the time at which
*E*_{k} is at a maximum and the time at which
*E*_{p} is at a minimum, the relationship between
*R*_{int} and α (Fig.
2, continuous line) is given by:
6
where α is expressed in rad. Equation 6 is obtained by integrating
*r*(*x*), as defined in Equation 5, and dividing the result by
2π. In the simulation, *R*_{int} is not equal to
*R*_{step} except when the curves are exactly out of phase
(α=0°) or exactly in phase (α=180°), or whenα
=±96.3° (Fig.
2). When the phase shift varies in the simulation as in human
walking (45°>α>-45°),
*R*_{int}≤*R*_{step} and both decrease with|α|
.

### Average recordings

To compare the energy transduction within the step in different subjects,
the step cycle was divided into four periods: the two periods with
*r*(*t*)=0 (*t*_{pk+} and
*t*_{pk-}) and the two with *r*(*t*)≠0
(Fig. 1). The mean values for
the four periods are given in Table
1 for each experimental condition. The abscissa of each of the two
phases with *r*(*t*)≠0 was normalized from zero to one, and
an average of *r*(*t*) was calculated at discrete intervals
along the normalized abscissa (0.01). The mean step cycle was then
reconstructed using on the abscissa, the mean values of the four time
intervals (Table 1, Figs
4,
6).

In some recordings, *t*_{pk+} and/or
*t*_{pk-} were zero, and the separation between the two periods
with *r*(*t*)≠0 was made using the minimum of
*r*(*t*) or, when oscillations where present (see
Fig. 7), the maximum and/or the
minimum of *E*_{p} and *E*_{k}. Often
*r*(*t*) failed to attain unity in spite of the fact that
*E*_{cg} attained a maximum or a minimum (i.e.|Δ
*E*_{cg}(*t*)|=0) because of the
discrete time periods used to calculate|Δ
*E*_{cg}(*t*)| and/or the
averaging of the curves (see Figs
3,4,5,6,7).

## Results

### Time course of energy recovery within the step of unloaded subjects

Typical recordings showing *r*(*t*) during an unloaded step
together with the simultaneous changes in *E*_{p},
*E*_{k} and *E*_{cg} are given in
Fig. 3A for a European subject
and in Fig. 3B for an African
subject. Similar to the trend shown by the simulation
(Fig. 1),
*r*(*t*)=0 during two periods. The first period,
*t*_{pk+}, occurs at the beginning of the lift of the centre of
mass, when both *E*_{p} and *E*_{k} increase
simultaneously as a result of positive work done by the muscular force. The
second period, *t*_{pk-}, occurs just after the maximum of
*E*_{p}, when both *E*_{p} and
*E*_{k} decrease simultaneously as a result of negative work
done by the muscular force. The succession of events, both in the simulation
and during the walking step, is therefore: (i) *t*_{pk+} to
begin the upward displacement and complete the acceleration forwards; (ii)
some transduction from *E*_{k} to *E*_{p} taking
place up to the end of the lift of the centre of mass, during a period
henceforth referred to as *t*_{tr,up}; (iii)
*t*_{pk-} to begin the downward displacement of the centre of
mass and complete the deceleration forwards; (iv) some transduction from
*E*_{p} to *E*_{k} taking place up to the end of
the descent, during a period henceforth referred to as
*t*_{tr,down}. Both *t*_{pk+} and
*t*_{pk-} start at the extremes of the vertical oscillation of
the centre of mass of the body.

A comparison of Figs 1 and
3 shows that, in contrast to
the simulation, the changes in *r*(*t*) during
*t*_{tr,up} and *t*_{tr,down} are not
symmetrical. During *t*_{tr,up}, when the body rides upwards on
the front leg and the point of application of force moves forward from heel
towards the toe of the supporting foot
(Elftman, 1939),
*r*(*t*) increases steeply to a plateau and then falls abruptly
to zero. Three peaks are usually observed on the plateau corresponding to a
more or less pronounced oscillation of *E*_{cg}. During
*t*_{tr,down}, when the body `falls forwards',
*r*(*t*) changes in a manner more similar to the simulation,
reaching a single peak and forming a bell-shaped curve.

In the simulation, *E*_{cg} attains one maximum (during the
lift) and one minimum (during the fall). During the walking step, in contrast,
*E*_{cg} usually attains two peaks: the first at the beginning
of the lift, the second at the end of the lift. The first peak of
*E*_{cg} occurs near the maximum of *E*_{k},
which coincides with the end of *t*_{pk+}; the second peak of
*E*_{cg} occurs near the maximum of *E*_{p},
which coincides with the beginning of *t*_{pk-}. The positive
increment in *E*_{cg} to the first peak (increment *a*)
corresponds to external positive work done by the muscular force mainly to
increase the kinetic energy of the centre of mass beyond the level attained as
a result of the decrement in potential energy. The end of increment *a*
occurs during the time of double contact, *t*_{dc}. The
positive increment in *E*_{cg} to the second peak (increment
*b*) corresponds to positive work done to complete the lift of the
centre of mass to a level greater than that attained as a result of the
decrement in kinetic energy. Increment *b* occurs during the time of
single contact, *t*_{sc}. The sum of these two positive
increments of *E*_{cg} (*a*+*b*) represents the
positive external work done at each step to translate the centre of mass of
the body in the sagittal plane (Cavagna et
al., 1963; Cavagna and
Margaria, 1966).

Average *r*(*t*) recordings, constructed as described in
Materials and methods, are given in Fig.
4A for unloaded European subjects (broken line), for unloaded
European women (thin continuous line) and for unloaded African women (thick
line). The area below the average *r*(*t*) recordings, divided
by the mean step period, is given by the curves in
Fig. 4B (to be compared with
the bottom graphs of Fig. 1).
These curves show that: (i) the relative amount of energy recovered during
*t*_{tr,up} is on average larger than that recovered during
*t*_{tr,down} in both European and African subjects; (ii) this
asymmetry is smaller in African women than in the European subjects as a
result of less complete pendular transduction during
*t*_{tr,up} and more complete transduction during
*t*_{tr,down}, which is a consequence of the more pronounced
`shoulder' on the *r*(*t*) recording at the beginning of the
descent of the centre of mass; (iii) the recovery at the end of the step
period, *R*_{int}, is equal in African women and in European
subjects; and (iv) no appreciable difference was found between all European
subjects (male and female) and the European women.

The mean values of *R*_{int} and *R*_{step}
(Equation 1) are given in Table
1. It can be seen that
*R*_{int}<*R*_{step} as predicted by the
simulation, both in the Europeans and in the African women. However, both
*R*_{int} and *R*_{step} measured during the
walking step are smaller than those predicted by the simulation
(Fig. 2).

### Effect of loading

Typical recordings showing *r*(*t*) within a step of walking
with a load, together with the simultaneous changes in *E*_{p},
*E*_{k} and *E*_{cg}, are given in
Fig. 5A for a European subject
and in Fig. 5B for an African
subject. The average *r*(*t*) recordings in
Fig. 6 compare the
load-carrying skills of African women and European subjects.

In all subjects, loading tends to decrease the duration of
*t*_{pk-} and, to a lesser extent, the duration of
*t*_{pk+}: the reductions are, however, not significant in
Europeans subjects whereas they are significant in African women, for whom
*t*_{pk-} decreases by approximately two-thirds
(Table 1). Since the step
period τ is not significantly decreased by loading (by only 2-3%, see
Table 1), the phase shiftsα
=360°*t*_{pk+}/τ andβ
=360°*t*_{pk-}/τ change in a manner similar to
*t*_{pk+} and *t*_{pk-}. Loading therefore tends
to increase the transduction between potential and kinetic energy by making
the two curves more exactly out of phase, particularly during the swing phase
(single-contact phase) of the step. As mentioned above, the effect is
significant in African women and not in European subjects.

Another effect of loading results from a change in the shape of the
potential and kinetic energy curves. This is shown by a more pronounced
`shoulder' of the *r*(*t*) record during the first part of the
descent of the centre of mass. This effect is also more pronounced in the
African women than in the European subjects
(Fig. 6).

Both these effects of loading tend to increase the fraction of the total
mechanical energy changes of the centre of mass that is recovered by the
pendular mechanism in the African women. This results in an increase in
*R*_{int} by the end of the step cycle compared with the
European subjects (Fig. 6). The
increase in *R*_{int} in the African women during load-carrying
is approximately equal to the increase in *R*_{step}
(Table 1). It should be
stressed that the effect of load-carrying occurs mainly during the swing phase
of the step, when (more frequently in African women) loading sometimes results
in a *t*_{pk-} value of zero with a continuous high levels of
transduction between potential and kinetic energy of the centre of mass
(Fig. 7).

## Discussion

### Normal walking

The present study provides (i) a new parameter, *R*_{int},
which summarizes the transduction between *E*_{p} and
*E*_{k} over the whole step cycle and (ii) the possibility of a
continuous analysis of such transduction during a walking step.

The time-average *R*_{int} is related but not equal to
*R*_{step}, previously determined from the total changes in
*E*_{p}, *E*_{kf} and *E*_{cg}
(see equation 1 in Cavagna et al.,
1976). As shown by the simulation, *R*_{int} and
*R*_{step} are affected in different ways by a phase shift
between sinusoidal curves representing *E*_{p} and
*E*_{k} (Fig.
2). By contrast, the experiments analyzed in the present study
suggest that *R*_{int} and *R*_{step} change in
a similar manner when a load is applied to the trunk during walking
(Table 1).

*R*_{int}, *R*_{step} and
*W*^{+}_{ext} were calculated for the 11 European
subjects of the present study during unloaded walking at different speeds
(Fig. 8). On average,
*R*_{int} is less than *R*_{step} up to
approximately 7 km h^{-1}, after which the trend is reversed, probably
as a result of a relative change in the amplitude and shape of the
*E*_{p} and *E*_{k} curves with the speed of
walking. Both *R*_{int} and *R*_{step} attain a
maximum at intermediate speeds: approximately 5 km h^{-1} for
*R*_{step} and 6 km h^{-1} for
*R*_{int}, whereas *W*^{+}_{ext} attains
a minimum at approximately 4 km h^{-1}. This result confirms that
maximal pendular transduction takes place at a speed higher than the most
economical speed of walking, as has become progressively more evident as more
data are collected (Willems et al.,
1995). The speed difference between the minimum of
*W*^{+}_{ext} and the maximum of
*R*_{step} is due to the fact that, when the speed increases
above 4 km h^{-1}, the increase in
*W*^{+}_{ext} from its minimum is smaller than the
continuous increase in
*W*^{+}_{v}+*W*^{+}_{f}; as a
consequence, the ratio
*W*^{+}_{ext}/(*W*^{+}_{v}+*W*^{+}_{f})
decreases and *R*_{step} increases (Equation 1). The same
argument is probably valid for *R*_{int} which, however, does
not have a simple relationship with *R*_{step}, not only for
the reasons explained by the simulation
(Fig. 2), but also because of
the changes in amplitude and shape of the *E*_{p} and
*E*_{k} curves with speed. In general, both
*R*_{step} and *R*_{int} represent an index of
the ability of the pendular mechanism to minimize the impact of
*W*^{+}_{v}+*W*^{+}_{f} on
*W*^{+}_{ext}. The result is that the minimum of
*W*^{+}_{ext} is attained at a lower speed, when and
because *W*^{+}_{v}+*W*^{+}_{f}
is smaller, in spite of the fact that the pendular mechanism works better at a
higher speed, when
*W*^{+}_{v}+*W*^{+}_{f} is
larger.

Our within-step analysis of pendular energy transduction shows that
muscular intervention may be divided into two components: (i) when the
transduction between *E*_{p} and *E*_{k} is zero
(i.e. during *t*_{pk+}, to increase both *E*_{p}
and *E*_{k} and during *t*_{pk}-, to decrease
both *E*_{p} and *E*_{k}), and (ii) during the
transduction between *E*_{p} and *E*_{k}, when
the muscles both release and absorb energy (during *t*_{tr,up}
and *t*_{tr,down}). The first component is due to a shift
between the *E*_{p} and *E*_{k} curves away from
being exactly out of phase (180°), whereas the second component is due to
a difference in the amplitude and shape of the two curves. It is now possible
to assess how energy recovery through pendular transduction is affected by the
phase shift between the *E*_{p} and *E*_{k}
curves and how it is affected by the difference in shape/amplitude of the two
curves. An example is discussed below for loaded compared with unloaded
walking.

An obvious reason for failure to recover energy using the pendular
mechanism is a phase shift of other than 180° between the potential and
kinetic energy curves. Why is the value of 180° not maintained? Although a
net input of energy during *t*_{pk+} to increase both
*E*_{p} and *E*_{k} is to be expected, to
overcome the energy lost by friction in the pendular motion, the necessity of
a net absorption of both *E*_{p} and *E*_{k}
during *t*_{pk-} is less clear. *t*_{pk+} occurs
mostly during the period of double contact and corresponds to the forward and
upward push of the back foot as it is about to leave the ground
(Cavagna and Margaria, 1966).
Mochon and McMahon (1980)
showed that the action of muscles during the double-support phase establishes
the initial conditions for the succeeding mainly ballistic phase of the step,
which takes place during the single contact. *t*_{pk-} on the
other hand occurs during the single-contact phase and corresponds to an
unexplained waste of energy due to the fact that the maximum of
*E*_{p} is attained before the minimum of
*E*_{k}.

During the periods when energy transduction does occur between
*E*_{p} and *E*_{k}, the failure of
*r*(*t*) to attain a value of unity implies that negative and
positive work is done by the muscular force to absorb or deliver energy
because the *E*_{p} and *E*_{k} curves are not
mirror images. During normal, unloaded walking, this failure is smaller during
*t*_{tr,up}, corresponding to the lift of the centre of mass,
than during *t*_{tr,down}, corresponding to the descent of the
centre of mass. This is unexpected because, as mentioned above, the lift is
initiated by the double-support phase of the step, whereas the descent is
initiated with the whole body pole-vaulting over the supporting leg in the
ballistic single-contact phase of the step, i.e. when the inverted pendulum
model should apply. The `square' shape of *r*(*t*) rising
abruptly to a plateau during *t*_{tr,up}, results in an
increase of approximately 60% in the total fractional energy recovered
(*R*_{int}). The less effective `triangular' shape of
*r*(*t*) during *t*_{tr,down} is due to a slower
rise to unity, during which the gravitational potential energy actively
absorbed by the muscles and elastic structures is greater than the
simultaneous increase in kinetic energy. The contrary is true for the shorter
period after the peak of *r*(*t*): energy must now be added to
increase *E*_{k} beyond the level attained due to the decrease
in *E*_{p} (Fig.
3).

### Walking with loads

One effect of loading is to improve the pendular transduction between
potential and kinetic energy of the centre of mass by making the changes in
*E*_{p} and *E*_{k} more exactly out of phase.
This is shown by the reduction in both *t*_{pk+} and
*t*_{pk-}, which is significant in the African women only
(Table 1). The reduction is
relatively larger for *t*_{pk-} than *t*_{pk+},
suggesting that the period during which both *E*_{p} and
*E*_{k} decrease as a result of negative work being done by the
muscular force is more easily reduced than the period during which energy is
added to the system. In fact, under some conditions, *t*_{pk-},
but not *t*_{pk+}, can disappear
(Fig. 7).

Of the two periods when an energy transduction occurs between
*E*_{p} and *E*_{k}, loading affects mainly
*t*_{tr,down}, when the body `falls forwards' on the supporting
leg (Figs 5,
6). Loading favors the
transduction of *E*_{p} to *E*_{k}, so that a
smaller amount of *E*_{p} has to be absorbed by the muscles
(compare the negative slopes of the *E*_{cg} curves
during *t*_{tr,down} in Figs
3 and
5). This is shown by a faster
increase in *r*(*t*) at the beginning of the descent of the
centre of mass, which is particularly evident in the African women
(Fig. 6). In the extreme case
illustrated in Fig. 7, loading
results both in the disappearance of *t*_{pk-} (discussed
above), and in a transduction between *E*_{p} and
*E*_{k} during the descent of the centre of mass similar to
that attained during the lift, resulting in a plateau at a high value of
*r*(*t*). It is quite possible that the differences described
between European subjects and African women in the present study may derive in
part from the location of the mass support: head-supported in the African
women and shoulder-supported in the European subjects.

### Concluding remarks

The present study throws more light on pendular energy transduction during
walking because it offers the possibility of investigating this energy
transduction at different phases of the gait cycle. One of the first outcomes
of this new analysis is a demonstration of the asymmetry between the upward
and downward phases of the pnedular oscillation of the centre of mass. The new
index *R*_{int}, designed to quantify pendular energy
transduction, confirms and extends the information given by the previously
used index, *R*_{step}. An application of this new approach is
the analysis of the effect of loading on the mechanics of walking: the phases
of the step mainly affected by loading can now be determined, even though the
mechanism of the observed changes is still unknown. Loading improves the
pendular transduction between *E*_{p} and
*E*_{k}, particularly during the single-contact ballistic phase
of the step. The improved pendular transduction is achieved because the
*E*_{p} and *E*_{k} curves become more exactly
out of phase because of a change in their relative shape, particularly at the
beginning of the descent of the centre of mass. Even if this mechanism occurs
at least to some extent in the European subjects, it is exploited fully by the
African women, with the result that the increase in the fraction of energy
recovered by pendular transduction over the whole step cycle in response to
loading is significant in the African women only.

- List of symbols
*E*_{cg}- total mechanical energy of the centre of mass:
*E*_{cg}=*E*_{k}+*E*_{p} *E*_{k}- kinetic energy of the centre of mass:
*E*_{k}=*E*_{kf}+*E*_{kv} *E*_{kf}- kinetic energy of forward motion of the centre of mass:
*E*_{kf}=0.5*MV*_{f}^{2} *E*_{kv}- kinetic energy of vertical motion of the centre of mass:
*E*_{kv}=0.5*MV*_{v}^{2} *E*_{p}- gravitational potential energy of the centre of mass
- g
- acceleration due to gravity
- M
- body mass
- r(t)
- instantaneous recovery of mechanical energy calculated from the
absolute value of the increments, both positive and negative, of
*E*_{p},*E*_{k}and*E*_{cg}during the step (Equation 2) - r(x)
- instantaneous recovery of mechanical energy calculated from the
absolute value of the derivative of the functions simulating
*E*_{p},*E*_{k}and*E*_{cg}during a cycle (Equation 5) - R
- recovery
*R*_{int}- mean value over one period of
*r*(*x*) (simulation) or*r*(*t*) (experimental data); for the simulation: and*R*_{int}(360°)=*R*_{int}; for the experimental data: and*R*_{int}(τ)=*R*_{int} *R*_{step}- recovery of mechanical energy calculated from the sum over one step of
the positive increments of
*E*_{p},*E*_{k}and*E*_{cg}(Equation 1) - t
- time
*t*_{dc}- fraction of the step period τ during which both feet are in contact with the ground (double contact)
*t*_{pk+}- difference between the time at which
*E*_{k}is maximum and the time at which*E*_{p}is minimum.*E*_{k}and*E*_{p}increase simultaneously during*t*_{pk+} *t*_{pk-}- difference between the time at which
*E*_{k}is minimum and the time at which*E*_{p}is maximum.*E*_{k}and*E*_{p}decrease simultaneously during*t*_{pk-} *t*_{sc}- fraction of the period τ during which one foot only contacts the ground (single contact)
*t*_{tr,down}- time of
*E*_{k}—*E*_{p}transduction during the descent of the centre of mass *t*_{tr,up}- time of
*E*_{k}—*E*_{p}transduction during the lift of the centre of mass *V*_{f}- instantaneous velocity of forward motion of the centre of mass
*V*_{v}- instantaneous velocity of vertical motion of the centre of mass
*W*_{ext}- external work done during each step calculated from the changes in
mechanical energy of the centre of mass,
*E*_{cg}=*E*_{p}+*E*_{kf}+*E*_{kv}.*W*^{+}_{ext}is the sum of the positive increments of*E*_{cg}during τ *W*_{f}- work calculated from the forward speed changes of the centre of mass
during each step.
*W*^{+}_{f}is the sum of the positive increments of*E*_{kf}during τ *W*_{k}- work calculated from the kinetic energy changes of the centre of mass
during each step.
*W*^{+}_{k}is the sum of the positive increments of*E*_{k}during τ *W*_{v}- work calculated from the potential energy changes of the centre of mass
during each step.
*W*^{+}_{v}is the sum of the positive increments of*E*_{p}during τ - α
- phase shift between the maximum of
*E*_{k}and the minimum of*E*_{p};α =360°*t*_{pk+}/τ, where*t*_{pk+}is the difference between the time at which*E*_{k}is a maximum and the time at which*E*_{p}is a minimum - β
- phase shift between the minimum of
*E*_{k}and the maximum of*E*_{p};β =360°*t*_{pk-}/τ, where*t*_{pk-}is the difference between the time at which*E*_{k}is a minimum and the time at which*E*_{p}is a maximum - τ
- step period, i.e. period of repeating change in the motion of the
centre of mass: τ=
*t*_{sc}+*t*_{dc}

## ACKNOWLEDGEMENTS

The authors would like to thank Dr Luigi Tremolada for his analysis of the simulation (Equations 4-6).

- © The Company of Biologists Limited 2002