Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill can reduce energy costs below normal

Abstract

Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a point-mass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energy-optimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a mass- spring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.

1. Introduction

While many modern humans mostly walk on stable, fixed surfaces such as tiled floors, concrete pavements and solid ground, humans can walk and run on a wide variety of surfaces with different compliances, damping and granularity, and do so with remarkable stereotypicality (e.g. [1,2]). In a famous public event, when pedestrians were allowed on to the shaky walking surface of the London Millennium Bridge [3,4], many walked in synchrony with each other and the bridge, with larger-than-normal step widths and therefore applied larger-than-normal sideways forces on the bridge. This eventually resulted in large lateral oscillations of the bridge, causing more and more people to walk the same way. Similar lateral oscillations have been observed in other footbridges as well [5–7]. Experiments with humans walking on laterally oscillating treadmills have shown a similar response, namely larger sideways forces and wider step widths [8]. In this article, we examine the mechanics of walking on a shaky bridge and walking on an externally shaken treadmill by deriving the energy-optimal motions of a simple mechanics-based mathematical model of a biped, and showing, perhaps surprisingly, that walking on such shaky surfaces can reduce walking energy requirements at high oscillation amplitudes.

A natural hypothesis is that humans change their walking gait on a shaky bridge to be more stable, i.e. not fall down. Wider step widths might improve sideways stability [9] and humans use sideways foot-placement to avoid falling sideways and recover from sideways perturbations [10,11]. Indeed, a few authors [7,8] have used ‘inverted pendulum models’ of lateral pedestrian motion with step-width-control-based stabilization to explain pedestrian-driven bridge oscillations. In these models, even though the pedestrians cannot influence the motion of the bridge (the bridge is assumed to oscillate sinusoidally with fixed amplitude and frequency), they perform mechanical work on the bridge; thus, the modelled pedestrian was found to act, effectively, as a negative damper on bridge motion, producing an additional sideways force at the bridge vibration frequency. In contrast to this stabilization hypothesis, Strogatz et al. [4] modelled the pedestrians as abstract phase oscillators with randomly distributed oscillation frequencies (close to the bridge’s natural frequency), coupled to the same mass-spring-damper modelling the London Millennium Bridge. Here, as the pedestrian count increased beyond a critical number, they synchronized with each other and the bridge, leading to a steady-state motion with large bridge oscillation amplitude. While these models provide insight into the causes for additional sideways forces and synchronization, none of these models have considered the fully coupled interaction of one or more humans with the bridge during forward walking.

In this article, we take an alternative approach to modelling the human response to surface motion. One of the most successful predictive theories of steady-state human locomotion is that humans move in a manner that minimizes the metabolic cost of walking. Experimental studies have shown that humans use the metabolically optimal speed, step length, step width, arm swing, gait transition speeds [12–16], etc. Energy-optimal motions in computational models of a biped have explained numerous features of human locomotion [17–25]. Remarkably, even with little practice, humans seem to find energy-optimal movement patterns even for novel tasks such as sideways walking [26], split-belt treadmill walking [27], and tasks with unusual frequency and step length constraints [28]. Therefore, we consider the hypothesis that humans move on shaky bridges and shaking treadmills in a manner that minimizes the metabolic energy cost of walking.

Here, we predict walking motions by numerical optimization, deriving the periodic motions that minimize a work-based metabolic cost of walking; we consider the modelling analogues of both walking on a shaky bridge and walking on an externally shaken treadmill, considering a range of speed and other parameter values. Our biped model, introduced in §2, is capable of arbitrarily complex walking motions [21,23], allowing for a wide range of pedestrian and bridge motions. We assume that the pedestrians walk in synchrony with the platform on which they walk and show that in some model parameter regimes, entrained walking on a shaken treadmill or a shaky bridge can have lower energy costs than normal walking on a non-moving surface. We examine the effect of the number of pedestrians, the bridge stiffness and the damping on the amplitude and frequency of oscillations, and suggest that metabolic energy minimization could be an alternative explanation of the unusual interaction between humans and the walking surface.

2. A three-dimensional biped model for walking on a moving surface

We consider a simple biped consisting of a point-mass upper body of mass m and two legs, as shown in figure 1, capable of three-dimensional movement. During each step, only one leg is in contact with the ground, while the other one can be swung arbitrarily. The legs are massless for dynamical purposes, so that leg swing does not affect the upper body, but have a mass m_foot to calculate the leg-swing-cost (as in [29,23]; see appendix A(b)). Both legs have point feet and their hip joints are coincident with the point-mass upper body, so that the model has zero hip-width. When a leg is in contact with the ground (that is, in ‘stance phase’), the stance-leg-knee can flex and extend, applying arbitrarily varying forces on the upper body, thereby changing the effective leg length. These motions are subject to a maximum leg length and leg-force constraint. It is further assumed that the foot does not slip relative to the walking surface during stance. Flight (when neither leg contacts the ground) and double-support or double-stance (when both legs are in contact with the ground) are not considered [21,29], as their absence seems optimal for this class of biped models at walking speeds [23]. Figure 1 shows two possible embodiments of the leg, one with a knee, one with a telescoping actuator [21]; the knee is just one way of accomplishing leg length changes. The model’s motion capability is independent of the embodiment [23].

Figure 1.

Biped model on a platform. The biped has a point-mass upper body and can move in all three directions: forward (X-direction), vertically (Y -direction) and laterally (Z-direction). The platform can only move laterally. Two possible embodiments of the leg are shown, one with a knee and one with a telescoping actuator.

In its leg force capability, the biped model is a generalization of the classic inverted pendulum walking model [21,30,31] and various spring mass models [32,33]. A planar version of this biped has been shown to discover both walking and running-like gaits when moving in a straight line while minimizing energy cost per unit distance [23]. Here, in addition to allowing three-dimensional movement for the centre-of-mass, we allow the walking surface to move laterally, i.e. perpendicular to walking direction (figure 1).

We consider two types of walking surfaces or ‘platforms’:

• — Infinite inertia platform, for which the platform can affect biped movement, but the biped cannot affect platform movement. The platform movement is externally specified, as in the case of a treadmill oscillated sideways, e.g. sinusoidally. Thus, we will also refer to this as the ‘shaken treadmill’ case.

• — Finite inertia platform, a one-degree-of-freedom mass-spring-damper system, whose motion can be influenced by the biped’s leg forces, as in equation (5). We only consider a lateral degree-of-freedom. Examples include light footbridges and walkways. Thus, we will also refer to this as the ‘shaky bridge’ case.

The differential equations of motion describing the motion of the biped model and the platform are given in appendix A(a), equations (2)–(4) for the biped and equations (5)–(6) for the two types of platforms.

The metabolic energy cost function is a sum of four components [23], described with equations in appendix A(b): (i) the resting metabolic rate, a constant cost per unit time, (ii) Stance work cost, proportional to a linear combination of the positive and negative work of the legs during stance, (iii) Stance force cost, proportional to integrated leg force during stance, and (iv) Swing leg cost, proportional to the work needed to move the swing leg to its next stance position.

We seek the walking motions, periodic over two walking steps (equal to one stride), that minimize the metabolic cost of forward progression. Thus, for either platform model, we parametrize the space of possible walking motions using finitely many unknowns, using initial conditions for the steps and describing the leg forces F(t) using piecewise linear functions [21]. We use numerical optimization to determine the values of these unknown parameters so as to produce net forward movement while minimizing an energy cost function, subject to leg length, leg force and periodicity constraints. See appendix A(c) for computational details (also [21,23]). For some calculations, we specify the speed and stride length—and consequently the stride frequency—whereas in other calculations, all such quantities are unknowns to be determined by the optimization. We specify speed and stride length when we wish to isolate the stance work and stance force cost, as all other cost terms are constant—independent of body motion—for a given speed and stride length.

For reference, we note that for this biped model, it has been shown [21,23] that the planar ‘inverted pendulum walking’ motion minimized this metabolic cost in the absence of any surface motion. In this optimal walking motion, each stance phase consists of an inverted pendulum-like motion of the body vaulting over a straight leg and the step-to-step transition accomplished using a push-off impulse by the trailing leg, followed by a heel-strike impulse by the leading leg. The stance work cost is entirely due to the positive and negative work performed during the push-off and heel-strike impulses. Of course, when the walking surface is moving, this planar walking motion will no longer be feasible.

Note that when the platform is moving, we do assume that the human walking motion is entrained to the platform’s motion so that the state of the whole system is periodic. For the infinite inertia platform, our basic calculation assumes equality of the stride period and platform period. For the finite inertia platform, our assumption does allow the bridge to perform multiple periods per human stride, but such motions were never optimal. Optimization calculations with periodicity of multiple human steps are more computationally expensive and were not considered. Further, we did not consider optimizations in which the system can be non-periodic; as described in the Discussion, such non-periodic optimization calculations are conceptually problematic or also computationally expensive.

In the following sections, all results are presented in terms of non-dimensional quantities, obtained by normalization using appropriate combinations of the mass of the body, the length of the fully extended leg and the acceleration due to gravity; the exact normalizations for non-dimensionalization are described in appendix A(a).

3. Infinite inertia platform: walking on oscillating treadmills

In this calculation, we couple the biped to an infinite inertia platform, sinusoidally oscillating sideways (equation (6)). As noted, this is a model of a person walking on an externally oscillated treadmill, with the person having no ability to affect the treadmill’s motion. We consider a range of platform oscillation amplitudes and frequencies. The walking stride frequency is assumed to match the platform oscillation frequency, so that the biped is entrained to the platform, i.e. during one walking stride, exactly one platform oscillation occurs. The initial position and velocity of the body centre of mass, the relative phase between the platform motion and the gait cycle, the position of the foot during the second step and the piecewise linear forces exerted by leg on the centre of mass are parameters of the optimization. The position of the foot during the first step is taken as the origin for the coordinate system.

For a fixed non-dimensional walking speed of 0.4 (about 1.2 m s⁻¹ for a person with leg length 0.95 m), we determine the optimal walking motion for a range of platform amplitudes and frequencies. The range of frequencies was chosen to be around the optimal non-dimensional stride frequency (around 0.4) for the biped while walking without any platform oscillation (planar walking). Figure 2a shows that the biped can reduce its metabolic cost of walking by appropriately coordinating the leg forces, the initial position and velocity of the centre of mass and the phase relative to the platform motion. In particular, we find that even for a non-dimensional oscillation amplitude of 0.03 (about 3 cm), there is a notable reduction in the model metabolic cost for a frequency beyond a non-dimensional frequency of 0.3. Within the frequency range explored here, higher platform oscillation amplitudes allow greater cost reduction for a broader range of frequencies. Electronic supplementary material, appendix figure S8, shows that the qualitative phenomenon of energy cost reduction persists for a range of forward speeds and model parameters.

Figure 2.

Energy-optimal walking on an oscillating infinite inertia platform. (a) Cost reduction with platform oscillation, (b) optimal body motions and (c) optimal body motions (perspective view). (a) Energy cost per unit distance versus oscillation frequency for 0, 0.03 and 0.07 non-dimensional oscillation amplitudes. Walking speed is fixed at 1.2 m s⁻¹ (non-dimensional speed=0.4). Thus, we see that oscillating platforms reduce walking energy cost for a range of oscillation frequencies. (b) Motion of the centre of mass and platform for 0.07 non-dimensional platform amplitude and non-dimensional frequency 0.4, denoted by a ‘star’ (⋆) in panel (a). The blue curve shows the corresponding position of the platform for each forward position of the centre of mass, the red curve represents the motion of the centre of mass, the green lines are the loci of the feet, with open circles showing the position of the foot at the heel-strike (red) and push-off (green). (c) The three-dimensional optimal body motions are shown in perspective view. The leg directions are shown as blue lines between the foot positions (filled black circles, moving sideways) and the centre of mass positions (black circles with filled grey). The two steps are reflection symmetric about a midline as seen in panel (b), even though not immediately apparent in this three-dimensional view.

 Figure 2b,c shows the body trajectory for a particular platform oscillation amplitude and frequency, indicating that the lateral body motion is smaller than the platform motion. Analogous to walking on a non-moving surface [21,23], we find that the optimal motion in the presence of platform motion is also an inverted pendulum-like motion with impulses corresponding to heel-strike and push-off—this is a non-planar inverted pendulum walking motion. Figure 2b,c shows such an inverted pendulum-like motion of the centre of mass; electronic supplementary material, appendix figure S1a, obtained by allowing arbitrary piecewise linear leg force changes, shows that the optimal motion displays the following features of inverted pendulum walking. The leg is always stretched close to the maximum limit, i.e. the biped walks with a straight knee during stance, with a large step width so as to manage the lateral platform movement. The push-off and heel-strike occur as brief impulses near the extremes of platform motion (as in [21]), enabling the instantaneous velocity change from one pendular step to the next; the push-off does positive work and the heel-strike does negative work. This optimal motion corresponds to a relative phase of about π/2 between the platform motion and the biped motion. When the platform moves leftward, the body remains to the left of the foot. As the body moves forward during a given stance phase, the foot (fixed relative to platform) moves inward towards the body from heel-strike to push-off.

For some oscillation parameters, we found that it is possible for the model to walk on an oscillating platform without any heel-strike or push-off impulses—in other words, smoothly transitioning from one inverted pendulum stance phase to the next. Such a motion would optimize the ‘stance work cost’ portion of our metabolic cost model, making this leg work equal zero; see electronic supplementary material, appendix figure S1b. But such a motion is not optimal for the full metabolic cost (i.e. not one of those found in figure 2a), because other components of metabolic cost (leg swing and stance force costs) are higher for this motion.

We find that the mechanism by which the oscillating platform reduces the cost of walking is by reducing the necessary push-off and heel-strike impulse magnitudes, thereby reducing the positive and negative stance leg work during these impulses. The variation in energy cost components can be seen in the electronic supplementary material, appendix figure S13, which shows that the decrease in the stance work cost more than compensates for a slight increase in stance force cost. By appropriately synchronizing the body motion with the platform motion, one can let the platform perform some of the mechanical work on the body that would otherwise have to be done by the legs. The leg work performed by the impulses are smaller because a smaller velocity direction change is required across the step-to-step transition than in normal walking. In particular, the out-of-plane motion of the centre of mass makes the sagittal-plane projection of the centre of mass trajectories much flatter than in normal walking—ultimately leading to the smaller velocity direction change across the step-to-step transition [34,35]. See the electronic supplementary material, video S1, for animations of these optimal walking motions.

4. Finite inertia platform: walking on shaky bridges

We consider a finite inertia platform that is a one degree-of-freedom spring-mass-damper system (as also in [4]). This model is a closer approximation to a bridge than an externally oscillated infinite inertia platform, because the pedestrian can actually influence the platform motion and this platform motion is not restricted to purely sinusoidal oscillations. Again, as for the infinite inertia case above, we first established that even when we allow arbitrary leg length changes and piecewise linear leg force changes, the optimal motion is a three-dimensional non-planar inverted pendular motion with push-off and heel-strike impulses, as shown in the electronic supplementary material, appendix figure S7. Therefore, the calculations described below assume the biped walks using exactly an inverted pendulum gait using the maximum leg length. This inverted pendulum simplification speeds up the optimizations and improves convergence. Push-off impulses are parameters of the optimization, but heel-strike impulses are computed, based on the motion of the centre-of-mass, assuming a plastic collision (as in [29]), to make the leg-length-rate after heel-strike exactly zero.

The stiffness, mass and damping of the platform are selected as equal to the corresponding modal quantities for the Millennium Bridge (as in [4]); see appendix A(a) for these and other human parameters used. Further, to establish more general trends, we also consider multiples of these stiffness and damping values, denoted 0.5×, 2×, etc. The unknown parameters for optimization are stride length, stride frequency, initial position and velocity of both the centre of mass and the platform, the position of the foot for the second step, and the magnitudes of push-off impulse; we do not constrain forward speed. Once again, the position of the foot during the first step is taken to be the origin of the coordinate system. The initial conditions of the platform serve the same purpose as the relative phase from the infinite-inertia calculation. Unlike the infinite-inertia platform, the platform oscillations are not constrained in any way here, in that there can be any number (even non-integer number) of oscillations per stride and the amplitude can be as large or as small as needed. This allows for solutions where there is no platform oscillation. For each stiffness and damping value, we perform the optimization calculations for a range of pedestrian counts N_pedestrians between 1 and 1000 or 1500. We model N_pedestrians pedestrians simply as one large pedestrian with N_pedestrians times the mass m of a single pedestrian, thus assuming that all N_pedestrians pedestrians walk in synchrony with each other. For reference, at N_pedestrians≈1600, the total pedestrian mass equals the bridge mass.

First, for stiffness and damping equal to that of the Millennium Bridge (1× damping curves in figure 3a,b), we find that a walking motion that substantially oscillates the platform is optimal when the pedestrian number increases beyond a critical level, about 750 pedestrians. The reduction in metabolic cost from the oscillating bridge is small. Nevertheless, these results show that if there are a large number of people walking in synchrony, they can potentially lower their metabolic cost by shaking the bridge. For lower numbers of pedestrians, the optimal walking motion is planar, without oscillating the bridge. Figure 4 shows the optimal body and foot motion for N_pedestrians=1000 and the London Millennium Bridge parameters; see also the electronic supplementary material, appendix figure S2a and video S2.

Figure 3.

Energy-optimal walking on the finite inertia platform shows platform oscillations reduced metabolic cost when there are sufficiently many pedestrians. (a) Metabolic energy per unit distance versus number of pedestrians for the exact Millennium Bridge stiffness and for different multiples of the Millennium Bridge damping. Lower damping and more pedestrians allow for greater reduction in the energy cost. (b) Platform oscillation amplitude versus number of pedestrians for damping in multiples of the Millennium Bridge damping and the exact Millennium Bridge stiffness. More pedestrians and lower damping lead to higher oscillation amplitudes. (c) Metabolic energy per unit distance versus number of pedestrians for stiffness in multiples of the Millennium Bridge stiffness and 0.5× the Millennium Bridge damping. Higher stiffness and more pedestrians allow for greater reduction in the energy cost. (d) Platform oscillation amplitude per unit distance versus number of pedestrians for stiffness in multiples of the Millennium Bridge stiffness and 0.5× the Millennium Bridge damping. Lower stiffness and more pedestrians allow for greater oscillation amplitudes. See the electronic supplementary material, appendix figures S3 and S5, for additional details such as optimal step widths, speed and stride frequency.

Figure 4.

Optimal walking motions for London Millennium Bridge parameters. (a) Perspective view and (b) top view (orthogonal projection) of the centre of mass trajectory for N_pedestrians≈1000, non-dimensional walking speed is close to 0.3 and non-dimensional stride frequency is about 0.29. Leg directions are shown between corresponding foot and centre of mass positions. The foot moves outward and then inward in a single step, as the body moves forward. See the electronic supplementary material, appendix figure S2, for another representation of the same information and electronic supplementary material, video S2, for an animation of this motion.

 Figure 3a,b also shows these trends for a range of damping values from 0× to 1× the Millennium Bridge damping, keeping the stiffness fixed at that of the Millennium Bridge (1× stiffness). As damping is decreased from 1×, the critical number of pedestrians required to make the oscillating platform solution optimal reduces; for instance, at 0.4× damping, the critical number is about 250 pedestrians (figure 3a,b). Independent of platform damping, increasing the number of pedestrians beyond the critical number leads to an increase in platform oscillation amplitude and a corresponding decrease in metabolic cost per unit distance; this decrease in metabolic cost is smaller for higher platform damping (figure 3a,b). For the oscillating solutions, the pedestrian walks with a large step width, which increases with platform oscillation amplitude and number of pedestrians (electronic supplementary material, appendix figures S3–S5). We also note that these qualitative trends persist for a large range of model parameters (foot mass, resting rate and stance force cost scaling) as shown in electronic supplementary material, appendix figure S9.

Fixing the damping to 0.5 times that of the Millennium Bridge and varying the platform stiffness, we see similar qualitative behaviour: for all stiffnesses, beyond a critical number of pedestrians, a larger number of pedestrians leads to oscillating platforms with a corresponding reduction in metabolic cost per unit distance. In figure 3c,d, for the number of pedestrians explored, we see that the metabolic energy cost per unit distance reduces with increasing stiffness from 0.75× to 1.25×; the metabolic cost increases with increasing stiffness from 1.25× to 1.5×. The critical number of people shows the opposite trend with changing stiffness. Higher stiffness leads to lower oscillation amplitudes. Overall, while the trends respond in a simple monotonic manner to changes in platform damping, they respond non-monotonically to changes in platform stiffness.

The principal mechanism for the metabolic cost reduction is the lowering of the stance work required of the legs, reflected in smaller magnitudes for the heel-strike impulse. The effect of oscillations on the individual components of the cost function can be seen in electronic supplementary material, appendix figures S10 and S11. The reduction in the required leg work is greatest when damping is zero, when an oscillating platform solution is optimal for even one pedestrian (see the electronic supplementary material, figure S4, for results of the zero damping case). When there is finite damping, the positive work at push-off should sufficiently exceed the negative work at heel-strike so as to compensate for the energy lost to the bridge’s structural damping. Again, the sideways motion of the body and the bridge reduces the necessary change in body velocity direction at the step-to-step transition, thereby decreasing the work required of the push-off and heel-strike impulses [35,34].

Why is there a critical pedestrian number for oscillating solutions to be better than planar solutions? The reason appears to be the presence of damping. In general, we find that increasing the number of pedestrians lowers the metabolic energy cost per unit distance in the oscillating platform solution. As noted, when there is considerable damping, net positive leg work needs to be performed to replace the energy lost in the bridge motion. This additional net positive work requirement (for a given platform amplitude) is distributed over all pedestrians, and consequently, for a large enough pedestrian count the oscillating solution becomes better than the planar solution.

The non-dimensional optimal speeds for the range of parameters considered appear to be between 0.3 and 0.45, which correspond to 0.95 and 1.4 m s⁻¹. The model parameters correspond to a slower-than-natural optimal speed for planar walking on a non-moving platform, but this optimal speed can be made to agree with normal human optimal speeds by choosing an appropriate resting metabolic rate, without changing any of the qualitative phenomena of energy cost reduction by a sideways shaking bridge.

For any given speed, the oscillating solution has lower cost than the planar solution only when the non-dimensional stride frequency is around 0.3 as demonstrated in the electronic supplementary material, appendix figure S12, which was obtained by solving the optimizations for a range of fixed walking speeds and stride lengths for 1000 pedestrians. This optimal stride frequency is also close to the bridge’s natural frequency (≈ 0.33).

5. Discussion

The simple biped model suggests that walking on a laterally oscillating platform could be more efficient than planar walking in terms of metabolic energy cost. For walking on the finite inertia platform (shaky bridge), we find that the optimal solution is inverted pendulum walking with stride frequency entrained to the platform oscillation frequency, with near 0 phase difference between the pedestrian and the platform. These results for the model are similar to observations made about humans walking on the London Millennium Bridge [3] as well as on laterally oscillating treadmills [36]. However, on the infinite inertia platform, the optimal for our model was closer to π/2 phase difference.

While we have shown that the bipeds can potentially reduce the energy cost by synchronizing their walking with the platform’s motion, we have not ruled out the possibility of some arbitrarily complex non-periodic motion of the system that has an even lower energy cost. We did not do optimizations allowing non-periodic motions for two complementary reasons—the same implicit reasons why almost every biped walking optimization has assumed periodicity. First, allowing the initial state to be different from the final state allows efficient forward movements that we would not call walking: for instance, turning off all the muscles and falling forward; or entirely using up some initially stored energy in the bridge to move forward, but only for a couple of steps. Such non-locomotor behaviours might be ruled by requiring persistent walking for sufficient distance, but then the resulting optimization over a large number of steps would be vastly more computationally intensive. We hope to consider optimizations with multiple independent pedestrians or with periodicity over multiple strides in future work.

For the models considered here, planar walking movement and zero step-width is optimal when the platform is not moving. However, humans walk with non-zero step width even in normal gait, perhaps because of having finite hip-widths [15]. Thus, normal human walking exerts a small sideways force even in the absence of platform oscillation. To understand the qualitative effects of having a minimum step width, we repeated the optimization calculations with the finite inertia platform while constraining the walking step width to be at least the normal human step width (about 0.15 leg lengths), so that even normal walking produced sideways leg forces for the model. Even with this non-zero step width constraint, one could have non-oscillating solutions if one half of the pedestrians were exactly out of sync with the other half, or if all pedestrians had random phase relative to each other. Such non-oscillating solutions were sub-optimal. This step-width constraint leads to solutions with oscillating platforms and synchronized pedestrians for all pedestrian counts (see the electronic supplementary material, appendix figure S6). For higher pedestrian numbers, the optimal solutions are identical to those obtained without the step width constraint and for lower pedestrian numbers, the optimal solution simply uses the minimum step width allowed (instead of zero, electronic supplementary material, figure S6). The finite step-width solutions have higher cost than the zero step-width solution with no oscillations, but as before, the cost decreases as the number of pedestrians increases (electronic supplementary material, figure S6).

Optimizing metabolic energy cost for this simple model can reproduce some aspects of human behaviour, however, given the simplicity of the model compared to a human, it may only predict qualitative aspects of phenomena while having quantitative differences with experiments. For example, the optimal solution discovered by this model for the exact parameters of the London Millennium Bridge needs a large number of pedestrians, about 750, for the onset of large bridge oscillations whereas published results are closer to 160. The use of periodicity constraints further prevents us from looking at the transient motion before the onset of steady state.

We assumed that human walkers will entrain to platform frequency, partly based on the observed steady-state motion of pedestrians on the London Millennium Bridge [3]. For the infinite inertia platform, the assumption that entrainment always occurs at steady state is probably reasonable for lateral oscillations near human stride frequencies, but may not hold true for oscillations much faster or slower than the stride frequencies, as found in [37] (see also [38] for entrainment studies with periodic torque perturbations to ankle, instead of surface oscillations). Without the assumption of entrainment, the energy optimization calculation must be over all non-periodic walking strategies, which makes the optimization intractable.

Our predictions that metabolic cost can be reduced by lateral motion can be directly tested in two kinds of experiments, analogous to the infinite and finite inertia cases we considered: (i) experiments with externally laterally oscillated treadmills with associated metabolic measurements [37,8], ensuring that the oscillations do not happen too slowly (as in [37]); (ii) experiments in which the lateral degree of freedom is a simple spring-mass-damper, which can be simulated, for example, by placing a sufficiently light treadmill on wheels facing sideways and connecting the treadmill to the wall using springs, bungee cords or elastic straps. In both these types of experiments, it would be of interest to test both the immediate dynamic and metabolic response of the human subject (within seconds and minutes), as well as the response after sufficient training (for a few hours, say). Energetics as the primary explanation for the Millennium Bridge phenomenon can be credible only if the humans entrain to optimal walking motions on the time-scale of less than a minute; there is some evidence for such fast optimization time-scales in human locomotion [26,28,39,40]. The experiments described above could also measure the forces exerted by the legs on the ground and the work performed by the legs; our models predict that the impact forces (force peaks in walking) will be reduced if there is appropriate entrainment to sideways movement. If impact forces are indeed reduced in experiment, one could use such laterally oscillated treadmills for low-impact workouts. We could use models and methods similar to those used here to study locomotion on other types of compliant surfaces, for instance, non-human primate locomotion on flexible tree branches [41,42] or humans running [1,43].

We have shown the theoretical possibility of metabolic cost reduction only when a large number of people act in synchrony to shake a bridge (at least for some bridge parameters). This result could be considered similar to the posited reduction in locomotor metabolic cost in other contexts such as birds flying in formation [44] and fish swimming in a school [45], although in these other contexts, the different individuals may benefit by different amounts unlike in our model.

We have focused on energy optimality as a theory for human walking strategies while on a moving surface. One complement to this hypothesis is the effect of neighbouring pedestrians due to space constraints (e.g. collision avoidance [36]) and other cognitive coupling between neighbours [46]. From a mechanical perspective, the most significant and plausible complement to the energy optimality hypothesis is the hypothesis that the observed human-platform dynamics is an emergent property of the controller that keeps human walking stable. While limited theoretical explorations [7,36] have found evidence for the stability hypothesis, a definitive understanding of the human-platform dynamics has to await a more detailed characterization of the controller that maintains stability during walking—an outstanding open problem in locomotion biomechanics [10,11,47]. Once we have this human walking controller, we could check if one or many biped models endowed with such a stabilizing controller, walking forward on a shaky bridge, automatically synchronize with each other and to the bridge, thereby causing large bridge oscillations. Given that energy optimality has been so successful in explaining other aspects of human movement and can at least explain some known qualitative features of humans on moving surfaces, we need careful studies that can distinguish energetic optimality from stability. For instance, it may superficially be thought that a wider step width is an indication that the humans are stability-challenged in the lateral direction; however, our calculations show that wider step width is energy optimal even when stability is not taken into account; thus, delineating the individual effects of energetics and stability might be non-trivial.

Appendix A

The appendix contains the following three sections:

• — Section (a) lists the equations of motion for all variants of the simple model described in figure 1 of the main manuscript. These variants are the parametrized-leg-force model and the inverted pendulum model, each of which may be coupled with an infinite or finite inertia platform.

• — Section (b) details each term in the cost function for the biped as described in §1 of the main manuscript.

• — Section (c) provides specific information about the ordinary differential equation (ODE) solver and optimization algorithms used in our calculations.

(a) Equations of motion

(i) Body motion

For the model shown in figure 1, there are two forces acting on the centre of mass during single stance, namely, the force along the leg F and the weight of the biped m_comg. The forward, vertical and lateral positions of the body centre of mass (CoM) and the stance foot in contact with the ground are represented as x, y and z, respectively, with appropriate subscripts. The leg length is ℓ, given by

$${\ell }^2={(x_{\mathrm{com}}-x_{\mathrm{foot}})}^2+{(y_{\mathrm{com}}-y_{\mathrm{foot}})}^2+{(z_{\mathrm{com}}-z_{\mathrm{foot}})}^2.$$ A 1

The equations of motion for the centre of mass are

$$\begin{array}{rl}{m}_{\mathrm{com}}{\ddot{x}}_{\mathrm{com}}& =F\frac{x_{\mathrm{com}}-x_{\mathrm{foot}}}{\ell },\end{array}$$ A 2

$$\begin{array}{rl}{m}_{\mathrm{com}}{\ddot{y}}_{\mathrm{com}}& =F\frac{y_{\mathrm{com}}-y_{\mathrm{foot}}}{\ell }-m_{\mathrm{com}}g\end{array}$$ A 3

$$\begin{array}{rl}\text{and}{m}_{\mathrm{com}}{\ddot{z}}_{\mathrm{com}}& =F\frac{z_{\mathrm{com}}-z_{\mathrm{foot}}}{\ell },\end{array}$$ A 4

which are applicable for all versions of the biped model.

(ii) Non-dimensionalization

Non-dimensionalization is performed by dividing by appropriate combinations of mass m_com, maximum leg length ℓ_max, and acceleration due to gravity g. In the following, we denote the non-dimensional analogues of dimensional quantities by adding an overbar; for instance, fore-aft position ${\overline{x}}_{\mathrm{com}}=x_{\mathrm{com}}/{\ell }_{\max }$, non-dimensional leg force $\overline{F}=F/(m_{\mathrm{com}}g)$, non-dimensional speed $\overline{v}=v/\sqrt{g{\ell }_{\max }}$ and non-dimensional time $\overline{t}=t\sqrt{g/{\ell }_{\max }}$. Energy cost per distance is normalized by mg, energy cost per unit time is normalized by $mg\sqrt{g{\ell }_{\max }}$, and energy per step is normalized by mgℓ_max.

(iii) Platform motion

When we have a finite inertia platform, we add another equation for the motion of the platform in the lateral direction:

$$m_{\mathrm{platform}}{\ddot{z}}_{\mathrm{platform}}+B_{\mathrm{platform}}{\dot{z}}_{\mathrm{platform}}+K_{\mathrm{platform}}{z}_{\mathrm{platform}}=-F\frac{z_{\mathrm{com}}-z_{\mathrm{foot}}}{\ell },$$ A 5

where K_platform and B_platform are the stiffness and damping of the platform in the lateral direction, respectively. For the infinite inertia platform, the platform motion is specified as sinusoidal, of the form:

$$z_{\mathrm{platform}}=A_{\mathrm{platform}}\sin ({\omega }_{\mathrm{platform}}t+\varphi ),$$ A 6

where A_platform is the amplitude, ω_platform is the frequency of oscillation, and ϕ is the phase difference between the biped stride and the platform oscillation.

(iv) Leg forces

We performed two different types of optimization calculations: (i) in which the leg forces are allowed to be arbitrary and (ii) in which the biped performs an inverted pendulum gait with push-off and heel-strike impulses. When the leg force is arbitrary, they are parametrized as piecewise linear F(t) to be computed by the optimizer. When the biped is assumed to perform an exact three-dimensional inverted pendulum motion, we use a differential algebraic equation formulation to enforce the leg length constraint. We do not fix the leg length, but rather constrain the motion of the centre of mass to be such that the leg length cannot change from the beginning of the motion. The second derivative of this leg length constraint (equation (A 1)) produces the following additional ODE:

$$\begin{array}{rl}& (x_{\mathrm{com}}-x_{\mathrm{foot}}){\ddot{x}}_{\mathrm{com}}+(y_{\mathrm{com}}-y_{\mathrm{foot}}){\ddot{y}}_{\mathrm{com}}+(z_{\mathrm{com}}-z_{\mathrm{foot}}){\ddot{z}}_{\mathrm{com}}\\ & =-{\dot{x}}_{\mathrm{com}}^2-{\dot{y}}_{\mathrm{com}}^2-{({\dot{z}}_{\mathrm{com}}^2-{\dot{z}}_{\mathrm{platform}}^2)}^2+{\ddot{z}}_{\mathrm{platform}}(z_{\mathrm{com}}-z_{\mathrm{foot}}).\end{array}$$ A 7

This equation, along with the equations (A 2)–(A 4) can be combined to determine the acceleration of the centre of mass and the force along the leg. For the finite inertia platform, we add equation (A 5) to these equations to get the lateral acceleration of the platform as well.

(v) Model parameters

For the finite inertia platform, the nominal values of mass, stiffness and damping are identical to those used in [4] as approximating the London Millennium Bridge: M=1.13×10⁵ kg, B_platform=1.1×10⁴ Nsm⁻¹, and K_platform=4.73×10⁶ Nm⁻¹. We also used human mass m=70 kg, maximum leg length ℓ_max=0.95 m and g=9.81 ms⁻¹.

(b) Metabolic cost model

The cost function contains the following four terms, resting metabolic rate, stance work cost, stance force cost and swing leg cost, defined as follows:

• (i) Resting metabolic rate. The experimentally determined metabolic cost of walking [48] for a human at a given speed of walking v is

  $$\mathrm{Metabolic}\hspace{0.17em}\mathrm{rate}\hspace{0.17em}\mathrm{per}\hspace{0.17em}\mathrm{unit}\hspace{0.17em}\mathrm{mass}=(2.2+1.155{\left(\frac{v}{1\hspace{0.17em}{\mathrm{ms}}^{-1}}\right)}^2)\mathrm{Watts}\hspace{0.17em}\mathrm{per}\hspace{0.17em}\mathrm{kg}.$$ A 8

  Of this, roughly 1.4 Watts per kilogram is the resting metabolic rate [48]. Converting this resting rate to the appropriate non-dimensional units and multiplying by the mass of the biped gives us the resting metabolic rate for the biped. The resting metabolic rate is key to obtaining a non-zero optimal speed [14]; given a finite resting rate, infinitesimal walking speeds would imply very large costs for travelling a given distance.
• (ii) Stance work cost. Humans are known to perform positive work at an average efficiency of approximately 25% (η_pos=0.25) and negative work at an efficiency of 120% (η_neg=1.2) [31,23]. We determine the stance cost as a linear combination of negative and positive work, W_neg and W_pos, done by the leg on the centre of mass, using the reciprocal efficiency as the weighing parameter.

  $$\mathrm{Stance}\hspace{0.17em}\mathrm{work}\hspace{0.17em}\mathrm{cost}=\frac{1}{{\eta }_{\mathrm{pos}}}{W}_{\mathrm{pos}}+\frac{1}{{\eta }_{\mathrm{neg}}}{W}_{\mathrm{neg}}.$$ A 9
  When the biped is controlled with arbitrary parametrized leg forces F(t), the positive and negative stance work is computed by integrating the positive and negative part of the power $P=F\dot{\ell }$ produced by the leg over the course of the motion:

  $$W_{\mathrm{pos}}=\int \lceil P{\rceil }^{+}\hspace{0.17em}\mathrm{d}t\text{and}{W}_{\mathrm{neg}}=\int \lceil -P{\rceil }^{+}\hspace{0.17em}\mathrm{d}t,$$ A 10

  where ⌈P⌉⁺ is the positive part: ⌈P⌉⁺=P when P≥0 and ⌈P⌉⁺=0 when P<0.
  For the inverted pendulum version of the biped, the work done by the leg on the centre of mass is entirely condensed into the push-off and heel-strike impulses; the expressions in equation (A 10) will evaluate to exactly zero during the inverted pendulum phase (leg length does not change, therefore the leg does zero work on the centre of mass). The work done by each of these impulses, which is entirely positive for push-off and entirely negative for heel-strike, equals the kinetic energy change produced by them in the pedestrian-platform system (see [35,31] or electronic supplementary material [26]):

  $$\begin{array}{rl}\mathrm{Kinetic}\hspace{0.17em}\mathrm{energy}\hspace{0.17em}\mathrm{change}& ={(\frac{m_{\mathrm{com}}}{2}{v}_{\mathrm{com}}^2+\frac{m_{\mathrm{platform}}}{2}{v}_{\mathrm{platform}}^2)}_{\mathrm{after}-\mathrm{impulse}}\\ & -{(\frac{m_{\mathrm{com}}}{2}{v}_{\mathrm{com}}^2+\frac{m_{\mathrm{platform}}}{2}{v}_{\mathrm{platform}}^2)}_{\mathrm{before}-\mathrm{impulse}}.\end{array}$$

• (iii) Swing leg cost. The cost of moving the swing leg is determined as the weighted sum of the positive and negative work needed to move the foot from its position during one stance phase to its next position, starting and ending at rest. For this work calculation, the velocity of the swing leg, v_swing, is taken as the ratio of distance between the positions of the foot in two consecutive steps and the step period. Both positive and negative work equal an effective leg kinetic energy $m_{\mathrm{foot}}{v}_{\mathrm{swing}}^2$ during swing; we use an effective foot mass m_foot=0.05m_com as in [26] so as to approximate the moment of inertia of the entire leg with the point-foot. Swing cost equals this kinetic energy scaled by the reciprocal efficiencies (1/η_pos+1/η_neg).

  This simplified swing cost we used does not go to zero nor is a minimum at the leg’s natural frequency, but instead monotonically increases with swing amplitude and frequency. So, we rule out the possibility of leg swing cost reduction by walking close to the passive leg swing frequency. It has been argued that in normal walking, humans walk at higher frequencies than passive leg swing (as do our models’ optima) so as to have smaller step lengths and smaller stance work costs [29]; that is, the presence of swing cost prevents the optimal stride frequency going to infinity—as large stride frequencies and small step lengths reduce the stance work cost.

• (iv) Stance force cost. This is a cost proportional to the time-integral of the force along the leg.

  $$\mathrm{Stance}\hspace{0.17em}\mathrm{force}\hspace{0.17em}\mathrm{cost}={\eta }_{\mathrm{force}}\int F\hspace{0.17em}\mathrm{d}t.$$ A 11

  We obtain the proportionality factor η_force by noting that subtracting the resting metabolic rate from equation (A 8) predicts a non-zero metabolic rate for zero walking speed. As a simplifying assumption, we attribute this difference, not explainable using stance or swing work, to the integral of the leg force. We use non-dimensional ${\overline{\eta }}_{\mathrm{force}}=0.026$. The stance force cost penalizes large step widths and large step lengths.

The sum of these four cost terms per unit distance is used as the objective function to be minimized in our optimization calculations. The non-dimensional version of this objective function is an estimate of the ‘cost of transport’ of walking [49,50].

(c) Details of the numerical optimization

As noted before, we performed two different types of optimizations, one in which the leg forces and length changes are allowed to be arbitrary (force-parametrized version) and one in which the walking is assumed to be inverted pendulum. We performed these two types of optimizations for both the infinite inertia platform and the finite inertia platform situations. Instead of describing all these four calculations, we describe two of them: how we minimize the metabolic energy-based cost function for a force-parametrized biped model walking on an infinite-inertia platform and for an inverted pendulum biped model walking on a finite-inertia platform. The other two optimizations, namely force-parametrized version for finite inertia and inverted pendulum version for infinite inertia are completely analogous. Overall, we use similar methods to those used in our previous work in biped optimizations [21,23]; see, especially, supplementary material of [23] for further technical details.

(i) Force-parametrized optimization for walking on an infinite inertia platform

For the force-parametrized biped model, for each calculation, we fix the step period t_step, walking speed v_step, the platform oscillation amplitude A, the platform oscillation period t_platform=2t_step by assumption of entrainment, from which we determine the platform oscillation frequency ω. The rest of the model is defined by the initial position and velocity of the centre of mass (x_com0,y_com0,z_com0) and $({\dot{x}}_{\mathrm{com}0},{\dot{y}}_{\mathrm{com}0},{\dot{z}}_{\mathrm{com}0})$ for each step, the position of the foot during the second step (x_foot2,0,z_foot2), the relative phase of the platform motion and the ϕ and 2N+2 elements of a piece-wise linear force curve F (N_segments+1 for each step). The position of the foot during the first step is the origin, without loss of generality. This gives us 2N_segments+17 parameters of a gait:

$$\mathbb{Z}=(t_{\mathrm{step}},v_{\mathrm{step}},\omega ,A,x_{\mathrm{CoM}0},y_{\mathrm{com}0},z_{\mathrm{com}0},{\dot{x}}_{\mathrm{com}0},{\dot{y}}_{\mathrm{com}0},{\dot{z}}_{\mathrm{com}0},\varphi ,x_{\mathrm{foot}2},z_{\mathrm{foot}2},F_j)$$

with j=1,…,2N_segments+2. As noted, five out of these 2N_segments+17 are fixed for each calculation and therefore, the rest 2N_segments+12 quantities are unknowns decided by the optimization. The gait is divided into 2N_segments grid segments, separated by the grid-points at which the leg forces are specified; the body motion in each segment is obtained by integrating the appropriate ODEs, with each integration using the end state of previous grid segment as the initial state.

This walking gait is subject to the following constraints:

• — A leg length inequality, ℓ≤ℓ_max, imposed by evaluating the leg length for at least three points in each grid segment. Since ℓ_max is used for normalization, we have ℓ_max=1.

• — Periodicity constraints requiring that the state at the end of the second step is equal to the state at the beginning of the first step except for a forward translation equal to the stride length (d_stride) in the x-direction: d_stride=2v_stept_step, x_com(2t_step)=x_com0+d_stride, y_com(2t_step)=y_com0, z_com(2t_step)=z_com0, ${\dot{x}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{x}}_{\mathrm{com}0}$, ${\dot{y}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{y}}_{\mathrm{com}0}$ and ${\dot{z}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{z}}_{\mathrm{com}0}$.

• — The forces are bounded 0≤F_j≤F_max for j=1,…,2N_segments+2 to ensure that the foot does not pull on the ground and the leg force magnitude is reasonable. We used ${\overline{F}}_{\max }=F/F_{\max }=4$; none of the qualitative results change if we used ${\overline{F}}_{\max }=3$ or 5.

• — The initial states and the position of the foot during the second step are bounded within a sufficiently large box. For example −l_max≤x_com0≤l_max. These bounds are never active at the optimal solution.

(ii) Inverted pendulum optimization for walking on a finite inertia platform

For each optimization with inverted-pendulum model on a finite inertia platform, we fix the mass m_platform, stiffness K_platform and damping constant B_platform for the platform and the number of pedestrians N_pedestrians. The rest of the model is defined by the step period t_step, the step length d_step, the initial position and velocity of the centre of mass (x_com10,y_com10,z_com10) and (${\dot{x}}_{\mathrm{com}10},{\dot{y}}_{\mathrm{com}10},{\dot{z}}_{\mathrm{com}10}$), the initial lateral position and velocity of the platform z_platform10 and ${\dot{z}}_{\mathrm{platform}10}$, the initial position and velocity of the centre of mass for the second step (x_com20,y_com20,z_com20) and (${\dot{x}}_{\mathrm{com}20},{\dot{y}}_{\mathrm{com}20},{\dot{z}}_{\mathrm{com}20}$), the initial lateral position and velocity of the platform during the second step z_platform20 and ${\dot{z}}_{\mathrm{platform}20}$, the position of the foot during the second step (x_foot2, 0, z_foot2), and the magnitudes of the four impulses I_push-off1, I_heel-strike1, I_push-off2 and I_heel-strike2. The foot position during the first step as the origin for the system. This gives us 28 parameters to define the gait $\mathbb{Z}$ = (m_platform, K_platform, B_platform, t_step, d_step, x_com10, y_com10, z_com10, ${\dot{x}}_{\mathrm{com}10}$, ${\dot{y}}_{\mathrm{com}10}$, ${\dot{z}}_{\mathrm{com}10}$, x_com20, y_com20, z_com20, ${\dot{x}}_{\mathrm{com}20}$, ${\dot{y}}_{\mathrm{com}20}$, ${\dot{z}}_{\mathrm{com}20}$, z_platform10, ${\dot{z}}_{\mathrm{platform}10}$, z_platform20, ${\dot{z}}_{\mathrm{platform}20}$, x_foot2, z_foot2, I_push-off1, I_heel-strike1, I_push-off2, I_heel-strike2), out of which four are fixed constants during a given optimization.

The gait is divided into two sections, one for each step. The initial point of a step is subject to a heel-strike impulse, which changes the velocity of the biped and the platform. This heel-strike impulse is calculated to ensure that the leg-length-rate after impulse is exactly 0 to allow for inverted pendulum walking. The resulting state is integrated using the ODE for the inverted pendulum model and finally this state is subject to a push-off impulse to produce the end state for the section. The heel-strike impulse needed to make leg-length-rate 0 at any point of time can be determined using the following equation:

$$I_{\mathrm{heel}\text{-}\mathrm{strike}}=-\frac{{\dot{x}}_{\mathrm{com}}(x_{\mathrm{com}}-x_{\mathrm{foot}})+{\dot{y}}_{\mathrm{com}}(y_{\mathrm{com}}-y_{\mathrm{foot}})+({\dot{z}}_{\mathrm{com}}-{\dot{z}}_{\mathrm{foot}})(z_{\mathrm{com}}-z_{\mathrm{foot}})}{{(x_{\mathrm{com}}-x_{\mathrm{foot}})}^2/m\ell +{(y_{\mathrm{com}}-y_{\mathrm{foot}})}^2/m\ell +{((m+m_{\mathrm{platform}})(z_{\mathrm{com}}-z_{\mathrm{foot}}))}^2/m{m}_{\mathrm{platform}}\ell }.$$ A 12

The optimization problem is solved subject to the following constraints:

• — A leg length equality, ℓ=ℓ_max imposed by evaluating the leg length at the beginning of each step; in non-dimensional terms, ℓ/ℓ_max=1. If the constraint is met at the beginning of the step, the fact that the impulses can only change velocity and that the motion is inverted-pendulum-like ensure that it shall be met till the end of the step.

• — Periodicity constraints requiring that the state at the end of the second step is equal to the state at the beginning of the first step except for a forward translation equal to the stride length d_stride in the x-direction: d_stride=2d_step. x_com(2t_step)=x_com10+d_stride, y_com(2t_step)=y_com10, z_com(2t_step)=z_com10, ${\dot{x}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{x}}_{\mathrm{com}10}$, ${\dot{y}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{y}}_{\mathrm{com}10}$, ${\dot{z}}_{\mathrm{com}}(2t_{\mathrm{step}})={\dot{z}}_{\mathrm{com}10}$, z_platform(2t_step)=z_platform10, and ${\dot{z}}_{\mathrm{platform}}(2t_{\mathrm{step}})={\dot{z}}_{\mathrm{platform}10}$.

• — Continuity constraints requiring that the state at the end of the first step is equal to the state at the beginning of the second step x_com(t_step)=x_com20, y_com(t_step)=y_com20, z_com(t_step)=z_com20, ${\dot{x}}_{\mathrm{com}}(t_{\mathrm{step}})={\dot{x}}_{\mathrm{com}20}$, ${\dot{y}}_{\mathrm{com}}(t_{\mathrm{step}})={\dot{y}}_{\mathrm{com}20}$, ${\dot{z}}_{\mathrm{com}}(t_{\mathrm{step}})={\dot{z}}_{\mathrm{com}20}$, z_platform(t_step)=z_platform20, and ${\dot{z}}_{\mathrm{paltform}}(t_{\mathrm{step}})={\dot{z}}_{\mathrm{platform}20}$.

• — The impulses are bounded 0≤I_j≤I_max to ensure that the foot does not pull on the ground and the impulse magnitude is reasonable. The impulse upper bound in never active in the optimal solutions, so its exact value is irrelevant.

• — The forces are bounded 0≤F_j≤F_max to ensure that the foot does not pull on the ground and the leg force magnitude is reasonable. This is done by calculating the force at sufficiently many intermediate point of the gait using the equations of motion. The upper bound on the force is never active in the optimal solutions, so its exact value is irrelevant.

(iii) Computational solution

The equations of motion were solved using ode45 in MATLAB with absolute and relative tolerances of at most 10⁻⁹ and also an equivalent constant step-size Runge–Kutta method for consistency. Optimization was performed with SNOPT [51,52], a robust constrained nonlinear optimization program; we used optimality and feasibility tolerances of 10⁻⁶. The parametrized-leg-force biped model used at piece-wise linear forces with at least 14 segments to simulate each step of walking.

The planar inverted pendulum walking solution is obtained robustly by the optimization from random initial seeds far away from the optimum, when the platform is constrained to not move (essentially a repetition of calculations elsewhere [21,23]). When the platform is moved or the bridge is allowed to move, to obtain the non-planar optimal solutions, we typically started the optimizations from initial seeds that are sufficiently perturbed versions of the planar solution. When we obtain optimal solutions for a sequence of parameter values, we use the optimal solution for the previous parameter value as the initial seed for the current parameter value—and then do a forward and backward sweep of the parameter values to pick the lower of the two optimal solutions. Numerical experiments suggested that there were at most two local minima, the planar walking optimum and the non-planar walking optimum, and sometimes only one; we did not see evidence for many more local minima. While we cannot rule out that there does not exist a lower global minimum, we have reliably demonstrated that lower-than-planar costs are achievable with non-planar walking.

Ethics statement

This work did not involve any active collection of human data, but only computer simulations of human behaviour.

Data accessibility

This work does not have any experimental data.

Author contributions

V.J. and M.S. conceived the mathematical models, interpreted the computational results and wrote the paper. V.J. implemented and performed most of the simulations and optimization calculations, partly in discussion with M.S. All authors gave final approval for publication.

Funding statement

This work was supported by National Science Foundation grant no. 1254842.

Conflict of interests

We have no competing interests.