## ABSTRACT

It is known that mechanical work to sustain walking is reduced, owing to a transfer of gravitational potential energy into kinetic energy, as in a pendulum. The factors affecting this transfer are unclear. In particular, the phase relationship between potential and kinetic energy curves of the center of mass is not known. In this study, we measured this relationship. The normalized time intervals α, between the maximum kinetic energy in the sagittal plane (*E*_{k}) and the minimum gravitational potential energy (*E*_{p}), and β, between the minimum *E*_{k} and the maximum *E*_{p}, were measured during walking at various speeds (0.5–2.5 m s^{−1}). In our group of subjects, α=β at 1.6 m s^{−1}, indicating that, at this speed, the time difference between *E*_{p} and *E*_{k} extremes is the same at the top and the bottom of the trajectory of the center of mass. It turns out that, at the same speed, the work done to lift the center of mass equals the work to accelerate it forwards, the *E*_{p}–*E*_{k} energy transfer approaches a maximum and the mass-specific external work per unit distance approaches a minimum.

## INTRODUCTION

where *M* is the mass of the body and *V*_{f,m}_{a}_{x} and *V*_{f,m}_{i}_{n}, respectively, are the maximum and minimum values of the forward velocity of the center of mass attained at each step. In order to maintain a constant average step speed, the loss in kinetic energy taking place at each step must be restored by reaccelerating the body forwards, and this requires work and energy expenditure.

Fortunately, as in pole vaulting, the mechanical energy remaining after the impact on the ground (Donelan et al., 2001, 2002a,b), and the work done during double contact by one leg against the other (Bastien et al., 2003), is stored during the lift (*S*_{v}) of the body as an increase in gravitational potential energy, Δ*E*_{p}=*M**g**S*_{v} (where ** g** is the acceleration of gravity). After the lift, the body ‘falls forwards’, restoring

*E*

_{kf}at the expense of

*E*

_{p}. In this way, some mechanical energy is conserved by a pendulum mechanism. Contrary to a pendulum, however, the two energy changes in walking are not exact mirror images of each other, with the consequence that mechanical work must be done by the muscles to maintain the motion of the center of mass in the sagittal plane.

*E*

_{p}and

*E*

_{kf}(Cavagna et al., 1976; Heglund et al., 1982; Cavagna et al., 2000; Massaad et al., 2007):

where *W*_{v} represents the amplitude of the *E*_{p} curve, i.e. the positive work (*W*_{v}=Δ*E*_{p}) done at each step to lift the center of mass; *W*_{f}=Δ*E*_{kf} represents the amplitude of the *E*_{kf} curve; and *W*_{ext} is the total positive external work actually done in each step to maintain the motion of the center of mass in the sagittal plane. In a frictionless pendulum, *W*_{ext}=0 and recovery=1.

In order to optimize the recovery of mechanical energy, the kinetic and the gravitational potential energy curves must have the same shape, be equal in amplitude and be opposite in phase, as in a pendulum. This study aimed to describe the phase relationship between gravitational potential energy *E*_{p} and the total kinetic energy *E*_{k}=*E*_{kf}+*E*_{kv}, where is the instantaneous kinetic energy of vertical motion (*V*_{v} is the vertical velocity of the center mass). To this aim, the time shift between the maximum kinetic energy and the minimum gravitational potential energy, and the time shift between the minimum kinetic energy and the maximum potential energy, were simultaneously measured, for the first time.

## MATERIALS AND METHODS

### Subjects and experimental procedure

The experiments were performed on five untrained male subjects (Table 1). Informed consent was obtained from all subjects. The study protocol was approved by the Ethics Committee of the University of Milan.

A total of 292 walks were analyzed during experiments performed on different days. Body mass values refer to the average of all days.

The subjects walked at different speeds across a 4 m×0.5 m force platform sensitive to the fore–aft (*F*_{f}) and vertical (*F*_{v}) components of the force impressed on it by the feet; the lateral component of the force was not considered. The characteristics of the platform were as described by Cavagna (1975). The platform had its surface at the level of the floor and was set 30 m from the beginning of a corridor 50 m long so that the subjects had plenty of space to reach a constant speed over the platform. The speed range considered in this study was from 0.5 m s^{−1} to 2.5 m s^{−1}. Two photocells set 2–3 m apart at the shoulder level of a subject were used to measure the average walking speed (*V*_{f}); we analyzed the steps recorded within the two photocells. Because the displacement of the center of mass within the body and the tilting of the trunk (Fenn, 1930) are small in comparison with the distance between the photocells, *V*_{f}, as measured, should not differ appreciably from the average forward speed of the center of gravity. The subjects wore gym shoes and walked over the platform on several different days. The time in which both feet were on the ground (*t*_{dc}) was determined using a small transmitter, carried at the waist, connected by wires to metal gauze patches glued to the soles of the shoes; for all subjects we used a complete heel–lateral–toe metal gauze, which is the most reliable to determine touch down and lift-off. When both feet were on the ground, the circuit was closed through the metallic surface of the platform and the transmitter operated, thus giving rise to a square signal at the output of a receiver (Cavagna et al., 1976). The *t*_{dc} signal was acquired together with the *F*_{f} and *F*_{v} tracings, and the photocell circuit output by an A/D board (PCI-MIO-16E-4, National Instruments, Austin, TX, USA), with a sample rate of 500 Hz. Acquired data were stored in memory for subsequent analysis.

### Analysis of force platform records

A custom LabVIEW software (version 7.1, National Instruments) (available from M.A.L. upon request) was used to calculate the forward and vertical velocities of the center of mass of the body, every 2 ms, by integrating the *F*_{f} and *F*_{v} tracings. The kinetic energies of forward and vertical motion (*E*_{kf} and *E*_{kv}) were calculated from the forward and vertical velocities. The potential energy (*E*_{p}) was calculated by integrating vertical velocity as a function of time to yield vertical displacement and multiplying this by the body weight. The total mechanical energy was calculated as *E*_{cg}=*E*_{kf}+*E*_{kv}+*E*_{p}. The positive external mechanical work was obtained by adding the increments in *E*_{cg} over an integer number of steps. The procedure involved in using force platforms records to measure external mechanical work has been described in detail by Cavagna, (1975). More specifically, the platform measures the vertical and horizontal forces during locomotion (fig. 1 of Cavagna, 1975), the integration of which yields the vertical and horizontal velocity changes. From the velocity changes, one can work out the vertical and horizontal kinetic energy, and through further integration of the vertical velocity, the center of mass displacement is calculated (fig. 2 of Cavagna, 1975), from which the potential energy can be worked out (fig. 3 of Cavagna, 1975). Once total kinetic energy and potential energy are determined, thus giving the total mechanical energy of the system, external work is calculated from the variation in it (fig. 3 of Cavagna, 1975).

In a perfect steady walk at flat level, the ratio between positive (*W*^{+}) and negative (*W*^{−}) work should be equal to one. In the 292 walks selected for analysis, =1.01±0.14, =1.03±0.14 and=1.06±0.28 (means±s.d., *N*=292).

where τ is the step period and *t*_{pk+} is the difference between the time at which *E*_{k}=*E*_{kf}+*E*_{kv} is maximum and the time at which *E*_{p} is minimum (solid vertical lines in Figs 1 and 2, thicker lines refer to maxima, thinner lines refer to minima). Positive values of α (maximum *E*_{k} following the minimum *E*_{p}) indicate that positive work is done during double contact to attain the maximum kinetic energy *E*_{k} and to lift the center of mass from its lowest point (Fig. 1). Negative values of α (maximum *E*_{k} preceding the minimum *E*_{p}; top record in Fig. 2) indicate that negative work is done during double contact to decelerate forward while attaining the lowest value of the vertical displacement. In conclusion, positive α values indicate positive work done during double contact, whereas negative values indicate negative work.

where *t*_{pk−} is the difference between the time at which *E*_{k} is minimum and the time at which *E*_{p} is maximum (dashed vertical lines in Figs 1 and 2, thicker lines refer to maxima, thinner lines refer to minima). Positive values of β (minimum *E*_{k} following the maximum *E*_{p}) indicate that negative work is done to decelerate forward while beginning the downward fall. In conclusion, positive β values indicate negative work done during midstance, whereas negative values indicate positive work.

It should be noted that the two curves of *E*_{p} and *E*_{k} have somewhat different shapes, so they are not symmetric (e.g. specular); for this reason, it is indeed tricky to define a single phase shift for the entire curves. However, the phase shift α that we define and study throughout this work is uniquely determined by its energetic role; namely, to be positive to accelerate forward and lift the body and vice versa. For this to be true, the only features of the *E*_{p} and *E*_{k} curves that are necessary to exploit are the neighborhoods of the extrema, i.e. the minimum *E*_{p} and the maximum *E*_{k}. Fortunately, these extrema are unique: *E*_{p} has just one minimum and *E*_{k} has just one maximum, the positions of which are compared with each other (and vice versa for the phase shift β).

In Fig. 3, the data collected as a function of walking speed were grouped in ∼0.14 m s^{−1} (0.5 km h^{−1}). The data points and the vertical segments represent the mean±s.d. in each speed interval. The values below the symbols indicate the number of items used to calculate the mean.

## RESULTS

Fig. 1 represents typical experimental records of the mechanical energy changes of the center of mass of the body: *E*_{p}, to sustain vertical displacement; *E*_{k}=*E*_{kf}+*E*_{kv}, the forward and vertical velocity changes; and *E*_{cg}, the combined displacement in the sagittal plane. As described in the Materials and Methods, the time intervals *t*_{pk+} and *t*_{pk−} are defined by the horizontal distance between the solid and dashed vertical lines, respectively. The position of the limbs during the step is shown by the stick figures at the bottom of the figure. The thick horizontal bar indicates the time of double contact. As shown, *t*_{pk+} takes place during double contact and *t*_{pk−} during single contact.

Fig. 2 shows records obtained at three walking speeds, increasing from bottom to top (part of the middle record is amplified in Fig. 1). It can be seen that, on average, *t*_{pk+} is greater than *t*_{pk−} at 0.70 m s^{−1} and that *t*_{pk+}≈*t*_{pk−} at 1.56 m s^{−1}, whereas at 2.29 m s^{−1}, *t*_{pk−} is nil and *t*_{pk+} is reversed.

The upper panel of Fig. 3 shows α, β and the ratio between forward and vertical work (*W*_{f}/*W*_{v}) as a function of speed. It appears that both α and β decrease with speed: α from positive to negative values, whereas β is always positive and reduces to zero at the highest speeds. β is less than α up to 1.6 m s^{−1} where it equals α (dashed vertical line). In addition, the ratio between work to sustain forward speed changes of the center of gravity (*W*_{f}) and its vertical displacement (*W*_{v}) increases with speed, attaining unity at the same speed at which α=β.

The bottom panel of Fig. 3 shows external work and recovery (Eqn 2) as a function of speed. Figs 1 and 2 show the energy tracings of one individual subject; hence, there is no need for normalizing the energy, which can be measured directly in joules. However, in Fig. 3, we average over all different subjects, who have different body masses and have traveled for different total distances during their walk; for this reason, we must normalize the energy expenditure (J) by body mass (kg) and distance traveled (m), thus obtaining the mass-specific external work done per unit distance, *W*_{ext}. The recovery attains a maximum of 0.66 at a speed of 1.46 m s^{−1}, in agreement with Cavagna et al. (1983). The minimum *W*_{ext} that is relevant for the energetics of the system is the local one attained at intermediate speed, and not the absolute minimum at 0.6 m s^{−1}, which is an artifact caused by the dragged motion of the limbs at very low speed. The local minimum *W*_{ext} and the maximum recovery are attained at a speed slightly less (0.14 m s^{−1}) than the speed at which α=β and *W*_{k}=*W*_{v} (dashed line).

Fig. 4 shows that the time of upward displacement (*t*_{up}) approximately equals that of downward displacement (*t*_{down}) for most walking speeds, except at speeds less than 0.9 m s^{−1}, at which *t*_{up}>*t*_{down}, and at speeds greater than ∼1.9 m s^{−1}, at which *t*_{down}>*t*_{up}. At the ‘optimal speed’ (dashed vertical line) *t*_{up}=*t*_{down}, the time of single contact equals ∼75% of τ.

## DISCUSSION

The most relevant feature of this study derives from the measurement of the phase shifts α and β between *E*_{p} and *E*_{k}. In particular, it is interesting that β=α≈10 deg at the speed (∼1.6 m s^{−1} in our group of subjects) at which *W*_{f}=*W*_{v}, the mass-specific external work per unit distance *W*_{ext} approaches a minimum and the recovery approaches a maximum. It follows that, at this optimal speed, the *E*_{p} and *E*_{k} curves are not exactly in opposite phase but are shifted by the same amount (β=α) at the top and the bottom of the trajectory of the center of mass. Furthermore, at this speed, the lift duration (*t*_{up}) equals the fall duration (*t*_{down}), whereas *t*_{up}>*t*_{down} at low speeds and *t*_{down}>*t*_{up} at high speeds (Fig. 4).

Given that the step length is the most natural scale of walking, we find it convenient to express the shifts α and β (measured in units of degrees) with two corresponding shifts (measured in units of lengths), by multiplying α and β by the step length, and dividing by 360 deg. Because the step length at the optimal speed was found to be, on average, 0.82 m, the shifts α (from the maximum *E*_{k} to the minimum *E*_{p}), at the bottom of the trajectory, and β (from the minimum *E*_{k} to the maximum *E*_{p}), at the top of the trajectory (Fig. 1), correspond in both cases to a length 0.82×10/360=0.023 m.

At the lowest speed, at which step length is, on average, 0.51 m, the shift during double contact at the bottom of the trajectory, from the maximum *E*_{k} to the minimum *E*_{p} (α≈46 deg in Fig. 3), corresponds to a length 0.51×46/360=0.065 m, whereas during single contact, at the top of the trajectory, from the minimum *E*_{k} to the maximum *E*_{p} (β≈20 deg in Fig. 3), it corresponds to a length 0.51×20/360=0.028 m*.*

At the highest speed, at which step length is, on average, 1 m, the shift during double contact, at the bottom of the trajectory, from the maximum *E*_{k} to the minimum *E*_{p} (α≈−28 deg in Fig. 3), corresponds to a length 1.00×(−28)/360=−0.078 m, whereas the shift during single contact at the top of the trajectory, from the minimum *E*_{k} to the maximum *E*_{p} (β≈−1 deg), corresponds to a length 1.00×(−1)/360=−0.002 m.

The expression ‘optimal speed’ should, in principle, refer to the speed at which the mass-specific metabolic energy expenditure per unit distance is at the minimum. This is not the case in the present study, as we define the optimal speed as that of the minimum mechanical energy expenditure, *W*_{ext}. Several studies showed that other mechanical work must contribute to the metabolic energy output: (1) the internal work as a result of the kinetic energy changes of the limbs relative to the center of mass (Cavagna and Kaneko, 1977), (2) the work to redirect the center of mass velocity from one inverted pendulum to the other (Donelan et al., 2001, 2002a,b), (3) the internal work done by one leg against the other during double contact (Bastien et al., 2003) and (4) the internal work done against frictional losses (Minetti et al., 2020). Furthermore, the optimal speed determined by the minimum *W*_{ext} differs between subjects (Fig. 3 shows the average of five subjects). The interesting (and new) finding here is that, at this speed, the time difference between *E*_{p} and *E*_{k} extremes is the same at the top and the bottom of the trajectory of the center of mass, i.e. *E*_{p} and *E*_{k} are shifted by the same amount at the top and bottom of the trajectory of the center of mass. This (together with the minimum external work and the maximum recovery) could prove to be true, independent of the absolute value of the optimal speed attained by different subjects. Further studies are required to determine how the optimal speed is changed in different groups of subjects (training, age, etc.).

In conclusion, the present study shows a new feature of the mechanics of walking: the shift between potential and kinetic energy of the center of mass being equal at the bottom and at the top of its trajectory; this occurs when the work done in forward and vertical direction are equal, the gravitational–kinetic energy transfer approaches a maximum and the mass-specific external mechanical work done per unit distance approaches a minimum (optimal walking speed). It must be pointed out that the optimal speed might differ between subjects according to their training.

## Acknowledgements

The authors thank Andrea Cavagna for discussions.

## Footnotes

**Author contributions**

Conceptualization: G.A.C.; Methodology: G.A.C., M.A.L.; Software: G.A.C., M.A.L.; Data curation: G.A.C., M.A.L.; Writing - original draft: G.A.C.; Writing - review & editing: G.A.C., M.A.L.; Visualization: G.A.C.; Supervision: G.A.C.

**Funding**

This study was supported by the Italian Ministero dell'Universita’ e della Ricerca.

## References

**Competing interests**

The authors declare no competing or financial interests.