# Social Force Model Calibration for Preschool Children Evacuations Based on Multiscenario Experiments

Authors: Qingjie Qi; Ruifeng She; Han Liu; Jingwen Zhang; Yue Zhang
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

The social force model has been widely used in simulating and analyzing pedestrian behavior. Like any agent-based models, the fidelity of the social force model largely relies on its numerical parameters that characterizes pedestrians’ behaviors. While parameters describing normal walking behaviors have been observed and calibrated in field experiments, those describing behaviors under abnormal and urgent circumstances have rarely been studied but are of practical significance in evaluating safety functionality of facilities, particularly those serving children or elders. Specifically aiming at providing a set of social force model parameters characterizing children’s behavior during evacuation, this study conducted evacuation experiments with preschool children under multiple emergency scenarios involving impaired-vision and flame scenarios and benchmarked against a normal scenario. A simulation-calibration framework is developed based on the social force model to calibrate evacuation behavior parameters by minimizing trajectory distance. The numerical approximation results indicate evident parameter disparities of preschool children from adults. This study can improve evacuation strategies and the designing/evaluation process of dedicated facilities layouts such as kindergarten corridors, activity rooms, and playgrounds.


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## Body

## 1. Introduction

Research on pedestrian evacuation has become a hot topic in the field of traffic safety because of a series of severe stampedes, such as the events in Jodhpur (2008), Mina (2015), Turin (2017), Mumbai (2017), and Mecca (2018). The behavior of individuals of different ages, under emergency situations, influences evacuation efficiency. Adults, in general, make good judgement for risks based on their own experience, while young children, due to the lack mental and physical maturity, are less competent in decision-making and reacting to danger. In the Victoria Hall disaster in the U.K., thousands of children ran over one another for toys, killing 183 of them in a stampede [1]. As the youngest age group starting to rely on day care service, preschool age children “Preschool age” refers to children under 5, and children age 5 who do not attend full-day kindergarten. This definition follows the Illinois Department of Children and Family Services’ Rules (https://www2.illinois.gov/dcfs/aboutus/notices/Documents/rules_408.pdf) may be the most vulnerable group during evacuation. Without systematic education, most preschool children have yet developed the capability to think and react to hazards. Moreover, many preschool facilities do not have evacuation plans and do not run practice drills [2, 3]. This exposes those children to danger in the event of an emergency. Given how unpredictably children may react to emergencies, it is of greater significance and responsibility to bring into consideration and improve evacuation efficiency in educational facilities. Such a need calls for a quantification tool to reproduce the children’s behavior during evacuation to evaluate any safety metric of a facility.Over the past few decades, various models and methods have been proposed to analyze the characteristics of pedestrian evacuation behavior. These include continuous models, network-based models and agent-based models. Hughes [4] was the first to propose a continuous pedestrian flow model that depicts pedestrian flow as continuous fluid flow from a macroscopic view. He introduced a potential function as the patronizing cost of traveling to a destination, and suggested that all cost-minimizing paths are parallel in opposite direction with the potential gradient. Huang et al. demonstrated that Hughes’ model reaches the dynamic user equilibrium, where pedestrians choose the fastest paths simultaneously [5]. Considering nonconvex or discontinuous domain, Guo et al. developed a network-based model for indoor obstacle environments [6]. Helbing et al. [7, 8] proposed a social force model, which describes pedestrian movement from a microscopic perspective. It can reproduce self-organizing crowd phenomena raised from the information and bottleneck oscillation effects in two-way flow. In the model, the pedestrians gravitate to their destination at a desired speed. Several psychological forces cause pedestrians to have varying accelerations. These forces include the attraction from the destination, the force between the pedestrians, and the force from obstacles. Seer et al. validated the social force model with complex real data [9]. Yuen and Lee extended the social force model and added overtaking behavior, which explains the phenomenon where pedestrians accelerate while attempting to pass those in front of them [10]. Qu et al. improved the social force model for stair scenes to analyze the rotation phenomenon [11]. In commercial implementations of the social force model, the parameters are calibrated against by maximizing likelihood or minimizing nonlinear square error between field experiment results and simulated motion [12–14]. Recently, Ci et al. integrated a chicken swarm optimization and K-means algorithm to select key-point for dynamic flow control [15]. Yuan et al. considered sample size and precision of data and found that Bayesian Logistic regression can identify significant risk factors when data sets are of different precision [16]. Yang et al. combined machine learning and statistical models to determine the relationship and mechanism between dynamic traffic flow characteristics and traffic safety [17].Pedestrian age is a key factor affecting the dynamics of motion. Frantzich et al. examined the evacuation behavior of college students at bottlenecks by changing corridor entrances [18]. Guo et al. studied the evacuation route selection of students with varying visual abilities [19]. Chen et al. conducted classroom experiments to analyze students’ preference for exits and the effect of grouping phenomena during evacuation [20]. Using child evacuation dynamics data from the Daycare Center, Larusdottir et al. found that children’s evacuation characteristics contrast adult evacuation features in many aspects (such as horizontal travel, climbing and descending stairs, and crossing entrances and exits) [21]. Cuesta et al. tested the number of evacuations, walking speed, route selection and evacuation arrival curves of 4–16 years-old children [22]. Cuesta et al. conducted five evacuation exercises for children aged 6 to 16, and compared the observed results with the results of an evacuation simulation [23]. Hamilton et al. conducted 12 full evacuations for children aged 4 to 12 in four schools, and analyzed evacuation data in horizontal and stair environments [24]. Najmanova et al. conducted evacuation experiments on children aged 3 to 6 and found that their behavior was affected most by age and cognitive routes [25]. Chen et al. used a questionnaire to survey children aged 8 to 12 regarding path selection. They found that children’s initial position, group behavior and follow-up behavior collectively influence evacuation path choice [26].While understanding how children perceive fire and smoke risk in their environments is critical, research on children’s evacuation behaviors in school is limited. Moreover, there is scant literature on the behavior of preschool children aged 4 to 5. This is the age group when children start to lose the full-time protection of their guardians, yet they are not old enough to be self-protective. Understanding the evacuation behavior of preschool children in dangerous environments, such as visual impairment in smoke and fire, is essential to providing safety measures for this vulnerable group. In addition, research on pedestrian behavior under smoke and flame conditions is scarce, due to both safety and ethical reasons. Although the social force model has proven effective in capturing pedestrian behavior characteristics, children’s behavior parameters under emergent evacuation have yet to be explored.In order to address the gap in the existing literature, we study the evacuation behavior of preschool children by conducting a group of experiments and developing a social force simulation tool. We design simulated scenarios that invoke reactions to emergencies while guaranteeing safety. For example, we design a scenario in which children wear dark sunglasses to mimic vision impaired by smoke. The preschool children’s group evacuations are recorded, and the social force model parameters with respect to trajectory closeness are calibrated with our simulation tool. We analyze children’s behavior characteristics and evacuation efficiency in various scenarios with the social force model’s quantized parameters.In Section2, the experiment’s procedure is discussed in detail. In Section 3, we implement the social force model to simulate the evacuation process and develop a numerical optimization procedure to determine the most suitable parameters, based on trajectory closeness. In Section 4, the simulation results in the above scenarios are analyzed to provide useful findings for the evacuation behavior of preschool children in emergency situations. Section 5 summarizes this study’s findings and discusses further prospects.


## 2. Evacuation Experiments

### 2.1. Experiment Descriptions

The experiment was conducted in an evacuation corridor in DW Preschool in Baoding, China. The dimensions of the corridor were 5.8 m∗1.3 m. Eighteen children (9 boys and 9 girls) from the K1 preschool class participated in the experiment. The children’s average age was 4.7 years. They had been studying in the K1 class for almost 11 months and therefore were familiar with the corridor environment. After the security monitoring room triggered the fire alarm, two teachers guided 18 children to evacuate from the classroom, to the corridor, and then to the stairs. We selected a section (4.0 m∗1.3 m) in the middle of the corridor to record videos to ensure the children’s speed at the beginning and end of the experimental section for a parallel experiment and comparison. Three 1080p Xiaomi security cameras were installed on the ceiling of the experimental section to monitor the corridor. The experiments started at 10 am and the indoor temperature of the preschool was 24 degrees Celsius.The experiment was approved by the children’s guardians and the school administration. For safety reasons, during the experiment, one teacher led at the front of the children’s team, and another teacher supervised at the end of the team and observed potential danger. In addition, two school security personnel monitored the experiment through the security cameras in order to take timely necessary countermeasures, as needed.


### 2.2. Scenario Design

We tested the preschool children’s behavioral responses in different scenarios, as illustrated in Figure1. The experimental scenarios were designed as follows:Figure 1
An Illustration of evacuation scenarios in a preschool corridor. (a) Scenario 1. (b) Scenario 2. (c) Scenario 5.
(a)(b)(c)Scenario 1.
A general emergency evacuation scenario. The emergency lights in the corridor flash during the evacuation process; there are no flames or smoke in the corridor.Scenario 2.
Smoke around the participants restricts their vision. If genuine smoke (e.g., solid carbon dioxide for stage usage) is released during the experiment, the substances produced may have an adverse effect on children. Actual combustion release, or even solid carbon dioxide used for stage effects, would have been harmful to the young participants. Therefore, we mimicked the cognition by having the children wear dark sunglasses to restrict their vision. The brightness and clarity seen through the sunglasses were similar to those of smoke.Scenario 3.
As in Scenario 1, the emergency lights in the corridor flash during the evacuation. The only difference is that the children wear dark sunglasses throughout the experiments.Scenario 4.
This is a scenario in which not only are the pedestrians’ fields of vision restricted by smoke, but their eyes also may be injured. Since fire smog is mostly toxic gas, it is likely to irritate the eyes, causing those who are affected to squint their eyes, either actively or passively. In this scenario, one of their dark sunglass lenses is covered with opaque paper to obscure the vision through that lens. In the experiment, the emergency lights in the corridor flashed throughout the evacuation process; the children wore the monocular dark sunglasses for the experiment.Scenario 5.
A scenario where there is a flame in the corridor. In a real emergency, the children’s evacuation reaction behavior in a fire scene would affect evacuation efficiency and safety. Since igniting a real flame would expose children to a threatening environment, we used a flame lamp placed in the corridor as a replacement. The flame lamp consisted of an electric fan at the bottom, and a red piece of satin on the top. When the flame lamp was powered, the fan would blow the satin upward to create a visual effect similar to a real flame. Scenario 3 is divided into the following three subcases, depending on the flame lamp arrangement’s size and density.Scenario 6.
This case tested children’s evacuation characteristics under small-scale flames. One flame lamp is arranged at the midpoint of the measured section. The children were instructed to evacuate in one direction in an environment with constantly flashing emergency lights.Scenario 7.
In this case, we increase the number of flame lamps and test the children’s reaction to a mass of flames. Four small flame lamps are arranged at the midpoint of the road section to be measured. The children are evacuated in one direction in an environment where the fire emergency lights blink constantly.Scenario 8.
In this case, the flame lamps are dispersed to test the children’s reaction to large-scale discrete flames. The testing section is divided into five equal parts, and four flame lamps are arranged at four equidistant points. The children are evacuated in one direction in an environment with constantly blinking emergency lights.


## 2.1. Experiment Descriptions

The experiment was conducted in an evacuation corridor in DW Preschool in Baoding, China. The dimensions of the corridor were 5.8 m∗1.3 m. Eighteen children (9 boys and 9 girls) from the K1 preschool class participated in the experiment. The children’s average age was 4.7 years. They had been studying in the K1 class for almost 11 months and therefore were familiar with the corridor environment. After the security monitoring room triggered the fire alarm, two teachers guided 18 children to evacuate from the classroom, to the corridor, and then to the stairs. We selected a section (4.0 m∗1.3 m) in the middle of the corridor to record videos to ensure the children’s speed at the beginning and end of the experimental section for a parallel experiment and comparison. Three 1080p Xiaomi security cameras were installed on the ceiling of the experimental section to monitor the corridor. The experiments started at 10 am and the indoor temperature of the preschool was 24 degrees Celsius.The experiment was approved by the children’s guardians and the school administration. For safety reasons, during the experiment, one teacher led at the front of the children’s team, and another teacher supervised at the end of the team and observed potential danger. In addition, two school security personnel monitored the experiment through the security cameras in order to take timely necessary countermeasures, as needed.


## 2.2. Scenario Design

We tested the preschool children’s behavioral responses in different scenarios, as illustrated in Figure1. The experimental scenarios were designed as follows:Figure 1
An Illustration of evacuation scenarios in a preschool corridor. (a) Scenario 1. (b) Scenario 2. (c) Scenario 5.
(a)(b)(c)Scenario 1.
A general emergency evacuation scenario. The emergency lights in the corridor flash during the evacuation process; there are no flames or smoke in the corridor.Scenario 2.
Smoke around the participants restricts their vision. If genuine smoke (e.g., solid carbon dioxide for stage usage) is released during the experiment, the substances produced may have an adverse effect on children. Actual combustion release, or even solid carbon dioxide used for stage effects, would have been harmful to the young participants. Therefore, we mimicked the cognition by having the children wear dark sunglasses to restrict their vision. The brightness and clarity seen through the sunglasses were similar to those of smoke.Scenario 3.
As in Scenario 1, the emergency lights in the corridor flash during the evacuation. The only difference is that the children wear dark sunglasses throughout the experiments.Scenario 4.
This is a scenario in which not only are the pedestrians’ fields of vision restricted by smoke, but their eyes also may be injured. Since fire smog is mostly toxic gas, it is likely to irritate the eyes, causing those who are affected to squint their eyes, either actively or passively. In this scenario, one of their dark sunglass lenses is covered with opaque paper to obscure the vision through that lens. In the experiment, the emergency lights in the corridor flashed throughout the evacuation process; the children wore the monocular dark sunglasses for the experiment.Scenario 5.
A scenario where there is a flame in the corridor. In a real emergency, the children’s evacuation reaction behavior in a fire scene would affect evacuation efficiency and safety. Since igniting a real flame would expose children to a threatening environment, we used a flame lamp placed in the corridor as a replacement. The flame lamp consisted of an electric fan at the bottom, and a red piece of satin on the top. When the flame lamp was powered, the fan would blow the satin upward to create a visual effect similar to a real flame. Scenario 3 is divided into the following three subcases, depending on the flame lamp arrangement’s size and density.Scenario 6.
This case tested children’s evacuation characteristics under small-scale flames. One flame lamp is arranged at the midpoint of the measured section. The children were instructed to evacuate in one direction in an environment with constantly flashing emergency lights.Scenario 7.
In this case, we increase the number of flame lamps and test the children’s reaction to a mass of flames. Four small flame lamps are arranged at the midpoint of the road section to be measured. The children are evacuated in one direction in an environment where the fire emergency lights blink constantly.Scenario 8.
In this case, the flame lamps are dispersed to test the children’s reaction to large-scale discrete flames. The testing section is divided into five equal parts, and four flame lamps are arranged at four equidistant points. The children are evacuated in one direction in an environment with constantly blinking emergency lights.


## 3. Social Force Model Validation

In order to gain insight from the preschool children’s evacuation experiment, and to make predictions for other cases, we implemented the social force model to generate predicted trajectories for each participants and in so doing calibrate the model parameters based on the experiment.


### 3.1. Model Description

The social force model describes a pedestrian’s reaction to his or her surroundings as the collective consequence of a set of psychological forces. Analogous to physical forces, these psychological forces guide theith pedestrian’s motion as per Newton’s 2nd law:(1)Fi⟶=mi⋅ai⟶=mi⋅dvi⟶dt,dvi=aidt=Fimidt,where F⟶ denotes the force vector. mi, ai⟶ and vi⟶ represent the inertia We write out mi as in equation (1) for completeness but omit it and assume its value to be 1 in the following formulations, because this quantity is mathematically redundant and can be absorbed by other parameters during calibration., the acceleration vector and the velocity vector, respectively. The net force consists of the psychological force as per the social force model and the physical forces when collision occurs. For safety considerations, we did not allow physical contact between the children in the preschool experiment. This also eliminated irrelevant terms in the model, which, while is a numerical simplification, enhances the fidelity of factors that are of causal significance. Several variants of the social force model have been proposed in the literature. We define net force as the summation of five parts:(2)F⟶=F⟶destination+F⟶circular+F⟶wall+F⟶obstacle+F⟶ellipticwλ,where wλ is a correction factor determining the relative weight (conceptually, attention) with which a pedestrian perceives lateral information (e.g., obstacles, congestions, and hazards) comparing to the driving force towards the destination. The included terms, with each depicting a type of driving force from the surroundings, are defined as follows.The first term is the attraction that steers the pedestrian towards its destination. It has the following form:(3)F⟶destination=dvi⟶dt≈vi0⋅ei0⟶t−vi⟶tτ,where vi⟶ represents the current velocity vector of pedestrian i; vi0 is the desired pedestrian speed; ei0⟶t is a unit vector indicating the direction from the current location of pedestrian i towards his or her destination; τ indicates the relaxation time within which the pedestrian wishes to steer.The other four terms are pointwise forces pointing towards the pedestrian from a surrounding object. TermF⟶circular and F⟶elliptic depicts two types of driving forces According to Johansson et al. [27], elliptic force considers the relative velocity of the pedestrian and the object. For further reading, see [27] for a discussion of the equations and parameters in which the notation is consistent with that of this study between pedestrians so as to avoid crownless [8]. Term F⟶wall and F⟶obstacles corresponds to the repulsive force from nonhazardous obstacles (e.g., walls, pillars) and hazards (e.g., fire), respectively. All these terms take a similar form with respective parameters:(4)F⟶ijdij=αe−dij/β.The chosen function form takes a larger value when the pedestrian is close to the object, and diminishes as they move further away. Collectively the schematic diagram of social forces in our model is illustrated in Figure2.Figure 2
Schematic diagram of social forces.


### 3.2. Model Estimation and Parameter Calibration

We developed a simulation subroutine in python implementing the social force model discussed in the previous section. Given the pedestrians’ initial conditions (i.e., initial position and velocity) and a set of parameters, we can simulate a series of motion. This subroutine produces a simulated trajectory, which is compared with the recorded trajectory. The closeness of trajectories, which is a temporal sequence of positions, is measured by the following equation:(5)fobjθ=∑i∈I∑t∈TΔtFxit−fxit,θ2+Fyit−fyit,θ2,where Fxit,Fyit is the recorded position coordinate at time t from our experiment. fxit,θ,fyit,θ represents the position coordinate at time t given a parameter set θ from the simulation. The terminating error, denoted as fobj, is given in equation (5), which is the least square error that measures the distance between the simulated trajectories against the experimented ones, and has a unit of m2⋅ped. Statistically speaking, we treat the position of pedestrians in each frame as samples from a temporally correlated random variable generator, fit=Fit+ξ, which centers at the supposedly correct experimented trajectories added by a zero-centered noise term. The proposed framework hence minimizes the variance of the noise term and yield an inference model that makes frame-to-frame predictions with high fidelity. The reference error fref is essentially the same as equation (5) with fit=0,∀i,t, which measures the total pedestrian-distance traveled and gives a comparable reference to the model variance. To minimize error functions lacking a closed form as a function of the measurement, our optimization process uses the SciPy.optimize python package to seek the set of parameters that minimizes the objective function.


## 3.1. Model Description

The social force model describes a pedestrian’s reaction to his or her surroundings as the collective consequence of a set of psychological forces. Analogous to physical forces, these psychological forces guide theith pedestrian’s motion as per Newton’s 2nd law:(1)Fi⟶=mi⋅ai⟶=mi⋅dvi⟶dt,dvi=aidt=Fimidt,where F⟶ denotes the force vector. mi, ai⟶ and vi⟶ represent the inertia We write out mi as in equation (1) for completeness but omit it and assume its value to be 1 in the following formulations, because this quantity is mathematically redundant and can be absorbed by other parameters during calibration., the acceleration vector and the velocity vector, respectively. The net force consists of the psychological force as per the social force model and the physical forces when collision occurs. For safety considerations, we did not allow physical contact between the children in the preschool experiment. This also eliminated irrelevant terms in the model, which, while is a numerical simplification, enhances the fidelity of factors that are of causal significance. Several variants of the social force model have been proposed in the literature. We define net force as the summation of five parts:(2)F⟶=F⟶destination+F⟶circular+F⟶wall+F⟶obstacle+F⟶ellipticwλ,where wλ is a correction factor determining the relative weight (conceptually, attention) with which a pedestrian perceives lateral information (e.g., obstacles, congestions, and hazards) comparing to the driving force towards the destination. The included terms, with each depicting a type of driving force from the surroundings, are defined as follows.The first term is the attraction that steers the pedestrian towards its destination. It has the following form:(3)F⟶destination=dvi⟶dt≈vi0⋅ei0⟶t−vi⟶tτ,where vi⟶ represents the current velocity vector of pedestrian i; vi0 is the desired pedestrian speed; ei0⟶t is a unit vector indicating the direction from the current location of pedestrian i towards his or her destination; τ indicates the relaxation time within which the pedestrian wishes to steer.The other four terms are pointwise forces pointing towards the pedestrian from a surrounding object. TermF⟶circular and F⟶elliptic depicts two types of driving forces According to Johansson et al. [27], elliptic force considers the relative velocity of the pedestrian and the object. For further reading, see [27] for a discussion of the equations and parameters in which the notation is consistent with that of this study between pedestrians so as to avoid crownless [8]. Term F⟶wall and F⟶obstacles corresponds to the repulsive force from nonhazardous obstacles (e.g., walls, pillars) and hazards (e.g., fire), respectively. All these terms take a similar form with respective parameters:(4)F⟶ijdij=αe−dij/β.The chosen function form takes a larger value when the pedestrian is close to the object, and diminishes as they move further away. Collectively the schematic diagram of social forces in our model is illustrated in Figure2.Figure 2
Schematic diagram of social forces.


## 3.2. Model Estimation and Parameter Calibration

We developed a simulation subroutine in python implementing the social force model discussed in the previous section. Given the pedestrians’ initial conditions (i.e., initial position and velocity) and a set of parameters, we can simulate a series of motion. This subroutine produces a simulated trajectory, which is compared with the recorded trajectory. The closeness of trajectories, which is a temporal sequence of positions, is measured by the following equation:(5)fobjθ=∑i∈I∑t∈TΔtFxit−fxit,θ2+Fyit−fyit,θ2,where Fxit,Fyit is the recorded position coordinate at time t from our experiment. fxit,θ,fyit,θ represents the position coordinate at time t given a parameter set θ from the simulation. The terminating error, denoted as fobj, is given in equation (5), which is the least square error that measures the distance between the simulated trajectories against the experimented ones, and has a unit of m2⋅ped. Statistically speaking, we treat the position of pedestrians in each frame as samples from a temporally correlated random variable generator, fit=Fit+ξ, which centers at the supposedly correct experimented trajectories added by a zero-centered noise term. The proposed framework hence minimizes the variance of the noise term and yield an inference model that makes frame-to-frame predictions with high fidelity. The reference error fref is essentially the same as equation (5) with fit=0,∀i,t, which measures the total pedestrian-distance traveled and gives a comparable reference to the model variance. To minimize error functions lacking a closed form as a function of the measurement, our optimization process uses the SciPy.optimize python package to seek the set of parameters that minimizes the objective function.


## 4. Results and Discussion

### 4.1. Experiment Results and Observations

We compared the evacuation process under different settings based on the video data collected from the experiments above. Experiments in each scenario were conducted twice. The evacuation times, measured from the first participant entering the filming region to the last one to leave, in each scenario, are shown in Figure3. The evacuation times were the same (23 seconds) in both trials in Scenario 1. The children’s vision was limited in the four experiments in Scenario 2. However, the evacuation times were less than those of Scenario 1. This may have been because children naturally fled regions where they were at unease. The evacuation in Scenario 4 (first experiment: 20 s, second experiment: 20 s) took longer than that of Scenario 3 (first experiment: 15 s, second experiment: 19 s). This was to be expected because the children’s vision was further restricted in Scenario 4, resulting in longer evacuation process. In Scenario 5, the evacuation times of the three cases a, b, and c decreased, respectively. In Scenario 6 in which only one flame lamp is placed, the two experiment evacuation times were 24 s and 23 s, respectively, which was similar the evacuation time in Scenario 1. In Scenario 7 in which four flame lamps were dispersed, the evacuation time for both trials were 20 s; in Scenario 8 in which four flame lamps were concentrated, the evacuation time for the two trials were 15 s and 18 s, respectively. Scenarios 7 and 8 have more flame lamps, and the will of the children to escape was stronger than in Scenario 6 in which there was only one flame lamp. Furthermore, the accumulation of flame lights in the environment in Scenario 8 may have suggested danger. Thus, the children hastened their escape.Figure 3
Scenario evacuation times.Figure4 shows children’s evacuation trajectories in different situations. In Scenario 1, since the children had good vision, and there were no obstructions or threatening flames, most of the children’s trajectories occupied the center of the corridor’s width. Most of the trajectories were located to the right of the road at the test start point, and then dispersed gradually. This shows that the children have trailing behaviors in the evacuation process. In emergency situations, a child will tend to follow the children in front of him or her. Therefore, we believe that the repulsive force of the children in front of the emergency evacuation to the children in the back is not significant.Figure 4
Observed trajectories inScenarios 1, 3 and 6.InScenario 2 where the children’s vision was limited, we found that they were more inclined to move near the wall, instead of walking down the center of the corridor, as in Scenario 1. There are two possible reasons for this phenomenon. Firstly, when preschool children are uncertain about the conditions ahead, it would be appealing for them to find a fixed reference, e.g., the wall, and follow it to avoid sudden impact with an unexpected object. Secondly, in cases of limited vision, children may lose balance and become dizzy. They need to be close to, or even touching, the wall to support and calm themselves.InScenario 6, since the flame lamp was located on the left side of the corridor against the wall, most of the children started to avoid the fire lamp when they were far away from it. In the experiment, two children approached the fire lamp before they quickly adjusted direction and proceeded along the right wall. This indicates that preschool children’s cognitive ability toward disaster is inferior to that of adults.


### 4.2. Model Calibration

The six scenarios were simulated to yield six sets of parameters that best describe the behavior under each scenario. The Nelder–Mead algorithm [28] is used in the optimization routine to minimize the objective function, which measures the least square error from the video benchmark. The objective function (5) has units in meters and can be interpreted as the average distance between the predicted trajectories and the real ones. The simulated trajectories are compared with the real ones every 0.25 seconds to evaluate the error, while the simulation takes a timestep of 0.025 seconds. The initial guess uses the values suggested by Seer et al. [9] and Johansson et al. [27] for λ, τ, αc, αe, βc and βe. For other parameters, the initial values are uniformly sampled within a reasonable range. Table 1. Presents the resultant parameters. The subscripts c, e, w and o refer to the circular force, the elliptic force, and the reactive force from walls and obstacles (fire in this case), respectively. The fobj value is the terminating error, and fref is the reference error, which is evaluated using the suggested parameters.Table 1
Estimated parameter values from the experiment scenarios.
134678Λ0.580.850.20.310.710.87Τ (s)0.440.730.630.890.510.56va (m/s)1.412.482.361.411.491.29VD (s)1.791.741.863.743.83.64αc (m/s2)0.611.20.69−0.53−0.140.45βc (m)3.787.312.691.971.513.66αe (m/s2)0.680.410.23−0.15−0.43−0.12βe (m)3.780.053.198.060.040.08αw (m/s2)0.49−0.42−0.880.17−0.010.04βw (m)3.17−2.29−1.486.533.64−3.38αo (m/s2)———0.30.750.48βo (m)———6.510.1312.36fobj0.110.490.710.610.820.58fref0.473.821.972.725.643.86Scenario 1 achieves the best approximation because its trajectories are smooth, and less forces are relevant. In Scenarios 3 and 4, the parameter λ shrinks when the fire has a potential irritation to children’s eyes. This indicates that children with compromised vision are likely to be disrupted by children in the rear. This may be caused by the psychological insecurity in vision—impaired evacuation. In addition, parameter τ decreases 13.7% from Scenarios 3 to 4, showing that stronger attraction steers children with inferior vision towards their destinations. Furthermore, the testers are attracted towards the walls when their vision is impaired. It is worth noting that the attraction force from the wall tends to be stronger when children’s vision is limited, which supports our hypothesis.Scenario 5 demonstrated that parameter λ increases along with the flame scale. This shows that, in a dangerous environment, children in the back push forward with more force than those in the front. Parameter τ in Scenarios 7 and 8 decreases 37.1% and 41.6%, respectively, compared to that of Scenario 6. This indicates that the children are more decisive when they are exposed to more dangerous environments. In comparison, the repulsive force from the fire lamps in Scenarios 6–8 dominate the other reactions, revealing their evasive nature toward danger. Admittedly, several of the resultant parameters fall outside of the suggested range in the literature [9]. This may suggest that preschool children’s evacuation behaviors are difference from adult behaviors for which the research is more mature. Overall, this calibration has yielded more suitable parameters for the social force model to describe preschool children’s reactions during emergency evacuation processes.


## 4.1. Experiment Results and Observations

We compared the evacuation process under different settings based on the video data collected from the experiments above. Experiments in each scenario were conducted twice. The evacuation times, measured from the first participant entering the filming region to the last one to leave, in each scenario, are shown in Figure3. The evacuation times were the same (23 seconds) in both trials in Scenario 1. The children’s vision was limited in the four experiments in Scenario 2. However, the evacuation times were less than those of Scenario 1. This may have been because children naturally fled regions where they were at unease. The evacuation in Scenario 4 (first experiment: 20 s, second experiment: 20 s) took longer than that of Scenario 3 (first experiment: 15 s, second experiment: 19 s). This was to be expected because the children’s vision was further restricted in Scenario 4, resulting in longer evacuation process. In Scenario 5, the evacuation times of the three cases a, b, and c decreased, respectively. In Scenario 6 in which only one flame lamp is placed, the two experiment evacuation times were 24 s and 23 s, respectively, which was similar the evacuation time in Scenario 1. In Scenario 7 in which four flame lamps were dispersed, the evacuation time for both trials were 20 s; in Scenario 8 in which four flame lamps were concentrated, the evacuation time for the two trials were 15 s and 18 s, respectively. Scenarios 7 and 8 have more flame lamps, and the will of the children to escape was stronger than in Scenario 6 in which there was only one flame lamp. Furthermore, the accumulation of flame lights in the environment in Scenario 8 may have suggested danger. Thus, the children hastened their escape.Figure 3
Scenario evacuation times.Figure4 shows children’s evacuation trajectories in different situations. In Scenario 1, since the children had good vision, and there were no obstructions or threatening flames, most of the children’s trajectories occupied the center of the corridor’s width. Most of the trajectories were located to the right of the road at the test start point, and then dispersed gradually. This shows that the children have trailing behaviors in the evacuation process. In emergency situations, a child will tend to follow the children in front of him or her. Therefore, we believe that the repulsive force of the children in front of the emergency evacuation to the children in the back is not significant.Figure 4
Observed trajectories inScenarios 1, 3 and 6.InScenario 2 where the children’s vision was limited, we found that they were more inclined to move near the wall, instead of walking down the center of the corridor, as in Scenario 1. There are two possible reasons for this phenomenon. Firstly, when preschool children are uncertain about the conditions ahead, it would be appealing for them to find a fixed reference, e.g., the wall, and follow it to avoid sudden impact with an unexpected object. Secondly, in cases of limited vision, children may lose balance and become dizzy. They need to be close to, or even touching, the wall to support and calm themselves.InScenario 6, since the flame lamp was located on the left side of the corridor against the wall, most of the children started to avoid the fire lamp when they were far away from it. In the experiment, two children approached the fire lamp before they quickly adjusted direction and proceeded along the right wall. This indicates that preschool children’s cognitive ability toward disaster is inferior to that of adults.


## 4.2. Model Calibration

The six scenarios were simulated to yield six sets of parameters that best describe the behavior under each scenario. The Nelder–Mead algorithm [28] is used in the optimization routine to minimize the objective function, which measures the least square error from the video benchmark. The objective function (5) has units in meters and can be interpreted as the average distance between the predicted trajectories and the real ones. The simulated trajectories are compared with the real ones every 0.25 seconds to evaluate the error, while the simulation takes a timestep of 0.025 seconds. The initial guess uses the values suggested by Seer et al. [9] and Johansson et al. [27] for λ, τ, αc, αe, βc and βe. For other parameters, the initial values are uniformly sampled within a reasonable range. Table 1. Presents the resultant parameters. The subscripts c, e, w and o refer to the circular force, the elliptic force, and the reactive force from walls and obstacles (fire in this case), respectively. The fobj value is the terminating error, and fref is the reference error, which is evaluated using the suggested parameters.Table 1
Estimated parameter values from the experiment scenarios.
134678Λ0.580.850.20.310.710.87Τ (s)0.440.730.630.890.510.56va (m/s)1.412.482.361.411.491.29VD (s)1.791.741.863.743.83.64αc (m/s2)0.611.20.69−0.53−0.140.45βc (m)3.787.312.691.971.513.66αe (m/s2)0.680.410.23−0.15−0.43−0.12βe (m)3.780.053.198.060.040.08αw (m/s2)0.49−0.42−0.880.17−0.010.04βw (m)3.17−2.29−1.486.533.64−3.38αo (m/s2)———0.30.750.48βo (m)———6.510.1312.36fobj0.110.490.710.610.820.58fref0.473.821.972.725.643.86Scenario 1 achieves the best approximation because its trajectories are smooth, and less forces are relevant. In Scenarios 3 and 4, the parameter λ shrinks when the fire has a potential irritation to children’s eyes. This indicates that children with compromised vision are likely to be disrupted by children in the rear. This may be caused by the psychological insecurity in vision—impaired evacuation. In addition, parameter τ decreases 13.7% from Scenarios 3 to 4, showing that stronger attraction steers children with inferior vision towards their destinations. Furthermore, the testers are attracted towards the walls when their vision is impaired. It is worth noting that the attraction force from the wall tends to be stronger when children’s vision is limited, which supports our hypothesis.Scenario 5 demonstrated that parameter λ increases along with the flame scale. This shows that, in a dangerous environment, children in the back push forward with more force than those in the front. Parameter τ in Scenarios 7 and 8 decreases 37.1% and 41.6%, respectively, compared to that of Scenario 6. This indicates that the children are more decisive when they are exposed to more dangerous environments. In comparison, the repulsive force from the fire lamps in Scenarios 6–8 dominate the other reactions, revealing their evasive nature toward danger. Admittedly, several of the resultant parameters fall outside of the suggested range in the literature [9]. This may suggest that preschool children’s evacuation behaviors are difference from adult behaviors for which the research is more mature. Overall, this calibration has yielded more suitable parameters for the social force model to describe preschool children’s reactions during emergency evacuation processes.


## 5. Conclusion

In this research, we designed experimental and modeling methods to investigate preschool children’s evacuation behavior in emergency scenarios. We try to restore the child’s response state in an emergency situation as much as possible using means that are not detrimental to the child’s health. For example, we mimicked the cognition by having the children wear dark sunglasses to restrict their vision because fire smog is a mostly toxic gas, and likely to irritate the eyes. Since igniting a real flame would expose children to a threatening environment, we used a flame lamp placed in the corridor as a replacement. The flame lamp consisted of an electric fan at the bottom and a red piece of satin on the top. The fan would blow the satin upward when the flame lamp was powered to create a visual effect similar to a real flame.We studied the children’s behavior in a normal scenario, in impaired vision scenarios and in flame scenarios. These features are summarized, as follows, according to their evacuation times and trajectories: (1) Evacuation times in impaired vision scenarios and with large-scale flames are shorter than the time needed in normal evacuation scenarios. When children’s vision is restricted, it takes them longer to escape. In scenarios where more flame lamps are present, children have stronger will to escape than when there are fewer lamps. Furthermore, flame concentration motivates the children to hasten their evacuation. (2) In normal evacuation, most of the children’s group travel trajectories occupy the center of the corridor’s width. We also observe that children employ trailing behaviors during evacuation. In the Smoke Scenario in which the children’s vision was limited, they were more inclined to move near the wall for guidance and support. In scenarios in which a flame was present, most children began avoiding the flame while they were distant from it. There is also latent reaction, suggesting a cognition to danger weaker than that of adults.Based on the characteristics of the children observed in these experiments, we implemented a social force model to simulate the evacuation of preschool children in a corridor. To formulate appropriate child behavior during evacuation, the force from the wall in the traditional social force model is changed into attractive, rather than repulsive force. We observed that psychological insecurity may be generated when children’s behavior is impaired. In addition, stronger forces towards the destination and the wall steer children with compromised vision. This corroborates our hypothesis. In Scenario 3, we observed that children ignore the impact from children in the rear and behave more decisively when they are convinced that larger flames are present. The repulsive force from the fire lamps demonstrates their natural aversion to danger. In this way, suitable parameters based on our improved social force model describe the preschool—aged group’s reaction in emergency evacuation by showing clear disparities from the parameters of adults. The calibrated model can then be embedded into the designing/evaluation process of dedicated facilities layouts such as kindergarten corridors, activity rooms, and playgrounds.Several simplifying assumptions in this paper could be relaxed in future research. For example, we only investigated children’s behavior under uniform smoke density distribution. This is a limitation of using sunglasses. It would be informative to study how dense smoke would affect route choice during evacuation, which involves latent cognition and senses other than vision. In addition, the social force model’s parameters for adults and students of other ages could be calibrated by additional experiments in the future. Moreover, experiments in a more complex interior space could be conducted to study children’s natural route choice behavior. For instance, cases where the exit is not visible, or in which multiple paths are present with the possibility of one being blocked could be studied. Under these circumstances, one may have to make dynamic judgement based on observing current conditions.



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