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The effect of overtaking behavior on unidirectional pedestrian flow
Safety Science 50 (2012) 1704–1714
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The effect of overtaking behavior on unidirectional pedestrian flow J.K.K. Yuen ⇑, E.W.M. Lee Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Available online 30 December 2011 Keywords: Pedestrian flow Evacuation Collective behavior Overtaking Social force model
a b s t r a c t We present a model of overtaking behavior that can be used to simulate unidirectional pedestrian flow in routine. All pedestrians have the ability to determine whether or not to overtake other pedestrians according to their desired velocity and position. Although existing models such as cellular automata models, lattice gas models, social force models, etc., can be used to predict evacuation performance, most of these models are either computationally inefficient or do not account for some crucial elements of human behavior in a moving crowd. Furthermore, these models use either empirical equations developed from experiments or mechanical system analogies to determine movement decisions. The pedestrian flow patterns simulated by these models may deviate significantly from reality. In reality, pedestrians walk at different velocities and pedestrians with a higher walking velocity are accustomed to overtaking other pedestrians with a lower walking velocity and this paper aims to mimic this behavior as the original social force model developed by Helbing et al. does not reflect this pattern of collective pedestrian behavior. In this paper, we propose modifications of the social force model that reflects how overtaking behavior operates in routine. The comparison of the pedestrian flow pattern between the original social force model and the modified social force models with the real data collected by the camcorder is also performed in order to demonstrate our modified social force model can be used to achieve reasonable simulations of overtaking behavior among pedestrians. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The past two decades have seen the development of a number of different kinds of evacuation models used to model complex crowd movement. These models provide important information in the analysis and design of channels, bridges, railway stations, shopping malls, and other public amenities. Microscopic approaches to pedestrian flow (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Helbing et al., 2000; Helbing and Molnar, 1995; Nagatani et al., 2001; Song et al., 2006; Zarboutis and Marmaras, 2004; Wolfram, 1986; Fukui and Ishibashi, 1999; Owen et al., 1996, 1997; Thompson and Marchant, 1995, 1996; Lo et al., 2004; Lo and Fang, 2000) have recently attracted considerable attention and existing evacuation modeling techniques have been comprehensively reviewed in a number of studies (Shannon et al., 1999; Gwynne et al., 1999; Kuligowski and Peacock, 2005). Existing microscopic models (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Helbing et al., 2000; Helbing and Molnar, 1995; Nagatani et al., 2001; Song et al., 2006; Zarboutis and Marmaras, 2004; Wolfram, 1986; Fukui and Ishibashi, 1999; Owen et al., 1996, 1997;
⇑ Corresponding author. Tel.: +852 3442 4317; fax: +852 2788 7612. E-mail address: [email protected] (J.K.K. Yuen). 0925-7535/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssci.2011.12.020
Thompson and Marchant, 1995, 1996; Lo et al., 2004; Lo and Fang, 2000) can be broadly divided into discrete space models and continuous space models (Gwynne et al., 1999; Kuligowski and Peacock, 2005). In early 1970s, Henderson (1971) has proposed that pedestrians can be treated like fluids, as the movement of the pedestrians behave very similar to gas particles, then Bradley (1993) has further describes the crowd movement of the pedestrians by using the Navier–Stokes equation under a high density circumstance. Both of the fluid dynamic models above provide the early framework of the continuous space models. For the fine network approach, the cellular automaton (CA) models (Wolfram, 1986) has been studied and applied in physics since 1980s which play an important role in the development of the discrete space models. Researchers like Blue and Adler (1999) and Fukui and Ishibashi (1999) have further applied the CA models in pedestrian movement simulation. Nowadays, a series of fine network approach models like EXODUS (Owen et al., 1996, 1997), SIMULEX (Thompson and Marchant, 1995, 1996), and SGEM (Lo et al., 2004; Lo and Fang, 2000), are applicable for evacuation simulation. Generally, discrete space models like the cellular automaton (CA) model (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Wolfram, 1986; Fukui and Ishibashi, 1999) and the lattice gas (LG) model (Nagatani et al., 2001) allow
J.K.K. Yuen, E.W.M. Lee / Safety Science 50 (2012) 1704–1714
pedestrians to move at or within a fixed node or grid and update the positions of pedestrians at discrete time intervals. In contrast, continuous space models like the social force model (Helbing et al., 2000; Helbing and Molnar, 1995) and part of the agent-based model (Zarboutis and Marmaras, 2004) allow pedestrians to move continuously within a pre-defined geometry. One drawback of both the CA and LG models is that they do not reflect reality by allowing pedestrians to move around in an unrestricted manner. Pedestrian behavior can be simulated successfully only when pedestrian motion is updated continuously. We therefore focus on continuous space models in this paper as most of them allow for finite interaction between pedestrians. Unlike the discrete space model that the compartment is divided into a fixed node or grid, by using the continuous space model, we can develop the geometry in a more precise way and the magnitude of the interaction force between the pedestrians and the boundaries can be clearly represented by defining it in terms of vectors. Helbing et al. (2000) and Helbing and Molnar (1995) introduced a stochastic model that incorporates ‘body force’ and ‘sliding force’ to describe the interactions between evacuees and boundaries in crowd conditions. Their specification is a highly renowned continuous space model that can successfully simulate the most typical forms of human behavior in evacuation conditions. We propose a modified version of this model to simulate overtaking behavior in a normal crowd movement environment. 1.1. The idea of overtaking In reality, pedestrians walk at different velocities. Pedestrians with a higher walking velocity are accustomed to overtaking other pedestrians with a lower walking velocity to maintain their own desired walking velocity. One example of this type of overtaking behavior can be observed at railway stations during peak hours, in which pedestrians rush along the platform and try to avoid colliding with each other when catching the train. The original social force model proposed in Helbing et al. (2000) shows the result of the ‘faster-is-slower effect’ due to the impatience shown in the event of an evacuation and how it reduces the chance of survival. This is because evacuees can become more motivated in an emergency situation. However, such behavior does not arise in a normal walking situation. Given that most of the existing pedestrian movement models fail to account for overtaking behavior and its common occurrence in normal crowd movement settings, there is a need to investigate this area in more detail. Moreover, competitive forms of behavior such as pushing and squeezing are less likely to occur as there are no or little time constraints in normal crowd movement conditions. Both of the above observations motivate us to modify the original social force model in a way that mimics the realistic movement of crowds in routine. Thompson proposed the possibility of overtaking in SIMULEX (Thompson and Marchant, 1995, 1996). It allows the assessing person to overtake the obstructing person by moving with two potential angles. Currently researchers are working on the collision avoidance behavior of pedestrians. Feurtey (2000) has simulated the collision avoidance behavior by introducing the disk of admissible displacements, it simplified the process of searching a collision-free path from a three-dimensional space to searching a collision-free point in a limited two-dimensional domain. Even the future position of all obstacles including wall and other pedestrians nearby is represented in a three dimensional space, by using the information of the position and speed of the obstacles, a safe trajectory can still be predicted by each agent from their current position. Chooramun et al. (2010) has proposed a hybrid model which combines the continuous, coarse and fine models to represent the discretization of space in circulation and evacuation models. In addition, the behaviors of wall and agent avoidance have
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been implemented in their hybrid model by computing a repulsive force between the pedestrians nearby, so that the pedestrians can maintain a desired interpersonal space between each other. Nevertheless, the model presented in Chooramun et al. (2010) can also address a range of additional behaviors such as cross-flows, merging flows, emergence of lane formation, and behavioral adaption in low and high density settings. This paper studies the unidirectional flow of pedestrians through a channel. We present a modified social force model that mimics the unidirectional flow of pedestrians through the channel by eliminating the unrealistic representation of competitive behavior reflected in the social force model. We also introduce the concept of overtaking behavior into the social force model to create a modified social force model that can be used to simulate normal crowd movement in a realistic manner. Fig. 1 illustrates overtaking behavior in a unidirectional channel. In this paper, the performances of the original social force model and our modified social force models will be compared with the pedestrian flow pattern recorded by camcorders in a real situation. 1.2. The original social force model We now provide a brief review of the original social force model. The social force model for pedestrian flow was first proposed by Helbing and Molnar (1995)). In Helbing et al. (2000), put forward another social force model that simulates panic situations. This model describes a mixture of socio-psychological and physical forces that influence behavior in a pedestrian flow. Pedestrians are normally driven by three forces: desire force FDi, social force FSi, and granular force FGi. The corresponding equations are:
F Di ¼ mi
v 0i ðtÞe0i ðtÞ v i ðtÞ si
ð1Þ
where mi is the mass of the ith pedestrian, vi is the actual velocity of pedestrian i, v 0i is the desired velocity of pedestrian i, e0i is the initial desired unit vector of pedestrian i and si is a pre-defined time interval;
F Si ¼
Np X jð–iÞ
Ai exp
ðr ij dij Þ nij Bi
ð2Þ
where Np is the total number of pedestrians in the geometry, dij = ||(ri rj)|| denotes the distance between pedestrians i and j, rij = (ri + rj) of the pedestrian radii ri and rj and, A and B are predefined constants that represent the strength and range of social interaction, and nij is the unit vector pointing from pedestrian j to pedestrian i;
F Gi ¼
Np X
kgðr ij dij Þnij þ kgðr ij dij ÞDv tji t ij
ð3Þ
jð–iÞ
where the granular force is determined by a body force kgðrij dij Þnij that counters body compression and a sliding force kgðrij dij ÞDv tji tij that impedes relative tangential motion, the tangential direction is represented by Dv tji ¼ ðv j v i Þ t ij and tij ¼ ðn2ij ; n1ij Þ, and k and j represent another two constants. In addition, the function g(x) is zero if pedestrians do not touch each other; otherwise, it is equal to the argument x. 1.3. Modifications to the social force model Seyfried et al. (2006) were the first to introduce the self-stopping mechanism for pedestrians in the social force model to prevent negative velocity among pedestrians caused by the large repulsion forces between them. They also modified the social force model to reproduce the empirical velocity density relation of pedestrian
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Either moving upward to overtake blocking
Or moving downward to overtake
Fig. 1. A schematic illustrating the action of overtaking.
movement for a one-dimensional system. Parisi et al. (2009) propose another self-stopping mechanism for pedestrians to prevent them from continuously pushing each other by introducing a respect area, the self-stopping mechanism will be triggered if a pedestrian entered the respect area of other pedestrians. The main advantage of the modifications made by Parisi et al. over those of Seyfried et al. is that they amend the desired velocity of pedestrians instead of changing their velocity. This gives pedestrians a selfslowing mechanism instead of the sudden stop action introduced by Seyfried et al. Lakoba et al. (2005) introduced three main modifications of the social force model that link the magnitude of the social force with pedestrian density. For different pedestrian orientations, Lakoba and his colleagues propose that weighting factors should be applied by multiplying the social repulsive forces at play between them. They also introduce a pedestrian memory effect that allows pedestrians to select their own desired direction to enhance the stability of the pedestrian flow. In addition, Zheng et al. (2002) employ the social force model in conjunction with a neural network and allow pedestrians to act as they wish in both normal and emergency situations. However, none of the studies outlined above modifies the social force model to allow pedestrians to overtaking other pedestrians. Work to modify the social force model in a manner that simulates overtaking behavior is therefore still somewhat limited. The rest of this paper is organized as follows. Section 2 describes the development of the overtaking sub-model. Case studies used to demonstrate the performance of the overtaking sub-model are detailed in Section 3. The results are discussed in Section 4 and our conclusions are presented in Section 5. 2. Model We consider the unidirectional flow of pedestrians in a channel. The idea of the visual angle and the modifications we propose to the original social force model are employed to simulate overtaking behavior throughout the pedestrian flow. The basic algorithm for the overtaking process is shown in Fig. 2. 2.1. Visual angle According to Eq. (2), the social repulsion forces between pedestrian i and j represent the interaction between pedestrian i and all other neighboring pedestrians, regardless of whether they are in front of or behind pedestrian i. Given that people are unlikely to notice anyone behind them unless the trailing person is very close and they can hear noise the trailing person makes, they usually pay less attention to what occurs behind them. In this situation, the
interaction between pedestrian i and all the neighboring pedestrians who are in front of pedestrian i is much more important than neighboring pedestrians behind pedestrian i For this reason, we assume that pedestrians pay less attention to what occurs behind them. The interaction between pedestrian i and neighboring pedestrians behind pedestrian i is set to zero and the angle of view, we call it visual angel is set to 180° instead of 120° or 270° for simplicity. The new social repulsion force between pedestrian i and pedestrian j can be determined by:
8 Np P ðr ij dij Þ > > nij Ai exp < Bi fij ¼ jð–iÞ > > : 0
if rjx r ix ð4Þ otherwise
This means that pedestrian i can interact only with pedestrians who are within his visual angle, as illustrated by Fig. 3. 2.2. Modification of the social force model There are generally two main factors that need to be considered in simulating overtaking behavior during crowd movement: (1) the direction of overtaking and (2) the degree of overtaking. 2.2.1. The direction of overtaking Fig. 1 shows a situation that if a pedestrian is in hurry and moving from left to right, he or she will move either upward or downward to avoid jamming. The decision on whether to move upward or downward is based on the determination of how to avoid both the social repulsion forces at play between pedestrians and the social boundary repulsion force exerted. Taking pedestrian i as the origin as illustrated in Fig. 3, the product of the social repulsion forces between pedestrian i and all neighboring pedestrians on the upward direction of the initial movement vector and the upward rational vector act as the upward social repulsion force Fup and the product of the social repulsion forces between pedestrian i and all neighboring pedestrians on the downward direction of the initial movement vector and the downward rational vector act as the downward social repulsion force Fdn. The upward social repulsion force and the downward social repulsion force can thus be represented as:
F up ¼
Np X
Ai exp
jð–iÞ
F dn ¼
Np X jð–iÞ
Ai exp
* ðr ij dij Þ nij up Bi
if r iy 6 rjy
* ðr ij dij Þ nij down Bi
if r iy 6 r jy
ð5Þ
ð6Þ
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Initialization of the Geometry & Pedestrians
1
Determine whether SRF exists between Pi & n.
N
End
Y
Determine whether the desired velocity of Pi > n
Y
N
End
Overtaking occurs
2
Determine the direction of overtaking
3
Determine the magnitude of the overtaking force
Add the overtaking force to the original social force model
Update and store the locations of the pedestrians Notes: SRF = social repulsion force Pi = Pedestrian i n = Pedestrian i
neighboring pedestrians
1
By applying the concept of the visual angle, the social repulsion force exists only between pedestrian i and all
neighboring pedestrians who are in front of him.
2
Pedestrian i will move towards the direction in which there is less social repulsion force and less social boundary
repulsion force. Fig. 2. Basic algorithm for the overtaking behavior process.
*
*
where up and down represent the upward and downward rational vector of the initial moving vector for the ith pedestrian, respectively. In addition, during the overtaking process, pedestrians seek space to overtake others and avoid getting hurt from the boundary. The social boundary repulsion force should also be included when the overtaking direction is considered. From Fig. 3, the lower boundary wall exerts an upward boundary repulsion force Fwup on pedestrian i and the upper boundary wall exerts a downward
boundary repulsion force Fwdn on pedestrian i. Therefore, the shorter the distance between the pedestrian and the boundary, the higher the social boundary repulsion force, and vice versa. The direction of the overtaking movement is determined by the social repulsion forces between pedestrians and the social boundary repulsion force exerted. If the sum of the total upward repulsion force and the downward boundary repulsion force is larger than the sum of the total downward repulsion force and the upward
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note: y1 > y2
y1
θi,
ut
y2
The i th pedestrian The neighboring pedestrians who can be seen by the i th pedestrian The neighboring pedestrians who cannot be seen by the i th pedestrian The social force exerted on the i th pedestrian by the i th neighboring pedestrian The upward social repulsion force acting on the i th pedestrian The downward social repulsion force acting on the i th pedestrian The upward boundary repulsion force acting on the i th pedestrian The downward boundary repulsion force acting on the i th pedestrian
θi,
The visual angle (180o) for the i th pedestrian
ut
The initial moving vector for the i tth pedestrian The boundary wall Fig. 3. A schematic illustrating the determination of the direction of overtaking.
boundary repulsion force, pedestrian i will move downward to avoid the larger upward repulsion force, and vice versa. Therefore, the overtaking direction can be described by
F Mi ¼
X
ðF up þ F dn Þ ðF wup þ F wdn Þ
ð7Þ
jðn–iÞ
F Mi
P 8 > < 1 if jðn–iÞðF up þ F dn Þ ðF wup þ F wdn Þ P 0 P ¼ > if ðF up þ F dn Þ ðF wup þ F wdn Þ < 0 :1
ð8Þ
jðn–iÞ
In mathematical order, the pedestrian will move downward if FMi is equal to 1 and will move upward if FMi is equal to 1.
2.2.2. The degree of overtaking The magnitude of the overtaking force is determined by a twodimensional Gaussian function (Eq. (9), the format of which is the product of the horizontal dimension Gaussian function and the vertical dimension Gaussian function shown in Eqs. (10) and (11), where q is the density of the specific area, a is a constant, x is the horizontal distance between pedestrians i and j, while y is the vertical distance between pedestrians i and j, lx and ly are set to zero as pedestrian i is located at the origin, and rx and ry are respectively related to a product of parameters and the desired velocity of pedestrian i. Fig. 4 shows the shape of the overtaking contour force between pedestrian i and j.
F Oi ¼
X a
jðn–iÞ
q
hðxÞ hðyÞ
ð9Þ
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ut
y
x
The i th pedestrian th The neighboring pedestrian j who can be seen by the i pedestrian
The overtaking contour force exerted on the i th pedestrian by the neighboring pedestrian Fig. 4. A schematic illustrating the overtaking force
1 ðx lx Þ2 hðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2r2x 2pr2x ðy ly Þ 1 hðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi exp 2r2y 2pr2y
2
! ð10Þ ! ð11Þ
2.2.3. The addition of overtaking force After establishing the direction and magnitude of the overtaking force, we can simulate overtaking behavior by adding the product of the magnitude and direction of the overtaking force to the original social force model as in Eq. (12). The general schematic of the overtaking contour force is illustrated in Fig. 4.
mi
dv i ¼ F Di þ F Si þ F Gi þ F Mi F Oi dt
ð12Þ
The performance of the above three models is compared with the pedestrian flow pattern recorded by camcorders in a real situation. All three models are initialized by the geometry and the initial positions of the pedestrians and their individual walking velocities are captured by the camcorders. Note that the walking velocity of each pedestrian is determined by dividing the total distance traveled by the time taken. The results predicted by the three models are analyzed both quantitatively and qualitatively. Quantitatively, the performances of the three models as mentioned are represented by the prediction errors between the models and the real situation. It enables us to compare the performance of the models directly. In addition, the trends of the pedestrian flow rate predicted by different models are also evaluated. Besides the quantitative analysis, we also conduct the qualitative analysis in Section 3.2. We compare the behaviors of the pedestrians simulated by the three models and investigate the snapshots from the real situation. Both qualitative and quantitative analyses ensure a less subjective comparison of the three models.
3. Simulation results and analysis Because the original social force model does not reflect the concept of the visual angle, a variation on the social force model that incorporates this concept is also developed for comparison with our proposed social force model described in Section 2 which also incorporates the visual angle. The following three models are therefore compared. 1. The original social force model. 2. A social force model incorporating the visual angle. 3. Our modified social force model.
3.1. Quantitative analysis Fig. 5 and Table 1 show the simulation results of the three models according to the pedestrian flow rates calculated as the time required for different numbers of pedestrians to leave the channel under consideration. One model is considered to be quantitatively more accurate than the others if the prediction error (in the form of the root-mean-square deviation) is smaller than that of the other models. The definition of RMSD is shown in (13) where n is the total ðrÞ ðmÞ number of pedestrians and t i and t i are respectively the real and
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12 11 10
Number of pedestrians leave
9 8 7 6 5 4 3 2
Real situation Original Social Force model (RMSD=0.38) Modified Social Force model+Visual Angle (RMSD=0.71) Original Social Force model+Visual Angle (RMSD=1.30)
1 0
8
9
10
11
12
13
14
15
16
17
18
Leaving time (s) Fig. 5. Pedestrians’ leaving time graph.
Table 1 The total leaving time of the pedestrians of the different models. Models
Total leaving time of the pedestrians (s)
Real situation Original social force model Modified social force model with visual angle Original social force model with visual angle
14.4 14.8 15.5 15.6
Table 2 The root-mean-square deviations of the different models. Models
Root-mean-square deviation
Original social force model Modified social force model with visual angle Original social force model with visual angle
0.38 0.71 1.30
modeled leaving times of the ith pedestrian. Table 2 summarized the root-mean-square deviation (RMSD) of the different models.
RMSD ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP 2 u n ðrÞ ðmÞ t i ti ti n
ð13Þ
Fig. 5 shows the results predicted by the three models describing the correlation between the number of pedestrians and the required time to leave the channel. In a microscopic point of view,
the performances of the three models are similar when the density in the compartment is low. Fig. 5 demonstrates that the pedestrians’ leaving time in real situation and the results predicted by the three models are similar when there are only two pedestrians leaving compartment). However, when the number of the leaving pedestrians increases, the deviation between the total leaving time predicted by the three models comparing to the real situation also increases. Fig. 5 depicts that the trend in the pedestrians’ leaving time predicted by the original social force model is generally closer to the real situation. As described in Section 1, the original social force model includes pushing and squeezing actions which seldom exist in a normal walking situation albeit the accuracy of the model. It can also be observed that the trend in the pedestrians’ leaving time predicted by the original social force model with the visual angle deviates from the real situation. It reveals that simply adding the visual angle to the original social force model may not be sufficient to predict pedestrian flow. Further fine-tuning of our modified social force model is therefore required. Table 2 shows the total leaving time of the pedestrians predicted by the three models and the real situation. In comparison to the original social force model (with visual angle), our modified social force model achieves closer to the real situation. It may conclude that taking account of overtaking behavior can improve the overall prediction of crowd movement. 3.2. Qualitative analysis In this part of the paper, we directly compare the behavior of pedestrians under the three models investigated by examining snapshots from the real situation with the simulated results.
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T=0s
T=3s
T=5s
T=7s
T = 10 s
Fig. 6a. Snapshots of actual pedestrians’ behavior at times 0 s, 3 s, 5 s, 7 s, and 10 s.
Fig. 6a shows the pedestrian flow pattern in the real situation. Figs. 6b–d show the simulation results of the original social force model, the original social force model with the visual angle, and the modified social force model, respectively, at different points in time. The solid circles represent the locations of pedestrians who have a higher desired walking velocity (higher than 1.0 m/s) and
the hollow circles represent those who have a lower desired walking velocity (lower than 1.0 m/s). Investigating the flow patterns of the pedestrians may enable us to depict their behavior. Fig. 6a illustrates the overtaking behavior of the pedestrians. According to the snapshots of the real situation, pedestrians with a higher walking velocity evade slower pedestrians in front of them
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Fig. 6b. Snapshots of pedestrians’ behavior simulated by the original social force model at times 2s, 4s, 6s, 8s, & 10s.
instead of pushing them and no jamming is observed as they move through the channel. The evading action observed demonstrates the existence of overtaking behavior in normal pedestrian flows. Fig. 6b shows that no overtaking takes place throughout the simulation. All of the pedestrians with a relatively high walking velocity push those in front of them instead of evading each other. Moreover, pedestrians with a lower walking velocity are pushed by those behind them and thus move somewhat faster than their desired walking velocity. This type of pushing behavior is seen as unrealistic in normal crowd movement situations. Fig. 6c shows only limited overtaking throughout the simulation. Pedestrians with a relatively high walking velocity can overtake slower pedestrians only in the upper part of the channel and are obstructed by slower pedestrians in the lower part of the channel.
Fig. 6c. Snapshots of pedestrians’ behavior simulated by the original social force model with visual angle at times 2 s, 4 s, 6 s, 8 s, and 10 s.
Overtaking is therefore observed only in the upper part of the channel. In contrast, jamming is observed in the lower part of the channel. It may therefore be concluded that the social force model incorporating the concept of the visual angle cannot be used to successfully simulate overtaking behavior throughout the simulation. Fig. 6d successfully shows overtaking behavior throughout the simulation in which pedestrians with a higher walking velocity overtake slower pedestrians in front of them by moving either up or down to evade slower pedestrians who block them. Moreover, no pushing behavior is seen in this figure, meaning pedestrians with a lower walking velocity can move at their desired walking velocity. Although the last two pedestrians have relatively high
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occurs in normal pedestrian movement. This can be observed in the captured behavior shown in Fig. 6a. Normally, all pedestrians move at their own desired walking velocity. Pedestrians who are in a hurry and are being obstructed by slower pedestrians will evade the slower pedestrians to continue moving rather than pushing the slower pedestrians. Unless someone has very limiting time constraints, generally speaking, seldom do people push or disturb others who are blocking their way, as this type of behavior is not common in our community. Moreover, the elderly, children, and disabled people move more slowly than other pedestrians. Pushing such pedestrians is highly unlikely in reality as people recognize that doing so may result in serious injury. The original social force model may therefore be unable to simulate such pedestrian flows. In comparison with the real situation, the performance achieved by adding the visual angle to the original social force model is less accurate than that achieved with our modified social force model. Our modified social force model can also be used to simulate overtaking behavior which cannot be simulated by the original social force model either with or without the visual angle. The results of this study support the conclusion that overtaking behavior among pedestrians can be closely simulated by our modified social force model. It is therefore possible to use our modified social force model to design buildings specifically for public areas (e.g. railway stations, shopping malls, tunnels, parks, etc.). Doing so will improve the sophistication and accuracy of the model, and improve the predictive capabilities. Therefore, we can simulate the walking of the elderly, children and disabled people in a more realistic manner which may improve the safety of the spatial design of the architecture.
5. Conclusion
Fig. 6d. Snapshots of pedestrians’ behavior simulated by the original social force model at times 2 s, 4 s, 6 s, 8 s, and 10 s.
walking velocities, they overtake others to a limited extent only and are not observed to push or engage in competitive behavior during the simulation. This is because no other pedestrian obstructs their movement, allowing them to move at their desired walking velocity and obviating the need to overtake others. 4. Discussion Although the original social force model is more accurate than our modified social force model in predicting the trend observed in pedestrians’ leaving time in the real situation, pushing behavior is frequently occur in the original social force model, it seldom
This paper reviewed the original social force model and existing modified versions of it. We then introduced a number of modifications to the model that make it valid for mimicking realistic crowd movement. The concept of the visual angle is proposed and implemented which minimizes pushing among pedestrians. We then propose a numerical algorithm that allows pedestrians to overtake slower pedestrians who are blocking them. The direction of the overtaking behavior is determined by considering the social repulsion between pedestrians and the repulsion exerted by the boundaries of the channel. The magnitude of the overtaking force is determined by a two-dimensional Gaussian function. Quantitative and qualitative analysis is presented of the original social force model, the original social force model incorporating the visual angle, and our modified social force model. While the results show that, compared to our model, the original social force model more closely reflects the trend in pedestrians’ leaving time observed in a real situation, this benefit is reduced by the fact that in the original model, pushing behavior frequently occurs which is seldom seen in normal pedestrian movement. Our modified model that allows for overtaking behavior throughout the simulation provides comparable results. Finally, we discuss the results obtained by the three models outlined above in detail and conclude that our modified social force model provides a reasonable representation of overtaking behavior among pedestrians in a normal crowd movement environment. As this research concentrates solely on the simulation of pedestrian flow in routine conditions, for those areas with high population densities (e.g. the entrance of the concern, stadium and the turnstiles at the railway station), by implementing the representation of overtaking, it benefits in modeling high-density crowds and eventually improves the safety. Further research will be carried out
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to investigate overtaking behavior in a variety of crowd movement settings. References Blue, V., Adler, J., 1999. Bi-directional emergent fundamental pedestrian flows from cellular automata microsimulation. Transportation and traffic theory. In: Ceder, A. (Ed.), Proceedings of the 14th International Symposium on Transportation and Traffic Theory (14th ISTTT Conference). Pergamon Press, Amsterdam, pp. 235–254. Bradley, G.E., 1993. A Proposed Mathematical Model for Computer Prediction of Crowd Movements and their Associated Risks. Engineering for Crowd Safety. Elsevier, Amsterdam, pp. 303–311. Burstedde, C., Klauckm, K., Schadschneiderm, A., et al., 2001. Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Physica A 295, 507–525. Chooramun, N., Lawrence, P.J., Galea, E.R., 2010. Implementing a hybrid space discretisation within an agent based evacuation model. In: Proceedings of the 5th International Conference on Pedestrian and Evacuation Dynamics, pp. 449–458. Feurtey, F., 2000. Simulating the Collision Avoidance Behaviour of Pedestrians, Thesis (Masters). The University of Tokyo, School of Engineering, Department of Electronic Engineering. Fukui, M., Ishibashi, Y., 1999. Self-organized phase transitions in CA-models for pedestrians. Journal of the Physical Society of Japan, 2861–2863. Gwynne, S., Galea, E.R., Owen, M., Lawrence, P.J., Filippidis, L., 1999. A review of the methodologies used in the computer simulation of evacuation from the built environment. Building and Environment 34, 741–749. Helbing, D., Molnar, P., 1995. Social force model for pedestrian dynamics. Physical Review E 51 (5), 4282–4286. Helbing, D., Farkas, I., Vicsek, T., 2000. Simulating dynamical features of escape panic. Nature 407, 487–490. Henderson, L.F., 1971. The statistics of crowd fluids. Nature 229, 381–383. Kirchner, A., Schadschneider, A., 2002. Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics. Physica A 312, 260–276. Kuligowski, E.D., Peacock, R.D., 2005. A Review of Building Evacuation Models. Technical Notes 1471. National Institute of Standards and Technology, pp. 1–30.
Lakoba, T.I., Kaup, D.J., Finkelstein, N.M., 2005. Modifications of the Helbing– Molnar–Farkas–Vicsek social force model for pedestrian evolution. Simulation 81, 339–352. Lo, S.M., Fang, Z., 2000. A spatial-grid evacuation model for buildings. Journal of Fire Science 18, 376–394. Lo, S.M., Fang, Z., Lin, P., Zhi, G.S., 2004. An evacuation model: the SGEM package. Fire Safety Journal 39, 169–190. Nagatani, T., Tajima, Y., Takimato, K., 2001. Scaling of pedestrian channel flow with a bottleneck. Physica A 294, 257–268. Owen, M., Galea, E.R., Lawrence, P.J., 1996. The exodus evacuation model applied to building evacuation scenarios. Journal of Fire Protection Engineering 8 (2), 65– 86. Owen, M., Galea, E.R., Lawrence, P.J., 1997. Advanced occupant behavioural features of the building EXODUS evacuation model. In: Proceedings of the Fifth International Symposium IAFSS, Melbourne Australia, pp. 795–806. Parisi, D.R., Gilman, M., Moldovan, H., 2009. A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions. Physica A 388, 3600–3608. Seyfried, A., Steffen, B., Lippert, T., 2006. Basics of modeling the pedestrian flow. Physica A 368, 121–238. Shannon, H.S., Robson, L.S., Guastello, S.J., 1999. Methodological criteria for evaluating occupational safety intervention. Safety Science 31, 161–179. Song, W., Xu, X., Wang, B.H., et al., 2006. Simulation of evacuation process using a multi-grid model for pedestrian dynamics. Physica A 363, 492–500. Thompson, P., Marchant, E. Wu, 1995. A computer model for the evacuation of large building populations. Fire Safety Journal 24, 131–148. Thompson, P., Marchant, E. Wu, 1996. Modelling evacuation in multi-storey buildings with SIMULEX. Fire Engineers Journal, 7–11. Wolfram, S., 1986. Statistical mechanics of cellular automata. Reviews of Modern Physics 55, 601–644. Zarboutis, N., Marmaras, N., 2004. Searching efficient plans for emergency rescue through simulation: the case of a metro fire. Cognition Technology and Work 6, 117–126. Zheng, M., Kashimori, Y., Kambara, T., 2002. A model describing collective behaviors of pedestrians with various personalities in danger situations. In: Proceedings of the 9th International Conference on Neural Information Processing, vol. 4, pp. 2083–2087.
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The effect of overtaking behavior on unidirectional pedestrian flow J.K.K. Yuen ⇑, E.W.M. Lee Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong Special Administrative Region
a r t i c l e
i n f o
Article history: Available online 30 December 2011 Keywords: Pedestrian flow Evacuation Collective behavior Overtaking Social force model
a b s t r a c t We present a model of overtaking behavior that can be used to simulate unidirectional pedestrian flow in routine. All pedestrians have the ability to determine whether or not to overtake other pedestrians according to their desired velocity and position. Although existing models such as cellular automata models, lattice gas models, social force models, etc., can be used to predict evacuation performance, most of these models are either computationally inefficient or do not account for some crucial elements of human behavior in a moving crowd. Furthermore, these models use either empirical equations developed from experiments or mechanical system analogies to determine movement decisions. The pedestrian flow patterns simulated by these models may deviate significantly from reality. In reality, pedestrians walk at different velocities and pedestrians with a higher walking velocity are accustomed to overtaking other pedestrians with a lower walking velocity and this paper aims to mimic this behavior as the original social force model developed by Helbing et al. does not reflect this pattern of collective pedestrian behavior. In this paper, we propose modifications of the social force model that reflects how overtaking behavior operates in routine. The comparison of the pedestrian flow pattern between the original social force model and the modified social force models with the real data collected by the camcorder is also performed in order to demonstrate our modified social force model can be used to achieve reasonable simulations of overtaking behavior among pedestrians. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The past two decades have seen the development of a number of different kinds of evacuation models used to model complex crowd movement. These models provide important information in the analysis and design of channels, bridges, railway stations, shopping malls, and other public amenities. Microscopic approaches to pedestrian flow (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Helbing et al., 2000; Helbing and Molnar, 1995; Nagatani et al., 2001; Song et al., 2006; Zarboutis and Marmaras, 2004; Wolfram, 1986; Fukui and Ishibashi, 1999; Owen et al., 1996, 1997; Thompson and Marchant, 1995, 1996; Lo et al., 2004; Lo and Fang, 2000) have recently attracted considerable attention and existing evacuation modeling techniques have been comprehensively reviewed in a number of studies (Shannon et al., 1999; Gwynne et al., 1999; Kuligowski and Peacock, 2005). Existing microscopic models (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Helbing et al., 2000; Helbing and Molnar, 1995; Nagatani et al., 2001; Song et al., 2006; Zarboutis and Marmaras, 2004; Wolfram, 1986; Fukui and Ishibashi, 1999; Owen et al., 1996, 1997;
⇑ Corresponding author. Tel.: +852 3442 4317; fax: +852 2788 7612. E-mail address: [email protected] (J.K.K. Yuen). 0925-7535/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssci.2011.12.020
Thompson and Marchant, 1995, 1996; Lo et al., 2004; Lo and Fang, 2000) can be broadly divided into discrete space models and continuous space models (Gwynne et al., 1999; Kuligowski and Peacock, 2005). In early 1970s, Henderson (1971) has proposed that pedestrians can be treated like fluids, as the movement of the pedestrians behave very similar to gas particles, then Bradley (1993) has further describes the crowd movement of the pedestrians by using the Navier–Stokes equation under a high density circumstance. Both of the fluid dynamic models above provide the early framework of the continuous space models. For the fine network approach, the cellular automaton (CA) models (Wolfram, 1986) has been studied and applied in physics since 1980s which play an important role in the development of the discrete space models. Researchers like Blue and Adler (1999) and Fukui and Ishibashi (1999) have further applied the CA models in pedestrian movement simulation. Nowadays, a series of fine network approach models like EXODUS (Owen et al., 1996, 1997), SIMULEX (Thompson and Marchant, 1995, 1996), and SGEM (Lo et al., 2004; Lo and Fang, 2000), are applicable for evacuation simulation. Generally, discrete space models like the cellular automaton (CA) model (Kirchner and Schadschneider, 2002; Burstedde et al., 2001; Blue and Adler, 1999; Wolfram, 1986; Fukui and Ishibashi, 1999) and the lattice gas (LG) model (Nagatani et al., 2001) allow
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pedestrians to move at or within a fixed node or grid and update the positions of pedestrians at discrete time intervals. In contrast, continuous space models like the social force model (Helbing et al., 2000; Helbing and Molnar, 1995) and part of the agent-based model (Zarboutis and Marmaras, 2004) allow pedestrians to move continuously within a pre-defined geometry. One drawback of both the CA and LG models is that they do not reflect reality by allowing pedestrians to move around in an unrestricted manner. Pedestrian behavior can be simulated successfully only when pedestrian motion is updated continuously. We therefore focus on continuous space models in this paper as most of them allow for finite interaction between pedestrians. Unlike the discrete space model that the compartment is divided into a fixed node or grid, by using the continuous space model, we can develop the geometry in a more precise way and the magnitude of the interaction force between the pedestrians and the boundaries can be clearly represented by defining it in terms of vectors. Helbing et al. (2000) and Helbing and Molnar (1995) introduced a stochastic model that incorporates ‘body force’ and ‘sliding force’ to describe the interactions between evacuees and boundaries in crowd conditions. Their specification is a highly renowned continuous space model that can successfully simulate the most typical forms of human behavior in evacuation conditions. We propose a modified version of this model to simulate overtaking behavior in a normal crowd movement environment. 1.1. The idea of overtaking In reality, pedestrians walk at different velocities. Pedestrians with a higher walking velocity are accustomed to overtaking other pedestrians with a lower walking velocity to maintain their own desired walking velocity. One example of this type of overtaking behavior can be observed at railway stations during peak hours, in which pedestrians rush along the platform and try to avoid colliding with each other when catching the train. The original social force model proposed in Helbing et al. (2000) shows the result of the ‘faster-is-slower effect’ due to the impatience shown in the event of an evacuation and how it reduces the chance of survival. This is because evacuees can become more motivated in an emergency situation. However, such behavior does not arise in a normal walking situation. Given that most of the existing pedestrian movement models fail to account for overtaking behavior and its common occurrence in normal crowd movement settings, there is a need to investigate this area in more detail. Moreover, competitive forms of behavior such as pushing and squeezing are less likely to occur as there are no or little time constraints in normal crowd movement conditions. Both of the above observations motivate us to modify the original social force model in a way that mimics the realistic movement of crowds in routine. Thompson proposed the possibility of overtaking in SIMULEX (Thompson and Marchant, 1995, 1996). It allows the assessing person to overtake the obstructing person by moving with two potential angles. Currently researchers are working on the collision avoidance behavior of pedestrians. Feurtey (2000) has simulated the collision avoidance behavior by introducing the disk of admissible displacements, it simplified the process of searching a collision-free path from a three-dimensional space to searching a collision-free point in a limited two-dimensional domain. Even the future position of all obstacles including wall and other pedestrians nearby is represented in a three dimensional space, by using the information of the position and speed of the obstacles, a safe trajectory can still be predicted by each agent from their current position. Chooramun et al. (2010) has proposed a hybrid model which combines the continuous, coarse and fine models to represent the discretization of space in circulation and evacuation models. In addition, the behaviors of wall and agent avoidance have
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been implemented in their hybrid model by computing a repulsive force between the pedestrians nearby, so that the pedestrians can maintain a desired interpersonal space between each other. Nevertheless, the model presented in Chooramun et al. (2010) can also address a range of additional behaviors such as cross-flows, merging flows, emergence of lane formation, and behavioral adaption in low and high density settings. This paper studies the unidirectional flow of pedestrians through a channel. We present a modified social force model that mimics the unidirectional flow of pedestrians through the channel by eliminating the unrealistic representation of competitive behavior reflected in the social force model. We also introduce the concept of overtaking behavior into the social force model to create a modified social force model that can be used to simulate normal crowd movement in a realistic manner. Fig. 1 illustrates overtaking behavior in a unidirectional channel. In this paper, the performances of the original social force model and our modified social force models will be compared with the pedestrian flow pattern recorded by camcorders in a real situation. 1.2. The original social force model We now provide a brief review of the original social force model. The social force model for pedestrian flow was first proposed by Helbing and Molnar (1995)). In Helbing et al. (2000), put forward another social force model that simulates panic situations. This model describes a mixture of socio-psychological and physical forces that influence behavior in a pedestrian flow. Pedestrians are normally driven by three forces: desire force FDi, social force FSi, and granular force FGi. The corresponding equations are:
F Di ¼ mi
v 0i ðtÞe0i ðtÞ v i ðtÞ si
ð1Þ
where mi is the mass of the ith pedestrian, vi is the actual velocity of pedestrian i, v 0i is the desired velocity of pedestrian i, e0i is the initial desired unit vector of pedestrian i and si is a pre-defined time interval;
F Si ¼
Np X jð–iÞ
Ai exp
ðr ij dij Þ nij Bi
ð2Þ
where Np is the total number of pedestrians in the geometry, dij = ||(ri rj)|| denotes the distance between pedestrians i and j, rij = (ri + rj) of the pedestrian radii ri and rj and, A and B are predefined constants that represent the strength and range of social interaction, and nij is the unit vector pointing from pedestrian j to pedestrian i;
F Gi ¼
Np X
kgðr ij dij Þnij þ kgðr ij dij ÞDv tji t ij
ð3Þ
jð–iÞ
where the granular force is determined by a body force kgðrij dij Þnij that counters body compression and a sliding force kgðrij dij ÞDv tji tij that impedes relative tangential motion, the tangential direction is represented by Dv tji ¼ ðv j v i Þ t ij and tij ¼ ðn2ij ; n1ij Þ, and k and j represent another two constants. In addition, the function g(x) is zero if pedestrians do not touch each other; otherwise, it is equal to the argument x. 1.3. Modifications to the social force model Seyfried et al. (2006) were the first to introduce the self-stopping mechanism for pedestrians in the social force model to prevent negative velocity among pedestrians caused by the large repulsion forces between them. They also modified the social force model to reproduce the empirical velocity density relation of pedestrian
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Either moving upward to overtake blocking
Or moving downward to overtake
Fig. 1. A schematic illustrating the action of overtaking.
movement for a one-dimensional system. Parisi et al. (2009) propose another self-stopping mechanism for pedestrians to prevent them from continuously pushing each other by introducing a respect area, the self-stopping mechanism will be triggered if a pedestrian entered the respect area of other pedestrians. The main advantage of the modifications made by Parisi et al. over those of Seyfried et al. is that they amend the desired velocity of pedestrians instead of changing their velocity. This gives pedestrians a selfslowing mechanism instead of the sudden stop action introduced by Seyfried et al. Lakoba et al. (2005) introduced three main modifications of the social force model that link the magnitude of the social force with pedestrian density. For different pedestrian orientations, Lakoba and his colleagues propose that weighting factors should be applied by multiplying the social repulsive forces at play between them. They also introduce a pedestrian memory effect that allows pedestrians to select their own desired direction to enhance the stability of the pedestrian flow. In addition, Zheng et al. (2002) employ the social force model in conjunction with a neural network and allow pedestrians to act as they wish in both normal and emergency situations. However, none of the studies outlined above modifies the social force model to allow pedestrians to overtaking other pedestrians. Work to modify the social force model in a manner that simulates overtaking behavior is therefore still somewhat limited. The rest of this paper is organized as follows. Section 2 describes the development of the overtaking sub-model. Case studies used to demonstrate the performance of the overtaking sub-model are detailed in Section 3. The results are discussed in Section 4 and our conclusions are presented in Section 5. 2. Model We consider the unidirectional flow of pedestrians in a channel. The idea of the visual angle and the modifications we propose to the original social force model are employed to simulate overtaking behavior throughout the pedestrian flow. The basic algorithm for the overtaking process is shown in Fig. 2. 2.1. Visual angle According to Eq. (2), the social repulsion forces between pedestrian i and j represent the interaction between pedestrian i and all other neighboring pedestrians, regardless of whether they are in front of or behind pedestrian i. Given that people are unlikely to notice anyone behind them unless the trailing person is very close and they can hear noise the trailing person makes, they usually pay less attention to what occurs behind them. In this situation, the
interaction between pedestrian i and all the neighboring pedestrians who are in front of pedestrian i is much more important than neighboring pedestrians behind pedestrian i For this reason, we assume that pedestrians pay less attention to what occurs behind them. The interaction between pedestrian i and neighboring pedestrians behind pedestrian i is set to zero and the angle of view, we call it visual angel is set to 180° instead of 120° or 270° for simplicity. The new social repulsion force between pedestrian i and pedestrian j can be determined by:
8 Np P ðr ij dij Þ > > nij Ai exp < Bi fij ¼ jð–iÞ > > : 0
if rjx r ix ð4Þ otherwise
This means that pedestrian i can interact only with pedestrians who are within his visual angle, as illustrated by Fig. 3. 2.2. Modification of the social force model There are generally two main factors that need to be considered in simulating overtaking behavior during crowd movement: (1) the direction of overtaking and (2) the degree of overtaking. 2.2.1. The direction of overtaking Fig. 1 shows a situation that if a pedestrian is in hurry and moving from left to right, he or she will move either upward or downward to avoid jamming. The decision on whether to move upward or downward is based on the determination of how to avoid both the social repulsion forces at play between pedestrians and the social boundary repulsion force exerted. Taking pedestrian i as the origin as illustrated in Fig. 3, the product of the social repulsion forces between pedestrian i and all neighboring pedestrians on the upward direction of the initial movement vector and the upward rational vector act as the upward social repulsion force Fup and the product of the social repulsion forces between pedestrian i and all neighboring pedestrians on the downward direction of the initial movement vector and the downward rational vector act as the downward social repulsion force Fdn. The upward social repulsion force and the downward social repulsion force can thus be represented as:
F up ¼
Np X
Ai exp
jð–iÞ
F dn ¼
Np X jð–iÞ
Ai exp
* ðr ij dij Þ nij up Bi
if r iy 6 rjy
* ðr ij dij Þ nij down Bi
if r iy 6 r jy
ð5Þ
ð6Þ
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Initialization of the Geometry & Pedestrians
1
Determine whether SRF exists between Pi & n.
N
End
Y
Determine whether the desired velocity of Pi > n
Y
N
End
Overtaking occurs
2
Determine the direction of overtaking
3
Determine the magnitude of the overtaking force
Add the overtaking force to the original social force model
Update and store the locations of the pedestrians Notes: SRF = social repulsion force Pi = Pedestrian i n = Pedestrian i
neighboring pedestrians
1
By applying the concept of the visual angle, the social repulsion force exists only between pedestrian i and all
neighboring pedestrians who are in front of him.
2
Pedestrian i will move towards the direction in which there is less social repulsion force and less social boundary
repulsion force. Fig. 2. Basic algorithm for the overtaking behavior process.
*
*
where up and down represent the upward and downward rational vector of the initial moving vector for the ith pedestrian, respectively. In addition, during the overtaking process, pedestrians seek space to overtake others and avoid getting hurt from the boundary. The social boundary repulsion force should also be included when the overtaking direction is considered. From Fig. 3, the lower boundary wall exerts an upward boundary repulsion force Fwup on pedestrian i and the upper boundary wall exerts a downward
boundary repulsion force Fwdn on pedestrian i. Therefore, the shorter the distance between the pedestrian and the boundary, the higher the social boundary repulsion force, and vice versa. The direction of the overtaking movement is determined by the social repulsion forces between pedestrians and the social boundary repulsion force exerted. If the sum of the total upward repulsion force and the downward boundary repulsion force is larger than the sum of the total downward repulsion force and the upward
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note: y1 > y2
y1
θi,
ut
y2
The i th pedestrian The neighboring pedestrians who can be seen by the i th pedestrian The neighboring pedestrians who cannot be seen by the i th pedestrian The social force exerted on the i th pedestrian by the i th neighboring pedestrian The upward social repulsion force acting on the i th pedestrian The downward social repulsion force acting on the i th pedestrian The upward boundary repulsion force acting on the i th pedestrian The downward boundary repulsion force acting on the i th pedestrian
θi,
The visual angle (180o) for the i th pedestrian
ut
The initial moving vector for the i tth pedestrian The boundary wall Fig. 3. A schematic illustrating the determination of the direction of overtaking.
boundary repulsion force, pedestrian i will move downward to avoid the larger upward repulsion force, and vice versa. Therefore, the overtaking direction can be described by
F Mi ¼
X
ðF up þ F dn Þ ðF wup þ F wdn Þ
ð7Þ
jðn–iÞ
F Mi
P 8 > < 1 if jðn–iÞðF up þ F dn Þ ðF wup þ F wdn Þ P 0 P ¼ > if ðF up þ F dn Þ ðF wup þ F wdn Þ < 0 :1
ð8Þ
jðn–iÞ
In mathematical order, the pedestrian will move downward if FMi is equal to 1 and will move upward if FMi is equal to 1.
2.2.2. The degree of overtaking The magnitude of the overtaking force is determined by a twodimensional Gaussian function (Eq. (9), the format of which is the product of the horizontal dimension Gaussian function and the vertical dimension Gaussian function shown in Eqs. (10) and (11), where q is the density of the specific area, a is a constant, x is the horizontal distance between pedestrians i and j, while y is the vertical distance between pedestrians i and j, lx and ly are set to zero as pedestrian i is located at the origin, and rx and ry are respectively related to a product of parameters and the desired velocity of pedestrian i. Fig. 4 shows the shape of the overtaking contour force between pedestrian i and j.
F Oi ¼
X a
jðn–iÞ
q
hðxÞ hðyÞ
ð9Þ
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ut
y
x
The i th pedestrian th The neighboring pedestrian j who can be seen by the i pedestrian
The overtaking contour force exerted on the i th pedestrian by the neighboring pedestrian Fig. 4. A schematic illustrating the overtaking force
1 ðx lx Þ2 hðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2r2x 2pr2x ðy ly Þ 1 hðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi exp 2r2y 2pr2y
2
! ð10Þ ! ð11Þ
2.2.3. The addition of overtaking force After establishing the direction and magnitude of the overtaking force, we can simulate overtaking behavior by adding the product of the magnitude and direction of the overtaking force to the original social force model as in Eq. (12). The general schematic of the overtaking contour force is illustrated in Fig. 4.
mi
dv i ¼ F Di þ F Si þ F Gi þ F Mi F Oi dt
ð12Þ
The performance of the above three models is compared with the pedestrian flow pattern recorded by camcorders in a real situation. All three models are initialized by the geometry and the initial positions of the pedestrians and their individual walking velocities are captured by the camcorders. Note that the walking velocity of each pedestrian is determined by dividing the total distance traveled by the time taken. The results predicted by the three models are analyzed both quantitatively and qualitatively. Quantitatively, the performances of the three models as mentioned are represented by the prediction errors between the models and the real situation. It enables us to compare the performance of the models directly. In addition, the trends of the pedestrian flow rate predicted by different models are also evaluated. Besides the quantitative analysis, we also conduct the qualitative analysis in Section 3.2. We compare the behaviors of the pedestrians simulated by the three models and investigate the snapshots from the real situation. Both qualitative and quantitative analyses ensure a less subjective comparison of the three models.
3. Simulation results and analysis Because the original social force model does not reflect the concept of the visual angle, a variation on the social force model that incorporates this concept is also developed for comparison with our proposed social force model described in Section 2 which also incorporates the visual angle. The following three models are therefore compared. 1. The original social force model. 2. A social force model incorporating the visual angle. 3. Our modified social force model.
3.1. Quantitative analysis Fig. 5 and Table 1 show the simulation results of the three models according to the pedestrian flow rates calculated as the time required for different numbers of pedestrians to leave the channel under consideration. One model is considered to be quantitatively more accurate than the others if the prediction error (in the form of the root-mean-square deviation) is smaller than that of the other models. The definition of RMSD is shown in (13) where n is the total ðrÞ ðmÞ number of pedestrians and t i and t i are respectively the real and
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12 11 10
Number of pedestrians leave
9 8 7 6 5 4 3 2
Real situation Original Social Force model (RMSD=0.38) Modified Social Force model+Visual Angle (RMSD=0.71) Original Social Force model+Visual Angle (RMSD=1.30)
1 0
8
9
10
11
12
13
14
15
16
17
18
Leaving time (s) Fig. 5. Pedestrians’ leaving time graph.
Table 1 The total leaving time of the pedestrians of the different models. Models
Total leaving time of the pedestrians (s)
Real situation Original social force model Modified social force model with visual angle Original social force model with visual angle
14.4 14.8 15.5 15.6
Table 2 The root-mean-square deviations of the different models. Models
Root-mean-square deviation
Original social force model Modified social force model with visual angle Original social force model with visual angle
0.38 0.71 1.30
modeled leaving times of the ith pedestrian. Table 2 summarized the root-mean-square deviation (RMSD) of the different models.
RMSD ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP 2 u n ðrÞ ðmÞ t i ti ti n
ð13Þ
Fig. 5 shows the results predicted by the three models describing the correlation between the number of pedestrians and the required time to leave the channel. In a microscopic point of view,
the performances of the three models are similar when the density in the compartment is low. Fig. 5 demonstrates that the pedestrians’ leaving time in real situation and the results predicted by the three models are similar when there are only two pedestrians leaving compartment). However, when the number of the leaving pedestrians increases, the deviation between the total leaving time predicted by the three models comparing to the real situation also increases. Fig. 5 depicts that the trend in the pedestrians’ leaving time predicted by the original social force model is generally closer to the real situation. As described in Section 1, the original social force model includes pushing and squeezing actions which seldom exist in a normal walking situation albeit the accuracy of the model. It can also be observed that the trend in the pedestrians’ leaving time predicted by the original social force model with the visual angle deviates from the real situation. It reveals that simply adding the visual angle to the original social force model may not be sufficient to predict pedestrian flow. Further fine-tuning of our modified social force model is therefore required. Table 2 shows the total leaving time of the pedestrians predicted by the three models and the real situation. In comparison to the original social force model (with visual angle), our modified social force model achieves closer to the real situation. It may conclude that taking account of overtaking behavior can improve the overall prediction of crowd movement. 3.2. Qualitative analysis In this part of the paper, we directly compare the behavior of pedestrians under the three models investigated by examining snapshots from the real situation with the simulated results.
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T=0s
T=3s
T=5s
T=7s
T = 10 s
Fig. 6a. Snapshots of actual pedestrians’ behavior at times 0 s, 3 s, 5 s, 7 s, and 10 s.
Fig. 6a shows the pedestrian flow pattern in the real situation. Figs. 6b–d show the simulation results of the original social force model, the original social force model with the visual angle, and the modified social force model, respectively, at different points in time. The solid circles represent the locations of pedestrians who have a higher desired walking velocity (higher than 1.0 m/s) and
the hollow circles represent those who have a lower desired walking velocity (lower than 1.0 m/s). Investigating the flow patterns of the pedestrians may enable us to depict their behavior. Fig. 6a illustrates the overtaking behavior of the pedestrians. According to the snapshots of the real situation, pedestrians with a higher walking velocity evade slower pedestrians in front of them
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Fig. 6b. Snapshots of pedestrians’ behavior simulated by the original social force model at times 2s, 4s, 6s, 8s, & 10s.
instead of pushing them and no jamming is observed as they move through the channel. The evading action observed demonstrates the existence of overtaking behavior in normal pedestrian flows. Fig. 6b shows that no overtaking takes place throughout the simulation. All of the pedestrians with a relatively high walking velocity push those in front of them instead of evading each other. Moreover, pedestrians with a lower walking velocity are pushed by those behind them and thus move somewhat faster than their desired walking velocity. This type of pushing behavior is seen as unrealistic in normal crowd movement situations. Fig. 6c shows only limited overtaking throughout the simulation. Pedestrians with a relatively high walking velocity can overtake slower pedestrians only in the upper part of the channel and are obstructed by slower pedestrians in the lower part of the channel.
Fig. 6c. Snapshots of pedestrians’ behavior simulated by the original social force model with visual angle at times 2 s, 4 s, 6 s, 8 s, and 10 s.
Overtaking is therefore observed only in the upper part of the channel. In contrast, jamming is observed in the lower part of the channel. It may therefore be concluded that the social force model incorporating the concept of the visual angle cannot be used to successfully simulate overtaking behavior throughout the simulation. Fig. 6d successfully shows overtaking behavior throughout the simulation in which pedestrians with a higher walking velocity overtake slower pedestrians in front of them by moving either up or down to evade slower pedestrians who block them. Moreover, no pushing behavior is seen in this figure, meaning pedestrians with a lower walking velocity can move at their desired walking velocity. Although the last two pedestrians have relatively high
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occurs in normal pedestrian movement. This can be observed in the captured behavior shown in Fig. 6a. Normally, all pedestrians move at their own desired walking velocity. Pedestrians who are in a hurry and are being obstructed by slower pedestrians will evade the slower pedestrians to continue moving rather than pushing the slower pedestrians. Unless someone has very limiting time constraints, generally speaking, seldom do people push or disturb others who are blocking their way, as this type of behavior is not common in our community. Moreover, the elderly, children, and disabled people move more slowly than other pedestrians. Pushing such pedestrians is highly unlikely in reality as people recognize that doing so may result in serious injury. The original social force model may therefore be unable to simulate such pedestrian flows. In comparison with the real situation, the performance achieved by adding the visual angle to the original social force model is less accurate than that achieved with our modified social force model. Our modified social force model can also be used to simulate overtaking behavior which cannot be simulated by the original social force model either with or without the visual angle. The results of this study support the conclusion that overtaking behavior among pedestrians can be closely simulated by our modified social force model. It is therefore possible to use our modified social force model to design buildings specifically for public areas (e.g. railway stations, shopping malls, tunnels, parks, etc.). Doing so will improve the sophistication and accuracy of the model, and improve the predictive capabilities. Therefore, we can simulate the walking of the elderly, children and disabled people in a more realistic manner which may improve the safety of the spatial design of the architecture.
5. Conclusion
Fig. 6d. Snapshots of pedestrians’ behavior simulated by the original social force model at times 2 s, 4 s, 6 s, 8 s, and 10 s.
walking velocities, they overtake others to a limited extent only and are not observed to push or engage in competitive behavior during the simulation. This is because no other pedestrian obstructs their movement, allowing them to move at their desired walking velocity and obviating the need to overtake others. 4. Discussion Although the original social force model is more accurate than our modified social force model in predicting the trend observed in pedestrians’ leaving time in the real situation, pushing behavior is frequently occur in the original social force model, it seldom
This paper reviewed the original social force model and existing modified versions of it. We then introduced a number of modifications to the model that make it valid for mimicking realistic crowd movement. The concept of the visual angle is proposed and implemented which minimizes pushing among pedestrians. We then propose a numerical algorithm that allows pedestrians to overtake slower pedestrians who are blocking them. The direction of the overtaking behavior is determined by considering the social repulsion between pedestrians and the repulsion exerted by the boundaries of the channel. The magnitude of the overtaking force is determined by a two-dimensional Gaussian function. Quantitative and qualitative analysis is presented of the original social force model, the original social force model incorporating the visual angle, and our modified social force model. While the results show that, compared to our model, the original social force model more closely reflects the trend in pedestrians’ leaving time observed in a real situation, this benefit is reduced by the fact that in the original model, pushing behavior frequently occurs which is seldom seen in normal pedestrian movement. Our modified model that allows for overtaking behavior throughout the simulation provides comparable results. Finally, we discuss the results obtained by the three models outlined above in detail and conclude that our modified social force model provides a reasonable representation of overtaking behavior among pedestrians in a normal crowd movement environment. As this research concentrates solely on the simulation of pedestrian flow in routine conditions, for those areas with high population densities (e.g. the entrance of the concern, stadium and the turnstiles at the railway station), by implementing the representation of overtaking, it benefits in modeling high-density crowds and eventually improves the safety. Further research will be carried out
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