A collective motion model based on two-layer relationship mechanism for bi-direction pedestrian flow simulation

Simulation Modelling Practice and Theory 84 (2018) 268–285

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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat

A collective motion model based on two-layer relationship mechanism for bi-direction pedestrian flow simulation Xin Qin a,b, Hong Liu a,b,∗, Hao Zhang a,b, Baoxi Liu a,b a b

School of Information Science and Engineering, Shandong Normal University, Jinan, China Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan, China

a r t i c l e

i n f o

Article history: Received 9 October 2017 Revised 15 March 2018 Accepted 16 March 2018

Keywords: Collective motion model Two-layer relationship mechanism Bi-direction pedestrian flow

a b s t r a c t In crowd dynamic, relations are existed among some pedestrians, which cause frequent interactions during evacuation, creating collective motion phenomena, such as the most common pattern of team-groups. Besides, collective behavior can make a beneficial effect on the evacuation process. Therefore, this paper proposes a collective motion model to simulate bi-direction pedestrian flow. First, a method of group vision sharing is proposed to help pedestrians learn the crowd around. Based on two-layer relationship mechanism proposed, aggregate force and collective collision avoidance force are added into the original social force formula. The aggregate force is the resultant of two forces, one is the attraction among the leader and team members, and the other one is that among members of groups due to the social relations. Simulation results show that the modified model can reproduce the team-groups collective pattern in real world bi-direction pedestrian flow, and can reduce the collision risk with regarding the group as collision avoidance unit. Furthermore, the evacuation efficiency is improved. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The rapid increases in pedestrians and crowded urban infrastructures have made it a significance to explore the pedestrian flow dynamic. Pedestrian traffic congestion brings great inconvenience to people’s travels. Moreover, travel delays result in great losses to the social economy and living quality of residents, especially in the public traffic of airplane and high-speed boarding process, a scientific guidance makes a great help to improve the efficiency [1]. In addition, in several large public places, such as the square in an assembly, crosswalk or subway station during rush hours [2], and a crowded concert scene, safe evacuation is a big issue in emergency cases, such as fire and gas leak. The lack of scientific and effective guidance for personnel evacuation results in chaos, which causes stampede and threaten the safety of the lives and properties of pedestrians. Traditional evacuation drills entail enormous manual labor, material resources, and financial resources, and simulating random events and the subconscious interests of pedestrians for their locations is difficult [3]. Using computer simulation technology for scene modeling, path optimization, and crowd movement behavior modeling to simulate the pedestrian flow in public places can contribute to the design of facilities and emergency evacuation guidance [4]. Pedestrian flow is a typical large crowd movement. In pedestrian flows [5,6], crowd movement includes one and bidirection, cross, reciprocating, and other forms [7]. The self-organization phenomena of pedestrian flow have distinctive

∗

Corresponding author at: School of Information Science and Engineering, Shandong Normal University, Jinan, China. E-mail address: [email protected] (H. Liu).

https://doi.org/10.1016/j.simpat.2018.03.005 1569-190X/© 2018 Elsevier B.V. All rights reserved.

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features, such as arching and clogging through the bottle neck and stripe formation of intersecting pedestrians in the bidirection flow [8,9], and some scholars came up with methods to observe the occurrence of phase transition [10]. Therefore, the application of evacuation models to simulate pedestrian flow, especially the most typical one—bi-direction pedestrian flow—has been the focus of many scholars. In the work of bi-direction pedestrian flow, the relationships among pedestrians which are complex and stable in real life should be taken into consideration. Relationships are usually composed of leading and social relationships, which make pedestrians walk together with a herd mentality and is like some biological cluster phenomena somehow [11–14]. On the basis of the factors above, leading relationship generates team effect and social relations causes small groups effect, these two effects often occur in the process of crowd evacuation. However, most of the existing crowd evacuation simulation models ignore the inter-individual relationships and regard an individual as an isolated particle, which prevent the realistic simulation of crowd evacuation. Thus, a further research on the details of force in team-group effect and in-depth understanding of its behavior and mechanism according to relationship are of vital importance to truly simulating bi-direction pedestrian flow. This paper proposes a modified social force model (MSFM) that is driven by the two-layer relationship mechanism and a collision avoidance strategy to simulate the self-organized phenomena in bi-direction pedestrian flow. The main contributions of this work are as follows: (a) (b) (c) (d)

A modified social force model is proposed to simulate the self-organized phenomena of bi-direction pedestrian flow. The combination of leading and social relationships forms the two-layer relationship mechanism. A method of group vision sharing is proposed to help pedestrians learn other pedestrians around. An aggregate force, including visual impact factor is added to the original formula, and the clustering method is introduced. (e) The collision in bi-direction pedestrian flow during intersection is reduced by a collective collision avoidance strategy which regards the group as an avoidance unit.

Based on the contributions above, the collective pattern of team-group is reproduced. Each group moves toward the target through the guidance of a leader. Meanwhile, team members gather into small groups according to their social relation. In addition, the lane formation of bi-direction pedestrian flow is reproduced under the collective collision avoidance strategy. A rich set of environmental attributes, specifically a real path environment, is considered. Thus, similar scenes are built to simulate the real scenes. The remainder of this paper is organized as follows. Section 2 mainly introduces related works on pedestrian flow simulation and social force model. Section 3 discusses the original social force model (OSFM). Section 4 presents the MSFM for bi-direction pedestrian flow simulation. Section 5 elaborates the model framework and implementation process. Section 6 shows the efficiency of the proposed model by experimenting on the effects of relationship density and relationship aggregation on the different density flows, and the validity of the model is verified by numerical comparison. The simulation results show that the MSFM can make the pedestrians in a team march with a leader and gather together in small groups to interact with each other, which can achieve orderly and efficient evacuation. The conclusion and future research focus are presented in Section 7. 2. Related work In the last decade, numerous simulation models, classified into two categories of microscopic and macroscopic [15], have been presented to elucidate the underlying dynamics of bi-direction pedestrian flow behaviors. Jiang et al. proposed a highorder macroscopic model, which shows that the traffic sonic speed and the group size influence on the width and number of lines [16]. However, macroscopic models cannot commendably describe the local details of pedestrian behavior. Generally, microscopic models mainly include the social force model (SFM) [17–19], lattice gas model (LGM) [20–23], and cellular automaton model (CAM) [24–27]. Using a microscopic pedestrian model, Guo et al. investigated the effects of the walkway corner and pedestrian preference to inside routes on pedestrian queue, trajectory, and flow-density relation [28]. From the individual’s perspective, the focus of microscopic models on the interaction between the individual and the environment can compensate for the deficiencies of the macroscopic ones to a certain extent. The three models above have been commonly used to simulate pedestrian flow. Based on CAM, Wang et al. came up with some rules for double-pedestrian teaming, which showed that team movement contributes to the significant corridor capacity effects and the type of teaming behavior influences the effects significantly [29]. Wei et al. proposed a novel pedestrian model that uses direction and collision gains to calculate the target position and extend the three-dimensional spaces of stairs [30]. Nowak and Schadschneider introduced an order parameter to analyze quantitatively four different states, i.e., free flow, disordered flow, lane formation, and gridlock [31]. Xue et al. introduced a concept of a dominant row, and quantify the evolution process of the lane formation to show steady separate lanes can form even in dense conditions [32]. Weng et al. simulated pedestrians counterflow with different walk velocities, and observed phase transition among freely moving phase, lane formation phase, and perfectly stopped phase with the range of entrance densities [33]. Tang proposed a model to investigate the effect of elementary students’ individual properties on the evacuation process [34]. Burstedde et al. proposed a 2-dimensional cellular automaton model shown the introduction of such a floor field is sufficient to model collective effects and self-organization encountered in pedestrian dynamics [35].

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Meanwhile, based on the LGM, the pattern formation and jamming transition are discussed by comparing the experimental result and simulation result in the model proposed by Isobe et al. [36], and Tajima et al. compared two models in this issue, it is investigated that branch flow joins the main flow at the junction in a T-shaped channel [37,38]. Muramatsu found a dynamical jamming transition from the freely moving state to the stopped state with the increase of density [39]. Work of Kuang showed the sub-conscious behavior plays a key role in reducing the occurrence of jam cluster [40]. Weng et al. divided the walk task of pedestrian into three basic behaviors: ‘‘move’’, ‘‘avoid’’, and ‘‘swirl’’, further found that the mean flow is always high if the corresponding mean velocity is high, and their phases also turn at the same critical total density [41]. Furthermore, Liang et al. built an extended small-grid LGM to select key parameters [42]. In the 90 s of the last century, Helbing proposed the classic social force model [43], which aroused concern among a few scholars to improve this model. This model simulates the evacuation phenomena of pedestrian flow using Newton mechanics from the perspective of individuals. Considering the interaction between the subjective intention of pedestrians-pedestrians and pedestrians-environment, the evacuation processes of “fast is slow,” “arching,” and so on are stimulated. However, this model simplifies the relation character among pedestrians to some extent [44], and it fails to guarantee that all pedestrians can avoid collision [45]. Meanwhile some discussion concerns the existence of oscillations in the movement [46]. Despite the shortcomings on relationship and group mechanism, SFM is an effective foundation in reproducing some crowd phenomena. On the basis of this original model, many scholars have proposed improved ones. Li et al. proposed a trace model, which is adaptive to the motions between followers and leaders, considering psychological factors [47]. Peijie et al. introduced the view radius of single pedestrian to the classic SFM to describe the pedestrian’s range of sight [48]. Parisi et al. proposed a self-stopping mechanism to prevent a simulated pedestrian from continuously pushing over the others [49]. Liu et al. proposed a method based on navigation knowledge and two-layer control mechanism which effectively solves the problem of microscopic models because each individual calculates the path and resolves the slow speed problem [50]. Yu et al. extended the repulsive force term of the SFM to reproduce crowd turbulence [51]. Several scholars focused on the steering behavior of pedestrians to study the relationship between the environment and the walking trajectory of pedestrians. Nasir et al. applied genetic algorithm to search for the optimum membership function parameters of the fuzzy model with the environmental stimuli, which are quantified using attractive and repulsive forces as input and angular change of direction as output [52]. However, most of the existing models for reproducing the self-organization phenomena of bi-direction pedestrian flow consider pedestrians as single isolated individuals or regard the group as a whole. Thus considerable friction and squeezing exist in the flow intersection. For further research, the details of force among the crowd should be captured because they can help accurately simulate and master the self-organization phenomena of the flow. In addition, they can reduce the risk of congestion injury and achieve rapid and orderly evacuation in dense crowds. 3. Discussion on the original social force model On the basis of physical mechanics, the forces of the OSFM proposed by Helbing [43] are attributed to the self-driving force generated by the desire of pedestrians to reach their destinations, the forces between pedestrians, and the forces provided by walls in the process of movement. According to Newton’s second law, the translation of acceleration and resultant force of pedestrians can be expressed by Formula 1.

mi

− → → − → − → d vi (t ) − = fi0 + fi j + fiw , dt w

(1)

j ( =i )

where

− → → − → v0i (t ) e0i − − vi (t ) 0 f i = mi ,

(2)

→ → − − → − − → − → − → fi j = fisj + fipj = Ai exp[(ri j − di j )/Bi ]ni j + kg(ri j − di j )ni j + κ g(ri j − di j )vtji ti j ,

(3)

− → → − → − → → − − → − −→ −→ p s fiw = fiw + fiw = Ai exp[(ri − diw )/Bi ]niw + kg(ri − diw )niw + κ g(ri − diw )( vi · tiw )tiw ,

(4)

ϕi

− → − → where mi is the quality of pedestrian i, vi is the actual velocity, and fi0 is the target driving force of pedestrian i, which − → is influenced by the target position. Pedestrian i prefers to move at a specific speed v0i in a certain e0i direction. ϕ i is the − → relaxation time of the pedestrian to adjust the walking speed to achieve vi . dij is the centroid distance of two pedestrians i − → − → − − → − → → 1 2 and j. ni j = (ni j , ni j ) = ( ri − r j )/di j is a normalized vector that individual j points to individual i, where ri and r j are the − → positions of pedestrians i and j, respectively. ti j = (−n2i j , n1i j ) is a tangential direction used to calculate the tangential velocity − → → → − − → − − → difference of two pedestrians expressed as vtji = ( v j − vi ) · ti j , the direction of ni j and ti j is shown in Fig. 7 in detail. It − → →  − can be used to maintain a safe distance between individuals to avoid excessive individual extrusion. fiw are j (=i ) f i j and w

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(a) Leader in forefront of the crowd

(c) Trajectories of the leader and social group

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(b) Small social groups in a team

(d) Collision avoidance with social group as a unit

Fig. 1. Team-groups pattern captured from video in daily life. (For interpretation of the references to color in this figure text, the reader is referred to the web version of this article.)

the forces that other individuals and walls exert upon individual i, respectively. Ai , Bi , k and κ are constants of the SFM, the values used in simulation are given in Section 6. Although the OSFM can simulate the behavior of a large-scale crowd to a great extent, this model also has obvious shortcomings. For instance, it considers individuals equally with no difference in forces between pedestrians, thereby allowing the crowd to evacuate loosely. However, complicated social relationships exist among persons in real life. In the process of flow evacuation, people with close relationships consider each other. This paper proposes a two-layer relationship mechanism to collection motion by adding aggregate force and collision avoidance repulsive force. The collective effect of pedestrians gathering in team-groups pattern can be realized, and the risks of collision can be reduced previously. Consequently, bi-direction pedestrian flow is simulated with higher authenticity. 4. Social force model driven by two-layer relationship mechanism 4.1. Two-Layer relationship mechanism In many existed models especially those which are used to simulate the bi-directional pedestrian flow, pedestrians are considered as isolated and equivalent individuals with no differences in force analysis, which is not consistent with the existence of social relationships in real life. Meanwhile, pedestrians who are more familiar with the surrounding environment can choose a better way and play the important role of a leader in a march. The video capture presented in Fig. 1 shows that leading relation and social relation are ubiquitous in real life. In these screenshots, there are many teams, and a typical team (each member has a yellow cap) composed of small social groups is captured, proving leader and social groups are existed. An open source path tracking tool named “Tracker” is used for free to deal with the video. By constructing the coordinate system of each frame in the video, the information of the pedestrian position is quantified as the coordinate information in the coordinate system, allowing us to exact coordinates and trajectories of pedestrians from the video. With this tool, several typical pedestrians such as the leader and family members are obtained and their trajectories of moving process are extracted. Usually, each team has a team leader in forefront of

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X. Qin et al. / Simulation Modelling Practice and Theory 84 (2018) 268–285 Table 1 Questionnaire results of pedestrians’ choice on walking with others in two cases.

The Leaders Role exist Family exist Friends exist Colleagues exist Strangers exist

In a relaxed environment (/person)

In emergency evacuation (/person)

987 10 0 0 817 372

974 843 421 47 27

Table 2 Gathering possibility of each relation degree in two-layer relationship mechanism. Relationship type

Gathering possibility

Leading relationship Social relationship Family Friend Colleague Stranger

(0.97–0.99] (0.84–1.00] (0.42–0.82] (0.04–0.38] [0.00–0.03]

the crowd with a distinct identity symbol (in Fig. 1, the symbol is a flag) or a leader temporary chosen according to his/her forefront position in a mainstream, and team members move in small social groups which are commonly made up of two or three pedestrians such as a father and his son during follow the leader. From their trajectories we can know that the trend of their trajectories is consistent, the leader has the route choice priority, and followers follow leader’s guidance. Besides, people with close social relations usually stand in a relatively near position or tend to walk together. What’ more, members in small social groups usually use social group as a unit to avoid collision with other pedestrians in counter flow. Therefore, according to the different degree of relations, the possibility of pedestrians gather together is different, so the possibility is quantified between 0 and 1 in this model. The range of possibility is determined according to a questionnaire survey results. In the questionnaire, 10 0 0 people were selected as the research object uniform containing 4 kinds of social relations. The willingness to keep movement consistent with other pedestrians of two-layer relationship is investigated under two cases: one is emergency evacuation case and the other one is relaxed walking case. In Table 1, questionnaire survey results show that 98.7% of people will choose to follow the guidance of leader in relaxed walking case when there is an explicitly leader role exist. In emergency, the ratio is 97.4%; In relaxed case, all pedestrians (100%), 81.7% and 37.2% of pedestrians choose to walk with families, friends and colleagues respectively. In emergency, 84.3% of people still choose to evacuate with their families, while the ratio of friend and colleague reduce to 42.1% and 4.7% respectively; Unless there are special circumstances, only 2.7% of people choose to travel with strangers. Finally, the interval of gathering probability is obtained. The relationships are classified and the possibilities are calculated according to the results of questionnaire shown in Table 2. The closer the relationship is, the higher the possibility is. The pedestrians are automatically divided into teams based on distance and relationships. Each team has a leader, and the relationships between the leader and the members in the same team are leader and follower and pedestrians gather into social groups according to gathering possibility. Thus, a dual relationship mechanism, where leaders act on social relations, is formed.

4.2. Introduction of group vision sharing Even if the influential factors are from the back of pedestrians in the OSFM, their perceptions of the surrounding environment are the same, which is not reasonable. In real life, vision plays an important role in the motion of people. Therefore, many scholars have introduced individual visual field into pedestrian movement. In the model proposed by Moussaïd et al., visual information is considered to make prediction of collision from the individual field [53]. However, people usually learn in a situation through visualizing and communicating with their peers to share environmental information naturally instead of observing only by isolated individual when their peers are around. This paper introduces the visual field and visual impact factor of pedestrians and a vision sharing within small groups. As shown in Fig. 2, pi and pj are in the same group, the − → gray semicircles indicate their visual field. Hi is the central axis of the individual visual field, which is in the same direction − → − → as vi . The ranges of this field is φ = 90◦ on the left and right of Hi . θ ij is the angle between the pedestrian moving direc− → − → tion ei (t ) and the unit vector ni j (t ) at moment t. R and r represents the radius of the visual field and pedestrian body size, respectively:

 λ=

0, 1



otherwise

θi j ≤

π 2

& & di j ≤ R

,

(5)

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Fig. 2. Visual field sharing of two individuals in the same group.

Fig. 3. Attraction between pedestrians based on relationships in the same group.

where λ is the visual factor function concerning the personal visual angle θ ij and distance dij between two pedestrians. In the field of pedestrian i, λ = 1 means that pedestrian j in the visual field of pedestrian i, otherwise λ = 0 means pedestrian j is out of the visual field of pedestrian i. The sharing of visual field between two peers in the same social group is a complementary to that of individuals, especially the expansion of visual angle transversely. When dij ≤ 4r, meaning pedestrians i and j who have close social relation are close enough in distance, it is reasonable to share the similar region information. As shown in Fig. 2, pedestrians i and j in red are two members of the same group. The other two pedestrians pk and pl are from the opposite direction, they are in the visual field of i, but pk is only in the visual field of i, not in the visual field of j. With group vision sharing method, the visual fields of pedestrian i and j are merged, sharing the local information of other pedestrians in the no overlapping parts. Thus, both i and j can know the position of the other two pedestrians pk and pl . The visual factor values are based on whether the influence of environmental factors is in the pedestrian visual field or not. On one hand, only group members who are in the visual field can produce the aggregation force to pedestrian i. On the other hand, when pedestrian from opposite flow occurs in the visual field of pedestrian i can produce a collective collision avoidance repulsive force on group of i. 4.3. Aggregate force based on the two-layer relationship mechanism 4.3.1. Aggregate force Under the two-layer relationship mechanism, the force formula of the OSFM is improved by adding aggregate force with visual factor, which means that two pedestrians will attract with each other in the flow if they have a relationship. The aggregate force increases with the increase of the gathering possibility, and the aggregate force formula is as follows:

−→ − → firel j = Ri j Ci exp[ (ri j − di j )/Di ]ni j ,

(6)

where Ci and Di are the aggregate parameters. Rij is the relationship gathering possibility between pedestrians i and j as mentioned in Table 2. Ci and Di indicates the attraction strength of individuals and the distance to guarantee the safety between the pedestrians, respectively. Aggregate force is weighed by relationship gathering possibility. The greater the aggregate force of the individuals, the faster and more closely they gather. The details of attraction based on relationships are shown in Fig. 3. 4.3.2. Clustering method In team-groups mechanism, the team leader is easily characterized by his forefront position of a crowd. Moussaïd et al. found that clusters in a crowd with no social relation have a lifetime between emergence and decay when at least one group

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member leave, alternating of stable and unstable state [54]. Nevertheless, small groups go through a process of aggregation then emergence a stable state based on social relations. What’s distinguished, because of the existence of social relations, once a small group is formed, a relatively stable internal state is formed relative to the outside, which consistents with the realistic phenomenon in daily life. To achieve the cluster of team-groups in simulation of the bi-direction pedestrian flow, the algorithm of the clustering process is described as follows: Where the value of parameter σ and τ are set by a statistical results extracted from real video data of the bi-direction pedestrian flow. In most cases, an isolated pedestrian can hardly keep moving within a distance of 4r over 3 s. This is because isolated pedestrians prefer to keep a comfort distance to other strangers in the case of plenty of space. 4.4. Collective collision avoidance force Collision avoidance behavior is one of critical factors that influence the pedestrian flow dynamics [55]. Through a continuous observation of the realistic pedestrian motion process in video, it is detected that pedestrians with social relations usually keep consistency of movement even faced with potential collision risk with other pedestrians in the opposite direction. In order to keep consistency with their peers, pedestrians take collective detour measures regarding groups as a whole. However, as for the bi-direction pedestrian flow, many of the existed models lack of previous collision avoidance strategy or just analyze individual collision avoidance while ignore the collective collision avoidance behavior in social life. Focus on individual, a pedestrian will predict the influence of those coming pedestrians he/she may encounter, and take in-advance collision avoidance behavior to decrease potential conflicts in the model proposed by Wang [56]. Based on aggregation force social groups are formed, further considering the social group as a whole, this paper proposes a collective collision avoidance force to the force formula of the OSFM, to avoid conflict from the counter flow with social group as a unit. Usually, the dynamics of a social group as whole is described by the behavior of its center of mass [57]. In order to describe the group dynamic, we choose the average position point of all group members as the center of mass to direct movement. The position of the center is quantized with coordinate Cgroup (x, y), and the calculation is shown in Formula 7, where Nm represents the number of group members:

Cgroup (x, y ) =

1 Nm



Nm  i=1

xi ,

Nm 



yi .

(7)

i=1

In Fig. 4(a), three social groups and an isolated individual (in order to analyze different conditions, here we assume it as a group made up with one pedestrian) are extracted form video with clustering method mentioned in 4.3.2. Several pedestrians who keep in a consistent movement state are divided into a group, the blue asterisk expresses the calculated group center, red circle and yellow circle represents pedestrian who is closest to the center and other member in the same group respectively. In Fig. 4(b), four pairs of trajectories of these groups in Fig. 4(a) are extracted from the video of realistic movement process. In each pair, the blue one is the trajectory of the group center and the other one is that of the closest member in the same group. Compared two trajectories in each social group, it is evidently to draw that if pedestrian exactly in the center position, two trajectories are coincided just as the third pairs reflected from the left hand order, else if no pedestrian in the center position, the closest pedestrian usually acts similar behavior with the center in a consistent pairs of trajectories. Thus, we pick pedestrian who is closest to the center as the decision-maker to represent their group as a whole. If a social group is only composed of two pedestrians, then the default choice is the pedestrian on the right side. In addition, there are relatively large fluctuations in the middle of the third and fourth pairs of trajectories, it is because the avoidance of other groups. This group behavior consistent with the realistic phenomenon in social life, people’s pattern of motion is regarding group as a unit to avoid collision from the outside pedestrians especially faced with other groups from opposite direction flow. Therefore, in order to keep the stability of group and motion consistency of members when faced with potential collision, we propose a collective collision avoidance force based on regarding a group as a unit. In this method, we assume two groups are in two teams with opposite moving direction. Pedestrian i and j are decision-makers of these two groups, respectively. For i, j is the nearest decision-maker located in the visual field of groupi :

−−−→

 − → fiajvoid = Ei exp ri j − di j /Fi ai j ,

(8)

where Ei and Fi are repulsive parameters. Ei indicates the repulsive strength of the individuals, and Fi indicates the distance − → to guarantee safety between the pedestrians. ai j is the avoidance force direction of decision-maker i under the effect of decision-maker j. The direction of collective collision avoidance force can be described in the following algorithm and Fig. 5: Each of the group members shares the same repulsive force to avoid collision so that stability and motion consistency are maintained within the group. In Fig. 6(a), two groups consist of three and five members are moving in opposite directions in simulation. Group framed in blue dashed circle move from left to right, and the other group framed in red dashed circle move from right to left. The blue and red arrows represent the direction of their movement, and the blue and red lines represent the trajectories of the two groups’ decision-maker, respectively. In Fig. 6(b), when the two groups faced with conflict, pedestrians will take collective avoidance measures. The total quality of the group with fewer personnel will be

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Fig. 4. Position and trajectories of three groups and an individual extracted from video. (For interpretation of the references to color in this figure text, the reader is referred to the web version of this article.)

Fig. 5. Avoid force between groups in two different direction flows.

smaller, according to the force formula, this group has bigger acceleration to avoid the other group more actively. As a result, smaller group has more trajectory deviation, and is more flexible in state changes and recovery, which consistent with real life collective crowd dynamic.

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Fig. 6. Collective collision avoidance of two groups in simulation. (For interpretation of the references to color in this figure text, the reader is referred to the web version of this article.)

Fig. 7. Details of the resultant force.

According to the aggregate force mentioned and the collective collision avoidance repulsive force above, for each pedestrian, the resultant force formula can be expressed as Formula 9, and its details are shown in Fig. 7:

mi

  − →  −→  −−−→ → − → − → d vi (t ) − avoid = fi0 + fi j + fiw + λ ∂1 firel + ∂ f , 2 j ij dt w j ( =i )

j ( =i )

(9)

j ( =i )

Where λ is the visual factor mentioned in Section 4.2 to express whether pedestrian j is in the visual field to bring effect −→ of the force on i. ∂ 1 and ∂ 2 are adjustment coefficients for the aggregation force firel and collective collision avoidance force j −−−→ a v oid fi j , respectively. The value of ∂ 1 is 1 when dij meet the condition of σ < dij ≤ 2R, otherwise the value is 0 as mentioned −→ in Algorithm 1, which means the effect of firel working during the clustering process. The value of ∂ 2 is 1 when meet the j → − → − 0 0 condition of ei · e j < 0, which means the moving direction of pedestrian i and j is opposite, otherwise the value is 0 as mentioned in Algorithm 2. In Fig. 7, except for the repulsive force and friction force from walls and other pedestrians according to OSFM, other forces in MSFM are illustrated by using pedestrian i as a example. Pedestrian i moves from left to right, and j is a member of the

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Algorithm 1 Clustering algorithm. Input: The size of crowd pnum, the number of teams teamnum, and the social relationship density of pedestrians ρ ; Output: The clustering state of the pedestrian evacuation process. Constraint: (1) Only one leader exists in a team; (2) Each team must contain four social relationships. Step 1: The position of the individuals with distance control and the relationships between individuals are initialized; Step 2: The gathering possibility is taken as standard to divide teams and groups, and the grades of social relationships among members in one team are established; Step 3: A pedestrian nearest to the target is picked as the leader in every team, and leadership is established; Step 4: Judge whether pedestrian i and j have merged into a group. If j has social relation with i, j move close than a distance of parameter σ = 4r (including the radius of the pedestrians themselves) to i in a minimum threshold time τ = 3s, assume that j and i clustered, then proceed to step 6; else if σ < dij ≤ 2R proceed to step 5. Repeat step 4 until one of the two pedestrians reached the exit; Step 5: The aggregate force is calculated according to the modified social force formula per piece of time until cluster into a group, then proceed to step 6; or reach the exit, end; Step 6: Merge the visual field of two pedestrians, and update the velocity of each member with the velocity of the group center (mentioned in 4.4), the movement of the individuals are driven by the modified social force model until reach the exit, end. Algorithm 2 Collective collision avoidance algorithm. Input: The position of decision-maker i and j; the target of decision-maker i and j. Output: The repulsive force of group collision avoidance. Constraint: (1) Pedestrians i and j are in different flow directions and pedestrian j is the nearest decision-maker to decision-maker i in its collective visual field; (2) Pedestrians i and j are decision-makers in their social groups. Step 1: A circle with lineij as the diameter is constructed, and cross-points of the perpendicular bisector of lineij and the circle named temp1 and temp2 are found. temp1 and temp2 are regarded as temporary target points. Step 2: If temp1 and temp2 are both in the direction toward the target of i, then proceed to step 3; else if only temp1 or temp2 is in the direction toward the target of pedestrian i, then proceed to step 4; else if the distance of pedestrians i and j is less than the diameter of the two particles, then proceed to step 5. Step 3: The unit tangent to point temp1 is regarded as the direction of the avoidance force, which means that the right way is preferred as shown in Fig. 5(a); Step 4: The unit tangent to point temp1 is picked as the direction of the avoidance force; else the unit tangent to point temp2 is picked shown in Fig. 5(b); Step 5: Pedestrians i and j take a step back, respectively, then return to Step 2; Step 6: Share the repulsive force with other members in group and group adjust direction according to the repulsion force collectively.

Fig. 8. Architecture of SFM driven by the two-layer relationship mechanism.

−→ same group as i, pedestrian k moves in the opposite direction with i and j. As for pedestrian i, firel is an attraction force j −−−→ − → which pedestrian j exerted on pedestrian i in the direction of ni j . Meanwhile, fikavoid is a repulsive force which pedestrian k − → exerted on pedestrian i in the direction of tik . 5. Model framework and implementation process Based on the above improvements of the SFM, this paper puts forward a MSFM driven by the two-layer relationship mechanism shown in Fig. 8. The concrete implementation process of the model framework is as follows: 5.1. Position and relationship initialization Many of the existing models have a defect that pedestrians who are close in relationship or in the same team are initialized far from each other, which causes detour phenomena by attraction and even leads to interspersed confusion among pedestrians, resulting in significantly reduced evacuation efficiency. In real life, people with close relationships usually stay together. Accordingly, in the proposed model, pedestrians with close relationships are distributed in the same team and guided by the same leader. The members of a team are evenly distributed in a certain area. Thus, based on distance control,

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the initialization region of both ends of the corridor is divided into m × n grids. m × n grids are merged into large areas with the same number of teams. Each team occupies a large area, and members of the team are randomly and uniformly initialized in the regional grid to ensure the distance between other members of the same team. Under the two-layer relationship mechanism, each team selects an individual nearest to the target as the leader of the team. In this paper, considering a great variety of flexible and changeable real crowd compositions, relationship density ρ is proposed to illustrate the proportion of people with social relation except strangers in the whole crowd. The method of extracting relationship density from real data is as follows. Extracting frames from the video of real evacuation, the total number of population is counted firstly. Then, identify the isolated individuals (i.e. strangers, they are relatively easy to identify because of their isolated movement) in the graph. Meanwhile, for bi-direction pedestrian flow, the total number of pedestrians and the crowd composition is invariable in the whole process of a movement. Therefore, the relationship density of a crowd is a fixed value in an evacuation process. Thus, the relationship density ρ of the crowd is calculated according to Formula (10):

ρ=

pnum − isonum , pnum

(10)

Where the total number of pedestrians is expressed aspnum, and the number of the isolated pedestrians who are strangers to others in a team is expressed as isonum. In order to explore the effect of relationship density on the evacuation efficiency and the formation of self-organization phenomena, for each flow scale, we take it into account to initialize social relationship. 5.2. Evacuation route planning and simulation In addition, route planning and choice reflect the behavior of pedestrians in evacuation [58,59]. Before beginning the process of movement, the individual surveys the surrounding environment, obtains the desired direction of movement, and calculates the repulsive force with the other individuals and the environment and the aggregate force between pedestrians in the same team. Then the resultant force is calculated using the Formula 9 to obtain individual velocity, acceleration and displacement. Thus, individual movement is realized. The modified model considers the two-layer relationship and the interaction between individuals in bi-direction pedestrian flows, and can show the movement tendency among the individuals in the process of individual movement. It compensates for the deficiency of the OSFM and improves the authenticity of crowd simulation. Finally, the motion data obtained from this method is introduced into the 3D simulation platform, and the real crowd evacuation simulation is demonstrated. 6. Simulation results and analysis To verify the effectiveness of the proposed method based on the theoretical model proposed in Section 3, a series of experiments are carried out using VS2012 + OSG as the experimental platform of the crowd evacuation simulation system. In this chapter, the modified model is tested and verified in contrast with other literature models and the real bi-direction pedestrian flow. Section 6.1 illustrates the contrast test, and Section 6.2 shows the simulation and real experiment results. The width of the doors and other environmental factor are significant to the utilization of the exit region [60]. These issues are considered, the simulation parameters are as follows: individual radius (r = 0.25 m), individual quality (m = 80 kg), angle of visual field (φ = 90◦ ), radius of visual field (R = 20 m), and the constants of the SFM (k = 1.2 × 105 kg · s−2 , κ = 2.4 × 105 kg · m−1 · s−1 , A = 2, 0 0 0 N, B = 0.08 m, C = 2, 0 0 0 N, D = 0.05 m, E = 2, 0 0 0 N, and F = 0.05 m). 6.1. Contrast test In this section, we carry out experiment for bi-direction pedestrian flow in a corridor with two exits on each side. The dimensions of the scenario is reduced in accordance with the actual scene of 56 × 22m2 . There is no obstacle in the corridor. There are walls on both sides and the right and left boundaries are open for pedestrians free to go out. For the sake of how relationship density affects the flow evacuation efficiency, and proves our modified model of this study is superior to the other kinds of simulation models. This section presents two contrast tests, the observation of the evacuation time on a variety of relationship density first, and the comparison of the modified model under the optimum relationship density with the OSFM, a modified LGM, a modified CAM proposed by Lu and the well-known model of Weidmann. Therefore, it can be drawn out that the modified model is more efficient in bi-direction pedestrian flow evacuation. 6.1.1. Influence of relationship density on evacuation efficiency According to Formula 10, the relationship density is the ratio of the number of families, friends and colleagues to the total crowd. In order to reflect the influence which different social structures make on the evacuation, and more accurately predict crowd evacuation in different social situations. Evacuation time is taken as the evaluation criteria, the density of the relationship varies from 0 to 1, and the change interval is 0.2. Taking 10 0, 20 0, and 30 0 individuals as examples, each density level is used for the experiment for 50 times. The average evacuation time under different densities is compared as shown in Fig. 9.

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279

Fig. 9. Effect of relationship density on evacuation.

The evacuation time decreases first and then increases with the increase of relationship density, and the optimal evacuation efficiency is reached at the density of 0.4 as shown in Fig. 9. Under the action of relationship aggregation, the motions of closely related individuals are more urgent due to the addition of distance control. Thus, their acceleration is greater than that of the individuals with low degrees of correlation. In the motion process, individuals with high degrees of relationship move faster than individuals with low degrees of relationship and are integrated into small groups. Under a leadership, the overall speed of the team increases to a certain extent. However, the number of individuals with social relations in the team increases with the increasing density. Consequently, the leader’s leading and gathering ability reaches the threshold. This phenomenon can be used to simulate real-life phenomenon. Furthermore, it can predict the real bi-direction pedestrian flow evacuation more accurately. 6.1.2. Comparison of five models under different pedestrian densities The fundamental diagrams can reflect characteristics of pedestrian flow, and empirical studies shows they follow nearly the same trend [61]. To fully verify that the modified model has a few advantages over the OSFM, LGM, the well-known model of Weidmann [62], and a modified CAM proposed by Lu et al. [63] in crowd evacuation simulation application to some extent, the fundamental diagrams of speed–density and flow–density relationships have been analyzed based on various experimental simulation data. According to the experimental results of the evacuation time–relationship density, 0.4 is chosen as the value of the relationship density, and the pedestrian density is from 0.5 to 10 pedestrians per square meter (p/m2 ). The average value of each group is obtained from the 50 experiments. The relationships of the average speed-density and average flow-density are shown in Fig. 10. In Fig. 10, the variation trend of our model simulation curve is consistent with empirical data of the existed models and reflects higher efficiency. For bi-direction pedestrian flow in corridor scenario as describe above, the following conclusions can be drawn from the experimental simulation data: 1. In Fig. 10(a), the speeds of two kinds of SFMs are faster than the LGM, model of Weidmann and Lu L-CAM. And the modified one has an overall faster speed than the other models, especially between 2–4p/m2 , which means in the same pedestrian density, MSFM has higher evacuation efficiency. This is because the pedestrians with relationships urgent to get together with the aggregate force, which results in a relatively high speed of motion. Meanwhile, the group collision avoidance force prevents pedestrians from constantly friction and collision. 2. At the beginning, the average speed decreases slowly with the increasing pedestrian density as shown in Fig. 10(a), and the modified model is obviously superior to the original one at the density of approximately 2p/m2 . As the density continues to increase, the gap between the two kinds of SFMs narrows because when density is around 5, which is a critical value, the room for clustering and avoidance strategy to work well is too small. 3. In Fig. 10(b), the flow rate increases first, suddenly drops, and finally arrive at a jam state that the flow decreases to zero. This is because when pedestrians density turn to high level, in the corridor scenario with two sides are closed by the wall, there is few room for the strategy of next step moving of LGM, Weidmann’s and Lu L-CAM dispatch, enormous location conflicts and force balance of SFM lead to deadlocks. 4. In Fig. 10(a) and (b), the speed as well as flow of MSFM is still high, when the other models have already arrived at a jam state. This is because under this range of density, there is enough room for pedestrians merge into team-group and carry out collective collision avoidance, so the speed and flow of MSFM still has a few speed and flow.

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Fig. 10. Effect of relationship density on the evacuation.

6.2. Simulation and real experiment results For the sake of validating the practicality and effectiveness of the modified model, a series of experiments are conducted, including literature simulation and real evacuation. Evacuation simulation is performed in VS2012 + OSG, and the evacuation paths are further analyzed in MATLAB2016 and in a realistic rendering platform. Meanwhile, the real evacuation is conducted from a radio shot by an unmanned aerial vehicle on campus. These experiments confirm that the reproduction of bi-direction pedestrian flow using the modified model closely conforms to the actual situation and provides efficient guidance. 6.2.1. Experimental results in simulation scene The pedestrian size is 200 for the comparative experiments of MSFM (here we set the density of the relationships to 0.4 for MSFM) with OSFM, and model simulated in our platform using the method proposed by Lee et al.[64]. Simulation results are as shown in Fig. 11. To show the experimental results more clearly, the process of evacuation is drawn using MATLAB2016. Shapes are used to distinguish relations, asterisk stands for leader, hollow circle represents stranger, diamond represents colleague, “x” represents friend, and point represents family relation. In Fig. 11(a)–(f), blue and red points represent right moving and left moving pedestrians respectively, all pedestrians are independent individuals. In Fig. 11(g)–(i), the same color represents the same team, pedestrians initialized in the left area move toward the right door, while pedestrians initialized in right area move to the opposite direction. Pedestrians of the OSFM move loosely at the beginning but come to a jam state when two flows interact as shown in Fig. 11(a)–(c). In Lee’s model shown in Fig. 11(d)–(f), ‘following effect’ and ‘evasive effect’ are used to reproduce clear lane formation, and pedestrians move in a long strip. However, pedestrians move in a ‘front and behind’ position state. This is because pedestrians are set to follow front people in the same direction and evasive people ahead in the opposite direction as an isolated people, lack of social relation interaction like the act of walking side by side, pedestrians walk as independent individuals. Meanwhile, in this process, pedestrians are set to follow or evasive people ahead without considering whether they are visible in their view. On the other hand, this ‘front and behind’ pattern saves the time which pedestrians use in clustering in our model. In the simulation of MSFM, the pedestrians with social relations gather together in small groups and move orderly under a leadership in closely linked teams as shown in Fig. 11(g)–(i). In addition, facing interaction, pedestrians avoid collision with group as a unit in advance, which can simulate the real world ‘team-groups’ dynamic while reproducing a good lane formation of the bi-direction pedestrian flows. The simulation results show that the modified model can reflect the effect of teams containing group aggregation better on the basis of reserving advantages of the original one, reproducing the self-organization phenomenon of the bi-direction pedestrian flow. What’s more, the modified model can prevent some intrinsic problems in the original models, e.g. the clustering method enhanced the organization of the crowd to reduce unrealistic oscillating motion and the collective collision avoidance prevents overlapping of particles. As a result, the evacuation simulation is more realistic. To better exhibit evolution process of the counter flow, the density profile of the collective motion process is presented. In Fig. 12, this paper further analyses the crowd dynamic by component density profile of above simulation with three models. We divide the simulation scene intom × ngrids, and calculate pedestrian density in every grid. As is shown in Fig. 12(a), no stable lane can be observed, and the average density of pedestrian is large when pedestrians from two directions interact.

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Fig. 11. Comparative experiments of three models. (For interpretation of the references to color in this figure text, the reader is referred to the web version of this article.)

Fig. 12. Density profile of three models.

This is because OSFM lack of previous collision avoidance strategy, which results in pedestrians constantly squeeze and friction. In Fig. 12(b), obvious lane can be observed in Lee’s model, high pedestrian density is concentrated at the center of the queue. However, simply move in a queue following manner results in an inadequate use of space. From Fig. 12(c), we can see lanes emerge in the pedestrian flow and but not very stable. This is due to the reason that pedestrians prefer to make full utility of the scene space, and pedestrians avoid collision using groups as units. Meanwhile, high density occurs in groups, which means the agglomeration of group members is relatively tight. On the premise that the members of the group maintain consistency of movement, it can reduce collision risk with other pedestrians out of the group, and the efficiency of evacuation will also be increased. 6.2.2. Experimental results in real scene To verify the authenticity and feasibility of the modified model in actual environments, we build a simulation scene according the real-life scene of a bi-direction pedestrian flow video as contrast. An unmanned aerial vehicle is used to take a record, size data of scene and pedestrians extracted from the video are reduced in the same proportion to model the simulation scene and pedestrians. According to the size data, the scene in this section is an open straight road and its dimension is 60m × 12m. The dynamic data including coordinates, instantaneous velocity and trajectories of pedestrians in their motion process in real-life scene are extracted by the tool of “Tracker”. In order to obtain better simulation results, according to velocity data acquired from video, the upper limit of pedestrian speed is set to 1.5 m/s, the other pedestrian parameters of MSFM are consistent with simulation in 6.2.1. The key frame of collective avoidance phenomenon in the

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Fig. 13. Bi-direction pedestrian flow in real-life scene and simulation. (For interpretation of the references to color in this figure text, the reader is referred to the web version of this article.)

process of pedestrian counter flow from real video are captured as shown in Fig. 13(a), (c) and (e). In order to make the simulation more intuitive as well as remove the influence of the unrelated pedestrians in the video, the pedestrians in the scene are simplified, and the simulation effect is shown in Fig. 13(b), (d) and (f). Fig. 13(a) is the initialization state of pedestrian (The video used for this research is intercepted from a continuous moving video, so that pedestrians have a steady initial velocity instead of accelerating from zero.), the pedestrian in the forefront position of a steam is the leader of a team as mentioned above. Combining with social relations, pedestrians move in team-group patterns. Fig. 13(c) and (e) are captures of a typical group which is made up by three pedestrians walking in the same direction. This group is marked in red box and faced with twice cases of collective collision avoidance during the entire moving process, and Fig. 13(b), (d) and (f) are simulation of the corresponding situation in Fig. 13(a), (c) and (e), respectively. In Fig. 13(b), (d) and (f), group members wear the same color clothes, and the typical group with three pedestrians is marked in red box as well. From the contrast, the team-group pattern and collective collision avoidance can be realistically reproduced in our simulation with MSFM. In order to reflect the real-life scene motion process intuitively, the trajectories of three pedestrians in the same group are extracted as shown in Fig. 14(a). Meanwhile, for more intuitive and clearer contrast, the simulation trajectories of three pedestrians in a group with MSFM are shown in Fig. 14(b). From Fig. 14(a) and (b), we can see, trajectories from simulation have a high similarity with trajectories exacted from real video data. At the same time, the bending of the two sets trajectories during the motion process is a good embodiment of the group’s collective avoidance. The trajectories from real data

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Fig. 14. Trajectories and velocities of a typical group made up by three pedestrians.

have relatively large volatility, which is due to the autonomy and randomness of human movement. While the features of the force in social force model determines pedestrians move in a relatively stable and smooth state in the absence of other pedestrian effects. Moreover, both sets trajectories reflect the tightness and consistency of group members in bi-direction pedestrian flow. For further analysis of the above two characteristics, the quantitative analysis of the tightness and movement tendency of the small groups was made. In order to facilitate analysis, the state of extracted data removed the acceleration process of the initial state of MSFM, and start timing from stable motion state. The first complete avoidance time period is 2–6 s, the second time is 6–10 s. The instantaneous velocity of the central pedestrian of this group is compared to analyze the dynamic motion characteristics in real-life scene and simulation. Because no pedestrian steps back in the observation, the general trend of movement is forward, so it is not significant to study the velocity component in the y direction. Thus, we make a contrast of velocity component in the x direction against moving time t, shown in Fig. 14(c). The variety trend of the velocity is similar, the red one is the real-life scene data and the blue one is the simulation data. Real-life scene data show the variety of real pedestrian is more random, this is because people in real life has more autonomy and it is difficult to maintain an absolute uniform motion even if the surrounding environment is constant. The simulation result varies similar trend as the real data, but the amplitude of the fluctuation is smoother, this is because the resultant force of pedestrians allows keeping a relative stable velocity when there is no other pedestrian interrupt in. Meanwhile, the disturbance of social force allows simulating the real motion. Both velocities have great fluctuation when make collective collision avoidance, which coincides with the trajectory data. On the basis of verifying the group motion consistency, average distance between centroid of members in one group is used to describe the stability of the group. Numerical results of real data and simulation are shown in Fig. 15. In Fig. 15, red line represents the real-life scene data, and the blue one represents numerical results of simulation with MSFM. In real-life bi-direction pedestrian flow, pedestrians have relatively high autonomy to change moving state, so that there is a great distance fluctuation in the process of motion. But the average distance is less than (0.6 m) to keep a tightness state during almost the entire movement process. Meanwhile, the blue line of simulation results shows that pedestrians under MSFM have lower autonomy and randomness, and the distance between centroid of members must meet the range of 2r– 4r, as mentioned in the clustering method above. So it is reasonable that the average distance is around 0.6 m, and is more stable than the real data. From the numerical results above, we can draw a conclusion that for both cases of real-life scene and simulation, the Euclidean distance between the centroid of members in one group keeps in a relatively stable state in bi-directional pedestrian flow. Thus, the tightness and motion consistence can be proved.

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Fig. 15. Average distance between centroid of members in one group. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

7. Conclusions To solve the problem of emergency evacuation and to reproduce the self-organization phenomenon of the bi-direction pedestrian flow, this paper combines leadership and social relationships to propose the two-layer relationship mechanism. Based on the force formula of the OSFM, aggregate force is added with the method of group vision sharing and relations. Thus, the pedestrians with close relationships can find each other and gather in small groups under the leadership of the leader, and more realistically simulate the evacuation behavior of the bi-direction pedestrian flow. Furthermore, the added group collision avoidance force prevents collision and friction when advancing toward intersections. This study introduces distance control to the team partition to avoid the phenomenon of the pedestrians with close relationships from gathering regardless of how far they are from each other and improve overall evacuation efficiency. Simulation results show that the population scale and the relationship of the different densities have a great impact on evacuation efficiency. The modified model can simulate closer team–groups effect and lane formation of bi-direction pedestrian flow in real life, which is of practical significance in actual evacuation. Based on the model proposed in this paper, many factors affect the bi-direction pedestrian flow, such as psychological factors and the complexity of the scene with obstacles. Future work should include continued research on these aspects. Acknowledgment This research is supported by the National Natural Science Foundation of China (61472232, 61272094 and 61572299) and by the project of Taishan scholarship. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.simpat.2018.03. 005. References [1] T.Q. Tang, Y.H. Wu, H.J. Huang, An aircraft boarding model accounting for passengers’ individual properties, Transp. Res. Part C 22 (2012) 1–16. [2] X. Chen, H. Li, J. Miao, A multiagent-based model for pedestrian simulation in subway stations, Simul. Modell. Pract. Theory 71 (2017) 134–148. [3] P.M. Kielar, A. Borrmann, Modeling pedestrians’ interest in locations: A concept to improve simulations of pedestrian destination choice, Simul. Modell. Pract. Theory 61 (2015) 47–62. [4] A. Sagun, D. Bouchlaghem, C.J. Anumba, Computer simulations vs. building guidance to enhance evacuation performance of buildings during emergency events, Simul. Modell. Pract. Theory 19 (2011) 1007–1019. [5] P.F.I. Casas, J. Casanovas, X. Ferran, Passenger flow simulation in a hub airport: an application to the Barcelona International Airport, Simul. Modell. Pract. Theory 44 (2014) 78–94. [6] L. Fu, W. Song, W. Lv, Multi-grid simulation of counter flow pedestrian dynamics with emotion propagation, Simul. Modell. Pract. Theory 60 (2016) 1–14. [7] D. Helbing, P. Molnár, I.J. Farkas, Self-organizing pedestrian movement, Environ. Plan. B 28 (2001) 361–383. [8] Y. Jiang, S. Zhou, F.B. Tian, Macroscopic pedestrian flow model with degrading spatial information, J. Comput. Sci. 10 (2015) 36–44. [9] S. Hoogendoorn, W. Daamen, Self-organization in pedestrian flow, in: Traffic and Granular Flow’03, Springer, Berlin, 2005, pp. 373–382. [10] J. Zhang, W. Klingsch, A. Schadschneider, Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions, J. Stat. Mech. (2011) P06004. [11] T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep. 517 (2012) 71–140. [12] M. Nagy, Z. Ákos, D. Biro, Hierarchical group dynamics in pigeon flocks, Nature 464 (2010) 890.

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