## Abstract

Although most of the literature on locomotion energetics and biomechanics is about constant-speed experiments, humans and animals tend to move at variable speeds in their daily life. This study addresses the following questions: *1*) how much extra metabolic energy is associated with traveling a unit distance by adopting acceleration/deceleration cycles in walking and running, with respect to constant speed, and *2*) how can biomechanics explain those metabolic findings. Ten males and ten females walked and ran at fluctuating speeds (5 ± 0, ± 1, ± 1.5, ± 2, ± 2.5 km/h for treadmill walking, 11 ± 0, ± 1, ± 2, ± 3, ± 4 km/h for treadmill and field running) in cycles lasting 6 s. Field experiments, consisting of subjects following a laser spot projected from a computer-controlled astronomic telescope, were necessary to check the noninertial bias of the oscillating-speed treadmill. Metabolic cost of transport was found to be almost constant at all speed oscillations for running and up to ±2 km/h for walking, with no remarkable differences between laboratory and field results. The substantial constancy of the metabolic cost is not explained by the predicted cost of pure acceleration/deceleration. As for walking, results from speed-oscillation running suggest that the inherent within-stride, elastic energy-free accelerations/decelerations when moving at constant speed work as a mechanical buffer for among-stride speed fluctuations, with no extra metabolic cost. Also, a recent theory about the analogy between sprint (level) running and constant-speed running on gradients, together with the mechanical determinants of gradient locomotion, helps to interpret the present findings.

- running economy
- speed oscillation

it is well known that the metabolic cost of transport of human walking (J·kg^{−1}·m^{−1}) is a U-shaped function of the progression speed, whereas running cost is speed independent (6). These and many other findings in the literature on human locomotion are based on constant-speed experiments on treadmills (14) although, in real life, human and animal gaits very often occur at variable speeds (11). When considering this feature of daily locomotion, one could be tempted to estimate the overall cost of transport by combining, in a sort of a discrete integral, the data obtained at fixed progression speeds, weighed for their duration in the accelerative/decelerative pattern. Particularly, that approach would underestimate the real metabolic cost of transport of running at oscillating speeds because, as mentioned above, the cost is speed independent. Whereas the cost of deceleration could be considered nil because the body kinetic energy could be dissipated as heat (which we know is not the case, Ref. 14), acceleration is expected to involve some extra mechanical/metabolic energy (as occurring in road vehicles).

A few years ago, Minetti and collaborators (9) showed that walking at a speed cyclically oscillating within a wide range (±2 km/h) resulted in the same cost of moving at the average constant speed. Mechanical measurements suggested that the human body behaves similarly to a hybrid car: the inherent mechanical energy increase and decrease occurring within a single stride in a constant speed sequence can be combined to be equivalent to a number of accelerating strides followed by the same number of decelerating strides, resulting in a similar energy balance at the end of the speed-oscillating cycle. In synthesis, as hybrid cars minimize the extra fuel consumption due to the oscillating speed in the urban environment by converting deceleration energy into acceleration (via electric energy), humans limit the extra cost of oscillating walking by exploiting the inherent within-stride energy fluctuation in multiple accelerating/decelerating strides. In the absence of a relevant anatomical structure capable of cumulatively storing deceleration energy of several steps for later use, during long-term oscillating speed cycles, the negative work is not done within the steps of the accelerative phase for it to be postponed to the overall decelerative phase, where conversely steps do not show any residual positive work, which is normally typical of constant-speed locomotion (9). In this way, the peculiar energy time course serves as a virtual storage/release system, allowing performance of long-term acceleration cycles at no extra metabolic cost.

Apart from describing a fundamental aspect of legged locomotion, the interest about oscillating-speed gaits is today boosted by the need to infer the metabolic consumption of sport activities such as soccer (12), rugby (5), and other field games, where subjects run at a variable speed. In particular, match analysis and other video techniques are devoted to sample players' movements, even in real time, from which the associated oxygen consumption could be indirectly estimated and the relevant training programmed.

In this study, we extend previous metabolic measurements on oscillating-speed walking (9) to running, with the aim to eventually establish the maximum acceleration/deceleration range at which the cost of transport does not deviate from that of the fixed average speed. The experiments have been conducted both on a variable-speed treadmill and overground to check the reliability of the laboratory-based methodology (noninertial reference system), previously confirmed for walking (9) and also in oscillating-speed running.

## MATERIALS AND METHODS

Ten female and ten male subjects (see Table 1 for anthropometric data) participated in the study, after having signed their informed consent. The investigation has been approved by the Ethical Committee of the University of Milan.

As in the previous study on walking, experimental sessions were organized both in the laboratory and in the field. Due to the complexity of the outdoor experimental protocol needed to test subjects in a truly inertial system, laboratory measurements were the first to be done. Running experiments were performed only by male subjects.

#### Laboratory session.

A motorized treadmill (Ergo LG; Woodway) was programmed to move at constant and oscillating speed, as described by linear ascending and descending speed ramps, each lasting 3 s. The average speed was 11 km/h, and the five oscillations were ±0, ±1, ±2, ±3, and ±4 km/h. After familiarization with the treadmill and the experimental protocols, the subjects ran for 5 min in each condition, chosen from a random sequence and separated by 10-min rest. Heart rate (HR), oxygen uptake (V̇o_{2}), and CO_{2} production were measured by a portable metabograph (k4b2; Cosmed) both at rest while standing and during exercise. The metabolic cost of transport (C) was calculated by dividing the net oxygen uptake, collected in the last minute of each condition, by the average progression speed. The final units, i.e., J·kg^{−1}·m^{−1}, were obtained by dividing by the subject mass and by converting ml O_{2} into J according to the measured Respiratory Quotient (RQ = CO_{2} production/O_{2} consumption).

In addition, 18 reflective markers were located on the most relevant joints of one subject to estimate the 3D path of the body center of mass (7), by means of motion-capture system (100 Hz; Vicon). The time course of that trajectory was used to infer the changes in mechanical energies (potential and horizontal/vertical kinetic and total) involved in running at the widest oscillating speed (see Fig. 4).

To obtain a complete set of speed-oscillation gaits on the same subjects and to slightly extend the oscillations originally investigated, we also replicated the above protocol for walking by following the procedure illustrated previously (9). The average speed chosen for walking was 5 km/h with 5 oscillations (±0.0, ±1.0, ±1.5, ±2.0, ±2.5 km/h).

#### Field measurements.

Subjects ran by following a green spot projected by a 532-nm, 300 mW laser (WickedLasers) on the soccer pitch of the San Siro “Meazza” Stadium in Milan (Fig. 1). The mechanical frame of a digitally motorized telescope (NextStar 4SE, Celestron) was programmed by custom software (LabView 8.6, National Instruments) as to move the attached laser from a height of about 40 m and describe a circular path with 68-m diameter (the width of the soccer pitch). At the average speed chosen, moving along that path was expected to generate a centrifugal acceleration of 0.27 m·s^{−2}, which was considered as uninfluential to the current dynamics with respect to straight-line running. The software controlled, via a RS-232 serial port, both azimuth and altitude of the two-step motors of the telescope as to project the laser dot at constant angular speed (11 ± 0 km/h) and at the other four oscillating speeds (Fig. 2). This was achieved by dividing the circular trajectory into 180 steps (2° each) and by setting the speed and duration of the linear translation between them. The 10 subjects, wearing protective laser goggles and equipped with the same portable metabograph used in the laboratory, completed four full circles for each condition. The assessment of the metabolic cost followed the same procedure as described above.

In both research environments, blood lactate (La) was sampled (Lactate Plus, Nova Biomedical) 4 min after each investigated condition to check the aerobic regime of running experiments.

#### Statistics.

All recorded data are presented as means ± SD in Table 2. For statistical analysis of V̇o_{2}, HR, La, RQ, and C, a one-way ANOVA for reapeated measures was performed (with a post hoc Bonferroni test) to check the effects of the five oscillation levels, both in walking and in pooled data. A two-way ANOVA for repeated measures was performed on running data (cost: variable of interest; within factors: acceleration level and environment, i.e., laboratory vs. field, with a post hoc Bonferroni test). The null hypothesis was rejected when *P* < 0.05.

## RESULTS

#### Walking.

V̇o_{2} and C increased as a function of the oscillation speed (see Table 2) by showing significant differences with respect to the constant-speed condition at all oscillation levels, and HR was significantly higher at the upper three levels; RQ values were found to be independent from the acceleration protocol.

#### Running.

Data are presented for nine male subjects because one of them reported blood lactate measurament higher than 4 mmol/l. Although moderately (but significantly) higher C values at the highest two oscillation levels (with respect to that at constant speed running) are shown, the data trend is very similar in both laboratory and field experiments. For this reason, we also present pooled data from the two environments in Table 2. It can be noted that the small increase of metabolic cost at high oscillation speeds is greater in the field condition, signaling that the noninertial bias of laboratory experiments, if any, could affect results only at high-acceleration running.

ANOVA revealed significantly higher HR values in the field condition, with no effects of the acceleration levels. The +10 bpm difference can be explained by the higher ambient temperature (31°C) during the field experiments with respect to the laboratory (25°C). It has been reported that a hot environment causes a decrease of central blood volume, with a parallel decrease of stroke volume, resulting in a higher HR with no change in V̇o_{2} (13).

## DISCUSSION

As discussed previously (9), acceleration/deceleration cycles should be theoretically associated with some extra metabolic expenditure. By considering the increases in kinetic energy of the body center of mass due to acceleration, the extra positive mechanical external work (Δ*W*^{+}_{EXT}, J·kg^{−1}·m^{−1}) was estimated (9) to exceed the one already associated with constant-speed running according to
*v* is half the speed oscillation (m/s) and δ*t* is the acceleration duration (s). The same applies to decelerations, where negative mechanical work has to be done to decrease the kinetic energy. Due to the fact that muscles consume metabolic energy to perform both work types, with different efficiencies (eff^{+} = 0.25, eff^{−} = 1.25) (1), we can expect an increase in running cost of transport [Δ*C* = 0.5(Δ*W*^{+}_{EXT}/eff^{+} + Δ*W*^{−}_{EXT}/eff^{−})] of

When expressed as extra energy expenditure (Δ*E* = *v*Δ*C*, mlO_{2}·kg^{−1}·min^{−1}), for an average speed of 11 km/h and an acceleration phase lasting 3 s, we obtain

As shown in Fig. 3, such a prediction greatly overestimates the pooled experimental data of running on the treadmill and in the field (Δ*C* = +49.4% rather than +6.3% at the widest speed oscillation). This trend indicates a substantial independency from the oscillation range. Although experimental sessions done in the laboratory were replicated in the field as a precaution against the noninertial frame of reference of the variable-speed treadmill (9), again we found no remarkable differences between the two conditions.

As in the previous analogous study (9), the present metabolic data of walking are almost constant up to a speed oscillation of ±2 km/h (Table 2).

The rationale to explain this phenomenon for walking has been based on the within-stride mechanical energy fluctuation occurring when moving at constant speed. We can expect that, during accelerative phases, mostly energy increases occur, the reverse being the case for decelerations. Therefore, by separately grouping the mechanical energy increases and decreases of constant-speed strides, we could obtain an equivalent sequence of strides, half of them in acceleration, half in deceleration. The two sequences should imply the same amount of postitive and negative work, thus the same overall metabolic energy expenditure (Fig. 4, *top*). Such an equivalence can be obtained only within a certain speed-change oscillation, and it is based on the physiological energy fluctuation range of the single stride at constant speed. For walking, it was concluded that ±2.5 km/h (and δ*t* = 3 s) was that limit. Again, in agreement with the previous findings (9), the energy expenditure of speed-oscillating walking (*v* ± Δ*v*, m/s) can be predicted by the proposed equation calculating the average metabolic cost at all the speeds experienced during the cycle, therefore neglecting the energy associated to the speed changes, as:

The explanation for running needs a more complex approach. The mechanical energy fluctuation within each stride is partly due to elastic energy stored and successively released by inert body structures, namely tendons, whose action does not affect the metabolic consumption of the runner (Fig. 4, *bottom*). However, the human body is not equiped with anatomical machinery devoted to store elastic energy on a long-term basis (as locusts do, for instance, Refs. 2 and 3), and the dynamics of that portion of the energy fluctuation cannot contribute to set cumulated mechanical energy increases in successive strides, as to correspond to the steady acceleration associated to the same (invariant) metabolic cost. It is necessary, then, to obtain an elastic-free estimate of the mechanical external work (*W*_{EXT}, J·kg^{−1}·m^{−1}) as
*C*_{R} is the measured metabolic cost of running at constant speed (i.e., the metabolic equivalent of the elastic-free mechanical work done), *eff* is the maximum expected muscle efficiency, and *W*_{INT} is the mechanical internal work, necessary to accelerate limbs with respect to the body center of mass during each locomotor cycle. This operation corresponds to removing the internal work from the maximum total mechanical work (= *C*_{R}*eff*) that muscles actually do (see Fig. 5). By substituting *C*_{R} with the metabolic cost measured in this study at a constant speed of 11 km/h, *eff* with 0.25 (1) and *W*_{INT} with values for that speed obtained previously (6), we obtain *W*_{EXT} = 0.83 J·kg^{−1}·m^{−1}. It should be noticed that the units for *W*_{EXT} (as for all the forms of the cost of transport) correspond to m·s^{−2}, thus the maximum acceleration can be estimated as:

Thus, we can say that the elastic-free external work done at constant-speed (11 km/h) running is equivalent to a speed excursion (acceleration) of ± 4.5 km/h performed in 3 s, and because of this no extra metabolic consumption should be expected within this boundary. This prediction slightly exceeds the widest speed oscillation here investigated, which was limited to 11 ± 4 km/h as to *1*) avoid anaerobic contribution to energy expenditure and *2*) ensure that, at the lowest speed, running could still be performed.

Recently, a new predictive framework has been proposed (4) to infer the metabolic expenditure of sprint running by suggesting the dynamical similarity between level accelerative strides and constant-speed strides on a steep incline. By using a regression of metabolic running cost collected on a wide range of gradients (10), those authors developed an equation that estimates the metabolic cost of running in acceleration by introducing the concepts of equivalent slope (ES) and equivalent mass (EM).

For the purposes of the present study, also decelerations need to be included in the metabolic prediction, thus the proposed equation for the cost of sprint running (*C*_{sr}, J·kg^{−1}·m^{−1}, Ref. 4),
*C*_{os}, J·kg^{−1}·m^{−1})
*Eq. 7* [*f*(*x*)], i.e.,
*x*, the metabolic cost, per each meter traveled, of running cycles of (half a meter in) acceleration and (half a meter in) deceleration. Because the mathematical form of *f*(*x*) is a polynomial function of the gradient (8):
*g*(*x*) (i.e., the term in brackets of *Eq. 8*) will retain just the even degree terms of the original polynomial included in *Eq. 7* (and Ref. 8), i.e.,

Fig. 6 shows *f*(*x*)·*EM*, namely the cost of ascending/accelerating, *f*(−*x*)·*EM*, i.e., the cost of descending/decelerating, and the combination of the two, *g*(*x*)·*EM* (i.e., *Eq. 8*), with the abscissas representing both the gradient and the speed oscillation. The equation for *C*_{os}, independent from the average running speed, predicts the cost of oscillating-speed running according to the equivalent slope theory of acceleration (4). Energy expenditure (*E*_{os}, mlO_{2}·kg^{−1}·min^{−1}) for a given average speed (*v*, m/s) can be estimated as

When these values are plotted together with experimental data (see Fig. 3) a very close match is found.

This encouraging comparison still does not explain, per se, the reason of the substantial lack of dependency of the metabolic cost on the excursion of speed oscillations. Actually, it has to be noticed that the computing framework proposed by di Prampero and collaborators (4) is only halfway driven by physics, the rest being based on experimental data (10). Those authors established a link between (level) running in acceleration and running on gradients (at constant speed). Thus, to understand the speed range independency of *C*_{os}, we need to refer to the mechanical determinants of the metabolic cost of gradient running.

A previous investigation (7), by measuring and analyzing the mechanical and metabolic features of gradient running (at constant speeds), explained the reasons for the cost decrease of downhill gradients and for the optimum gradient occurring at −10% of incline. After having removed from the negative and positive work the parts attributable to elastic energy storage and release, respectively, it was noted that those phenomena were the effects of *1*) a residual negative external work in uphill gradients (up to +15%) and of positive external work in downhill gradients (up to −15%), and *2*) the fivefold difference between the muscle efficiency for positive and negative work (see their Fig. 6). For speed-oscillating running, the metabolic equivalent of the positive and negative mechanical external work of matched gradients (−5% and +5%, −10% and +10%) has to be represented as column stacks, together with the one for the internal work (Fig. 7). Further subdivision of each component (horizontal dashed lines) shows the amount of the (metabolically converted) mechanical work measured during uphill and downhill running (shown as up and down, respectively). It can be observed that, despite the partitioning of positive external work (dark gray bar segments) into uphill (acceleration) and downhill (deceleration) parts of the cycle different at all gradients, the sum of the two is almost constant. This applies also to the negative external work (white bar segments), whereas, for internal work, we have both the constancy of the total amount and its partitioning. By looking at the total estimated metabolic cost, as obtained by converting all the forms of mechancal work (7) according to the different efficiencies, we obtain that a combination of −5% and +5% gradients implies a +2.9% change in the metabolic cost with respect to level running. The same calculation leads to +15.8% when a gradient combination of −10% and +10% is simulated. These two figures compare well with the increases found by adapting the predictions from di Prampero (4) to the present experimental protocol, for ES = ±5% (corresponding to Δ*v* = ±2.6 km/h in 3 s) and ES = ±10% (corresponding to Δ*v* = ±5.3 km/h in 3 s), which result as +3.3% and +13.3%, respectively.

The illustrated rationale again confirms the functionally independency of *C*_{os} within a given speed-change excursion of acceleration/deceleration cycles. When we replace, in Fig. 7, up with acceleration and down with deceleration, it is clearly apparent that the constancy of *C*_{os} is due, in addition to the different efficiencies of positive and negative work, to the residual deceleration during acceleration phases and the reverse, similarly to what happens in gradient running.

In conclusion, similarly to walking, the substantial constancy of the metabolic cost of running at speed oscillations up to a given span (±4 km/h or ± 1.1 m/s in 3-s ramps) seems to be explained by the inherent mechanical energy fluctuation normally occurring when moving at constant speed (see *Eq. 6* and dashed line in Fig. 3).

In other words, differently from wheeled vehicles, where the absence of energy fluctuations at constant speed makes any speed changes very expensive (hybrid cars were invented to mitigate this problem), legged-body dynamics (ground constraints) makes locomotion expensive even at constant speed, but this allows the use of the same fuel when (limited) speed oscillations are issued. Apart from the relevance of these findings in fundamental biomechanics and bioenergetics of locomotion, we expect that the proposed equations will be used to estimate the metabolic energy expenditure in a variety of field activities and sports, where moderate-speed excursions are present.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

Author contributions: A.E.M. conception and design of research; A.E.M. interpreted results of experiments; A.E.M., P.G., E.S., and D.C. prepared figures; A.E.M. drafted manuscript; A.E.M., P.G., E.S., and D.C. edited and revised manuscript; A.E.M., P.G., E.S., and D.C. approved final version of manuscript; P.G., E.S., and D.C. performed experiments; P.G., E.S., and D.C. analyzed data.

## ACKNOWLEDGMENTS

The authors thank P. Barletta at M-I Stadio Srl and all the subjects who took part to the experiments for their availability. The authors are also grateful to R. Telli and G. Pavei for assistance with the data collection.

- Copyright © 2013 the American Physiological Society