Abstract
The avoidance behavior of pedestrians was characterized in the present paper by simulating the movement of crowds in both unidirectional and bidirectional pedestrian flow. A phase change of alternative lane formation observed in real bidirectional pedestrian flows has been studied, where pedestrians tended to evade individuals in counterflow and simultaneously keep a certain distance from each other in the uniform pedestrian flow when the counterflow disappeared. What is more, the comparison between the effect of evading and pushing behavior on evacuation has been investigated in the room egress scenario. Additionally, the evading and overtaking behavior of fast pedestrians have also been simulated in heterogeneous crowds. The performance of the proposed model was compared to the experimental data and the results obtained using other evacuation models. Numerical results showed that both the phase change of alternative lane formation in bidirectional pedestrian flow and the effective evading behavior in unidirectional pedestrian flow were conductive to reduce the evacuation time of pedestrian crowds. Even though pushing behavior of fast pedestrians seemed to improve the flow through the wide exit, it might lead to the panic and other negative effect on the crowds, such as crowds trample. The proposed model in this paper could provide a theoretical basis for the pedestrian crowd management during emergency evacuation.
1. Introduction
Nowadays, the gathering of pedestrian crowds becomes widespread in public places, such as stations, stadiums, shopping malls, etc. [1]. Thus, understanding human behaviors and defining behavioral rules in emergency conditions are essential issues in human safety assessment [2].
A wave of models of pedestrian dynamics, inspired by pioneering models, such as the social force model [3] and the cellular automaton model [4], has been developed in the past years. For example, the combination of an interaction force between pedestrians and the navigation force of the guider was introduced in an extended social force model to study the influence of guiders in the case of emergency evacuation [5]. And the effects of guider type, guider number, guider distribution, and strategy on evacuation were discussed in a multigrid cellular automaton model [6]. In order to reduce congestion time, a modified two-layer social force model was proposed to simulate a group gathering process recoded in video studies [7]. And the concept of panic coefficient was put forward in order to depict the effect of pedestrian panic psychology on the evacuation process [8]. In addition, a thermostatted kinetic theory [9] was newly employed for describing the pedestrian crowd dynamics into a metro station with multiple exits. The external force field was coupled to trigger the evacuation conditions in the nonequilibrium crowd systems under panic conditions [10, 11]. A preference of pedestrian for a specific gate among exits in the metro station could be determined by a constant nonuniform external force. And the pedestrian’s distributions at each exit were analyzed in different conditions. However, the interaction between individuals during their way to the destination could not be understood well, especially in a heterogeneous crowd. For example, how is about the collision avoidance behavior in the case of one pedestrian dodging an arrested one in the traffic.
The collision avoidance mechanisms in the pedestrian crowds has been widely studied since it is one of critical factors that drive both individual behavior and the formation of collective patterns in the crowd, such as overtaking behavior in unidirectional flow [12] and the lane formation phenomenon in counterflow [13].
Overtaking is a common phenomenon in reality [14], where pedestrians with high speed tend to evade and overtake other pedestrians with low speed [15]. Besides, a personal preference of the direction to evade each other on the left-hand side or on the right-hand side is also required for achieving the effective overtaking [16]. For example, the ‘social norm’ (in Japan) of avoiding on the left and moving on the left when overtaken to give space to the individual who has the overtaking play was implemented by rotating the pedestrians’ velocity vector counterclockwise by a certain angle [17]. The choice of the direction could also be conducted by considering the combined influence from the surroundings [12] or directly picking the shortest path to the target [18].
Lane formation of pedestrians is a well-known pedestrian collective phenomenon where individuals avoid the occupants from the counterflow and share the available space by forming lanes of uniform walking directions [19], which could enhance traffic efficiency by reducing the need for avoidance strategies among individuals in head-on configurations [20]. Various efforts [21–24] have been conducted to analyze the lane formation phenomena [25] in pedestrian counterflow, considering the impacts of psychological influence [26], completion and cooperation [27], and swapping behavior [28] as well as the bottleneck in a corridor [29]. The following effect and evasive effect were considered to mainly contribute to the lane formation [30, 31]. And it was found that avoidance behavior is more relevant [32].
The empirical observations in this work showed that pedestrians in bidirectional pedestrian flow tend also to evade other occupants in the same unidirectional flow and keep a certain distance from them when there is sufficient walking space available. Thus the pedestrians experience spatial separation, aggregation, and reseparation of alternative lanes through individual interactions with the counterflow. These repetitious phase changes are commonly found in real bidirectional pedestrian flows in reality and thus merit the further study. That dynamic collective pedestrian pattern derived from the collision avoidance, however, gained less attention from all above models.
In this work, we consider a corridor in a real situation with a bidirectional pedestrian flow. The lane formation and the phase change of the pedestrian lanes have been simulated. Additionally, the proposed model could also be used to describe the evading and overtaking behavior in a nature way by the introduced collision avoidance mechanism.
2. Establishment of the Model
2.1. Movement Model
The movement of agent is generally relevant to the freedom of both translational degrees and rotational degrees in reality. The social force model and shoulder rotation model [33, 34] were employed as the starting point for pedestrian movement presented in this work. The motion of a pedestrian can be described by the combination of a driving force, that reflects the pedestrian’s internal motivation to move in a given direction at a certain desired speed, and repulsive forces describing the effects of interactions with other pedestrians and boundaries such as walls or obstacles in streets [16]. And simultaneously, the target motive angle of the body is also changed so that the agent tries to move shoulder along the wall and the crowd tried to avoid colliding with the agent [33]. The position and movement of each pedestrian evolve depending on the equation of motion:where is the position of pedestrian . The velocity of pedestrian at time is given by . is the mass of pedestrian . The pedestrians’ internal motivation to walk towards their target at the desired walking speed is reflected by the motive force:
where is the velocity of pedestrian and the relaxation time parameter sets the strength of this motive force. The repulsive forces and in (1) describe the effects of interactions with other pedestrians and obstacles, respectively. is a small random fluctuation force. The agent-agent interaction force has two parts, the agent-agent contact force and the agent-agent social force. The contact force turns on only when pedestrians are in contact considering both the elastic force and frictional force. The social force is used to keep reasonable distances from other pedestrians. And the agent-wall interaction force could be treated similarly.
In the shoulder rotation model, agents try to avoid collisions by adjusting their walking directions and by rotating their bodies to move shoulder first. Equations (1) and (2) describe the translational degrees of freedom of the agents, and the rotational degrees of freedom could be treated similarly. The elliptical cross-sectional shape of the agent body is approximated by three circles, and the rotational function of the elliptical body is correspondingly added [34]. Each pedestrian has his own rotational equation of motion given bywhere , , and are, respectively, the torques of the motive, contact, and psychological social forces. is the moment of inertia, and is the body angle. By the combination of the social force model and the shoulder rotational model, pedestrians can change their body angles in response to the interaction forces in (1) and then adjust their body angle to face the target again according to the motive torque. More details regarding the interaction forces and the corresponding moments are found in [33, 34].
2.2. Collision Avoidance Model
The collision avoidance model proposed in this work is a modification of an optimization-based overtaking model [14] by considering the additional effect of counterflow in the crowds. A short-range area is used as the visual area for each agent, which let pedestrians receive information regarding their surroundings before the decision-making of collision avoidance. The area is divided into three overlapping sectors , , and (see Figure 1), each covering a wide sector around a desired walking direction. The vectorial direction set is , pointing to the left side, straight ahead, and to the right side, respectively. Straight ahead means always the direction of the preferred velocity in (2). Thus, the pedestrians can detect the occupancy in their visual areas and choose the optimal walking direction in accordance with the sector with the fewest occupants.
We assume that any occupant in the crowds within a sector will decrease the score of that sector:where is the score of sector with . is the score corresponding to occupants of the same unidirectional flow, and score corresponds to occupants of the bidirectional pedestrian flow in the crowds.
The score depends on the location and moving velocity of each occupant in the same flow:where is a constant. The inner products can be used to weight the scores of the sectors depending on the velocities the agents. In our model, each occupant in a sector reduces the score of that sector, such that . The absolute value of the inner products is used to ensure that influence on the unidirectional pedestrian flow. is the velocity of pedestrian . is the vectorial direction of the desired velocity . And is the skin-skin distance between agents and . is employed in the denominator to avoid an undefined value. in (5) is introduced to describe the human sensitivity to the occupancy of each sector. Here, is a reference speed and is the sensitivity parameter. is a parameter to help decide whether to prefer the detour or the shortest path. For instance, for a pedestrian, who therefore gains a lower score with when , and evades the front sector by choosing another higher score sector. On the other hand, an agent with tends to neglect the impact of the neighboring pedestrian agents in the front sector.
Similarly, each occupant of the counterflow within a given sector decreases the score of that sector, so that the current agent could avoid them by choosing the direction with less counterflow:where and are constants. And in this way, pedestrians could evade the occupants in the counterflow.
Additional, the direct-path preference [18] and the right-hand preference [35] were also considered in this collision avoidance model for direction and , respectively.where is the direct-path score with a positive value for the direction. And is a constant. The symbol represents a binary variable, equal to one if and zero otherwise. , i.e., , is used to specify the straight-ahead direction .where is the score for the left or right direction. and are used to activate the walking directions and , respectively, in a manner similar to . is a parameter used to assign the side preferences of pedestrians. For example, ()>0 holds for pedestrians who prefer the right-hand side, and vice versa. Thus, the preferences of pedestrians for the binary directions (i.e., the left or right side) also depend on their velocities and parameter.
Depending on the combination of the utility in each sector and the sector scores based on the direction preferences, the optimal walking direction with the highest total score, , will be selected from among the three directions , times per second on average. , and are used to weight the trade-off among the three evaluation criteria, where .
3. Case Study
3.1. Dynamic Lane Formation in the Counterflow
3.1.1. Pedestrian Flow Pattern in Reality
Figure 2 shows the pedestrian lane formation phenomenon recorded in a real bidirectional pedestrian flow. The measurement area recorded in the video is a corridor in a shopping mall in China. The data were collected with a HERO5 Black camera (firmware version $‘HD5.02.01.50.00’$). Empirical observations show that pedestrians experience spatial separation, aggregation and reseparation of alternative lanes through individual interactions with the counterflow. Pedestrians tend to evade the pedestrians from the counterflow and try to maintain a certain distance from pedestrian in the same unidirectional flow before and after the interaction with the counterflow.
The performance of the proposed model was compared with the pedestrian flow patterns recorded in a real situation and with the simulation results obtained using a counterflow-based active decision model [31], i.e., the Evac model.
The two models were initialized with the geometry and initial positions of the pedestrians and their individual walking velocities as captured by the camcorders. The walking velocity of each pedestrian was determined by dividing the total distance travelled by the time taken. We specified the parameters of the proposed model as follows: the parameters of the extended social force model were chosen in accordance with [33].
For the collision avoidance model part, the constants , , , and were used to calculate the utility of waking space in each sector. The sensitivity was represented by the parameter and the reference velocity . Since the walking speed of each pedestrian involved in the counterflow (see Figure 2) to be studied was generally less than (around ), the parameter was equal to to achieve . Additionally, the constants for the direction preferences were set to and . The weighting parameters were specified as , , and .
3.1.2. Simulation Results
The pedestrian flow patterns of lane formation in the bidirectional pedestrian flow were obtained using both the proposed model and the Evac model [31]; see Figures 3 and 4, respectively.
The observed pedestrian lane formation behavior was reproduced by both models. However, the pedestrian flow pattern of alternative pedestrian lane formation in the bidirectional pedestrian flow after the interaction of the counterflows that was predicted by the proposed model agreed better with that recorded in reality. Pedestrians in a uniform pedestrian flow tended to evade other pedestrians ahead of them while keeping a certain distance from each other when there was sufficient walking space. By contrast, in the Evac model, pedestrians preferred to follow agents travelling in the same direction and thus formed a single-file line.
The simulation results obtained with the two counterflow models are also shown in Figure 5 and Table 1 in the form of pedestrian flow rates calculated as the time required for a given number of pedestrians to walk rightward through the corridor. The root-mean-square deviation RMSD [12] was employed to perform a quantitative analysis of the proposed model. As displayed in Figure 5, red circles indicate the experimental data recorded by the camera. Black pluses indicate the simulation results obtained with the proposed model. Blue triangles indicate the simulation results of the Evac model. The results of each approach were averaged over 20 simulations, and the bars represent the standard deviations.
It could be found from the comparison among the results in Figure 5 that the pedestrians’ leaving times as predicted by the two models and observed in the real situation are similar when there is only one pedestrian leaving the corridor. However, the deviations of the leaving times predicted by the two counterflow models relative to the real situation increase with an increasing number of leaving pedestrians. Generally, the trend in the pedestrian leaving times predicted by the proposed model was closer to the real situation.
One model can be considered to be quantitatively more accurate than another if its prediction error RMSD [12].
where is the total number of pedestrians and and are the real and modelled leaving times, respectively, of the pedestrian. The values and the total leaving times of pedestrians leaving from the right end of the corridor were summarized in Table 1. Results in the table showed that, in comparison to the Evac counterflow model, the proposed counterflow model achieved results closer to the real observations, with a small prediction error of RMSD=0.3. It thus implied that the proposed model can be considered to be quantitatively more accurate than Evac counterflow model.
Through the above combination of qualitative and quantitative analyses (Figures 3 and 5 and Table 1), it may thus be concluded that the proposed model can well predict the lane formation phenomenon and the observed phase change. Additionally, it could be found in Figure 5 that the phase change of the lane formation could contribute to less evacuation time of pedestrians; on the other hand, the herd and pushing behavior led to more evacuation time by unrealistic collision among individuals when there was enough walking space.
3.2. Evading Behavior and Pushing Behavior
Generally, individuals try to find more walking space available to evade other pedestrians in front in normal circumstances or push them directly for quick egress in an emergency situation. These behaviors commonly affect apparently evacuation results in different ways. In this section, varying combinations of weighting parameters and have been used to describe the overtaking and pushing behaviors of fast pedestrians in the heterogeneous crowds, and the corresponding effects on evacuation have been discussed. With big and small , fast pedestrians could give priority to overtaking others by avoiding the collision. On the other hand, fast pedestrians with small weighting parameter and big parameter prefer going forward and pushing occupants in front directly. The first scene simulated heterogeneous pedestrian crowd evacuation from a room with a single exit; see Figure 6. The individuals marked in black color were fast pedestrians who walked at higher walking speed; on the other hand, the ones marked in red color were slow pedestrians.
Given the fact that the limited flow could be due to a forced capacity reduction, i.e., the limit bottleneck width, the egress simulations were conducted in the room with different exit widths. The number of pedestrians who crossed the exit over time was recorded; thus the flow of the pedestrian crowds through the room exit could be calculated by [36].
The total number of pedestrians in the heterogeneous crowds simulated was and the proportions of fast and slow pedestrians in the crowds were the same. Walking speed 1.25m/s and 0.76m/s [37] were chosen for the fast and slow individuals respectively in the room evacuation simulation. It is reasonable to assume that the pedestrians with high walking speed (e.g., higher than 1.0 m/s) prefer to evade and overtake the obstructions in front of them. On the other hand, pedestrians with lower speed (e.g., lower than 1.0 m/s) prefer the right-hand traffic and moving straight ahead without much detour in a normal walking situation. Thus, parameter was equal to in the simulations so that could be used to depict the heterogeneity in the crowds [14]. Based on the above assumptions, the flow of the heterogeneous pedestrian crowds was obtained through different exit widths with different sets of weighting parameters , , and ; see Figure 7.
As illustrated in Figure 7, the flow value predicted by the proposed model with different sets of weight parameters displayed the increment with the increasing width of the exit. And the higher degree of the pushing behavior in the heterogeneous pedestrian crowds, such as in the condition of , leaded to a lower pedestrian flow rate in the condition of narrow exit width, such as when the width was less than . That is because the pushing behavior leaded to the strong interactions among fast and slow individuals, which cause easily the crowds jamming near the narrow bottleneck. However, high degree of pedestrians pushing behavior surprisingly caused a higher flow in the room with a wide exit. The reason may be that some slow movers pushed by fast pedestrians were forced to accelerate up to a higher speed even than their own desired speed.
Figure 8 showed that the intensive interactions clearly among the individuals near the exit occurred with a low density, where the slow individuals may even be push out of the exit, or slow mover might have to wait behind the exit due to the strong repulsive effect from fast pedestrian flow. It could be found that the slow pedestrian marked with the rectangle could only left the room much too late, who was just initialized next to the door at the beginning of the evacuation. The simulation results were in accordance with the previous effort [38] that pedestrians hinder each other due to strong competition for the unoccupied target sites near the exit and an ordered outflow was inhibited. Thus, even though the kind of offensive pushing behavior in simulation improved the flow rate through the wider exit, it might lead to the panic and other negative effect on the crowds, such as crowds trample, due to the strong interactions among individuals in the heterogeneous crowds.
3.3. Evading Behavior and Overtaking Behavior
Pedestrians with a higher walking velocity are accustomed to overtaking other pedestrians with a lower walking velocity to maintain their own desired walking velocity [12]. Thus we assume that fast pedestrians prefer to evade the individuals walking slowly in front, especially in sparse pedestrian crowds. The scenario in which we performed the analyses was a group of 6 individuals in a sparse heterogeneous pedestrians group in the room with exit width ; see Figure 9. In order to achieve more feasible possibility for the real application, the fast pedestrians have been initialized behind the group of slow pedestrians in the room. Six pedestrians aligned in a matrix group and the walking direction of the unidirectional pedestrian flow in the room was from left to right. The parameters of the proposed model and the types of pedestrians introduced in the above section were kept. But the set of the weighting parameters and was employed. The initialized positions of the pedestrians were described by the horizontal distance between pedestrians in a line. The vertical distance between the fast and slow pedestrians in the alternative column was . And the distance between the fast pedestrians marked in black color and the wall was also fixed.
The simulation results were obtained under the proposed model in comparison with two previous pedestrian evacuation models, Evac model [31] and the social force model which referred to the movement model introduced above, as shown in Figures 10 and 11. The graph displayed the number of individuals left the room over time in different scenarios with different horizontal space and , respectively.
The simulation results in Figures 10 and 11 showed that the time spent by first three pedestrians, i.e., the fast pedestrian group under the proposed model in this work, was less than that by the two other pedestrian evacuation models. Especially, the evacuation time in Figure 11 cost by the first three pedestrians evacuated was obviously shorter when . Thus, the comparisons demonstrated that the 3 fast pedestrians governed by the proposed model with weighting parameters and in the sparse heterogeneous pedestrian crowds could evade and overtake the slow occupants in the front effectively when there was enough walking space. And the pushing behavior and the strong interactions among individuals in emergency situation, see in Figure 8, have been avoided in the normal situation. On the other hand, in the other two models, the pedestrians with lower walking velocities were pushed by the fast movers behind them frequently and thus move somewhat faster than their desired walking velocity. That was also the reason that the 3 slow pedestrians governed by the proposed model in this work showed even longer time for completing the egress than that by the 3 slow pedestrians under other models. That is, the proposed model with the introduced collision avoidance could avoid unreasonable collision in the sparse heterogeneous crowds.
Additionally, the comparison between the results in Figures 10 and 11 showed that the longer horizontal distance between the slow pedestrians provided more consecutive spaces available for fast pedestrians at perpendicularly beside and the diagonally in front. Therefore, the fast pedestrians could maintain high mobility by evading and overtaking the slow pedestrians in the front with fewer interactions. As shown in Figure 12, the fast pedestrians under the proposed model started to evade actively and overtake the slow pedestrians at time instead of following or pushing them. Consequently, the effective evading and overtaking behavior contributed to more competitiveness of the fast pedestrians when there was enough walking space. And the simulation results showed that the evading and overtaking behavior of the fast individuals are strongly affected by the formation of the pedestrians in the front, i.e., the walking space available among and around them.
4. Conclusions
In order to demonstrate the characterization of collision avoidance in pedestrian crowds, this paper proposed a pedestrian evacuation model based on the empirical observations to reproduce the dynamic pedestrian lane formation in the counterflow, the phase change of the lane formation, pushing behavior, and overtaking behavior in the heterogeneous crowds. The results revealed that the combination of pedestrians’ evading behaviors and direction preferences can be used to simulate effectively the different collision avoidance behaviors which contributed to the pedestrian dynamic and self-organization phenomena in both unidirectional and bidirectional pedestrian crowds.
Since the phase change of the lane formation could reduce the evacuation time of the crowds and improve the evacuation efficiency, accounting for the phase changes of alternative pedestrian lanes in the pedestrian counterflow can improve the overall prediction of crowd movement in pedestrian flow. Furthermore, simulation results showed that pedestrians hinder each other due to strong competition for the unoccupied target sites near the exit and an ordered outflow was inhibited. Additionally, the evading and overtaking behavior of fast individuals were strongly affected by the formation of the pedestrians in the front, i.e., the walking space available among and around them. What is more, even though the offensive pushing behavior might improve the flow rate through the wide exit, it also leaded to the strong interactions among individuals and may cause the crowds panic and other negative effect on the crowds, such as crowds trample. The results showed that the proposed model might provide the more reasonable evacuation strategies for pedestrian facilities and serve as a reference for facility design.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Tis research was sponsored by the National Natural Science Foundation of China (Grant no. 51509060) and Special project from China Ministry of Industry and Information Technology (Grant no. KY10100170137).