Universal power law governing pedestrian interactions
UUniversal power law governing pedestrian interactions
Ioannis Karamouzas, Brian Skinner, and Stephen J. Guy Department of Computer Science and Engineering, University of Minnesota, USA Materials Science Division, Argonne National Laboratory, USA (Dated: December 4, 2014)Human crowds often bear a striking resemblance to interacting particle systems, and this hasprompted many researchers to describe pedestrian dynamics in terms of interaction forces and po-tential energies. The correct quantitative form of this interaction, however, has remained an openquestion. Here, we introduce a novel statistical-mechanical approach to directly measure the interac-tion energy between pedestrians. This analysis, when applied to a large collection of human motiondata, reveals a simple power law interaction that is based not on the physical separation betweenpedestrians but on their projected time to a potential future collision, and is therefore fundamentallyanticipatory in nature. Remarkably, this simple law is able to describe human interactions acrossa wide variety of situations, speeds and densities. We further show, through simulations, that theinteraction law we identify is sufficient to reproduce many known crowd phenomena.
In terms of its large-scale behaviors, a crowd of pedes-trians can look strikingly similar to many other collec-tions of repulsively-interacting particles [1–4]. These sim-ilarities have inspired a variety of pedestrian crowd mod-els, including cellular automata and continuum-based ap-proaches [5–8], as well as simple particle or agent-basedmodels [9–15]. Many of these models conform to a long-standing hypothesis that humans in a crowd interactwith their neighbors through some form of “social po-tential” [16], analogous to the repulsive potential ener-gies between physical particles. How to best determinethe quantitative form of this interaction potential, how-ever, has remained an open question, with most previousresearchers employing a simulation-driven approach.Previously, direct measurement of the interaction lawbetween pedestrians has been confounded by two primaryfactors. First, each individual in a crowd experiences acomplex environment of competing forces, making it dif-ficult to isolate and robustly quantify a single pairwiseinteraction. Secondly, a pedestrian’s motion is stronglyinfluenced not just by the present position of neighboringpedestrians, but by their anticipated future positions [17–21], a fact which has influenced recent models [22–25].Consider, for example, two well-separated pedestrianswalking into a head-on collision (Fig. 1a). These pedestri-ans typically exhibit relatively large acceleration as theymove to avoid each other, as would result from a largerepulsive force. On the other hand, pedestrians walkingin parallel directions exhibit almost no acceleration, evenwhen their mutual separation is small (Fig. 1b).Here, we address both of the aforementioned factorsusing a data-driven, statistical mechanics-based analysisthat accounts properly for the anticipatory nature of hu-man interactions. This approach allows us to directly androbustly measure the interaction energy between pedes-trians. The consistency of our measurements across avariety of settings suggests a simple and universal lawgoverning pedestrian motion.To perform our analysis, we turn to the large col- lections of recently published crowd datasets recordedby motion-capture or computer vision-based techniques.These datasets include pedestrian trajectories from sev-eral outdoor environments [26, 27] and controlled lab set-tings [28] (a summary of datasets is given in the Sup-plemental Material [29]). To reduce statistical noise,datasets with similar densities were combined together,resulting in one
Outdoor dataset comprising 1,146 trajec-tories of pedestrians in sparse-to-moderate outdoor set-tings, and one
Bottleneck dataset with 354 trajectoriesof pedestrians in dense crowds passing through narrowbottlenecks. In analyzing these datasets, our primarytool for quantifying the strength of interactions betweenpedestrians is the statistical-mechanical pair distributionfunction, denoted g .As in the typical condensed matter setting [30], here wedefine the pair distribution function g ( x ) as the observedprobability density for two pedestrians to have relativeseparation x divided by the expected probability den-sity for two non-interacting pedestrians to have the sameseparation. In general, the probability density for non-interacting pedestrians cannot be known a priori , since itdepends on where and how frequently pedestrians enterand exit the environment. However, for large datasets weare able to closely approximate this distribution by sam-pling the separation between all pairs of pedestrians thatare not simultaneously present in the scene (and there-fore not interacting). As defined above, small values ofthe pair distribution function, g ( x ) (cid:28)
1, correspond tosituations where interactions produce strong avoidance.If the Cartesian distance r between two pedestrianswere a sufficient descriptor of their interaction, we wouldexpect the shape of the pair distribution function g ( r ) tobe independent of all other variables. However, as canbe seen in Fig. 1c, g ( r ) has large, qualitative differenceswhen the data is binned by the rate at which the twopedestrians are approaching each other, v = − dr/dt . Inparticular, pedestrians with a small rate of approach aremore likely to be found close together than those that are a r X i v : . [ phy s i c s . s o c - ph ] D ec r [m] g ( r ) υ ≤
1 1 < υ ≤ υ > 2 τ [s] g ( τ ) υ ≤
1 1 < υ ≤ υ > 2 (d)(a) (b)(c) FIG. 1. Analysis of anticipation effects in pedestrian motion.(a) Two pedestrians react strongly to avoid an upcoming colli-sion even though they are far from each other (path segmentsover an interval of 4 s are shown as colored lines, with ar-rows indicating acceleration). (b) In the same environment,two pedestrians walk close to each other without any rela-tive acceleration. (c) The pair distribution function g as afunction of inter-pedestrian separation r shows very differentbehavior when plotted for pedestrian pairs with different rateof approach v = − dr/dt . Units of v are m/s. (d) In con-trast, when g is computed as a function of time-to-collision, τ , curves corresponding to different v collapse onto each other. approaching each other quickly (as evidenced by the sep-aration between the curves at small r ). A particularlypronounced difference can be seen for the curve corre-sponding to small v , where the large peak suggests atendency for pedestrians with similar velocities to walkclosely together.While the distance r is not a sufficient descriptor ofinteractions, we find that the pair distribution functioncan, in fact, be accurately parameterized by a single vari-able that describes how imminent potentially upcomingcollisions are. We refer to this variable as the time-to-collision , denoted τ , which we define as the duration of time for which two pedestrians could continue walkingat their current velocities before colliding. As shown inFig. 1d, when the pair distribution function is plottedas a function of τ , curves for different rates of approachcollapse onto each other, with no evidence of a separatedependence of the interaction on v . Even when binnedby other parameters such as the relative orientation be-tween pedestrians, there is no significant difference be-tween curves (see the Supplemental Material [29]). Thisconsistent collapse of the curves suggests that the sin-gle variable τ provides an appropriate description of theinteraction between pedestrians.This pair distribution function, g ( τ ), describes the ex-tent to which different configurations of pedestrians aremade unlikely by the mutual interaction between pedes-trians. In general, situations with strong interactions(small τ ) are suppressed statistically, since the mutualrepulsion between two approaching pedestrians makes itvery unlikely that the pedestrians will arrive at a situ-ation where a collision is imminent. This suppressioncan be described in terms of a pedestrian “interactionenergy” E ( τ ). In particular, in situations where the av-erage density of pedestrians does not vary strongly withtime, the probability of a pair of pedestrians having time-to-collision τ can be assumed to follow a Boltzmann-likerelation, g ( τ ) ∝ exp[ − E ( τ ) /E ]. Here, E is a character-istic pedestrian energy, whose value is scene-dependent.This use of a Boltzmann-like relation between g ( τ ) and E ( τ ) amounts to an assumption that the systems beingconsidered are at, or near, statistical equilibrium. In ouranalysis, this assumption is motivated by the observa-tion that the intensive properties of the system in eachof the datasets (e.g., the average pedestrian density andwalking speed) are essentially time-independent. If thistime-independence is taken as given, a Boltzmann-likerelation follows as a consequence of entropy maximiza-tion. By rearranging this relation, the interaction energycan be expressed in terms of g ( τ ) as: E ( τ ) ∝ ln [1 /g ( τ )] . (1)A further, self-consistent validation of Eq. (1) is providedbelow.Figure 2 plots the interaction law defined by Eq. (1)using the values for g ( τ ) derived from our two aggre-gated pedestrian datasets. It is worth emphasizing thatthese two datasets capture very different types of pedes-trian motion. The pedestrian trajectories in the Outdoor dataset are generally multi-directional paths in sparse-to-moderate densities, with pedestrians often walking ingroups or stopping for brief conversations. In contrast,trajectories in the
Bottleneck dataset are largely unidi-rectional, with uniformly high density, and with littlestopping or grouping between individuals.Remarkably, despite their large qualitative differences,both datasets reveal the same power-law relationship un-derlying pedestrian interactions. For both datasets, the E ( τ ) [ a r b . un i t s ] τ [s] −1 −2 −1 E ( τ ) [ a r b . un i t s ] τ [s] Fit − E( τ ) ∝ τ −2 OutdoorBottleneck (a)(b) t t FIG. 2. (a) The interaction energy computed from the dense
Bottleneck dataset and from the more sparse
Outdoor dataset(inset). The overall constant k is normalized so that E (1) = 1.Both datasets fit well to a power law up to a point marked t , beyond which there is no discernible interaction. Solidlines shows the fit to the data and colored regions show theircorresponding 95% confidence interval ( Bottleneck , R = 0 . Outdoor , R = 0 . interaction energy E shows a quadratic falloff as a func-tion of τ , so that E ( τ ) ∝ /τ over the interval where E is well-defined. For smaller values of τ (less than ∼ τ , on the otherhand, the observed interaction energy quickly vanishes,suggesting a truncation of the interaction when the time-to-collision is large. We denote the maximum observedinteraction range as t ( Bottleneck: t ≈ . Outdoor: t ≈ . t does not, by itself, indicate the in-trinsic interaction range between pedestrians, since in-teractions between distant, non-neighboring pedestriansare screened by the presence of nearest-neighbors, as inother dense, interacting systems [30, 31]. For a crowdwith density ρ , the characteristic “screening time” canbe expected to scale as the typical distance betweennearest neighbors, ρ − / , divided by the mean walk- ing speed u . Such scaling is indeed consistent with thetrend observed in our data, with the denser Bottleneck dataset ( ρ = 2 . − , u = 0 .
55 m/s) demonstrating asmaller value of t than the sparser Outdoor dataset( ρ = 0 .
27 m − , u = 0 .
86 m/s) [32]. While the large- τ behavior in our datasets is therefore dominated byscreening, we can use the largest observed values of t to place a lower bound estimate on the intrinsic rangeof unscreened interactions (which we denote as τ ). Thisestimate suggests that an appropriate value is τ ≈ − τ , we infer from the data thefollowing form of the pedestrian interaction law: E ( τ ) = kτ e − τ/τ . (2)Here, k is a constant that sets the units for energy.To demonstrate the general nature of the identified in-teraction law, we performed simulations of pedestriansthat adapt their behavior according to Eq. (2) via force-based interactions. In particular, the energy E ( τ ) di-rectly implies a natural definition of the force F experi-enced by pedestrians when interacting: F = −∇ r (cid:18) kτ e − τ/τ (cid:19) , (3)where ∇ r is the spatial gradient. A full analytical ex-pression for this derivative is given in the SupplementalMaterial [29].For the purposes of simulation, each pedestrian is alsogiven a driving force associated with its desired directionof motion, following Ref. 9. The resulting force model issufficient to reproduce a wide variety of important pedes-trian behaviors, including the formation of lanes, archingin narrow passages, slowdowns in congestion, and antic-ipatory collision avoidance (Fig. 3). Additionally, thesimulated pedestrians match the known fundamental di-agram [33] of speed-density relationships for real humancrowds and qualitatively capture the empirical behaviorof g ( r ) depicted in Fig. 1c [29].Our simulations also reproduce the anticipatory powerlaw described by Eq. (2), as shown in Fig. 4. In con-trast, simulations generated by distance-based interac-tion forces fail to show a dependence of E on τ (Fig. 4).Other, more recent models of pedestrian behavior alsocannot consistently capture the empirical power-law re-lationship (see the Supplemental Material [29]). The abil-ity of our own simulations to reproduce E ( τ ) also pro-vides a self-consistent validation of our use of the Boltz-mann relation to infer the interaction energy from data.Interesting behavior can also be seen when Eq. (3) isapplied to walkers propelled forward in the direction oftheir current velocity without having a specific goal (as (d)(a)(c) (b) (e) Initial state Steady state
FIG. 3. Stills from simulations of agents following the forcelaw derived from Eq. (2). In figures (a)-(d), agents are rep-resented as cylinders and color-coded according to their goaldirection. The simulated agents display many emergent phe-nomena also seen in human crowds, including arching aroundnarrow passages (a), clogging and “zipping” patterns at bot-tlenecks (c), and spontaneously self-organized lane formation(b and d). Figure (e) depicts a simulation of agents without apreferred goal direction (arrows represent the agents’ currentorientations). The agents’ interactions lead to large-scale syn-chronization of their motion. Further simulation details aregiven in the Supplemental Material [29]. implemented, for example, in Ref. 34). In such cases,complex spatio-temporal patterns emerge, leading even-tually to large scale synchronization of motion. An exam-ple is illustrated in Fig. 3e, where a collection of pedestri-ans that is initialized to a high energy state with manyimminent collisions settles over time into a low energystate where pedestrians move in unison. This result isqualitatively similar to observed behavior in dense, non-goal-oriented human crowds [35], and is reminiscent ofthe “flocking” behavior seen in a variety of animal groups[36–40]. A detailed study of such collective behaviors,however, is outside the scope of our present work.While the model implied by Eq. (3) is widely applica-ble, it may not be sufficient on its own to capture certaincrowd phenomena, such as the shock waves and turbulent
FIG. 4. Inferred interaction energy E ∝ ln(1 /g ) as a func-tion of time-to-collision τ for different simulations, obtainedusing the anticipatory force described by Eq. (3), and thedistance-dependent force described in Ref. 9 (inset). For sim-ulations with strictly distance-dependent interactions, the in-ferred interaction energy does not show a dependence on τ .In contrast, simulations following our model closely matchthe observed empirical power law for E ( τ ). Shaded regionsdenote average energy values ± one standard deviation. flows that have been reported to occur in extremely highdensity crowds [41]. In such very dense situations, satu-rating effects such as finite human reaction time becomerelevant, and these may alter the quantitative form of theinteraction in a way that is not well-captured by our time-to-collision-based analysis. Augmenting our result withan additional close-ranged component of the interactionmay give a better description of these extremely densescenarios, and is a promising avenue for future work.To conclude, our statistical mechanics-based analysisof a large collection of human data has allowed us toquantify the nature and strength of interactions betweenpedestrians. This novel type of analysis opens new av-enues for studying the behavior of humans using real lifedata. The data we have analyzed here reveals the ex-istence of a single anticipatory power law governing themotions of humans. The consistency of this law across avariety of scenarios provides a new means to understandhow pedestrians behave and suggests new ways to eval-uate models of pedestrian interactions. Further, theseresults suggest a general quantitative law for describinghuman anticipation that may extend to other studies ofhuman behavior, which may therefore be amenable to asimilar type of analysis.Complete simulation source code, along with videosand links to data used in this study, can be foundat our companion webpage: http://motion.cs.umn.edu/PowerLaw . We would like to thank Anne-H´el`eneOlivier, Alex Kamenev, Julien Pettr´e, Igor Aranson, Di-nesh Manocha, and Leo Kadanoff for helpful discussions.We also acknowledge support from Intel and from Uni-versity of Minnesota’s MnDRIVE Initiative on Robotics,Sensors, and Advanced Manufacturing. Work at ArgonneNational Laboratory is supported by the U.S. Depart-ment of Energy, Office of Basic Energy Sciences undercontract no. DE-AC02-06CH11357. [1] D. Helbing, P. Moln´ar, I. Farkas, and K. Bolay, Env.Plan. B , 361 (2001).[2] M. Moussa¨ıd, E. G. Guillot, M. Moreau, J. Fehrenbach,O. Chabiron, S. Lemercier, J. Pettr´e, C. 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Ioannis Karamouzas, Brian Skinner, and Stephen J. Guy Department of Computer Science and Engineering, University of Minnesota, USA Materials Science Division, Argonne National Laboratory, USA (Dated: December 4, 2014)
EXPERIMENTAL DATASETS
Our findings draw from datasets published by three different research groups. These datasets are illustrated inFig. S1 and summarized in Table S1. The experiments labeled b250 combined and b4 combined are described inRefs. 1 and 2, and are combined to comprise the
Bottleneck dataset. These experimental trials were recorded at25 fps using multiple cameras, with trajectories automatically tracked and corrected for perspective distortion [1, 3].The experiments involved participants walking through a 4 m-long corridor that has a bottleneck of width of 2.5 mor 1 m. The remaining four datasets were combined to comprise the
Outdoor dataset. The three datasets labeledcrowds zara01, crowds zara02, and students003 were recorded at 25 fps using a single camera and the trajectoriesof the pedestrians were manually tracked and post-processed to minimize errors and correct the distortion due topixel noise and the camera perspective [4]. The dataset denoted seq eth was recorded at 2.5 fps and tracked in asemi-automatic process [5]. In addition to their different settings, the two datasets types,
Bottleneck and
Outdoor ,are also distinguished from each other by their different pedestrian densities (see Table S1), while within each typethe densities are similar.Matlab was used to process the corresponding 2D positional data of the pedestrians’ trajectories after applying alow-pass second order Butterworth filter to remove noise and reduce oscillation effects (zero phase shift; 0.8 normalizedcutoff frequency for the four outdoor datasets and 0.24 normalized cutoff frequency for the two bottleneck datasets).In all datasets and for each time instant, we infer the instantaneous velocity of each pedestrian using a discretederivative. To estimate the time-to-collision τ , which indicates when (if ever) a given pair of pedestrians will collideif they continue moving with their current velocities, we assume that each pedestrian can be modeled as a disc witha fixed radius. In our analysis, we used a radius of 0 . τ ) samples from the Outdoor datasets and 177,672 samples from the
Bottleneck datasets. This number of samples was sufficient to draw statistically significant conclusions about thepower law governing pedestrian interactions. While all results presented here correspond to an assumed radius of0 . FIG. S1. Pedestrian trajectories for four of the six datasets examined in this paper, colored by time-averaged density from low(dark red) to high (white) values. (A-B) are from sparse, outdoor environments with largely bi-directional flows. (C) is froma moderately dense environment with multi-directional flow. (D) is from controlled experiments where a dense crowd walksthrough a narrow constriction. Horizontal and vertical axes label the distance from the center of the scene, in meters. a r X i v : . [ phy s i c s . s o c - ph ] D ec TABLE S1. Characteristics of the datasets analyzed. The first two listed datasets were grouped together to comprise the
Bottleneck dataset. The remaining four datasets define the combined
Outdoor dataset. The average density reported in thetable is computed using the generalized definition of Edie [6].
Dataset Type Description Flow Location No. Density Citation DataPed ( / m ) files Participants 176 2.328 [1, 2] [7]b250 Lab navigating Uni- J¨ulich,combined setting through a 2.5 m directional Germanywide bottleneckParticipants 178 2.665 [1, 2] [7]l4 Lab navigating Uni- J¨ulich,combined setting through a 1.5 m directional Germanywide bottleneckcrowds Outdoor Pedestrians Bi- Nicosia, 148 0.206 [4] [8]zara01 setting interacting at a directional Cyprusshopping streetcrowds Outdoor Pedestrians Bi- Nicosia, 204 0.267 [4] [8]zara02 setting interacting at a directional Cyprusshopping streetstudents003 Outdoor Students Multi- Tel Aviv, 434 0.391 [4] [8]setting interacting at a directional Israeluniversity campusseq eth Students 360 0.148 [5] [9]Outdoor interacting outside Bi- Z¨urich,setting the ETH main directional Switzerlandbuilding
DETAILED DESCRIPTION OF THE PAIR-DISTRIBUTION FUNCTION
For a given variable x that describes the separation between two pedestrians, the pair distribution function g ( x )indicates the degree to which a pair separation x is made unlikely by the interactions between pedestrians. Specifically, g ( x ) is defined by g ( x ) = P ( x ) /P NI ( x ) , where P ( x ) is the probability density function for the relative separation x between pedestrians in the dataset, and P NI ( x ) is the probability density function for x that would arise, hypothetically,if pedestrians were non-interacting. For large separation values we expect that pedestrians do not influence each other,and therefore lim x →∞ g ( x ) = 1 .While there is no way to remove the interactions between pedestrians in the dataset, we propose the followingapproach to closely approximate the distribution P NI ( x ) . We begin by randomly permuting time information betweendifferent pedestrians, so that each moment in a pedestrian’s trajectory has its instantaneous position and velocitypreserved, but is assigned a randomly permuted time. The resulting “time-scrambled” dataset maintains the samespatially-averaged density at any given instant and the same time-averaged flow rate of pedestrians across any locationin the scene as in the original dataset. However, the pedestrian positions in the time-scrambled dataset are uncorrelatedwith each other at any given value of the scrambled time, since they are drawn from trajectories at different real times.Creating a probability density function of the scrambled data gives P NI ( x ) , allowing us to define g ( x ) = P ( x ) /P NI ( x ) .Figure 1c of the main text shows the result of this process using for the variable x the Cartesian distance r betweenpedestrians, while Fig. 1d shows the result when the probability density functions are computed for the time-to-collision τ . STATISTICAL METHODSAnalysis of similarity between g ( r ) and g ( τ ) curves We analyzed the effect that the relative velocity between pedestrians has on the distance- and time-to-collision-basedpair distribution functions by conducting separate one-way ANOVAs for g ( r ) and g ( τ ) respectively on the Outdoor dataset. All tests were performed in STATISTICA (version 8.0) with significance levels set to 5%. To estimate g ( r )and g ( τ ), we used intervals of 0.04 m and 0.04 s, respectively, and clustered pairs of pedestrians into three categoriesaccording to their rate of approach v = − dr/dt (Fig. 1c and Fig. 1d) for values of r < τ < g ( r ) has a significant dependence on the rate of approach,[ F (2 , . , P < . g ( τ ), the pair distribution function does not vary for different values of v , [ F (2 , . , P = 0 . τ is a sufficient descriptor of pedestrian interactions. Power-law fit for E ( τ ) Regarding the power-law fit shown in Fig. 2 of the main text, for both the
Outdoor and the
Bottleneck datasets,we estimated g ( τ ), and subsequently the interaction energy E ( τ ), using intervals of 0.01 s. In both datasets, due totracking errors and statistical noise, the energy is only well-defined over a finite interval, with the computed energyvalues fluctuating around a maximum observed energy at very small values of τ and becoming indistinguishable fromnoise at large values of τ . To estimate the lower τ boundary, we first clustered the data into bins of 0.2 s, and used aseries of t-tests between successive bins to determine the first two bins with significantly different interaction energies.In the Outdoor dataset, the analysis revealed a significant difference in E ( τ ) between [0.2 s, 0.4 s) and [0.4 s, 0.6 s)indicating a value of τ = 0 . t (38) = 6 . , P < . Bottleneck dataset,the first two bins already exhibit a statistically significant difference in energy [ t (38) = 10 . , P < . τ . To estimate the upper boundary value, t , we conductedseparate ANOVA tests and determined the first three successive clusters for which the interaction energy does notvary. In the Outdoor dataset, [2.2 s, 2.4 s) denotes the first bin that has the same energy as its two subsequent ones,indicating t = 2 . F (2 ,
57) = 1 . , P = 0 . Bottleneck dataset, the corresponding bin is [1.2 s, 1.4 s)resulting in the estimate t = 1 . F (2 ,
57) = 1 . , P = 0 . E ( τ ) follows a power law. A linear fit of log E vs. log τ with bisquareweighting reveals an exponent of 2 . ± .
123 for the
Outdoor dataset and 2 . ± .
192 for the
Bottleneck dataset.As can be seen in Fig. 2b, the interaction energy in both datasets can be well modeled with an exponent of 2[ t (174) = 0 . , P = 0 .
42 for the
Outdoor and t (106) = 0 . , P = 0 .
865 for the
Bottleneck ]. We note that, for visualclarity, the data in Fig. 2a and Fig. 2b are down-sampled, showing E ( τ ) samples every 0.02 s and 0.03 s respectively. PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES
Figure 1c of the main text demonstrates that the Cartesian distance r between two pedestrians is not a sufficientdescriptor of their interaction, as it cannot account for the dependence of the pair distribution function g ( r ) onthe pedestrian rate of approach v . Here we show that even the full two-dimensional displacement vector r cannotadequately parameterize pedestrian interactions. In other words, we show that the empirical behavior of the pairdistribution function is inconsistent with any form of the interaction that depends only on relative spatial coordinates. r [m] g ( r , θ = ) υ ≤
11 < υ ≤ υ > 2 FIG. S2. The pair distribution function g in polar coordinates ( r, θ ) for pedestrians in the Outdoor dataset, plotted for thefixed angle θ = 0. g ( r, θ = 0) shows a significant dependence on the rate v at which the pedestrians approach each other,indicating that the displacement vector r is not a sufficient descriptor of pedestrian interactions. τ [s] g ( τ ) ≤ θ ≤ π /3 π /3 < θ ≤ π /32 π /3 < θ ≤ π r [m] g ( r ) Bottleneck Outdoor BA FIG. S3. Appropriateness of the time-to-collision variable ( τ ) as a descriptor of pedestrian-pedestrian interactions. (A) Thepair distribution function g ( τ ) as a function of time-to-collision τ for different values of the relative orientation θ betweeninteracting pedestrians in the Outdoor dataset. The different g ( τ ) curves collapse onto each other, suggesting that, in additionto the velocity-independence demonstrated in Fig. 1d of the main text, g ( τ ) is also independent of the pedestrians’ relativeorientation. (B) The pair distribution function g as a function of the distance r between pedestrian pairs that are not on acollision course (i.e., pairs that have undefined τ ). For both the Bottleneck and the
Outdoor datasets, there is no evidence ofany repulsion beyond r ≈ . τ alone provides an appropriateparameterization of pedestrian interactions. The displacement vector r connecting two pedestrians can be parameterized by the vector norm r and the angle θ between r and a given pedestrian’s current heading. We find that, for a given fixed value of θ , the pair distributionfunction g ( r, θ ) shows a significant dependence on the pedestrian rate of approach v . This is shown explicitly for θ = 0in Fig. S2. Similar conclusions can also be drawn for different values of θ , suggesting that the displacement vector r alone cannot accurately quantify pedestrian interactions. ORIENTATION-INDEPENDENCE OF g ( τ ) In Fig. 1d of the main text, it is shown that g ( τ ) is independent of the rate v at which pedestrians approach other.Here we show that g ( τ ) is also independent of the relative orientation between pedestrians.Figure S3A plots g ( τ ) for different values of the angle θ , defined (as above) as the angle between a pedestrian’svelocity vector and the displacement vector connecting the two interacting pedestrians. As in Fig. 1d, the curves for g ( τ ) corresponding to different θ collapse onto each other. This suggests that the interaction between pedestrians ascaptured by τ is independent of the pedestrians’ relative orientation. ABSENCE OF INTERACTION FOR UNDEFINED τ Figure 2 of the main text presents the interaction energy for pedestrian pairs with finite time-to-collision τ . Herewe demonstrate that for pedestrian pairs that are not on a collision course (i.e., for pairs with undefined τ ), there isno evidence of any finite interaction beyond a short-ranged exclusion.Figure S3B shows the pair distribution function g ( r ) plotted only for pedestrian pairs with undefined τ . For the Bottleneck dataset, g ( r ) ≈ r & . . Outdoor dataset, a finite repulsive interaction,corresponding to g ( r ) <
1, also appears only at r . . g ( r ) at r ≈ . ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE
Equation (2) of the main text defines the interaction energy of a pair of pedestrians with finite time-to-collision τ .Within a force-based simulation model, this energy is directly related to the force F ij experienced by the pedestrian i due to the interaction with another pedestrian j . In particular, F ij = −∇ x ij E ( τ ) = −∇ x ij (cid:16) kτ − e − τ/τ (cid:17) , (S1)as in Eq. (3) of the main text. Here, τ is understood to be a function of the relative displacement x ij = x i − x j between pedestrians and their relative velocity v ij = v i − v j .At any given simulation step, we estimate τ by linearly extrapolating the trajectories of the pedestrians i and j based on their current velocities. Specifically, a collision is said to occur at some time τ > R i and R j , respectively, intersect. If no such time exists, the interaction force F ij is .Otherwise, τ = b −√ da , where a = k v ij k , b = − x ij · v ij , c = k x ij k − ( R i + R j ) , and d = b − ac . By substituting τ into Eq. (S1), the interaction force can be written explicitly as: F ij = − " ke − τ/τ k v ij k τ (cid:18) τ + 1 τ (cid:19) v ij − k v ij k x ij − ( x ij · v ij ) v ij q ( x ij · v ij ) − k v ij k ( k x ij k − ( R i + R j ) ) . (S2)A complete simulation is produced by combining this interaction force, and a similar force associated with repulsionfrom static obstacles, with a driving force. C++ and Python code of our complete force-based simulation model areavailable at http://motion.cs.umn.edu/PowerLaw/ . SIMULATION RESULTS
We tested the anticipatory interaction law via computer simulations using the derived force-based model. Toapproximate the behavior of typical humans, the preferred walking speeds of the pedestrians were normally distributedwith an average value of 1 . ± . k = 1 . τ = 3 s as the default parametervalues of the interaction forces. [See Eq. (S2)].Details of the simulations are listed below: • Evacuation : 150 pedestrians exit a room (10 m wide ×
24 m long) through a narrow doorway. Due to the re-stricted movement of the pedestrians, arch-like blockings are formed near the exit, leading to clogging phenomenasimilar to the ones observed in granular media [11, 12]. See Fig. 3a. • Hallway : 300 pedestrians cross paths while walking from opposite ends of an open hallway that is 20 m wide.The pedestrians dynamically form lanes of uniform walking directions to efficiently resolve collisions. See Fig. 3b. • Bottleneck : 150 pedestrians start in a 5 m-wide waiting area and have to pass through a 5 m-long bottleneck ofvariable width (1 m − . • Crossing : Two groups, of 40 pedestrians each, cross paths perpendicularly. The pedestrians prefer to slow downand let others pass rather than deviate from their planned courses. As such, homogeneous clusters of pedestriansemerge within the two groups, leading to the formation of diagonal line-shaped patterns [14]. See Fig. 3d. • Collective motion : 750 pedestrians are placed in an enclosed square area of size 40 ×
40 m, and at each timestep are propelled forward in the direction of their current velocity without having a specific goal. Pedestriansare initially given random orientations, and after a long enough time they spontaneously form a vortex patternin which all pedestrians are walking in unison. See Fig. 3e.Overall, as can be seen in Fig. 3 of the main text, the derived force law described in Eq. (S2) is able to reproduce awide variety of collective phenomena. We also used the generalized definitions of flow, speed, and density suggestedby Edie [6] to measure the density-dependent behavior that the agents exhibit in several of the simulations described τ [s] E ( τ ) [ a r b . un i t s ] Fit − E( τ ) ∝ τ −2 HallwayBottleneckEvacuation −1 ] S peed [ m / s ] HallwayBottleneckEvacuationWeidmann
A B E( τ ) ∝ τ −2 FIG. S4. Analysis of simulations generated using the interaction force described in Eq. (S2). (A) Speed-density relationseen in various simulations. The results are obtained using the generalized definitions of flow, speed and density of Edie [6] byclustering the data into bins of 0 . . The corresponding fundamental diagram is compared to measurements of Weidmann[10]. (B) The trajectories of simulated pedestrians reveal the same power-law interaction that we identify from real humancrowd data (see Fig. 2 of the main text). above. Each simulation area was divided into two-dimensional cells measuring 0 . × . g ( r ). This is shown in Fig. S5A, where simulation results for g ( r ) are plotted for different values of thepedestrian rate of approach v . The strong dependence of g ( r ) on v is consistent with what we observe in the Outdoor dataset (Fig. 1c). The peak in g ( r ) for small v can be mainly attributed to spontaneous lane formation betweenpedestrians walking in the same direction. Simulation approaches based on distance-dependent forces show somewhatdifferent behavior of g ( r ), as shown in Fig. S4B, with a weaker dependence on v . Such distance-dependent forcesimulations also fail to show a strong dependence of g on the time-to-collision (as depicted in Fig. S6A).Importantly, in addition to reproducing known crowd phenomena and flows, our simulations also reproduce theempirical behavior of g ( τ ) described in this Letter. Figure S4B shows that the inferred pedestrian interaction energy E ( τ ) ∝ ln(1 /g ( τ )) in the hallway, bottleneck (2 . r [m] g ( r ) r [m] g ( r ) υ ≤
11 < υ ≤ υ > 2 0 < υ ≤
11 < υ ≤ υ > 2 A B
FIG. S5. The pair distribution function g ( r ) in the hallway simulation generated using (A) the interaction force describedin Eq. (S2), and (B) the distance-dependent force described in [11]. The variable v denotes the rate of approach betweenpedestrians, and is measured in units of m/s. See also Fig. 1c of the main text. FIG. S6. The inferred interaction energy E ( τ ) ∝ ln(1 /g ( τ )) as a function of time-to-collision ( τ ) for simulations obtained using(A) the distance-dependent model described in [11], (B) the behavioral heuristics model proposed in [16], (C) the least-effortmodel proposed in [15], and (D) our derived anticipatory force model. In all figures, colored lines indicate average energy valuesevery 0.05 s and shaded regions denote ± one standard deviation. NOTE
C++ and Python implementation of the anticipatory force model is available athttp://motion.cs.umn.edu/PowerLaw/, along with videos demonstrating simulation results. We also providelinks to the data used to derive the power law of human interactions. [1] A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht, and W. Klingsch, Transp. Sci. , 395 (2009).[2] A. Seyfried, M. Boltes, J. K¨ahler, W. Klingsch, A. Portz, T. Rupprecht, A. Schadschneider, B. Steffen, and A. Winkens, in Pedestrian and Evacuation Dynamics 2008 , edited by W. Klingsch, C. Rogsch, A. Schadschneider, and M. Schreckenberg(Springer Berlin Heidelberg, 2010) pp. 145–156.[3] M. Boltes, A. Seyfried, B. Steffen, and A. Schadschneider, in
Pedestrian and Evacuation Dynamics 2008 , edited byW. Klingsch, C. Rogsch, A. Schadschneider, and M. Schreckenberg (Springer Berlin Heidelberg, 2010) pp. 43–54.[4] A. Lerner, Y. Chrysanthou, and D. Lischinski, Computer Graphics Forum , 655 (2007).[5] S. Pellegrini, A. Ess, K. Schindler, and L. Van Gool, in IEEE International Conference on Computer Vision (2009) pp.261–268.[6] L. C. Edie, in
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