Update and extension of the ‘equivalent slope’ of speed-changing level locomotion in humans: a computational model for shuttle running

Unsteady locomotion in terms of both non-linear trajectory and speed changes is a common observation in the everyday life of humans and animals, as well as in sport activities of bipeds and quadrupeds. The scientific interest regarding mechanical (manoeuvrability, static and dynamic stability, performance) and metabolic sustainability of those gait changes with respect to linear/constant speed locomotion has been met so far by only a few papers (e.g. Alexander, 2002; Wilson et al., 2018).

Moving at fluctuating speed in humans has received attention only in the last two decades. In particular, the metabolic cost and mechanics of walking and running at imposed unsteady speed were studied (Minetti et al., 2001, 2013), but only for small–intermediate accelerations/decelerations about the average speed. Interest in the metabolic implications of unsteady locomotion increased with attempts to infer the players' effort in sports such as soccer and rugby, where locomotion during the match is far from occurring at constant speed. In addition, relevant speed changes are supposed to be associated with high metabolic cost, but the experimental protocol capable of reliably measuring this came up against the problem of steady-state condition. For this reason, an alternative had to be found, and ‘the equivalent slope’ concept came to hand.

The concept of ‘equivalent slope’ (ES) has previously been used in the physiomechanics of cycling and sprint running. In cycling, the ‘rolling resistance equivalent slope, a very shallow downhill gradient at which the negative potential energy changes balance the work necessary to overcome the rolling resistance’ was suggested as a handy laboratory tool to estimate tyre friction on the ground (Ardigò et al., 2003). The concept was to convert the deceleration of coasting down (on the level) into a downhill slope at which the bicycle+rider could passively remain stationary on a treadmill (thus moving at constant speed). At that gradient, gravity provides a net forward component that equates the rolling resistance effect when moving on the level. Very conveniently, the coefficient of rolling resistance corresponds to the tangent of that slope angle.

Recent work (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished) on the mechanics and energetics of shuttle running, an activity incorporating large accelerations and even larger decelerations, and the fact that sprint running also involves accelerations (up to 7 m s⁻²) well beyond the indicated limits, both encouraged us to update and further develop the previous predictive tools (di Prampero et al., 2005; Minetti et al., 2002) in order: (1) to enhance inference reliability in an extended range of accelerations and (2) to also make metabolic predictions for decelerated running.

The extension to a wider range of accelerations benefits from recent metabolic results (Giovanelli et al., 2016) of uphill, constant-speed running up to i=+0.84 that, according to Eqn 2, corresponds to level accelerations up to 8.24 m s⁻².

The aim of the present study was to build a conceptual/computational framework allowing us to infer the metabolic demand of accelerated/decelerated running, based either on the ES analogue (see above) or on the ultimate meaning of the cost of transport (see below), which is immediately applicable to shuttle runs in humans, but is also potentially transferable to unsteady animal locomotion.

The accuracy of the newly proposed equation(s) for C has been tested by computing the ratio, in the range −0.45≤i≤+0.45, between Eqns 4 and 3 with the negative and positive gradient branches of Eqn 1, respectively, resulting in an average value of 0.9993. Curves for Eqns 3 and 4 are also plotted in Fig. 1.

The present mathematical approach (Eqns 4 and 3) to data fitting is more modellistic than descriptive (Eqn 1), and it comes from a suggestion about the efficiency of locomotion introduced in the first half of the last century (Margaria, 1938). In synthesis, there has to be a minimum cost (C_min, J kg⁻¹ m⁻¹) for gradient running (and walking), which relates to coping with the inevitable work to sustain overall positive (uphill) and negative (downhill) changes in potential energy of the body (W_vert, J kg⁻¹ m_vert⁻¹, where m_vert is the vertical metres travelled).

The linear components of Eqns 3 and 4 are plotted in Fig. 1 as straight lines.

Eqns 9 and 11, alone or in combination, allow us to infer the metabolic cost of just accelerated running (as in sprints), in structured sequences of acceleration and deceleration bouts (shuttle running), and in sporting activities where complex combinations of accelerated and decelerated running irregularly occur (as in soccer, rugby, basketball, baseball, etc.).

A relevant comment is on the mechanical resemblance of the ES when applied to different motion activities. When referring to just the overall centre of mass, the mechanics of speed changing on a level can be converted into a constant speed at a given, equivalent (uphill or downhill) slope. This is the case of the ES of bicycles (Ardigò et al., 2003), invoked to easily estimate tyre rolling resistance, where the subject even refrains from pedalling on the downhill gradient, which makes the bike stationary on the treadmill (this is also a pure rolling resistance measurement as no air drag is acting on the subject).

It is possible that legged species such as humans would adopt different values of these variables, and thus would be expected to have a different W_int, when moving in the two ‘equivalent’ conditions. In particular, inferring the metabolism of level locomotion from gradient experiments where effort included the metabolic equivalent of a different (total) mechanical work value could introduce some bias. Moving on slopes, particularly uphill running, is related to a much slower progression speed and a higher stride frequency (Minetti et al., 1994). Even the maintenance of the speed independency of the metabolic cost of transport on slopes cannot help in this respect, and a check for W_int should be done for the two compared conditions. This is the case for extreme accelerations (or decelerations) such as during a 100 m sprint, where ES is so high that speed and stride frequency (but also stride length) of a manageable metabolic experiment on a gradient could potentially lead to an underestimation of W_int.

Another important precaution is to consider all other determinants of the total mechanical work in the activity under investigation. Air drag, for example, can be relevant at the highest speed of the acceleration phase of sprint running (but not in ES for rolling resistance or slope running in the lab). In contrast, for shuttle running, Eqns 9 and 11 can take care of the metabolic equivalent of the accelerative and decelerative phases, respectively, but the energy required to rotate the body at speed inversion is not included in the prediction.

This new mathematical model, based on the revamped concept of ES, can be implemented in a new activity logger aimed at detecting and monitoring daily and physical activity with an improved analysis of energy expenditure. Unlike the present computational scheme, where acceleration and deceleration phases in shuttle running have been modelled as exponential functions of time, activity monitors (with GPS) would start from the continuous daily recording of body geolocation, from which instantaneous speed and acceleration would be obtained and fed into the described model (Eqns 9 and 11).

The model is based on the assumption that the accelerative and decelerative phases of a maximal shuttle run (SR) of different section (leg) distances are portions of the same patterns exhibited when performing a very long SR (say 20+20 m). In a short-distance SR (for instance, a 10+10 m SR), speed will reach a lower maximum value than in a longer SR, but its rise from zero and descent to zero (in a single leg) will follow the same exponential pattern of the much longer-lasting acceleration and deceleration of the longest SR. This tendency also comes from a recent biomechanical analysis of SRs from 5+5 to 20+20 m (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished), and, just for the acceleration phase, also from 20 m sprint experiments on the same subjects (to allow comparison, 20+20 m SR shows accelerative phases of about 13 m).

In the computational model, it seemed useful to represent on the time axis a deceleration phase first (t<0), followed by the acceleration (t≥0), so that we could represent in the same graphic frame (one leg of) SRs of very different lengths (see Fig. 2 legend) and accommodate the iterative process to determine the maximum speed for each of them (⁠⁠).

The computational model has, as inputs, data from long-distance SRs (say 20+20 m): v_max, τ_af and τ_df. These three parameters were obtained as follows: v_max and τ_af came as unknowns of a non-linear regression of Eqn 13, fitting speed data in the acceleration phase, while τ_df was the only unknown parameter of a non-linear regression of Eqn 13 where v_max was imposed (as one of the results from the other regression), fitting just deceleration data (see  Appendix 1).

The time course of distance, speed, deceleration/acceleration, ES, EM and C are based on the new equations developed in this paper, as explained in Fig. 2 legend. Then, depending on the SR distance of interest (s_SR, actually just one leg, 10 m for, say, 10+10 m SR), an iterative process finds the relevant average C, the average apparent efficiency of muscle positive work and other outcomes such as the maximum speed reached.

The computational process starts from the (obvious) fact that the maximum speed reached at the end of the acceleration phase is the same as that at the start of the deceleration phase. The tentative value for starts from v_max and decreases it by small amounts; for each of them (which in Fig. 2 corresponds to lowering the horizontal dashed lines): (1) deceleration and acceleration durations are calculated (as intersections with the two exponential curves for speed, and marked by two vertical dashed lines in Fig. 2), (2) deceleration and acceleration distances are obtained (as values on the s curves, Fig. 2, corresponding to the two identified time intervals) and (3) summed to obtain the total inferred SR distance (single leg), which (4) is compared with the s_SR by using a small threshold; (5) if the estimated distance is higher than the goal, the tentative value for v_max,SR is lowered and a new iteration starts.

At the end of the process, the average metabolic cost of that SR (C_SR, J kg⁻¹ m⁻¹), due just to speed changes (dv) of the body centre of mass, can be obtained by the Δs-weighted mean of the C curve within the iteratively ‘established’ time frame of that specific SR (i.e. within each pair of vertical dashed lines in Fig. 2). The entire computational process was designed using Labview Programming Language (v13, National Instruments, Austin, TX, USA); the software ran on a MacBook Pro (Apple, Cupertino, CA, USA) laptop computer. Also, early stages of the mathematical model were tested by using Grapher (Apple).

The software algorithm also provides an estimate of the average ‘apparent’ efficiency of the positive mechanical work during each SR, based on the ES values during SRs. Level running (at constant speed) is associated with an equal amount of positive and negative work of the body centre of mass (due to equal excursions of potential and kinetic increases and decreases). That ‘external’ work could be called ‘apparent’ as some of the positive (negative) work is not generated by muscles: instead, it comes ‘at no metabolic cost’ from mechanical strain energy released by (the amount previously stored, again at no metabolic cost, in) tendons. When running on uphill (downhill) slopes, positive (negative) work becomes predominant, and beyond the slope range of ±0.35, gradient running shows a monotonic increase (decrease) of the body centre of mass [thus, just positive (negative) work is done], which impairs the chances of storing (releasing) elastic energy in (from) tendons (Minetti et al., 1994).

Therefore, with respect to the slope, the apparent efficiency of positive mechanical work is assumed to be: (a) 0 for downhill slopes steeper than −0.35, (b) an increasing value up to 0.80 at i=0 (in that range and in the next, different mixes of positive and negative work are performed), which is the maximum apparent efficiency of positive muscle work in level running at the maximum speed of SRs (Cavagna and Kaneko, 1977), (c) a decreasing value down to 0.25 (maximum muscle efficiency) at a slope equal to +0.35, (d) beyond which 0.25 would be constant as only positive work is done (see Fig. 3).

In Fig. 2, the time course of apparent efficiency (eff, which depends on ES) is shown, within the relevant time frame, as a thick white curve. From the calculated average value along one SR leg [eff_mean, assumed to be the same for the other leg(s)], we could infer how much mechanical energy saving via the elastic mechanism has occurred (zero for eff_mean≈0.25 and the maximum possible when approaching 0.75–0.80).

Predictions of average C_SR (on the level) rely, as mentioned, on gradient running (Minetti et al., 1994; Giovanelli et al., 2016) where W_int was the result of different values of the crucial variables involved (see Eqn 12). Thus, estimates of C_SR based on the ES concept have to be corrected by adding the metabolic equivalent of the ‘extra’ W_int that would fill the gap toward the actual W_int of SR events. To do this, we compared single-stride W_int data obtained during a 20 m sprint (G.P., P. Zamparo, N. Fuji, T. Otsu, N. Numazu, A.E.M. and A. Monte, unpublished) to the average values measured in steady running experiments at different gradients (Minetti et al., 1994). To allow a functionally meaningful estimation of the W_int gap between the two conditions, data measured on slopes were time aligned according to when, during the sprint, the ES value was close to the investigated uphill gradient.

Also, the metabolic energy of body turning at the end of SR legs is not included in the model based on Eqns 9 and 11. Here, three approaches are available, as detailed below: (i) estimating the minimum mechanical and metabolic cost to twist the body 180 deg about the vertical axis in the same time interval of the SR experiments, (ii) searching the literature to find the metabolic cost of running in circles of small radius, and (iii) inspecting studies about the metabolic cost of changing direction in running.

Input parameters for the computational model were: v_max=7.175 m s⁻¹, τ_af=0.803 s⁻¹ and τ_df=0.472 s⁻¹ (data source for the quoted regressions is P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished, maximal 20+20 m SR only); their coefficient of variation (CV) ranges from 7.7% to 10.9%.

Model-estimated maximum speed of SRs (⁠⁠) was compared with the (average value of) measured maximum speed, at the different SR distances, and resulted in an overestimation of 4.96±1.64%.

The values of C_AR and C_DR, i.e. the main determinants of C_SR, depend on the prevalence of high values of (positive or negative, respectively) ES during the SR leg. As short-distance SRs are travelled at higher acceleration and deceleration than long-distance SRs, average ES is high. It has to be mentioned, though, that within the time course of each SR leg, very short distances are travelled at the end of the braking phase and at the beginning of the accelerative phase. Thus, despite the high ES values of those phases, the influence of their related high cost on average C_AR and C_DR (units are J kg⁻¹ m⁻¹ travelled) is mitigated by the distance-based weighted mean performed along each phase time axis.

When simulated acceleration/deceleration patterns from the measured average v_max, τ_af and τ_df and their associated ES values were fed into the relationship between slope and efficiency shown in Fig. 3, the average apparent efficiency of positive work (by assuming a fixed eff−/eff+ ratio of 5) was found to be 35.3%, 29.9%, 23.2% and 17.5% for 20+20, 15+15, 10+10 and 5+5 m shuttles, respectively. Those values are shown together with the ones experimentally obtained in Fig. 6.

The proposed model, which includes a revision of previous equations and an extension to decelerations of the ES for speed-changing running (di Prampero et al., 2005) together with the cost of the additional internal work and of body turning, closely predicts (dashed curve in Fig. 5) the metabolic cost of SRs at different distances (Buglione and di Prampero, 2013; Zamparo et al., 2015), a motor activity with bouts of maximal increases/decreases of running speed. Although the general trend of experimental C of SRs is paralleled by the new ES predictions, the overall underestimation resulting from just considering the cost of speed changes (dv) suggests, for each specific motor act, that all the potential sources of metabolic extra cost should be considered. As already argued above, there is a subtle bias potentially embedded in the ES metaphor: the (steady) running C at each slope (which is almost speed independent) incorporates the metabolic cost of mechanical internal work that depends on a submaximal stride frequency. When trying to infer C for sprint running or, as here, in maximal SRs, the W_int increase due to a much higher stride frequency has to be considered. In the proposed case, the cost of turning has been estimated but there are other metabolic components, such as isometric contractions and stabilizing co-contractions, that were not. These could be the reasons for the underestimation of C_SR at the shortest leg distance.

In addition to the metabolic cost prediction, the model combines the ES concept with biomechanical findings about the mix of positive/negative work in running on different gradients (Minetti et al., 1994) and predicts values of apparent efficiency of muscle positive work, at the different SR lengths, which closely resembles (see Fig. 6) the measured values (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished). Experimental data and predictions suggest that SRs shorter than 15+15 m do not exploit the elastic energy storage/release typical of the ‘landing–take-off sequence’ operated by tendons, as witnessed by efficiency values compatible with muscle activity only (as already found in 5+5 m SR by Zamparo et al., 2016). Recently, it has been proposed, based on a combination of experiments and modelling, that elastic energy stored in tendons at the beginning of a maximal sprint is remarkable (Lai et al., 2016). This stored energy, different from that coming from an otherwise wasted potential and kinetic energy of the body centre of mass in the typical running bounce, derives from part of the positive work done by muscle contraction to explosively accelerate the body in the first few steps in sprinting. It is likely that, in this case, muscle power enhancement, rather than the muscle work saving, is the main goal and it is achieved through ‘muscle power amplification’ (Galantis and Woledge, 2003). Because of the central role and the (net) amount of muscle positive work in the early stages of sprinting, we expect that the decreased apparent efficiency at short SRs (caused by the prevalence of high ES of the initial strides), which is part of our results, remains compatible with those recent findings.

Limitations due to extrapolation of Eqn 8 at very high slopes (or ES) have been improved by Eqns 9 and 11, which now completes the prediction range by including negative gradients (or ES); this allows us to safely infer the metabolic cost of running acceleration and deceleration for a speed change from −8.24 to 8.24 m s⁻², corresponding to ES from −84% to +84%. Although the upper limit is very close to the maximum acceleration ability of humans, this is not true for the lower limit: previous papers dealing with the maximum negative power during drop landing (Minetti et al., 1998) and with kinetic analysis of speed changes during maximal SRs of different lengths (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished) show that, as expected from the force–velocity relationship of muscle contraction, eccentric performance is also much higher than concentric performance in in vivo, complex motor acts. Thus, as Giovanelli and co-authors (2016) extended the metabolic cost measurements of running to much higher uphill gradients than previously (from +45% to +84%), the same should be done for downhill running at (equivalent) slopes up to almost double (×1.7, in absolute terms) the actual uphill limit (namely, from −45% to −143%).

Approximations in the mathematical and computational model can affect overall predictions. Although the new equations for C (Eqns 3 and 4) very closely fit experimental data for gradient running (Minetti et al., 2002; Giovanelli et al., 2016), their asymptotic trend, particularly for steep downhill slopes, can be improved by investigating a wider gradient range (as mentioned) and the theoretically non-linear relationship between ‘unavoidable’ metabolic cost and gradient (see  Appendix 2).

The inset graph in Fig. 2 reveals that the assumption ruling the simulation, i.e. that short-distance SRs use ‘truncated’ portions of acceleration and deceleration phases displayed at the maximum investigated distance, generates speed time courses with a sharp peak in between. This slightly differs from the experimental patterns (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished). As the inaccuracy pertains to a zone where small accelerations and decelerations (and the related ES values) occur, we are confident that a potential correction (spline) would not alter the conclusion from the obtained predictions.

Unfortunately, no easy rule of thumb for decelerated running allows us to replace Eqn 11, mostly because of the complex mix between positive and negative mechanical work, and their efficiencies, occurring at negative slopes/ES (Minetti et al., 1994).

As shown in this paper and in the previous literature (di Prampero et al., 2005; Osgnach et al., 2010; Gaudino et al., 2013, 2014; Minetti et al., 2013; Coutts et al., 2015; Kempton et al., 2015; di Prampero et al., 2015), the ES concept can be profitably used in estimating the metabolic cost of speed-changing running. Other than shedding light on the metabolic effects of remarkable speed oscillation in a gait, where inherent velocity changes occur even at constant speed, the present revised methodology has potentially wide application in the monitoring of activity in daily life (submaximal speed changes) and exercise physiology, particularly in sport activities where unsteady locomotion is prevalent (submaximal and maximal speed changes). Also, the time course of metabolic power (=instantaneous cost×instantaneous speed) can be calculated starting from one of the proposed computation frameworks (the ‘rule of thumb’ equations) to work out the maximum metabolic performance required in prey/predator settings. The next step would be the development of a model for speed-changing walking, with predictions compared with the experimental measurements published so far in the literature.

The 3D coordinates of each body segment, sampled at 100 Hz by a 35-camera system (Vicon Oxford Metrics, Oxford, UK) in a recent study on SR kinematics (P. Zamparo, G.P., A. Monte, F. Nardello, T. Otsu, N. Numazu, N. Fujii and A.E.M., unpublished), allowed us to obtain the displacement speed of the body centre of mass. Speed data from acceleration and deceleration phases of just the first leg in maximal 20+20 m SRs were analysed. The acceleration phase was fitted according to Eqn 13 by means of a non-linear regression model where both v_max and τ were estimated. The deceleration phase was fitted according to the same equation, by means of a non-linear regression model where v_max was imposed (as a result from the previous regression) and τ was estimated. This statistical strategy was adopted, first, to better capture the experimental trend of running speed to reach the maximum value ‘in acceleration’ and, second, to avoid speed discontinuity of SR leg pattern as reconstructed from Eqns 3 and 4. Such a granted continuity later allowed the computational algorithm to iteratively find, for each SR distance, the correct acceleration and deceleration timing compatible with that distance (see ‘Computational model of shuttle running’, above).

The proposed regression equations (Eqns 3 and 4) incorporate a linear function of gradient i (see below) and an exponential component. The former, as mentioned, accounts for the inspection trends in Fig. 1 at steep gradients and their role in representing the minimum gravitational work that has to be done. The latter has been conceived to represent the deviation of C from the two linear components at gradients in the range −0.35<i<+0.35, where C probably also reflects the metabolic equivalent of other mechanical determinants as the cost of mixed positive and negative work, which tends to disappear outside that gradient range (Minetti et al., 1994).

The approximation i=sin(atani) holds only in the range −0.45<i<+0.45, with a tendency of Eqn 5 to even deviate consistently (−16% at i=+0.80) from Eqn 4. For the aims of this paper, devoted to improving the predictive efficacy of Eqn 1 at even steeper uphill slopes and to extend the ES model to the estimation of metabolic cost of running decelerations, the suggested linearized regression C=ai+be^ci can be regarded as an approximate model incorporating part of the effects of the minimum gravitational work that has to be done. We leave the deepening of understanding about the discrepancy between observed C at very steep gradients and Eqn 5 to future investigations.

The authors are indebted to Paola Zamparo and Andrea Monte for enriching discussions on the outcomes from the model/methodology.