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Physica A 389 (2010) 515526

Contents lists available atScienceDirect

Physica A

journal homepage:www.elsevier.com/locate/physa

A microscopic pedestrian-simulation model and its application tointersecting flows

Ren-Yong Guo a, S.C. Wong b,, Hai-Jun Huang a, Peng Zhang c, William H.K. Lam da School of Economics and Management, Beijing University of Aeronautics and Astronautics, Beijing 100191, Chinab Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Chinac Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, Chinad Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

a r t i c l e i n f o

Article history:

Received 31 January 2009Received in revised form 24 September2009Available online 9 October 2009

Keywords:

Pedestrian simulationPedestrian experimentIntersecting flowsModel validation and calibration

a b s t r a c t

We develop a microscopic pedestrian-simulation model in which pedestrian positions areupdated at discrete time steps. At each time step, each pedestrian probabilistically selects

a direction of movement from a predetermined set according to a logit-type functionthat considers the dynamics of other pedestrians around, and then selects a step sizethat satisfies a certain distribution. We perform a number of field experiments on realintersecting pedestrian flows with four different angles. We then validate and calibrate

the model using sample data on the deviation angles, step velocities, and velocitydensityrelations obtained from the experiments.

2009 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, numerous studies have examined the behavior and properties of pedestrian traffic on streets and insidebuildings. Pedestrian traffic has been studied using models [17]and using empirical or experimental investigations withvideo analysis [813]. In general, two types of method are used to model pedestrian flow. The first, which is usuallyapplied to large crowds, involves treating the crowd as a whole, usually as a fluid or continuum, that responds to localinfluences[3,14,15]. The second, which is more suitable for microscopic models, treats pedestrians as discrete individualsin a computer simulation. Existing microscopic models can be broadly classified into three categories: continuous, discrete,and semicontinuous.

Continuous models use differential equation systems to describe the continuous movement of pedestrians in space andtime, and include the social force [1,16]and optimal control theory models [4]. Usually, the resultant group of equations

must be solved by numerical methods (e.g., the Euler method or the RungeKutta method). The time step size used in theiteration cannot be too long or too short, as too long a step size leads to illogical pedestrian movement and too short a sizerequires a vast amount of computation time. In this class of model, the movement of pedestrians is completely rationaland determined, yet in reality a pedestrian faced with the same obstacles at different times may make different movements.Discrete models include the lattice gas [10,17,18] and cellular automatonmodels[6,7,19,20]. In this class of model, space andtime are discretized to approximate the real movement of pedestrians. Most discrete models, however, cannot accuratelycompute the distance and speed of pedestrian movement [18], and are ill suited for simulating two streams of populationsobliquely intersecting because space is discretized into square lattices. In semicontinuous models [5,21], in contrast, thespace occupied by pedestrians is continuously evolving, but time is measured by intervals.

Corresponding author. Tel.: +852 2859 1964.E-mail address:[email protected](S.C. Wong).

0378-4371/$ see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2009.10.008
http://www.elsevier.com/locate/physahttp://www.elsevier.com/locate/physamailto:[email protected]://dx.doi.org/10.1016/j.physa.2009.10.008http://dx.doi.org/10.1016/j.physa.2009.10.008mailto:[email protected]://www.elsevier.com/locate/physahttp://www.elsevier.com/locate/physa

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516 R.-Y. Guo et al. / Physica A 389 (2010) 515526

Fig. 1. Possible movements of a pedestrian.

Some microscopic models aim to simulate real dynamic scenarios, whereas others aim to understand self-organizeddynamic patterns. The validity of these models, however, is unknown, and even if some prove to be valid, they are difficultto calibrate. In real life, pedestrians move in two-dimensional spaces, and their behavior is complex and easily affected bytheir surroundings. Hence, the task of validating and calibrating pedestrian models is more difficult than it is for vehicle flowmodels[22].

Studies have been conducted into the characteristics of unidirectional and bidirectional counter pedestrian flows[2,9,13,2325]. In reality, twostreams of pedestrians canobliquely or perpendicularlyintersect at different anglesin additionto meeting head on, yet there are few studies that deal with this issue [11,26,27].

In this paper, a semicontinuous model is developed in which pedestrian space is continuous and pedestrians position isupdated at discrete time intervals. The model is able to calculate normal pedestrian distributions in space and pedestrianmovements at normal step frequencies over time, and can be used to simulate two streams of pedestrians obliquelyintersecting. Rules governing the selection of movement directions and step size guarantee that the model can be used toaccurately compute the distance and speed of pedestrian movement. In the model, pedestrians are not treated as particlesand the sizes of their bodies are considered. The model is thus suitable for simulating the movement of dense crowds. In themodel, pedestrians select their direction of movement according to a logit-based discrete choice principle.

A number of pedestrian experiments are presented in which two streams of students are asked to walk through twodesignated walkways that are bounded by traffic cones and intersect at four different angles. The model is then validatedand calibrated using the distributions of the deviation angles and the step velocities of pedestrian movements from theexperiments in cases when there are no other pedestrians close to a given pedestrian, as well as the relation of the velocityof pedestrians in the reference stream (one of the two intersecting streams) against the density of the reference stream,and the total density of pedestrians and intersecting angles observed in the experiments. Numerical simulations run by themodel with calibrated parameters show that the model is able to calculate changing trends in the velocity with the densityof the reference stream, the total density of pedestrians and intersecting angles, and also the lane and group walk behaviorof pedestrians.

The remainder of this paper is organized as follows. In Section 2, we formulate the semicontinuous pedestrian simulationmodel. In Section3,we describe the simulation and experimental scenarios. The validation and calibration of the model arepresented in Section4.Section5concludes the paper.

2. Model description

In the model, the movement of each pedestrian is decided by a combination of the pedestrians movement directionand step size at each time step dt. A pedestrian first determines his or her movement direction, and then moves a certaindistance in that direction. The current position and possible movement direction of a pedestrian are shown in Fig. 1.In the

figure, pedestrian n, which is denoted by the gray circle, is moving in the desired direction en (i.e., toward a destination).The radius of pedestrian nis rn. It is assumed that the movement of this pedestrian is affected by the movement of othersand by obstacles in the surrounding circular area, the radius of which is Rn= 8rn(seeFig. 1). When the distance betweentwo pedestrians is relatively large, there are few interactions between them. Therefore, to reduce computing time, it isassumed that only when the center point of a pedestrian is in the circular area, this pedestrian may affect the behavior ofthe pedestrian under consideration. The pedestrian can move along direction dni (i= 1, . . . , 12)or remain stationary. Theangles ni (i = 1, . . . , 12)between the twelve directions and the desired direction enare , 3/4, /2, 3 /8, /4, /8, 0,/8, /4, 3/8, /2, and 3/4, respectively.

The probability of moving along one of the twelve directions is denoted by Pni (i = 1, . . . , 12) and the probabilityof remaining stationary by Pn0 . The effect that pedestrian m has on the movement of pedestrian n in directions towardpedestrianm is illustrated inFig. 2.If the direction dni of pedestriann is in the radial cone between radials l1and l2, whichparallel linesl3 and l4, respectively, then movement in that direction will be affected. The effect of a wall on pedestrian ncan be defined in a similar fashion. If the distance between a wall and pedestrian n is less than Rn rn, then the wall willaffect the movement of pedestriannin the directions toward the wall. As shown inFig. 3,the movement of pedestriannindirectionsdn8,d

n9,d

n10,d

n11, andd

n12in the radial cone between radialsl1 and l2 will be affected by the wall. If the movement

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R.-Y. Guo et al. / Physica A 389 (2010) 515526 517

Fig. 2. Actions of two pedestrians.

Fig. 3. Actions of a pedestrian encountering a wall.

of pedestriann in a direction is affected by another pedestrian or a wall, then the latter imposes a repulsive action on theformer in that direction.

The following logit-based formula is used to compute the movement probabilityPni .

Pni =exp

max{0, cos ni }

m

fmni W

fWni

1 +12

k=1exp

max{0, cos nk }

m

fmnk W

fWnk

, i = 1, . . . , 12, (1)

Pn0= 11 +

12k=1

exp

max{0, cos nk }

m

fmnk W

fWnk

, (2)

where the deviation parameter (>0) represents the strength of the effect of deviation from the desired direction. A higher value indicates a preference towards a smaller deviation, which means that the pedestrian is more likely to move indirections that are closer to the desired direction. ni is the angle between directiond

ni and the desired directionen. Eq.(1)

indicates that pedestrians have a greater probability of moving in directions closer to the desired direction. fmni andfWn

i

are the repulsive actions of pedestrian m and wall W, which affect the movement of pedestrian n in directiondni towardspedestrianmand wallWrespectively according to the aforementioned rules.

fmni is given by

fmni = (max

{0, cos

} + )(max

{0, cos

} +)

max{dmn rm rn, },

if the directiondni is in the radial

cone between radialsl1andl2,0, otherwise,

(3)

where the intensity parameters are > 0, > 0, and > 0; is the angle between the desired direction of pedestriannand the direct path to pedestrianm; is the angle between the desired direction of pedestrian m and the direct path topedestriann;dmn denotes the distance between pedestriansm and n; andrm andrn are the radii of pedestrians m and n,respectively (seeFig. 2). Parameter is a small positive number (0.00001 in this study) to ensure that the denominator islarger than zero.

Eq. (3) indicates that the greater the distance between pedestrians m and n, the smaller the repulsive action thatpedestrian m imposes on pedestrian n . When the distance between two pedestrians approaches the sum of their radii,this action is very large. This guarantees that pedestrian ndoes not move in a direction closer to pedestrian m. The intensityparameter reflects the extent of this repulsive action, where the larger the value, the stronger the action. The second termin the numerator implies that pedestriannis more affected by pedestrians in front, and also that pedestrians in a direction

that deviates more greatly from pedestrian ns desired direction will have less effect on pedestriann. When pedestrian nmoves in a direction far from a pedestrian, the pedestrian has a minimum affect on pedestrian n, and the minimum effect

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Fig. 4. Illustration of the movement of pedestriannfrom one position to another for a given time step. The gray circle represents the pedestrian and thedashed line circle the possible movement of the pedestrian.

is reflected by parameter. For instance, inFig. 2,if the desired direction of pedestrian nise2n, then pedestrianmwill have

less effect on pedestrian n compared with the case that the desired direction of pedestriann is e1n. The third term in thenumerator indicates that the degree of influence that pedestrian m has on pedestrian n is also determined by the desireddirection of pedestrian m. When pedestrian m moves in a direction far from pedestrian n, pedestrian m has a minimum affecton pedestriann, and the minimum effect is reflected by parameter . For instance, inFig. 2,if pedestrianmintends to movein direction e3m, which is far from pedestrian n, then the effect on pedestrian n will be minimal. Here, the angle between

directione1m and the direct path to pedestrian n is equal to that between directione2mand the direct path to pedestrian n.

In both cases, as the desired direction of pedestriannis e1n, pedestrianm has identical effect onn. However, when the two

pedestrians target the same position, the update rule for this situation holds that the movement of pedestrian n will bedifferent in both cases.

Similarly,fWni is given by

fWni =

(max{0, cos } + )(1 + )max{dWn rn, }

, if the directiondni is in the radial cone between radials l1andl2,

0, otherwise,(4)

where is the angle between the desired direction en of pedestrian n and the direct path to wall W anddWndenotes thedistance between pedestriannand the wall (seeFig. 3).

Once the direction of movement has been determined, the pedestrian selects a step size. There are four possible choicesof step size sn: n, 2n/3, n/3,and0. nis the free step size and satisfies a lognormaldistribution (asobtained by sample datafrom field experiments). A longer step is the most preferred among the four possibilities. It is necessary to guarantee thatafter a position update, the position of a pedestrian does not overlap the positions of other pedestrians. We thus regulate the

pedestrian selection of step size so that points ci(i = 1, 2, 3) (see Fig. 4) do not overlap areas occupied by others. The anglesbetween the direction from the center of the dashed circle in Fig. 4 to pointsci(i = 1, 2, 3) and the movement direction are /4, 0 and /4, respectively. The coordinates of these points are given by

c1= xn + sn(cos , sin ) + rn(cos sin , cos + sin )/

2,

c2= xn + (sn + rn)(cos , sin ), (5)c3= xn + sn(cos , sin ) + rn(sin + cos , sin cos )/

2,

wherexn is the coordinate of the center of the circle currently occupied by pedestrian nand denotes the angle betweenthe movement direction and the positive horizontal axis.

At each time step, the positions of all of the pedestrians are updated at the same time. If, after the movement ofpedestriansn and m, the distance between them is less than(rn+ rm) sin(3 /8), then we consider the two pedestriansto be targeting the same position, in which case their positions will overlap. We then assume that one of them will move

and the other will remain in position, and that which pedestrian takes which action is random.

3. Simulations and experiments with intersecting pedestrian flows

3.1. Simulation scenario

We apply our model to simulate the scenario shown inFig. 5.The scenario involves two walkways of the same size thatintersect at angleand have overlapping centers. The length and width of the two walkways are L and W, respectively.In the simulation, L= 15 m, W= 3 m, the time step dt= 0.2 s, the pedestrian radius rn= 0.2 m, and the intersectingangle= /4, /2, 3 /4, and . The intersection between the overlapping areas of the two intersecting walkways andthe central square area in the horizontal walkway is referred to as the region of interest (ROI) (seeFig. 5). At each of thefirst 60 time steps, a pedestrian is generated in a random position unoccupied by others at the entrance boundary of thehorizontal and oblique walkways with the probabilities Ph andPo, respectively. The density of pedestrians in the ROI can

thus be controlled by adjusting these two values. When a pedestrian on one of the two walkways moves in the desireddirection and arrives at the exit boundary, then that pedestrian is removed from the walkway. A detailed description of the

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Fig. 5. Simulation scenario with flows intersecting at different angles.

Fig. 6. Flow chart of the model run.

model run used to simulate the scenario is shown in Fig. 6.In the figure,Tis a counter,Ntis the total number of pedestriansin the two walkways, andNgandNlare the numbers of pedestrians generated and leaving at each time step respectively.

Only the data on pedestrian movement in the ROI are considered. Further, to ensure more accurate analysis of thevelocitydensity relation of the intersecting streams, only the data collected when the two streams of pedestrians arecompletely mixed together are used. The start time of a complete mix is defined as the time at which both two streamsof pedestrians have just crossed the corresponding center line of the ROI (see Fig. 5), and the end time is defined as the timeat which all of the pedestrians in one stream have passed the center line.

At each time step, the velocity of a pedestrian is calculated by dividing the projection of the displacement in the desireddirection at the time step by the incremental time step. Note that the velocity has a negative value if the direction ofdisplacement is contrary to the pedestrians desired direction. The velocity of the reference stream (which is one of the

two pedestrian streams) is obtained by taking the average velocity of every pedestrian in the reference stream in the ROIat a given time step. The step velocity of a pedestrian is obtained by dividing the pedestrians step size by the incremental

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a b

c d

Fig. 7. Experimental scenario for flows intersecting at different angles: (a)= /4, (b)= /2, (c)= 3 /4, and (d)= .

time step. Letr andcdenote the density of the reference stream and conflict stream in the ROI, respectively. The totaldensity is thent= r+ c. The densities for the two streams are calculated by dividing the number of pedestrians in thecorresponding stream in the ROI by the area of the ROI.

3.2. Experiment description

We carried out a group of experiments on intersecting pedestrian flows. Fig. 7 shows the experimental scenario forintersecting flows when the angleis set to /4, /2, 3/4, and . Students were recruited and asked to walk along twospecially designated walkways bounded by traffic cones and variously intersecting at the four intersecting angles. The twowalkways were of the same size and their centers overlapped. For the angles = /2 and , the length and width of thewalkways were 14 m and 3 m, respectively. For the angles = /4 and 3 /4, the length and width of the walkways were16 m and 3 m, respectively.

To carry out further analyses of pedestrian behavior after the experiments, the coordinate trajectory of each pedestrianhad to be known. The two pedestrian streams on the different walkways were distinguished from each other by blue andgreen hats[26]to provide us with a more efficient method of coordinate acquisition. For the angles /4, /2, 3/4, and , 24, 23, 24, and 18 experiments were conducted, respectively. At the start of each experiment, the two teams of students

were placed in waiting areas at the ends of the two walkways. The total pedestrian population in each team ranged from24 to 90, and the number of pedestrians present in each stream was controlled. The two streams were then asked to beginwalking along the walkways, and their movements were recorded with a digital video camera. The camera was positionedto overlook the centerline of the ROI of the horizontal walkway.

After the completion of the walking experiment, the video recording was converted into an image sequence. We sampleda sequence of frames at a time interval of 0.2 s. These images were then segmented by the image segmentation technologydeveloped in [28]to identify the color of the hats worn by the pedestrians. By scanning and extracting the colors of thesegmented images, individual hats were identifiedand theirimagecoordinatesobtained.However, the automaticcoordinateacquisition technology used is not very efficient, and does not always provide accurate results. Hence, additional manualcoordinate acquisition was carried out, with students being recruited to identify the positions of the hats in the images. Theimage coordinates were then transformed into real-world coordinates using the method developed in Ref. [29]. The real-world coordinates of individual pedestrians were linked from frame to frame and the pedestrian trajectories thus obtained.The details of the experimental setup can be found in Ref. [30].

The definitions of the reference stream velocity, the pedestrian step velocity, and the density of the two streams were thesame as those used in the simulation scenario. The experiments were performed with a homogeneous type of pedestrian

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(only students) and the diversity of the individuals involved was not considered. It is thus likely that some of the pedestriansin the experiments walked at a relatively faster speed than pedestrians in real life might.

4. Validation and calibration of the model

4.1. Calibration procedure

The calibration of microscopic pedestrian models is a problem to which a definitive solution has yet to be found. Acommonly used method is to constantly modify the parameters and components of the model in question until a satisfactoryapproximation between the simulation results and the fundamental flow-density or velocitydensity diagram is achieved.For example, Berrou et al. [31] calibrated the Legion simulation model by comparing the distribution of pedestrian flow anddensity fluctuations at bottlenecks. Brogan and Johnson [32] used three evaluation metricsthe distance error, area error,and speed errorto compare the paths generated by their pedestrian model to the observed paths to calibrate their model.Johansson et al.[33] introduced a method to calibrate microscopic pedestrian-simulation models using video trajectorydata, and applied the method to the social force model. Robin et al.[21]proposed a microscopic pedestrian model based ondiscrete choice modeling, and calibrated it by the maximum likelihood estimation on a real dataset of pedestriantrajectories.It must be noted, however, that the behavior of pedestrians varies not only with their physical characteristics, but also withtheir purpose and surrounding environment. Thus, the aforementioned methods should be evaluated assuming a givenpurpose and environment.

As shown by the results of the numerical simulations in Section4.3,the model proposed here can calculate the changing

trend in the velocity of the reference stream using the variablesr,t, and obtained in the field experiments. It is thuspossible to adjust the model parameters so that the relation between the velocity and the three variables obtained by themodel fits those obtained in the experiment. Here, we calibrate the proposed model using a heuristic method, the processof which is given as follows.

Step 1. The parameter and the distribution of the free step size nare determined using the sample data on deviation angleand step velocity from the field experiments for the case in which there are no other pedestrians and obstacles in the circulararea around a pedestrian.

Step2. Based on the relation of the velocity of the reference stream to the variablesr,t, andin the field experiments,the parameters,andare calibrated.

A detailed description of the calibration of the model is presented in the following section.

4.2. Calibration of the deviation parameters and free step size

Let the probability of pedestrian n moving in direction i when there are no other pedestrians and obstacles in thesurrounding circular area bePni (). This can then be denoted as

Pni () =exp

max{0, cos ni }

1 +

12k=1

exp

max{0, cos nk } , i = 1, . . . , 12. (6)

From the experimental data, 1119 sample points for the deviation angle and corresponding step velocity when there areno other pedestrians or obstacles in the circular area around a pedestrian are obtained. Let the frequency of movement indirectioni (=1, . . . , 12)for the pedestrians in the experiments when there are no other pedestrians and obstacles in thesurrounding circular area be

Pi .Fig. 8shows the

Pi -values obtained from these sample points. The deviation directions are

divided into twelve sectors with centers specified by the twelve directions. If a deviation direction is in a sector of one of thetwelve directions, then the deviation is classified as being in that direction.

We calibrate parameterby solving the following optimization problem.

min0

12i=1

Pni ()

Pi2

. (7)

By solving the optimization problem, we have the optimal solution = 36.27 and the minimum value of 0.000049.Fig. 8shows the probability of movement in the twelve directions obtained by Eq. (6)when= 36.27. The figure showsthat the frequencies and probabilities are veryclose, and thus in the numerical simulations the parameter is takenas 36.27.

From these sample points, the probability densities of step velocities for the case in which there are no other pedestriansand obstacles in the circular area around a pedestrian can be obtained. Fig. 9 displays the probability densities andcorresponding fitted curve. The probability densities are fitted by the lognormal distribution, which has the density function

p(y) = 1

y2exp

(lny )2

22 , y> 0,

0, y 0, (8)

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Fig. 8. Frequencies of movement in twelve directions from the experimental data and the probabilities of movement in the twelve directions from thesimulation data for the case in which there are no other pedestrians and obstacles in the circular area around a pedestrian.

Fig. 9. Probability densitiesand corresponding fittedcurveof step velocity from the experimentaldatafor thecasein whichthere areno other pedestriansand obstacles in the circular area around a pedestrian.

with a location parameter and scale parameter > 0. The mean, variance, and parameters and of the fitteddistribution are 1.04, 0.05, 0.0172, and 0.2124, respectively. We thus set the free step size of the pedestrians in the modelas a random variable that satisfies the lognormal distribution.

4.3. Calibration of the other parameters

In the experiments, the velocity of the reference stream and corresponding randt-values were recorded, and 14,388,16,263, 16,316, and 12,674 sample points obtained for the intersecting angles /4, /2, 3/4, and , respectively.Fig. 10

shows the pseudo-color plots delineating the mean velocity of the reference stream against the densities randtfor thefour intersecting angles obtained from the experiments. The r tspace is divided into small square areas of 0.2 0.2and the mean velocity is the average velocity of the reference stream when the corresponding (r, t)is in a square. Thefigure shows that for different rand values, the mean velocity as a function of the total density t is generally decreasing.This is similar to the change in pedestrian velocity in unidirectional or bidirectional counterflows in studies in which thetotal density was obtained by field experiments and numerical simulations [2,3436]. Increasing the density of either thereference or conflict stream in the ROI results in less free space for pedestrian movement, and hence the mean velocitydeclines. When the values oft andare fixed, the mean velocity generally increases when r is increasing.Fig. 10alsoindicates that the relation between the mean velocity and the intersecting angle is quadratic. As the intersecting angle increases from /4 to /2, the mean velocity declines, and as increases from /2 to , the mean velocity generallyincreases. This demonstrates that two perpendicular intersecting streams have a greater effect on each other than streamscrossing at other angles.

Fifty simulations were conducted for each set of parameters( , , )and each intersecting angle. In each simulation,

the generation probabilitiesPh and Pofor the pedestrians on the two walkways took values from the sets {0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7, 0.8, 0.9, 1.0} and {0.2, 0.3, 0.4, 0.5, 0.6}, respectively. For each set of parameters ( , , ), the velocity of the

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Fig.10. Pseudo-color plots delineating the mean velocity of the reference stream against densities rand tfor the four intersecting angles obtained fromthe experiments.

reference stream and corresponding r andt-values were also recorded. The t

rspace is divided into small square

areas of 0.2 0.2. v rij (i= 1, 2, 3, 4)are the average velocities of the reference streams in the experiments for the fourintersecting angles, as the corresponding (r, t)is in square j. In the simulation, velocityvrij ( , , )can also be similarlydefined for a set of parameters( , , ).Mi (i = 1, 2, 3, 4)is the set of square areas in which the numbers of sample pointsobtained from both the experiments and the simulations are more than 0 for the four intersecting angles. Our target is toobtain a set of parameters( , , )such that the following uni-objective optimization problem is optimized

min(,,)

4i=1

1|Mi|

jMi

vrij ( , , ) v rij

2 , (9)that is, the velocitydensity relation obtained by the simulations is close to that obtained by the experiments as soon aspossible. In fact, it is difficult to obtain the optimal solution to the problem for large search space, because of the non-uniqueness property of the optimal solution and the non-monotonicity of velocity with variables ( , , ). Therefore, and

we only obtain a satisfactory solution(, , )= (0.5, 0.000015, 0.05), for which the corresponding value is 0.1802.Fig. 11shows the pseudo-color plots delineating the mean velocity of the reference stream against densities randt forthe four intersecting angles obtained from the simulation with the calibrated parameters. The figure shows that thechangingtrend of the mean velocity with the variablesr,tandbecomes more obvious.

Fig. 12displays the typical stages of the intersecting flows for the four intersecting angles at time steps 55, 65, and75 obtained from the simulation with the calibrated parameters. It can be seen that for all four different angles, lanes ofpedestrians walking in the same direction occur. The lanes in each stream are discontinuous and are often broken into bythe opposite stream. This suggests several people in the same direction exhibit group walking in a line or side by side. Insuch groups, the pedestrian in front or at the edge of the lane protects the pedestrians at the rear or on the inside, who arethus less affected by upstream rivals.

5. Conclusions

This paper proposes a microscopic pedestrian-simulation model in which the movement of each pedestrian is decidedby a combination of movement direction and step size. Each pedestrians movement direction is determined by that

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Fig. 11. Pseudo-color plots delineating the mean velocity of the reference stream against densities rand tfor the four intersecting angles obtained fromthe simulations.

pedestrians desired direction and the dynamics of surrounding pedestrians. The model also considers a pedestrians size,and is thus suitable for simulating the movement of dense crowds. The model is simpler in form than other microscopicmodels. We present a group of intersecting pedestrian flow experiments in which groups of students were asked to walkon designated walkways. The coordinates of their trajectories were obtained, and the proposed model was calibrated usingsample data obtained from the experiments. Numerical simulations using the model with calibrated parameters indicatethat the model is able to calculate the velocitydensity relation from the experiments, and the lane and group walkingbehavior of pedestrians.

Note that the proposed model differs from the discrete choice model proposed in Ref. [5] on several counts. First,althoughboth modelsconsider thefactors that affect thedirection of movement of pedestrians,including the desired directionand theposition of other pedestrians, the equations used to calculate the effect are different. Second, in the model in [5] pedestriansare considered as particles andtheir sizes are not considered, andthus themodel is not suitable for simulating the movementof dense crowds. In contrast,our model considerspedestrian size andensures that thepositions of pedestrians do notoverlapafter position updates. Third, in the model in [5], pedestrians probabilistically and simultaneously select their movement

direction and step size from two preset sets that comprise 11 and 3 elements, respectively, and thus have 33 possibleselection combinations. In our model, pedestrians first select one of 13 preset directions and then determine their stepsize from a set of 4 elements. Our model thus requires fewer computations than the model in [ 5].

In the social force model, the movement of pedestrians is completely rational and determined, and the model is thusunable to calculate the distribution of the step velocity and the movement probability described by our model.

Because more factors are involved and pedestrian space is continuous, the new model requires more computing time,compared with the lattice gas and cellular automaton models. In addition, the model is suitable to formulate pedestriansnon-panic movements, and push and bump among pedestrians are not considered. Whether the model or a modifiedversion can calculate other phenomena in intersecting pedestrian streams is a question that we will address in futureresearch.

Acknowledgements

We thank the anonymous reviewers for their constructive comments. The work described in this paper was jointlysupported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project

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= = =

= /43 = /43 = /43

= /2 = /2 = /2

= /4 = /4 = /4

Fig. 12. Typical stages of the intersecting flows for the four intersecting angles at time steps 55, 65, and 75.

No.: HKU7183/06E), the National Natural Science Foundation of China (70521001, 70629001), the National Basic ResearchProgram of China (2006CB705503), and the University of Hong Kong (10207394).

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