Reducing gravity takes the bounce out of running

In gravity below Earth normal, a person should be able to take higher leaps in running. We asked ten subjects to run on a treadmill in five levels of simulated reduced gravity and optically tracked center of mass kinematics. Subjects consistently reduced ballistic height compared to running in normal gravity. We explain this trend by considering the vertical takeoff velocity (defined as maximum vertical velocity). Energetically optimal gaits should balance energetic costs of ground-contact collisions (favouring lower takeoff velocity), and step frequency penalties such as leg swing work (favouring higher takeoff velocity, but less so in reduced gravity). Measured vertical takeoff velocity scaled with the square root of gravitational acceleration, following energetic optimality predictions and explaining why ballistic height decreases in lower gravity. The success of work-based costs in predicting this behaviour challenges the notion that gait adaptation in reduced gravity results from an unloading of the stance phase. Only the relationship between takeoff velocity and swing cost changes in reduced gravity; the energetic cost of the down-to-up transition for a given vertical takeoff velocity does not change with gravity. Because lower gravity allows an elongated swing phase for a given takeoff velocity, the motor control system can relax the vertical momentum change in the stance phase, so reducing ballistic height, without great energetic penalty to leg swing work. While it may seem counterintuitive, using less “bouncy” gaits in reduced gravity is a strategy to reduce energetic costs, to which humans seem extremely sensitive. Summary Statement During running, humans take higher leaps in normal gravity than in reduced gravity, in order to optimally balance the competing costs of stance and leg-swing work.

vertical takeoff velocity 1 . Lost energy must be recovered through muscular work to maintain a periodic gait, 48 and so an energetically-optimal gait will minimize these losses. If center-of-mass kinetic energy loss were the 49 only source of energetic cost, then the optimal solution would always be to minimize vertical takeoff velocity. 50 However, such a scenario would require an infinite stepping frequency as V approaches zero (Alexander, 51 1992;Ruina et al., 2005), as step frequency (ignoring stance time and air resistance) is f = g/(2V ), where 52 g is gravitational acceleration.

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Let us suppose there is an energetic penalty that scales with step frequency, as E freq ∝ f k ∝ g k /V k , 54 where k > 0. Such a penalty may arise from work-based costs associated with swinging the leg, which 55 are frequency-dependent (k = 2;Alexander, 1992;Doke et al., 2005), or from short muscle burst durations 56 recruiting less efficient, fast-twitch muscle fibres (k ≈ 3; Kram and Taylor, 1990;Kuo, 2001). Notably, this 57 penalty increases with gravity, since the non-contact duration will be shorter for any given takeoff velocity 58 in higher gravity. The penalty also has minimal cost when V is maximal; smaller takeoff velocities require 59 more frequent steps, which is costly. Therefore, the two sources of cost act in opposite directions: collisional 60 loss promotes lower takeoff velocities, while frequency-based cost promotes higher takeoff velocities.

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If these two effects are additive, then it follows that the total cost per step is when ∂Etot ∂V = 0. Solving the latter equation for V yields 68 V * ∝ g k/(k+2) (2) 69 as the unique critical value. Here the asterisk denotes a predicted (optimal) value. Since E tot approaches 70 infinity as V approaches 0 and infinity (Eqn 1), the critical value must be the global minimum in the domain 71 V > 0. As k > 0, it follows from Eqn 2 that the energetically-optimal solution is to reduce the vertical 72 takeoff velocity as gravity decreases.

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The observation 2 of He et al. (1991) that V ∝ √ g implies k = 2, a finding consistent with frequency costs 74 arising from the work of swinging the limb (Alexander, 1992;Doke et al., 2005). However, their empirical 75 assessment of the relationship used a small sample size, with only four subjects. We tested the prediction of 76 Eqn 2 by measuring the maximum vertical velocity over each running stride, as a proxy for takeoff velocity, 77 in ten subjects using a harness that simulates reduced gravity. We also measured the maximum vertical 78 displacement in the ballistic phase to verify whether the counter-intuitive observation of lowered ballistic 79 center-of-mass height in hypogravity, as exemplified in Movie 1, is a consistent feature of reduced gravity 80 running.

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We asked ten healthy subjects to run on a treadmill for two minutes at 2 m s -1 in five different gravity levels the study protocol and informed consent was obtained from all subjects. Leg length for each subject was 89 measured during standing from the base of the shoe to the greater trochanter on one leg.

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Due to the unusual experience of running in reduced gravity, subjects were allowed to acclimate at their 91 leisure before indicating they were ready to begin each two-minute measurement trial. In each case, the 92 subject was asked to simply run in any way that felt comfortable. Data from 30 to 90 s from trial start 93 were analyzed, providing a buffer between acclimating to experimental conditions at trial start and initiating 94 slowdown at trial end.

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Implementation and measurement of reduced gravity 96 Gravity levels were chosen to span a broad range. Of particular interest were low gravities, at which the 97 model predicts unusual body trajectories. Thus, low levels of gravity were sampled more thoroughly than 98 others. The order in which gravity levels were tested was randomized for each subject, so as to minimize 99 sequence conditioning effects.

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For each gravity condition, the simulated gravity system was adjusted in order to modulate the force 101 pulling upward on the subject. In this particular harness, variations in spring force caused by support spring 102 stretch during cyclic loading over the stride were virtually eliminated using an intervening lever (see Figs 3 103 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. desired upward force, given subject weight and targeted effective gravity. Using this system, the standard 107 deviation of the upward force during a trial (averaged across all trials) was 3% of the subject's Earth-normal 108 body weight.

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Achieving exact target gravity levels was not possible since the lever's moment arm is limited by discrete 110 force increments (approximately 15 N). Thus, each subject received a slight variation of the targeted gravity 111 conditions, depending on their weight. A real-time data acquisition system allowed us to measure tension 112 forces at the gravity harness and calculate the effective gravity level at the beginning of each new condition.

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mounted to a C-shaped steel hook connecting the tensioned cable and harness. The strain gauge signal 115 was passed to a strain conditioning amplifier (National Instruments SCXI-1000 amp with SCXI-1520 8-

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Vertical takeoff velocities were identified as local maxima in the vertical velocity profile. Filtering and 130 differentiation errors occasionally resulted in some erroneous maxima being identified. To rectify this, first 131 any maxima within ten time steps of data boundaries were rejected. Second, the stride period was measured 132 as time between adjacent maxima. If any stride period was 25% lower than the median stride period or less,

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the maxima corresponding to that stride period were compared and the largest maximum kept, with the 134 other being rejected. This process was repeated until no outliers remained.

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Position data used to determine ballistic height were processed with a 4 th -order low-pass Butterworth 136 filter at 9 Hz cutoff. Ballistic height was defined as the vertical displacement from takeoff to the maximum 137 height within each stride. No outlier rejection was used to eliminate vertical position data peaks, since the 138 filtering was slight and no differentiation was required. If a takeoff could not be identified prior to the point 139 of maximum height within half the median stride time, the associated measurement of ballistic height was 140 rejected; this strategy prevented peaks from being associated with takeoff from a different stride.

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Statistical methods

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Takeoff velocities and ballistic heights were averaged across all gait cycles in each trial for each subject. To 143 test whether ballistic height varied with gravity, a linear model between ballistic height and gravitational 144 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. Since the proportionality coefficient between V * and √ g is unknown a priori, we derived its value from a least 147 squares best fit of measured vertical takeoff velocity against the square root of gravitational acceleration, 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 impulsive model with k = 2 predicts that the ballistic height should remain constant (dash line in Fig.   172 2A). This constant value agrees with empirical data at low g, but exhibits increasing error towards normal 173 g.

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We defined "takeoff" as occurring when the net force on the body was null and velocity was maximal; 175 however, this does not equate to the moment when the stance foot leaves the ground. After the point 176 of maximal velocity, upward ground reaction forces decay to zero. During this time, the net downward 177 acceleration on the body is less than gravitational acceleration. Thus, the body travels higher than would 178 be expected if maximal velocity corresponded exactly to the point where the body entered a true ballistic 179 phase, as in the model (Fig. 1). 180 We can account for the missing impulse with the spring-mass model. This model describes the kinematics 181 and dynamics of running well (McMahon and Cheng, 1990;He et al., 1991;Blickhan and Full, 1993), and 182 3 As long as stride frequency is greater than natural frequency, which is very likely the case for the present study (Appendix A) provides a way to estimate stance time from takeoff velocity (though it lacks the ability to predict takeoff velocity; McMahon and Cheng, 1990). Notably, correcting the prediction V ∝ √ g with spring-mass model estimates of finite stance yields the following relationship for ballistic height (Appendix B): where ω 0 is the natural angular frequency of vertical oscillation, and A is a constant in the relationship 186 V = A √ g. Note that Eqn 3 is linear in g, and approaches the predictions from the impulsive model alone  The present results indicate that the cost of step frequency is a key factor in locomotion. Although the 216 exact value of the optimal takeoff velocity depends on both frequency-based penalties and collisional costs, 217 the former penalties change with gravity while the latter do not (Fig. 4). The collisional cost landscape 218 is independent of gravity because the final vertical landing velocity is alone responsible for the lost energy.

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Regardless of gravitational acceleration, vertical landing speed must equal vertical takeoff speed in the model; 220 so a particular takeoff 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; 222 thus, the frequency-based cost curves are decreased as gravity decreases (Fig. 4). Frequency-based costs, 223 particularly limb-swing work, appear to be an important determinant of the effective movement strategies 224 available to the motor control system. Their apparent influence warrants further investigation into the extent 225 of their contribution to metabolic expenditure.

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While the present study corroborates others in finding that a work-based cost (k = 2) predicts locomotion 227 well (Alexander, 1980(Alexander, , 1992Hasaneini et al., 2013), other authors have favoured a higher-order "force/time" the relationship between takeoff velocity and swing cost changes; this allows the subject to settle on a lower 240 stance cost, whose relationship to takeoff velocity does not change as a function of gravity (Fig. 4). These 241 trends can be explained simply from muscular work, and do not rely on any independent force-magnitude 242 cost.

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The model presented in this article is admittedly simple and makes unrealistic assumptions, including 244 impulsive stance, no horizontal muscular force, non-distributed mass, and a simple relationship between The dataset supporting this article has been uploaded as part of the supplementary material (Table S1).
He, J. P., Kram, R. and McMahon, T. A. (1991). Mechanics of running under simulated low gravity. In the impulsive model of running, a point mass bounces off vertical, massless legs during an infinitesimal stance phase. As the horizontal velocity U is conserved, the vertical takeoff velocity V dictates the step frequency and stride length. Smaller takeoff velocities (light grey) result in more frequent steps that incur an energetic penalty, while larger takeoff velocities (dark grey) reduce the frequency penalty but increase losses during stance. The small box represents a short time around stance that is expanded in panel B. (B) We assume that the center-of-mass speed at landing is equal to the takeoff speed. The vertical velocity V and its associated kinetic energy are lost during an impulsive foot-ground collision of infinitessimally short duration. The lost energy must be resupplied through muscular work. Horizontal acceleration is assumed small and is neglected in the model. . Each data point is a mean value measured in one subject (ten subjects total) across multiple steps (n ≥ 50) during a one-minute period at a given gravity level. For both panels, if error bars (twice the s.e.m.) are smaller than the markers, then they are not shown. Data used for creating these graphics are given in Table S1.  Figure 3: A log-log plot of vertical takeoff velocity against gravitational acceleration shows that the impulsive model yields the best fit when E freq ∝ f 2 . The least squares linear fit is shown in red as a solid line, with 95% confidence interval as a grey area. The linear fit exhibits R 2 = 0.70 and a slope of 0.47 ± 0.09 (best estimate ± 95% CI, N = 50), which is not significantly different from the predicted slope of 0.5 for k = 2 (black solid line), where k is the exponent relating frequency to cost (E freq ∝ f k ). Both k = 1 and k = 3 (shallow and steep dashed lines, respectively) yield predicted slopes (0.33 and 0.60, respectively) that lie outside the 95% CI, indicating that a work-based swing cost at k = 2 is a superior fit to the data, while a simple linear frequency cost (k = 1) and an approximate force/time cost (k = 3, see Kuo 2001) do not represent these data well. Data points are from ten subjects running at five gravity conditions each, and each point is the mean of at least 64 takeoffs measured during each trial.  Fig. 2B. Labels of gravity levels (g) are placed over the colours they represent. The collisional cost curve (E col = mV 2 /2, black dot-dash line) does not change with gravity, while the frequency-based energetic cost curve (E freq , dotted lines) is sensitive to gravity, leading to an effect on total energy per step (E tot , solid lines). In lower gravity, a runner can stay in the air longer for a given takeoff velocity, so the associated frequency-based cost goes down. However, the cost of collisions at that same velocity is unchanged, since it depends only on the velocity itself. The relaxation of frequency-based cost allows the runner settle on a lower, optimal takeoff velocity (yellow stars) with both a lower frequency-based and collisional cost, compared to higher gravity.  Figure A1: Vertical takeoff velocity scales with the square root of gravitational acceleration times leg length during running. The least squares fit for the model given by Eqn A4 is shown as a dashed line. The fit exhibits R 2 = 0.745, using all fifty data points. Error bars (twice standard error of the mean takeoff velocity measured during a trial, n ≥ 64) are smaller than the marker size.

Figure legends
Appendices not act directly on their limbs. Consequently, while their center of mass might experience reduced weight, 389 the limbs swing under the influence of normal gravity. It is prudent to check how this affects the predictions 390 of the impulsive model.

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The work required to swing a limb is (Doke et al., 2005) where I is the moment of inertia of the limb about the hip, f is the frequency of oscillation and f n is the 392 natural frequency (equal to g/l for a simple pendulum, where l is leg length Therefore, the assumption that f > f n very likely holds in this case.

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The leg moment of inertia about the hip scales approximately as I ∝ ml 2 , where m is body mass and l 402 is the leg length (Winter, 2009). Assuming f = g/(2V ), and invoking E tot = E col + E swing , we have where B is some proportionality constant. To achieve the energetically optimal takeoff velocity, we take the 404 derivative of Eqn A2 with respect to V , yielding 405 where we note that any dependence on f n has disappeared. However, there is a new dependence on l. Solving

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Eqn A3 for V * , we find Empirical V is plotted against gl in Fig. A1  B Ballistic height corrections from the spring-mass model 414 We seek to predict the vertical center-of-mass displacement achieved between takeoff (maximum vertical 415 velocity) and the maximum height during the flight phase. We know the maximum height from ballistics to 416 be v 2 /(2g), where v is the vertical velocity at toe-off. However, we do not know the displacement between 417 takeoff and toe off, nor do we know how to relate the velocity at takeoff to the velocity at toe-off. Both of 418 these unknowns could be calculated using the ground reaction force during stance, but this was not measured 419 empirically.

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Instead, we can rely on the spring-mass model, which gives a decent approximation of the ground reaction 421 forces assuming the velocity at toe-off and natural angular frequency (ω 0 ) are given (McMahon and Cheng, 422 1990). In our case, the toe-off velocity is unknown, but the spring-mass model allows us to relate it to the 423 maximum vertical velocity, which can in turn be predicted by the impulsive model. ω 0 is defined as k/m,

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where k is the "spring" stiffness and m is mass. k is not actually the tendon stiffness, but is the virtual 425 stiffness generated by the motor control system during stance (Farley and Ferris, 1998;Donelan and Kram, 426 2000); that is, the muscle and tendon forces combine to generate ground reaction forces as if there were 427 one linear spring acting on the center of mass. The complicated interplay between muscles, tendons and 428 energetics makes the angular frequency hard to predict.

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Fortunately, the vertical spring stiffness is held more-or-less constant through changes in gravity (He 430 et al., 1991), so we can use the empirically derived value 5 of ω 0 ∼ 18 rad s -1 . It remains simply to find the 431 displacement between takeoff and toe-off, and the vertical toe-off velocity, in terms of the vertical takeoff 432 velocity and gravity. 433 We follow McMahon and Cheng (1990) in assuming a point-mass body of mass m and massless legs. 434 We assume that the ground reaction force is well-approximated by the compression of a spring with angular 435 frequency ω 0 . For simplicity, we use a hopping model, which assumes that a person exhibits a small excursion 436 angle (i.e. θ ∼ 0). The leg length minus resting length is r, and so the dynamics of the system arë r + ω 2 0 r + g = 0,

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The instantaneous velocity is thusṙ

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Eqns B1-B3 are valid for 0 ≤ t ≤ t s , where t s = (2π − 2 arctan(Gr))/ω 0 is the stance period, and 441 we have introduced the non-dimensional Groucho number Gr ≡ vω 0 /g (McMahon and Cheng, 1990). For 442 t s < t < t s + 2v/g, the body is in a ballistic phase. 443 We can now determine the timing and magnitude of the peak vertical velocity. Let t * correspond to any 444 time at which a maximum speed is acheived. Since Eqn B3 is continuous and periodic, local maxima and 445 minima in velocity must satisfyr = 0. Therefore, from Eqn B1, Combining Eqn B4 with B2 and solving for 0 ≤ t * ≤ t s yields t * = arctan(Gr −1 ) + nπ /ω 0 , n = 0, 1 447 corresponding to the points of maximal speed during stance. The second point (n = 1), corresponds to the 448 time at which maximal velocity is achieved, t m = arctan(Gr −1 ) + π /ω 0 . (B5)

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In the main manuscript, we define the ballistic height (H) as the vertical displacement from the time of 454 maximal vertical velocity to the maximum height achieved during a stride, that is, 455 We need only insert Eqns B4 and B6 into Eqn B7 to find .

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Note that the first term is identical to the prediction of the impulsive model (i.e. V = v), while the second 457 term gives a correction from the spring mass model, due to finite stance time. Since we have established 458 that V = A √ g, the prediction for H in terms of g alone is