Reducing gravity takes the bounce out of running

Under normal circumstances, why do humans and animals select particular steady gaits from the myriad possibilities available? One theory is that the chosen gaits minimize metabolic energy expenditure (Alexander and Jayes, 1983; Ruina et al., 2005). To test this theory, one can subject organisms to abnormal circumstances. If the gait changes to a new energetic optimum, it can be inferred that energetics also govern gait choice under normal conditions (Bertram and Ruina, 2001; Long and Srinivasan, 2013; Selinger et al., 2015).

One ‘normal’ gait is the bipedal run, and one abnormal circumstance is that of reduced gravity. Movie 1 demonstrates the profound effect reducing gravity has on running kinematics. A representative subject runs at 2 m s^–1 in both Earth-normal and simulated lunar gravity (approximately one-sixth of Earth-normal). The change in kinematics is apparent: the gait in normal gravity involves pronounced centre-of-mass undulations compared with the near-flat trajectory of the low-gravity gait. Although centre-of-mass vertical excursions during stance are known to decrease in reduced gravity (Donelan and Kram, 2000), we observed that the height achieved in the flight phase also decreases. This gait modification seems paradoxical: in reduced gravity, people are free to run with much higher leaps. Instead, they seem to flatten their gait. Why should this be?

A simple explanation posits that the behaviour is energetically beneficial. To explore the energetic consequences of choosing to run with lower leaps in reduced gravity, we first considered the impulsive model of running, following Rashevsky (1948) and Bekker (1962), which treats a human runner as a point mass body bouncing off rigid vertical limbs (Fig. 1). Stance is treated as an inelastic, impulsive collision with the ground. In reality, stance occurs in finite time, and elastic mechanisms exist. However, the inelastic approximation is remarkably productive in explaining gait choice (Ruina et al., 2005). When we use the term ‘energetic cost of collisions’, we are generally referring to non-recoverable energy loss during stance resulting from some interaction of the centre of mass with the ground (Bertram and Hasaneini, 2013). Such losses may arise from damping, active negative work or discontinuous velocity profiles. In any case, modelling these interactions as an inelastic collision provides a simple estimation of the net cost.

During this collision, all vertical velocity is lost while horizontal velocity is conserved (Fig. 1B). The total kinetic energy lost per step is therefore E_col=mV²/2, where m is the runner's mass and V is their vertical take-off velocity. Lost energy must be recovered through muscular work to maintain a periodic gait, and so an energetically optimal gait will minimize these losses. If centre-of-mass kinetic energy loss were the only source of energetic cost, then the optimal solution would always be to minimize vertical take-off velocity. However, such a scenario would require an infinite stepping frequency as V approaches zero (Alexander, 1992; Ruina et al., 2005), as step frequency (ignoring stance time and air resistance) is f =g/(2V), where g is gravitational acceleration.

Let us suppose there is an energetic penalty that scales with step frequency, as E_freq∝f^k∝g^k/V^k, where k>0. Such a penalty may arise from work-based costs associated with swinging the leg, which are frequency dependent (k=2; Alexander, 1992; Doke et al., 2005), or from short muscle burst durations recruiting less efficient, fast-twitch muscle fibres (k≈3; Kram and Taylor, 1990; Kuo, 2001). Notably, this penalty increases with gravity, as the non-contact duration will be shorter for any given take-off velocity in higher gravity. The penalty also has minimal cost when V is maximal; smaller take-off velocities require more frequent steps, which is costly. Therefore, the two sources of cost act in opposite directions: collisional loss promotes lower take-off velocities, whereas frequency-based cost promotes higher take-off velocities.

The observation of He et al. (1991) that implies k=2, a finding consistent with frequency costs arising from the work of swinging the limb (Alexander, 1992; Doke et al., 2005). In reality, He et al. (1991) measured vertical speed at initial foot contact, but for the impulsive model in its simplest form, this is indistinguishable from take-off velocity. Their empirical assessment of the relationship used a small sample size, with only four subjects. We tested the prediction of Eqn 2 by measuring the take-off velocity over each running stride in 10 subjects using a harness that simulates reduced gravity. [In this paper, we are taking the vertical take-off velocity as the maximum vertical velocity during the gait cycle, following Cavagna (2006).] We also measured the maximum vertical displacement in the ballistic phase to verify whether the counterintuitive observation of lowered ballistic centre-of-mass height in hypogravity, as exemplified in Movie 1, is a consistent feature of reduced-gravity running.

We asked 10 healthy subjects to run on a treadmill for 2 min at 2 m s^–1 in five different gravity levels (0.15, 0.25, 0.35, 0.50 and 1.00 g_e, where g_e is 9.8 m s^–2). A belt speed of 2 m s^–1 was chosen as a comfortable, intermediate jogging pace that could be accomplished at all gravity levels. Reduced gravities were simulated using a harness-pulley system similar to that used by Donelan and Kram (2000), but differing in the use of a spring-pendulum system to generate near-constant force for a large range of motion. Hasaneini et al. (2017 preprint) provide more details of the apparatus. The University of Calgary Research Ethics Board approved the study protocol and written informed consent was obtained from all subjects. Leg length for each subject was measured during standing from the base of the shoe to the greater trochanter on one leg.

Owing to the unusual experience of running in reduced gravity, subjects were allowed to acclimate at their leisure before indicating they were ready to begin each 2-min measurement trial. In each case, the subject was asked simply to run in any way that felt comfortable. Data from 30 to 90 s from trial start were analyzed, providing a buffer between acclimating to experimental conditions at trial start and initiating slowdown at trial end.

Gravity levels were chosen to span a broad range. Of particular interest were low gravities, at which the model predicts unusual body trajectories. Thus, low levels of gravity were sampled more thoroughly than others. The order in which gravity levels were tested was randomized for each subject, so as to minimize sequence conditioning effects.

For each gravity condition, the simulated gravity system was adjusted in order to modulate the force pulling upward on the subject. In this particular harness, variations in spring force caused by support-spring stretch during cyclic loading over the stride were virtually eliminated using an intervening lever (see figs 3 and 4 in Hasaneini et al., 2017 preprint). The lever moment arm was adjusted in order to set the upward force applied to the harness, and was calibrated with a known set of weights prior to all data collection. A linear interpolation of the calibration was used to determine the moment arm necessary to achieve the desired upward force, given the subject weight and targeted effective gravity. Using this system, the standard deviation of the upward force during a trial (averaged across all trials) was 3% of the subject's Earth-normal body weight.

Achieving exact target gravity levels was not possible because the lever's moment arm is limited by discrete force increments (approximately 15 N). Thus, each subject received a slight variation of the targeted gravity conditions, depending on their weight. A real-time data acquisition system allowed us to measure tension forces at the gravity harness and calculate the effective gravity level at the beginning of each new condition. The force-sensing system consisted of an analog strain gauge (Micro-Measurements CEA-06-125UW-350, Wendell, NC, USA), mounted to a C-shaped steel hook connecting the tensioned cable and harness. The strain gauge signal was passed to a strain conditioning amplifier (National Instruments SCXI-1000 amp with SCXI-1520 eight-channel universal strain gauge module connected with SCXI-1314 terminal block, Austin, TX, USA), digitized (NI-USB-6251 mass termination) and acquired in a custom virtual instrument in LabVIEW (National Instruments). The tension transducer was calibrated with a known set of weights once before and once after each data collection trial to correct for modest drift error in the signal. The calibration used was the mean of the pre- and post-experiment calibrations.

A marker was placed at the lumbar region of the subject's back, approximating the position of the centre of mass. Each trial was filmed at 120 Hz using a Casio EX-ZR700 digital camera (Casio Computer Co., Ltd, Shibuya, Tokyo, Japan). The marker position was digitized in DLTdv5 (Hedrick, 2008). Position data were differentiated using a central differencing scheme to generate velocity profiles, which were further processed with a fourth-order low-pass Butterworth filter at 7 Hz cut-off. The vertical take-off velocity was defined as the maximum vertical velocity during each gait cycle (V in Fig. 1). This definition corresponds to the moment at the end of stance where the net vertical force on the body is null, in accordance with a definition of take-off proposed by Cavagna (2006).

Vertical take-off velocities were identified as local maxima in the vertical velocity profile. Filtering and differentiation errors occasionally resulted in some erroneous maxima being identified. To rectify this, first any maxima within 10 time steps of data boundaries were rejected. Second, the stride period was measured as time between adjacent maxima. If any stride period was 25% lower than the median stride period or less, the maxima corresponding to that stride period were compared and the largest maximum was kept, with the other being rejected. This process was repeated until no outliers remained.

Position data used to determine ballistic height were processed with a fourth-order low-pass Butterworth filter at 9 Hz cut-off. Ballistic height was defined as the vertical displacement from take-off to the maximum height within each stride. No outlier rejection was used to eliminate vertical position data peaks, as the filtering was slight and no differentiation was required. If a take-off could not be identified prior to the point of maximum height within half the median stride time, the associated measurement of ballistic height was rejected; this strategy prevented peaks from being associated with take-off from a different stride.

Take-off velocities and ballistic heights were averaged across all gait cycles in each trial for each subject. To test whether ballistic height varied with gravity, a linear model between ballistic height and gravitational acceleration was fitted to the data using least-squares regression, and the validity of the fit was assessed using an F-test. A linear model was also tested for log(V) against log(g) using the same methods. Because the proportionality coefficient between V* and is unknown a priori, we derived its value from a least-squares best fit of measured vertical take-off velocity against the square root of gravitational acceleration, setting the intercept to zero. Given a minimal correlation coefficient of 0.5 and sample size of 50, a post hoc power analysis yields statistical power of 0.96, with type I error margin of 0.05. Data were analyzed using custom scripts written in MATLAB (v. 2016b, MathWorks, Natick, MA, USA).

Data from all trials are shown in Fig. 2. Ballistic height increases with gravity (linear versus constant model, P=4×10^–4, R²=0.24, N=50; Fig. 2A), validating the counterintuitive result exemplified in Movie 1 as a consistent feature of running in hypogravity.

Take-off velocity also increases with gravitational acceleration (Fig. 2B), and a least-squares fit of Eqn 2 using k=2 follows empirical measurements well (R²=0.73, N=50). Other values of k were also tested (Fig. 3). If the impulsive model is accurate, then the best-fit slope of a scatter plot of log(V) against log(g) should correspond to k/(k+2) (Eqn 2), that is, slopes of 0.33, 0.50 or 0.60 for k=1, 2 and 3, respectively. Only the slope predicted by k=2 falls within the 95% confidence interval of the least squares slope (0.47±0.09; Fig. 3).

A best fit at k=2 implies a frequency-based cost arising primarily from the work of leg swing. However, because only the centre of mass is offloaded by the harness, the natural frequency of limb swing remains unchanged for all target gravity levels (Donelan and Kram, 2000). Because metabolic energy of swing is minimal at natural frequency (Doke et al., 2005), it is necessary to adjust the predictions from the impulsive model (Appendix 1). An adjusted model exhibits a fit with R²=0.745 (N=50; Fig. A1), only marginally better than the simple model with k=2 (R²=0.73; Fig. 2B). The predictions do not change greatly, because time spent in the air is affected by gravity, and more air time requires less work to swing the legs, regardless of natural frequency [as long as stride frequency is greater than natural frequency, which is very likely the case for the present study (Appendix 1)].

The impulsive model with k=2 predicts that the ballistic height should remain constant (dashed line in Fig. 2A). This constant value agrees with empirical data at low g, but exhibits increasing error towards Earth-normal g.

We defined ‘take-off’ as occurring when the net force on the body was null and velocity was maximal; however, this does not equate to the moment when the stance foot leaves the ground. After the point of maximal velocity, upward ground reaction forces decay to zero. During this time, the net downward acceleration on the body is less than gravitational acceleration. Thus, the body travels higher than would be expected if maximal velocity corresponded exactly to the point where the body entered a true ballistic phase, as in the model (Fig. 1).

Human runners lower the height achieved in the ballistic phase as gravity decreases. This adaptation requires pronounced modification of the take-off velocity, as maintaining the latter parameter in all conditions would result in substantially increased ballistic height in reduced gravity. Why human runners would modify their gait so greatly was initially unclear.

A simple work-based model of energetic cost explains the trends well. The fit in Fig. 2B exhibits an R² value of 0.73, indicating that a simple energetic model can explain over two-thirds of the variation in maximum vertical velocity resulting from changes in gravity. Human runners seem to be sensitive to these energetic costs and adjust their take-off velocity accordingly. However, the model has its limitations, and an accounting of finite stance (which was initially neglected in the model) was necessary to explain the trend of increasing ballistic height with gravity. Despite the updated model matching the general trend of the data, the slope in Eqn 3 is reduced compared with the empirically derived slope.

The use of the external lumbar point as a centre of mass approximation may explain some of the remaining difference between Eqn 3 and observation. At lower gravity, the body maintained a relatively erect, rigid posture (as exemplified by Movie 1), and so the lumbar marker likely follows the centre of mass closely. However, at higher gravity, the legs move through larger excursions and the torso exhibits slight rotation, making the lumbar estimate less accurate. At normal gravity, Slawinski et al. (2004) showed that the lumbar point overestimates vertical oscillations of the flight phase (by less than 1 cm), though their trials were at a high belt speed (5 m s⁻¹). If the same results hold in our case, we would expect that the measured ballistic height in Fig. 2A should be slightly lower at higher levels of gravity, reducing the actual slope and possibly improving the agreement to Eqn 3. Future work could use a multisegment model to improve centre of mass and ballistic height measurements, but such a technique is unlikely to reverse the trend of increasing ballistic height with gravitational acceleration.

The present results indicate that the cost of step frequency is a key factor in locomotion. Although the exact value of the optimal take-off velocity depends on both frequency-based penalties and collisional costs, the former penalties change with gravity while the latter do not (Fig. 4). The collisional cost landscape is independent of gravity because the final vertical landing velocity is alone responsible for the lost energy. Regardless of gravitational acceleration, vertical landing speed must equal vertical take-off speed in the model, so a particular take-off velocity will have a particular, unchanging collisional cost.

However, taking off at a particular vertical velocity results in greater flight time at lower levels of gravity; thus, the frequency-based cost curves are decreased as gravity decreases (Fig. 4). Frequency-based costs, particularly limb-swing work, appear to be an important determinant of the effective movement strategies available to the motor control system. Their apparent influence warrants further investigation into the extent of their contribution to metabolic expenditure.

Although the present study corroborates others in finding that a work-based cost (k=2) predicts locomotion well (Alexander, 1980, 1992, Hasaneini et al., 2013), other authors have favoured a higher-order ‘force/time’ cost (Kuo, 2001; Doke et al., 2005; Doke and Kuo, 2007). Interestingly, a higher-order model in frequency cost (k=3) did not fit the present data; however, our simple model with k=3 only approximates the force/time cost in the swing phase, and does not account for a rate cost during stance. Further research must be done to distinguish the predictive value of work-based cost to its alternatives; however, for the present results, a work-based model is sufficient, at least for take-off velocity.

The present results challenge the notion that metabolic cost of running is determined largely by the cost of generating force during stance (Kram and Taylor, 1990; Arellano and Kram, 2014), purportedly supported by the observation that metabolic cost is proportional to gravity (Farley and McMahon, 1992). According to the best-fit model presented here, the net cost (Eqn 1) at optimal take-off velocity (Eqn 2) is expected to increase in proportion to gravitational acceleration [that is, E_tot(V*)∝g], as Farley and McMahon (1992) observed. The cost of vertical acceleration of the centre of mass can decrease as gravity is reduced only because the relationship between take-off velocity and swing cost changes; this allows the subject to settle on a lower stance cost, whose relationship to take-off velocity does not change as a function of gravity (Fig. 4). These trends can be explained simply from muscular work, and do not rely on any independent force-magnitude cost.

The model presented in this article is admittedly simple and makes unrealistic assumptions, including impulsive stance, no horizontal muscular force, non-distributed mass, and a simple relationship between step frequency and energetic cost. Further, horizontal accelerations will incur a larger portion of energetic losses as horizontal speed increases (Willems et al., 1995), and the trade-off between swing and stance costs may change. The present model would not be able to anticipate any such trend, as it has no dependence on horizontal speed. Future investigations could evaluate work-based costs using more advanced optimal control models (Srinivasan and Ruina, 2006; Hasaneini et al., 2013), eliminating some of these assumptions and allowing for an investigation into horizontal speed dependence. Despite its simplicity, the impulsive model with work-based swing cost is able to correctly predict the observed trends in take-off velocity with gravity, and demonstrates that understanding the energetic cost of both swing and stance is crucial to evaluating why the central nervous system selects specific running motions in different circumstances.

Although many running conditions are quite familiar, running in reduced gravity is outside our general experience. Surprisingly, releasing an individual from the downward force of gravity does not result in higher leaps between foot contacts. Rather, humans use less bouncy gaits with slow take-off velocities in reduced gravity, taking advantage of a reduced collisional cost while balancing a stride-frequency penalty.

Cost of swing work in partial reduced gravity

The experimental apparatus (Hasaneini et al., 2017 preprint) unloads a subject's centre of mass, but does not act directly on their limbs. Consequently, while their centre of mass might experience reduced weight, the limbs swing under the influence of normal gravity. It is prudent to check how this affects the predictions of the impulsive model.

Ballistic height corrections from the spring-mass model

We seek to predict the vertical centre-of-mass displacement achieved between take-off (maximum vertical velocity) and the maximum height during the flight phase. The vertical displacement from toe-off is v²/(2g), where v is the vertical velocity at toe-off. However, we do not know the displacement between take-off and toe-off, nor do we know how to relate the velocity at take-off to the velocity at toe-off. Both of these unknowns could be calculated using the ground reaction force during stance, but this was not measured empirically.

Instead, we can rely on the spring-mass model, which gives a decent approximation of the ground reaction forces assuming the velocity at toe-off and natural angular frequency (ω₀) are given (McMahon and Cheng, 1990). In our case, the toe-off velocity is unknown, but the spring-mass model allows us to relate it to the maximum vertical velocity, which can in turn be predicted by the impulsive model. ω₀ is defined as ⁠, where k is the ‘spring’ stiffness and m is mass. k is not actually the tendon stiffness, but is the virtual stiffness generated by the motor control system during stance (Farley and Ferris, 1998; Donelan and Kram, 2000); that is, the muscle and tendon forces combine to generate ground reaction forces as if there were one linear spring acting on the centre of mass. The complicated interplay between muscles, tendons and energetics makes the angular frequency hard to predict.

Fortunately, the vertical spring stiffness is held more-or-less constant through changes in gravity (He et al., 1991), so we can use the empirically derived value of ω₀∼18 rad s⁻¹. It remains simply to find the displacement between take-off and toe-off, and the vertical toe-off velocity, in terms of the vertical take-off velocity and gravity. [The value of ω₀ was calculated by taking the average value of vertical stiffness data in fig. 7 of He et al. (1991), dividing by average subject mass from the same study and taking the square root. ω₀=18 rad s⁻¹ falls within all the error bars of fig. 7, and so seems representative of the natural angular frequency at all levels of gravity.]

We thank Art Kuo, Jim Usherwood, David Lee and Allison Smith for comments on earlier drafts, as well as two anonymous reviewers, who provided constructive insights that greatly improved the manuscript.