An analysis of the rebound of the body in backward human running

Subjective evidence suggests that backward running provides a way to recover from injuries occurring in normal running while maintaining training. In order to substantiate this anecdotal evidence biomechanical aspects have been investigated aimed at measuring the differences between forward and backward running in the kinematics of body segments (Bates et al., 1986; Devita and Stribling, 1991), the ground reaction force and impulse (Threlkeld et al., 1989; Devita and Stribling, 1991), the energy expenditure (Flynn et al., 1994; Wright and Weyand, 2001), the lower limb joint moments of force and joint muscle powers (Devita and Stribling, 1991; Flynn and Soutas-Little, 1993) and the changes in foot–ground contact time (stance time) as a function of speed (Wright and Weyand, 2001). A common observation in all these studies is that the step frequency and the energy expenditure are greater in backward running than in forward running. The greater step frequency has been suggested to be due to a shorter stance time (Threlkeld et al., 1989) and a shorter flight time caused by a reduced vertical impulse with a similar swing/stance ratio (Threlkeld et al., 1989; Devita and Stribling, 1991). The peak vertical force during backward running was found to be lower than the peak vertical force in forward running (Threlkeld et al., 1989). The shorter stance time together with a greater muscle activation were considered to be the cause of the greater energy expenditure in backward running (Wright and Weyand, 2001). A recent study showed that the greater energy expenditure in backward running is due to a lower efficiency of positive work production (Cavagna et al., 2011), but the characteristics of the rebound resulting in this lower efficiency have not been investigated. In particular the determinants of the step frequency in backward running are not clear.

Factors that determine the more economic choice of step frequency in forward running are: (i) tuning the step frequency to the natural frequency of the bouncing system, and (ii) choosing a step frequency that minimizes the total aerobic-limited step average power within the limits set by the anaerobic-limited push average power (Cavagna, 2010). The first strategy is usually adopted at low running speeds and abandoned for the second strategy at high running speeds by increasing the average upward acceleration above 1 g during the lower part of the vertical oscillation of the centre of mass. This results in a step frequency lower than the natural frequency of the system owing to a relatively greater duration of the upper part of the oscillation when the downward acceleration cannot exceed 1 g, but allows a lower step average power expenditure to reset the limbs at each step. In old age the first strategy is followed: the average upward acceleration never exceeds 1 g, the lower part of the oscillation equals that of the upper part and the step frequency is similar to the natural frequency of the system at all speeds. This allows development of a lower force during the push, but the increase in step frequency results in a greater power outlay to reset the limbs at each step (Cavagna et al., 2008a).

The apparent similarity between the characteristics of backward running hitherto reported in the literature and the mechanics of running in old age, i.e. a greater step frequency and energy expenditure, a lower vertical ground reaction force and a shorter flight time, suggests that the first of the two strategies mentioned above is adopted in backward running. However, the relationship between step frequency and natural frequency of the bouncing system in backward running is not known.

The rationale of the present study was to identify: (i) the changes in motion of the centre of mass of the body associated with the greater step frequency and energy expenditure found in backward running, and (ii) the effect of these changes in motion on the stretch–shorten cycle of muscle–tendon units during the rebound of the body. Is the greater step frequency due to a shorter duration of both the lower and the upper part of the oscillation of the centre of mass, as in old age? In other words, does backward running provide a smother ride with a lower force exerted for each step on the ground as some studies suggest? How does the reversal of motion in backward running affect mechanical energy storage and release during the rebound of the body? To answer these questions, we studied subjects (experienced in backward running) during forward and backward running on soil at different speeds, and measured: (i) the amplitude and the duration of the vertical displacement of the centre of mass during stance, during the aerial phase and during the lower and upper part of the oscillation, (ii) the maximal upwards acceleration attained during the step, (iii) the within-step transduction between kinetic energy and gravitational potential energy of the centre of mass during the lift and the fall, (iv) the orientation in the sagittal plane of the impulse given by the ground reaction force during the push and the brake, and (v) the stiffness of the bouncing system.

The results described in this study were obtained on the same runs used previously (Cavagna et al., 2011) to determine negative–positive work durations and the external mechanical work done to maintain the motion of the centre of mass. As described below, the analysis was expanded in this study to obtain further novel information on the differences in the rebound of the centre of mass in forward and backward running (Figs 1, 2, 3, 4, 5, 6). For the reader's convenience, the first part of the Materials and methods section briefly describes the experimental procedure used previously (Cavagna et al., 2011). The second part describes the additional measurement procedures made in the present study.

Experiments were carried out on subjects with on average 6 years of weekly backward running training (seven males and two females). The characteristics of the male subjects were: age 43.3±6.2 years, height 1.72±0.06 m and mass 69.3±4.4 kg (means ± s.d., N=7). The characteristics of the female subjects were: age 45.0±5.7 years, height 1.60±0.00 m and mass 54.1±7.6 kg (means ± s.d., N=2). Informed, written consent was obtained from each subject. The experiments were performed according to the Declaration of Helsinki.

The subjects ran forwards and backwards on a 50 m long indoor track that had built into it, at floor level 30 m from one end of the track, a strain gauge force platform sensitive to the forward and vertical component of the force exerted by the feet on the ground. The force platform dimensions (4 m long and 0.5 m wide) were large enough to avoid subjects altering their step length in the attempt to hit it. Two photocells placed ∼3 m apart alongside the platform were used to measure the average running speed. Mechanical details of the force platform and the procedures involved in using platform records have been thoroughly described in a previous study (Cavagna, 1975).

The runners were asked to achieve and maintain a constant average speed over the platform. Subsequent runs started from opposite sides of the track. The speed range was 3–17 km h^–1 (0.83–4.72 m s^–1). Subjects were asked to voluntarily decrease their running speed to low values, below the metabolic walk–run transition, which has been estimated to be about 6.5 km h^–1 in backward running (Terblanche et al., 2003) and about 7.4–8.1 km h^–1 in forward running (Margaria, 1976; Terblanche et al., 2003). Nevertheless, the mechanics of running, and not that of walking, was maintained even at the lowest speeds. In fact, contrary to walking, the gravitational potential energy and the kinetic energy of forward motion of the centre of mass oscillated in phase during the steps analysed in this study, with a negligible transduction between them calculated as percentage recovery (Cavagna et al., 1976): 0.9±1.1% (mean ± s.d., N=107) in backward running and 1.9±1.8% (mean ± s.d., N=104) in forward running. Expanding the range of running speeds to low values expands the information on the relative amount of energy absorbed and restored by muscle versus tendon within muscle–tendon units at different durations of their stretch–shorten cycle (Cavagna, 2009).

The photocell signals and the platform signals, proportional to the ground reaction forces, were acquired at a rate of 500 Hz through a dedicated DAQ board (PCI MIO 16E, National Instruments, Austin, TX, USA) on a microcomputer. Custom LabVIEW (version 7.1, National Instruments) software programs were developed for data acquisition and subsequent analysis.

Previous studies (e.g. Cavagna et al., 2008b) describe in detail the procedure followed to determine from the platform signals the instantaneous forward velocity V_f(t), the instantaneous vertical velocity V_v(t), the kinetic energy of forward motion E_kf(t)=0.5M_bV_f(t)² (where M_b is the mass of the body), the kinetic energy of vertical motion E_kv(t)=0.5M_bV_v(t)², the vertical displacement of the centre of mass S_v(t), the gravitational potential energy E_p(t)=M_bgS_v(t) (where g is the acceleration due to gravity), the translational kinetic energy in the sagittal plane E_k(t)=E_kf(t)+E_kv(t) and the total mechanical energy E_cm(t)=E_k(t)+E_p(t). More details on this are provided in our previous paper (Cavagna et al., 2011).

The step period τ and the vertical oscillation of the centre of mass S_v were divided into two parts: a lower part taking place when the vertical force measured by the force platform is greater than the body weight (t_ce and S_ce), and an upper part taking place when the vertical force is smaller than body weight (t_ae and S_ae). The step period and the vertical displacement were also divided, according to tradition, into the fractions taking place during the ground contact time (t_c and S_c) and during the aerial time (t_a and S_a). The measurement procedure and physical meaning of the S_v fractions are described in previous studies (e.g. Cavagna et al., 2008a).

Note that when E_p(t) and E_k(t) increase and decrease in phase during the step no transduction can occur between them, the sum of their changes equals the change of E_cm(t), and r(t)=0. When, in contrast, the changes of E_p(t) and E_k(t) are equal and have opposite sign the transduction between them is complete, ΔE_cm(t)=0 and r(t)=1. The cumulative value of energy recovery, R_int(t), resulting from the instantaneous E_p–E_k transduction during the step, was measured from the area below the r(t) record divided by the step period: R_int(t)=(∫₀^tr(u)du)/τ. At the end of the step R_int(τ)=R_int (Cavagna et al., 2008b).

The mass-specific vertical stiffness, k/M_b, is given by the slope of the relationship between vertical acceleration and vertical displacement of the centre of mass in the range corresponding to the amplitude of the oscillation of the spring-mass system, i.e. from its equilibrium position (F_v=M_bg and vertical acceleration a_v=0) to its maximal deformation, S_ce, attained at the lowest point of the trajectory of the centre of mass of the body when a_v is at a maximum. In this study we chose to measure the mass-specific vertical stiffness as k/M_b=a_v,mx/S_ce (Cavagna et al., 2008a). The maximal upward acceleration a_v,mx was measured by interpolation between the oscillations of the time derivative of V_v(t) using a quadratic least-squares fit (LabVIEW waveform peak detection VI). For comparison, k/M_b was also measured from the slope of a least-squares linear fit of the a_vversus S_v relationship from the equilibrium position to S_ce: (i) including data points obtained during both downward and upward displacement of the centre of mass (k/M_b up and down fit), and (ii) including data points obtained during the upward displacement only to avoid the oscillation in the a_vversus S_v record following the impact with the ground (k/M_b up fit). Furthermore the mass-specific vertical stiffness was measured as k/M_b half-period=(π/t_ce)² on the assumption that t_ce represents the half-period of the resonant frequency of the oscillating system f_s, i.e. that f_s=(k/M_b)^0.5/(2π)=1/(2t_ce).

Average values measured over the whole speed range were: k/M_b=a_v,mx/S_ce=499.384±192.621 (s^–2, k/M_b up and down fit=454.051±148.174 (s^–2), k/M_b up fit=482.144±171.079 (s^–2), k/M_b half-period=491.778±185.593 (s^–2) in backward running (means ± s.d., N=107); and k/M_b=a_v,mx/S_ce=351.769±84.020 (s^–2), k/M_b up and down fit=308.496±67.493 (s^–2), k/M_b up fit=344.270±73.406 (s^–2), k/M_b half-period=347.821±84.992 (s^–2), in forward running (means ± s.d., N=104).

The similarity of the mass-specific vertical stiffness measured as a_v,mx/S_ce and as (π/t_ce)² suggests that the duration of the lower part of the vertical oscillation of the centre of mass, t_ce, may indeed be considered to represent the half-period of the bouncing system.

We chose to use k/M_b=a_v,mx/S_ce as a measure less affected by: (i) the noise of the a_vversus S_v record following the impact with the ground, (ii) the assumption that the stiffness measured during the upward displacement only equals that during the upward and downward displacement, and (iii) the assumption that t_ce represents the half-period of the oscillating system. However, the mean values given above show that the conclusions reached in this study would not change using either of the methods described.

The data collected as a function of the running speed were grouped into the following speed intervals: 3 to <4 km h^–1, 4 to <5 km h^–1...16 to <17 km h^–1 for backward running, and 3 to <4 km h^–1, 4 to <5 km h^–1...16 to <17.5 km h^–1 for forward running. The data points in Figs 1, 3 and 6 represent the mean ± s.d. in each of the above speed intervals and the numbers near the symbols in Fig. 1 give the number of items (N) in the mean. In addition, means (±s.d.) of the data obtained over the whole speed range (N=107 for backward running and N=104 for forward running) are given. When comparing the means of different variables within a subject group, with the same N value at a given running speed, a paired samples t-test was used to determine whether the means were significantly different. When comparing the means of different variables between two subject groups having different N values, a two-sample t-test assuming unequal variance was used. The values of P refer to the two-tail comparison (Microsoft Excel for Mac version 11.6.2).

The step period (τ, upper panel) and the vertical displacement of the centre of mass during each step (S_v, lower panel) are given as a function of the running speed in Fig. 1 (black squares) with their fractions corresponding to the lower part of the oscillation (t_ce and S_ce, red circles) and to the upper part of the oscillation (t_ae and S_ae, blue circles). The fractions taking place during the ground contact phase (t_c and S_c, red dotted lines) and during the aerial phase (t_a and S_a, blue dotted lines) are also given for comparison. Left panels refer to running backwards, right panels to running forwards. A comparison of the left and right panels in Fig. 1 reveals the following.

1. In backward running, as in forward running, the step period τ =t_ce+t_ae decreases with speed. The decrease in τ is due in both cases to a decrease in duration of the lower part of the oscillation t_ce with a small change in duration of the upper part t_ae. In backward running t_ce decreases more rapidly with speed than in forward running, resulting in a sharper decrease in τ and, as a consequence, in a sharper increase in step frequency f=1/τ.

2. In backward running, as in forward running, the duration of the lower part of the vertical oscillation is about equal to that of the upper part at the lowest running speed indicating that the rebound is symmetric, i.e. t_ce≈t_ae. Because of the sharper decrease of t_ce in backward running, the difference between t_ce and t_ae, i.e. the on-off ground asymmetry of the rebound t_ae/t_ce (Cavagna, 2009), increases with speed more rapidly in backward running than in forward running. It is interesting to note that in both cases a division of the step based on the ground contact and aerial times (t_c and t_a, dotted lines in Fig. 1) reveals the largest asymmetry just when t_ce and t_ae show that the rebound is symmetric.

3. In backward running, as in forward running, the vertical displacement of the centre of mass of the body, S_v, increases with speed to a maximum of ∼0.08 m that is attained at ∼2 m s^–1 in backward running and at ∼3.5 m s^–1 in forward running. Subsequently, S_v decreases with speed more rapidly in backward running than in forward running. Similarly to t_ae and t_ce, the corresponding vertical displacements S_ae and S_ce diverge earlier with increasing speed in backward running than in forward running.

Fig. 2 shows the vertical force exerted on the force platform F_v and the corresponding vertical acceleration of the centre of mass of the body for one subject during backward and forward running steps at three different speeds. Note that the oscillation following heel strike during the increment of F_v in forward running is absent or markedly attenuated in backward running (Threlkeld et al., 1989), and that F_v falls sharply to zero in backward running whereas the aerial phase is attained through a gradual decrease of F_v in forward running. These observations are consistent with the finding that the landing–takeoff asymmetry is reversed in backward running (Cavagna et al., 2011). Note also that the mean vertical force exerted during stance (horizontal continuous lines) is on average greater in backward running than in forward running.

Fig. 3 shows the maximal upward acceleration a_v,mx attained at each step in backward running and in forward running. The straight lines are linear fits of all the data: backward a_v,mx,b (m s^–2)=8.199+2.904V_f (m s^–1) (R=0.778) and forward a_v,mx,f (m s^–2)=6.271+2.664V_f (m s^–1) (R=0.836). Averaged over the whole speed range, a_v,mx is ∼18% greater when running backwards (a_v,mx,b=15.682±3.742 m s^–2versus a_v,mx,f=13.248±3.464 m s^–2, P=1.874×10^–6).

In backward running the gravitational potential energy E_p and the kinetic energy of forward motion E_kf are even more precisely in phase than in forward running (see ‘Experimental protocol’ section in Materials and methods). It follows that the E_p–E_k transduction takes place essentially between E_p and E_kv. Fig. 4 shows that the E_p–E_k transduction attained at the end of the step R_int is greater in backward running than in forward running (R_int,b=0.421±0.067 versus R_int,f=0.360±0.080, P=1.385×10^–8) due to a larger E_k into E_p transduction during the lift. In forward running the E_p–E_k transduction is greater during the descent than during the lift (0.208±0.038 versus 0.152±0.048, P=7.244×10^–34), whereas in backward running the E_p–E_k transduction is on average slightly greater during the lift than during the descent (0.216±0.031 versus 0.205±0.038, P=1.916×10^–7).

The momentum of the centre of mass in the vertical direction, M_bV_v, is plotted as a function of the momentum of the centre of mass in the forward direction, M_bV_f, during the push (increment of M_bV_f) and during the brake (decrement of M_bV_f) in Fig. 5. Momentum in the vertical and forward directions is calculated from the corresponding kinetic energies E_kv and E_kf. Note that increments and decrements of momentum, M_bΔV, equal increments and decrements of the impulse, FΔt, where F is the ground reaction force. The red section of each plot indicates M_bV_v changes due to a vertical impulse with a vertical force exerted on the ground greater than body weight, resulting in an increase of M_bV_v from zero to its maximum (vertical lines) during the push and in a decrease of M_bV_v from its maximum to zero during the brake. The blue section of each plot indicates M_bV_v changes due to a vertical impulse with a vertical force less than body weight, resulting in a decrease of M_bV_v during the push and in an increase of M_bV_v during the brake. The light blue section indicates the aerial phase when M_bV_f is constant. The fraction of the push during which M_bV_v increases simultaneously with M_bV_f is given below each plot by the ratio a_f,v/a_f. The fraction of the brake during which M_bV_v and M_bV_f simultaneously decrease is given by the ratio d_f,v/d_f.

A comparison of the tracings obtained during the push in backward running (left column in Fig. 5) with those obtained during the brake in forward running (right column) shows that a vertical force greater than body weight is exerted over most of the push (∼85%) in backward running whereas it is exerted over most of the brake (∼90%) in forward running. In general, the data in Fig. 5 indicate that in backward running the impulse of the ground reaction force is directed more vertically during the push than during the brake, i.e. a_f,v/a_f>d_f,v/d_f, whereas in forward running the impulse of the ground reaction force is directed more vertically during the brake than during the push, i.e. d_f,v/d_f>a_f,v/a_f. Average values obtained from the nine subjects running at a slow, intermediate and high speed are: a_f,v/a_f=0.81±0.04 and d_f,v/d_f=0.65±0.10 at the slow backward running speed of 5.06±0.45 km h^–1versus a_f,v/a_f=0.37±0.11 and d_f,v/d_f=0.85±0.10 at the slow forward running speed of 5.08±0.58 km h^–1 (N=9); a_f,v/a_f=0.85±0.03 and d_f,v/d_f=0.74±0.09 at the intermediate backward running speed of 9.17±0.69 km h^–1versus a_f,v/a_f=0.53±0.07 and d_f,v/d_f=0.93±0.06 at the intermediate forward running speed of 9.13±0.66 km h^–1 (N=9); a_f,v/a_f=0.85±0.09 and d_f,v/d_f=0.79±0.09 at the high backward running speed of 14.59±1.43 km h^–1versus a_f,v/a_f=0.69±0.08 and d_f,v/d_f=0.93±0.03 at the high forward running speed of 15.60±0.97 km h^–1 (N=9).

The upper panels of Fig. 6 show the mass-specific vertical stiffness, k/M_b=a_v,mx/S_ce. It can be seen that in backward running the stiffness is on average greater than in forward running. The bottom panels in Fig. 6 show the resonant frequency of the bouncing system f_s=(k/M_b)^0.5/(2π) (dashed lines) together with the freely chosen step frequency f=1/τ (continuous line). In backward running as in forward running the natural frequency of the system f_s is on average greater than the step frequency f. Both f_s and f increase with speed more steeply in backward running than in forward running.

The present study confirms that, at a given speed, the step frequency is higher in backward running than in forward running (Devita and Stribling, 1991; Wright and Weyand, 2001) and that the stance time, i.e. the time of foot contact on the ground (t_c in Fig. 1) is shorter in backward running than in forward running (Threlkeld et al., 1989; Wright and Weyand, 2001). However, stance time was reported to be the same fraction of stride duration in backward running as in forward running and the vertical impulse given to the body by the ground reaction force was reported to be smaller in backward running than in forward running, with a shorter flight time explaining the higher step frequency (Threlkeld et al., 1989; Devita and Stribling, 1991). These conclusions are not supported by the present results.

What follows shows that the backward running step is characterized not only by a greater peak vertical acceleration attained during stance a_v,mx (Fig. 3) but also by a greater average vertical acceleration and vertical momentum gained during the lower part of the oscillation of the centre of mass.

Eqn 3 shows that the mean acceleration upward during the lower part of the vertical oscillation ā_v,ce is greater in backward running than in forward running because: (i) as shown in Fig. 1 t_ce,f>t_ce,b (t_ce,f=0.172±0.021 s versus t_ce,b=0.149±0.026 s, P=1.986×10^–11), and (ii) ā_v,ae,b>ā_v,ae,f: in fact the mean acceleration downward during the upper part of the vertical oscillation ā_v,ae attains the maximum value of 1 g during the aerial phase t_a and the ratio t_a/t_ae is greater in backward running than in forward running (t_a,b/t_ae,b=0.531±0.202 versus t_a,f/t_ae,f=0.377±0.251, P=1.866×10^–6).

The greater vertical momentum impressed on the body in backward running during the lower part of the vertical oscillation of the centre of mass causes a greater duration of the upper part of the oscillation relative to the lower part, i.e. a greater on–off ground asymmetry of the rebound t_ae/t_ce (t_ae,b/t_ce,b=1.271±0.184 versus t_ae,f/t_ce,f=1.097±0.173, P=2.105×10^–11).

Alternatively, according to the more common approach used in the literature, we find that in backward running the aerial time t_a is increased relative to the contact time t_c (t_a,b/t_c,b=0.475±0.251 versus t_a,f/t_c,f=0.296±0.237, P=2.502×10^–7). The greater ratio t_a/t_c in backward running shows that the stance time t_c is a smaller fraction of the step period τ. A lower t_c/τ ratio requires a greater average vertical force exerted during stance to maintain the average vertical force over the whole step cycle equal to body weight.

Note that t_ae>t_ce whereas t_a<t_c because the duration of the upper part of the oscillation of the centre of mass t_ae includes a fraction of the contact time t_c (Blickhan, 1989).

In conclusion, the peak vertical acceleration a_v,mx (Fig. 3), the mean vertical acceleration and the vertical momentum gained during the lower part of the oscillation of the centre of mass (Eqns 3 and 4) are all greater in backward running than in forward running with the consequence that the backward running step is characterized by a greater on–off ground asymmetry of the rebound t_ae/t_ce.

During human running at low and intermediate speeds, as well as in animal hopping, running and trotting, the duration of the brake after landing is shorter than that of the push before takeoff. As during level running at a constant speed the momentum lost equals the momentum gained, the mean force exerted during the brake must be greater than that developed during the push, i.e. t_brake<t_push and (‘hard landing and soft takeoff’) (Cavagna et al., 2008b; Cavagna and Legramandi, 2009).

In backward running the landing–takeoff asymmetry is reversed, i.e. t_brake>t_push and (‘soft landing and hard takeoff’) (Cavagna et al., 2011). The present study shows that the reversed landing–takeoff asymmetry in backward running is also evidenced by a different trend of the E_p–E_k transduction during the step (R_int in Fig. 4) and by a different orientation of the ground reaction force during the push and the brake (Fig. 5).

As pointed out in a previous study (Cavagna et al., 2008b), the E_p–E_k transduction has a different physiological meaning during the downward displacement and during the lift of the centre of mass.

The E_p into E_k transduction during the downward displacement can be viewed as a mechanism exploiting gravity to passively increase the vertical downward velocity and as a consequence the kinetic energy. The E_p into E_k transduction precedes the negative work phase of the step, provides kinetic energy to be stored within muscle–tendon units during the subsequent brake and requires a high force to decelerate the body downwards. This mechanism applies to forward running as shown by the large increment of R_int during the downward displacement (Fig. 4) and by the finding that almost the entire brake takes place with a ground reaction force having a vertical component greater than body weight (Fig. 5).

The E_k into E_p transduction during the lift follows the positive work phase of the step and increases with the push average power developed against gravity before takeoff. In fact, the greater the vertical push the greater the increment in kinetic energy of vertical motion and its subsequent transduction into gravitational potential energy. This mechanism applies to backward running as shown by a large increment of R_int during the lift (Fig. 4) and by the finding that most of the push takes place with a ground reaction force having a vertical component greater than body weight (Fig. 5).

In conclusion, the reversed use of the lever system of the limbs in backward running results not only in the reversal of the muscle groups involved during the brake and the push (Devita and Stribling, 1991) and the reversal of the magnitude of the force and displacement vectors during negative and positive work (Cavagna et al., 2011) but also in the reversal of the E_p–E_k transduction during the step and in the brake–push direction of the ground reaction force during stance.

The step frequency, f=1/(t_ae+t_ce)=1/(t_a+t_c), is greater in backward running than in forward running because of a shorter duration of the lower part of the oscillation, t_ce, with a similar duration of the upper part of the oscillation, t_ae (Fig. 1). Alternatively, the greater step frequency in backward running can be considered to be due to a decrease of the contact time t_c (Wright and Weyand, 2001) compensated only in part by an increase of the flight time t_a (t_a,b=0.098±0.038 s versus t_a,f=0.072±0.049 s, P=2.389×10^–5). The shorter contact time is probably the consequence of a reduced overall range of motion at the hip and knee joints (Bates et al., 1986).

The shorter duration of the lower part of the oscillation, t_ce, is the expression of an increased vertical stiffness, a_v,mx/S_ce≈(π/t_ce)² (Fig. 6, upper panels), resulting in a greater resonant frequency of the bouncing system, f_s=(a_v,mx/S_ce)^0.5/(2π)≈1/(2t_ce) (Fig. 6, lower panels). In other words, bouncing steps are more frequent in backward running than in forward running because the bounce takes place on a more rigid system.

The increase in stiffness in backward running may be due to the lack of flexion of the knee, usually used in forward running as a shock absorber after landing. The knee joint, already flexed at touch down, maintains a nearly isometric/fixed position during the initial stance phase and extends at the end of the stance phase to propel the body upwards and forwards (Flynn and Soutas-Little, 1993; Devita and Stribling, 1991).

As mentioned in the Introduction of this study, the similarity between the characteristics of backward running hitherto reported in the literature with that of running in old age (a greater step frequency and energy expenditure with a lower vertical ground reaction force and a shorter flight time) suggested a similar mechanism was involved, i.e. a smother ride at the expense of a greater energy expenditure to reset the limbs more times per minute. The present results show that this is not the case. The average upward acceleration during the lower part of the oscillation never exceeds 1 g during running by old men whereas it is greater than 1 g in backward running with the consequence that the duration of the upper part of the oscillation is greater than that of the lower part, i.e. t_ae>t_ce (Fig. 1), and the step frequency f=1/(t_ae+t_ce) is lower than the frequency of the system f_s≈1/(2t_ce) (Fig. 6). Backward running therefore does not provide as smooth a ride as running in old age, but the higher step frequency may indeed require greater power to reset the limbs more times per minute (Cavagna et al., 2011).

In this study, all of the between-gait comparisons were made at the same absolute speed, i.e. at relatively faster backward running speeds versus relatively slower forward running speeds. If backward and forward running are compared at an equivalent speed, such as the fastest top speed, the conclusions are different. Specifically, the duration of contact on the ground is shorter in backward running than in forward running at the same absolute speed (Fig. 1), but attains the same value in the two conditions at top speed (Weyand et al., 2010). The vertical stiffness is greater in backward running than in forward running at the same absolute speed (Fig. 6), but is possibly similar in the two conditions at top speed as suggested by: (i) the similar contact duration in the two gaits (Weyand et al., 2010) and (ii) the similar natural frequency of the bouncing system f_s at the highest speeds of backward running in the present study (∼4.8 Hz in Fig. 6) and at the highest speeds of forward running in a previous study [∼5.0 Hz in fig. 2 of Cavagna et al. (Cavagna et al., 1988)].

In a catapult, elastic energy is stored slowly during deformation of an elastic structure and subsequently released rapidly during recoil of the same structure. In spite of losses due to elastic hysteresis, the work returned per unit time, i.e. the power output, is greater than the power input. As pointed out by Alexander, ‘catapults are power amplifiers’ (Alexander, 1988) and some animals, particularly jumping insects, use the same principle. We think that the catapult principle also applies to backward running, although with some peculiarities and a much lower power amplification than that observed in jumping insects. As in a catapult, and opposite to forward running, the duration of negative work, when muscle–tendon units are stretched and elastic energy is stored, exceeds the duration of positive work, when muscle–tendon units shorten and elastic energy is released [see fig. 2 of Cavagna et al. (Cavagna et al., 2011)]. Because during running on the level at a constant speed the negative work done during the brake equals the positive work done during the push, the difference in brake–push duration translates into a power output during the push greater than the power input during the brake. According to the data collected in a previous study (Cavagna et al., 2011), brake/push power=1.157±0.164 (N=104) in forward running whereas push/brake power=1.149±0.151 (N=107) in backward running. In backward running, however, unlike a catapult, positive work has to be done over a shorter distance than negative work, requiring a greater force for the same amount of work (Cavagna et al., 2011). Furthermore, whereas in a catapult elastic energy is stored and released by the same structure, this is not necessarily the case in backward running. In particular, the extensors of the knee, which are responsible for a large fraction of the power output during the push (Devita and Stribling, 1991), shorten from a state of quasi-isometric contraction without previous stretching (Flynn and Soutas-Little, 1993). In other words, in backward running some energy has to be produced ex novo without previous loading.

The described differences in the stretch–shorten cycle taking place during the rebound of the body in backward and forward running have three possible physiological consequences: (i) a greater energy expenditure in backward running due to the lower efficiency of positive work production (Cavagna et al., 2011), (ii) a reduced risk of muscle damage after landing during the brake in backward running because muscle–tendon units are lengthened at a slower rate as a result of the longer duration of the negative work phase, and (iii) enhanced elastic energy storage in forward running as a consequence of the higher force attained during stretching when elastic structures are loaded, whereas the contribution of the contractile component is enhanced in backward running because positive work has to be produced with a greater average force and some of it without previous loading. The forward running mechanism based on elastic energy storage and recovery is more efficient, but subject to a greater risk of muscle fibre damage during fast muscle stretching following impact with the ground, whereas the backward running mechanism is more expensive, less efficient but probably safer.

 
• a_v

  vertical acceleration of the centre of mass

 
• a_v,mx

  maximal vertical acceleration of the centre of mass

 
• E_cm

  mechanical energy of the centre of mass

 
• E_k

  kinetic energy of the centre of mass

 
• E_kf

  kinetic energy of forward motion of the centre of mass

 
• E_kv

  kinetic energy of vertical motion of the centre of mass

 
• E_p

  gravitational potential energy of the centre of mass

 
• f

  step frequency

 
• f_s

  resonant frequency of the system

 
• F_v

  vertical force

 
• 

  mean force exerted during the brake

 
• 

  mean force developed during the push

 
• M_b

  body mass

 
• r(t)

  time course of the transduction between gravitational potential energy, E_p(t), and translational kinetic energy in the sagittal plane, E_k(t)

 
• R_int

  cumulative energy recovery resulting from the instantaneous E_k–E_p transduction

 
• S_a

  vertical displacement of the centre of mass during t_a

 
• S_ae

  vertical displacement of the centre of mass during the upper part of the oscillation

 
• S_c

  vertical displacement of the centre of mass during t_c

 
• S_ce

  vertical displacement of the centre of mass during the lower part of the oscillation

 
• S_v

  vertical displacement of the centre of mass during each step

 
• t_a

  aerial time

 
• t_ae

  duration of the upper part of the oscillation

 
• t_brake

  duration of the brake

 
• t_c

  ground contact time

 
• t_ce

  duration of the lower part of the oscillation

 
• t_push

  duration of the push

 
• V_f

  forward velocity

 
• V_v

  vertical velocity

 
• τ

  step period

The authors wish to thank Giuseppe Angeli, President of the Italian Retro-running Association, for organizing the experimental sessions and the nine athletes for coming to Milan from all over Italy in their holidays to run on the force plate.