Introduction

The bouncing step of running is often divided into a contact time, during which the foot is on the ground, and an aerial time, during which the body is airborne. This division is inappropriate when analysing the apparent elastic bounce of the body in physical terms. In fact, an aerial time may not occur at all in the bouncing step: during trotting and in human running at very low speeds the vertical oscillation of the centre of mass of the body may take place entirely in contact with the ground (Cavagna et al. 1988). Even in the presence of an aerial phase, the vertical displacement of the centre of mass of the body below and above the instant the foot contacts and leaves the ground bears no relation with the amplitude of the oscillation. Both in the spring-mass model of running (Blickhan 1989) as in the real running step, the vertical oscillation takes place above and below an equilibrium position where the vertical force equals body weight.

The apparent elastic bounce of the body therefore has been divided into two parts: a part taking place when the vertical force exerted on the ground is greater than body weight (lower part of the oscillation, called effective contact time t ce) and a part taking place when this force is smaller than body weight (upper part of the oscillation, called effective aerial time, t ae). Note that, according to this division, the lower part of the oscillation takes place always in contact with the ground, whereas the upper part of the oscillation includes a fraction of the contact time and may or may not include an aerial time. Note also that t ce always represents the half period of the oscillation, both in the absence and in the presence of an aerial phase, whereas t ae exceeds the half period of the oscillation in the presence of an aerial phase. At low speeds of human running the duration of the lower part of the vertical oscillation, t ce, is about equal to that of the upper part, t ae. This was called a symmetric rebound. At high speeds the duration of the upper part of the oscillation is greater than that of the lower part, i.e. t ae > t ce. This was called an asymmetric rebound (Cavagna et al. 1988).

Further analysis of the bouncing step in human running revealed a difference between the events taking place during compression of the ‘elastic’ system after ‘landing’, when muscle–tendon units are stretched beyond the equilibrium position during the fall, and during recoil of the system before ‘takeoff’, when muscle–tendon units shorten to the equilibrium position during the lift. This difference, called landing–takeoff asymmetry, results in duration of negative work, t brake, shorter than that of positive work, t push (Cavagna 2006).

The landing–takeoff asymmetry must not be confused with the asymmetry of the rebound. Aim of this note is to give a quantitative definition of the two asymmetries, explain their physiological meaning and suggest a possible relation between the two.

Asymmetry of the rebound

During running on the level, the vertical momentum lost and gained during t ce (lower part of the oscillation) must equal the vertical momentum lost and gained during t ae (upper part of the oscillation), i.e.

$$ \overline{a}_{{{\text{v}},{\text{ce}}}} t_{\text{ce}} = \overline{a}_{{{\text{v}},{\text{ae}}}} t_{\text{ae}} $$
(1)

where \( \overline{a}_{\text{v,ce}} \) and \( \overline{a}_{\text{v,ae}} \) are the average vertical accelerations of the centre of mass during the effective contact time t ce and the effective aerial time t ae, respectively. According to Eq. 1, a symmetric rebound (i.e. t ce = t ae) implies the same average vertical acceleration during the lower and the upper part of the vertical oscillation of the centre of mass (i.e. \( \overline{a}_{{{\text{v}},{\text{ce}}}} \) = \( \overline{a}_{{{\text{v}},{\text{ae}}}} \)).

At low running speeds this condition is met: \( \overline{a}_{{{\text{v}},{\text{ce}}}} \) and \( \overline{a}_{\text{v,ae}} \) increase similarly with speed (see Fig. 5 of Schepens et al. 1998), t ae ~ t ce (Fig. 1a) and the step frequency equals the natural frequency of the bouncing system, f s = 1/(2t ce) (Cavagna et al. 1988). Note that the natural frequency of the bouncing system calculated as f s = 1/(2t ce) coincides, as a first approximation, with the frequency calculated as f s = (k/M)1/2/(2π), where M is the body mass and k/M is the mass specific vertical stiffness (Cavagna et al. 1988). The vertical stiffness, k/M, increases with the speed of locomotion in all the animals tested in previous studies (Cavagna et al. 1988; Farley et al. 1993). In humans k/M, and as a consequence the effective contact time, t ce (Fig. 1), remain about constant up to 8–11 km h−1 (Cavagna et al. 1988, 2008b; Schepens et al. 1998). Tuning the step frequency to the natural frequency of the bouncing system during running at low speeds (less than ~11 km h−1) results in a minimum of metabolic energy expenditure and in a maximum of the efficiency of conversion of stored chemical energy into positive mechanical work (measured as the sum of the mechanical work to sustain the motion of the centre of mass relative to the surroundings and of the limbs relative to the centre of mass) (Cavagna et al. 1997).

Fig. 1
figure 1

a The on–off ground asymmetry of the rebound, indicated by a duration of the upper part of the vertical oscillation of the centre of mass at each step greater than that of the lower part, i.e. by t ae > t ce, depends on the amount of force impressed to the body in the lower part of the oscillation during both positive and negative work. b The landing–takeoff asymmetry, indicated by a duration of positive work greater than that of negative work, i.e. by t push > t brake, depends on the difference in force, lower during positive work and greater during negative work. This difference would be nil in an elastic bounce. It can be seen that the on–off ground asymmetry of the rebound increases with running speed, whereas the landing–takeoff asymmetry decreases with speed. This suggests that the increase in force with speed, which increases the on–off ground asymmetry, decreases the landing–takeoff asymmetry approaching an elastic bounce by privileging the role of tendons relative to muscle within muscle–tendon units. Data points and vertical bars represent the mean and SD (N = 12–30) of 359 measurements made on ten adult subjects, eight males and two females (weight 65.3 ± 8.4 kg, height 1.75 ± 0.06 m, age 28 ± 9.8 years) during running on the level at different speeds (Cavagna 2006)

Full size image

With increasing running speed this ideal condition is no longer met: t ce decreases with increasing speed whereas t ae remains about constant (Fig. 1a). The decrease of t ce with running speed can be explained as follows. It has been shown that the angle swept by the leg during the ground contact time increases continuously with speed in trotting and hopping animals (Farley et al. 1993). As explained in the first two paragraphs of this “Note”, the ground contact time exceeds the half period of the oscillating system, t ce. In humans, the fraction of the step length, L ce, taking place during the lower part of the oscillation, i.e. during t ce, tends to a constant value at high running speeds (Cavagna et al. 1988; Schepens et al. 1998). This is possibly due to anatomical and/or functional factors limiting the angle swept by the body structures connecting centre of mass and point of contact on the ground during t ce. It follows that with increasing running speed, V f, the half period of the oscillation, which is t ce ~ L ce/V f, necessarily decreases.

In order to maintain a symmetric rebound, t ae should decrease with speed similarly to t ce, but his would imply \( \overline{a}_{\text{v,ae}} \) = \( \overline{a}_{{{\text{v}},{\text{ce}}}} \) (Eq. 1). In young and adult humans, however, \( \overline{a}_{\text{v,ce}} \) increases with running speed beyond 1g, whereas \( \overline{a}_{\text{v,ae}} \) cannot increase beyond 1g (note that \( \overline{a}_{\text{v,ae}} \) would attain a maximum value of 1g if t ae would equal the aerial time). As a consequence, a greater duration of t ae relative to t ce is necessary to compensate for the lower acceleration during t ae relative to t ce (Eq. 1). This translates into an asymmetric rebound, i.e. t ae > t ce (Fig. 1a), and in a step frequency, f = 1/(t ce + t ae) lower than the apparent natural frequency of the bouncing system f s = 1/(2t ce). The asymmetry of the rebound therefore is an on–off ground asymmetry, expression of the amount of force impressed to the body during the lower part of the oscillation when both positive and negative works are done.

Factors that determine the choice of the step frequency at a given running speed are: (1) tuning the step frequency f to the natural frequency of the system f s (Cavagna et al. 1997), and (2) choosing a step frequency that minimize the total (external plus internal) aerobic-limited step-average power, within the limits set by the anaerobic-limited push-average power (Cavagna et al. 1991). As mentioned above, the first strategy is usually adopted at low running speeds when t ce ~ t ae (Fig. 1), with the consequence that f ~ f s, and is abandoned for the second strategy at high running speeds when t ce < t ae (Fig. 1), with the consequence that f < f s. By decreasing the step frequency below the natural frequency of the bouncing system, the on–off ground asymmetry of the rebound has the physiological advantage to limit the increase in the power spent to reset the limbs at each step, and, with it, the total step-average mechanical power (Cavagna et al. 1991). Running with high, long leaps at a low frequency (large on–off ground asymmetry) would be a convenient strategy to adopt at high running speeds, provided that enough push-average power is at disposal to allow these leaps. In the old age, muscular force is reduced (Doherty 2003) and this strategy is not adopted: during running, the average upward acceleration \( \overline{a}_{\text{v,ce}} \) never exceeds 1g. The step frequency is tuned to the natural frequency of the system over the whole speed range. This allows development of a lower force during the push, but the greater increase in step frequency with speed results in a more sharp increase of the power spent to reset the limbs at each step (Cavagna et al. 2008b).

Landing–takeoff asymmetry

When running on the level at a constant speed, the momentum lost during negative work equals the momentum gained during positive work:

$$ \bar{F}_{\text{brake}} t_{\text{brake}} = \bar{F}_{\text{push}} t_{\text{push}} $$
(2)

Figure 1b shows that t push > t brake up to about 14 km h−1, above this speed t push = t brake (Cavagna 2006).

According to Eq. 2, the difference in work duration, t push > t brake, i.e. the landing–takeoff asymmetry, implies an average force exerted by muscles during the brake after landing, greater than that exerted during the push before takeoff, i.e. \( \bar{F}_{\text{brake}} \) > \( \bar{F}_{\text{push}} \). The landing–takeoff asymmetry increases with \( \bar{F}_{\text{brake}} \)/\( \bar{F}_{\text{push}} \) and is expression of the difference in force exerted during negative and positive work. Note that in the bounce of an elastic structure with no hysteresis, \( \bar{F}_{\text{brake}} \) = \( \bar{F}_{\text{push}} \). The landing–takeoff asymmetry therefore is expression of the deviation of the muscle–tendon units from an elastic structure.

Experiments on isolated muscle specimens (e.g. Phillips et al. 1991) and in vivo on humans (e.g. Klass et al. 2005) showed that in the old age muscular force is reduced during shortening whereas is preserved or scarcely reduced during stretching. The effect of this change of muscle contractile properties with age is consistent with the finding that during running t push is greater in old subjects than in young subjects, whereas t brake is similar in old and young subjects (see Fig. 4 of Cavagna et al. 2008a). This finding suggests that the landing–takeoff asymmetry of running derives from the different response of muscle to stretching and shortening.

Possible relation between the asymmetry of the rebound and the landing–takeoff asymmetry

Figure 1 shows that in human running the discrepancy between t ae and t ce increases with speed, whereas the discrepancy between t push and t brake decreases with speed. Apparently, the increase of the vertical push with speed, making the on–off ground rebound asymmetric, has an opposite effect on the landing–takeoff asymmetry: the greater the force during the rebound, the smaller its difference between stretching and shortening. This finding may be explained as follows.

In muscle–tendon units, muscle and tendon are two structures in series subjected to the same force at their extremities. When the unit is stretched during negative work, the lengthening of the two structures will depend on their stiffness: the lower the stiffness the greater the lengthening. The stiffness of muscle is greater the greater its activation and the resulting force (see, e.g. Fig. 2 of Morgan 1977). At low running speeds, muscle activation and force are lower than at high speeds. It follows that muscle lengthening during negative work, and its subsequent shortening during positive work, will be greater at low than at high speeds. In fact, short-range compliance of the muscle–tendon unit decreases with increasing force, i.e. with muscle activation, falling asymptotically towards tendon compliance (Morgan 1977).

According to the force–velocity relation of muscle contractile component, the force exerted during stretching by each active muscle fibre is greater than during shortening, i.e. muscle exhibits a large hysteresis in its stretch–shorten cycle. Tendon, on the contrary, approaches an elastic structure with small hysteresis (Alexander 2002). It is therefore reasonable to assume that the landing–takeoff asymmetry, i.e. t push > t brake due to \( \bar{F}_{\text{brake}} \) > \( \bar{F}_{\text{push}} \), is greater at low than at high running speeds due to a contribution of muscle to the length change of the muscle–tendon unit, which is greater at low than at high speeds.

It is known that the efficiency of positive work production during running and hopping increases with speed suggesting that the role played by elasticity increases with the velocity of stretching and shortening muscle–tendon units (Alexander and Vernon 1975; Cavagna and Kaneko 1977). This finding is in qualitative agreement with the decrease of the landing–takeoff asymmetry with running speed.

In conclusion, increasing the vertical push with increasing running speed, increases the on–off ground asymmetry of the rebound, but, at least in human running, decreases the landing–takeoff asymmetry possibly by privileging the role of tendons relative to muscle within muscle–tendon units. This may result in a more elastic rebound explaining the greater efficiency of positive work production observed at high running speeds. Direct measurements of tendon versus muscle length change within muscle–tendon units with increasing running speed may confirm the indirect approach used here.