Thousands of people died and more were injured in crowd stampedes over the past 100 years around the world.^[1] Cities are densely populated nowadays and the collective behaviors seem to occur more frequently, such as traveling by subway, shopping in malls, and watching sports games in stadia. Hence, how to guarantee the safety and comfort of pedestrian motion and improve the efficiency of public space has become an urgent problem to be solved.

Pedestrian dynamics refers to the study of how and where crowds form and move, which usually studies the dynamic features of pedestrian flows. The study on pedestrian dynamics has attracted considerable attention of researchers in fields of physics, transportation, and more. Some good pedestrian models have been put forward with the advance of research. In general, pedestrian models are divided into three categories: (i) macroscopic models, where pedestrians are studied as a group without considering the interactions among pedestrians and pedestrian flows are analogous to gases and fluids in some sense;^[2] (ii) microscopic models, where every pedestrian is treated as an individual and the interactions among pedestrians are considered;^[3] (iii) mesoscopic models, which inherit from both macroscopic models and microscopic models.^[4] The overview of some representative pedestrian models can be found in Refs. [5]–[7]. Pedestrian models can describe the individual and group behaviors and reproduce the empirical scenarios, so they can help researchers to analyze the walking behaviors and dynamic features of pedestrian flows. In the previous investigations, arch-like blockings in bottlenecks, lane formation in bidirectional flows, and other self-organization phenomena have been observed and studied by various approaches.^[1,8] Furthermore, the escape panic^[9,10] and the emergency evacuation process under normal conditions and from a smoke-filled or afire room^[11–15] have been studied by using multiple pedestrian models.

Recently, intersecting flows become one of the topics of most concern because they are practically unavoidable and likely to cause clogging and accidents. The stripe formation is observed when two pedestrian flows intersect, but no stable pattern exists when three or more pedestrian flows intersect.^[1,8] Zhang et al. proposed a two-dimensional model based on the cellular automata model in order to simulate pedestrians’ crossing behaviors and the simulation results showed that this model can be applied successfully.^[16] Saegusa et al. studied the intersecting flows under a periodic boundary condition by using the lattice gas model, and found that the overshooting in the fundamental diagram is attributed to the unstable congestion.^[17] Guo et al. put forward a new microscopic model which is similar to the discrete choice model, and conducted several real pedestrian experiments with different intersecting angles to calibrate it.^[18] Besides, pedestrians’ avoidance behaviors in crossing flow were investigated in Ref. [19].

On the basis of the previous studies, little research has been reported about the dynamic features of intersecting flows with different cross angles, so we aim to find the direction of the stripes, the influence of the cross angles on dynamic features, and the moving details of pedestrians at intersections in this paper. The model we choose is the social force model, which is a continuous microscopic model. It is universally accepted that self-organization phenomena are the consequence of interactions, which are considered in the social force model. By using the social force model, we can reproduce the stripes formation successfully. To reflect the effect of the cross angles, we set up several simulation scenarios in which only the cross angles are different, and then investigate the mean speeds and the fluctuations of speed. By comparing the mean speeds and the standard deviations, we can obtain whether the cross angles affect the motion of pedestrians and what the effects are. As for the moving details, we adopt the approach that picks some individuals and then snapshots them for each second in a time period when they are at intersections, then we find the common regularities.

The organization of this paper is as follows. In Section 2, we will introduce the social force model. Section 3 presents the simulation scenarios. Section 4 shows the simulation results and their analysis. Section 5 is the conclusion summarizing the whole paper.

Based on socio-psychological studies that both socio-psychology and physical forces affect the motion of pedestrians, Helbing et al. introduced the social force model to investigate the pedestrian movement, which is particularly suited for describing the occurrence of pressure in crowds.^[9]

The social force model considers the influence of the neighboring pedestrians and the environment on individuals’ movement in crowds. Pedestrians are driven by: the desired force , the interaction force f_ij between pedestrians i and j, the interaction force f_iW between pedestrian i and wall W. The mathematical formula of the social force model is expressed as

Generally, pedestrians have certain destinations, so each pedestrian has a desired velocity and a desired direction. In practice, the actual speed and the desired speed are different due to the interference of obstacles and other neighboring pedestrians. The desired force expresses pedestrians’ willingness to achieve the desired velocity. Pedestrian i likes to move with a certain desired speed in a certain direction , and therefore tends to adapt his or her actual velocity v_i with a certain characteristic time τ_i. So the desired force is given by

Pedestrian i prefers to keep a distance from other pedestrian j, so there exists an interaction force f_ij. This force describes not only the psychological tendency to stay away from others, but also the physical force that occurs when the pedestrians touch each other. The interaction force between pedestrians i and j is given by

When d_ij is smaller than r_ij, the pedestrians touch each other and compressions emerge. In this situation, there exist two extra forces. The term kg(r_ij − d_ij)n_ij reflects the body compression force, which can neutralize the compression of the bodies. The term reflects the sliding friction force, which blocks the tangential motion. k and κ are large positive constants.

The interaction force between pedestrian i and wall W is treated analogously as

The subway stations, malls, stadia, or other public places are likely to show intersecting pedestrian flows. Intersections of pedestrian flows are one of the major problems that affect the safety and efficiency of pedestrian movement and impossible to avoid in practice, so it is meaningful to know how people behave at intersections and the effect of the cross angles.

Real-life experiment is unexpected and dangerous, so there is an urgent need to set up a reliable model to simulate the pedestrian motion. The parameters in the social force model are specified in Table 1, which is referred to in Ref. [9].

Table 1.

Table 1.

Table 1. Parameters of the social force model. . Symbol Meaning Value m pedestrian mass 80 kg A strength of repulsive interaction 2000 N B range of repulsive interaction 0.08 m k coefficient of body compression 1.2 × 105 kg⋅ s−2 κ coefficient of sliding friction 2.4 × 105 kg ⋅ m−1 ⋅ s−1

Table 1. Parameters of the social force model. .

We study the intersecting flows which consist of two components of pedestrians moving into an area whose size is 50 m × 50 m. The schematic diagram of the simulation scenarios is shown in Fig. 1. Three simulation scenarios are built with different cross angles, which are 60°, 90°, and 120°, named scenario 1, scenario 2, and scenario 3, respectively. For each scenario, the flow consists of two components of pedestrians; one component moves from west to east, and the other component moves from south to northeast or north or northwest. We use red hollow circles to represent the pedestrians going eastward and blue hollow circles to represent the pedestrians who walk in the other direction.

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of the simulation scenarios. Hollow circles of different colors represent pedestrians walking in different directions, which are e1 and e2. θ represents the cross angle. The dashed square in the center of the area is the statistical area.

In each scenario, the initial speed of pedestrians on both sides is 0. On each side, the distance between pedestrians’ initial positions and the entrance is 1 m, which is designed in order to allow the pedestrians to enter the area with their normal walking state. The width of pedestrians’ initial position line is 4 m, that is, the initial positions are (−1 m, 13–17 m), (13–17 m, −1 m) in scenario 1, (−1 m, 23–27 m), (23–27 m, −1 m) in scenario 2, and (−1 m, 13–17 m), (23–27 m, −1 m) in scenario 3. Also, the initial number of pedestrians and the inflow rate of the two streams are exactly the same in the three scenarios. The initial number is 50, and the inflow rate is set to 5 p/s. The positions and speeds of pedestrians are updated after each time step, and time step ΔT is set to 0.01 s.

The parameters of the social force model in these scenarios, namely, the desired speed, reaction time, and body size, are identical. According to Ref. [20], the parameters are set as follows. Reaction time τ satisfies a uniform distribution between 0.2 s and 0.5 s with interval Δτ = 0.05 s. The desired speed is subject to a normal distribution, the mean of which is 1.35 m/s and the standard deviation is 0.2. The radius uniformly distributes between 0.2 m and 0.3 m with interval Δr = 0.025 m. The parameters are presented in Table 2.

Table 2.

Table 2.

Table 2. Parameter settings in different scenarios. . Simulation scenario Reaction time τs Time interval Δτ/s Desired speed (m/s) Standard deviation σv Radius r/m Radius interval Δr/m Scenario 1 0.35 0.05 1.35 0.2 0.25 0.025 Scenario 2 0.35 0.05 1.35 0.2 0.25 0.025 Scenario 3 0.35 0.05 1.35 0.2 0.25 0.025

Table 2. Parameter settings in different scenarios. .

The simulation time of each scenario is 150 s, and each scenario runs 10 times.

To ensure the accurate analysis of dynamic characteristics at intersections, only the data collected when the two streams of pedestrians intersect completely are used, so we define a statistical area of size 10 m × 10 m, which is the central dashed square shown in Fig. 1. The selection of the location of the statistical area is essential. The coordinates of the statistical area’s vertexes are selected as follows: In scenario 1, S₁ (23 m, 22 m), S₂ (33 m, 22 m), S₃ (33 m, 12 m), S₄ (23 m, 12 m); in scenario 2, S₁ (20 m, 30 m), S₂ (30 m, 30 m), S₃ (30 m, 20 m), S₄ (20 m, 20 m); in scenario 3, S₁ (11 m, 20 m), S₂ (21 m, 20 m), S₃ (21 m, 10 m), S₄ (11 m, 10 m). Only the data in the statistical area are considered. The dynamic characteristics of intersecting pedestrian flows can be analyzed by the mean speed and the fluctuation of speed. According to the observations, stripes are formed after 20 s, so we calculate the mean speed in the statistical area from 40 s to 150 s when the stripes are basically stable.

By using the social force model, we have investigated the direction of self-organized stripes, presented and analyzed the dynamic features of the intersecting flows with three typical cross angles, and found the way that pedestrians move at intersections. We have confirmed the empirical observations in simulations that pedestrians form stripes in a self-organized way when two pedestrian streams intersect and the direction of the stripes is perpendicular to the sum of the directional vectors of the two pedestrian flows. The simulation results show that greater cross angle results in more conflicts which are characterized by low mean speed and higher level of fluctuation. Greater cross angle is more likely to cause blockings and have adverse effects on the passing of pedestrians. What is more, we found that pedestrians move with stripes and can extend along the stripes. This phenomenon is regarded as an optimal way to pass the intersections quickly.