The rebound of the body during uphill and downhill running at different speeds

When running on the level at a constant average speed, the mechanical energy of the centre of mass of the body (COM) oscillates throughout the step like a spring-mass system bouncing on the ground (Cavagna et al., 1976). During the rebound of the body, the muscle–tendon units (MTU) of the supporting lower limb undergo a stretch–shortening cycle during which part of the mechanical energy of the COM is absorbed during the negative work phase to be restored during the next positive work phase (Cavagna et al., 1988).

When running on the level, the upward and downward movements of the COM are equal, and the positive and negative work done each step to sustain its movements relative to the surroundings are equal (i.e. ⁠). When running on a slope, muscles are compelled to produce or dissipate energy to increase or decrease the potential energy of the COM (DeVita et al., 2008). In this case, the spring-mass model is not suitable anymore because the ratio between positive and negative work increases or decreases with the slope of the terrain (Minetti et al., 1994). Several aspects of the mechanics of human running uphill and downhill have been studied these last decades: e.g. the muscular work done, the energy consumed and the muscular efficiency at different slopes (Minetti et al., 1994), the net muscular moment and power at the hip, knee and ankle during uphill running (Roberts and Belliveau, 2005), and the possible elastic energy storage and recovery in the arch and Achilles' tendon while running uphill and downhill (Snyder et al., 2012).

To our knowledge, the change in the bouncing mechanism while running on a slope at different speeds has never been analysed. Indeed, this mechanism will be affected by the ratio between positive and negative work. Between 2.2 and 3.3 m s⁻¹, this ratio changes with the slope of the terrain, but not with the speed of progression (Minetti et al., 1994). However, to the best of our knowledge, this ratio has never been measured while running on a slope at higher speeds.

To analyse the bouncing mechanism, we have measured the three components of the ground reaction force (GRF) of 10 subjects running on an inclined treadmill (from −9 to +9 deg, in 3 deg increments) at 10 different speeds (from 2.2 to 5.6 m s⁻¹). From these curves, we have analysed (1) the different periods of the running steps, (2) the vertical movements of the COM and (3) the energy fluctuations of the COM in order to understand how the bouncing mechanism of running deviates from the spring-mass model with slope and speed. We believe that three aspects of this mechanism will be affected.

First, from a mechanical point of view, a spring-mass system bouncing vertically on the ground oscillates around an equilibrium point at which the vertical component of the GRF (F_v) is equal to body weight (BW) (Cavagna et al., 1988). This is true whatever the slope; indeed, when running in steady state, the average vertical velocity of the COM (⁠⁠) does not change from one step to the next. Consequently, the average vertical acceleration of the COM and the average force ⁠. The step period (T) can thus be divided into two parts: the first during which F_v>BW (t_ce), taking place during the contact of the foot on the ground, and the second during which F_v≤BW (t_ae), taking place during both ground contact and the aerial phase. The period t_ce corresponds to the half period of the oscillation of the bouncing system. During t_ae, the bouncing model is not valid when the body leaves the ground.

When running on a flat terrain at speeds up to ∼3.1 m s⁻¹, t_ce≈t_ae (symmetric step). As speed increases above 3.1 m s⁻¹, t_ae becomes progressively greater than t_ce (asymmetric step). This asymmetry arises from the fact that, when speed increases, the average vertical acceleration during t_ce (i.e. ⁠) becomes greater than the acceleration of gravity (g) whereas during t_ae, cannot exceed 1 g. Consequently, a longer t_ae is necessary to dissipate and restore the momentum lost and gained during t_ce. At a given speed for a given t_ce, the asymmetric step requires a greater ⁠; however, the increase in the step period due to a longer t_ae results in a smaller internal power necessary to reset the limbs at each step (Cavagna et al., 1988).

When running on a positive slope, muscles are performing more positive than negative work. We expect that in order to contain the increase in muscular forces during t_ce, the step remains symmetric above 3.1 m s⁻¹ and T becomes shorter than during running on the level. In contrast, when running on a negative slope, muscles are performing more negative work. Because muscles are able to develop higher forces during eccentric contractions, we expect that T is tuned to contain the internal power, rather than the force during t_ce. Consequently, T should become longer than during running on the level and the step should become asymmetric below 3.1 m s⁻¹.

Second, when running on a flat terrain, the momentums lost and gained over a step are equal, and =0. Consequently, the upwards and downwards vertical displacements of the COM (S⁺ and S⁻) are equal. On the contrary, when running on a slope, and S⁺≠S⁻. We expect that the dissimilarity between S⁺ and S⁻ will be the main factor affecting the bouncing mechanism. We also expect that S⁻ in uphill and S⁺ in downhill running will progressively disappear and with it the amount of energy that can potentially be stored in the MTU.

Third, on a flat terrain, the energy that is due to the vertical movements of the COM (E_v) and the energy that is due to its horizontal movements (E_f) are fluctuating in phase (Cavagna et al., 1976), and the negative and positive work done are approximately equal: Δ⁺ E_f≈Δ⁻ E_f and Δ⁺ E_v≈Δ⁻ E_v. However, when running on a slope, Δ⁺ E_f≈Δ⁻ E_f but Δ⁺ E_v≠Δ⁻ E_v. We expect that (1) because of this imbalance, the fluctuations of E_f and E_v are no longer in phase and, consequently, (2) an energy exchange occurs between these two curves. Therefore, we have evaluated the duration of the positive and negative work phases and the energy transduction between E_f and E_v during the period of contact t_c (Cavagna et al., 2008a).

We hypothesize that the three aspects described here above will jeopardize the bouncing mechanism of running and in turn, the possibility to store and release elastic energy into the elastic structures of the lower limb. However, we also hypothesize that the disappearance of this mechanism is intended to restrain the increase in or that is due to slope.

Finally, we propose a model that describes the vertical oscillations of the COM during t_ce while running on a slope. The aim of this model is to understand how all of the lower-limb muscles are tuned to generate the basic oscillation of the bouncing system. The spring-mass system bouncing on the ground that models running on the level (Blickhan, 1989; Cavagna et al., 1988) cannot describe running on a slope because energy must be added or dissipated. Therefore, we have incorporated an actuator in the spring-mass model that will generate a force proportional to V_v and produce or absorb energy during contact.

Ten recreational runners (three females and seven males) participated in the study (age: 31.8±8.3 years, mass: 68.8±10.2 kg, height: 1.78±0.07 m, mean±s.d.). Informed written consent was obtained from each subject. The studies followed the guidelines of the Declaration of Helsinki, and the procedures were approved by the Ethics Committee of the Université catholique de Louvain.

Subjects ran on an instrumented treadmill at seven different inclinations: 0, ±3, ±6 and ±9 deg. To neutralize the effect of learning and muscle fatigue, half of the subjects started with an inclination of 0 deg that was increased and the other half with an inclination of 9 deg that was decreased. At each slope, subjects ran at 10 speeds presented in a different order (8, 10, 11, 12, 13, 14, 15, 16, 18 and 20 km h⁻¹, corresponding to 2.22, 2.78, 3.06, 3.33, 3.61, 3.89, 4.17, 4.44, 5.00 and 5.56 m s⁻¹, respectively). Note that for downhill running, the speed of the belt was limited by the manufacturer to 5.0 m s⁻¹. At each slope, half the subjects started with the belt turning forwards to simulate uphill running and the other half with the belt turning backwards to simulate downhill running.

Data were recorded during a period of 3–10 s, depending of the speed and the inclination of the treadmill. Between 6 and 36 steps per trial were recorded; a total of 15,760 steps were analysed. On a +6 deg slope, one subject could not run at 5.6 m s⁻¹, and on a +9 deg slope, one subject could not run at 5.0 m s⁻¹ and three could not run at 5.6 m s⁻¹.

where θ is the angle between horizontal and the tread surface. The electrical motor was instrumented with an optical angle encoder to measure the speed of the belt (V_belt). The average speed of the belt over a stride (⁠⁠) differed by 2.8±1.4% (mean±s.d.) from the chosen speed and the instantaneous V_belt did not change by more than 5% of ⁠.

The treadmill contained its own signal conditioning system: the GRF signals were amplified, low-pass filtered (4-pole Bessel filter with a −3 dB cut-off frequency at 200 Hz) and digitized by a 16-bit analog-to-digital converter at 1000 Hz. This system was connected to a PC via ethernet using TCP/IP (Genin et al., 2010). Acquisition and data processing were performed using custom-built software (LABVIEW 2010, National Instruments, Austin, TX, USA, and MATLAB 2013, MathWorks, Natick, MA, USA).

Steps were divided according to the F_v–time curves (Fig. 1): a step started and ended when F_v became greater than BW. The effective contact time (t_ce) was the period during which F_v≥BW, and the effective aerial time (t_ae) was the period during which F_v<BW (Cavagna et al., 1988). The time of contact (t_c) was the period during which F_v>10 N, and the aerial phase (t_a) was the period during which F_v≤10 N. The step duration was then calculated as T=t_ce+t_ae.

The acceleration (a), velocity (V) and displacement (S) of the COM and the external work done (W_ext) were computed from the GRF using a method similar to that of Gosseye et al. (2010). Therefore, this method is only explained in brief.

These computations were done over strides (i.e. two steps, starting on the right foot). The fore-aft, lateral and vertical accelerations of the COM relative to the reference frame of the laboratory were calculated by dividing F_f, F_l and by the body mass m (where is the average vertical force over a stride). In theory, (as explained in the Introduction); therefore, strides were analysed only if was within 5% of BW.

The time curves of the three components of a were integrated numerically to determine the fore-aft (V_f) lateral (V_l) and vertical (V_v) velocity of the COM, plus an integration constant, which was set on the assumption that the average velocity of the COM over a stride was equal to for V_f, to zero for V_l and for V_v.

The vertical displacement of the COM (S) was then computed by time integration of V_v. The upward and downward displacements of the COM were divided according to t_c (⁠and ⁠) and t_a (⁠and ⁠). The minimum vertical displacement (S_min) necessary to overcome the slope of the terrain each step can be computed by ⁠, where is the average running speed over the step and is the step length.

The external work (W_ext) is the work necessary to move the COM relative to the surroundings plus the work done on or by the environment (Willems et al., 1995). It was measured from the work done by the GRF (Gosseye et al., 2010).

where represents the variation of V_belt around ⁠. Because these terms represented less than 3% of W_ext, they were neglected in this study.

The positive and negative work over one step (⁠⁠, ⁠, ⁠, and ⁠, ⁠, ⁠, ⁠) were computed as the sum of the positive and negative increments of the E_f, E_l, E_v and E_ext curves, respectively (Fig. 2). Because W_l represents less than 1.5% of W_ext, it is not presented separately in the results. The work per step was computed as the work over the stride divided by two.

where b is a constant depending on the vertical velocity of the COM at touch down and c is the actuator coefficient. The coefficient c is positive when running uphill and negative when running downhill. In this way, the power developed by the actuator force (⁠⁠) is positive in uphill running, showing that the actuator works like a motor giving energy to the body, and negative in downhill running, showing that the actuator works like a damper absorbing energy (see insets in Fig. 7A).

where a_v(i), V_v(i) and S(i) were the experimental data at instant i. In this way, n equations were produced, where n is the number of samples during contact. This set of equations resulted in an over-constrained system, from which the constant (b/m), the mass-specific stiffness (k/m) and the actuator coefficient (c/m) were computed by a regression analysis using singular value decomposition that minimizes the sum of squared errors.

For running on the level, we also compared the values obtained from Eqns 7 and 8. In Eqn 8, c/m is close to zero and k/m differs by 1.8±2.1% (mean±s.d., n=2440) from Eqn 7.

The goodness of the spring-actuator-mass model was assessed by computing (1) the variance explained by the regression model (r²), and (2) the root mean square error (RMSE), which expresses the agreement between the measured and the computed values of the GRF.

Data were grouped into speed–slope classes. To obtain one value per subject in each class, the steps of the same subject in a same class were averaged. The mean and standard deviation of the runner population were then computed in each class (grand mean). A two-way ANOVA with post hoc Bonferroni correction (PASW Statistics 19, SPSS, IBM, Armonk, NY, USA) was performed to assess the individual and interaction effects of speed and slope on the calculated variables (P-values were set at 0.05).

The mass-specific external work done per unit distance is plotted in Fig. 3A. In the fore-aft direction, the work done to move the COM horizontally (W_f) increases similarly with speed at any slope (P=0.995). Moreover, because subjects move at a constant speed, ⁠.

Therefore, the imbalance between and on a slope is only due to a modification in and ⁠. During uphill running, increases, while decreases and tends to disappear above ∼4.4 m s⁻¹ at +6 deg and above ∼3.3 m s⁻¹ at +9 deg. On a negative slope, tends to disappear above ∼5.0 m s⁻¹ at −6 deg and above ∼4.2 m s⁻¹ at −9 deg.

Note that the ratio changes with slope (P<0.001), but is independent of speed (P=0.17): equals 0.5 at 0 deg, but decreases monotonically below 0.5 on a negative slope and increases monotonically above 0.5 on a positive slope (Fig. 3B).

The step period (T) is given as a function of running speed in Fig. 4A. The slope has a significant effect on the step period (P<0.001): T decreases when running uphill, whereas it increases when running downhill. The effect of slope on T is more marked on positive than on negative slopes, though this effect is significant at all slopes (Bonferroni post hoc, P<0.01). The slope does not affect the effective contact time t_ce, except at +9 deg (Bonferroni post hoc, P<0.01). Thus, the change in T is mainly due to a change in t_ae (P<0.001), which in turn, is largely due to a change in t_a.

When running on the level, the step is symmetric (i.e. t_ae≈t_ce) at speeds up to ∼2.8 m s⁻¹ (Bonferroni post hoc, P<0.01). These results are consistent with those reported in the literature (Cavagna et al., 1988; Schepens et al., 1998). When running uphill, t_ae≈t_ce in a larger range of speeds because the mean vertical acceleration during t_ce (⁠⁠) is kept smaller than 1 g at higher speeds than on the level (Fig. 4B): up to ∼3.3 m s⁻¹ at +3 deg, to ∼3.9 m s⁻¹ at +6 deg and to 5.6 m s⁻¹ at +9 deg. When running downhill, the difference between t_ae and t_ce tends increase because becomes >1 g at speeds lower than those on the level. Consequently, the range of speeds at which t_ae≈t_ce becomes narrower, i.e. below ∼2.2 m s⁻¹ at −6 and −9 deg.

On the level, the upward (S⁺) and downward (S⁻) displacements of the COM over one step are equal (Fig. 5). On a slope, the difference between S⁺ and S⁻ increases with inclination (P<0.001).

When running uphill, S⁺ increases with slope and speed. At slow speeds, the displacement upwards during the aerial phase (⁠⁠) is almost nil, whatever the slope. When speed increases, represents a greater part of S⁺. The displacement downwards (S⁻) decreases when slope and speed increase: S⁻ disappears above 5.0 m s⁻¹ at +6 deg and above 3.9 m s⁻¹ at +9 deg. Note that the reduction of S⁻ is first due to a reduction of ⁠. At +6 deg and +9 deg, is almost nil at all speeds, suggesting that V_v at touchdown is close to zero.

When running downhill, the opposite phenomenon is observed: S⁻ increases and S⁺ decreases with slope and speed. The effect of slope and speed is more marked on S⁻ when running downhill than on S⁺ when running uphill. When running downhill, represents a significant part of S⁻, whereas is almost nil whatever the slope and speed, indicating that V_v at take-off is close to zero.

When running on a slope, muscles are compelled to modify the ratio between concentric and eccentric contraction. Consequently, the timing of negative and positive work production during t_c is modified (Fig. 6A). The time during which muscles perform negative external work (t_brake) is extended during downhill running (P<0.001) while the duration of the positive work phase (t_push) is increased during uphill running (P<0.001). Because at a given speed, t_c does not change significantly with slope (Fig. 6A), the modification of t_brake is compensated by an opposite change in t_push.

The relative time of occurrence of the minimum of E_f (E_f,min) changes little with slope [see Figs 2 (arrows) and 6B]. On the contrary, the minimum of E_v (E_v,min) appears relatively earlier during contact in uphill running and later in downhill running. The effect of the slope on the relative time of occurrence of E_v,min is more accentuated when speed increases. Between E_f,min and E_v,min, the E_f and E_v curves (grey zone in Fig. 6B) are out of phase and an energy exchange can occur from E_f to E_v when E_v,min precedes E_f,min, and from E_v to E_f when E_f,min precedes E_v,min. This exchange of energy allows recovery of a significant amount of energy (i.e. %R>10%) only on steep slopes and at speeds >3.6 m s⁻¹ (Fig. 6C).

When running on a slope, the lower-limb muscles can be modelled as a mass mounted on a spring in parallel with an actuator that generates a muscular driving force proportional to the vertical velocity of the COM and produces (uphill) or absorbs (downhill) energy during t_ce. The overall mass-specific stiffness k/m and the actuator coefficient c/m generated by the lower-limb muscles (Fig. 7B) are modified by both the slope of the terrain and the speed of progression (P<0.001). Compared with at 0 deg, k/m decreases on a positive slope and increases on a negative slope. At a given slope, k/m increases with velocity, though the effect of speed is greater on negative than positive slopes (P<0.001).

During level running, the complex musculoskeletal system of the lower limb behaves like a single linear spring and c/m is nil. Running uphill requires additional energy to overcome the slope. In this case, c/m is positive and increases with the mechanical demands, i.e. with increasing slope but also with increasing speed. On the contrary, running downhill requires dissipation of energy. Consequently, c/m is negative and its absolute value increases when slope becomes steeper and when speed increases. Note also that the effect of speed on c/m is greater on negative than on positive slopes (P<0.001).

This model describes the basic vertical oscillation of the COM during running on a slope (Fig. 7A) without taking into account the first peak in the F_v curve as a result of foot slap (Alexander et al., 1986; Schepens et al., 2000) and the vibrations of the treadmill motor (Fig. 1); the r² of the least square method is always >0.63 and RMSE ranges between 120 and 440 N (Table 1).

This study was intended to help understand how the bouncing mechanism is modified when the slope of the terrain becomes positive or negative. Our results show that the bouncing mechanism still exists on shallow slopes and progressively disappears when the slope increases. This mechanism disappears earlier on positive than on negative slopes and earlier at high speeds than at slow speeds. In this section, we will discuss how the step period, the vertical movement of the COM and the energy fluctuation of the COM affect the bouncing mechanism of running. We will also show that these variables are tuned to contain the increase in the positive or negative muscular work and power that is due to slope. We finally discuss the quality and limits of the mechanical model describing the vertical movement of the COM during slope running.

Minetti et al. (1994) show that on a positive slope the step period T decreases significantly, whereas on a negative slope T tends to increase (though this increase is not significant). Our results confirm that, at a given speed, T is shorter when running uphill than when running on the level. On the contrary, during downhill running, our results show a significant increase in T, even if this change is less marked than on a positive slope. This discrepancy with the results of Minetti and colleagues may be explained by the fact that the number of steps analysed here (n=15,760 steps) is higher than that in Minetti's study (n=418 steps).

In our study, we observe that t_c (Fig. 6A) and t_ce (Fig. 4A) at a given speed do not change significantly with slope, in both downhill and uphill running (except at +9 deg at the highest speeds). The change in the step period T is thus mainly due to a change in t_a, which in turn changes t_ae.

As shown by Cavagna et al. (1991), running on flat terrain with a long effective aerial phase is a convenient strategy to decrease the average power developed over the step, provided that muscles are able to develop enough power during the push. Our results suggest that the changes in t_ae are intended to contain the additional muscular power due to the slope.

When running uphill as compared with running on the level, T decreases because t_ae is reduced. Because t_ce is not modified by slope, the step remains symmetric (i.e. t_ce≈t_ae) at higher speeds than on flat terrain. These results are similar to those obtained when running on the level in hyper-gravity (Cavagna et al., 2005): at 1.3 g, the rebound remains symmetric up to 4.4 m s⁻¹.

At a given speed and for a given t_ce, a symmetric rebound results in a shorter step than does an asymmetric rebound. As a result, the minimal height (S_min) that the COM must gain each step is smaller, the impact against the ground is reduced and the force and power during the push are decreased (Cavagna et al., 1991). However, decreasing T results in a greater internal power to move the limb segments relative to the COM.

Snyder and Farley (2011) have observed that the optimal step period at which the oxygen consumption is minimal does not change for slopes between −3 and +3 deg. However, these authors show that the freely chosen T decreases slightly between 0 and 3 deg. This decrease in T might be a strategy to contain the mechanical power during the push.

The changes observed in uphill running are similar to those observed in old men running (Cavagna et al., 2008b). In older subjects, the average upward acceleration (⁠⁠) is lower than in younger ones, leading to a symmetric rebound. According to these authors, the lower force attained during contact by the old subjects may be explained in part by the loss of muscular strength (e.g. Doherty, 2003). Similarly, the strategy adopted while running uphill could be due to the limits set by the muscular strength and/or power. Furthermore, in uphill running, the GRF vector is farther from the leg joint centres (DeVita et al., 2007). Thus, the slope of the terrain alters the muscle mechanical advantage by creating longer lever arms, and leads to higher joint torques and power outputs (Roberts and Belliveau, 2005). Therefore, decreasing T might be a beneficial strategy to limit the muscular moments and to reduce the mechanical load applied to the MTU.

At the opposite, when running downhill, T increases as compared with running on the level because t_ae increases. Because t_ce is not modified, steps are asymmetric (i.e. t_ce<t_ae) at lower speeds than on a flat terrain. These results are similar to those obtained when running in unweighted conditions (Sainton et al., 2015): when BW is reduced by 40%, t_c remains unchanged while t_a is increased.

The step asymmetry has the physiological advantage to limit the internal power. However, this asymmetry requires a greater and, consequently, a greater power during t_ce. The choice of this strategy in downhill running may be due to the difference in force exerted during negative and positive work phases: during downhill running, muscles contract mainly eccentrically, allowing the development of higher forces. Moreover, in downhill running, runners land with the leg more extended (Leroux et al., 2002), resulting in reduction of the lever arms of the GRF about the lower-limb joints and thus of the net muscular moments.

When running uphill at a given speed, to maintain the bouncing mechanism similar to level running, the runners should increase to overcome the slope, without changing (Fig. 3A). However, the runner limits by reducing ⁠. The opposite phenomenon is observed during downhill running, where is limited by reducing ⁠. Because the work done to accelerate and decelerate the COM forwards (W_f) does not change with slope, the change in and is due to a change in and ⁠, which in turn is mainly due to a modification of S⁺ and S⁻ during the step (Fig. 5).

When running uphill, the upward displacement of the COM each step (S⁺) increases with slope and speed because S_min increases. In contrast, the downward displacement (S⁻) decreases. At low speeds and on shallow slopes, S⁻ is still present because the muscular power at disposal during the push is great enough to increase S⁺ beyond S_min. In this way, the presence of S⁻ allows the storage of elastic energy into the MTU to be restored during the next positive work phase. On the contrary, at high speeds and on steep slopes, the power during the push approaches the maximal performance of the runner. Therefore, S⁺ is maintained close to S_min; consequently, S⁻ and (Fig. 3A) are almost nil and no rebound is possible.

When running downhill, S⁻ increases faster with slope and speed than S⁺ in uphill running (Fig. 5); these changes in S⁻ are mainly due to a change in the downward fall of the COM during t_a (⁠⁠). In contrast, the effect of slope and speed on S⁺ on a negative slope is less marked than the changes in S⁻ on a positive slope. Consequently, when running downhill, the possibility of elastic storage increases with speed and slope. Though, as mentioned by Snyder and Farley (2011) and Snyder et al. (2012), the recoil is limited because S⁺ and are small.

Furthermore, the large observed in downhill running results in a high V_v at touchdown, which in turn causes greater GRF. According to Zelik and Kuo (2012), the vibrations of the soft tissues induced by the impact could reduce the muscular work done by dissipating energy.

Running is thought to employ a spring-mass mechanism by which interactions between the COM and the ground allow storage and release of elastic energy in the MTU. In vivo measurements of muscle–tendon interaction have highlighted the influence of the stretch–recoil of tendons on the power output of muscles during running (Ishikawa and Komi, 2008; Roberts et al., 2007).

On the level at low and intermediate speeds, t_push>t_brake (Fig. 6A), although ⁠. This difference in time is due to the greater muscular force exerted during the eccentric phase. The fact that t_push>t_brake thus suggests an important contribution of the contractile machinery to the MTU length change and to the work production (Cavagna, 2006). At higher speeds (i.e. above ∼3.9 m s⁻¹), t_push=t_brake, which suggests that when muscle activation is progressively augmented with increasing speed, the MTU length change is mainly due to tendon length change. The work contribution by the contractile machinery is thus progressively substituted by elastic storage and recovery by tendons. The t_push/t_brake ratio may therefore be an expression of the deviation of the MTU's response from that of an elastic structure. On a slope, the change in the ratio between and most likely affects the interaction between muscle and tendons during the stretch-shortening cycle (Roberts and Azizi, 2011).

When running uphill, the linear increase of changes the partitioning of t_c into t_push and t_brake (Fig. 6A). In order to contain the average muscular power required during the push, t_push increases with slope and t_push>t_brake. This suggests that the MTU length change is mainly due to a shortening of the contractile machinery and that less elastic energy is stored – as in old men running on the level (Cavagna et al., 2008a). As proposed by Roberts and Azizi (2011), it could be that on a positive slope, MTUs work like power amplifiers: the energy produced during the muscular contraction is stored at a low pace in the tendons to be released at a higher pace.

In downhill running when slope becomes steeper, in order to limit the power during the brake, t_brake increases and t_push becomes shorter. The higher GRF observed in downhill running could favour the role of the tendon relative to that of muscle: muscle would perform a quasi-isometric contraction and the energy would be stored in the tendon during rapid stretch, to be dissipated later by a slower lengthening of the muscle (Roberts and Azizi, 2011).

The partition between t_push and t_brake changes because the minimum of E_v appears increasingly earlier in uphill running and increasingly later in downhill running, whereas the minimum of E_f occurs always more or less in the middle of the contact period (Fig. 6B and arrows in Fig. 2). For this reason, when slope increases, a phase shift between E_f and E_v emerges and an energy transduction between these two forms of energy can occur.

When running uphill, the lower limb acts like a pole in athletics (Schade et al., 2006). During the first part of contact (i.e. period between the two arrows in the upper panels of Fig. 2), the COM loses horizontal velocity, while it gains height and vertical velocity. During this phase, there is an energy transduction between E_f and E_v. The fact that E_ext decreases shows that the loss in E_f is greater than the gain in E_v; part of the E_ext lost is stored in the elastic element of the MTU to be released during the second part of contact to increase the kinetic and potential of the COM. Note that the same phenomenon during the running step preceding the jump over an obstacle was described by Mauroy et al. (2013).

When running downhill, the energy transduction occurs during the second part of the contact, when E_v is decreasing while E_f increases. In this case, the potential energy lost is used to accelerate the COM forward.

In level running, the energy recovered through an exchange between E_v and E_f (Eqn 6) is negligible (%R<5%), like in a spring-mass system (Cavagna et al., 1976). When running at high speed on steep slopes, the rebound of the body deviates from a spring-mass system. However, the energy transduction allows recovering only a small amount of energy: at best %R≈20% in uphill running and %R≈10% in downhill running.

In this study, we propose a model that reproduces in first approximation the basic oscillation of a spring-mass actuator (see inset in Fig. 7A). The aim of this model is to understand how all of the lower-limb muscles are tuned to generate an overall leg stiffness and leg actuation to sustain the vertical oscillation of the bouncing system.

Running on a slope is not a pure elastic phenomenon because energy must be added or released at each step. Therefore, our model includes an actuator placed parallel to the spring. This actuator generates a force proportional to V_v; a linear force–velocity relationship was chosen because functional tasks that involve all lower-limb joints show a quasi-linear relationship (Bobbert, 2012; Rahmani et al., 2001), rather than the classical hyperbolic relationship described in isolated muscles (Hill, 1938). The coefficient c/m is positive during uphill running, showing that the actuator adds energy to the system, and negative during downhill running, because energy is dissipated at each step. As speed and slope increase, the discrepancy between the upward and downward displacement of the COM increases, the rebound of the COM deviates from a spring-mass system and the coefficient c/m increases.

On a slope, k/m decreases when running uphill and increases when running downhill. On a positive slope, in order to maintain the symmetry of the step, remains ≤1 g (Fig. 4B). This lower acceleration is most likely obtained by reducing the stiffness of the contractile elements of the MTU and the overall k/m becomes smaller than that on the level (Fig. 7B).

Note also that on positive slopes, the limbs at impact are more flexed to prevent pitching backward (Birn-Jeffery and Higham, 2014; Leroux et al., 2002). On the level, runners adopting a more crouched posture, similar to Groucho running (McMahon et al., 1987), present lower vertical GRF and consequently a smaller k/m. In uphill running, the more flexed posture during stance could explain, at least in part, the smaller k/m.

On a negative slope, the step is asymmetric at most speeds and slopes because >1 g (Fig. 4B). When acceleration increases, muscles fibres oppose a progressively greater force to stretching and the contractile machinery becomes stiffer than the tendon. Subsequently, the overall k/m becomes greater than on the level. Furthermore, increasing k/m minimizes the lowering of the COM during contact (Fig. 5). This strategy might serve as an intrinsic safety mechanism to limit the risk of MTU damage after landing (DeVita et al., 2008).

Our simple spring-actuator-mass model predicts the F_v–time curve, though it does not take into account the first peak in the F_v curve, which is due to foot slap (Alexander et al., 1986; Schepens et al., 2000). To estimate the contribution of the foot collision on the shape of the F_v–time curve, we used a Fourier series analysis (Clark and Weyand, 2014). This analysis decomposes the F_v signal into low- and high-frequency components. According to Clark and Weyand (2014), the high-frequency components are mainly due to the acceleration of the lower limb during the impact phase. In our study, we observed that the variance accounted for by these high-frequency components increases at high speeds and on steep negative slopes (see Table S1). This observation supports the idea that the decrease in goodness-of-fit indexes (Table 1) is due to a higher contribution of the foot’s interaction with the ground.

These last results corroborate those of Clark and Weyand (2014) obtained in sprint running; they show that at swift speeds, the GRF waveform deviates from the simple spring-mass model pattern, most likely because of the greater importance of the foot–ground collision. Therefore, Clark and Weyand (2014) propose a model including two masses and springs to take into account the interaction of the lower-limb segments with the ground. Our spring-actuator-mass model could thus be refined by adding a second spring mass representing the foot and shank, although kinematic data of these segments are needed to feed this implemented model.

Cavagna et al. (1991) suggested that running with a long aerial phase limits the step-average power as long as muscles are able to develop enough power during the push and/or the brake. Our results support this hypothesis.

When running uphill at a given speed, the average external power developed during the positive work phase seems to be the limiting factor. Actually, when slope increases, in order to keep a long aerial time t_a, the vertical velocity of the COM at take-off should increase because the minimal vertical displacement (S_min) increases. This would require a greater power during the push. As a matter of fact, this power is limited by: (1) reducing t_a and thus the step period T, (2) increasing the duration of the push (t_push) at the expense of the duration of the brake (t_brake) and (3) reducing the downward displacement of the COM. As a result, uphill running deviates from a bouncing mechanism as speed and slope increases.

When running downhill, the average external power developed during the negative work phase seems to be the limiting factor. Indeed, despite a lower vertical velocity at take-off, t_a – and thus T – increases with slope and speed because the ballistic fall of the COM increases. A longer t_a increases the external power developed during the brake because the energy to be dissipated after touchdown is greater. In spite of better muscular performance during eccentric than concentric contraction, the power during the brake is limited by (1) increasing t_brake at the expense of t_push and (2) reducing the upward displacement of the COM. Consequently, the bouncing mechanism during downhill running gradually disappears as speed and slope increase.

The authors thank Profs D. De Jaeger and V. Legat for their advice.