Running at a constant speed seems like a series of repetitive, identical strides, but it is not. There are small variations in each stride. Self-inflicted 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 step-by-step 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.

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 step-to-step 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 low-dimensional 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 passive-elastic 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 spring-mass 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 spring-like 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 step-to-step 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 spring-mass 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 human-derived controller on a simple mathematical model of a biped (Srinivasan, 2011), showing that this biped model runs without falling down, despite incessant noise-like perturbations, large external perturbations, and on uneven terrain.

A human-derived 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.

The step-to-step variability during running appears superficially random. We show that this variability contains low-dimensional 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 fore-aft, and $z$ is vertical. The results we present are for the highest running speed and we discuss speed-dependence of our results separately. Unless otherwise specified, all quantities and results in equations and figures are non-dimensionalized using body mass $m$, acceleration due to gravity $g$, and leg length ${\mathrm{\ell }}_{\max }$. Forces are normalized by $mg$, distances by ${\mathrm{\ell }}_{\max }$, speeds by $\sqrt{g{\mathrm{\ell }}_{\max }}$, time by $\sqrt{{\mathrm{\ell }}_{\max }/g}$, impulses by $m\sqrt{g{\mathrm{\ell }}_{\max }}$, etc.

Figure 1

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Fore-aft and sideways impulses independently control center of mass motion

The step-to-step 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).

Figure 2

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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 fore-aft ground reaction impulses over a step (Figure 2c) are well-predicted 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 fore-aft impulse depends primarily on the fore-aft velocity ${\dot{y}}_a$. Thus, it appears that the control in the fore-aft and sideways directions are independent or decoupled. Pooled over all subjects, the best-fit linear model for the sideways impulse $P_x$ is:

(1) ${\displaystyle \begin{array}{l}\mathrm{L}\mathrm{e}\mathrm{f}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_x=-1.03\mathrm{\Delta }{\dot{x}}_a,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=0.55,\mathrm{a}\mathrm{n}\mathrm{d}\\ \mathrm{R}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_x=-1.07\mathrm{\Delta }{\dot{x}}_a,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=0.53,\end{array}}$

and that for the fore-aft GRF impulse $P_y$ is:

(2) ${\displaystyle \begin{array}{l}\mathrm{L}\mathrm{e}\mathrm{f}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_y=-0.72\mathrm{\Delta }{\dot{y}}_a,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=0.32,\mathrm{a}\mathrm{n}\mathrm{d}\\ \mathrm{R}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_y=-0.72\mathrm{\Delta }{\dot{y}}_a,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=0.33,\end{array}}$

as in Figure 3. All coefficients in Equations (1) and (2) are significant at ${\displaystyle p<{10}^{-4}}$. Both sideways and fore-aft impulses depend negligibly on vertical position deviations, so that including ${\displaystyle z_a}$ in the regression increases the ${\displaystyle R^2}$ values by less than $0.02$.

Figure 3

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Within-step vertical impulse modulations control vertical position

The control of vertical position is qualitatively different from that of control in the fore-aft and sideways directions, as we cannot use net vertical impulse for vertical position control due to the impulse-momentum 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 impulse-momentum 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, $P_z={\int }_0^{T_{\mathrm{step}}}{F}_z(t)𝑑t={\int }_0^{T_{\mathrm{step}}}mg𝑑t=mg{T}_{\mathrm{step}}$. 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 $(F_z/m-g)$ twice, the change in vertical position $z(T_{\mathrm{step}})-z(0)$ over a step is given by:

(4) $z(T_{\mathrm{step}})-z(0)={\displaystyle \frac{g}{6}}\left({(T_{\mathrm{step}}-t_{\mathrm{peak}})}^2-t_{\mathrm{peak}}^2\right).$

Figure 4

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If the step was symmetric about mid-stance ($t_{\mathrm{peak}}=T_{\mathrm{step}}/2$), 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 first-half impulse and increase the second-half 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 experimentally-derived 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 ${\mathrm{\Delta }{P}_z|}_{T_{\mathrm{step}}/2}^{T_{\mathrm{step}}}$:

(5) ${\displaystyle \mathrm{L}\mathrm{e}\mathrm{f}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_z{|}_0^{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}/2}=-2.5\mathrm{\Delta }{z}_a\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Delta }{P}_z{|}_{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}/2}^{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}}=+2.5\mathrm{\Delta }{z}_a\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=0.35,\mathrm{a}\mathrm{n}\mathrm{d}}$

(6) ${\displaystyle \mathrm{R}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}:\mathrm{\Delta }{P}_z{|}_0^{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}/2}=-2.3\mathrm{\Delta }{z}_a\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\Delta }{P}_z{|}_{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}/2}^{T_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}}=+2.3\mathrm{\Delta }{z}_a\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{R}^2=\mathrm{0.30.}}$

We see that a positive ${\displaystyle \mathrm{\Delta }{z}_a}$ corresponds to a decrease in the first-half 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:

(7) ${\displaystyle \mathrm{\Delta }{\ell }_{\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}}=0.3\mathrm{\Delta }{z}_a,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}p<{10}^{-4}\mathrm{a}\mathrm{n}\mathrm{d}{R}^2=\mathrm{0.25.}}$

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 step-up perturbation, so reducing the landing leg length reduces trip likelihood.

Impulse control is achieved by phase-dependent 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 phase-dependent 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 phase-dependent sensitivities of the GRFs (Figure 5) to the corresponding predictor variable in $({\dot{x}}_a,{\dot{y}}_a,z_a)$.

Figure 5

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The phase-dependent 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 fore-aft velocity perturbation, the fore-aft GRF is modulated so that there is a net negative force on the body over the next step (Figure 5b). The sensitivity of the fore-aft 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 ${\displaystyle {\varphi }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}<0.5}$) than during the acceleration phase (roughly ${\displaystyle {\varphi }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}>0.5}$).

Swing foot re-positioning 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 re-positioning may happen during this flight phase.

Figure 6

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Simple models of running capture GRFs and phase-dependent 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 point-mass 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 Hill-type muscles (Figure 7b). See Materials and methods for how the nominal running motion and the feedback controllers are specified for the models.

Figure 7

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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 phase-dependent ground reaction force feedback gains for the models are qualitatively similar to the phase-dependent gains inferred from experiment (Figure 8b), again, despite not explicitly fitting the shape of these phase-dependent 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.

Figure 8

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Human-derived 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 fore-aft, 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.

Figure 9

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The inputs to the feedback controller (${\dot{x}}_a,{\dot{y}}_a,z_a$) do not include the absolute sideways and fore-aft position $(x_a,y_a)$ of the runner. Therefore, the controller does not correct position perturbations (station-keeping). A sideways or fore-aft push to the model results in convergence to the nominal running motion, except for a sideways or fore-aft position offset (Figure 9c).

Non-zero leg work for energy-changing perturbations

Figure 10 illustrates the leg work-loop for the unperturbed run (net zero work) and when positive perturbations are applied to sideways and fore-aft velocities, and vertical positions. All such positive perturbations result in net negative work on the first step after the perturbation, reflected in the work-loops 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 energy-neutral spring-mass model of running (Ghigliazza et al., 2005; Biewener and Daley, 2007; Srinivasan and Holmes, 2008).

Figure 10

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Explaining variability: muscle-driven running model does not fall despite noise

To simulate the step-to-step 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 muscle-driven 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 noise-like 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 muscle-driven model is robust to motor noise presumably because of the intrinsic stabilizing properties of force-length and force-velocity relations (Hogan, 1984; Jindrich and Full, 2002).

Figure 11

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We have mined the step-to-step variability in human running to show how humans modulate leg forces and foot placement to run stably. We then used these data-derived 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 Raibert-like controller used in early running robots (Raibert, 1986) in that the foot placement opposes velocity deviations with no sideways-fore-aft 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, finite-time 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 Newton-Euler 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 physics-based 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 fore-aft foot placement and fore-aft 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 fore-aft perturbation. While station-keeping was not prioritized over a single step, it may occur on a slower time-scale with a multi-step 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 subject-to-subject 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 100-fold 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.

Figure 12

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Our model predicts that when a runner starts at a higher-than-normal height at flight apex, or equivalently, encounters a step-down, 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 step-downs (Daley and Biewener, 2006; Grimmer et al., 2008; Müller et al., 2012; Qiao and Jindrich, 2012). Conversely, step-ups decrease touch-down angle, as predicted (Birn-Jeffery 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 step-to-step 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 work-loops with net mechanical work in Figure 10. Energy-conservative spring-like leg behavior does not allow such net mechanical work and can achieve only partial asymptotic stability, not being able to handle energy-changing perturbations (as noted by (Ghigliazza et al., 2005)). Indeed, it is generally thought that even the spring-mass-like steady state center of mass motion in running is due to considerable muscle action and has been termed pseudo-elastic (Ruina et al., 2005) or pseudo-compliant (McN. Alexander, 1997). Remarkably, energy-optimal running movements in models with no leg springs produce similar spring-mass-like 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 spring-mass 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 mid-stance.

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 fore-aft 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 re-weighting on a slow time-scale (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, non-point-mass upper bodies and legs with masses, the simple point-mass 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 force-velocity 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 force-velocity relation, which may be sufficient to produce damping-like and stabilizing muscle behavior when activated, but this damping behavior may be accentuated by a more realistic nonlinear force-velocity 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 non-invasive: 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 non-athletes, the young and the elderly, and adults with and without movement disorders. Once such differences are well-characterized, 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.

We collected running data by conducting human subject experiments, obtained linear models to characterize the control strategies hidden in the step-to-step variability, and performed dynamic simulations using the inferred controller on a mathematical model of the runner.

All anonymized raw human running data (1GB) from this study are available at the following Dryad link: https://doi.org/10.5061/dryad.1nt24m0

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Author details

1. Nidhi Seethapathi

   1. Mechanical and Aerospace Engineering, The Ohio State University, Columbus, United States
   2. Department of Bioengineering, University of Pennsylvania, Philadelphia, United States

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Investigation, Visualization, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   snidhi@seas.upenn.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-5159-9717

2. Manoj Srinivasan

   Mechanical and Aerospace Engineering, The Ohio State University, Columbus, United States

   Contribution

   Conceptualization, Data curation, Software, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Project administration, Writing—review and editing

   For correspondence

   srinivasan.88@osu.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-7811-3617

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