Walking crowds on a shaky surface: Stable walkers discover1

Millennium Bridge oscillations with and without pedestrian synchrony2

Varun Joshi and Manoj Srinivasan

Mechanical and Aerospace Engineering, Ohio State University

3

Abstract4

Why did the London Millennium Bridge shake when there was a big enough crowd walking on it?5

What features of human walking dynamics when coupled to a shaky surface produce such shaking?6

Here, we use a simple biped model capable of walking stably in 3D to examine these questions.7

We simulate multiple such stable bipeds walking simultaneously on a bridge, showing that they8

naturally synchronize under certain conditions, but that synchronization is not required to shake9

the bridge. Under such shaking conditions, the simulated walkers increase their step-widths and10

expend more metabolic energy than when the bridge does not shake. We also find that such bipeds11

can walk stably on externally shaken treadmills, synchronizing with the treadmill motion for a range12

of oscillation amplitudes and frequencies, sometimes performing net positive work on the treadmill.13

Our simulations illustrate how interactions between (idealized) bipeds through the walking surface14

can produce emergent collective behavior that may not be exhibited by just a single biped.15

Introduction16

Large lateral oscillations of bridges and walkways due to pedestrian interactions have been observed on17

the London Millennium Bridge, Changi Mezzanine Bridge, and many others [1]. These oscillations have18

been studied by observing humans as they walk on bridges [2, 3] and laterally oscillating treadmills19

[4, 1], by modeling walkers using non-biological oscillators [5, 6] or in-place stepping models without20

forward walking [7, 8, 6]. Here, we perform 3D simulations of multiple walkers to investigate how21

pedestrian-induced oscillations emerge from the dynamics of a simplified but mechanistically-based22

model of a human walking on a shaky bridge.23

1

We explore these pedestrian-induced oscillations by considering principles underlying how humans walk.24

Minimization of metabolic energy can approximately explain a variety of human walking behaviours25

[9]. We previously showed that walking in synchrony with lateral bridge oscillations lowers the walking26

energy cost, suggesting the usefulness of synchrony [10]. Here, in contrast, we use the human preference27

to walk without falling down (being stable) to model the observed phenomena [8]. The human body is28

a top-heavy object, often modeled as an inverted pendulum [9], and is generally unstable without active29

feedback control. We show that the dynamics of feedback-stabilized walking bipeds, when coupled with30

the bridge’s dynamics, can produce bridge oscillations and pedestrian synchrony.31

Methods32

Biped model and walking mechanics. We model a walking human using a classical biped model33

[9] with a point-mass upper body with mass m and massless legs with point-feet (figure 1a). This biped34

uses planar inverted pendulum walking as the “nominal periodic motion,” i.e., the motion when external35

perturbations are absent. This planar motion implies that a single pedestrian can walk without shaking36

the bridge, thus removing finite step widths as a mechanism for bridge oscillation. This allows us to37

examine the stability response in isolation. Each step of the biped is an inverted pendulum motion:38

the point-mass body vaults over the point-foot with constant leg length `0 (figure 1b). Transitions39

between one inverted pendular phase and the next are accomplished by a push-o↵ impulse Ipush�o↵ by40

the trailing leg and a heel-strike impulse Iheel�strike by the leading leg. At any moment, exactly one41

leg touches the ground (the stance leg). The massless swing leg has no dynamics; the foot is simply42

placed at the appropriate location for the next step. This inverted pendular walking is parameterized43

by forward walking speed and step frequency. We use these two parameters to explore diversity in44

pedestrian properties.45

While the nominal motion is planar and periodic, the biped can move in 3D. The biped is capable of 3D46

inverted pendulum walking, pivoting about foot positions (xfoot, yfoot) anywhere on the ground plane,47

while maintaining constant leg length `0. After the motion is initiated, the biped can a↵ect further48

motion by choosing the subsequent foot positions and push-o↵ impulses. The heel-strike impulse is49

determined by the condition that the leg length rate after heel-strike must be zero to be consistent50

with the inverted pendulum motion. See Supplementary Information for equations, simulation details,51

modelling rationale, and discussion of these simplifying assumptions.52

2

foot position ( xfoot

, yfoot

, 0)

walking direction

Point-mass body(x, y, z)

stance leg(actuated)

swing leg(no dynamicsexcept foot placement)

Bridge/platform can onlymove laterally ( X direction )

X

Y

Z

Bridge/Walking platform

Body can move in 3D3D inverted pendulum motion of step i

3D inverted pendulum motion of step i+1

Step-to-step transition mediatedby push-off impulse& heel-strike impulse

a) Minimal biped on a shaky surface

c) Simulating a large crowd with N pedestrians with fewer simulated bipeds, each with correspondingly higher mass

b) 3D inverted pendular walking motion

Constantleg length

Constantleg length

foot position ( x

foot, y

foot)

step width

N = 400 pedestrianswith

m = 1 human mass

P = 80 simulated bipedsm = 5 human masses each(80 x 5 = 400 pedestrians)

P = 4 simulated bipedsm = 100 human masses each(4 x 100 = 400 pedestrians)

is simulatedusing

(for instance)

or

Figure 1: Biped model. a) Point-mass biped with massless legs. The walking platform can only oscillatelaterally. b) Inverted pendulum walking. c) Simulating a large crowd with fewer simulated bipeds.

Feedback control to enable stable walking. Inverted pendulum walking without feedback con-53

trol is unstable: the biped cannot recover from sideways pushes if the feet remain at their nominal54

positions. To stabilize this biped in the presence of perturbations, we implement a feedback controller55

for foot-placement and push-o↵ impulse based on human behaviour and other simple models [11].56

When perturbed, the biped changes its foot position and push-o↵ impulse from their nominal val-57

ues (xfoot,nominal, yfoot,nominal, Ipush-o↵,nominal) by (�xfoot, �yfoot, �Ipush-o↵). These changes are linear58

functions of perceived deviations in center-of-mass state from nominal (xerror, xerror, yerror). Specifi-59

cally, the fore-aft foot position change is: �yfoot = yfoot � yfoot,nominal = k1yerror. The sideways foot60

position change is: �xfoot = xfoot � xfoot,nominal = k2xerror + k3xerror. The push-o↵ impulse change is:61

�Ipush-o↵ = Ipush-o↵ � Ipush-o↵,nominal = k4yerror. For foot placement, the biped steps in the direction62

of the perceived perturbation [11], stepping rightwards if it is falling rightwards and taking a longer63

step if moving too fast forward (and similarly, stepping leftwards when falling leftward, etc.). We allow64

3

the foot to cross-over to the contralateral side as needed. The biped uses a smaller push-o↵ impulse if65

going too fast.66

While on a shaking surface, we assume the perceived sideways velocity error xerror is weighted equally67

between velocity relative to earth (absolute) and velocity relative to the walking surface: xerror =68

(xerror, absolute + xerror, relative)/2. This represents humans weighting vision and proprioceptive sensors69

to produce a coherent body state estimate.70

Modeling a shaky bridge and a shaken treadmill. The shaky bridge is modelled as a mass that71

can move only sideways, attached to the world with a spring and damper with values corresponding72

to the London Millennium Bridge [5]. Given the bridge’s finite mass, the bipeds can a↵ect the bridge73

motion. We also consider walking on an externally shaken treadmill with sinusoidal sideways oscillations74

of prescribed frequencies and amplitudes [1, 12]. In this case, the biped cannot a↵ect the walking surface75

motion.76

Simulating a large crowd. We wish to simulate hundreds of pedestrians on the bridge. For compu-77

tational tractability, we use a smaller number of simulated bipeds P and vary the individual biped mass78

to simulate a larger group of N “e↵ective” pedestrians with equal total mass. For example, to simulate79

N = 400 e↵ective pedestrians, we use P = 80 bipeds, each with 5 human masses. This represents 8080

groups of 5 pedestrians each, where all the 5 pedestrians in one group walk in synchrony, but the 8081

groups need not be synchronized. This simplification is justified because the steady state bridge and82

pedestrian motions depend on N but not P (figure 2a).83

Results84

In all simulations, the initial phase di↵erence between the walkers and the bridge was random and had85

no e↵ect on any of the steady state results here. Supplementary figures S3-S5 illustrate bridge-shaking86

biped behavior for a range of model parameters.87

The biped is stable on solid ground. The biped’s planar inverted pendulum walking motion was88

stable on non-moving ground: the maximum ‘Floquet multiplier’ had absolute value of 0.4634 (< 189

4

implies stability). Thus, the biped returns asymptotically to periodic inverted pendulum walking after90

any small discrete perturbation in 3D space. (Large-enough perturbations can make the biped fall.) In91

the limit of large bridge mass or slow imposed ground motions, the biped’s behavior approaches that on92

non-moving ground. When the biped is coupled to a shaky bridge or when on a shaken platform, the93

system changes substantially in a manner that produces the dynamical behavior below. Supplementary94

Figure S2 shows how the biped falls in a few steps if the feedback controller is turned o↵ (zero gains);95

if the gains are reversed in sign, the biped falls in fewer steps.96

Identical walkers shake the bridge if su�ciently many. The bridge oscillation amplitude de-97

pends on the e↵ective number of pedestrians N (figure 2a-b). We find three regimes of steady-state98

behaviour: (1) For N less than 150, the bridge comes to rest at steady-state. (2) For N between 15099

and 350, the steady state motion of the bridge is oscillatory with multiple frequency components. (3)100

For N of 350 and above, the steady-state motion of the bridge is oscillatory, matching the pedestrian101

stride-frequency. The bridge approaches these oscillatory steady states even if the initial bridge motion102

is arbitrarily small (Supplementary figures S3). Figure 2a,c show bridge oscillations decay or grow in103

di↵erent regimes. When the bridge oscillates, the bipeds adopt a greater step width.104

Identical walkers synchronize. We quantify synchrony between pedestrians using the ‘order pa-105

rameter’ ([5] and Supplementary Appendix, section S6). An order parameter of one implies perfect106

pedestrian synchrony with phase locking: all bipeds begin and end each step at the same time. An107

order parameter value of zero implies that pedestrians are far from being phase-locked. For identical108

walkers, we found two regimes: for N less than 150, pedestrians never synchronized with each other.109

At N of 150 and above, pedestrians synchronize with each other at steady-state, with the time taken110

to achieve synchrony decreasing with increasing N (figure 2d).111

Non-identical walkers do not synchronize but do shake the bridge. Real humans are not112

identical. We simulated multiple bipeds with di↵erent nominal step frequencies, drawn from a normal113

distribution with a standard deviation of 0.02 (non-dimensional) about the mean frequency of our114

nominal walker. Such dissimilar bipeds never synchronized even at large N . However, despite not115

synchronizing, the bipeds shake the bridge at N of about 150 and above (figure 2c,d).116

5

Stea

dy

stat

e p

latf

orm

RM

S p

osi

tio

n

0 100 200 300 400Number of pedestrians N

0

0.01

0.02

0.03

0.04

a) Bridge steady-state behavior does not depend on number of simulated groups Bridge has a steady state oscillation for large N

No oscillations

Quasi-periodicor long-periodic

oscillations

1:1 entrained& synced periodicoscillations

b)

g) Steady-state phase difference variation with frequency

platform oscillation frequency

f ) Steady-state phase difference variation with amplitude

Walking on a shaken treadmill

Walking on a shaky bridge

e) Shaking the bridge does not lower the energy cost of walking

pedestrian synced

ped

estr

ian

un

syn

ced

0.27 0.29 0.31

ped

estr

ian

un

syn

ced

100

90

80

70

stea

dy-

stat

e p

has

e d

iffer

ence

ped

estr

ian

un

syn

ced

platform oscillation amplitude

pedestrian synced

ped

estr

ian

falls

stea

dy-

stat

e p

has

e d

iffer

ence 100

90

80

70

0.02 0.04 0.06

ped

estr

ian

falls

ped

estr

ian

un

syn

ced

Bri

dg

e la

tera

l dis

pla

cem

ent

0.08

-0.08

0

0 30 60 90 120 150 180

Time

0.08

-0.08

0

240 pedestrians

0.08

-0.08

0

400 pedestrians

2 groups 4 groups 6 groups 80 groups

80 pedestrians

80 pedestrians

240 pedestrians

400 pedestriansO

rder

par

amet

er

0 50 100 150 200 250

0

1

0

1

0

1

Similar bipeds Dissimilar bipeds

d) Evolution of walking phase synchrony

Time

Similar bipeds Dissimilar bipeds

c) Evolution of bridge lateral movement (using 8 biped groups)

80 pedestrians

240 pedestrians

400 pedestrians

Bri

dg

e la

tera

l dis

pla

cem

ent

0 100 200 300 400

Time

0.06

-0.06

0

0.06

-0.06

0

0.06

-0.06

0

met

abo

lic e

ner

gy

co

st p

er u

nit

dis

tan

ce

step number

0.405

0.365

0.385

2000 100

400 pedestrians

80 pedestrians

240 pedestrians

Figure 2: Walking on a shaky bridge. a) Platform oscillation with P = 2, 4, 6, and 80 groups of pedestriansrepresenting an equivalent number N = 80, 240, or 400 pedestrians. The steady-state is independent of P ,barring time-o↵sets due to random initial phase. We see decaying oscillations for low N , oscillations with multi-step periodicity for intermediate N and two-step periodic oscillations for large N . b) Platform steady-stateoscillation amplitude (root-mean-squared position of the steady-state motion) as a function of N , showing threequalitatively di↵erent regimes. c) Bridge motion when the bipeds (P = 8) are identical and non-identical.d) Order parameter variation showing that identical biped synchronize but non-identical bipeds do not. SeeSupplementary Videos for walking animations. e) The energy cost of walking goes up when the pedestriansshake the bridge, comparing the 400 pedestrian case (shaking) to the 80 pedestrian case (no shaking). Walkingon a shaken treadmill. The steady-state phase di↵erence as a function of f) platform oscillation amplitudeand g) platform oscillation frequency. Pedestrians entrain to platform oscillations for some frequencies andamplitudes. All quantities non-dimensional.

6

Biped entrains to a shaken treadmill if the amplitude and frequency are right. When on117

a shaken treadmill, the biped synchronizes with treadmill motion for a small range of amplitudes and118

frequencies. When synchronized, the steady-state phase di↵erence between the treadmill and biped119

motions depend on treadmill oscillations (Figure 2e-f). For a given treadmill frequency, synchrony120

occurs at a narrow range of amplitudes: lower amplitudes fail to create synchrony and higher amplitudes121

cause the biped to fall (Figure 2f). Similarly, synchrony only occurs for a narrow range of oscillation122

frequencies close to the nominal step frequency of the biped (Figure 2g).123

Energy cost increases when the bridge shakes. In a previous study, we had argued that humans124

could reduce their walking metabolic cost by optimally walking on a shaking bridge [10]. That model125

ignored stability considerations. Estimating the metabolic cost for gaits observed in this current study,126

we find that the metabolic cost is lowest when the biped walks on solid ground and increases when the127

bridge starts shaking (see figure 2e). We hypothesize that the short time-scale response of the humans128

will be governed by the constraint that they should not fall (thus increasing the cost), but if the129

subjects continued to walk on shaky bridges, they may learn to walk energy-e�ciently [10], modifying130

their controller to be simultaneously energy-e�cient and stable, perhaps using optimal feedback control131

[13].132

Discussion133

Our simulations suggest that walking stability might contribute to pedestrian induced lateral oscil-134

lations of bridges. While our bipeds could synchronize to each other on bridges, synchrony was not135

necessary for lateral oscillations. Our finding that bridge oscillations can occur with or without syn-136

chrony agrees with observations of people walking on bridges [2, 3]. We also find synchrony with137

externally shaken treadmills for a range of oscillation amplitudes and frequencies. Previous shaken-138

treadmill experiments [12, 1] did not find such synchronization but used amplitudes and frequencies139

where our model also found no synchronization. The human and bridge models used here are simplifi-140

cations. Future work could use more complex biped models, more elaborate stabilizing controllers, and141

incorporate pedestrian-collision avoidance and visually-based step-synchronizing tendencies of humans.142

In summary, we have focused on a bio-inspired feedback controller for the biped, which produces bridge143

oscillations. It is conceivable that there exist non-human-like controllers that stabilizes locomotion but144

7

do not shake the bridge.145

Competing interests. None.146

Funding. Supported by NSF CMMI grant 1538342.147

Author Contributions. Both authors conceived the models, interpreted the results, and wrote the148

paper. VJ devised all simulations. All authors approved the publication.149

Ethics statement. No human experiments.150

Data availability. Not applicable.151

References152

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induced lateral forces on footbridges. J Sound Vib. 2011;330:1265–1284.154

[2] Dallard P, Fitzpatrick T, Flint A, Le Bourva S, Low A, Ridsdill Smith RM, et al. The London155

Millennium Footbridge. The Structural Engineer. 2001;79:17–33.156

[3] Brownjohn JMW, Fok P, Roche M, Omenzetter P. Long span steel pedestrian bridge at Singapore157

Changi Airport - Part 2: Crowd loading tests and vibration mitigation measures. Struct Eng.158

2004;82:28–34.159

[4] Carroll SP, Owen JS, Hussein MFM. Reproduction of lateral ground reaction forces from visual160

marker data and analysis of balance response while walking on a laterally oscillating deck. Eng161

Struct. 2013;49:1034–1047.162

[5] Strogatz SH, Abrams DM, McRobie A, Eckhardt B, Ott E. Theoretical mechanics: crowd syn-163

chrony on the Millennium Bridge. Nature. 2005;438:43–4.164

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[6] Belykh I, Jeter R, Belykh V. Foot force models of crowd dynamics on a wobbly bridge. Science165

Advances. 2017;3(11):e1701512.166

[7] Bocian M, Macdonald JHG, Burn JF. Biomechanically inspired modelling of pedestrian-induced167

forces on laterally oscillating structures. J Sound Vib. 2012;331:3914–3929.168

[8] Macdonald JHG. Lateral excitation of bridges by balancing pedestrians. Proc R Soc A. 2008;p.169

1055–1073.170

[9] Srinivasan M. Fifteen observations on the structure of energy minimizing gaits in many simple171

biped models. J R Soc Interface. 2011;8:74–98.172

[10] Joshi V, Srinivasan M. Walking on a moving surface: energy-optimal walking motions on a173

shaky bridge and a shaking treadmill can reduce energy costs below normal. Proc R Soc A.174

2015;471:20140662.175

[11] Wang Y, Srinivasan M. Stepping in the direction of the fall: the next foot placement can be176

predicted from current upper body state in steady-state walking. Biol lett. 2014;10:1–9.177

[12] Bocian M, Burn JF, Macdonald JHG, Brownjohn JMW. From phase drift to synchronisation178

pedestrian stepping behaviour on laterally oscillating structures and consequences for dynamic179

stability. J Sound Vib. 2017;392:382–399.180

[13] Todorov E, Jordan MI. Optimal feedback control as a theory of motor coordination. Nature181

neuroscience. 2002;5:1226.182

9

Supplementary Appendix to:

Walking crowds on a shaky surface: Stable walkers discoverMillennium bridge oscillations with and without pedestrian synchrony

Varun Joshi and Manoj Srinivasan

Department of Mechanical and Aerospace Engineering

The Ohio State University

S1 Rationale for biped model selection

Independent of the literature on pedestrian-induced bridge oscillations, numerous previous studieshave considered the mechanics of bipedal walking and running, some focusing on energetics and somefocusing on stability and control [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Further mathematicalmodels proposed with the motivation of understanding interactions with a bridge include mathematicalabstractions of human locomotion [16, 17], linear inverted pendulums which assume that humans staynearly vertical [18, 19] and more complicated walkers with springs and dampers that use ad-hoc externalforces to maintain stability [20]. The inverted pendulum model used here is just one of many suchmodels, but we feel that it provides a better approximation to actual human motion while remainingcomputationally tractable.

The inverted pendulum model is certainly (and intentionally) an extreme simplification of the complexdynamics that attend the human body and indeed the human walking motion. It is perhaps thesimplest walking model that captures key features of walking stability, specifically the instability dueto inverted pendulum-like body dynamics and stabilization using foot placement. Further, as notedin the main manuscript, our weighting of absolute and relative sideways velocity error in the footplacement controller promotes stability despite surface motions and models humans weighting di↵erentsensors to produce a coherent state estimate [21]. In addition to simplifying the human body tohaving a point-mass upper body and massless legs with no swing dynamics, the walking motion itselfis simplified in a manner that there is no double stance phase and there is no change in direction. Oursimulated walkers have a preferred straight line walking direction, whereas in a real crowd, walkers mayoccasionally change direction to avoid another pedestrian, a feature we hope to incorporate in futurework.

Previous models used to simulate interactions of bipeds with a bridge were 2D models restricted to thefrontal plane and they did not have the ability to walk forward or the ability to fall to the ground (akey aspect of walking stability). It is known that fore-aft and sideways walking dynamics are indeedcoupled [8]. A biped that is stabilized in the sideways direction may still fall forward, so it may not beaccurate to ignore the forward walking dynamics.

1

S2 Body equations of motion

For the model shown in figure 1 of main manuscript, there are two forces acting on the center of massduring single stance, namely, the force along the leg F and the weight of the biped mg, where m isthe biped mass, and g is acceleration due to gravity. The lateral, forward and vertical positions of thebody center of mass (CoM) are given by x, y and z respectively and the stance foot position in contactwith the ground is given by (xfoot, yfoot, zfoot). The leg length is `, given by,

`2 = (x� xfoot)2 + (y � yfoot)

2 + (z � zfoot)2 . (1)

where zfoot = 0. The equations of motion for the center of mass are:

mx = Fx� xfoot

`, (2)

my = Fy � yfoot

`, and (3)

mz = Fz � zfoot

`�mg, (4)

which are applicable for all versions of the biped model. Recall that non-dimensionalization is performedby dividing by appropriate combinations of mass m, inverted pendulum leg length `0, and accelerationdue to gravity g. In the following, we denote the non-dimensional analogues of dimensional quantitiesby adding an overbar; for instance, fore-aft position y = y/`0, non-dimensional leg force F = F/(mg),non-dimensional speed v = v/

pg`0, and non-dimensional time t = t

pg/`0. The origin for all position

measurements is the current stance-foot position.

S3 Platform equations of motion

To simulate the bridge, we add another equation for the motion X(t) of the bridge in the lateraldirection:

Mxplatform +Bxplatform +Kxplatform = �F · x� xfoot`

(5)

where K and B are the sti↵ness and damping of the platform in the lateral direction respectively, andM is the appropriate modal mass of the bridge.

To simulate an externall shaken treadmill, the platform motion is specified as sinusoidal, of the form:

xplatform = Aplatform sin (!platformt+ �) (6)

where Aplatform is the amplitude, !platform is the frequency of oscillation, and � is the phase di↵erencebetween the biped stride and the platform oscillation.

S4 Leg force equations

The biped is assumed to perform an exact 3D inverted pendulum motion, we use a di↵erential algebraicequation formulation to enforce the leg length constraint. We do not fix the leg length, but ratherconstrain the motion of the center of mass to be such that the leg length cannot change from the

2

beginning of the motion. The second derivative of this leg length constraint (equation 1) produces thefollowing additional ODE:

(x� xfoot) x+ (y � yfoot) y + (z � zfoot) z = � (x� xplatform)2 � y2 � z2 + xplatform (x� xfoot) (7)

This equation, along with the equations 2-4 can be combined to determine the acceleration of thecenter of mass and the force along the leg. For the finite inertia platform, we add equation 5 to theseequations to get the lateral acceleration of the platform as well.

S5 Impulse equations

At the end of each biped stance phase a push-o↵ impulse is applied to coupled system along the leg ofthe biped. These impulses directly a↵ect the momentum of the biped CoM and the platform.

mxpost-push-o↵ = mxpre-push-o↵ + Ipush-o↵ · x� xfoot`0

, (8)

mypost-push-o↵ = mypre-push-o↵ + Ipush-o↵ · y � yfoot`0

(9)

mzpost-push-o↵ = mzpre-push-o↵ + Ipush-o↵ · z � zfoot`0

, (10)

In the finite inertia case, we also use equation 11.

mplatformxplatform,post-push-o↵ = mplatformxplatform,pre-push-o↵ + Ipush-o↵ · xfoot � x

`0(11)

After the push-o↵, we solve for the value of the heel-strike impulses for each biped that would makethe leg-length-rate of the biped 0, thus insuring continued inverted pendular motion, we apply theheel-strike impulse similar to equations 8 - 10.

S6 Order parameter equations

In order to compute the order parameter we first define the phase of a biped. We use step-length andCoM position to determine phase:

� = 2⇡y

yfoot(12)

Where both y and yfoot are measured with the stance-foot as the origin. The magnitude of the orderparameter is computed as follows [22]:

r =

������1

P

PX

j=1

ei�j

������(13)

where �j is the phase of the jth biped and P is the total number of simulated bipeds.

3

S7 Simulation parameters

We use k1 = 1, k2 = 2 , k3 = 2 and k4 = �0.5 for the non-dimensional control gains in our foot-placement and push-o↵ feedback controllers. The non-dimensional biped speed is set to be 0.3.

For the finite inertia platform, the nominal values of mass, sti↵ness, and damping are identical to thoseused in [16] as approximating the London Millennium Bridge: M = 1.13 ⇥ 105 kg, B = 1.1 ⇥ 104

Nsm�1, and K = 4.73⇥ 106 Nm�1.

For non-dimensionalization, we used human mass m = 70 kg, maximum leg length `0 = 1 m, andg = 9.81 ms�1. Small changes to these values do not a↵ect the qualitative results presented in thisarticle.

S8 Metabolic cost model

The metabolic cost function contains the following four terms, identical to those defined by Joshi andSrinivasan [4]:

1. the resting metabolic rate, the metabolic rate while not moving, about 1.4 Watts per kg [23].The cost of locomotion is, as below, over and above this resting cost.

2. the stance work cost, a work-based metabolic cost term, a linear combination of the positiveand negative work performed by the leg. We use an average positive work e�ciency of 25%(⌘pos = 0.25) and negative work e�ciency of 120% (⌘neg = 1.2) [1, 2]. The work is done by thebiped only during the push-o↵ and heel-strike impulses.

3. the swing leg cost is the the cost of moving the swing leg from stance phase to the next, againquantifies by a linear combination of the positive and negative work as before.

4. and the stance force cost, the integral of the force along the leg.

The sum of these four cost terms, per unit distance, is non-dimensionalized to estimate of the ‘cost oftransport’ of walking [24, 25].

S9 Mathematical remarks on entrainment

Resonance. The classic notion of a resonance is a phenomenon in linear dynamical systems, in whicha structure (such as a bridge) with a particular natural frequency of passive vibration fn is excitedby some external periodic force with frequency close to fn. In such cases, the applied external forceis not generally a↵ected by the motion of the system. The phenomena considered here cannot fullybe understood using ideas and intuition drawn from resonance. First, in our bridge-pedestrian case,the pedestrians and the bridge, and there is two-way coupling between the two. The pedestrians area↵ected by the bridge and vice versa. Further, If the pedestrians walk with zero step width initially,which was our assumption for simplicity, there is no initial sideways forcing on the bridge. So it is

4

the instability of this planar periodic motion that gives rise to the steady state bridge motion. Suchinstability-driven forcing of structures and having multiple regimes of oscillation versus no oscillationare not part of the phenomenology of resonance.

Phase response curves. The formal mathematics of entrainment and synchrony usually invoke thenotion of ‘phase response curves’, which characterize how an asymptotically stable oscillator such as ahuman walker is a↵ected by a discrete external perturbation such as the sudden and brief movement ofthe ground or a discrete pull from the outside [26, 22]. Then, a continuous or periodic external forcingsuch as due to a bridge can be considered as a sum over such brief forces, and the resulting response ofthe bipedal walker by integrating over the forcing function (by ‘convolving’ the forcing and the phaseresponse curve). The technically correct use of the phase response curves requires that the bridge andthe pedestrians are weakly coupled. Such weak coupling assumption is a good approximation when afew pedestrians are coupled to the massive bridge, shaking slightly, so that the bridge can a↵ect theperson, but the person cannot a↵ect the bridge; the same is true for the response to a slightly shakentreadmill. However, once we have hundreds of pedestrians on the bridge or if the bridge or treadmillmoves substantially, the weak coupling assumption may not be accurate. The use of phase responsecurves for the mathematical analysis of human-bridge coupling is beyond the scope of this article.

Limiting cases. As alluded to in the previous paragraph, independent of the use of the phaseresponse curves, it may be insightful to consider some limiting cases of the coupled pedestrian-humansystem. Given that the entrainment occurs only when the pedestrian mass becomes large enough, itmay be useful to consider the limit of infinitesimal bridge mass. But the limit of truly zero bridge masswith non-zero sti↵ness and zero damping leads to a mathematically singular limit that does not allowinverted pendular walking. This is because the push-o↵ and heel-strike impulses from the human willcreate infinite sideways bridge speeds due to the zero bridge mass and damping. It may be possible toconsider a bridge with zero mass, but with non-zero damping and sti↵ness, but this limit is not muchmore analytically tractable and also requires some simulation.

S10 An impulsively stabilized inverted pendulum

We have used a mechanistically-motivated feedback controller for human walking to show that bridge-shaking naturally emerges from these dynamics coupled to the bridge. We hypothesize that the primarydynamical mechanism needed for the bridge to shake sideways is some ‘feedback controller’ that correctsa sideways fall by applying a restoring sideways force against the substrate roughly periodically. Wenow provide a second example supporting this hypothesis.

Consider an inverted pendulum ‘standing’ on a platform (such as a bridge), attached to the worldthrough a spring (figure S7a). The inverted pendulum is stabilized by an ankle torque impulse, appliedonce every T time units. The stabilizing ankle torque impulse Pankle uses a proportional-derivativecontroller, proportional to the inverted pendulum angle and angular velocity: Pankle = �kp✓ � kd✓.The linearized coupled equations of motion for the bridge and the inverted pendulum are of the form:

Mxplatform +Kxplatform + Cxplatform = F, and mx+ F = 0

where m is the mass of the point-mass in the inverted pendulum, F is the horizontal interactionforce between the pendular mass and the bridge due to passive pendular dynamicss, and angle ✓ =

5

(x � xplatform)/`. To simulate the e↵ect of multiple synchronized inverted pendula, we can scale themass m, the interaction force F , and the gains kp and kd by the number of pendula N . For low N ,the vertical position of the inverted pendulum is stable (✓ = 0) due to the feedback controller, but forlarge N , the vertical position becomes unstable giving rise to platform oscillations (figure S7b), quitesimilar to those observed in the walking simulations described in the main manuscript.

A related problem is related to the art of slack-lining, in which a person walks on a thin elasticband tied between two poles [27], in which one observes uncontrolled oscillations of the feet and theelastic band when the feedback control has a delay between sensing and actuation, rather like theimplicit delay between successive actions during walking or the impulsively stabilized walker describedabove. Similarly, continuous (non-impulsive) stabilization of inverted pendulum using a traditional PDcontroller goes unstable at large delay in the feedback [28].

Number of pedestrians N Number of pedestrians N

Bridge oscillation peaks for last 20 seconds (”steady state”)

0

0.02

0.04

0.06

100 200 300 4000

Identical bipeds

100 200 300 4000

0.02

0.04

0.06

0

Non-identical bipeds

Bri

dg

e o

scill

atio

n p

eaks

Bri

dg

e o

scill

atio

n p

eaks

Bridge motion is not two-step periodic Bridge motion is two-step periodic

Figure S1: An “orbit diagram” illustrating complex motions of the bridge and dependenceof bridge “amplitude” on N . At each pedestrian number N , we perform a long simulation. Fromthis long simulation, we obtained the ‘peaks’ of the bridge oscillations (i.e. the maximum rightwardbridge deviations from its static position) from the last 20 seconds of the simulation. We plot all ofthese oscillation peaks as a function of N . If all oscillation peaks are identical, we obtain a single pointper N , as in the N > 350 regime for identical bipeds – corresponding to the bridge having the sameperiod as two human steps. If the oscillation peaks are di↵erent, as in a multi-step periodic motion ora non-periodic motion, the points represent the various oscillation peaks that the bridge goes through.

6

Y

X

Z

Y

X

Z

bridge shakes laterally

bridge shakes laterally

walking direction (forward)

walking direction (forward)

biped falls down

biped falls down

a) Control gains with reversed signs

N = 400, P = 1, Bridge initial motion = 1e-4

N = 400, P = 1, Bridge initial motion = 1e-4

b) Controller switched off: control gains set to zero

Figure S2: Bad controllers make the biped fall. Biped body trajectory is shown for two di↵erent“bad” feedback controllers, and both these controllers results in the biped falling in a few steps. a)Controller with all feedback gains in the model are reversed in sign. b) Controller with all feedbackgains set to zero. Both simulations start with the same small initial bridge oscillation. We notice thatthe biped with the zero gain controller falls later than the biped with the reversed-sign-gains controller.

7

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

initial perturbation 10-2

initial perturbation 10-9

initial perturbation 10-4

a) changes to the initial platform perturbation only affect the onset of steady-state oscillations

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

initial perturbation 10-2

initial perturbation 10-9

initial perturbation 10-4

b) with 4 bipeds being simulated changing the initial conditions even with random initial biped phases only affects the transient motion of the bridge

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

1

0

initial perturbation 10-2

initial perturbation 10-9

initial perturbation 10-4

c) the order parameter at steady-state is also the same

0.5

N = 400, P = 1

N = 400, P = 4

N = 400, P = 4

Figure S3: Steady state is independent of bridge and pedestrian initial conditions. a) Thesteady state bridge oscillations do not depend on initial conditions, but smaller initial bridge deviationimplies a longer time duration to reach near steady-state. b) The bridge’s steady state does not dependon the initial conditions of pedestrians, whereas the transients do depend on the initial conditions. c)The P = 4 bipeds simulated synchronize despite random pedestrial initial conditions and di↵erentinitial bridge deviations.

8

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

gain 1.9 mass 450

gain 2.1 mass 360

a) Changing gain k2 changes the number of pedestrians

required to shake the bridge

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

b) Changing gain k3 changes the stead-state oscillation amplitude by a small amount

gain 1.9 mass 400

gain 2.1 mass 400

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

c) Changing gain k1 changes the steady-state oscillation amplitude

gain 0.9 mass 400

gain 1.1 mass 400

350

bri

dge

sidew

ays

moti

on

time

0 100 25020015050 300

0.08

-0.08

0

0.04

-0.04

d) Changing gain k4 changes the steady-state oscillation amplitude

gain -0.4 mass 400

gain -0.6 mass 400

Figure S4: Bridge-shaking behavior is seen for a range of feedback gain values. Steady statebridge oscillations are shown for a range of feedback gain parameters, suggesting some robustness ofthe observed behavior to chosen gain values. However, as noted in Figure S2, this robustness does notextend to large changes in gains, such as reversing their signs or setting them equal to zero.

9

bri

dge

sidew

ays

moti

on

time

400

0.08

-0.08

0

0.04

-0.04

bri

dge

sidew

ays

moti

on

time0 100 200 300

0.08

-0.08

0

0.04

-0.04

c) Higher speeds need a smaller mass to produce two step synchronized oscillationstake more time to produce these oscillations

speed 0.31 mass 300speed 0.32 mass 250

speed 0.36 mass 80

bri

dge

sidew

ays

moti

on

time

0.08

-0.08

0

0.04

-0.04

speed 0.28 mass 600speed 0.29 mass 500speed 0.30 mass 400

b) For much higher speeds we can get close to two step periodic results without exact periodicity

a) Orbit diagrams for different walking speeds

4000 100 200 300

4000 100 200 300

d) The biped with the selected control gains is only stable for speeds as low as 0.28

Number of pedestrians N

00

0.02

0.04

0.06

100 200 300 4000

speed 0.30

Bri

dge

osc

illa

tion p

eaks

Number of pedestrians N

0

0.02

0.04

0.06

125 2500

speed 0.32

Bri

dge

osc

illa

tion p

eaks

Number of pedestrians N

0.02

0.04

0.06

150 300 450 6000

speed 0.28

Bri

dge

osc

illa

tion p

eaks

Bridge motion is not two-step periodic Bridge motion is two-step periodic

Figure S5: Bridge-shaking behavior is seen for a range of speeds. a) The bridge amplitudeas a function of e↵ective pedestrian number is shown as an orbit diagram for three di↵erent speeds.The same three qualitative regimes are seen for the di↵erent speeds, albeit with di↵erent ranges foreach regime. b, c, d) Bridge motion is shown for di↵erent speeds and correspondingly numbers ofpedestrians to qualitatively illustrate that the bridge-shaking behavior is seen for a range of speeds.For larger speeds, for given controller gains, the bridge settles into a periodic motion for a smallernumber of pedestrians.

10

bri

dge

sidew

ays

moti

on

time

0.02

-0.02

0

0.01

-0.01

0 40 100806020 120

bip

ed s

idew

ays

GR

F

time

0 40 100806020 120

0.04

-0.04

0

0.02

-0.02

a) Pedestrian lateral ground reaction force for the simulation

b) Treadmill lateral motion for the simulation

Figure S6: Lateral reaction forces on a shaken platform. The plots are from simulations of asingle biped walking on a treadmill that shakes sinusoidally with the following parameters: pedestrianforward walking speed = 0.5 (non-dimensional) = 1.56 m/s platform lateral shaking amplitude = 0.01(non-dimensional) = 1 cm, and platform lateral shaking frequency = 0.3 (non-dimensional) = 0.94 Hz.The ground reaction forces have two dominant frequencies, resulting in a ‘beating’ oscillation as in [29].For other parameters, when the person entrains to the platform one-to-one, the lateral reaction forcesare simply periodic at the shaking frequency, without the beating character.

11

Platform mass M

Spring k

θ

Ankle torque impulse

Inverted pendulum“stander”

-0.01

0

0.01

-0.1

0

0.1

pla

tfo

rm

dis

pla

cem

en

tp

latf

orm

d

isp

lace

me

nt

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

N = 200

N = 300

Platform amplitude decreases for N below a threshold

Platform amplitude increases for N below a threshold

Platform shaking

No platform shaking

a) Inverted pendulum standing with periodic stabilizing impulses

b) Standing model shows similar qualitative behavior to walking models

xplatform

x

Figure S7: Impulsive inverted pendulum standing. a) An inverted pendulum pivots about its“ankle” on a platform. For consistency, the platform has the same properties as the Millennium Bridgeand the inverted pendulum has the same mass and length as the biped models discussed earlier. Theimpulsive torques are applied at the frequency of normal human walking. b) When we scale thependulum properties (mass, feedback gains, and interaction forces) by the number of bipeds N , weobtain decaying platform oscillations for low N (e.g., N = 200) and growing platform oscillations forlarger N (e.g., N = 300).

12

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[2] M. Srinivasan, “Fifteen observations on the structure of energy minimizing gaits in many simplebiped models,” J R Soc Interface, vol. 8, pp. 74–98, 2011.

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[5] S. Song and H. Geyer, “Generalization of a muscle-reflex control model to 3D walking,” P Ann

Int IEEE EMBS, no. Eec 0540865, pp. 7463–7466, 2013.

[6] A. Seyfarth, H. Geyer, and H. Herr, “Swing-leg retraction: a simple control model for stablerunning,” Journal of Experimental Biology, vol. 206, no. 15, pp. 2547–2555, 2003.

[7] S. Kuindersma, F. Permenter, and R. Tedrake, “An e�ciently solvable quadratic program forstabilizing dynamic locomotion,” in Robotics and Automation (ICRA), 2014 IEEE International

Conference on, pp. 2589–2594, IEEE, 2014.

[8] Y. Wang and M. Srinivasan, “Stepping in the direction of the fall: the next foot placement canbe predicted from current upper body state in steady-state walking.,” Biol lett, vol. 10, pp. 1–9,2014.

[9] C. E. Bauby and A. D. Kuo, “Active control of lateral balance in human walking,” Journal of

biomechanics, vol. 33, no. 11, pp. 1433–1440, 2000.

[10] M. Kim and S. H. Collins, “Once-per-step control of ankle push-o↵ work improves balance in athree-dimensional simulation of bipedal walking,” IEEE Trans Robot, pp. 1–13, 2017.

[11] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant leg behaviour explains basic dynamics ofwalking and running,” Proceedings of the Royal Society of London B: Biological Sciences, vol. 273,no. 1603, pp. 2861–2867, 2006.

[12] M. A. Daley and J. R. Usherwood, “Two explanations for the compliant running paradox: reducedwork of bouncing viscera and increased stability in uneven terrain,” Biology Letters, vol. 6, no. 3,pp. 418–421, 2010.

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[14] F. C. Anderson and M. G. Pandy, “Dynamic optimization of human walking,” Journal of biome-

chanical engineering, vol. 123, no. 5, pp. 381–390, 2001.

[15] M. Ackermann and A. J. Van den Bogert, “Optimality principles for model-based prediction ofhuman gait,” Journal of biomechanics, vol. 43, no. 6, pp. 1055–1060, 2010.

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[16] S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, “Theoretical mechanics:crowd synchrony on the Millennium Bridge.,” Nature, vol. 438, pp. 43–4, 2005.

[17] I. Belykh, R. Jeter, and V. Belykh, “Foot force models of crowd dynamics on a wobbly bridge,”Science advances, vol. 3, no. 11, p. e1701512, 2017.

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[19] M. Bocian, J. H. Macdonald, and J. F. Burn, “Biomechanically inspired modeling of pedestrian-induced vertical self-excited forces,” Journal of Bridge Engineering, vol. 18, no. 12, pp. 1336–1346,2013.

[20] Q. Yang, J. Qin, and S. Law, “A three-dimensional human walking model,” Journal of Sound and

Vibration, vol. 357, pp. 437–456, 2015.

[21] S. Hwang, P. Agada, T. Kiemel, and J. J. Jeka, “Dynamic reweighting of three modalities forsensor fusion,” PLoS ONE, vol. 9, p. e88132, 2014.

[22] S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in popula-tions of coupled oscillators,” Physica D, vol. 143, pp. 1–20, 2000.

[23] A. Bobbert, “Energy expenditure in level and grade walking,” Journal of Applied Physiology,vol. 15, no. 6, pp. 1015–1021, 1960.

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14

Paper PedestrianModel

Controller 3D BridgeModel

MultiplePedestrians

Pedestrian re-sponse to bridgemotion

Belykh2017[17]

Phase os-cillator

Phase cou-pling withbridge

No Phase oscil-lator whichresponds topedestrianmotion

Yes and pedes-trians canhave di↵erentparameters

Phase can change asa function of bridgemotion

Strogatz2005[16]

Phase os-cillator

Phase cou-pling withbridge

No Phase oscil-lator whichresponds topedestrianmotion

Yes Phase can change asa function of bridgemotion

Bocian2013[19]

Invertedpendulum

Impulsemaintainslinearmomentum

No Verticallyshaking bridgewith sinusoidalmotion

No Step-timing canchange in responseto bridge motion

Bocian2012[30]

Linearinvertedpendulum

Hof footplacementmodel

No Multiplebridge vibra-tion modesdriven bypedestrianmotion

Yes Lateral foot-positioncan change in re-sponse to bridge mo-tion

JoshiSrini-vasan2014[4]

Point-mass withtelescop-ing legs

Energyminimiz-ing feed-forwardoptimaltrajectory

Yes Lateral spring-mass-dampermodel

No Just periodic motion

JoshiSrini-vasan2018

InvertedPendulum

Linearcontrollerbasedon hu-man foot-placementwith an ad-hoc linearpush-o↵impulse law

Yes Lateral spring-mass-damperbridge

Yes and pedes-trians canhave di↵erentparameters

Step-length, step-width and push-o↵impulse can changein response to bridgemotion

Table S1: Features of some previous articles on 3D walking or interaction with bridge surface, showinghow our article is distinct from them.

15