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A cell-based study on pedestrian acceleration and overtaking in a transfer station corridor
Physica A 392 (2013) 1828–1839
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A cell-based study on pedestrian acceleration and overtaking in a transfer station corridor Xiangfeng Ji a,∗ , Xuemei Zhou b,1 , Bin Ran c,a,2 a
School of Transportation, Southeast University, No. 2 Sipailou, Xuanwu District, Nanjing, Jiangsu Province, China
b
School of Transportation Engineering, Tongji University, No. 4800 Caoan, Jiading District, Shanghai, China
c
Department of Civil and Environmental Engineering, University of Wisconsin, Madison, United States
article
info
Article history: Received 28 February 2012 Received in revised form 20 November 2012 Available online 3 January 2013 Keywords: Pedestrian Cell Force Fuzzy logic Acceleration Overtaking Micro-simulation
abstract Pedestrian speed in a transfer station corridor is faster than usual and sometimes running can be found among some of them. In this paper, pedestrians are divided into two categories. The first one is aggressive, and the other is conservative. Aggressive pedestrians weaving their way through crowd in the corridor are the study object of this paper. During recent decades, much attention has been paid to the pedestrians’ behavior, such as overtaking (also deceleration) and collision avoidance, and that continues in this paper. After sufficiently analyzing the characteristics of pedestrian flow in transfer station corridor, a cell-based model is presented in this paper, including the acceleration (also deceleration) and overtaking analysis. Acceleration (also deceleration) in a corridor is fixed according to Newton’s Law and then speed calculated with a kinematic formula is discretized into cells based on the fuzzy logic. After the speed is updated, overtaking is analyzed based on updated speed and force explicitly, compared to rule-based models, which herein we call implicit ones. During the analysis of overtaking, a threshold value to determine the overtaking direction is introduced. Actually, model in this paper is a twostep one. The first step is to update speed, which is the cells the pedestrian can move in one time interval and the other is to analyze the overtaking. Finally, a comparison between the rule-based cellular automata, the model in this paper and data in HCM 2000 is made to demonstrate our model can be used to achieve reasonable simulation of acceleration (also deceleration) and overtaking among pedestrians. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Hubs, especially transfer stations, are the critical nodes of the transportation infrastructure of a city and pedestrians are the main entities in them. Therefore, it is important to understand pedestrian behaviors, which are significant for the planning and design of transfer stations, especially during peak hours when there are massive pedestrian flows. Good mathematical models can support infrastructure designers and planners to optimize their plans. Starting from 1937, different researchers with different backgrounds have developed many models to mimic pedestrian movement. According to the different descriptions of pedestrian behaviors, models of pedestrian simulation are divided into two categories, which are macroscopic models and microscopic models. Henderson (1971) [1] presented the earliest
∗
Corresponding author. Tel.: +86 25 83795356; fax: +86 25 83795356. E-mail addresses: [email protected] (X. Ji), [email protected] (X. Zhou), [email protected], [email protected] (B. Ran).
1 Tel.: +86 21 69583001; fax: +86 21 69583001. 2 Tel.: +86 25 83795356; fax: +86 25 83795356. 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.12.016
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macroscopic model for the pedestrian flow, which analyzed pedestrian flow in analogy to fluid. Hughes (2002) [2] establishes a two-dimensional macroscopic model to simulate pedestrian flow and attains the fundamental diagram. Besides, Xia et al. (2009) [3] and Dogbe (2010) [4] also attain similar results. In the domain of the microscopic models, there exist a social force model, cellular automata model and some related improved models. The social force model, first presented by Helbing and Molnar (1995) [5], took pedestrians as self-driving particles. Movements of pedestrians are decided by self-driving force, repulsive forces from other pedestrians and obstacles. Later, Helbing et al. (2001, 2005, 2007) [6–8] improved this model from different aspects. Blue and Adler (1998) [9] first used the cellular automata model to mimic pedestrian movements and they simulated unidirectional flow (1998) [9] and bidirectional flow (2001) [10]. During recent decades, studies on pedestrian behavior have attracted lots of researchers. Hoogendoorn and Bovy (2004) [11] researched the pedestrian route choice based on the utility theory under uncertainty and solved this problem with dynamic programming. Antonini et al. (2006) [12] presented the pedestrian discrete choice model, which included movement direction and movement speed. The final utility was decided by expected direction, expected speed, actual location, actual speed, and so on. Antonini et al. verified the model with measurement data. Lo et al. (2006) [13] researched the exit choice problem during the evacuation in closed space based on game theory. After this, many researchers solved the collisions problems based on game theory, such as Tanimoto et al. (2010) [14] and Asano et al. (2009) [15]. Asano et al. (2009) [15] mimicked the collision avoidance base on pure strategy Nash equilibrium and solved it by the augment allocation algorithm. Asano et al. (2010) [16] integrated pedestrian route choice and collision avoidance based on the paper in 2009. Besides, overtaking can be found frequently in reality, especially in transfer stations, where different pedestrians may have different speeds. However, studies about overtaking are implicit ones to a large extent and explicitly overtaking analysis is scarce. Thompson et al. (1995) [17] use SIMULEX to analyze the possibility of pedestrians overtaking obstacle pedestrians with two potential angles. Blue and Adler (1998) [9] use the rule-based cellular automata to analyze the overtaking possibility. Herein we call these rule-based ones implicit overtaking. Yuen et al. (2012) [18] present a modified social force model to mimic the overtaking behavior of unidirectional pedestrians and make a comparison between the original social force model and the modified one with the real data to achieve reasonable simulations of overtaking behavior among pedestrians. This one is an explicit one. All the models can be further divided into discrete models and continuous models. Generally speaking, discrete models like cellular automata allow pedestrians to move according to some rules. Herein we call them rule-based models. In contrast, continuous models like the social force model allow pedestrians to move according to forces within defined geometry. Herein we call them force-based models. From the above-mentioned discussion, studies on pedestrian flow concerns pedestrians’ behaviors much more during recent decades. Overtaking and collision-avoidance are two main aspects. In this paper, overtaking in unidirectional flow is considered and collision-avoidance and the extension to bidirectional flow are left for future research. Actually, this paper can be a building block for the complex and detailed pedestrian flow analysis. In this paper, pedestrians are divided into two categories. The first one is aggressive, and the other is conservative. Aggressive pedestrians weaving their way through a crowd in a corridor is the study object of this paper. After sufficiently analyzing the characteristics of pedestrian flow in a transfer station corridor, a cell-based model is presented in this paper, including the acceleration (also deceleration) and overtaking analysis. Acceleration (also deceleration) in a corridor is fixed according to Newton’s Law and then speed calculated with a kinematic formula is discretized into cells based on the fuzzy logic. After speed is updated, overtaking is analyzed based on updated speed and force explicitly, compared to rule-based models, which herein we call implicit ones. During the analysis of overtaking, a threshold value to determine the overtaking direction is introduced. Actually, the model in this paper is a two-step one. The first step is to update speed, which is the cells a pedestrian can move in one time interval and the other is to analyze the overtaking. Finally, a comparison between the rule-based cellular automata in Blue and Adler (1998) [9], the model in this paper and data in HCM 2000 is made to demonstrate our model can be used to achieve reasonable simulation of acceleration (also deceleration) and overtaking among pedestrians. Although some models which detail the forces between pedestrians have been presented, such as Chraibi et al. (2009) [19] and Chraibi et al. (2010) [20], our model has some differences from those as follows. Firstly, pedestrians are divided into two categories, which are aggressive and conservative. The study object is aggressive pedestrians’ weaving their way through a crowd. Secondly, the above-mentioned models are continuous models and have low calculation efficiency, while our model is cell-based and has good calculation efficiency. Thirdly, our model analyzes the overtaking behavior of unidirectional pedestrians according to updated speed and force explicitly with a threshold value to fix the overtaking direction, compared to cellular automata models. Finally, during the process of force analysis, this paper introduces a new parameter called familiarity and the anisotropy of a pedestrian is also analyzed, so the force analysis in this paper is an improved one. Therefore, this paper is a new one as far as we know. The rest of this paper is organized as follows. Section 2 mainly analyzes characteristics of unidirectional pedestrian flow in a transfer station corridor. Next, the acceleration (also deceleration) analysis model is established, which includes model analysis, force analysis, definition of forces and then speed-updating analysis based on fuzzy logic is done in Section 4. In Section 5, the overtaking analysis is made based on the force explicitly with the introduction of a threshold value to determine the overtaking direction. In the next section, a comparison between the cellular automata model in Blue and Adler (1998) [9], the model in this paper and real data in HCM 2000 is made, and some results are attained. Finally, some conclusions are obtained in Section 7.
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Horizontal
Vertical Fig. 1. Route choice in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2. Pedestrian flow characteristics analysis Movements of pedestrians in a transfer station corridor are influenced by macroscopic or microscopic factors, a complex process interacting with the with environment. Hoogendoorn et al. [10] presented a three-level theory of pedestrian micro behaviors, which are strategic level, tactical level and operational level. This paper uses this theory in the analysis of characteristics of unidirectional pedestrian flow in corridors. (a) Strategic level Behaviors at the strategic level reflect pedestrians’ macroscopic decisions, such as buying tickets, transferring and so on. The object of this paper is pedestrian flow in a transfer station corridor. The macroscopic decision of the strategic level is to transfer. Therefore, pedestrians will transfer to the desired destination under their experience or the guidance of signs. (b) Tactical level At the tactical level, pedestrians will attain an activity chain from the activity set decided in the strategic level. In stations, the transfer behaviors will be constrained by time and space as follows. Time constraint Generally speaking, pedestrians will go to destinations directly without some unnecessary activities during peak hours. In particular, pedestrians will speed up under time pressure and the phenomenon of overtaking will appear. Space constraint Route choice of pedestrians is an important part of the tactical level. The transfer corridor can be divided into different segments vertically and in different times, pedestrians maybe walk in different segments horizontally as follows in Fig. 1. The green and red arrow lines are two possible routes in this paper. Because of a block from facilities and other pedestrians, pedestrians cannot get full information about other pedestrians and the environment. Therefore, pedestrians have to move according to their experience or following others. Route choice in this paper can also be called lane changing. (c) Operational level Behaviors at the operational level include some micro behaviors, such as collision avoidance, speed features, moving direction and so on. Moving direction When there are many pedestrians in the transfer corridor, pedestrians will feel significant pressure from behind, so moving backwards is impossible during peak hours in general. Speed features Speeds in transfer stations are somewhat larger than normal speeds. Besides, for the relatively closed space and constantly changing environment, pedestrians are likely to speed up or slow down frequently. In particular, when pedestrians feel the stress of time, they will speed up. Collision avoidance Collision avoidance mainly includes two parts, which are avoidance from other pedestrians and avoidance from obstacles. Pedestrians will move in the shortest route or in the shortest time under the premise of no collisions. 3. Acceleration analysis 3.1. Model analysis The transfer corridor in this paper is discretized into cells with side length of 0.4 m (1.3ft), which is the typical space occupied by a pedestrian in a dense crowd as in Burstedde et al. (2001) [21]. As described in Section 2, different pedestrians may have different speeds and overtaking is frequently found in a transfer corridor. Pedestrians are divided into two classes in this paper. One is aggressive and the other is conservative. Pedestrians who are aggressive are assigned with a maximum
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Fig. 2. Pedestrian moving directions or the visual angle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Different situations where pedestrians move forward 2 cells.
rate of two cells per time interval (about 0.8 m (2.6ft) per time interval); pedestrians who are conservative are assigned with a maximum rate of one cell per time interval (about 0.4 m (1.3ft) per time interval). The time interval is 0.3 s. Those who are aggressive are nearly running, and the fact that they weave their way through the crowd is the study object of this paper. Besides, pedestrians can react to the front situation and can speed up or slow down stochastically in different front density. In particular during the peak hours, there is massive pedestrian flow, so moving backwards is relatively impossible. Therefore, the moving direction at the operational level of pedestrians in this paper which is also called the visual angle in Yuen et al. (2012) [18] is shown as follows in Fig. 2 where the bold black line presents walls of the corridor and the corridor is discretized into small cell squares with side length 0.4 m. The yellow arrow stands for the desired direction, the dotted blue arrow is the visual angle in Ref. [18] and the green solid arrow is the moving direction, or the visual angle based on Assumption 1 in this paper. Assumption 1. Pedestrians move in the domain of the desired direction, including forward, upper left and upper right (stated in the moving direction of the pedestrian). That is to say pedestrians’ moving left or moving right in the abstract corridor is unmeaningful. In this paper, we call this inertia, which is pedestrians’ subconsciousness to make themselves closer to the destination in the next time interval. Assumption 2. Unlike vehicles, pedestrians can go, stop or turn in a much smaller time than vehicles. So in this paper, if pedestrians move forward two cells in one time interval, the following three situations are equivalent in Fig. 3. Generally speaking, pedestrians are inclined to choose the way of least consumption. So the probability of choosing a different moving direction during overtaking is different, which is determined by the situations around. Assumption 3. Like vehicles, pedestrians have a different reaction to stimulus from the front and behind. This is anisotropy. 3.2. Force analysis Acceleration (also deceleration) of one pedestrian is based on Newton’s law. The forces exerted on pedestrians include self-driving forces of pedestrians, repulsive forces between pedestrians, repulsive forces between pedestrians and obstacles and friction forces among pedestrians and obstacles. Besides, the force perpendicular to pedestrian moving direction cannot affect a pedestrian’s acceleration and deceleration. Therefore, force decomposed along the moving direction should be considered during the analysis of acceleration or deceleration.
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Fig. 4. Situations where friction forces exist in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Each pedestrian has expectation to the destination, so each pedestrian has a self-driving force. However, not every pedestrian has a repulsive force and friction force. Only when one pedestrian moves close to another or an obstacle, will the repulsive forces appear. And the friction forces appear in a stricter situation. Only when one pedestrian is adjacent to another or an obstacle, will the friction forces appear and the friction forces depend on the repulsive forces. The situations where friction forces exist are as follows in Fig. 4. In Fig. 4, the green circles denote the situation where repulsive forces exist and the red circles denote the situation where friction forces exist. The model developed in this paper is actually a two step one which includes acceleration (also deceleration) analysis and overtaking analysis. The realization of overtaking depends on lane changing. Force and speed are all vectors. When two vectors are orthogonal to each other, they have no effect on each other. Therefore, during the acceleration (also deceleration) analysis, the force should be decomposed to the direction of speed, which is horizontal, and during the overtaking analysis, the force should be decomposed to the vertical direction. 3.3. Definitions of forces Qualitative analysis in Section 3.2 is not enough for the simulation. Therefore, a quantitative analysis on the pedestrian self-driving forces, repulsive forces among pedestrians and obstacles and friction forces among pedestrians and obstacles is presented. (a) Self-driving force of pedestrians The self-driving force denotes the force pedestrians exert on themselves in order to get to destinations. This paper adopts the definition in literature [5]. F⃗idr v = mi
Vi0 − ∥V⃗i ∥
τi
⃗e0i .
Where, mi denotes the mass of pedestrian i; Vi0 denotes the desired speed of pedestrian i; V⃗i denotes the real speed of pedestrian i; ⃗ e0i denotes the desired moving direction of pedestrian i; τi denotes the lag time of pedestrian i. (b) Repulsive forces between pedestrians Repulsive forces between pedestrians not only relate to the distance between pedestrians, but also relate to the relative speeds between pedestrians. The larger the distance is, the smaller the repulsive force is. Yet, the larger the relative speed is, the larger the repulsive force is. At the same time, the repulsive force is also influenced by the kindred and friendship. The more familiar pedestrians are, the smaller the repulsive force is. In this paper, this is shown with a so-called familiarity parameter ρ . The repulsive forces are calculated in this paper according to the following formula in the vector form or norm form. Where, mi denotes the mass of pedestrian i; V⃗i denotes the speed of pedestrian i; V⃗j denotes the speed of pedestrian j; Vij
⃗i denotes the position of denotes the projection of the relative speed between pedestrian i and j on the direction of ⃗ eij ; R ⃗j denotes the position of pedestrian j. In general, pedestrians have different senses to the repulsive forces pedestrian i; R from different directions and will have a stronger sense to the repulsive force from forward direction than backward. That is anisotropy, which is shown with a parameter ω in this paper. rep
F⃗ij
= −mi ρω
Vij2
∥R⃗ij ∥
⃗eij ⃗e0i .
Or,
Vij2 ⃗eij sin(θ ). ∥ ⃗ ∥ = −mi ρω ∥R⃗ij ∥ rep Fij
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Fig. 5. Angle of φ and θ .
Where, ω = λ + (1 − λ) ∗ cos(ϕ/2)(0 ≤ λ ≤ 1, 0 ≤ ϕ ≤ π ) Vij = (V⃗i − V⃗j ) ∗ ⃗ eij
⃗ij R ⃗eij = ⃗ Rij ⃗ij = R⃗j − R⃗i . R The followings are definitions of λ, ρ, ϕ and θ .λ represents the anisotropy of a pedestrian. If pedestrians are familiar with each other, the value of ρ is less than 1. The more familiar pedestrians are, the smaller value ρ is. ϕ represents a function of radian φ which takes shape from the moving direction of a pedestrian to the direction of repulsive force exerted on the pedestrian in an anti-clockwise rotation, and θ is the included angle of the direction of repulsive force andthe vertical line. Angles of φ and θ are shown in Fig. 5 and ϕ = π −φ . In this paper, the φ can be a value from the value set of 0, π4 , π2 , 34π , π .
and the θ can be a value from the value set 0, π4 , π2 , 34π , π .
(c) Repulsive force between pedestrian and obstacle Pedestrians and obstacles, such as walls, must keep a certain distance apart and the repulsive forces between them are analogous to the repulsive forces between pedestrians. That is to say the obstacles can be regarded as pedestrians with a speed value of 0. 2 Viobs
rep
F⃗iobs = −mi ω
∥R⃗iobs ∥
⃗eiobs ⃗e0i ,
Or,
2 Viobs rep ⃗ ⃗eiobs ∥Fiobs ∥ = −mi ω sin(θ ). ⃗ ∥Riobs ∥ Parameters in the above formula are similar to parameters in the formula of Section 3.3(b) (d) Friction forces among pedestrians and obstacles When two pedestrians are adjacent to each other or the pedestrian is adjacent to obstacles, such as walls, friction forces will exist. According to Hooke’s law, the friction force can be calculated according to the following formula. friction
F⃗ij
= µF⃗ij .
Where F⃗ij denotes vertical pressure or the component force of repulsive force in the direction vertical to the pedestrian moving direction. µ denotes the friction parameter. In summary, the multi-force model of pedestrian movement is based on Newton’s law, which is as formula (1). Where, rep F⃗ij denotes the repulsive force pedestrian j exerts on pedestrian i; N denotes the number of pedestrians who exert repulsive friction
rep
forces on pedestrian i; F⃗iobs denotes the repulsive force an obstacle exerts on pedestrian i; obs denotes obstacle; F⃗ik denotes the friction forces between pedestrians and obstacles; M denotes the total number of pedestrians and obstacles ⃗i denotes adjacent to pedestrian i; F⃗idr v denotes the self-driving force of pedestrian i; mi denotes the mass of pedestrian i; a the acceleration of pedestrian i. F⃗i =
N j̸=i
rep
F⃗ij
+
obs
rep
F⃗iobs +
M
friction
F⃗ik
+ F⃗idr v = mi a⃗i .
k̸=i
Actually, pedestrian mass can be removed from the above formula by dividing mi .
(1)
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Fig. 6. fuzzy triangular and membership function.
4. Speed-updating analysis Pedestrians’ acceleration calculated by formula (1) in Section 3.3 is continuous, so speeds in the next time interval determined by current speed and acceleration should also be continuous. However, the model proposed in this paper is cell-based. So in this paper, we further discretize the speed based on fuzzy logic. As is well known, the history of fuzzy logic began with the simple idea conceived by Lotfi Zadeh in the early 1960s that classes of objects that do not have precisely defined criteria of membership are prevalent when we deal with the real world. Traffic engineers have applied this theory into different domains of traffic, mainly into route choice models and travel demand. Henn (2000) [22] proposed a new route choice model taking account the imprecisions and the uncertainties lying in the dynamic choice process based on a fuzzy subset. A comparison is made between the fuzzy choice model and LOGIT model and the same results are found. Ghatee et al. [23] proposed a new traffic assignment model based on fuzzy level of travel demand and with this model, network planners are able to estimate the number of travelers on network links. During the modeling of pedestrians, Castro. et al. (2011) [24] defined a fuzzy rule that permits to identify whether an object is in danger because another could hit him and predicted the object collisions. Llorca (2011) [25] provided a collision avoidance system (CAS) for autonomous vehicles, focusing on pedestrian collision avoidance. The collision avoidance maneuver is performed using fuzzy controllers for the actuators that mimic human behavior and reactions, along with a high-precision GPS, which provides the information needed for the autonomous navigation. Definition 1. Fuzzy sets. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function µA : X → [0, 1] and µA (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X , where X is called universe, and A is fuzzy subset of X . One of the most important steps toward using fuzzy logic for problem-solving is representing the problem in fuzzy terms. This process is call conceptualization in fuzzy terms. The terms have linguistic values. Definition 2. Linguistic values, also called fuzzy labels, have semantic meaning and can be expressed numerically by their membership functions. Definition 3. The process of representing a linguistic variable into a set of linguistic values is called fuzzy quantization. Two parameters must be defined for the quantization procedure: (1) the number of fuzzy labels and (2) the form of the membership functions for each of the fuzzy labels. In this paper, the fuzzy label is pedestrian speed the membership function is shown as follows in Fig. 6. Assumption 4. Unlike vehicles, pedestrians can change their moving direction in a very short time interval, much smaller than the simulation time interval. Therefore, during the process of calculating the speed in next time interval, the direction is ignored as if the speed is a scalar and the direction of pedestrian speed in next time interval is determined in overtaking analysis. According to kinematic formula vt = v0 + at, where vt is pedestrian speed in next time interval, v0 is the current speed, t is the time interval and a is obtained by formula (1) in Section 3.3.
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Fig. 7. Fuzzy preference indices.
As aforementioned, pedestrians can react to the front situation and can speed up or slow down stochastically in different pedestrian densities in front. In this paper, this is called stochastic acceleration and deceleration and is considered implicitly in the fuzzy logic. In this paper, the method of Teodorovic et al. (1990) [26] is adopted to analyze the cells a pedestrian can move in the next time interval based on the factor vt . Their method is applied to solve the problem of traffic assignment between two alternative roads on a highway network. And this is the first time that this method has been applied to such models in this paper as far as we know. Fuzzy set of speed As schematically portrayed in Fig. 6, the vt is the factor whose definition is introduced as a fuzzy number, and then one pedestrian’s partition of input space according to his/her perception can be defined as ‘Much Less’ (ML), ‘Less’ (LS), ‘Equal’ (EQ), ‘Greater’ (GT) and ‘Much Greater’ (MG), where vtU , vtM and vtL correspond to the upper, middle and lower values of a fuzzy triangular number and where uv (vtU ) = 0, uv (vtM ) = 1, and uv (vtL ) = 0. The fuzzy triangular and membership functions for the partitions can be defined as follows.
µML (v) = µLS (v) =
1 0
vt ≤ vtL otherwise
M (vt − vt )/(vtM − vtL ) 0
vtL ≤ vt ≤ vtM otherwise
(vt − vtL )/(vtM − vtL ) vtL ≤ vt ≤ vtM µEQ (v) = (vtU − vt )/(vtU − vtM ) vtM ≤ vt ≤ vtU 0 otherwise M U M (vt − vt )/(vt − vt ) vtM ≤ vt ≤ vtU µGT (v) = 0 otherwise 1 vt ≥ vtU µMG (v) = 0 otherwise. Fuzzy preference indices On the other hand, the output space, which corresponds to the preference indices (PI) assigned to each alternative in a pairwise comparison. The preference indices (PI) are labeled as ‘Very Weak’ (VW), ‘Weak’ (WK), ‘Equal’ (EQ), ‘Strong’ (ST) and ‘Very Strong’ (VS) to represent the preference values in a pairwise comparison between any two alternatives with respect to vt . Next, we introduce the pedestrian’s preference indices P1 and P2 for any two alternatives, which represent pedestrian’s degree of preference. Obviously, P1 and P2 satisfy the following relationships. 0 ≤ P1 ≤ 1,
0 ≤ P2 ≤ 1 ,
P1 + P2 = 1.
The preference for a particular moving cell can be expressed by an adjective in this paper, such as ‘Very Weak’ (VW), ‘Weak’ (WK), ‘Equal’ (EQ), ‘Strong’ (ST) and ‘Very Strong’ (VS) as shown in Fig. 7. A set of ‘if–then’ rules that underlies the fundamental relationship between two fuzzy perceptions is established which can be used to describe the moving cell when one pedestrian is making a direct comparison as follows. If vt = ML, then P1 = VS; If vt = LS, then P1 = ST ; If vt = EQ , then P1 = EQ ; If vt = GT , then P1 = WK ; If vt = MG, then P1 = VW . Let us consider, for example, the rule ‘‘If vt = LS, then P1 = ST ’’. The implied relation between fuzzy variables vt and P1 can be expressed in terms of the Cartesian product of the fuzzy sets LS and ST . Let us denote by R the Cartesian product of sets LS and ST . The relationship is defined as follows. R(x, y) = LS × ST = min {µLS (x), µST (y)} ,
∀x, y.
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Upper - left
Upper - right
Lower - left
Lower - right
Fig. 8. Partition of space.
Fig. 9. Diagrammatic sketch of included angle.
Now we can find the value of the fuzzy variable P1 , given the fuzzy value of vt . The fuzzy set P1 is P1 = maxX
{min[µvt (x), min(µLS (x), µST (y))]}, ∀y.
The value of the preference index which suggests the pedestrian’s preference must be determined. There are two rules as to how to choose this value. One may choose the value with the highest membership grade, or the center of gravity of the membership function. In this paper, we choose the highest membership grade for calculating the value of the preference index. Once P1 is known, P2 is calculated as P2 = 1 − P1 . 5. Overtaking analysis In Yuen et al. (2012) [18], there are two main factors that should be considered in analyzing pedestrian overtaking behaviors. The first one discusses the overtaking direction like green arrows in Fig. 3 and the other one discusses the overtaking degree, which is the magnitude of the overtaking force. In Ref. [18], the overtaking direction of one pedestrian is determined by the forces in visual angle in Fig. 2. That is to say the forces behind the pedestrian are ignored. In this paper, we improve the model by including the anisotropy of pedestrians in Assumption 3. The space around one pedestrian is divided into four subspaces with a horizontal line and a vertical line in Fig. 8. The four subspaces are upper-right (UR), upper-left (UL), lower-right (LR) and lower-left (LL). Pedestrians have different reactions to the stimuli. That is anisotropy. Forces like self-driving force and friction force do not affect pedestrians’ overtaking, because they are horizontal and the overtaking appears vertically, which is perpendicular. The repulsive forces from pedestrians in the four subspaces are calculated as follows. FUR =
m
UR Frep cos(θ )
FUL =
i=1
FLR =
k
n
UL Frep cos(θ )
i =1 LR Frep cos(θ )
FLL =
i =1
l
LL Frep cos(θ ).
i=1
Where FUR is the repulsive force of pedestrians in subspace upper-right and the rest can be deduced by analogy. The repulsive force is calculated as in Section 3.2 and should be decomposed in the vertical direction. The angle θ is the included angle of repulsive force direction and the vertical n lineULin Fig. 9. m UR Therefore, Fupper = λFUL + (1 − λ)FUR = λ i=1 Frep cos(θ ) + (1 − λ) i=1 Frep cos(θ ) and Flower = λFLL + (1 − λ)FLR =
λ
LR LL Frep cos(θ ) + (1 − λ) i=1 Frep cos(θ ). Where Fupper and Flower are the resultant force of repulsive force from upper and lower, respectively and λ denotes the anisotropy of the pedestrian.
l
i=1
k
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If |Fupper − Flower | ≤ δ , pedestrians will stay on the current route; if Fupper − Flower > δ , pedestrians will choose to overtake from the lower; if Fupper − Flower < −δ , pedestrians will choose to do this from the upper, where δ is the threshold value and it is vital enough that should be carefully determined in the simulation analysis. The value of δ is not fixed and it is determined by the pedestrian’s speed in next time interval and the empty cells ahead as δ = f (speednext , cellsempty ). When the empty cells ahead can afford the speed, the value of is large enough and vice versa. 6. Simulation and analysis This cell-based model is run with pedestrians distributed randomly on a lattice 40 cells long and 10 cells wide and the periodic boundary is adopted. That is to say the cells are connected as a closed loop and curvature effect is removed. Relaxation time of pedestrians who are aggressive is 0.3 s and that of conservative pedestrians is 0.5 s. Because the simulation time is 0.3 s, that is to say aggressive pedestrians move first and conservative ones move later, which is common in reality. The desired speeds of conservative pedestrians are Gaussian distributed with mean value 1.34 m/s and standard deviation 0.26 m/s as in Helbing and Molnar (1995) [5] and those of aggressive ones are double that. The model was run for 6000 time intervals with the first 1000 time intervals to initialize pedestrian movement. 6.1. Revisit of local rules of Blue and Adler Local rules in Blue and Adler (1998) [9] are simply revisited as follows, where pn denotes pedestrians numbered from 1 to n, v(pn ) denotes the speed of the pedestrian, vmax (pn ) denotes the maximum desired speed of the pedestrian and gap(pn ) denotes the number of empty cells ahead. Anyone who is interested in the full version can refer to Blue and Adler (1998) [9]. Rule1a: IF the lane immediately to the left is beyond the lattice boundary, OR the cell immediately to the left is occupied by another pedestrian, OR the cell immediately to the left is free but the cell two lanes over to the left is within the lattice and occupied by a pedestrian, THEN assign the cell to the left to be occupied. Rule1b: IF the lane immediately to the right is beyond the lattice boundary, OR the cell immediately to the right is occupied by another pedestrian, OR the cell immediately to the right is free but the cell two lanes over to the right is within the lattice and occupied by a pedestrian, THEN assign the cell to the right to be occupied. Rule 2: IF the lane immediately to the right is assigned to be occupied, AND the cell immediately to the left is assigned to be occupied, THEN assign pedestrian pn to the current lane, ELSE go to Rule 3. Rule 3: IF there exists a gap that is uniquely maximal, THEN assign pn to that lane. Rule 4: IF there exist equal maximum gaps in two or more lanes, apply one of the three tie-breaking rules to determine assignment for pn : 4a. Three-way tie: Randomly determine lane assignment by using 80/10/10 split between current lane and two adjacent lanes. 4b. Two-way tie between adjacent lanes: Randomly determine lane assignment by using 50/50 split. 4c. Two-way tie between current lane and single adjacent lane: Randomly determine lane assignment by using 50/50 split. Rule 5: IF the current gap is less than or equal to the maximum speed for the pedestrian, THEN set the current speed of the pedestrian to the gap size [v(pn ) = gap(pn )], ELSE set the current speed of the pedestrian to the maximum speed [v(pn ) = vmax (pn )]. 6.2. Statement of local rule Each interval of the simulation is based on a two-step parallel update of the pedestrians. During the first step, value of speed which is the cells pedestrian can move in next time interval of pedestrian is fixed and during the second step, the moving direction is fixed, which also can be called overtaking direction, or lane changing. The overall rule is shown as follows. The count of interval is k. Rule 1: Set k = 1. Forces and acceleration are calculated with the formulas in Section 3.3 and the pedestrian speed, which is cells to move in next time interval are determined as in Section 4. Rule 2: Overtaking or stay in the current. 2.1 If cells ahead of pedestrian can afford the speed, the pedestrian will stay in the current lane; 2.2 If the cells ahead of pedestrian cannot afford the speed, overtaking appears. The direction of overtaking is determined in Section 5. 2.3 If both of the direction of 2.1 and 2.2 cannot afford the speed, update the speed as min(cellsahead , speedcurrent ), where cellsahead is the empty cells ahead in both the current lane and the overtaking direction fixed in 2.2 and speedcurrent is the updated speed in Rule 1. Rule 3: Collision avoidance. 3.1 If two or more conservative pedestrians walk to the same cell, or two or more aggressive ones walk to the same cell, choose one pedestrian to move according to uniform distribution; 3.2 If aggressive and conservative pedestrians walk to the same cell, choose an aggressive one to move. Rule 4 Set k = k + 1. And recycle the rule 1–3 until the maximum time intervals.
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Fig. 10.
6.3. Simulation and discussion The fundamental diagram of pedestrian flow is the target for determining the effectiveness of the cell-based model to adequately portray macroscopic behavior. Level of service criteria for pedestrian flows are defined in the HCM and are based on the relationship of three variables: space, flow rate, and walking speed. Space is the inverse of density and relates area of walkway to number of pedestrians. Space is typically measured in square feet per pedestrian. Flow rate, measured in pedestrians per minute per foot of width, is analogous to volume. Average speed is the rate of movement given in feet per minute. The fundamental relationship is given as Flow rate = average speed / space = average speed × density. Different distributions of aggressive pedestrians are simulated in this paper, from 5% to 50% with a step of 5%. And results of 15% of aggressive pedestrians fit the curves best. Fundamental diagrams are as follows in Fig. 10, where the unit is changed to feet. In the left figure of Fig. 10, abscissa axis is density which is scaled in pedestrians per square foot and axis of ordinates is speed which is scaled in feet per minute. In the right figure of Fig. 10, abscissa axis is also density and axis of ordinates is flow which is scaled in pedestrians per minute per foot of width. Fig. 10 indicates that model in this paper is a realistic representation of macroscopic flows. The maximum flow of 24.3 P/min per ft. of width is very close to the maximum capacity of 25 P/min per ft. of width indicated in the HCM 2000 [27] and 24.5 P/min per. of width in Blue and Adler (1998) [9]. However, when the density increases to a large value, flow of model in this paper decreases more quickly, because large density hinders pedestrians’ weaving their way through a crowd. So does the speed, when the density increases to a large value, speed of movement in this paper is smaller than that in Blue and Adler (1998) [9].
7. Conclusion In this paper, a cell-based model considering two kinds of pedestrians, aggressive and conservative, is proposed. Aggressive pedestrians’ weaving their way through crowd in a corridor is the study object of this paper. The acceleration and overtaking analysis is done here. The acceleration is fixed according to Newton’s law and speed calculated with kinematic formula is discretized in to cells based on fuzzy logic. The overtaking is done explicitly with force and a threshold value is introduced to determine the overtaking direction. The model in this paper appears capable of representing the pedestrians’ behavior, including acceleration and overtaking and the rule set in this paper can capture the aggregate behavior in a reasonable distribution of aggressive pedestrians, compared to the model in Blue and Adler (1998) [9] and the real data in HCM 2000. The model in this paper can be a building block for more complex and detailed models, such as models for bidirectional pedestrian flow with explicit acceleration and overtaking analysis and models combining explicit acceleration, overtaking and collision analysis. Besides, some parameters are presented here, and a sensitivity of these parameters can also be done in the future work. The proposed model here can be an expansion of the theoretical models in pedestrian flow analysis. In practical use, it can evaluate the design of the corridor and also can be guidance for the design and construction of the corridor.
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Acknowledgments Special thanks go to the anonymous reviewers for their great suggestions. This study is supported by the National Basic Research Program of China (973 Program-No. 2012CB725400) and a former draft is done in Tongji University, China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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A cell-based study on pedestrian acceleration and overtaking in a transfer station corridor Xiangfeng Ji a,∗ , Xuemei Zhou b,1 , Bin Ran c,a,2 a
School of Transportation, Southeast University, No. 2 Sipailou, Xuanwu District, Nanjing, Jiangsu Province, China
b
School of Transportation Engineering, Tongji University, No. 4800 Caoan, Jiading District, Shanghai, China
c
Department of Civil and Environmental Engineering, University of Wisconsin, Madison, United States
article
info
Article history: Received 28 February 2012 Received in revised form 20 November 2012 Available online 3 January 2013 Keywords: Pedestrian Cell Force Fuzzy logic Acceleration Overtaking Micro-simulation
abstract Pedestrian speed in a transfer station corridor is faster than usual and sometimes running can be found among some of them. In this paper, pedestrians are divided into two categories. The first one is aggressive, and the other is conservative. Aggressive pedestrians weaving their way through crowd in the corridor are the study object of this paper. During recent decades, much attention has been paid to the pedestrians’ behavior, such as overtaking (also deceleration) and collision avoidance, and that continues in this paper. After sufficiently analyzing the characteristics of pedestrian flow in transfer station corridor, a cell-based model is presented in this paper, including the acceleration (also deceleration) and overtaking analysis. Acceleration (also deceleration) in a corridor is fixed according to Newton’s Law and then speed calculated with a kinematic formula is discretized into cells based on the fuzzy logic. After the speed is updated, overtaking is analyzed based on updated speed and force explicitly, compared to rule-based models, which herein we call implicit ones. During the analysis of overtaking, a threshold value to determine the overtaking direction is introduced. Actually, model in this paper is a twostep one. The first step is to update speed, which is the cells the pedestrian can move in one time interval and the other is to analyze the overtaking. Finally, a comparison between the rule-based cellular automata, the model in this paper and data in HCM 2000 is made to demonstrate our model can be used to achieve reasonable simulation of acceleration (also deceleration) and overtaking among pedestrians. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Hubs, especially transfer stations, are the critical nodes of the transportation infrastructure of a city and pedestrians are the main entities in them. Therefore, it is important to understand pedestrian behaviors, which are significant for the planning and design of transfer stations, especially during peak hours when there are massive pedestrian flows. Good mathematical models can support infrastructure designers and planners to optimize their plans. Starting from 1937, different researchers with different backgrounds have developed many models to mimic pedestrian movement. According to the different descriptions of pedestrian behaviors, models of pedestrian simulation are divided into two categories, which are macroscopic models and microscopic models. Henderson (1971) [1] presented the earliest
∗
Corresponding author. Tel.: +86 25 83795356; fax: +86 25 83795356. E-mail addresses: [email protected] (X. Ji), [email protected] (X. Zhou), [email protected], [email protected] (B. Ran).
1 Tel.: +86 21 69583001; fax: +86 21 69583001. 2 Tel.: +86 25 83795356; fax: +86 25 83795356. 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.12.016
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macroscopic model for the pedestrian flow, which analyzed pedestrian flow in analogy to fluid. Hughes (2002) [2] establishes a two-dimensional macroscopic model to simulate pedestrian flow and attains the fundamental diagram. Besides, Xia et al. (2009) [3] and Dogbe (2010) [4] also attain similar results. In the domain of the microscopic models, there exist a social force model, cellular automata model and some related improved models. The social force model, first presented by Helbing and Molnar (1995) [5], took pedestrians as self-driving particles. Movements of pedestrians are decided by self-driving force, repulsive forces from other pedestrians and obstacles. Later, Helbing et al. (2001, 2005, 2007) [6–8] improved this model from different aspects. Blue and Adler (1998) [9] first used the cellular automata model to mimic pedestrian movements and they simulated unidirectional flow (1998) [9] and bidirectional flow (2001) [10]. During recent decades, studies on pedestrian behavior have attracted lots of researchers. Hoogendoorn and Bovy (2004) [11] researched the pedestrian route choice based on the utility theory under uncertainty and solved this problem with dynamic programming. Antonini et al. (2006) [12] presented the pedestrian discrete choice model, which included movement direction and movement speed. The final utility was decided by expected direction, expected speed, actual location, actual speed, and so on. Antonini et al. verified the model with measurement data. Lo et al. (2006) [13] researched the exit choice problem during the evacuation in closed space based on game theory. After this, many researchers solved the collisions problems based on game theory, such as Tanimoto et al. (2010) [14] and Asano et al. (2009) [15]. Asano et al. (2009) [15] mimicked the collision avoidance base on pure strategy Nash equilibrium and solved it by the augment allocation algorithm. Asano et al. (2010) [16] integrated pedestrian route choice and collision avoidance based on the paper in 2009. Besides, overtaking can be found frequently in reality, especially in transfer stations, where different pedestrians may have different speeds. However, studies about overtaking are implicit ones to a large extent and explicitly overtaking analysis is scarce. Thompson et al. (1995) [17] use SIMULEX to analyze the possibility of pedestrians overtaking obstacle pedestrians with two potential angles. Blue and Adler (1998) [9] use the rule-based cellular automata to analyze the overtaking possibility. Herein we call these rule-based ones implicit overtaking. Yuen et al. (2012) [18] present a modified social force model to mimic the overtaking behavior of unidirectional pedestrians and make a comparison between the original social force model and the modified one with the real data to achieve reasonable simulations of overtaking behavior among pedestrians. This one is an explicit one. All the models can be further divided into discrete models and continuous models. Generally speaking, discrete models like cellular automata allow pedestrians to move according to some rules. Herein we call them rule-based models. In contrast, continuous models like the social force model allow pedestrians to move according to forces within defined geometry. Herein we call them force-based models. From the above-mentioned discussion, studies on pedestrian flow concerns pedestrians’ behaviors much more during recent decades. Overtaking and collision-avoidance are two main aspects. In this paper, overtaking in unidirectional flow is considered and collision-avoidance and the extension to bidirectional flow are left for future research. Actually, this paper can be a building block for the complex and detailed pedestrian flow analysis. In this paper, pedestrians are divided into two categories. The first one is aggressive, and the other is conservative. Aggressive pedestrians weaving their way through a crowd in a corridor is the study object of this paper. After sufficiently analyzing the characteristics of pedestrian flow in a transfer station corridor, a cell-based model is presented in this paper, including the acceleration (also deceleration) and overtaking analysis. Acceleration (also deceleration) in a corridor is fixed according to Newton’s Law and then speed calculated with a kinematic formula is discretized into cells based on the fuzzy logic. After speed is updated, overtaking is analyzed based on updated speed and force explicitly, compared to rule-based models, which herein we call implicit ones. During the analysis of overtaking, a threshold value to determine the overtaking direction is introduced. Actually, the model in this paper is a two-step one. The first step is to update speed, which is the cells a pedestrian can move in one time interval and the other is to analyze the overtaking. Finally, a comparison between the rule-based cellular automata in Blue and Adler (1998) [9], the model in this paper and data in HCM 2000 is made to demonstrate our model can be used to achieve reasonable simulation of acceleration (also deceleration) and overtaking among pedestrians. Although some models which detail the forces between pedestrians have been presented, such as Chraibi et al. (2009) [19] and Chraibi et al. (2010) [20], our model has some differences from those as follows. Firstly, pedestrians are divided into two categories, which are aggressive and conservative. The study object is aggressive pedestrians’ weaving their way through a crowd. Secondly, the above-mentioned models are continuous models and have low calculation efficiency, while our model is cell-based and has good calculation efficiency. Thirdly, our model analyzes the overtaking behavior of unidirectional pedestrians according to updated speed and force explicitly with a threshold value to fix the overtaking direction, compared to cellular automata models. Finally, during the process of force analysis, this paper introduces a new parameter called familiarity and the anisotropy of a pedestrian is also analyzed, so the force analysis in this paper is an improved one. Therefore, this paper is a new one as far as we know. The rest of this paper is organized as follows. Section 2 mainly analyzes characteristics of unidirectional pedestrian flow in a transfer station corridor. Next, the acceleration (also deceleration) analysis model is established, which includes model analysis, force analysis, definition of forces and then speed-updating analysis based on fuzzy logic is done in Section 4. In Section 5, the overtaking analysis is made based on the force explicitly with the introduction of a threshold value to determine the overtaking direction. In the next section, a comparison between the cellular automata model in Blue and Adler (1998) [9], the model in this paper and real data in HCM 2000 is made, and some results are attained. Finally, some conclusions are obtained in Section 7.
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Horizontal
Vertical Fig. 1. Route choice in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
2. Pedestrian flow characteristics analysis Movements of pedestrians in a transfer station corridor are influenced by macroscopic or microscopic factors, a complex process interacting with the with environment. Hoogendoorn et al. [10] presented a three-level theory of pedestrian micro behaviors, which are strategic level, tactical level and operational level. This paper uses this theory in the analysis of characteristics of unidirectional pedestrian flow in corridors. (a) Strategic level Behaviors at the strategic level reflect pedestrians’ macroscopic decisions, such as buying tickets, transferring and so on. The object of this paper is pedestrian flow in a transfer station corridor. The macroscopic decision of the strategic level is to transfer. Therefore, pedestrians will transfer to the desired destination under their experience or the guidance of signs. (b) Tactical level At the tactical level, pedestrians will attain an activity chain from the activity set decided in the strategic level. In stations, the transfer behaviors will be constrained by time and space as follows. Time constraint Generally speaking, pedestrians will go to destinations directly without some unnecessary activities during peak hours. In particular, pedestrians will speed up under time pressure and the phenomenon of overtaking will appear. Space constraint Route choice of pedestrians is an important part of the tactical level. The transfer corridor can be divided into different segments vertically and in different times, pedestrians maybe walk in different segments horizontally as follows in Fig. 1. The green and red arrow lines are two possible routes in this paper. Because of a block from facilities and other pedestrians, pedestrians cannot get full information about other pedestrians and the environment. Therefore, pedestrians have to move according to their experience or following others. Route choice in this paper can also be called lane changing. (c) Operational level Behaviors at the operational level include some micro behaviors, such as collision avoidance, speed features, moving direction and so on. Moving direction When there are many pedestrians in the transfer corridor, pedestrians will feel significant pressure from behind, so moving backwards is impossible during peak hours in general. Speed features Speeds in transfer stations are somewhat larger than normal speeds. Besides, for the relatively closed space and constantly changing environment, pedestrians are likely to speed up or slow down frequently. In particular, when pedestrians feel the stress of time, they will speed up. Collision avoidance Collision avoidance mainly includes two parts, which are avoidance from other pedestrians and avoidance from obstacles. Pedestrians will move in the shortest route or in the shortest time under the premise of no collisions. 3. Acceleration analysis 3.1. Model analysis The transfer corridor in this paper is discretized into cells with side length of 0.4 m (1.3ft), which is the typical space occupied by a pedestrian in a dense crowd as in Burstedde et al. (2001) [21]. As described in Section 2, different pedestrians may have different speeds and overtaking is frequently found in a transfer corridor. Pedestrians are divided into two classes in this paper. One is aggressive and the other is conservative. Pedestrians who are aggressive are assigned with a maximum
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Fig. 2. Pedestrian moving directions or the visual angle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Different situations where pedestrians move forward 2 cells.
rate of two cells per time interval (about 0.8 m (2.6ft) per time interval); pedestrians who are conservative are assigned with a maximum rate of one cell per time interval (about 0.4 m (1.3ft) per time interval). The time interval is 0.3 s. Those who are aggressive are nearly running, and the fact that they weave their way through the crowd is the study object of this paper. Besides, pedestrians can react to the front situation and can speed up or slow down stochastically in different front density. In particular during the peak hours, there is massive pedestrian flow, so moving backwards is relatively impossible. Therefore, the moving direction at the operational level of pedestrians in this paper which is also called the visual angle in Yuen et al. (2012) [18] is shown as follows in Fig. 2 where the bold black line presents walls of the corridor and the corridor is discretized into small cell squares with side length 0.4 m. The yellow arrow stands for the desired direction, the dotted blue arrow is the visual angle in Ref. [18] and the green solid arrow is the moving direction, or the visual angle based on Assumption 1 in this paper. Assumption 1. Pedestrians move in the domain of the desired direction, including forward, upper left and upper right (stated in the moving direction of the pedestrian). That is to say pedestrians’ moving left or moving right in the abstract corridor is unmeaningful. In this paper, we call this inertia, which is pedestrians’ subconsciousness to make themselves closer to the destination in the next time interval. Assumption 2. Unlike vehicles, pedestrians can go, stop or turn in a much smaller time than vehicles. So in this paper, if pedestrians move forward two cells in one time interval, the following three situations are equivalent in Fig. 3. Generally speaking, pedestrians are inclined to choose the way of least consumption. So the probability of choosing a different moving direction during overtaking is different, which is determined by the situations around. Assumption 3. Like vehicles, pedestrians have a different reaction to stimulus from the front and behind. This is anisotropy. 3.2. Force analysis Acceleration (also deceleration) of one pedestrian is based on Newton’s law. The forces exerted on pedestrians include self-driving forces of pedestrians, repulsive forces between pedestrians, repulsive forces between pedestrians and obstacles and friction forces among pedestrians and obstacles. Besides, the force perpendicular to pedestrian moving direction cannot affect a pedestrian’s acceleration and deceleration. Therefore, force decomposed along the moving direction should be considered during the analysis of acceleration or deceleration.
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Fig. 4. Situations where friction forces exist in this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Each pedestrian has expectation to the destination, so each pedestrian has a self-driving force. However, not every pedestrian has a repulsive force and friction force. Only when one pedestrian moves close to another or an obstacle, will the repulsive forces appear. And the friction forces appear in a stricter situation. Only when one pedestrian is adjacent to another or an obstacle, will the friction forces appear and the friction forces depend on the repulsive forces. The situations where friction forces exist are as follows in Fig. 4. In Fig. 4, the green circles denote the situation where repulsive forces exist and the red circles denote the situation where friction forces exist. The model developed in this paper is actually a two step one which includes acceleration (also deceleration) analysis and overtaking analysis. The realization of overtaking depends on lane changing. Force and speed are all vectors. When two vectors are orthogonal to each other, they have no effect on each other. Therefore, during the acceleration (also deceleration) analysis, the force should be decomposed to the direction of speed, which is horizontal, and during the overtaking analysis, the force should be decomposed to the vertical direction. 3.3. Definitions of forces Qualitative analysis in Section 3.2 is not enough for the simulation. Therefore, a quantitative analysis on the pedestrian self-driving forces, repulsive forces among pedestrians and obstacles and friction forces among pedestrians and obstacles is presented. (a) Self-driving force of pedestrians The self-driving force denotes the force pedestrians exert on themselves in order to get to destinations. This paper adopts the definition in literature [5]. F⃗idr v = mi
Vi0 − ∥V⃗i ∥
τi
⃗e0i .
Where, mi denotes the mass of pedestrian i; Vi0 denotes the desired speed of pedestrian i; V⃗i denotes the real speed of pedestrian i; ⃗ e0i denotes the desired moving direction of pedestrian i; τi denotes the lag time of pedestrian i. (b) Repulsive forces between pedestrians Repulsive forces between pedestrians not only relate to the distance between pedestrians, but also relate to the relative speeds between pedestrians. The larger the distance is, the smaller the repulsive force is. Yet, the larger the relative speed is, the larger the repulsive force is. At the same time, the repulsive force is also influenced by the kindred and friendship. The more familiar pedestrians are, the smaller the repulsive force is. In this paper, this is shown with a so-called familiarity parameter ρ . The repulsive forces are calculated in this paper according to the following formula in the vector form or norm form. Where, mi denotes the mass of pedestrian i; V⃗i denotes the speed of pedestrian i; V⃗j denotes the speed of pedestrian j; Vij
⃗i denotes the position of denotes the projection of the relative speed between pedestrian i and j on the direction of ⃗ eij ; R ⃗j denotes the position of pedestrian j. In general, pedestrians have different senses to the repulsive forces pedestrian i; R from different directions and will have a stronger sense to the repulsive force from forward direction than backward. That is anisotropy, which is shown with a parameter ω in this paper. rep
F⃗ij
= −mi ρω
Vij2
∥R⃗ij ∥
⃗eij ⃗e0i .
Or,
Vij2 ⃗eij sin(θ ). ∥ ⃗ ∥ = −mi ρω ∥R⃗ij ∥ rep Fij
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Fig. 5. Angle of φ and θ .
Where, ω = λ + (1 − λ) ∗ cos(ϕ/2)(0 ≤ λ ≤ 1, 0 ≤ ϕ ≤ π ) Vij = (V⃗i − V⃗j ) ∗ ⃗ eij
⃗ij R ⃗eij = ⃗ Rij ⃗ij = R⃗j − R⃗i . R The followings are definitions of λ, ρ, ϕ and θ .λ represents the anisotropy of a pedestrian. If pedestrians are familiar with each other, the value of ρ is less than 1. The more familiar pedestrians are, the smaller value ρ is. ϕ represents a function of radian φ which takes shape from the moving direction of a pedestrian to the direction of repulsive force exerted on the pedestrian in an anti-clockwise rotation, and θ is the included angle of the direction of repulsive force andthe vertical line. Angles of φ and θ are shown in Fig. 5 and ϕ = π −φ . In this paper, the φ can be a value from the value set of 0, π4 , π2 , 34π , π .
and the θ can be a value from the value set 0, π4 , π2 , 34π , π .
(c) Repulsive force between pedestrian and obstacle Pedestrians and obstacles, such as walls, must keep a certain distance apart and the repulsive forces between them are analogous to the repulsive forces between pedestrians. That is to say the obstacles can be regarded as pedestrians with a speed value of 0. 2 Viobs
rep
F⃗iobs = −mi ω
∥R⃗iobs ∥
⃗eiobs ⃗e0i ,
Or,
2 Viobs rep ⃗ ⃗eiobs ∥Fiobs ∥ = −mi ω sin(θ ). ⃗ ∥Riobs ∥ Parameters in the above formula are similar to parameters in the formula of Section 3.3(b) (d) Friction forces among pedestrians and obstacles When two pedestrians are adjacent to each other or the pedestrian is adjacent to obstacles, such as walls, friction forces will exist. According to Hooke’s law, the friction force can be calculated according to the following formula. friction
F⃗ij
= µF⃗ij .
Where F⃗ij denotes vertical pressure or the component force of repulsive force in the direction vertical to the pedestrian moving direction. µ denotes the friction parameter. In summary, the multi-force model of pedestrian movement is based on Newton’s law, which is as formula (1). Where, rep F⃗ij denotes the repulsive force pedestrian j exerts on pedestrian i; N denotes the number of pedestrians who exert repulsive friction
rep
forces on pedestrian i; F⃗iobs denotes the repulsive force an obstacle exerts on pedestrian i; obs denotes obstacle; F⃗ik denotes the friction forces between pedestrians and obstacles; M denotes the total number of pedestrians and obstacles ⃗i denotes adjacent to pedestrian i; F⃗idr v denotes the self-driving force of pedestrian i; mi denotes the mass of pedestrian i; a the acceleration of pedestrian i. F⃗i =
N j̸=i
rep
F⃗ij
+
obs
rep
F⃗iobs +
M
friction
F⃗ik
+ F⃗idr v = mi a⃗i .
k̸=i
Actually, pedestrian mass can be removed from the above formula by dividing mi .
(1)
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Fig. 6. fuzzy triangular and membership function.
4. Speed-updating analysis Pedestrians’ acceleration calculated by formula (1) in Section 3.3 is continuous, so speeds in the next time interval determined by current speed and acceleration should also be continuous. However, the model proposed in this paper is cell-based. So in this paper, we further discretize the speed based on fuzzy logic. As is well known, the history of fuzzy logic began with the simple idea conceived by Lotfi Zadeh in the early 1960s that classes of objects that do not have precisely defined criteria of membership are prevalent when we deal with the real world. Traffic engineers have applied this theory into different domains of traffic, mainly into route choice models and travel demand. Henn (2000) [22] proposed a new route choice model taking account the imprecisions and the uncertainties lying in the dynamic choice process based on a fuzzy subset. A comparison is made between the fuzzy choice model and LOGIT model and the same results are found. Ghatee et al. [23] proposed a new traffic assignment model based on fuzzy level of travel demand and with this model, network planners are able to estimate the number of travelers on network links. During the modeling of pedestrians, Castro. et al. (2011) [24] defined a fuzzy rule that permits to identify whether an object is in danger because another could hit him and predicted the object collisions. Llorca (2011) [25] provided a collision avoidance system (CAS) for autonomous vehicles, focusing on pedestrian collision avoidance. The collision avoidance maneuver is performed using fuzzy controllers for the actuators that mimic human behavior and reactions, along with a high-precision GPS, which provides the information needed for the autonomous navigation. Definition 1. Fuzzy sets. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function µA : X → [0, 1] and µA (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X , where X is called universe, and A is fuzzy subset of X . One of the most important steps toward using fuzzy logic for problem-solving is representing the problem in fuzzy terms. This process is call conceptualization in fuzzy terms. The terms have linguistic values. Definition 2. Linguistic values, also called fuzzy labels, have semantic meaning and can be expressed numerically by their membership functions. Definition 3. The process of representing a linguistic variable into a set of linguistic values is called fuzzy quantization. Two parameters must be defined for the quantization procedure: (1) the number of fuzzy labels and (2) the form of the membership functions for each of the fuzzy labels. In this paper, the fuzzy label is pedestrian speed the membership function is shown as follows in Fig. 6. Assumption 4. Unlike vehicles, pedestrians can change their moving direction in a very short time interval, much smaller than the simulation time interval. Therefore, during the process of calculating the speed in next time interval, the direction is ignored as if the speed is a scalar and the direction of pedestrian speed in next time interval is determined in overtaking analysis. According to kinematic formula vt = v0 + at, where vt is pedestrian speed in next time interval, v0 is the current speed, t is the time interval and a is obtained by formula (1) in Section 3.3.
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Fig. 7. Fuzzy preference indices.
As aforementioned, pedestrians can react to the front situation and can speed up or slow down stochastically in different pedestrian densities in front. In this paper, this is called stochastic acceleration and deceleration and is considered implicitly in the fuzzy logic. In this paper, the method of Teodorovic et al. (1990) [26] is adopted to analyze the cells a pedestrian can move in the next time interval based on the factor vt . Their method is applied to solve the problem of traffic assignment between two alternative roads on a highway network. And this is the first time that this method has been applied to such models in this paper as far as we know. Fuzzy set of speed As schematically portrayed in Fig. 6, the vt is the factor whose definition is introduced as a fuzzy number, and then one pedestrian’s partition of input space according to his/her perception can be defined as ‘Much Less’ (ML), ‘Less’ (LS), ‘Equal’ (EQ), ‘Greater’ (GT) and ‘Much Greater’ (MG), where vtU , vtM and vtL correspond to the upper, middle and lower values of a fuzzy triangular number and where uv (vtU ) = 0, uv (vtM ) = 1, and uv (vtL ) = 0. The fuzzy triangular and membership functions for the partitions can be defined as follows.
µML (v) = µLS (v) =
1 0
vt ≤ vtL otherwise
M (vt − vt )/(vtM − vtL ) 0
vtL ≤ vt ≤ vtM otherwise
(vt − vtL )/(vtM − vtL ) vtL ≤ vt ≤ vtM µEQ (v) = (vtU − vt )/(vtU − vtM ) vtM ≤ vt ≤ vtU 0 otherwise M U M (vt − vt )/(vt − vt ) vtM ≤ vt ≤ vtU µGT (v) = 0 otherwise 1 vt ≥ vtU µMG (v) = 0 otherwise. Fuzzy preference indices On the other hand, the output space, which corresponds to the preference indices (PI) assigned to each alternative in a pairwise comparison. The preference indices (PI) are labeled as ‘Very Weak’ (VW), ‘Weak’ (WK), ‘Equal’ (EQ), ‘Strong’ (ST) and ‘Very Strong’ (VS) to represent the preference values in a pairwise comparison between any two alternatives with respect to vt . Next, we introduce the pedestrian’s preference indices P1 and P2 for any two alternatives, which represent pedestrian’s degree of preference. Obviously, P1 and P2 satisfy the following relationships. 0 ≤ P1 ≤ 1,
0 ≤ P2 ≤ 1 ,
P1 + P2 = 1.
The preference for a particular moving cell can be expressed by an adjective in this paper, such as ‘Very Weak’ (VW), ‘Weak’ (WK), ‘Equal’ (EQ), ‘Strong’ (ST) and ‘Very Strong’ (VS) as shown in Fig. 7. A set of ‘if–then’ rules that underlies the fundamental relationship between two fuzzy perceptions is established which can be used to describe the moving cell when one pedestrian is making a direct comparison as follows. If vt = ML, then P1 = VS; If vt = LS, then P1 = ST ; If vt = EQ , then P1 = EQ ; If vt = GT , then P1 = WK ; If vt = MG, then P1 = VW . Let us consider, for example, the rule ‘‘If vt = LS, then P1 = ST ’’. The implied relation between fuzzy variables vt and P1 can be expressed in terms of the Cartesian product of the fuzzy sets LS and ST . Let us denote by R the Cartesian product of sets LS and ST . The relationship is defined as follows. R(x, y) = LS × ST = min {µLS (x), µST (y)} ,
∀x, y.
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Upper - left
Upper - right
Lower - left
Lower - right
Fig. 8. Partition of space.
Fig. 9. Diagrammatic sketch of included angle.
Now we can find the value of the fuzzy variable P1 , given the fuzzy value of vt . The fuzzy set P1 is P1 = maxX
{min[µvt (x), min(µLS (x), µST (y))]}, ∀y.
The value of the preference index which suggests the pedestrian’s preference must be determined. There are two rules as to how to choose this value. One may choose the value with the highest membership grade, or the center of gravity of the membership function. In this paper, we choose the highest membership grade for calculating the value of the preference index. Once P1 is known, P2 is calculated as P2 = 1 − P1 . 5. Overtaking analysis In Yuen et al. (2012) [18], there are two main factors that should be considered in analyzing pedestrian overtaking behaviors. The first one discusses the overtaking direction like green arrows in Fig. 3 and the other one discusses the overtaking degree, which is the magnitude of the overtaking force. In Ref. [18], the overtaking direction of one pedestrian is determined by the forces in visual angle in Fig. 2. That is to say the forces behind the pedestrian are ignored. In this paper, we improve the model by including the anisotropy of pedestrians in Assumption 3. The space around one pedestrian is divided into four subspaces with a horizontal line and a vertical line in Fig. 8. The four subspaces are upper-right (UR), upper-left (UL), lower-right (LR) and lower-left (LL). Pedestrians have different reactions to the stimuli. That is anisotropy. Forces like self-driving force and friction force do not affect pedestrians’ overtaking, because they are horizontal and the overtaking appears vertically, which is perpendicular. The repulsive forces from pedestrians in the four subspaces are calculated as follows. FUR =
m
UR Frep cos(θ )
FUL =
i=1
FLR =
k
n
UL Frep cos(θ )
i =1 LR Frep cos(θ )
FLL =
i =1
l
LL Frep cos(θ ).
i=1
Where FUR is the repulsive force of pedestrians in subspace upper-right and the rest can be deduced by analogy. The repulsive force is calculated as in Section 3.2 and should be decomposed in the vertical direction. The angle θ is the included angle of repulsive force direction and the vertical n lineULin Fig. 9. m UR Therefore, Fupper = λFUL + (1 − λ)FUR = λ i=1 Frep cos(θ ) + (1 − λ) i=1 Frep cos(θ ) and Flower = λFLL + (1 − λ)FLR =
λ
LR LL Frep cos(θ ) + (1 − λ) i=1 Frep cos(θ ). Where Fupper and Flower are the resultant force of repulsive force from upper and lower, respectively and λ denotes the anisotropy of the pedestrian.
l
i=1
k
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If |Fupper − Flower | ≤ δ , pedestrians will stay on the current route; if Fupper − Flower > δ , pedestrians will choose to overtake from the lower; if Fupper − Flower < −δ , pedestrians will choose to do this from the upper, where δ is the threshold value and it is vital enough that should be carefully determined in the simulation analysis. The value of δ is not fixed and it is determined by the pedestrian’s speed in next time interval and the empty cells ahead as δ = f (speednext , cellsempty ). When the empty cells ahead can afford the speed, the value of is large enough and vice versa. 6. Simulation and analysis This cell-based model is run with pedestrians distributed randomly on a lattice 40 cells long and 10 cells wide and the periodic boundary is adopted. That is to say the cells are connected as a closed loop and curvature effect is removed. Relaxation time of pedestrians who are aggressive is 0.3 s and that of conservative pedestrians is 0.5 s. Because the simulation time is 0.3 s, that is to say aggressive pedestrians move first and conservative ones move later, which is common in reality. The desired speeds of conservative pedestrians are Gaussian distributed with mean value 1.34 m/s and standard deviation 0.26 m/s as in Helbing and Molnar (1995) [5] and those of aggressive ones are double that. The model was run for 6000 time intervals with the first 1000 time intervals to initialize pedestrian movement. 6.1. Revisit of local rules of Blue and Adler Local rules in Blue and Adler (1998) [9] are simply revisited as follows, where pn denotes pedestrians numbered from 1 to n, v(pn ) denotes the speed of the pedestrian, vmax (pn ) denotes the maximum desired speed of the pedestrian and gap(pn ) denotes the number of empty cells ahead. Anyone who is interested in the full version can refer to Blue and Adler (1998) [9]. Rule1a: IF the lane immediately to the left is beyond the lattice boundary, OR the cell immediately to the left is occupied by another pedestrian, OR the cell immediately to the left is free but the cell two lanes over to the left is within the lattice and occupied by a pedestrian, THEN assign the cell to the left to be occupied. Rule1b: IF the lane immediately to the right is beyond the lattice boundary, OR the cell immediately to the right is occupied by another pedestrian, OR the cell immediately to the right is free but the cell two lanes over to the right is within the lattice and occupied by a pedestrian, THEN assign the cell to the right to be occupied. Rule 2: IF the lane immediately to the right is assigned to be occupied, AND the cell immediately to the left is assigned to be occupied, THEN assign pedestrian pn to the current lane, ELSE go to Rule 3. Rule 3: IF there exists a gap that is uniquely maximal, THEN assign pn to that lane. Rule 4: IF there exist equal maximum gaps in two or more lanes, apply one of the three tie-breaking rules to determine assignment for pn : 4a. Three-way tie: Randomly determine lane assignment by using 80/10/10 split between current lane and two adjacent lanes. 4b. Two-way tie between adjacent lanes: Randomly determine lane assignment by using 50/50 split. 4c. Two-way tie between current lane and single adjacent lane: Randomly determine lane assignment by using 50/50 split. Rule 5: IF the current gap is less than or equal to the maximum speed for the pedestrian, THEN set the current speed of the pedestrian to the gap size [v(pn ) = gap(pn )], ELSE set the current speed of the pedestrian to the maximum speed [v(pn ) = vmax (pn )]. 6.2. Statement of local rule Each interval of the simulation is based on a two-step parallel update of the pedestrians. During the first step, value of speed which is the cells pedestrian can move in next time interval of pedestrian is fixed and during the second step, the moving direction is fixed, which also can be called overtaking direction, or lane changing. The overall rule is shown as follows. The count of interval is k. Rule 1: Set k = 1. Forces and acceleration are calculated with the formulas in Section 3.3 and the pedestrian speed, which is cells to move in next time interval are determined as in Section 4. Rule 2: Overtaking or stay in the current. 2.1 If cells ahead of pedestrian can afford the speed, the pedestrian will stay in the current lane; 2.2 If the cells ahead of pedestrian cannot afford the speed, overtaking appears. The direction of overtaking is determined in Section 5. 2.3 If both of the direction of 2.1 and 2.2 cannot afford the speed, update the speed as min(cellsahead , speedcurrent ), where cellsahead is the empty cells ahead in both the current lane and the overtaking direction fixed in 2.2 and speedcurrent is the updated speed in Rule 1. Rule 3: Collision avoidance. 3.1 If two or more conservative pedestrians walk to the same cell, or two or more aggressive ones walk to the same cell, choose one pedestrian to move according to uniform distribution; 3.2 If aggressive and conservative pedestrians walk to the same cell, choose an aggressive one to move. Rule 4 Set k = k + 1. And recycle the rule 1–3 until the maximum time intervals.
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Fig. 10.
6.3. Simulation and discussion The fundamental diagram of pedestrian flow is the target for determining the effectiveness of the cell-based model to adequately portray macroscopic behavior. Level of service criteria for pedestrian flows are defined in the HCM and are based on the relationship of three variables: space, flow rate, and walking speed. Space is the inverse of density and relates area of walkway to number of pedestrians. Space is typically measured in square feet per pedestrian. Flow rate, measured in pedestrians per minute per foot of width, is analogous to volume. Average speed is the rate of movement given in feet per minute. The fundamental relationship is given as Flow rate = average speed / space = average speed × density. Different distributions of aggressive pedestrians are simulated in this paper, from 5% to 50% with a step of 5%. And results of 15% of aggressive pedestrians fit the curves best. Fundamental diagrams are as follows in Fig. 10, where the unit is changed to feet. In the left figure of Fig. 10, abscissa axis is density which is scaled in pedestrians per square foot and axis of ordinates is speed which is scaled in feet per minute. In the right figure of Fig. 10, abscissa axis is also density and axis of ordinates is flow which is scaled in pedestrians per minute per foot of width. Fig. 10 indicates that model in this paper is a realistic representation of macroscopic flows. The maximum flow of 24.3 P/min per ft. of width is very close to the maximum capacity of 25 P/min per ft. of width indicated in the HCM 2000 [27] and 24.5 P/min per. of width in Blue and Adler (1998) [9]. However, when the density increases to a large value, flow of model in this paper decreases more quickly, because large density hinders pedestrians’ weaving their way through a crowd. So does the speed, when the density increases to a large value, speed of movement in this paper is smaller than that in Blue and Adler (1998) [9].
7. Conclusion In this paper, a cell-based model considering two kinds of pedestrians, aggressive and conservative, is proposed. Aggressive pedestrians’ weaving their way through crowd in a corridor is the study object of this paper. The acceleration and overtaking analysis is done here. The acceleration is fixed according to Newton’s law and speed calculated with kinematic formula is discretized in to cells based on fuzzy logic. The overtaking is done explicitly with force and a threshold value is introduced to determine the overtaking direction. The model in this paper appears capable of representing the pedestrians’ behavior, including acceleration and overtaking and the rule set in this paper can capture the aggregate behavior in a reasonable distribution of aggressive pedestrians, compared to the model in Blue and Adler (1998) [9] and the real data in HCM 2000. The model in this paper can be a building block for more complex and detailed models, such as models for bidirectional pedestrian flow with explicit acceleration and overtaking analysis and models combining explicit acceleration, overtaking and collision analysis. Besides, some parameters are presented here, and a sensitivity of these parameters can also be done in the future work. The proposed model here can be an expansion of the theoretical models in pedestrian flow analysis. In practical use, it can evaluate the design of the corridor and also can be guidance for the design and construction of the corridor.
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Acknowledgments Special thanks go to the anonymous reviewers for their great suggestions. This study is supported by the National Basic Research Program of China (973 Program-No. 2012CB725400) and a former draft is done in Tongji University, China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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