Abstract
Humans can run without falling down, usually despite uneven terrain or occasional pushes. Even without such external perturbations, intrinsic sources like sensorimotor noise perturb the running motion incessantly, making each step variable. Here, using simple and generalizable models, we show that even such small steptostep variability contains considerable information about strategies used to run stably. Deviations in the center of mass motion predict the corrective strategies during the next stance, well in advance of foot touchdown. Horizontal motion is stabilized by total leg impulse modulations, whereas the vertical motion is stabilized by differentially modulating the impulse within stance. We implement these humanderived control strategies on a simple computational biped, showing that it runs stably for hundreds of steps despite incessant noiselike perturbations or larger discrete perturbations. This running controller derived from natural variability echoes behaviors observed in previous animal and robot studies.
https://doi.org/10.7554/eLife.38371.001eLife digest
Running at a constant speed seems like a series of repetitive, identical strides, but it is not. There are small variations in each stride. Selfinflicted errors in the forces generated by the muscles, or misperceptions from the senses, may cause these tiny imperfections. Uneven terrain or other outside forces, like a push, can also cause changes in a running stride. People must correct for these small changes as they run to avoid falling down. The only way to correct errors in a stride is by changing the force exerted on the ground by the leg.
Now, Seethapathi and Srinivasan document stepbystep how people correct for small imperfections in their running stride to avoid falling. In the experiments, eight people ran on a treadmill at three different speeds while the motion of their torso and each foot was measured along with the forces of each foot on the treadmill. Seethapathi and Srinivasan found that these runners corrected for minor deviations by changing where each foot lands and how much force each leg applies to the treadmill. The runners placed their foot at a different position on each step and these varying foot positions could be predicted by the errors in the body movement between steps. These errors in body movement could also be used to predict how the runners would change the forces applied by their legs on each step. Imperfections in the stride were mostly corrected within the next step. Errors that would cause the runner to fall to the side were corrected more quickly than errors in forward or backward motion. Seethapathi and Srinivasan incorporated these corrective strategies into a computer simulation of a runner, resulting in a simulated runner that did not fall even when pushed.
These findings may inform the design of robots that run more like humans. They may also help create robotic exoskeletons, prosthetic legs and other assistive devices that help people with disabilities move more fluidly and avoid falling.
https://doi.org/10.7554/eLife.38371.002Introduction
Human running is often modeled as being periodic (Blickhan and Full, 1993; Seyfarth et al., 2002; Srinivasan and Holmes, 2008). But running is not exactly periodic, even on a treadmill at constant speed. Body motion during running varies from step to step (Cavanagh et al., 1977; Belli et al., 1995; Jordan et al., 2007; Jordan and Newell, 2008). This steptostep variability could be due to internal perturbative sources like muscle force noise and sensory noise (Warren et al., 1986; Harris and Wolpert, 1998; Osborne et al., 2005) or small external perturbations (e.g. visual field inhomogeneity, small ground imperfections). To run without falling, the body’s ‘running controller’ must prevent the effects of these small perturbations from growing too large. Here, we provide an experimentally derived lowdimensional characterization of this control that reveals how humans run without falling down.
One classic modeling paradigm for running control assumes that the human leg behaves like a linear spring (Blickhan, 1989; McMahon and Cheng, 1990; Blickhan and Full, 1993). This paradigm has been used to argue how passiveelastic properties may reduce muscle work needed for locomotion (Alexander and Vernon, 1975; Alexander, 1990) and has been useful in examining locomotion in a simplified setting. Variants of these springmass running models have demonstrated stable running (Seyfarth et al., 2002; Seipel and Holmes, 2005; Ghigliazza et al., 2005; Geyer et al., 2006; Srinivasan and Holmes, 2008; Englsberger et al., 2016). These models have been successful in fitting the average center of mass motion during running (Blickhan and Full, 1993; Geyer et al., 2006; Srinivasan and Holmes, 2008). However, understanding running stability requires understanding how deviations from the average motion are controlled. It has been previously recognized that springlike leg mechanics cannot explain how deviations from the average motion are controlled and eventually attenuated (e.g. Ghigliazza et al., 2005; Biewener and Daley, 2007; Maus et al., 2015). Here, we examine the role of active muscle control in running stability, using more general models of human locomotion rooted in Newtonian mechanics (Srinivasan, 2011).
One way of characterizing the running controller is to apply perturbations (for instance, pushes or pulls or sudden changes in terrain) and examine how the body recovers from the perturbations (Van Woensel and Cavanagh, 1992; Daley and Biewener, 2006; Qiao and Jindrich, 2014; Riddick and Kuo, 2016). Instead of such external perturbations, here, we use the naturally occurring steptostep variability (Hurmuzlu and Basdogan, 1994; Maus et al., 2015) to characterize the controller. Previous attempts at examining such variability for controller information focused only on walking (Hurmuzlu and Basdogan, 1994; Wang and Srinivasan, 2012; Wang, 2013; Wang and Srinivasan, 2014) or considered variants of the springmass model (Maus et al., 2015). Here, we directly characterize the control in terms of how humans modulate their leg force magnitude and direction during running. The only way to control the center of mass motion is for the leg to systematically change the forces and the impulses it applies on the ground. We uncover how such center of mass control is achieved. We then implement this humanderived controller on a simple mathematical model of a biped (Srinivasan, 2011), showing that this biped model runs without falling down, despite incessant noiselike perturbations, large external perturbations, and on uneven terrain.
A humanderived controller such as the one proposed here could inform monitoring devices to quantify running stability or fall likelihood (O'Loughlin et al., 1993), or could help understand running movement disorders. Further, implementing such controllers into robotic prostheses and exoskeletons (Dollar and Herr, 2008; Shultz et al., 2015) will allow the human body to interact more ‘naturally’ with the device, rather than having to compensate for an unnatural controller. Some running robots have demonstrated stable running, using a variety of control schemes (Raibert, 1986; Chevallereau et al., 2005; Tajima et al., 2009; Nelson et al., 2019). But these robots fall short of human performance and versatility. Understanding human running may lead to better running robots.
Results
The steptostep variability during running appears superficially random. We show that this variability contains lowdimensional structure, specifically containing information about control strategies involved in running stably. We implement these strategies into a feedback controller, thereby stabilizing a simple mathematical model of a biped, and make further predictions.
We measured body motion and ground reaction forces (GRFs) of humans running for hundreds of steps on a treadmill at three speeds: 2.5, 2.7 and 2.9 m/s (see Materials and methods). Each running step consists of a ‘flight phase’ with neither foot on the ground, and a ‘stance phase’ with one foot on the ground. Figure 1 shows the coordinate system and sign convention: $x$ is sideways, $y$ is foreaft, and $z$ is vertical. The results we present are for the highest running speed and we discuss speeddependence of our results separately. Unless otherwise specified, all quantities and results in equations and figures are nondimensionalized using body mass $m$, acceleration due to gravity $g$, and leg length ${\mathrm{\ell}}_{\mathrm{max}}$. Forces are normalized by $mg$, distances by ${\mathrm{\ell}}_{\mathrm{max}}$, speeds by $\sqrt{g{\mathrm{\ell}}_{\mathrm{max}}}$, time by $\sqrt{{\mathrm{\ell}}_{\mathrm{max}}/g}$, impulses by $m\sqrt{g{\mathrm{\ell}}_{\mathrm{max}}}$, etc.
A hypothesized controller structure for stable running
During flight phase, the body center of mass moves in a nearly parabolic trajectory and the runner has no control over this parabolic motion (as the aerodynamic forces generated by the person are negligible, unlike birds). From Newton’s second law, it follows that the only way to control the center of mass motion is to modulate the total ground reaction force components during stance phase, when the leg is in contact with the ground. However, there are infinitely many ways to modulate the ground reaction forces to control the center of mass motion. Here, we examine how the ground reaction forces are modulated in response to center of mass state deviations during the previous flight apex (Figure 1). A flight apex is defined as when the center of mass height $z$ is maximum. The center of mass position and velocity at flight apex are denoted by $({x}_{a},{y}_{a},{z}_{a})$ and $({\dot{x}}_{a},{\dot{y}}_{a},{\dot{z}}_{a})$, respectively. Because the vertical velocity at flight apex ${\dot{z}}_{a}=0$ by definition, ${\dot{z}}_{a}$ is not considered as an explanatory variable. The absolute horizontal position $({x}_{a},{y}_{a})$ on the treadmill changes with a much slower timescale than other variables. Therefore, our default set of explanatory variables is $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$. We will include the horizontal position $({x}_{a},{y}_{a})$ when we comment later on ‘station keeping’.
Foreaft and sideways impulses independently control center of mass motion
The steptostep variability in the center of mass state at flight apex over hundreds of steps is shown in Figure 2a. To be stable, the runner needs to prevent this motion variability from growing without bound. As noted, the only way to control this motion is by using the ground reaction forces (GRFs). Consequently, the ground reaction force components over the stance phase are also variable (Figure 2b).
The net effect of the ground reaction forces on the center of mass velocity over a stance phase is captured by the force impulse, namely, the integral of the force. The variability in the sideways and foreaft ground reaction impulses over a step (Figure 2c) are wellpredicted by the variability in the center of mass state $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$ at the previous flight apex (Figure 3). Moreover, the sideways impulse depends primarily on the sideways velocity ${\dot{x}}_{a}$ and the foreaft impulse depends primarily on the foreaft velocity ${\dot{y}}_{a}$. Thus, it appears that the control in the foreaft and sideways directions are independent or decoupled. Pooled over all subjects, the bestfit linear model for the sideways impulse ${P}_{x}$ is:
and that for the foreaft GRF impulse ${P}_{y}$ is:
as in Figure 3. All coefficients in Equations (1) and (2) are significant at $p\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}{10}^{4}$. Both sideways and foreaft impulses depend negligibly on vertical position deviations, so that including $z}_{a$ in the regression increases the $R}^{2$ values by less than $0.02$.
Almost deadbeat: impulses correct horizontal velocity mostly within a step
The linear models for the foreaft and sideways impulses in Equations (1) and (2) have a simple interpretation. The $\mathrm{\Delta}{\dot{x}}_{a}$ coefficient of about $1$ in Equation (1) (that is, $\mathrm{\Delta}{P}_{x}\approx \mathrm{\Delta}{\dot{x}}_{a}$) implies that sideways velocity deviations are completely corrected in one step, on average (over all steps and all subjects). This correction could have been done over many steps, as would be the case if the coefficient were $0.5$, say. But humans seem to exhibit a ‘onestep deadbeat controller’ on average for sideways velocity deviations (the term deadbeat control refers to when state deviations decay to zero in a finite amount of time). Of course, this singlestep correction is not perfect. An ${R}^{2}$ value of about $0.55$ suggests that the system overcorrects or undercorrects deviations for any given step.
Analogously, the coefficient of $0.72$ in Equation (2) suggests that about 72% of a forward velocity deviation is corrected in a single step, on average. While this is not strictly ‘deadbeat control’, it results in only ${(10.72)}^{2}=0.08$ of a perturbation remaining after two steps, and ${(10.72)}^{3}=0.02$ of a perturbation after three steps, indicating rapid control.
Apextoapex maps also show fast decay of center of mass deviations
We corroborate the above findings regarding perturbation decay with the ‘apextoapex maps’: that is, linear models that describe the relation between deviations in the state at one flight apex and those at the next flight apex. The righttoleft map from the state ${S}_{\mathrm{right}}={[{\dot{x}}_{a},{\dot{y}}_{a},{z}_{a}]}_{\mathrm{right}}$ at an apex preceding a right stance to the state at the next flight apex (preceding a left stance) is, approximately:
where the superscript $}^{\ast$ indicates that the coefficient is not significantly different from zero ($p\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}0.05$). The lefttoright matrix ${K}_{\mathrm{L}\to \mathrm{R}}$ is similar to ${K}_{\mathrm{R}\to \mathrm{L}}$, except for the sign changes due to mirrorsymmetry. The matrix product of ${K}_{\mathrm{L}\to \mathrm{R}}$ and ${K}_{\mathrm{R}\to \mathrm{L}}$ — Jacobians of the Poincare map (Hurmuzlu and Basdogan, 1994; Guckenheimer and Holmes, 2013; Maus et al., 2015) — quantify how apex state deviations grow or decay over one stride (two steps). The eigenvalues of this matrix product were all less than one in absolute value, indicating a stable periodic motion. The largest eigenvalue was $0.14$, indicating that at most $14$% of a perturbation remains after a stride on average in any direction.
The low value of ${K}_{\mathrm{R}\to \mathrm{L}}(1,1)$, not significantly different from zero, suggests that a purely sideways velocity perturbation gets corrected essentially over one step on average, consistent with the sideways impulse control (Equation 1). Similarly, the value ${K}_{\mathrm{R}\to \mathrm{L}}(2,2)=0.27$ suggests that 73% of a forward velocity deviation is corrected in one step, consistent with the foreaft impulse control (Equation 2). Finally, the $(3,3)$ element of the step map (Equation 3) suggests that less than 50% of a deviation in vertical position ${z}_{a}$ remains after a step. See (Maus et al., 2015) for a detailed Floquet analysis of human running including more state variables, complementing the simplified version here.
Withinstep vertical impulse modulations control vertical position
The control of vertical position is qualitatively different from that of control in the foreaft and sideways directions, as we cannot use net vertical impulse for vertical position control due to the impulsemomentum considerations below. A flight apex occurs when the center of mass vertical velocity is zero. So, the net vertical impulse between two consecutive flight apexes is also zero (as it equals the change in vertical momentum, according to the impulsemomentum equation). Therefore, changing the net vertical impulse over a stance phase will not accomplish any meaningful control in the vertical direction. However, we will show that by differentially modulating the vertical impulse within one stance phase, we can change the vertical position (${z}_{a}$) from one flight apex to the next, without changing the net impulse.
To show this most simply, consider infinitesimal flight phases and a stance phase from $t=0$ to $t={T}_{\mathrm{step}}$. The total impulse ${P}_{z}$ due to the vertical ground reaction force ${F}_{z}(t)$ equals that due to gravity, which is given by, ${P}_{z}={\int}_{0}^{{T}_{\mathrm{step}}}{F}_{z}(t)\mathit{d}t={\int}_{0}^{{T}_{\mathrm{step}}}mg\mathit{d}t=mg{T}_{\mathrm{step}}$. For a triangular stance force (Figure 4) with peak force ${F}_{\mathrm{peak}}$ at ${t}_{\mathrm{peak}}$, we get ${F}_{\mathrm{peak}}=2mg$. Then, by integrating the vertical acceleration $({F}_{z}/mg)$ twice, the change in vertical position $z({T}_{\mathrm{step}})z(0)$ over a step is given by:
If the step was symmetric about midstance (${t}_{\mathrm{peak}}={T}_{\mathrm{step}}/2$), there is no vertical position change over a step ($z({T}_{\mathrm{step}})=z(0)$). The flight apex vertical position on the next step $z({T}_{\mathrm{step}})$ can be changed by changing ${t}_{\mathrm{peak}}$ relative to ${T}_{\mathrm{step}}/2$ (Figure 4). For example, if $z(0)$ at one flight phase was greater than its nominal value and the runner wishes to reduce it, this simple model predicts that the runner will decrease the firsthalf impulse and increase the secondhalf impulse; doing this is equivalent to delaying ${t}_{\mathrm{peak}}$ relative to ${T}_{\mathrm{step}}/2$ (as in Figure 4). This prediction is in agreement with the following experimentallyderived linear relations for the first half vertical impulse from $t=0$ to ${T}_{\mathrm{step}}/2$, namely ${\mathrm{\Delta}{P}_{z}}_{0}^{{T}_{\mathrm{step}}/2}$, and the second half vertical impulse from $t={T}_{\mathrm{step}}/2$ to ${T}_{\mathrm{step}}$, namely ${\mathrm{\Delta}{P}_{z}}_{{T}_{\mathrm{step}}/2}^{{T}_{\mathrm{step}}}$:
We see that a positive $\mathrm{\Delta}{z}_{a}$ corresponds to a decrease in the firsthalf vertical impulse and an increase in the second half vertical impulse. In addition to the vertical impulse, the landing leg length is also modulated in response to vertical flight apex deviations. Regressing the leg length $\mathrm{\ell}$ at the beginning of stance with the flight apex state, we found that this landing leg length is mostly a function of the vertical position at flight apex:
Thus, a downward position deviation at flight apex would result in landing with a shorter leg length than nominal (e.g. via flexed knee or ankle). A downward position deviation is analogous to a sudden stepup perturbation, so reducing the landing leg length reduces trip likelihood.
Impulse control is achieved by phasedependent force modulations
The linear models above tell us how deviations from nominal motion at flight apex are corrected grossly over the next stance. But they do not tell us how the forces are modified continuously throughout a stance phase. The variability of the GRF components $({F}_{x},{F}_{y},{F}_{z})$ depend on the ‘phase’ of the stride cycle, specifically, the time fraction ${\varphi}_{\mathrm{stance}}$ of stance (Figure 2b). To explain this phasedependent force variability within a single step, we compute the phase dependent sensitivity of $({F}_{x},{F}_{y},{F}_{z})$ to the center of mass state as follows. For each output, say ${F}_{x}$, we divide the stance duration into 20 phases and compute a linear model for ${F}_{x}$ at each of those phases, all with $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$ as inputs. We refer to the coefficients in these linear models as a function of the phase ${\varphi}_{\mathrm{stance}}$ as the phasedependent sensitivities of the GRFs (Figure 5) to the corresponding predictor variable in $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$.
The phasedependent sensitivity of sideways GRF to ${\dot{x}}_{a}$ shows that ${F}_{x}$ is decreased over the whole step to correct a positive sideways velocity deviation at flight and that a majority of this correction occurs during the middle of stance (Figure 5a). Similarly, in response to a positive foreaft velocity perturbation, the foreaft GRF is modulated so that there is a net negative force on the body over the next step (Figure 5b). The sensitivity of the foreaft force ${F}_{y}$ is more in the first half of stance than during the second half of stance, being modulated more during the deceleration phase (roughly ${\varphi}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}0.5$) than during the acceleration phase (roughly ${\varphi}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}0.5$).
Foot placement control: step in the direction of the fall
Placing the foot relative to the body allows a runner to modulate the leg force direction and thus the GRF components. The foot position $({x}_{f},{y}_{f})$ relative to center of mass position at the beginning of stance phase $({x}_{s},{y}_{s})$ is predicted by the previous flight apex state as described by the following equations. Specifically, sideways foot placement is described by the following equations:
The foreaft foot placement is described by the following equations:
That is, a sideways velocity perturbation to the body results in the foot being placed further along the direction of the perturbation. So, a rightward perturbation results in a more rightward step. Analogously, a forward velocity perturbation results in the foot being placed further forward relative to the body. As with the impulses, again, there is no significant coupling between sideways and foreaft directions. Foreaft foot placement modulation also depends on vertical position deviations, in a manner that the runner lands with a steeper leg when landing from a higher flight apex ${z}_{a}$. Such dependence of landing leg angle on vertical position is analogous to behavior in terrainchange experiments (Daley and Biewener, 2006; Müller et al., 2012; Qiao and Jindrich, 2012; BirnJeffery and Daley, 2012), as discussed in detail later. We speculate that using foot placement based on center of mass state may be an efficient way to affect the center of mass motion, compared to, say, changing the leg force magnitudes and leg joint torques after the foot touches down (Clark, 2018).
Swing foot repositioning happens during flight, just before foot touchdown
One possibility is that the foot placement deviations are achieved early on during the swing phase and this deviation is preserved during swing until the foot touchdown. However, this does not appear to be the case. Figure 6 shows the fraction of foot placement variance predicted by the swing foot state over the previous step. Less than 10% of the eventual foot placement is predicted by the swing foot at the beginning of flight phase (Figure 6). The explanatory power of the swing foot rises rapidly during the flight phase from less than 10% to a 100% when it becomes the next stance foot, suggesting that most swing foot repositioning may happen during this flight phase.
Center of mass state predicts future foot placement before the foot state does
At the beginning of flight phase (and earlier), the center of mass state is a vastly better predictor of the next foot placement than the swing foot itself (Figure 6). We can predict the foot placement using the center of mass state better than just the relative swing foot state until about 100 ms before foot touchdown. The explanatory power of the center of mass remains flat during flight. This flatness is likely because center of mass state follows a parabolic path during flight and thus accumulates no new variation. This lag between the explanatory power of the center of mass and the foot suggests that the error information in the center of mass state is yet to be fully incorporated into the swing foot repositioning until the flight phase. During the brief flight phase, when the swing foot’s explanatory power increases, information from center of mass state is transferred to the foot, presumably via some mixture of feedback control and feedforward dynamics.
Continuous stance state feedback, stationkeeping, and running speed do not significantly affect stance control
As an alternative to control based on discrete monitoring of deviations at the previous flight apex state, we considered a ‘continuous control’ model. Specifically, we obtained linear models for the GRFs based on the current center of mass state during stance $(\dot{x},\dot{y},z)$. These linear models did not differ significantly in the fraction of GRF variance explained, compared to the apexbased control model ($p=0.94$). In the linear models above, adding the sideways and foreaft apex body position $({x}_{a},{y}_{a})$ to the explanatory variables improves the ${R}^{2}$ values by less than $1.5\%$. Thus, the runners did not prioritize controlling their position relative to the treadmill (stationkeeping). Further, the regression coefficients for $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$ did not vary significantly across the three running speeds ($p\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}0.05$, paired ttest).
Approximate leftright symmetry in the control
The running control gains have approximate bilateral (leftright) symmetry. The gains that couple sideways direction variables and either foreaft or vertical direction variables have mirrorsymmetry (see Equations 1,2,8,9). That is, these gains for the left leg’s stance are the negatives of corresponding gains for the right leg’s stance. On the other hand, gains that couple one sideways variable with another sideways variable, or one foreaft variable with another foreaft variable, are the same for the left and right legs without any such sign changes. This mirror symmetry in running control likely follows from the approximate mirror symmetry in body physiology about the sagittal plane and was also found in walking (Wang and Srinivasan, 2014; Ankaralı et al., 2015). This symmetry suggests the lack of a substantially dominant limb for running control, in contrast to the asymmetry and limb role differentiation that occurs in some other tasks (Peters, 1988).
Simple models of running capture GRFs and phasedependent GRF modulations
We now show that the experimentally derived control strategies described above are sufficient to control the running dynamics of a simple mathematical model of a biped. We consider a biped with pointmass upper body and massless telescoping legs capable of generating arbitrary force profiles (unlike a spring). We considered two versions of this biped model (Figure 7a), one with direct control of the leg force and another that produces leg forces via Hilltype muscles (Figure 7b). See Materials and methods for how the nominal running motion and the feedback controllers are specified for the models.
We find that the models’ ground reaction forces are similar to experimental data despite not explicitly matching the curves (Figure 8a). Further, we find that the phasedependent ground reaction force feedback gains for the models are qualitatively similar to the phasedependent gains inferred from experiment (Figure 8b), again, despite not explicitly fitting the shape of these phasedependent gains. This shows that these simple models can not only capture the average motion during running, but also how the runner responds to deviations from the average motion.
Humanderived controller stabilizes a minimal model of bipedal running
The simple models’ running motions are not stable without the controller: an arbitrarily small perturbation makes it diverge from the original running motion. With the foot placement and leg force controller turned on, the running motion is asymptotically stable. Figure 9a–d shows the model recovering from foreaft, sideways, and vertical perturbations at flight apex. It is a mathematical theorem that a stable periodic motion that can reject perturbations at one phase (say, flight apex) can reject perturbations at any phase (Guckenheimer and Holmes, 2013). So, it follows that our model rejects perturbations at any phase.
The inputs to the feedback controller (${\dot{x}}_{a},{\dot{y}}_{a},{z}_{a}$) do not include the absolute sideways and foreaft position $({x}_{a},{y}_{a})$ of the runner. Therefore, the controller does not correct position perturbations (stationkeeping). A sideways or foreaft push to the model results in convergence to the nominal running motion, except for a sideways or foreaft position offset (Figure 9c).
Nonzero leg work for energychanging perturbations
Figure 10 illustrates the leg workloop for the unperturbed run (net zero work) and when positive perturbations are applied to sideways and foreaft velocities, and vertical positions. All such positive perturbations result in net negative work on the first step after the perturbation, reflected in the workloops with net negative area within them. Such net positive or negative leg work is clearly necessary to recover from perturbations that change the total mechanical energy of the runner, as was recognized in prior discussions of the energyneutral springmass model of running (Ghigliazza et al., 2005; Biewener and Daley, 2007; Srinivasan and Holmes, 2008).
Explaining variability: muscledriven running model does not fall despite noise
To simulate the steptostep variability in real human running, we added ‘noise’ to our foot placement and leg forces (for the direct force control model) or muscle activations (for the muscledriven model) and simulated the biped models for a few hundred steps. This noise is meant to model the phenomenon that intended muscle forces tend to deviate from actual muscle forces due to motor noise (Harris and Wolpert, 1998). We find that while the direct leg force control model falls down, the runner with muscles does not fall down for hundreds of steps despite the noise. The stable motion of the center of mass in the presence of noiselike perturbations is shown in Figure 11a. The variability in the center of mass state at flight apex for the model (Figure 11b) as a result of the simulated noisy control is qualitatively similar to the variability found in experiment (Figure 2a). The model is also able to run without falling despite vertical position perturbations at flight apex, which are equivalent to uneven terrain. Thus, even though the model was derived using data on horizontal ground, it is capable of running robustly on uneven terrain. The muscledriven model is robust to motor noise presumably because of the intrinsic stabilizing properties of forcelength and forcevelocity relations (Hogan, 1984; Jindrich and Full, 2002).
Discussion
We have mined the steptostep variability in human running to show how humans modulate leg forces and foot placement to run stably. We then used these dataderived control strategies on a biped model, demonstrating robustness to discrete perturbations and persistent motor noise.
We have shown that humans use foot placement or leg angle control in a manner that they step in the direction of the perturbation, thereby directing the leg force so as to oppose the perturbation. This result provides an empirical basis for ad hoc assumptions about leg angle control made in previous running models (Seyfarth, 2003; Ghigliazza et al., 2005; Peuker et al., 2012). The foot placement controller derived from running data is qualitatively similar to the classic Raibertlike controller used in early running robots (Raibert, 1986) in that the foot placement opposes velocity deviations with no sidewaysforeaft coupling, but differs in that it has a dependence on vertical position perturbations. This makes such robotic controllers inadvertently biomimetic. Humans use similar foot placement control in walking, stepping in the direction of the perturbation (Hof et al., 2010; Wang and Srinivasan, 2014). Previous work had shown that appropriate foot placement is used in running ostriches while turning (Jindrich et al., 2007), running humans in cutting maneuvers (Besier et al., 2003), and turning while walking (Patla et al., 1999).
Some past work on inferring stability from variability focused on kinematic measures of variability such as Floquet multipliers, finitetime Lyapunov exponents (Dingwell et al., 2001) and long term correlations in walking and running variability (Hausdorff et al., 1995; Jordan et al., 2006; Kaipust et al., 2012). Such measures can provide discriminative diagnostic measures (Kaipust et al., 2012), but do not attempt to provide a causal narrative about how locomotion is controlled. Our approach here, rooted in NewtonEuler mechanics, is able to discover potential causal strategies underlying locomotion stability, and by extension, could inform treatment of pathological unstable movements in addition to diagnosis. Other past studies have used variants of the principal component analysis (Cappellini et al., 2006; Maus et al., 2015) to demonstrate that the intrinsic variability in human locomotion may reside in a lower dimensional manifold (Cappellini et al., 2006; Chang et al., 2009; Yen et al., 2009; Dingwell et al., 2010; Maus et al., 2015). Here, by focusing on how the center of mass is controlled through forces, we have implicitly used a physicsbased dimensionality reduction to examine the dominant control strategies.
While our work relies on linear regressions from data, the basic physics relating the inputs and outputs in these models suggest a natural causal account. This causal account, based on simplifying modeling assumptions, ignores the effect of variables not considered here. Our goal here was to delineate the explanatory power of controller descriptions based on center of mass state. To identify the effect of perturbations of other possibly relevant state variables (such as trunk attitude and angular velocity), we may need to either independently perturb these state variables or show that the natural variability in such variables is not significantly correlated with the center of mass state.
The gain relating sideways foot placement and sideways velocity deviation was about 2.5 times greater than the gain relating foreaft foot placement and foreaft velocity deviation; a similar factor of 3 was found in walking (Wang and Srinivasan, 2014), perhaps reflecting the greater sideways instability of a biped without foot placement control (Ghigliazza et al., 2005). Also consistent with lower control authority and a greater fall propensity in the sideways direction, we find that the recovery from a sideways perturbation is faster than from a foreaft perturbation. While stationkeeping was not prioritized over a single step, it may occur on a slower timescale with a multistep controller, not considered here.
The results we have presented have been for data pooled over all subjects. Performing the regressions for data from individual subjects indicates that the dominant terms in the inferred controllers are similar for all subjects; the subjecttosubject variability in the estimated control gains are shown in Figure 12. Figure 12 illustrates how the accuracy of an estimated control gain depends on the number of strides used for regression. For such linear regressions, the error estimate (standard deviation) is expected to decrease with ${N}_{\mathrm{stride}}$ like $1/{N}_{\mathrm{stride}}^{2}$, so that a factor of 10 decrease in error requires a 100fold increase in sample size (Wang and Srinivasan, 2012; Hamilton, 1994). This dependence on ${N}_{\mathrm{stride}}$ may guide selection of sample sizes for future experimental designs.
Our model predicts that when a runner starts at a higherthannormal height at flight apex, or equivalently, encounters a stepdown, the runner lands with a steeper leg angle (Figure 9d). Such behavior has been observed in humans and bipedal running birds running with large unforeseen or visible stepdowns (Daley and Biewener, 2006; Grimmer et al., 2008; Müller et al., 2012; Qiao and Jindrich, 2012). Conversely, stepups decrease touchdown angle, as predicted (BirnJeffery and Daley, 2012). This behavior has been attributed to swing leg retraction just before foot contact (Seyfarth, 2003), but our foot placement controller captures this phenomenon despite not having explicit leg swing dynamics. While the terrain perturbations in the aforementioned experiments were large (5–20 cm), our model is based on data with tiny steptostep deviations (vertical position ${z}_{a}$ s.d. 5 mm). This agreement indicates that humans may use qualitatively similar control strategies for large external perturbations and small intrinsic perturbations. Such foot placement control has also been used to control robots running on uneven terrain (Hodgins and Raibert, 1991).
It is expected that any running controller that achieves asymptotic stability will need to perform net mechanical work in response to perturbations that decrease or increase the body’s mechanical energy (Ghigliazza et al., 2005; Srinivasan and Holmes, 2008; Maus et al., 2015). Our results are consistent with such expectation, as illustrated by the workloops with net mechanical work in Figure 10. Energyconservative springlike leg behavior does not allow such net mechanical work and can achieve only partial asymptotic stability, not being able to handle energychanging perturbations (as noted by (Ghigliazza et al., 2005)). Indeed, it is generally thought that even the springmasslike steady state center of mass motion in running is due to considerable muscle action and has been termed pseudoelastic (Ruina et al., 2005) or pseudocompliant (McN. Alexander, 1997). Remarkably, energyoptimal running movements in models with no leg springs produce similar springmasslike center of mass trajectories (Srinivasan, 2011), with leg muscles performing equal amounts of positive and negative work.
A previous article (Maus et al., 2015) fit running data to variants of the springmass model, allowing the spring stiffness and spring length to change during stance, and showing that constant values for these parameters cannot fit running data. Here, we have used a simpler model to directly describe the control of stance leg force or activation (Figure 8). Such direct control of leg force or activation is perhaps more parsimonious than the simultaneous control of two variables, namely, spring stiffness and length. We have shown that humans modulate GRF continuously over the whole stance phase for control (Figure 5); Maus and colleagues (Maus et al., 2015) assumed, for simplicity, an instantaneous finite energy input at midstance.
The stabilizing responses we have characterized in this study are likely due to a mixture of feedforward dynamics and active neurally mediated feedback control. When we use the term "control" here, we implicitly refer to this mixture. It is hard to rigorously separate the roles of feedforward and feedback control without recording motor neuronal outputs and how these outputs interact with the properties of muscles. Nevertheless, we can determine the feasibility of feedback control by checking whether there is enough time for feedback control, given typical neuromuscular latencies. Our typical flight phase durations are greater than or about roughly equal to the typical short or middle latencies in reflex or feedback loops involving vestibular (Fitzpatrick et al., 1994; Iles et al., 2007) or proprioceptive mechanisms (Pearson and Collins, 1993; Sinkjær et al., 1999). This suggests feasibility of feedback based on flight phase or late stance phase information regarding center of mass state. While we have focused on the control of stance based on the previous flight apex, we have found that equivalent controllers based on the center of mass state at the end of previous stance have similar predictive ability (Figure 6), thus allowing more time for neural feedback. Specifically, the lag between the information in the center of mass state and the swing foot state regarding future foot placement is about 0.1 s for sideways placement and about twice that for foreaft foot placement, suggesting sufficient time for neurally mediated feedback control of foot placement (Figure 6).
Center of mass state or other body state information needed for feedback control could be estimated by the nervous system by integrating sensory signals from vision (Patla, 1997), proprioceptive sensors (especially when the foot is on the ground (Sainburg et al., 1995)), and vestibular sensors (Angelaki and Cullen, 2008), potentially in combination with predictive internal models (Wolpert et al., 1995; Cullen, 2004). In future work, repeating the calculations herein (for instance, Figure 6) for experiments that systematically block or degrade (say, by adding sensory noise) one or more of these sensors may tell us the relative contributions of these sensors to running control. We speculate that most available relevant sensory information is used, perhaps analogous to an optimal state estimator (Kuo, 2005; Srinivasan, 2009), and degrading one sensor may result in sensory reweighting on a slow timescale (Carver et al., 2006; Assländer and Peterka, 2014). Such experiments may also help explicitly distinguish the effects of sensory and motor noise, which we have implicitly combined here into a single residual term in the linear regressions.
In this work, we have obtained a running controller with simplifying assumptions. Because humans have extended feet, nonpointmass upper bodies and legs with masses, the simple pointmass model may not capture all aspects of the running data (Bullimore and Burn, 2006; Srinivasan and Holmes, 2008). Further, we have made simplifying assumptions about muscle architecture, muscle properties (linear forcevelocity relations), and muscle activation, which are meant to capture the main qualitative dynamical features of muscles, rather than model them quantitatively precisely. For instance, we used a linear forcevelocity relation, which may be sufficient to produce dampinglike and stabilizing muscle behavior when activated, but this damping behavior may be accentuated by a more realistic nonlinear forcevelocity relation.
Future work will also involve obtaining controllers for more complex biped models and muscle models with different control architectures, which, for instance, might include feedback control based on not just the center of mass state, but the states of individual body segments. We have focused on linear relations between state deviations and control, as this is naturally suited for small deviations and perturbations that our data contains. In future work, we hope to examine the range of perturbation sizes for which this linear description is accurate by comparing this linear control to responses in experiments with larger perturbations, also inferring nonlinear descriptions should they improve predictive capability. We will also examine other control architectures, for instance, more explicitly incorporating state estimation and considering continuous control of motor outputs based on an estimated state, partly correcting for neural latencies using internal models.
The methods used here are simple and noninvasive: they can be replicated to study running stability and control in other animals, or indeed, other approximately periodic tasks such as flapping flight and swimming. These methods are suitable for analyzing differences in different populations like athletes and nonathletes, the young and the elderly, and adults with and without movement disorders. Once such differences are wellcharacterized, this information could be used, say, in a rehabilitation setting to track progress from a controller in the presence of a movement disorder to a more healthy controller, and to design rehabilitation robots that assist in this progress.
Materials and methods
We collected running data by conducting human subject experiments, obtained linear models to characterize the control strategies hidden in the steptostep variability, and performed dynamic simulations using the inferred controller on a mathematical model of the runner.
Experimental methods
Request a detailed protocolThe protocols were approved by the Ohio State University Institution Review Board and subjects participated with informed consent. Eight subjects, three female and five male (age 25.0 ±5 years, weight 66.8 ±7 kg, height 1.8 ±0.14 m, leg length 1.05 ±0.08 m, mean ± s.d.) ran on a splitbelt treadmill at three constant speeds: 2.5, 2.7 and 2.9 m/s, presented in random order. Each speed had 2075 ± 67 strides across all subjects (one stride = two steps) with subjects running about 3.5 min on average. Subjects wore a loose safety harness that did not constrain their motion. Threedimensional ground reaction forces and moments on each belt of the treadmill were recorded by separate sixaxis load cells (Bertec Inc, 1000 Hz). Body segment motion was measured using markerbased motion capture (Vicon T20, 100 Hz) with four reflective markers on each foot and on the torso.
Calculating input state variables during flight apex
Request a detailed protocolWe define flight apex as when the center of mass velocity reaches its peak height ($\dot{z}=0$). The input to the running controller is drawn from the center of mass state at flight apex, namely position $({x}_{a},{y}_{a},{z}_{a})$ and velocity $({\dot{x}}_{a},{\dot{y}}_{a},{\dot{z}}_{a})$. Unless otherwise specified, we use the flight apex state $({\dot{x}}_{a},{\dot{y}}_{a},{z}_{a})$ as inputs in our linear models. The vertical velocity ${\dot{z}}_{a}$ at flight apex is zero by definition and hence not included as an input. The center of mass velocities are obtained by integrating the center of mass accelerations, that is, the massnormalized net force on the body: $\ddot{x}={F}_{x}/m$, $\ddot{y}={F}_{y}/m$, and $\ddot{z}={F}_{z}/mg$, where ${F}_{x},{F}_{y}$ and ${F}_{z}$ are the measured ground reaction forces on the body. To obtain the integration constants, we assume that the mean velocity and acceleration over the whole trial are zero, because the person does not translate appreciably in the lab frame over a trial. To remove the slow integration drift in the center of mass velocity, we used a highpass filter with a frequency cutoff equal to an eighth of mean step frequency (Luinge and Veltink, 2005; Schepers et al., 2009). Changing this highpass filter cutoff to a twentieth of the step frequency instead, or using a piecewiselinear detrending over 20 steps, do not affect any of this article’s conclusions. This is because the stabilitycritical timescales are much shorter. We ignored air drag here, because including it changed the velocities by less than 10^{5} ms^{1}, which is much smaller than the steptostep variability. We use a weighted mean of four markers, roughly at the sacral level, as an approximation of the center of mass position (Gard et al., 2004; Wang and Srinivasan, 2014; Perry and Srinivasan, 2017).
Calculating the output control variables during stance
Request a detailed protocolWe assume that the following variables are used to control the runner: GRFs, foot placement, and the landing leg length. Stance phases are identified as when the vertical GRF exceeds a threshold value to account for measurement noise (${F}_{\mathrm{z}}\phantom{\rule{thinmathspace}{0ex}}>\phantom{\rule{thinmathspace}{0ex}}30$ N). The corresponding stance duration is ${T}_{\mathrm{stance}}$. The GRF impulses $({P}_{x},{P}_{y},{P}_{z})$ for each step are obtained by integrating the GRF components over the stance phase (${P}_{x}={\int}_{0}^{{T}_{\mathrm{stance}}}{F}_{x}\mathit{d}t$, etc). In addition to considering how GRF control occurs grossly over one step, we also consider GRF control throughout stance as a function of stance phase fraction ${\varphi}_{\mathrm{stance}}$. Each stance phase is divided into $n$ bins of duration ${T}_{\mathrm{stance}}/n$. To approximate how the GRFs changes with the stance phase fraction ${\varphi}_{\mathrm{stance}}$, we used the binned averages of the GRF in each of $n=20$ bins.
Linear regressions between the outputs and the inputs
Request a detailed protocolWe compute the mean values of the inputs over all steps in each trial and obtain deviations from these means $(\mathrm{\Delta}{\dot{x}}_{a},\mathrm{\Delta}{\dot{y}}_{a},\mathrm{\Delta}{z}_{a})$. Similarly, we compute the deviations from the means of the output variables $\mathrm{\Delta}F({\varphi}_{\mathrm{stance}})$, $\mathrm{\Delta}P$, and $\mathrm{\Delta}({x}_{f}{x}_{s},{y}_{f}{y}_{s})$. We use ordinary least squares regression to obtain linear models between the inputs and the outputs and report significant coefficients. Specifically, we have $\mathrm{\Delta}\mathrm{Output}=J\cdot \mathrm{\Delta}\mathrm{Input}$, where the Jacobian matrix $J$ represents the matrix of coefficients in the linear model. Each element of the matrix $J$ quantifies the sensitivity of an output variable to small changes in a corresponding input variable, as inferred from the data and subject to the simplifying model assumptions. These sensitivity coefficients could be interpreted as partial derivatives, such as: $\partial {T}_{\mathrm{stance}}/\partial {\dot{x}}_{\mathrm{a}}$, $\partial {F}_{y}({\varphi}_{\mathrm{stance}})/\partial {\dot{y}}_{\mathrm{a}}$, and so on. Unless otherwise specified, the results presented are based on deviations of all subjects pooled together as one dataset, but we find that the models of individual subjects’ data are qualitatively similar (as indicated in Figure 12). The coefficients for the right leg and left leg are computed separately, to accommodate sign changes due to symmetry about the sagittal plane.
Regressions with phasedependent inputs
Request a detailed protocolIn addition to the regressions described above using the flight apex state as the predictor, we used the center of mass state $(\mathrm{\Delta}{\dot{x}}_{a},\mathrm{\Delta}{\dot{y}}_{a},\mathrm{\Delta}{z}_{a})$ at different phases over the previous step to predict each of the stance phase outputs. Specifically, for each stance phase output, we performed $n=20$ regressions, each using the center of mass state at one of the $n=20$ equally spaced gait phases over the previous step, where one full step is defined as starting and ending at a touchdown. This analysis allows us to investigate the predictive ability of the center of mass state at different phases. For these phasedependent regressions, in addition to using the center of mass state as the predictor, we repeated the calculations using the swing foot state (position and velocity relative to the center of mass), so as to compare the different predictive abilities as in Figure 6.
Implementing the dataderived control on a minimal mathematical biped
Request a detailed protocolWe consider two simple models of running, similar in spirit to previous models in terms of simplicity (Blickhan and Full, 1993; Geyer et al., 2006; Srinivasan and Holmes, 2008), but generalized such that the leg forces are not constrained by ad hoc springlikeleg assumptions (Srinivasan, 2011). Instead, the biped controller details are inferred from our experimentally obtained linear models. Both biped models have a pointmass upper body and massless legs (Srinivasan and Ruina, 2006; Srinivasan, 2011), that can change effective leg length during stance by modulating the leg force (Figure 7a). During flight phase, the pointmass body undergoes parabolic free flight. The legs can apply forces on the upper body during stance phase. The two models, dubbed ‘direct force control model’ and ‘muscle control model’ differ in how the leg force is produced and controlled. For the muscle control model, we use a Hill muscle model with forcelength and forcevelocity relations (Figure 7b, c and d). The 3D equations of motion of the pointmass biped are: $m\ddot{x}={F}_{\mathrm{l}\mathrm{e}\mathrm{g}}\cdot (x{x}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}})/\ell$, $m\ddot{y}={F}_{\mathrm{l}\mathrm{e}\mathrm{g}}\cdot (y{y}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}})/\ell$, and $m\ddot{z}=mg+{F}_{\mathrm{l}\mathrm{e}\mathrm{g}}\cdot (z{z}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}})/\ell$, where ${F}_{\mathrm{leg}}$ is the scalar leg force, $({x}_{\mathrm{foot}},{y}_{\mathrm{foot}},{z}_{\mathrm{foot}})$ is the foot position with ${z}_{\mathrm{foot}}=0$ on flat terrain and $\ell =\sqrt{(x{x}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}}{)}^{2}+(y{y}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}}{)}^{2}+(z{z}_{\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}}{)}^{2}}$ is the leg length from body to foot. In the ‘direct force control model’, the object of control is the leg force ${F}_{\mathrm{leg}}$ during stance phase. In the ‘muscle control model’, the object of control is the muscle activation ${a}_{\mathrm{muscle}}$, which is converted to muscle force via the forcelength and forcevelocity equations of Hilltype muscles (Figure 7a–b). See (Zajac, 1989; Srinivasan and Ruina, 2006; Srinivasan, 2011) for more detailed equations of motion and muscle model equations.
Both models have two terms in their control: (1) a feedforward or ‘nominal’ term, that depends only on the average or desired periodic motion and (2) feedback modification of the control in response to state deviations at flight phase. The model’s leg force or muscle activation is modeled as a twoterm sine series of the form ${A}_{1}\mathrm{sin}(2\pi t/2{T}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}})+{A}_{2}\mathrm{sin}(2\pi t/{T}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}})$, as shown in Figure 7e. By changing the relative weights of ${A}_{1}$ and ${A}_{2}$, the shape of the leg force profile can be changed from being symmetric about the peak force to being asymmetric, with the peak force preceding or following midstance. We parameterize the running motion using stance duration ${T}_{\mathrm{stance}}$, flight duration ${T}_{\mathrm{flight}}$, 2D foot placement $({x}_{\mathrm{foot}},{y}_{\mathrm{foot}})$, 3D initial conditions for stance $(x(0),y(0),z(0))$, and the coefficients of the twoterm sine series (${A}_{1}$ and ${A}_{2}$). We solve for these variables to obtain a periodic running motion that accurately match the forward speed, step period, step width, and peak leg force from experimental data (Figure 8a) by using an optimization procedure (Srinivasan and Holmes, 2008; Srinivasan, 2011) that enforces a constraint satisfaction tolerance of less than ${10}^{6}$. The runner leaves the ground when it reaches the maximum leg length, but the nominal leg length at landing is assumed to be shorter (95%) than the maximum leg length, as seen in running data (Voloshina and Ferris, 2015). We enforce that left and right stances are mirror symmetric. Unlike previous simple running models, our model’s nominal periodic motion has nonzero step width and a stance phase that is asymmetric about midstance. This asymmetric stance is due to unequal landing and takeoff leg lengths, and the asymmetry of the leg force or muscle activation about midstance.
The foot placement control for the models are based on the experimentally derived control and given by the linear model in Equations 8 and 9. The leg force feedback control based on apex body state, for the direct force control model, has gains as shown in Figure 8b. The muscle control model’s feedback control gains are also shown superimposed in Figure 8b. These control gains were derived for the two models by modifying the Fourier coefficients for the force and muscle activations respectively, so that the linear map from one apex to the next is best matched to that from data (Equation 3). While there are infinitely many controllers, even for this simple biped model, that can approximate the apextoapex map, our simplifying assumptions constrains the space of controllers to produce a unique fit. The leg forces and muscle activations are rectified, so that they never become negative despite feedback control (Blum et al., 2017). The foot placement control and leg force feedback control are activated only when the apex state deviates from nominal.
To obtain a running simulation over many steps, we break up each step into three phases: flight from apex to beginning of stance, the stance phase, and flight from the end of stance to flight apex. The control actions for the next stance are chosen at flight apex. As previously defined for the experimental data, the flight apex is when $\dot{z}$ becomes zero. In some cases, if the vertical velocity is downward when a stance phase ends ($\dot{z}\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}0$), there is no flight ‘apex’ and the controller uses the end of stance state instead of flight apex state as input. The end of flight and thus, the beginning of stance, are determined as the moment when the distance between the body and the target foot position is exactly equal to the landing leg length. The leg length at landing is also controlled based on flight apex state, based on the linear model in Equation 7. At flight apex, if the distance to the next foot position is less than the target landing leg length, the runner immediately goes into stance.
Such a simulation, when started from initial conditions exactly on the nominal periodic motion, results in a perfectly periodic motion when there are no further perturbations. We then resimulated the two pointmass running models for hundreds of steps, in the presence of noisy foot placements and leg forces or muscle activations with steptostep variability. To model the noise in foot placement, we computed the ‘desired’ foot placement based on the center of mass state at flight apex (Equations 89) and then added a deviation drawn from a normal distribution, whose variance equals the foot placement variance not explained by Equation 8. Similarly, we incorporated imprecise control of leg forces or muscle activation in the following manner: for each stance phase, once the leg force $F(t)$ (for model1) or muscle activation $a(t)$ (for model2) is determined based on the center of mass state at the previous flight apex, we ‘corrupt’ these functions by a multiplicative noise term, so that the actual leg force or muscle activation is $F(t)(1+\u03f5)$ or $a(t)(1+\u03f5)$ respectively, where $\u03f5$ is drawn from a normal distribution with variance equal to the unexplained steptostep variability in leg force magnitude. Thus, we use the unexplained variance in the foot placement and leg forces from experimental regressions as a simple model of the intrinsic noise in active control.
Data availability
All anonymized raw human running data (1GB) from this study are available at the following Dryad link: https://doi.org/10.5061/dryad.1nt24m0

DryadData from: Steptostep variations in human running reveal how humans run without falling.https://doi.org/10.5061/dryad.1nt24m0
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Decision letter

David LentinkReviewing Editor; Stanford University, United States

Andrew J KingSenior Editor; University of Oxford, United Kingdom

Andrew BiewenerReviewer; Harvard, United States

Greg SawickiReviewer; NC State University, United States
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Steptostep variations in human running reveal how humans run without falling" for consideration by eLife. Your article has been reviewed by Andrew King as the Senior Editor, a Reviewing Editor, and two reviewers. The following individuals involved in review of your submission have agreed to reveal their identity: Andrew Biewener (Reviewer #1) and Greg Sawicki (Reviewer #2).
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
Summary:
How do humans stabilize themselves while running? This study of human running exploits the information contained in the natural steptostep variability of the movement of the center of mass of the body as well as the ground reaction forces under the foot to 'reverseengineer' the control algorithm. Using welldesigned experiments and mathematical modelling, the authors provide key insights into how changes in foot placement and ground reaction force are modulated to deal with noise and perturbations to run stably. Interestingly, humans make these adjustments sidetoside within a stride, while foreaft adjustments are made over several strides. Previous studies explained the role of elastic storage in the legs to run efficiently, the present findings shed new light on the central role of active muscle control to run stably. This finding is of broad interest to roboticists, engineers, and biomechanists interested in human and animal locomotion over terrain. The new information on how humans control their gait may be used to develop better controllers for powered exoskeletons and prostheses as well as humanoid robots.
Essential revisions:
1A) The authors tend to overstate the novelty of their findings in suggesting that past work based on a passive springmass running model cannot account for stable running. Past work based on springmass running mechanics was developed to argue for how passiveelastic properties may reduce the work of muscles needed to control leg and body movement. But these past studies do not claim that muscle work per se is unnecessary. As the author's own work (Srinivasan, 2011) and earlier papers (Alexander, 1997; Ruina et al., 2005) have noted (in addition to others – e.g. Biewener and Daley, 2007), springlike leg behavior can result from an inelastic limb, in which muscle negative work is performed followed by equal positive work. The authors should revise to more accurately represent that understanding and approach of past workers on running dynamics in relation to their own work. We understand that some of these earlier studies may have overstated their implications towards dynamic stability during running. However, we encourage the authors to focus on their new data and model and let that speak towards the contribution of their study. Examples are listed below:
1B) The majority of those who have modeled steady running as a springmass system with springlike leg dynamics do not suggest that this passive model is an accurate representation of how running is controlled, or that it would necessarily result in stable periodic motion following a perturbation. The authors' telescoping or single knee joint musclecontrolled leg model which shows net leg work (either negative or positive) is unsurprising in this regard and an expected result. Please better reflect that past workers who analyze running as springmass dynamics implicitly or explicitly recognize that some muscle work must be done to control the timevarying motions of a runner's CoM state for stable periodic motion.
1C) Introduction "Here, in contrast, we eschew springmasslike assumptions and characterize the control in terms of how humans modulate their leg force magnitude and direction." This is somewhat misleading in that springmass assumptions previously made by other workers were to explore the extent to which running (trotting, hopping) can be explained by passive dynamics, while recognizing that some amount of muscle work to stabilize running (or to go uphill/downhill, change direction) is necessary. The majority of prior workers do not claim that running is purely passive and does not require muscle work for control and stability. Please reflect the understanding and work in the field more accurately here.
1D) Subsection “Humanderived controller stabilizes a minimal model of bipedal running”: "Running stably cannot be purely passive and involves active leg work." Please revise so this does not overstate the significance of the authors' modeling and findings.
1E) Discussion section: The pseudoelastic behavior of a runner's leg, which in fact could be entirely based on actuation (negative work followed by equivalent positive work) – as citations to Ruina et al., 2005 and Alexander, 1997 demonstrate, as well as the author's own past work (Srinivasan, 2011).
2) There is a large literature dedicated to understanding variability in gait (e.g., Dingwell's GEM; Chang's UCM; Hausdorff's longterm noise correlations; Laquaniti's principle components analysis etc.). These approached are predominantly based on characterizing the variability in kinematic measures as a way to assess 'stability', but they cannot say much about the underlying control and this has limited their impact on understanding control of gait mechanics more fundamentally. The approach presented here is a longoverdue, a more formal linear systems ID framework grounded in NewtonEuler mechanics. It may be worth acknowledging this and comparing/contrasting the utility of these two approaches (e.g., diagnosis vs. prediction) without overly criticizing previous approaches. E.g. by focusing one or a couple of sentences in the discussion on the key new abilities that add to previous approaches and how they could be used together in future studies.
3) The modeling approach taken here is framed in the form of a linear system ID, which is OK for small perturbations, in this case step to step variation around an average behavior. Figure 8 starts to address this in simulation, but how well does the approach hold for more explicit, larger perturbations at different phases of the gait where nonlinearities might become more important? What is the plan to deal with this? Please clarify this, or the underlying limitations, in the discussion.
4) Please discuss the potential physiological mechanisms underlying the extracted controller, and suggest experiments that could be done to elucidate them if you have a new perspective on this. E.g., the authors mention a combination of feedforward and feedback processes but do not elaborate much on how vision, vestibular, proprioception, and cerebellar internal model may contribute.
5) Can this approach be applied to smaller data sets from single individuals as well? How variable is the controller across people and how much data is necessary to extract a good model?
6) Introduction "Numerous running robots have demonstrated stable periodic running, using a variety of control schemes (Raibert, 1986; Chevallereau et al., 2005; Tajima et al., 2009)." If robots are moving stably and periodically, as stated here, why then isn't human running and other animal running not stable and periodic as per the authors' lead statement (Introduction)? The earlier statement is made to argue the importance of muscle actuation for control to achieve truly stable periodic motion. Clearly both animals and robots need to do some muscle work to control movement. This inconsistency in phrasing needs to be addressed.
7) The regressions for equations 1and 2 should be shown (at least in supplemental materials).
8) Subsection “Impulse control is achieved by phasedependent force modulations”: "compute the phasedependent sensitivity of the GRFs to…" Exactly how are these phasedependent sensitivities computed? (The yaxis label for these needs to be defined/explained more clearly.) This is explained in part (subsection “Linear regressions between the outputs and the inputs”) but could be made more clear and explicit when showing and referring to these patterns in Figure 4.
9) Figure 5: Swingfoot reposition and CoM state predictability of foot position – how these are determined/calculated needs to be made clear. Results are stated but their basis is not shown.
10) Figure 6: The forcevelocity and forcelength relations shown and implemented in the authors' model may be overly simple. FV is an inverse hyperbolic not linear relationship and the lengthening side of FV is much more skewed than the authors' model. (Also consider Zajac, 1989). How would a more realistic FV relationship (inverse hyperbolic for shortening and a more steep and rapid leveling off of force in relation to lengthening) affect the authors' model?
https://doi.org/10.7554/eLife.38371.022Author response
Essential revisions:
1A) The authors tend to overstate the novelty of their findings in suggesting that past work based on a passive springmass running model cannot account for stable running. Past work based on springmass running mechanics was developed to argue for how passiveelastic properties may reduce the work of muscles needed to control leg and body movement. But these past studies do not claim that muscle work per se is unnecessary. As the author's own work (Srinivasan, 2011) and earlier papers (Alexander, 1997; Ruina et al., 2005) have noted (in addition to others – e.g. Biewener and Daley, 2007), springlike leg behavior can result from an inelastic limb, in which muscle negative work is performed followed by equal positive work. The authors should revise to more accurately represent that understanding and approach of past workers on running dynamics in relation to their own work. We understand that some of these earlier studies may have overstated their implications towards dynamic stability during running. However, we encourage the authors to focus on their new data and model and let that speak towards the contribution of their study. Examples are listed below:
Thank you for these constructive remarks. We do agree with the remarks by the reviewers that many authors including those that have used and pioneered the springmass model paradigm recognized the limitations and simplifications of springmass models over the years, while perhaps some authors may have taken the metaphor far as the reviewers agree. We also agree that initially spring mass models were inspired by how passive elastic properties may help reduce the muscle work needed for running and have been valuable in thinking about these tasks. So as the reviewers suggest, we have focused more on our results rather than prior springmasslike models. Specifically, we have made the following edits to reduce or remove our critique of springmasslike models and/or more explicitly attribute its weakness to prior authors (which we had perhaps done more implicitly in the prior version via parenthetical citations). We have listed many of these edits below, which are in addition to the additional ones listed in response to comments 1B, 1C, 1D, and 1E.
Original: “The classic modeling paradigm for running control assumes that the human leg behaves like a linear spring (Blickhan, 1989; McMahon and Cheng, 1990; Blickhan and Full, 1993).”
Revised: “One classic modeling paradigm for running control assumes that the human leg behaves like a linear spring (Blickhan, 1989; McMahon and Cheng, 1990; Blickhan and Full, 1993). This paradigm has been used to argue how passiveelastic properties may reduce muscle work needed for locomotion (Alexander and Vernon, 1975; Alexander, 1990) and has been useful in examining different aspects of locomotion in a simplified setting.”
Original: “These models are usually based on fitting to the average periodic center of mass motion during 38 running (Blickhan and Full, 1993; Geyer et al., 2006; Srinivasan and Holmes, 2008). However, understanding running stability requires understanding how deviations from the average motion 40 are controlled.”
Revised: “These models have been successful in fitting the average periodic center of mass motion during running (Blickhan and Full, 1993; Geyer et al., 2006; Srinivasan and Holmes, 2008). Understanding running stability requires understanding how deviations from the average motion are controlled.”
Original: “Springlike leg mechanics cannot explain how deviations from the average motion are controlled (Maus et al., 2015).”
Revised: “It has been previously recognized that springlike leg mechanics cannot explain how deviations from the average motion are controlled and are eventually attenuated (e.g., Ghigliazza et al., 2005; Biewener and Daley, 2007; Maus et al., 2015).”
Original: “Here, we use more general models of human locomotion, rooted in Newtonian mechanics, to implement running control, without making any springlike parametric assumptions (Srinivasan 2011).”
Revised: “Here, we examine the role of active muscle control in running stability, using more general models of human locomotion rooted in Newtonian mechanics (Srinivasan, 2011).”
Edited version: In the following sentence, we have removed the reference to springmass model.
“We uncover how such center of mass control is achieved. We then implement this humanderived controller on a simple mathematical model of a biped (Srinivasan, 2011), showing that this biped model runs without falling down, despite incessant noiselike perturbations, large external perturbations, and on uneven terrain.”
Revised: “Here, we have used a simpler `nonparametric' model to directly describe the control of stance leg force or activation (Figure 8).”
1B) The majority of those who have modeled steady running as a springmass system with springlike leg dynamics do not suggest that this passive model is an accurate representation of how running is controlled, or that it would necessarily result in stable periodic motion following a perturbation. The authors' telescoping or single knee joint musclecontrolled leg model which shows net leg work (either negative or positive) is unsurprising in this regard and an expected result. Please better reflect that past workers who analyze running as springmass dynamics implicitly or explicitly recognize that some muscle work must be done to control the timevarying motions of a runner's CoM state for stable periodic motion.
Thank you for these remarks. In our previous writing of the paragraph, we weren’t clear enough that we were attributing some of these reasoning to prior researchers, although we had cited them parenthetically. We have rewritten the relevant paragraph to hopefully reflect this prior understanding better:
“It is expected that any running controller that achieves asymptotic stability will need to perform net mechanical work in response to perturbations that decrease or increase the body’s mechanical energy (Ghigliazza et al., 2005; Srinivasan and Holmes, 2008; Maus et al., 2015). Our results are consistent with such expectation, as illustrated by the workloops with net mechanical work in Figure 11. Energyconservative springlike leg behavior does not allow such net mechanical work and can achieve only partial asymptotic stability, not being able to handle energychanging perturbations (as noted by Ghigliazza et al. (2005)).”
1C) Introduction "Here, in contrast, we eschew springmasslike assumptions and characterize the control in terms of how humans modulate their leg force magnitude and direction." This is somewhat misleading in that springmass assumptions previously made by other workers were to explore the extent to which running (trotting, hopping) can be explained by passive dynamics, while recognizing that some amount of muscle work to stabilize running (or to go uphill/downhill, change direction) is necessary. The majority of prior workers do not claim that running is purely passive and does not require muscle work for control and stability. Please reflect the understanding and work in the field more accurately here.
Thank you for these remarks. We have now edited the following sentence:
“Here, in contrast, we eschew springmasslike assumptions and characterize the control in terms of how humans modulate their leg force magnitude and direction.” to “Here, we directly characterize the control in terms of how humans modulate their leg force magnitude and direction during running.”
1D) Subsection “Humanderived controller stabilizes a minimal model of bipedal running”: "Running stably cannot be purely passive and involves active leg work." Please revise so this does not overstate the significance of the authors' modeling and findings.
Thank you. As suggested, we have edited the original subsection “Running stably cannot be purely passive and involves active leg work” to be more ‘illustrative’ of prior expectations, more explicitly acknowledging various authors on the topic.
“Nonzero leg work for energychanging perturbations.
Figure 11 illustrates the leg workloop for the unperturbed run (net zero work) and when positive perturbations are applied to sideways and foreaft velocities, and vertical positions. All such positive perturbations result in net negative work on the first step after the perturbation, reflected in the workloops with net negative area within them. Such net positive or negative leg work is clearly necessary to recover from perturbations that change the total mechanical energy of the runner, as was recognized in prior discussions of the energyneutral springmass model of running (Ghigliazza et al., 2005; Biewener and Daley, 2007; Srinivasan and Holmes, 2008).”
1E) Discussion section: The pseudoelastic behavior of a runner's leg, which in fact could be entirely based on actuation (negative work followed by equivalent positive work) – as citations to Ruina et al., 2005 and Alexander, 1997 demonstrate, as well as the author's own past work (Srinivasan, 2011).
Thank you for these remarks. We have now deleted the sentence that previously was” Our results suggest…”, as described in response to reviewer comment 1B.
Regarding the seventh paragraph of the Discussion section, we assume that the reviewers are agreeing with us and simply drawing our attention to our paragraph regarding prior discussions of pseudoelastic leg behavior in running. We have left this paragraph mostly intact, editing it down slightly as follows:
“Indeed, it is generally thought that even the springmasslike steady state center of mass motion in running is due to considerable muscle action, and has been termed pseudoelastic (Ruina et al., 2005) or pseudocompliant (Alexander, 1997). Remarkably, energyoptimal running movements in models with no leg springs produce similar springmasslike center of mass trajectories (Srinivasan, 2011), with leg muscles performing equal amounts of positive and negative work.”
2) There is a large literature dedicated to understanding variability in gait (e.g., Dingwell's GEM; Chang's UCM; Hausdorff's longterm noise correlations; Laquaniti's principle components analysis etc.). These approached are predominantly based on characterizing the variability in kinematic measures as a way to assess 'stability', but they cannot say much about the underlying control and this has limited their impact on understanding control of gait mechanics more fundamentally. The approach presented here is a longoverdue, a more formal linear systems ID framework grounded in NewtonEuler mechanics. It may be worth acknowledging this and comparing/contrasting the utility of these two approaches (e.g., diagnosis vs. prediction) without overly criticizing previous approaches. E.g. by focusing one or a couple of sentences in the discussion on the key new abilities that add to previous approaches and how they could be used together in future studies.
Thank you for these remarks. We have now added a new paragraph that show how our methods can complement these other techniques:
“Some past work on inferring stability from variability focused on kinematic measures of variability such as Floquet multipliers, finitetime Lyapunov exponents (Dingwell et al., 2001) and longterm correlations in walking and running variability (Hausdorff et al., 1995; Jordan et al., 2006; Kaipust et al., 2012). Such measures can provide discriminative diagnostic measures (Kaipust et al., 2012), but do not attempt to provide a causal narrative about how locomotion is controlled. Our approach here, rooted in NewtonEuler mechanics, is able to discover potential causal strategies underlying locomotion stability, and by extension, could inform treatment of pathological unstable movements in addition to diagnosis. Other past studies has used variants of the principal component analysis (Cappellini et al., 2006; Maus et al., 2015) to demonstrate that the intrinsic variability in human locomotion may reside in a lower dimensional manifold (Cappellini et al., 2006; Maus et al., 2015, Dingwell et al., 2010; Chang et al., 2009; Yen et al., 2009; Maus et al., 2015). Here, we used a physicsbased dimensionality reduction to examine the dominant control strategies, by focusing on how the center of mass is controlled through forces.”
3) The modeling approach taken here is framed in the form of a linear system ID, which is OK for small perturbations, in this case step to step variation around an average behavior. Figure 8 starts to address this in simulation, but how well does the approach hold for more explicit, larger perturbations at different phases of the gait where nonlinearities might become more important? What is the plan to deal with this? Please clarify this, or the underlying limitations, in the discussion.
Thank you for these remarks. We have now added the following remarks in the Discussion section:
“We have focused on linear relations between state deviations and control as it is naturally suited for small deviations and perturbations that our data contains. In future work, we hope to examine the range of perturbation sizes for which this linear description is accurate by comparing this linear control to responses to experiments with larger perturbations, also inferring nonlinear descriptions should they improve predictive capability.”
4) Please discuss the potential physiological mechanisms underlying the extracted controller, and suggest experiments that could be done to elucidate them if you have a new perspective on this. E.g., the authors mention a combination of feedforward and feedback processes but do not elaborate much on how vision, vestibular, proprioception, and cerebellar internal model may contribute.
Thank you for these remarks. We have now added the following paragraph in the Discussion section, addressing the sensory integration that may be needed to accomplish feedback control:
“Center of mass state or other body state information needed for feedback control could be estimated by the nervous system by integrated sensory signals from vision (Patia, 1997), proprioceptive sensors (especially when the foot is on the ground (Sainburg et al., 1995)), and vestibular sensors (Angelaki and Cullen, 2008), potentially in combination with predictive internal models (Walpert et al., 1995; Cullen, 2004). In future work, repeating the calculations herein (for instance, Figure 5) for experiments that systematically block or degrade (say, by adding sensory nose) one or more of these sensors may well tell us the contributions of theses sensors to running control. We speculate that the most available relevant sensory information is used, perhaps analogous to an optimal state estimator (Kuo, 2005; Srinivasan, 2009) and degrading one sensor may result in sensory reweighting on a slow timescale (Carver et al., 2006; Assländer and Peterka, 2014).”
5) Can this approach be applied to smaller data sets from single individuals as well? How variable is the controller across people and how much data is necessary to extract a good model?
Thank you for this question. We have now added a completely new illustrative figure (Figure 12) that provides information on these questions. We have also added the following paragraph in the Discussion section, remarking on these issues.
“The results we have presented have been for data pooled over all subjects. Performing the regressions for data from individual subjects indicates that the dominant terms in the inferred controllers are similar for all subjects; the subject to subject variability in the estimated control gains are shown in Figure 12a. Figure 12b illustrates how the accuracy of an estimated control gain depends on the number of strides used for regression. For such linear regressions, the error estimate (standard deviation) is expected to decrease with N_{stride} like 1∕N^{2}_{stride}, so that a factor of 10 decrease in error requires a 100fold increase in sample size (Wang and Srinivasan, 2012; Hamilton, 1994). This dependence on N_{stride} may guide selection of sample sizes for future experimental designs.”
6) Introduction "Numerous running robots have demonstrated stable periodic running, using a variety of control schemes (Raibert, 1986; Chevallereau et al., 2005; Tajima et al., 2009)." If robots are moving stably and periodically, as stated here, why then isn't human running and other animal running not stable and periodic as per the authors' lead statement (Introduction)? The earlier statement is made to argue the importance of muscle actuation for control to achieve truly stable periodic motion. Clearly both animals and robots need to do some muscle work to control movement. This inconsistency in phrasing needs to be addressed.
Thank you for pointing to the inconsistency in the wording. We have now edited as follows, which hopefully removes the inconsistency.
"Some running robots have demonstrated stable running (that is, running without falling down) using a variety of control schemes (Raibert, 1986; Chevallereau et al., 2005; Tajima et al., 2009; Nelson et al., 2019)."
That is, we have edited the sentence by (1) removing the word ‘periodic’ when referring to robot running and (2) also clarified what we mean by ‘stable robot running’, that is, ‘running without falling down.’ By stable running, we don’t mean that the motion is perfectly periodic, but just that motion does not result in a fall. We hope our edits has addressed the perceived inconsistency between this sentence and the earlier sentence about human running not being perfectly periodic.
7) The regressions for equations 1and 2 should be shown (at least in supplemental materials).
Thank you for this suggestion. We have now included a new Figure 3 in the article that shows this regression, included below with its caption.
“of the stride cycle, specifically, the time fraction Ø_{stance} of stance (Figure 2b). To explain this phasedependent force variability within a single step, we compute the phase dependent sensitivity of (F_{x}, Fy, F_{z}) to the center of mass state as follows. For each output, say F_{x}, we divide the stance duration into 20 phases and compute a linear model for F_{x}at each of those phases, all with (ẋ_{a}, ẏ_{a}, z_{a}) as inputs. We refer to the coefficients in these linear models as a function of the phase Ø_{stance} as the phasedependent sensitivities of the GRFs (Figure 5) to the corresponding predictor variable in (ẋ_{a}, ẏ_{a}, z_{a}).”
8) Subsection “Impulse control is achieved by phasedependent force modulations”: "compute the phasedependent sensitivity of the GRFs to…" Exactly how are these phasedependent sensitivities computed? (The yaxis label for these needs to be defined/explained more clearly.) This is explained in part (subsection “Linear regressions between the outputs and the inputs”) but could be made more clear and explicit when showing and referring to these patterns in Figure 4.
9) Figure 5: Swingfoot reposition and CoM state predictability of foot position – how these are determined/calculated needs to be made clear. Results are stated but their basis is not shown.
Thank you for these remarks. We have now added the following additional paragraph in the Materials and methods section.
“In addition to the regressions described above using the flight apex state as the predictor, we used the center of mass state (Δẋ_{a}, Δẏ_{a}, Δz_{a}) at different phases over the previous step to predict each of the stance phase outputs. Specifically, for each stance phase output, we performed 20 regressions, each using the center of mass state at one of 20 equally spaced gait phases over the previous step, where one full step is defined as starting and ending at a touchdown. This analysis allows us to investigate the predictive ability of the center of mass state at different phases. For these phasedependent regressions, in addition to using the center of mass state as the predictor, we repeated the calculations using the swing foot state (position and velocity relative to the center of mass), so as to compare the different predictive abilities as in Figure 6.”
We have also expanded the figure caption from
“Figure 6. Swing foot control before foot placement.The fraction of sideways foot placement (panel a) and foreaft foot placement (panel b) variance at beginning of stance predicted by the center of mass (CoM) state or swing foot state during the previous one step (flight and stance). To produce this figure, a sequence of linear models were built for predicting the foot placement based on the center of mass state or swing foot state during different phases through the previous step. We plot the R2 value corresponding to these linear models (that is, fraction of variance explained) as a function of the gait phase used for the prediction; the gait phase is represented as the fraction of a step starting from beginning of previous stance phase. The solid and dashed lines represent right and left foot placements respectively.”
10) Figure 6: The forcevelocity and forcelength relations shown and implemented in the authors' model may be overly simple. FV is an inverse hyperbolic not linear relationship and the lengthening side of FV is much more skewed than the authors' model. (Also consider Zajac, 1989). How would a more realistic FV relationship (inverse hyperbolic for shortening and a more steep and rapid leveling off of force in relation to lengthening) affect the authors' model?
Thank you for these remarks. We have now added the following sentences so as to comment on this model simplification:
“Further, we've made simplifying assumptions about muscle architecture, muscle properties (linear forcevelocity relations), and muscle activation, which are meant to capture the primary qualitative dynamical features of muscles, rather than model them quantitatively precisely. For instance, the linear forcevelocity relation may be sufficient to produce dampinglike muscle behavior when activated, but this damping behavior may be accentuated by a nonlinear forcevelocity relation. Future work will also involve obtaining controllers for more complex biped models and muscle models.”
https://doi.org/10.7554/eLife.38371.023Article and author information
Author details
Funding
National Science Foundation (NSF CMMI grant 1254842)
 Manoj Srinivasan
Schlumberger Foundation
 Nidhi Seethapathi
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Andy Ruina for useful comments on an early draft. This work was supported by NSF CMMI grant 1254842 and a Schlumberger Foundation Faculty for the Future fellowship.
Ethics
Human subjects: The protocols were approved by the Ohio State University Institutional Review Board under protocol number 2012H0032. All subjects participated with informed consent.
Senior Editor
 Andrew J King, University of Oxford, United Kingdom
Reviewing Editor
 David Lentink, Stanford University, United States
Reviewers
 Andrew Biewener, Harvard, United States
 Greg Sawicki, NC State University, United States
Publication history
 Received: May 15, 2018
 Accepted: February 6, 2019
 Version of Record published: March 19, 2019 (version 1)
Copyright
© 2019, Seethapathi and Srinivasan
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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