a study of pedestrian stepping behaviour for crowd simulation · 2016. 12. 10. · michael j....

Transportation Research Procedia 2 ( 2014 ) 282 – 290

2352-1465 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of Department of Transport & Planning Faculty of Civil Engineering and Geosciences Delft University of Technologydoi: 10.1016/j.trpro.2014.09.054

Available online at www.sciencedirect.com

ScienceDirect

The Conference in Pedestrian and Evacuation Dynamics 2014 (PED2014)

A study of pedestrian stepping behaviour for crowd simulationMichael J. Seitz a,b,∗, Felix Dietrich a,b, Gerta Köster a

aMunich University of Applied Sciences, Lothstr. 64, 80335 Munich, GermanybTechnische Universität München, Boltzmannstr. 3, 85747 Garching, Germany

Abstract

Pedestrian stepping behaviour has been widely ignored in crowd simulation models. Yet, the continuous motion of pedestriantorsos is the result of decisions about discrete steps. In this contribution we discuss biomechanical pedestrian stepping behaviourand present arguments that show its importance. In addition, we analyse empirical data from controlled experiments to better un-derstand how pedestrians make their steps. The results show the dependence of step length on speed in different walking situations,like walking sideways or climbing stairs. Independently of whether subjects walked ahead, sideways or backwards in the plane,increasing speed goes along with increasing step lengths.

c© 2014 The Authors. Published by Elsevier B.V.Peer-review under responsibility of PED2014.

Keywords: stepping behaviour; pedestrian dynamics; crowd simulation; empirical

1. Introduction

The principle of human locomotion is bipedalism. Although this seems a trivial statement, it has largely beenignored in pedestrian and crowd simulations. Bipedalism is an important feature of human kind that has come a longway in evolution (Tanner (1981); Schmitt (2003); Richmond et al. (2001)). Many fields of study are concerned withit. Some prominent examples are the evolution of bipedalism in biology, pathological movement in medicine, andkinematics and kinetics of human movement in biomechanics. Finally, in computational biomechanics simulationmodels of human walking are developed (Yamazaki et al. (1979)), focusing on individuals, not collective systems,such as crowds.The study of stepping behaviour has proven to be important for safety research. Examples are the excitation of

bridges (Strogatz et al. (2005); Macdonald (2009)), stumbling and falling (Nenonen (2013)), or the coefficient offriction necessary for turning around a corner (Fino and Lockhart (2014)). Especially important is the modellingof stair ascend and descend, since many large buildings comprise a lot of stairwells that have to be considered inevacuation scenarios (Galea et al. (2008)).

∗ Corresponding author. Tel.: +49-89-1265-3762; fax: +49-89-1265-3780.E-mail address: [email protected]

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of Department of Transport & Planning Faculty of Civil Engineering and Geosciences Delft University of Technology

The Conference on Pedestrian and Evacuation Dynamics 2014 (PED2014)

283 Michael J. Seitz et al. / Transportation Research Procedia 2 ( 2014 ) 282 – 290

Crowd simulation models that model individuals, such as the Social Force Model (Helbing and Molnár (1995);Helbing et al. (2000)), originally focused on scenarios with many pedestrians and their interactions. Mostly they donot consider individual steps or biomechanical movement, but rather forces that stem from the social and physicalcontact between pedestrians. A more recent model (Moussaı̈d et al. (2010)) separates the psychological from thephysical layer and thereby allows for modelling techniques closer to modern psychological research, such as cognitiveheuristics (Gigerenzer et al. (1999)). The Optimal Steps Model (Seitz and Köster (2012, 2014)) is based on stepwisemovement and employs it for numerical discretisation. Biomechanics could be regarded as an additional microscopiclayer in simulation models interacting with and connecting both the psychological and physical layer.At first, we discuss some aspects of biomechanics to start off a discourse in the community of crowd and pedestrian

dynamics research (Sec. 2). In the next section we present video analysis techniques for the investigation of stepwisemotion and results from a series of controlled experiments (Sec. 3). Finally we discuss the results of the experimentsand the possible future of crowd modelling with biomechanical aspects (Sec. 4).

2. Biomechanical background

In this section we discuss some general concepts well known in biomechanics and results from recent investigationsthat show how important the field is for pedestrian safety. According to Winter (2009): “The Biomechanics ofhuman movement can be defined as the interdiscipline that describes, analyses, and assesses human movement.” Itsapplications mostly lie in medicine and fields of study close to it. Therefore many topics might not be of direct interestfor pedestrian and crowd simulations, such as detailed models for neural control of movement. Nevertheless, all ofthis affects observed pedestrian behaviour to some extent.There are several different scales in the study of human biomechanics. At the highest level one might consider

kinematics, the mere study of movement, such as the position and velocity of limbs. In kinetics the forces necessaryfor this movement are investigated. For instance, the forces a single muscle applies on a tendon might be of interesthere. Also studies of muscle and joint mechanics and neural control are part of the biomechanics of human movement.Winter (2009) gives an overview on these topics.An issue that has direct relevance for pedestrian safety is the coefficient of friction necessary to turn around a

corner. Fino and Lockhart (2014) found that it is higher than a safety guideline suggests. Others investigate theforces acting during stair ascend and descend (McFadyen and Winter (1988)). These kind of findings could also beconsidered for the sudden change of direction and constraints of movement in crowd simulations.Two directions of investigation can be distinguished. First, forward models try to predict movement based on a

model of the forces and limbs acting in the human body. Second, inverse models try to reconstruct the forces thatlead to an observed movement. The latter has the advantage that the input parameters can be observed. Forces insidemuscles, which are necessary for forward models, on the other hand, cannot be observed (Winter (2009)). Here,validation is easier since the outcome is predicted movement, which can be compared to observations.Pedestrian simulation could benefit from both approaches. Forward models could be used to predict the stepping

behaviour and constraints on possible movement. Inverse modelling may be employed for the prediction of headswaying based on the analysis of gait. The specific requirements on the simulation determine the level of detailnecessary. In general, however, pedestrian models must be designed properly to allow the use of biomechanicalmovement models (see discussion in Sec. 4).Computer simulation is not new to biomechanics (Yamazaki et al. (1979)). Models vary greatly in complexity. For

instance, pedestrians are simulated as an inverted pendulum (Kuo (2007); Macdonald (2009)), or with a detailed modelof muscles. The latter is computationally complex (Martin and Schmiedeler (2014)) and therefore not yet suitable forthe simulation of many pedestrians. Somewhere in between are segments models (Koopman et al. (1995)). Mostlyeither the frontal (Macdonald (2009)), also called coronal plane, or the sagittal plane (Martin and Schmiedeler, 2014)is considered, which means the models are two-dimensional or planar. The choice of plane, frontal or sagittal, hasto be made based on the requirements in the application. This choice differs from current models of more than onepedestrian, which mostly use the top-down view or transverse plane.The excitation of bridges was assumed to be the result of pedestrians synchronising their stepping behaviour (Stro-

gatz et al. (2005)). Yet, a simple biomechanical frontal plane model of pedestrians predicts the excitation due topedestrians solely balancing their weight during the single-stance phase (Macdonald (2009)). This example also


Fig. 1. Set-ups for part three of the controlled experiments. Obstacles were indicated with masking tape in experiments (a) and (c). In experiment(b) tables were used in addition to the masking tape. Subjects had to walk around the indicated obstacle. Only one subject walked at a time so thatno interaction took place among them. In (a) the obstacle is a corner of 90 degrees, in (b) the obstacle is the same corner with tables, and in (c) theobstacle is a straight line making subjects turn around 180 degrees.

shows how pedestrian stepping behaviour can have a direct impact on building structures and thus must be consideredin civil engineering.

3. Controlled experiments

We conducted a series of experiments with twelve subjects, all computer science students, in the hall of a universitybuilding. Each of the participants had to walk individually, therefore no interactions among them took place duringthe experiment. The experimental area was video recorded from an oblique position above (see Sec. 3.1).In part one, the subjects had to walk directly from one point to another without obstacles on the way. In each

experiment they were instructed to walk slowly, normally and fast, but always at a speed they felt comfortable with,that is, the speed was self-selected. This part of the experiment was aimed at step length and speed measurement. Atfirst they had to walk forwards, then backwards and sideways. In a fourth experiment they had to walk forwards, stopat a verbal command and then resume walking after a second verbal command. The resulting stepping behaviour isanalysed in Sec. 3.2.In part two, subjects had to walk up and down a stairway. They were instructed to walk normally and then, in a

second round, fast. Here the focus lies on how many stairs they took with one step at different speeds. The results ofthis experiment are presented in Sec. 3.3.In part three, the same instruction as in part one was given concerning walking speeds, but now participants had

to walk around an obstacle. Three experiments were conducted (see Fig. 1): first, the obstacle was a corner with a90 degree angle indicated by white masking tape on the ground; second, the same obstacle was complemented withtables forming the same corner but with some height; third, they had to walk around a straight line on the ground,again indicated with white masking tape. The observed trajectories and distances to the obstacle are discussed inSec. 3.4.

3.1. Video analysis

The objective of the video analysis is to extract the exact positions of pedestrians’ feet after each step. At first,the position of the toes when touching the ground is marked manually in the video footage. Whenever the toes aremarked, the time is stored as well. This is done with the original video footage without transformations to facilitatethe process of marking the steps. Therefore, the coordinates do not represent the actual position in the plane, but theposition of the pixel in the video footage. The positions have to be projected according to the spatial distortion to getthe actual positions in the plane. For this, a large square area was marked on the floor that facilitates the creation of aprojection matrix.Whenever pedestrians’ heads are tracked, the distance to the floor has to be considered. With our approach, the

height of subjects does not pose a problem, since the positions of the feet that are marked are also located in the plane.Therefore, this analysis can be applied with very steep viewing angles. The feet have to be visible throughout thetracking, so that they can be exactly located. This poses a problem in crowds or situations where the feet are coveredby other objects. In controlled experiments, however, a setting can be chosen that matches these requirements. Aslanted camera position from above is advantageous for the visibility of the feet.


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Fig. 2. Left: Definition of the pedestrian’s position (c) with the coordinates of the left foot (a) and the right foot (b). Position (c) is the arithmeticmean of the two feet’s coordinates and has the same distance (d) to both of them. Right: Trajectory of a pedestrian walking from left to right in acontrolled experiment. The left foot’s positions are shown in red, the right foot’s positions in blue and the resulting pedestrian position in black.)

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Fig. 3. Contour-plots for the position after the next step for walking ahead (a), walking backwards (b), sideways (c) and stopping and resumewalking (d). The current position is indicated with a red cross at the origin (x,y) = (0, 0). The last step was taken in the direction of y, that is, in thedirection of the vector (x,y) = (0, 1). In (d) subjects even sometimes took steps backwards in order to equilibrate when stopping suddenly.

We use the data generated by tracking the feet to obtain the subject’s position. For this, we consider the pedestrianto be located in the middle between the two positions of his or her feet at the moment when the subject’s toes touchthe ground. This means we use the arithmetic mean of the left and right foot as position and the time when the frontfoot touches the ground (see Fig. 2, left).The trajectory of the pedestrian’s motion is then obtained by connecting the pedestrian’s positions. The left and

right steps are not observable in this trajectory, but the stepwise movement is still present as the interpolation of bothfeet (see Fig. 2, right). The resulting data can be used for comparison with or calibration of simulation models.

3.2. Step length and speed

There is a well known dependence of step length on speed of motion in the plane for pedestrians (Kirtley et al.(1985); Grieve and Gear (1966); Fukagawa et al. (1995)). The relationship can also be predicted with biomechanicalmodels (Kuo (2001)). In a previous study (Seitz and Köster (2012)) a linear dependency for walking and a non-lineardependency for jogging and running has been found. These are all investigations for forwards walking. Furthermore,


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Fig. 4. Step length and speed scatter plots with linear regressions for walking forwards (a), backwards (b), sideways (c), walking forwards andstopping in between (d). For (a) the linear model seems accurate, but for (b)–(d) the linearity could be questioned because of the limited statisticaldispersion in speed.

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Fig. 5. Relative frequencies of step lengths in histograms for walking forwards (a), backwards (b), sideways (c), walking forwards and stopping inbetween (d). For both, walking backwards and sideways, smaller steps were observed than for walking ahead.

so far only relatively fast walking speeds have been observed. In real environments pedestrians might walk slower andin certain situations sideways or backwards. Here we complement the data from the previous study with slower speedsand present observations for subjects walking sideways and backwards. Additionally, we conducted one experimentwhere participants had to stop and resume walking. Fig. 3 depicts the general stepping behaviour of subjects duringthese experiments.Fig. 4 shows the observed relations between speed and step length, which are all significant (t-test, p < 0.01). For

experiments (b)–(d) the linearity of the relation is questionable due to the limited statistical dispersion of the datain speed. For normal walking ahead however, the linear model with a slope of 0.27 and an intercept of 0.37 seemsaccurate. Slightly different coefficients have been found in the previous study (Seitz and Köster (2012)). Puttingtogether all data for forwards walking from the previous and this study, we obtain a linear model with slope 0.27 andintercept 0.38, which is almost identical to the linear model we find when only using the new data.Fig. 5 shows histograms of the step lengths for the same experiments. Step lengths are smaller when subjects

walked backwards (mean step length = 0.52 m) or sideways (mean step length = 0.42 m) than when they walked for-wards (mean step length = 0.75 m), which seems plausible given the particular movement. Furthermore, pedestriansnaturally move slower when walking sideways or backwards, thus leading to smaller steps as well. Some subjectscrossed their legs when they walked sideways, which allowed them to move faster. Others jumped, that is, both feetwere off the ground for some time, also resulting in faster motion. The latter can be observed for jogging and runningahead as well.

3.3. Stair ascend and descend

In this section, we analyse the speed and trajectories of pedestrians on stairs. We use a stairway with 24 stairs intotal, where the seventh stair is a small plateau (1.52 m deep). Each normal stair is 0.3 m deep. The difference inheight between each stair is 0.165 m, so that the total height difference is 3.96 m. Twelve participants were asked toseparately walk up the stairway with their normal speed, walk down again, walk up the stairway fast and finally walkdown fast again. All participants had to finish the experiment before the next experiment started.


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Fig. 6. Trajectories of subjects moving downstairs (left to right). The scale is given in meters. Individual stairs of the stairway are shown as vertical,black lines. The plateau is visible at the abscissa = 6. Note that the image projection was performed in the plane spanned by the eighth and 24thstair of the staircase. Together with the steep viewing angle, this reduces the depth of the plateau relative to the other stairs compared to a directtop-down view. Subjects kept to the middle of the stair and did not use the railing.

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The middle of the stairway was used by most of the subjects in both directions (see Fig. 6). Nobody used therailing. Fig. 7 shows the subjects’ speeds for every stair. Here we measure speed in stair per second, because speed inthe plane would neglect the height. Note that due to the plateau on the seventh stair, the number of stairs per secondis reduced to zero when subjects walked only on the plateau. The speed does not decrease towards the end of thestairway, implying that the subjects did not slow down even when climbing upstairs. When asked to move fast, theysubstantially increased their speed.Fig. 8 shows the percentages of zero, one and two stairs when walking upstairs (top, (a) and (c)) and downstairs

(bottom, (e) and (g)). When asked to climb fast, subjects sometimes took two stairs at a time. Zero stairs per secondwere taken on the plateau, since most subjects placed both feet on it and then proceeded with the eighth stair of thestairway. Histograms ((b),(d),(f ),(h)) of Fig. 8 show the speed in stairs per second. The difference between walkingupstairs (b) and downstairs (f ) at normal speed is small, the difference between subjects walking fast or slow is large.Subjects who climbed upstairs fast (d) were slower than many subjects climbing downstairs fast (h).We compute the mean speed of the subjects on the whole stairway (total distance covered / total time). The mean

speeds of all subjects in the experiments are 0.62 m/s for normal ascend, 1.22 m/s for fast ascend, 0.65 m/s for normaldescend and 1.46 m/s for fast descend.In summary, subjects ascending the stairway at normal speed are slower by about 50% than walking at normal

speed in the plane. When told to move fast, they reach their normal walking speed in the plane. Ascending is onlyslower than descending when climbing faster than usual. Note that this might not be true for pedestrians of a differentage group, such as children or older persons.


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Fig. 8. Histograms for stair ascend (a)–(d) and descend (e)–(h). The number of stairs taken with one step while ascending with normal self-selectedspeed is shown in (a). The distribution of stairs per second is shown in (b). In (c) and (d) the same is shown for self-selected fast ascend. In thesecond row (e)–(h) the same data is given for stair descend.

3.4. Obstacle distances

At corners pedestrians have to adjust their walking behaviour: a higher coefficient of friction is needed with higherspeeds at corners than with lower speeds (Fino and Lockhart (2014)). In this part we present the resulting trajectoriesfrom controlled experiments with individual pedestrians walking around a corner. The corner is indicated with whitemasking tape or additional tables with a height of 76 cm (see Fig. 1). The trajectories are shown in Fig. 9 (a) and (d)for a 90 degree corner with masking tape, Fig. 9 (b) and (e) for the same corner with tables and Fig. 9 (c) and (f ) fora straight line of masking tape, which subjects had to walk around and thus make a turn by 180 degrees.For each set-up both walking directions, walking from left to right and right to left, are shown. Some differences

can be noted, especially with the corner that was indicated with masking tape: pedestrians keep a greater distanceto the obstacle after passing the corner. This could be explained by inertia when changing direction of motion at thecorner. The effect also seems present in the experiment with tables in Fig. 9 (b) and (e), but is less pronounced.Comparing the experiments with masking tape and tables as obstacles, subjects clearly kept greater distances to

the tables than to the white line on the ground. In Fig. 9 (a) and (d) one can even see that subjects actually walkedover the line and thus the trajectories intersect with the obstacle. This is not possible with obstacles of some height,especially since the depicted trajectories represent the center of the pedestrians’ bodies.Another important observation is that pedestrians keep less distance to the corner than to the side of the obstacle.

This is clearly visible in Fig. 9 (a) and (d) for the 90 degree corner with masking tape, but even more extreme in Fig. 9(c) and (f ) with the straight line. It is also present in the experiment with tables in Fig. 9. An explanation could be, thatpedestrians do not want to get too close to obstacles, but they also want to minimise travel time, which saves energyfrom a biological point of view. Furthermore, it might also be a direct motivation to reach the target sooner. Thisconflict of keeping distance and minimising travel time becomes present at the corner, where the obstacle is skirtedclosely presumably in order to save time.

4. Discussion

We discussed some aspects of biomechanics in order to encourage crowd modelling researchers to study this field.Computational biomechanical models exist but are computationally complex (Martin and Schmiedeler (2014)) andtherefore might not all be suitable for crowd simulation. However, there are simple models that have proven to beuseful for the investigation of collective behaviour and the impact on the environment (Macdonald (2009)).


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Fig. 9. Trajectories of single pedestrians walking around a corner indicated by white masking tape on the ground or tables (also see Fig. 1). Thelabels of the axes are given in meters. In the top row (a)–(c) subjects walked from left to right, and in the bottom row (d)–(f ) from right to left.Trajectories were determined by tracking the position of the feet. In (a) and (d) the corner was indicated with white masking tape. One can observean asymmetry: subjected started close to the tape and then kept a slightly greater distance after the corner. In (b) and (e) the same corner wasrepresented by masking tape and tables of 76 cm height. The asymmetry is present as well. Furthermore, subjects kept a greater distance in general.In (c) and (f ) subjects walked around a line indicated by white masking tape on the ground.

The video footage analysis used for this contribution might lead to further applications. Tracking feet instead ofheads has the advantage of more detailed information about the movement process. This information can be usedfor explicit modelling of pedestrian motion based on stepping. For instance, the process of climbing stairs stronglydepends on stepping behaviour, which is an important issue in safety science. Additionally, the swaying of heads is achallenge in video tracking and requires a better understanding of biomechanical human movement.The series of controlled experiments supports previous findings (Seitz and Köster (2012)) that have shown a linear

dependency of step length on walking speed. The linear model was refined and can be used for crowd simulationmodels that make use of this relation, such as the Optimal Steps Model (Seitz and Köster (2012)). Other walkingbehaviours, such as walking sideways or backwards, also show a significant relation between step length and speed.The investigation of walking on stairs showed how the stepping behaviour depends on speed and also how speeddiffers in ascend and descend. We found a clear difference in trajectories when pedestrians are skirting obstacles ofdifferent heights. Furthermore subjects accepted smaller distances to the obstacle when walking around a corner thanwhen they walked along the side.Crowds are immensely complex systems. Many different mechanisms contribute to crowd movement. Some of

the most important ones are contact forces, body movement, cognitive processes and social interaction. These areinvestigated in their respective domains of physics, biomechanics, psychology and social psychology. We propose amodular approach to modelling crowd dynamics, that is, separating the different layers. This would make it easier forexperts from the respective fields to contribute to simulation models. Furthermore, it would reduce complexity to amore manageable level, which might facilitate advances in modelling and validation.

Acknowledgements

We thank Maik Boltes, Mohcine Chraibi and Daniel Salden from Forschungszentrum Jülich for advice on videoanalysis techniques. This work was partially funded by the German Federal Ministry of Education and Researchthrough the project MultikOSi on assistance systems for urban events – multi criteria integration for openness and


safety (Grant No. 13N12824). The authors gratefully acknowledge the support by the Faculty Graduate Center Ce-DoSIA of TUM Graduate School at Technische Universität München, Germany. Support from the TopMath GraduateCenter of TUM Graduate School at Technische Universität München, Germany, and from the TopMath Program atthe Elite Network of Bavaria is gratefully acknowledged.

References

Fino, P., Lockhart, T.E., 2014. Required coefficient of friction during turning at self-selected slow, normal, and fast walking speeds. Journal ofBiomechanics 47, 1395–1400. doi:10.1016/j.jbiomech.2014.01.032.

Fukagawa, N.K., Wolfson, L., Judge, J., Whipple, R., King, M., 1995. Strength is a major factor in balance, gait, and the occurrence of falls. TheJournals of Gerontology Series A: Biological Sciences and Medical Sciences 50A, 64–67. doi:10.1093/gerona/50A.Special Issue.64.

Galea, E., Sharp, G., Lawrence, P., Holden, R., 2008. Approximating the evacuation of the world trade center north tower using computersimulation. Journal of Fire Protection Engineering 18, 85–115. doi:10.1177/1042391507079343.

Gigerenzer, G., Todd, P.M., A.B.C. Research Group, 1999. Simple Heuristics That Make Us Smart. Oxford University Press.Grieve, D.W., Gear, R.J., 1966. The relationships between length of stride, step frequency, time of swing and speed of walking for children and

adults. Ergonomics 9, 379–399. URL: http://www.tandfonline.com/doi/abs/10.1080/00140136608964399.Helbing, D., Farkas, I., Vicsek, T., 2000. Simulating dynamical features of escape panic. Nature 407, 487–490.Helbing, D., Molnár, P., 1995. Social Force Model for pedestrian dynamics. Physical Review E 51, 4282–4286.Kirtley, C., Whittle, M.W., Jefferson, R., 1985. Influence of walking speed on gait parameters. Journal of Biomedical Engineering 7, 282–288.Koopman, B., Grootenboer, H.J., de Jongh, H.J., 1995. An inverse dynamics model for the analysis, reconstruction and prediction of bipedal

walking. Journal of Biomechanics 28, 1369–1376. doi:10.1016/0021-9290(94)00185-7.Kuo, A.D., 2001. A simple model of bipedal walking predicts the preferred speed-step length relationship. Journal of Biomechanical Engineering

123, 264–269.Kuo, A.D., 2007. The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Human Movement Science

26, 617–656. doi:10.1016/j.humov.2007.04.003.Macdonald, J.H., 2009. Lateral excitation of bridges by balancing pedestrians. Proceedings of the Royal Society A: Mathematical, Physical and

Engineering Science 465, 1055–1073. doi:10.1098/rspa.2008.0367.Martin, A.E., Schmiedeler, J.P., 2014. Predicting human walking gaits with a simple planar model. Journal of Biomechanics 47, 1416–1421.

doi:10.1016/j.jbiomech.2014.01.035.McFadyen, B.J., Winter, D.A., 1988. An integrated biomechanical analysis of normal stair ascent and descent. Journal of Biomechanics 21,

733–744. doi:10.1016/0021-9290(88)90282-5.Moussaı̈d, M., Perozo, N., Garnier, S., Helbing, D., Theraulaz, G., 2010. The walking behaviour of pedestrian social groups and its impact on

crowd dynamics. PLoS ONE 5, e10047. doi:10.1371/journal.pone.0010047.Nenonen, N., 2013. Analysing factors related to slipping, stumbling, and falling accidents at work: Application of data mining methods to finnish

occupational accidents and diseases statistics database. Applied Ergonomics 44, 215–224. doi:10.1016/j.apergo.2012.07.001.Richmond, B.G., Begun, D.R., Strait, D.S., 2001. Origin of human bipedalism: The knuckle-walking hypothesis revisited. American Journal of

Physical Anthropology 116, 70–105. doi:10.1002/ajpa.10019.Schmitt, D., 2003. Insights into the evolution of human bipedalism from experimental studies of humans and other primates. Journal of Experi-

mental Biology 206, 1437–1448. doi:10.1242/jeb.00279.Seitz, M.J., Köster, G., 2012. Natural discretization of pedestrian movement in continuous space. Physical Review E 86, 046108.

doi:10.1103/PhysRevE.86.046108.Seitz, M.J., Köster, G., 2014. How update schemes influence crowd simulations. Journal of Statistical Mechanics: Theory and Experiment

(accepted).Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B., Ott, E., 2005. Theoretical mechanics: Crowd synchrony on the millennium bridge.

Nature 438, 43–44. doi:10.1038/438043a.Tanner, N.M., 1981. On Becoming Human. Cambridge University Press.Winter, D.A., 2009. Biomechanics and Motor Control of Human Movement. 4th ed., John Wiley & Sons.Yamazaki, N., Ishida, H., Kimura, T., Okada, M., 1979. Biomechanical analysis of primate bipedal walking by computer simulation. Journal of

Human Evolution 8, 337–349. doi:10.1016/0047-2484(79)90057-5.

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