Inflow process of pedestrians to a confined space

To better design safe and comfortable urban spaces, understanding the nature of human crowd movement is important. However, precise interactions among pedestrians are difficult to measure in the presence of their complex decision-making processes and many related factors. While extensive studies on pedestrian flow through bottlenecks and corridors have been conducted, the dominant mode of interaction in these scenarios may not be relevant in different scenarios. Here, we attempt to decipher the factors that affect human reactions to other individuals from a different perspective. We conducted experiments employing the inflow process in which pedestrians successively enter a confined area (like an elevator) and look for a temporary position. In this process, pedestrians have a wider range of options regarding their motion than in the classical scenarios; therefore, other factors might become relevant. The preference of location is visualized by pedestrian density profiles obtained from recorded pedestrian trajectories. Non-trivial patterns of space acquisition, e.g., an apparent preference for positions near corners, were observed. This indicates the relevance of psychological and anticipative factors beyond the private sphere, which have not been deeply discussed so far in the literature on pedestrian dynamics. From the results, four major factors, which we call flow avoidance, distance cost, angle cost, and boundary preference, were suggested. We confirmed that a description of decision-making based on these factors can give a rise to realistic preference patterns, using a simple mathematical model. Our findings provide new perspectives and a baseline for considering the optimization of design and safety in crowded public areas and public transport carriers.


I. INTRODUCTION
Human crowding in public areas is still a significant issue in social and engineering science [1,2]. To design comfortable and safe urban spaces, it is important to understand the nature of the interactions among pedestrians and their consequences. However, due to the lack of such knowledge, many inefficient or even disastrous situations are still found, e.g., serious overcrowding in transportation systems in urban areas and crowding during disasters, leading to a stampede. Meanwhile, relatively recent attempts have partly succeeded in modeling the collective phenomena of pedestrians [3][4][5][6], including spontaneous lane formation in a bidirectional flow [7][8][9][10], crowd turbulence in overcrowded areas [11,12], and formation of stripe patterns in intersecting flows [5,13]. In addition to experimental [9,[14][15][16][17][18] and empirical [19][20][21] studies, several types of mathematical models that facilitate simulations and analyses of central phenomena have been developed (see [3][4][5][6] for reviews). Such approaches have also contributed to the understanding of how walking people recognize and react to their environment.
In this paper we study the inflow process of pedestrians that was recently proposed as a model process for understanding the passenger entrance behavior to elevators, buses, trains, etc. [22,23]. The inflow process is defined by successive entry of pedestrians into a confined area and their subsequent dwelling. While entering, each pedestrian performs a complex decision-making process perceiving the current situation in the room (e.g., the distribution of persons), anticipating the behavior of subsequently entering persons and planning to the exit from the room. These stimuli and knowledge are evaluated for the decision, taking into account social conventions. Among pedestrians, there is competition to optimize the final location with respect to easy and fast exit under the limitation for avoiding overcrowding to secure comfort. Classical studies of pedestrian dynamics focused on motion in corridors and bottlenecks to investigate flow capacities or jamming at high densities [3][4][5][6]. In such scenarios, the decision process is negligible, because pedestrians are not given the freedom of choice (e.g., changing the direction or their locations) because of predefined destinations in the setup of the experiment or restrictions on the motion at high densities. In the inflow process, no predefined destinations are given. Our goal is to reveal the modes of interactions among pedestrians that become visible through the decisions. The distribution of pedestrians' positions could result from the interplay among several factors. A dominant mechanism could be based on the concept of personal space, the area individuals maintain around themselves into which others cannot intrude without causing discomfort [24][25][26].
Note that the effect of personal space has been integrated into computational models in various contexts in pedestrian dynamics [22,[27][28][29].
To understand such human behavior, further experimental and empirical studies are needed. Recent behavioral experiments on the inflow process [23,31] succeeded in revealing that pedestrians prefer the areas near the boundaries, which had been predicted using a cellular automata model [22]. In addition, Liu et al. [30,31] extended the experiments described in Ref. [23] by varying the pedestrian number and the motivation for future exiting.
The number of pedestrians, i.e., the final pedestrian density in the area is considered to be a significant factor in pedestrian behavior because the main mechanism of the inflow process is expected to be strongly related to the avoidance reactions against other pedestrians. If the final density is unknown or changes during the experimental trials, subjects might decide their final positions based on an incorrect understanding of the future situations. To easily control this factor, we restrict ourselves to considering a fixed pedestrian number and area size in this study. Instead, we consider the geometry of the room as a variable in our experiment. As minor changes in the geometry of the area, we focus on (i) differences in entrance position and (ii) presence of an obstacle. A previous study [22] suggested that the entrance position strongly affects the pedestrian distribution pattern. In fact, these experimental settings allowed us to obtain robust average behavior and analyze the underlying decisionmaking. Using pedestrian trajectory detection [32] and density estimation [33] techniques developed in the context of pedestrian dynamics, we attempt to visualize the behavior and the resulting decisions of pedestrians in such situations, thereby providing an insight into the anticipation and evaluation of future situations regarding their personal space.
In this study, we arranged a basic experiment of the inflow process (Fig. 1). Test subjects were asked to enter a confined area, and temporarily stay there as described in Sec. II. On the basis of the obtained results, we report their interpretations (Sec. III). Details of techniques and analyses used are summarized in Appendix. Finally we discuss the implications and limitations of the results in Sec. IV.

II. EXPERIMENTAL DESIGN
Controlled experiments have been conducted in the Research Center for Advanced Science and Technology (RCAST) at The University of Tokyo, Japan. A total of 25 male participants were recruited from students of The University of Tokyo, who were paid for participation.
The participants were strangers to each other. The experiment was conducted for three different room structures (see Fig. 1c): (i) With the entrance in the middle without obstacles ("normal scenario"; N); (ii) With the entrance in the middle with an obstacle in front ("obstacle scenario"; O); (iii) With the entrance near a corner but without an obstacle ("corner scenario"; C). The obstacle scenario "O" is included because it has been suggested for evacuation processes of pedestrians that an obstacle facilitates faster exiting by impeding clogging [5,34]. However, its (potentially, adverse) effect on the inflow process is still unknown. In addition, the obstacle models the effects of hand-rail poles in buses or trains, which is important within the transportation engineering context.
In each trial, 25 participants were asked to enter an area (3.6 m× 3.6 m) set up on the ground, which was enclosed within walls with a height of 2.0 m ( Fig. 1a and Fig. 1b).
A PVC pipe (φ0.11 m; 2.0 m high) was used as an obstacle. The initial positions of the pedestrians were marked on the ground to regulate the inflow speed, and their entrance order was randomized after each round. They were instructed not to hurry but behave as they would when boarding an elevator. The area was covered with a blue sheet to cover the support structure of the walls and the obstacle, which were sufficiently thin. The width of the entrance was 0.6 m. This value was chosen not to let the participants see inside the area before entrance in order to control the information about the configuration of pedestrians inside.
The experimental area was recorded with a digital camera (SONY HDR-SR7; recording format 1080i60) mounted at a window on the 4th floor (21 m high) of a neighboring building.
The participants were equipped with caps and black shirts for video tracking, and their trajectories were collected using tracking software (PeTrack [32]). We asked the participants to remain in the area for a while after the final participant stopped walking and then asked them to exit the room through the entrance for the next trial. We repeated nine trials for each scenario. Prior to the experiment, we conducted a few test trials using the normal scenario to let the participants know the size of the area, speed of walking, etc., to prevent them from being bewildered in an unfamiliar environment. Owing to these test trials, we observed no apparent difference in their behavioral patterns across experimental trials.

III. RESULTS
A. Qualitative analysis of collective characteristics This behavior could be interpreted as the process of reducing the discomfort from not being able to see the next pedestrians entering the area. This potentially allows avoidance motion in case of conflicts. As explained, the personal space of pedestrians is fixed from the boundary of the area, and therefore, the direction for the boundary contains less uncertainty, whereas stimuli requiring a reaction are mainly expected from the direction of the entrance.
Furthermore, pedestrians who entered the space first and fixed their positions are face-toface with incoming pedestrians, potentially leading to discomfort at eye contact. It is known that more personal space is necessary when a person is confronted by others [35]. Therefore, the alignment of head directions supports a more efficient use of space, with respective of the level of comfort.
Note that across the three scenarios, we observed no significant difference in the time required for 25 pedestrians to enter the area.

B. Qualitative analysis of individual decisions
Next, we study pedestrians' choice of location. By introducing polar coordinates as shown in Fig. 1e, the location of pedestrian i is expressed as (r i , θ i ). For the corner scenario, the origin of these coordinates is in the corner where the entrance is located, i.e., at (x, y) = (1.8, 0). We also define r max (θ i ) for each direction θ i , which denotes the distance between the entrance and the wall in that direction. To evaluate the positions of pedestrians, we use  This clearly illustrates that pedestrians fill the area from the boundary walls to the entrance.
This could be interpreted as anticipative behavior because the positions near boundaries are generally farther away from the entrance (exit). The normalized distance for N and O cases approximately follow the same curve. Meanwhile, in scenario C, the deviation of the normalized distance for the first four pedestrians is larger than for the other two scenarios, which reflects the inhomogeneity in the pedestrian distribution, which we will discuss next. preference for the top-middle boundary contributes to the large deviations in the normalized distance (Fig. 3a). These qualitative differences

C. Interpretation and theoretical description
As we have seen, the decision for which position to take is made based on anticipation.
We discuss the factors that could affect the location choice of pedestrians to interpret the results based on the following assumptions. First, the decision is made by considering an expected ideal configuration based on the knowledge of the total number of persons. In this article, pedestrians anticipate an ideal uniform distribution, upon which they base their evaluation of each location. Second, a trade-off between finding a desirable location and minimizing costs to reach there is considered, such that an option that reconciles these factors is realized.
Each factor is assumed as follows. For comfortableness, pedestrians prefer positions where less disturbance is expected. Here, we consider avoidance factors against two types of interferences that are suggested by the occupation order shown in Fig. 2 and Fig. 3b.
The first is (i) flow avoidance, i.e., the desire to avoid any dynamical interference into the private space caused by succeeding pedestrians passing by to reach their desired positions.
This factor should be salient near the entrance, as schematically shown in Fig. 4a.
The other factor is psychological pressure in the (expected) final state. The degree of discomfort caused by the presence of other persons is known to strongly depend on the distance between them (i.e. the local density) [36]. In this measure, the boundaries, in particular corners, are preferable because the level of discomfort is lower due to fewer neighbors ( Fig. 4b; see also Appendix D 2). Hence we refer to the second factor as (ii) boundary preference. Similar behavior has been reported in other situations, e.g., seat preference in trains [37] and classrooms [38].
The presence of flow avoidance is supported by the fact that in scenario C, the area in front of the entrance is less preferred, even though it is near a boundary (Fig. 3b). Furthermore, the boundary preference is clearly seen in the same figure as the middle area, which is not directly affected by the inflow, is less preferred than the boundaries.
Additional contributions are (iii) distance cost and (iv) angle cost. The distance cost takes into account the effort that has to be made to reach a comfortable position. A position becomes more unattractive if it is farther away from the entrance. It is natural to assume that this distance factor is isotropic as shown in Fig. 4c. The dependence on the distance is less obvious and not necessarily linear. Distance cost explains why the area that is most distant from the entrance (bottom-left) is not preferred in scenario C, although it should be most attractive from the perspective of flow avoidance and boundary preference (Fig. 3b).
The angle cost takes into account the effort necessary when changing the direction of motion. When a pedestrian selects a position corresponding to a large angle |θ i |, he or she has to immediately change the walking direction to directly reach there. This leads to additional energy consumption and consequently discomfort. During the experiments, in many cases the participants avoided immediate turning but instead chose a longer but more comfortable path following an arc with a gradual angle variation. In addition, the approach to such positions requires immediate decision-making. Any delay might increase the walking distance. This can be clearly observed in the experiment (see Fig. 3c). Therefore it is reasonable to believe that the angle cost is especially salient for large angles, as shown in Fig. 4d. This factor explains why the preference exhibits distinctive patterns in the areas near the two corners, i.e., the top-left and bottom-right corners in scenario C (Fig. 3b).
These two areas have the same conditions of distance from the entrance and boundary preference. Also, the flow avoidance factor should not be effective in these two areas.
We suggest that these intuitively plausible factors (i)-(iv) predominantly affect the pedestrians decision-making behavior. For a demonstration, we simply superposed these factors for scenarios N and C (Fig. 4e and Fig. 4f), choosing parameter values such that realistic patterns were qualitatively reproduced. We want to emphasize that this interpretation by the four factors is at least capable of reproducing realistic patterns and might be useful for understanding the underlying human collective behavior. It is not designed to quantitatively predict each pedestrian's decision. Thus, the functions and parameters used in Fig. 4a-Fig. 4d should be considered as rough qualitative estimates. Furthermore it needs to be critically checked whether these four factors are additive or not. However, this formulation contributes rich insights to the understanding of the inflow process and serves as starting point for further investigations of this problem.

IV. DISCUSSIONS
We reported our experimental findings on the inflow process for three different scenarios, i.e., with two different entrance positions and with an obstacle. We observed that the position of the entrance had a significant impact on the pedestrians' choice of location, whereas the obstacle did not largely affect pedestrian movement. In the experiments, we intentionally selected the number of participants and the area size (i.e., the average final density) such that inter-pedestrian interactions were of intermediate strength. Although its validity should be further investigated, our description of the results using the four factors suggests expected preference patterns in different scenarios, which could be considered as a base-line reference for future studies. For example, when only a small number of pedestrians enter the area (and they know that), expected pressure will be small, which would result in approximately no preference for boundaries. Moreover, flow avoidance would be less important, making positions near the entrance more attractive. This becomes obvious in the limiting case of a single passenger in an elevator (see Fig. 5 for example visualizations). Meanwhile, when the area is very large, the preference for the boundary would be outweighed by the distance cost, and the area close to the entrance would be preferred (see Fig. 6 for example visualizations).
Furthermore, for people in cultures with high tolerance against interference of personal space, the boundary preference factor might become weaker, which should be investigated via cross-cultural experiments. Also, the presence of different types of individuals (e.g., females and children) and social groups (e.g., families and friends) [21,39,40] may affect their boundary preference. Therefore, such differences might be understood through the four factors, although our experimental results are based on limited scenarios. In reality, other attractions are often observed. For example, in trains and buses in rush hours, exit doors are more attractive for passengers who attempt to leave first, which has been experimentally confirmed [30,31]. Such behavioral differences may appear based on the destination of each passenger. In future studies, for which our findings are useful for providing a base-line case, such a trade-off should also be investigated. Furthermore, the alignment of head direction to an entrance could be diverted by attractive objects. When there is a common understanding of the next motion (e.g., leaving through a separate exit) passengers are attracted to that direction and align their heads accordingly.
Detailed models of pedestrian motion inevitably require to take unobservable internal states of humans into account, e.g., for decision processes and interactions between pedestrians. Our findings suggest some principles in evaluating the current and future states around each individual. Pedestrians try to avoid interference in their personal space, considering the cost that depends on physical constraints (distance, angle, etc.). Such future anticipation cannot be captured by one of the current paradigms of pedestrian motion that is described by a driving force to a predefined target and local avoidance [41][42][43], but requires models that include decision processes, like most cellular automata [44]. For this reason, the inflow process has not received much attention in pedestrian dynamics research until now.
A theory of space evaluation of pedestrians could provide a missing link for better understanding pedestrian motion. In addition, it could potentially lead to important applications for designing comfortable and safe facilities. In our experiment, the shape of the area was suitable for attaining a uniform pedestrian distribution. Meanwhile, space in mass transport systems (e.g. trains and buses) is sometimes not open due to the presence of seats and hand rails that could distort or blind pedestrians' space recognition, and lead to inefficient inhomogeneous distributions and local overcrowding. In our experiments, an additional obstacle had only a small effect on pedestrian trajectories, while leaving the system dynamics mostly unaffected. However, the final distribution and the location preference could change if the obstacle were placed at a more obstructive location (e.g. just in front of the entrance), or if several obstacles were placed in the area. This is a topic of future research.

Appendix A: Trajectory detection
To extract the trajectory of each individual person, the software PeTrack was used [32].
After deinterlacing the image, the lens distortion was removed. Then, the extrinsic cal-  (Fig. 1d). see also Ref. [33]). Let A i be the area of each Voronoi cell for pedestrian i (who is in the experimental area). The density at a point (x, y) is defined as First, we divided the experimental area into squares of side 0.2 m. For each square j, the time-averaged density (ρ j ′ ) was calculated using the Voronoi diagram for each trial. In this procedure, we used the trajectory data of eight frames per second between the entrance times of the first and final pedestrians. Then after normalizing the average values to neutralize the difference in entrance time (ρ j ′ →ρ j such that the sum over the experimental area becomes a constant value j 0.2 2ρ j = 25 2 for every trial , they were averaged over trials. If the area is uniformly used and pedestrian inflow rate is constant, the time averaged density is calculated as ρ j = 1 2 25 3.6 2 = 0.965 ped/m 2 .
Appendix C: Deviation of walking trajectory from the shortest path We define s i as the walking distance of pedestrian i from the entrance to the final position.
If a pedestrian directly moves to his final position, it coincides with the shortest distance Here r i (t) is the position vector of pedestrian i at time t, and t f is the time when a pedestrian reached the final position, which was collected using the velocity threshold condition, t f = min{t > t e ||ṙ i (t)| < 0.15 m/sec} in practice (Fig. 3c). Here, t e represents the pedestrian's entrance time. Using these variables, the degree of the walking path deviation is defined as which is, by definition, always larger than 1.

Appendix D: Schematic representations of location preference
To capture the elusive features of pedestrian interactions, we attempt to illustrate the factors that determine the choice of locations, and thereby reproduce Fig. 3b by considering a simple superposition where P, F, B, D and A are abstract cost functions for the location preference, the gain by the flow avoidance, the boundary preference, the distance cost and the angle cost, respectively.
These factors are defined to increase as the location becomes more desirable for pedestrians.
Of course, these factors are not necessarily additive, but it is natural to consider that P is a monotonically increasing function of F, W, D and A. As the simplest form, we tentatively adopt this definition of P in this article.

Flow avoidance
To express the flow avoidance we assume a Gaussian function (Fig. 4a) where f , (x 0 , y 0 ), and (f 1 , f 2 ) represent a positive constant, the location of the entrance and the widths of flow avoidance, respectively. Considering the direction of flow, it is natural to presume f 1 < f 2 . Pedestrians might anticipate other pedestrians' choice, which would affect this flow avoidance. However, it is a formidable challenge to include such feedback. Thus we here forbear to go into detail and instead define F a priori. Since this factor only reduces the preference near the entrance, the conclusion in this study is not significantly affected by a specific choice of this function.

Boundary preference
When describing the psychological pressure between pedestrians, it is natural to consider a function g that decays with the distance between two pedestrians (Fig. D4). In the uniform pedestrian distribution preconceived by a pedestrian, the expected number of pedestrians in an area dA at position r ′ ∈ D (D : experimental area) is denoted byρdA, whereρ is the normalized density (pedestrians per area). This area provides pressure against position r, whose magnitude is g(|r − r ′ |)ρdA. Thus, the total (expected) psychological pressure on position r is obtained by taking the sum of this factor for all the positions (r ′ ) in the experimental area. For simplicity we take the limit of dA → 0, which yields Because the area out of the boundaries does not contribute to this value, the areas of boundaries become more preferable (see Fig. 4a). Thus, this factor explains pedestrians' preference to the boundaries. The pedestrian density is normalized asρ = 1 in this study (for 25 pedestrians in a 3.6 m× 3.6 m area). Here, we tentatively defined the discomfort function as g(x) = b(1 + x) −2 (with a positive constant b), avoiding a singularity at x = 0 because pedestrians do not normally fear being physically overlapped (at the densities considered here). It could also be defined with the singularity, for which case, the integral should be performed except for human's body area |r − r ′ | < r 0 (body radius). Both definitions produce qualitatively the same preference, and thus we adopted the simpler one because our interest is not in finding an exact description of the factor. r r͛ r r͛ Degree of discomfort r r͛ FIG. 7. Closeness and degree of discomfort. In this article we assumed the following function: g(x) = b(1 + x) −2 , where x represents the distance between the two individuals (see also Eq. (D3)).

Distance cost
The distance cost should be a monotonically increasing function of |r|. For simplicity we assume the linear expression with a positive constant d (Fig. 4c).

Angle cost
For the angle cost we selected a function that nonlinearly decreases with the absolute value of θ: (Fig. 4d) A(r) = −a|θ| α .
Here a and α are positive constants. In this article, we tentatively assume α = 3.
We emphasize that the absolute values of F, B, D, and A are not essential, but their relative strengths are important. As the exact forms of these functions cannot be easily determined, we used simple functions that satisfy the requirements discussed in Sec. III C.
In Fig. 4, we selected the parameter values as (f, f