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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

Video 1

Download asset

Video 2

Download asset

Video 3

Download asset

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

Download asset Open asset

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

Download asset Open asset

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

Download asset Open asset

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

1. 1. Seethapathi N
   2. Srinivasan M

   (2018) Dryad

   Data from: Step-to-step variations in human running reveal how humans run without falling.

   https://doi.org/10.5061/dryad.1nt24m0

1. 1. Alexander RM

   (1990) Three uses for springs in legged locomotion

   The International Journal of Robotics Research 9:53–61.

   https://doi.org/10.1177/027836499000900205

   • Google Scholar
2. 1. Alexander RM
   2. Vernon A

   (1975) The mechanics of hopping by kangaroos (Macropodidae)

   Journal of Zoology 177:265–303.

   https://doi.org/10.1111/j.1469-7998.1975.tb05983.x

   • Google Scholar
3. 1. Angelaki DE
   2. Cullen KE

   (2008) Vestibular system: the many facets of a multimodal sense

   Annual Review of Neuroscience 31:125–150.

   https://doi.org/10.1146/annurev.neuro.31.060407.125555

   • Google Scholar
4. 1. Ankaralı MM
   2. Sefati S
   3. Madhav MS
   4. Long A
   5. Bastian AJ
   6. Cowan NJ

   (2015) Walking dynamics are symmetric (enough)

   Journal of the Royal Society Interface 12:20150209.

   https://doi.org/10.1098/rsif.2015.0209

   • Google Scholar
5. 1. Assländer L
   2. Peterka RJ

   (2014) Sensory reweighting dynamics in human postural control

   Journal of Neurophysiology 111:1852–1864.

   https://doi.org/10.1152/jn.00669.2013

   • Google Scholar
6. 1. Belli A
   2. Lacour JR
   3. Komi PV
   4. Candau R
   5. Denis C

   (1995) Mechanical step variability during treadmill running

   European Journal of Applied Physiology and Occupational Physiology 70:510–517.

   https://doi.org/10.1007/BF00634380

   • Google Scholar
7. 1. Besier TF
   2. Lloyd DG
   3. Ackland TR

   (2003) Muscle activation strategies at the knee during running and cutting maneuvers

   Medicine & Science in Sports & Exercise 35:119–127.

   https://doi.org/10.1097/00005768-200301000-00019

   • Google Scholar
8. 1. Biewener AA
   2. Daley MA

   (2007) Unsteady locomotion: integrating muscle function with whole body dynamics and neuromuscular control

   Journal of Experimental Biology 210:2949–2960.

   https://doi.org/10.1242/jeb.005801

   • Google Scholar
9. 1. Birn-Jeffery AV
   2. Daley MA

   (2012) Birds achieve high robustness in uneven terrain through active control of landing conditions

   The Journal of Experimental Biology 215:2117–2127.

   https://doi.org/10.1242/jeb.065557

   • Google Scholar
10. 1. Blickhan R

    (1989) The spring-mass model for running and hopping

    Journal of Biomechanics 22:1217–1227.

    https://doi.org/10.1016/0021-9290(89)90224-8

    • Google Scholar
11. 1. Blickhan R
    2. Full RJ

    (1993) Similarity in multilegged locomotion: bouncing like a monopode

    Journal of Comparative Physiology A 173:509–517.

    https://doi.org/10.1007/BF00197760

    • Google Scholar
12. 1. Blum KP
    2. Lamotte D’Incamps B
    3. Zytnicki D
    4. Ting LH

    (2017) Force encoding in muscle spindles during stretch of passive muscle

    PLOS Computational Biology 13:e1005767.

    https://doi.org/10.1371/journal.pcbi.1005767

    • Google Scholar
13. 1. Bullimore SR
    2. Burn JF

    (2006) Consequences of forward translation of the point of force application for the mechanics of running

    Journal of Theoretical Biology 238:211–219.

    https://doi.org/10.1016/j.jtbi.2005.05.011

    • Google Scholar
14. 1. Cappellini G
    2. Ivanenko YP
    3. Poppele RE
    4. Lacquaniti F

    (2006) Motor patterns in human walking and running

    Journal of Neurophysiology 95:3426–3437.

    https://doi.org/10.1152/jn.00081.2006

    • Google Scholar
15. 1. Carver S
    2. Kiemel T
    3. Jeka JJ

    (2006) Modeling the dynamics of sensory reweighting

    Biological Cybernetics 95:123–134.

    https://doi.org/10.1007/s00422-006-0069-5

    • Google Scholar
16. 1. Cavanagh PR
    2. Pollock ML
    3. Landa J

    (1977) A biomechanical comparison of elite and good distance runners

    Annals of the New York Academy of Sciences 301:328–345.

    https://doi.org/10.1111/j.1749-6632.1977.tb38211.x

    • Google Scholar
17. 1. Chang Y-H
    2. Auyang AG
    3. Scholz JP
    4. Nichols TR

    (2009) Whole limb kinematics are preferentially conserved over individual joint kinematics after peripheral nerve injury

    Journal of Experimental Biology 212:3511–3521.

    https://doi.org/10.1242/jeb.033886

    • Google Scholar
18. 1. Chevallereau C
    2. Westervelt ER
    3. Grizzle JW

    (2005) Asymptotically stable running for a Five-Link, Four-Actuator, planar bipedal robot

    The International Journal of Robotics Research 24:431–464.

    https://doi.org/10.1177/0278364905054929

    • Google Scholar
19. Thesis

    1. Clark BL

    (2018)

    Energetic efficiency and stability in bipedal locomotion: 3D walking and energy-optimal perturbation rejection. PhD thesis.

    The Ohio State University.

    • Google Scholar
20. 1. Cullen KE

    (2004) Sensory signals during active versus passive movement

    Current Opinion in Neurobiology 14:698–706.

    https://doi.org/10.1016/j.conb.2004.10.002

    • Google Scholar
21. 1. Daley MA
    2. Biewener AA

    (2006) Running over rough terrain reveals limb control for intrinsic stability

    PNAS 103:15681–15686.

    https://doi.org/10.1073/pnas.0601473103

    • Google Scholar
22. 1. Dingwell JB
    2. Cusumano JP
    3. Cavanagh PR
    4. Sternad D

    (2001) Local dynamic stability versus kinematic variability of continuous overground and treadmill walking

    Journal of Biomechanical Engineering 123:27–32.

    https://doi.org/10.1115/1.1336798

    • Google Scholar
23. 1. Dingwell JB
    2. John J
    3. Cusumano JP

    (2010) Do humans optimally exploit redundancy to control step variability in walking?

    PLOS Computational Biology 6:e1000856.

    https://doi.org/10.1371/journal.pcbi.1000856

    • Google Scholar
24. Conference

    1. Dollar AM
    2. Herr H

    (2008) Design of a quasi-passive knee exoskeleton to assist running

    IEEE/RSJ International Conference on Intelligent Robots and Systems. pp. 747–754.

    https://doi.org/10.1109/IROS.2008.4651202

    • Google Scholar
25. Book

    1. Efron B
    2. Tibshirani RJ

    (1994)

    An Introduction to the Bootstrap

    CRC press.

    • Google Scholar
26. 1. Englsberger J
    2. Kozlowski P
    3. Ott C
    4. Albu-Schaffer A

    (2016) Biologically inspired deadbeat control for running: from human analysis to humanoid control and back

    IEEE Transactions on Robotics 32:854–867.

    https://doi.org/10.1109/TRO.2016.2581199

    • Google Scholar
27. 1. Fitzpatrick R
    2. Burke D
    3. Gandevia SC

    (1994) Task-dependent reflex responses and movement illusions evoked by galvanic vestibular stimulation in standing humans

    The Journal of Physiology 478:363–372.

    https://doi.org/10.1113/jphysiol.1994.sp020257

    • Google Scholar
28. 1. Gard SA
    2. Miff SC
    3. Kuo AD

    (2004) Comparison of kinematic and kinetic methods for computing the vertical motion of the body center of mass during walking

    Human Movement Science 22:597–610.

    https://doi.org/10.1016/j.humov.2003.11.002

    • Google Scholar
29. 1. Geyer H
    2. Seyfarth A
    3. Blickhan R

    (2006) Compliant leg behaviour explains basic dynamics of walking and running

    Proceedings of the Royal Society B: Biological Sciences 273:2861–2867.

    https://doi.org/10.1098/rspb.2006.3637

    • Google Scholar
30. 1. Ghigliazza RM
    2. Altendorfer R
    3. Holmes P
    4. Koditschek D

    (2005) A simply stabilized running model

    SIAM Review 47:519–549.

    https://doi.org/10.1137/050626594

    • Google Scholar
31. 1. Grimmer S
    2. Ernst M
    3. Gunther M
    4. Blickhan R

    (2008) Running on uneven ground: leg adjustment to vertical steps and self-stability

    Journal of Experimental Biology 211:2989–3000.

    https://doi.org/10.1242/jeb.014357

    • Google Scholar
32. Book

    1. Guckenheimer J
    2. Holmes PJ

    (2013)

    Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

    Springer Science & Business Media.

    • Google Scholar
33. Book

    1. Hamilton JD

    (1994)

    Time Series Analysis (2)

    Princeton, NJ: Princeton University Press.

    • Google Scholar
34. 1. Harris CM
    2. Wolpert DM

    (1998) Signal-dependent noise determines motor planning

    Nature 394:780–784.

    https://doi.org/10.1038/29528

    • Google Scholar
35. 1. Hausdorff JM
    2. Peng CK
    3. Ladin Z
    4. Wei JY
    5. Goldberger AL

    (1995) Is walking a random walk? evidence for long-range correlations in stride interval of human gait

    Journal of Applied Physiology 78:349–358.

    https://doi.org/10.1152/jappl.1995.78.1.349

    • Google Scholar
36. 1. Hodgins JK
    2. Raibert MN

    (1991) Adjusting step length for rough terrain locomotion

    IEEE Transactions on Robotics and Automation 7:289–298.

    https://doi.org/10.1109/70.88138

    • Google Scholar
37. 1. Hof AL
    2. Vermerris SM
    3. Gjaltema WA

    (2010) Balance responses to lateral perturbations in human treadmill walking

    Journal of Experimental Biology 213:2655–2664.

    https://doi.org/10.1242/jeb.042572

    • Google Scholar
38. 1. Hogan N

    (1984) Adaptive control of mechanical impedance by coactivation of antagonist muscles

    IEEE Transactions on Automatic Control 29:681–690.

    https://doi.org/10.1109/TAC.1984.1103644

    • Google Scholar
39. 1. Hurmuzlu Y
    2. Basdogan C

    (1994) On the measurement of dynamic stability of human locomotion

    Journal of Biomechanical Engineering 116:30–36.

    https://doi.org/10.1115/1.2895701

    • Google Scholar
40. 1. Iles JF
    2. Baderin R
    3. Tanner R
    4. Simon A

    (2007) Human standing and walking: comparison of the effects of stimulation of the vestibular system

    Experimental Brain Research 178:151–166.

    https://doi.org/10.1007/s00221-006-0721-2

    • Google Scholar
41. 1. Jindrich DL
    2. Smith NC
    3. Jespers K
    4. Wilson AM

    (2007) Mechanics of cutting maneuvers by ostriches (Struthio camelus)

    Journal of Experimental Biology 210:1378–1390.

    https://doi.org/10.1242/jeb.001545

    • Google Scholar
42. 1. Jindrich DL
    2. Full RJ

    (2002)

    Dynamic stabilization of rapid hexapedal locomotion

    The Journal of Experimental Biology 205:2803–2823.

    • Google Scholar
43. 1. Jordan K
    2. Challis JH
    3. Newell KM

    (2006) Long range correlations in the stride interval of running

    Gait & Posture 24:120–125.

    https://doi.org/10.1016/j.gaitpost.2005.08.003

    • Google Scholar
44. 1. Jordan K
    2. Challis JH
    3. Newell KM

    (2007) Speed influences on the scaling behavior of gait cycle fluctuations during treadmill running

    Human Movement Science 26:87–102.

    https://doi.org/10.1016/j.humov.2006.10.001

    • Google Scholar
45. 1. Jordan K
    2. Newell KM

    (2008) The structure of variability in human walking and running is Speed-Dependent

    Exercise and Sport Sciences Reviews 36:200–204.

    https://doi.org/10.1097/JES.0b013e3181877d71

    • Google Scholar
46. 1. Kaipust JP
    2. Huisinga JM
    3. Filipi M
    4. Stergiou N

    (2012) Gait variability measures reveal differences between multiple sclerosis patients and healthy controls

    Motor Control 16:229–244.

    https://doi.org/10.1123/mcj.16.2.229

    • Google Scholar
47. 1. Kuo AD

    (2005) An optimal state estimation model of sensory integration in human postural balance

    Journal of Neural Engineering 2:S235–S249.

    https://doi.org/10.1088/1741-2560/2/3/S07

    • Google Scholar
48. 1. Luinge HJ
    2. Veltink PH

    (2005) Measuring orientation of human body segments using miniature gyroscopes and accelerometers

    Medical & Biological Engineering & Computing 43:273–282.

    https://doi.org/10.1007/BF02345966

    • Google Scholar
49. 1. Maus H-M
    2. Revzen S
    3. Guckenheimer J
    4. Ludwig C
    5. Reger J
    6. Seyfarth A

    (2015) Constructing predictive models of human running

    Journal of the Royal Society Interface 12:20140899.

    https://doi.org/10.1098/rsif.2014.0899

    • Google Scholar
50. 1. McMahon TA
    2. Cheng GC

    (1990) The mechanics of running: how does stiffness couple with speed?

    Journal of Biomechanics 23:65–78.

    https://doi.org/10.1016/0021-9290(90)90042-2

    • Google Scholar
51. 1. McN. Alexander R

    (1997) Invited editorial on “Interaction of leg stiffness and surface stiffness during human hopping”

    Journal of Applied Physiology 82:13–14.

    https://doi.org/10.1152/jappl.1997.82.1.13

    • Google Scholar
52. 1. Müller R
    2. Ernst M
    3. Blickhan R

    (2012) Leg adjustments during running across visible and camouflaged incidental changes in ground level

    Journal of Experimental Biology 215:3072–3079.

    https://doi.org/10.1242/jeb.072314

    • PubMed
    • Google Scholar
53. Book

    1. Nelson G
    2. Saunders A
    3. Playter R

    (2019)

    The PETMAN and Atlas Robots at Boston Dynamics

    In: Goswami PV A, editors. Humanoid Robotics: A Reference. Springer. pp. 169–186.

    • Google Scholar
54. 1. O'Loughlin JL
    2. Robitaille Y
    3. Boivin J-F
    4. Suissa S

    (1993) Incidence of and risk factors for falls and injurious falls among the Community-dwelling elderly

    American Journal of Epidemiology 137:342–354.

    https://doi.org/10.1093/oxfordjournals.aje.a116681

    • Google Scholar
55. 1. Osborne LC
    2. Lisberger SG
    3. Bialek W

    (2005) A sensory source for motor variation

    Nature 437:412–416.

    https://doi.org/10.1038/nature03961

    • Google Scholar
56. 1. Patla AE

    (1997) Understanding the roles of vision in the control of human locomotion

    Gait & Posture 5:54–69.

    https://doi.org/10.1016/S0966-6362(96)01109-5

    • Google Scholar
57. 1. Patla AE
    2. Adkin A
    3. Ballard T

    (1999) Online steering: coordination and control of body center of mass, head and body reorientation

    Experimental Brain Research 129:0629–0634.

    https://doi.org/10.1007/s002210050932

    • Google Scholar
58. 1. Pearson KG
    2. Collins DF

    (1993) Reversal of the influence of group ib afferents from plantaris on activity in medial gastrocnemius muscle during locomotor activity

    Journal of Neurophysiology 70:1009–1017.

    https://doi.org/10.1152/jn.1993.70.3.1009

    • Google Scholar
59. 1. Perry JA
    2. Srinivasan M

    (2017) Walking with wider steps changes foot placement control, increases kinematic variability and does not improve linear stability

    Royal Society Open Science 4:160627.

    https://doi.org/10.1098/rsos.160627

    • Google Scholar
60. 1. Peters M

    (1988) Footedness: asymmetries in foot preference and skill and neuropsychological assessment of foot movement

    Psychological Bulletin 103:179–192.

    https://doi.org/10.1037/0033-2909.103.2.179

    • Google Scholar
61. 1. Peuker F
    2. Maufroy C
    3. Seyfarth A

    (2012) Leg-adjustment strategies for stable running in three dimensions

    Bioinspiration & Biomimetics 7:036002.

    https://doi.org/10.1088/1748-3182/7/3/036002

    • Google Scholar
62. 1. Qiao M
    2. Jindrich DL

    (2012) Task-Level strategies for human Sagittal-Plane running maneuvers are consistent with robotic control policies

    PLOS ONE 7:e51888.

    https://doi.org/10.1371/journal.pone.0051888

    • Google Scholar
63. 1. Qiao M
    2. Jindrich DL

    (2014) Compensations during unsteady locomotion

    Integrative and Comparative Biology 54:1109–1121.

    https://doi.org/10.1093/icb/icu058

    • Google Scholar
64. Book

    1. Raibert MH

    (1986)

    Legged Robots That Balance

    MIT press.

    • Google Scholar
65. Conference

    1. Riddick RC
    2. Kuo AD

    (2016)

    Shortcomings of thinking like a spring loaded inverted pendulum (SLIP) model: mechanical compensation to fore-aft perturbations in human running

    Dynamic Walking Conference 2016.

    • Google Scholar
66. 1. Ruina A
    2. Bertram JEA
    3. Srinivasan M

    (2005) A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition

    Journal of Theoretical Biology 237:170–192.

    https://doi.org/10.1016/j.jtbi.2005.04.004

    • Google Scholar
67. 1. Sainburg RL
    2. Ghilardi MF
    3. Poizner H
    4. Ghez C

    (1995) Control of limb dynamics in normal subjects and patients without proprioception

    Journal of Neurophysiology 73:820–835.

    https://doi.org/10.1152/jn.1995.73.2.820

    • Google Scholar
68. 1. Schepers HM
    2. van Asseldonk E
    3. Buurke JH
    4. Veltink PH

    (2009) Ambulatory estimation of center of mass displacement during walking

    IEEE Transactions on Biomedical Engineering 56:1189–1195.

    https://doi.org/10.1109/TBME.2008.2011059

    • Google Scholar
69. 1. Seipel JE
    2. Holmes P

    (2005) Running in three dimensions: analysis of a Point-mass Sprung-leg model

    The International Journal of Robotics Research 24:657–674.

    https://doi.org/10.1177/0278364905056194

    • Google Scholar
70. 1. Seyfarth A
    2. Geyer H
    3. Günther M
    4. Blickhan R

    (2002) A movement criterion for running

    Journal of Biomechanics 35:649–655.

    https://doi.org/10.1016/S0021-9290(01)00245-7

    • Google Scholar
71. 1. Seyfarth A

    (2003) Swing-leg retraction: a simple control model for stable running

    Journal of Experimental Biology 206:2547–2555.

    https://doi.org/10.1242/jeb.00463

    • Google Scholar
72. 1. Shultz AH
    2. Lawson BE
    3. Goldfarb M

    (2015)

    Running with a powered knee and ankle prosthesis

    IEEE Transactions on Neural Systems and Rehabilitation Engineering : A Publication of the IEEE Engineering in Medicine and Biology Society 23:403–412.

    • Google Scholar
73. 1. Sinkjær T
    2. Andersen JB
    3. Nielsen JF
    4. Hansen HJ

    (1999) Soleus long-latency stretch reflexes during walking in healthy and spastic humans

    Clinical Neurophysiology 110:951–959.

    https://doi.org/10.1016/S1388-2457(99)00034-6

    • Google Scholar
74. 1. Srinivasan M

    (2009) Optimal speeds for walking and running, and walking on a moving walkway

    Chaos: An Interdisciplinary Journal of Nonlinear Science 19:026112.

    https://doi.org/10.1063/1.3141428

    • Google Scholar
75. 1. Srinivasan M

    (2011) Fifteen observations on the structure of energy-minimizing gaits in many simple biped models

    Journal of the Royal Society Interface 8:74–98.

    https://doi.org/10.1098/rsif.2009.0544

    • Google Scholar
76. 1. Srinivasan M
    2. Holmes P

    (2008) How well can spring-mass-like telescoping leg models fit multi-pedal sagittal-plane locomotion data?

    Journal of Theoretical Biology 255:1–7.

    https://doi.org/10.1016/j.jtbi.2008.06.034

    • Google Scholar
77. 1. Srinivasan M
    2. Ruina A

    (2006) Computer optimization of a minimal biped model discovers walking and running

    Nature 439:72–75.

    https://doi.org/10.1038/nature04113

    • Google Scholar
78. Conference

    1. Tajima R
    2. Honda D
    3. Suga K

    (2009)

    Fast running experiments involving a humanoid robot

    2009 IEEE International Conference on Robotics and Automation. pp. 1571–1576.

    • Google Scholar
79. 1. Van Woensel W
    2. Cavanagh PR

    (1992) A perturbation study of lower extremity motion during running

    International Journal of Sport Biomechanics 8:30–47.

    https://doi.org/10.1123/ijsb.8.1.30

    • Google Scholar
80. 1. Voloshina AS
    2. Ferris DP

    (2015) Biomechanics and energetics of running on uneven terrain

    Journal of Experimental Biology 218:711–719.

    https://doi.org/10.1242/jeb.106518

    • Google Scholar
81. Thesis

    1. Wang Y

    (2013)

    System identification around periodic orbits with application to steady state human walking. PhD thesis

    The Ohio State University.

    • Google Scholar
82. Conference

    1. Wang Y
    2. Srinivasan M

    (2012) System identification and stability analysis of steady human walking and the swing leg dynamics

    ASME 2012 5th Annual Dynamic Systems and Control Conference Joint with the JSME 2012 11th Motion and Vibration Conference. pp. 19–23.

    https://doi.org/10.1115/DSCC2012-MOVIC2012-8663

    • Google Scholar
83. 1. Wang Y
    2. Srinivasan M

    (2014) Stepping in the direction of the fall: the next foot placement can be predicted from current upper body state in steady-state walking

    Biology Letters 10:20140405.

    https://doi.org/10.1098/rsbl.2014.0405

    • Google Scholar
84. 1. Warren WH
    2. Young DS
    3. Lee DN

    (1986) Visual control of step length during running over irregular terrain

    Journal of Experimental Psychology: Human Perception and Performance 12:259–266.

    https://doi.org/10.1037/0096-1523.12.3.259

    • Google Scholar
85. 1. Wolpert D
    2. Ghahramani Z
    3. Jordan M

    (1995) An internal model for sensorimotor integration

    Science 269:1880–1882.

    https://doi.org/10.1126/science.7569931

    • Google Scholar
86. 1. Yen JT
    2. Auyang AG
    3. Chang Y-H

    (2009) Joint-level kinetic redundancy is exploited to control limb-level forces during human hopping

    Experimental Brain Research 196:439–451.

    https://doi.org/10.1007/s00221-009-1868-4

    • Google Scholar
87. 1. Zajac FE

    (1989)

    Muscle and tendon properties models scaling and application to biomechanics and motor

    Critical Reviews in Biomedical Engineering 17:359–411.

    • Google Scholar

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

• 4,911

  views

• 449

  downloads

• 29

  citations

Views, downloads and citations are aggregated across all versions of this paper published by eLife.