Introduction

Crowd dynamics is a research direction that spans multiple disciplines and fields, aiming to explore the movement characteristics of pedestrians in various situations and their potential coordination mechanisms. These include self-organizing behavior1, self-organizing queuing effects2,3, self-organizing synchronization effects4, leadership influence5, robot-guided evacuation6, non-adaptive behaviors7, game theory models8, and the use of machine vision for crowd density estimation and evacuation simulation9. During the evacuation process, the self-organizing characteristics, pace synchronization, and coordination behaviors exhibited by the crowd contribute to enhancing evacuation efficiency and safety. Currently, research in crowd dynamics primarily utilizes two approaches: experiments and numerical simulations. Among these, evacuation drills are the most common data collection method, and several related studies have been conducted10. For example, studies have investigated the impact of obstacles at exits on evacuation efficiency11, crowd evacuation experiments under restricted visibility12,13, counterflow experiments14, and crowd evacuation experiments on staircases in public spaces15. Additionally, research focusing on optimizing evacuation routes to reduce evacuation risks has also garnered widespread attention16,17,18,19,20. In terms of numerical simulations, various models have been developed, such as the modified social force model21, pedestrian dynamics-based simulation models22,23,24, heterogeneous crowd evacuation influence models25,26, and deep neural network models for predicting pedestrian evacuation27. Research has also considered pedestrian flow as fluid motion28. In29,30, grid units were used to represent spatial clusters, employing a lattice gas model to simulate pedestrian flow characteristics and conduct crowd evacuation simulations in hallways. Moreover, emergency evacuation simulations in construction sites have also been studied31. In recent years, research in crowd dynamics has encompassed various aspects such as self-organizing behavior, queuing effects, and psychological characteristics. Furthermore, the development of diverse simulation models based on pedestrian dynamics provides new perspectives for analyzing crowd movement and behavioral characteristics. These studies lay an important theoretical foundation and practical guidance for improving evacuation efficiency and safety.

With the increasing number of high-rise buildings and their increasingly complex structures, stairs have become an important means for people to move in vertical space during emergencies such as fires and earthquakes (e.g., in large factories32, stairways in scenic areas33, subway stations34,35, residential buildings36, and stadiums37). However, current research on crowd evacuation still primarily focuses on horizontal movement38. In comparison to horizontal movement, evacuating on stairs involves both horizontal and vertical displacement, making it more dangerous and more likely to result in stampedes and congestion39. In recent years, the study of stair evacuation has attracted the attention of many researchers40. Chen et al. found that children are more likely to be involved in stair-related stampede accidents than adults and face greater risks41. Shi et al. studied the evacuation of people of different age groups on stairs and found that age has a significant nonlinear impact on the speed of people moving up and down stairs21. Hashempour et al. analyzed the evacuation movements of children and adolescents on stairs, discovering that there is a significant difference in the average speed of teenagers going upstairs between genders, while the difference in average speed among children is relatively small42. Zhang et al. developed a model for evacuating using escalators, which showed that the evacuation efficiency of pedestrians using escalators is significantly higher than that of regular stairs, providing new ideas for evacuations in public spaces like large shopping malls43. Deng et al. built a three-stage layered model for calculating emergency evacuation times to analyze the evacuation process of people on stairs, providing a theoretical basis for emergency evacuation plans in high-rise buildings44. Latosinski et al. studied the impact of various parameters on the evacuation efficiency of firefighters climbing stairs, finding that the weight of carried equipment, walking distance, gender, age, body mass index, and the use of handrails all significantly affect evacuation speed45. Pouw et al. anonymously tracked the stair trajectories of millions of pedestrians in Eindhoven Central Station, analyzing the movement characteristics of pedestrians under unconstrained conditions and illustrating the changes in walking dynamics as they transition from stairs to landing46. Additionally, Cao et al. discovered that the synchronization mechanism triggered by people during vertical evacuation not only facilitates an orderly evacuation process but also improves evacuation efficiency; conversely, disordered evacuations can lead to inefficiency and increase the risk of congestion and stampedes47. Overall, although existing research primarily focuses on horizontal movement, studies on vertical evacuation on stairs are gradually increasing, providing new ideas and methods to enhance the safety and evacuation efficiency of high-rise buildings.

In many cases, interacting pedestrians in a crowd spontaneously adjust their pace. At a certain moment, the step frequency and phase of the leading and following pedestrians are equal, triggering a synchronization mechanism, a phenomenon known as synchronization. This synchronized behavior makes the entire evacuation process orderly4. Wang et al. observed two types of step length synchronization in single-file experiments and analyzed the impact of step length synchronization on step length48. Lu et al. discovered through single-file evacuation experiments that pedestrians’ step lengths are sensitive to social distance markers; after social distance markers are set, the reduction in pedestrians’ step lengths occurs faster than that of their step frequency49. Yanagisawa et al. introduced background music in single-file experiments and found that the rhythm of the music could reduce heterogeneity between individuals, thereby inducing synchronization during the evacuation process50. Pimentel et al. conducted a series of evacuation experiments in corridors and overpasses to observe pedestrian synchronization. However, their experimental video footage indicated that synchronization mechanisms were not triggered within the observed density range, even though the lateral sway of pedestrians increased with density during the experiment51. The oscillation that occurred on the day of the Millennium Bridge opening in London in 2000 significantly heightened interest in crowd synchronization52. More in-depth research on the synchronization of group dynamics during vertical evacuations (especially as a self-organized coordination effect between individuals) still requires attention. In the currently limited human self-organization experimental studies, tracking and detecting synchronization is not directly based on specific foot movements but rather on the trajectories of head movements53. Since single-file experiments involve the purest longitudinal forces, it allows for the calculation of various parameters such as pedestrians’ step lengths, and the single-file setup also helps improve evacuation efficiency54. Zeng et al. studied the walking dynamics of normal and fast-moving crowds in corridors using single-file experiments and found that pedestrians tend to exhibit stop-and-go behaviors when evacuating at normal walking speeds, while at faster walking speeds, all pedestrians have higher motivation to move, delaying stop-and-go behaviors55. Wang et al. conducted single-file evacuation experiments with middle and primary school students in a smoke-filled building during a fire drill and found that different age groups adapted differently to the evacuation environment56. Haghani et al. discovered that having pedestrians line up during an emergency significantly improves evacuation efficiency; conversely, when the evacuation pattern shifts from individual to group, the evacuation efficiency declines57. Tan et al. analyzed the dynamic characteristics of lateral movement in pedestrians using single-file evacuation experiments and found that the speed of pedestrians exhibits periodic fluctuations during lateral movement, with corresponding spatiotemporal diagrams showing serrated or wavy patterns58. Cao et al. designed single-file evacuation experiments for pedestrians under different visibility conditions, discovering that density and visibility significantly impact evacuation efficiency, and that under low visibility, pedestrians are prone to stop-and-go behaviors13. Overall, the current research findings provide an important theoretical foundation for understanding crowd behavior in emergencies, especially regarding improving evacuation efficiency and safety, highlighting the critical roles of self-organization and synchronization mechanisms. These advancements offer new insights and methods for future optimization of crowd evacuation and emergency management. This paper primarily conducts a single-file evacuation experiment for ascending and descending stairs, extracting pedestrian footprint samples to compute and analyze the self-organizing behavior of the crowd, termed “synchronization behavior.”

This paper primarily designs a single-file vertical evacuation experiment. The results revealed that pedestrians exhibit efficient collective synchronized movement, enhancing evacuation effectiveness. Visual analysis of the experimental data showed that once pedestrians trigger the synchronization mechanism and move in similar uniform linear motion, synchronization continues until the crowd movement ceases. Additionally, a characteristic density that most effectively triggers the synchronization mechanism was identified and its value calculated.

Methods

This experiment aims to study the formation process of synchronization mechanisms in vertical evacuation and to determine the characteristic density value that most easily triggers synchronization. It also analyzes the benefits that synchronization mechanisms provide to collective pedestrian movement. A small deviation from a straight line on stairs facilitates the triggering of the synchronization mechanism among pedestrians. This observation inspired us to conduct a single-file vertical evacuation experiment to investigate the triggering mechanism of synchronization. Therefore, this paper designs a vertical single-file evacuation experiment to investigate the triggering of synchronization mechanisms during vertical evacuation and the benefits they offer to collective human movement.

The experiment was conducted twice, in April and September 2023, at Sichuan Normal University in China, in accordance with relevant ethical regulations at the university and current laws in China. Approval for this research was obtained from the Ethics Committee of Sichuan Normal University. Participants received material rewards for their involvement. While they were not informed about the purpose of the experiment, they were briefed on the procedures. Informed consent was obtained from all participants.

As shown in Fig. 1(a), the experimental site is the staircase in the Sichuan Normal University cafeteria. To avoid interference from other people, the experiment was scheduled for one hour after the lunch period, ensuring that no one else would affect the process and that the lighting was better, minimizing the impact of insufficient light on pedestrian evacuation. This approach helps in obtaining the purest vertical stepping data. The staircase has 15 steps, each 0.148 m high and 0.30 m wide, with a total inclined length of 4.8 m, a vertical height of 2.24 m, and an inclination angle of approximately 27.85°(Fig. 1b). A total of 60 participants were randomly recruited (33 female, 27 male; average age 20 years, average height 166.87 cm).

Fig. 1
figure 1

Experimental scene diagram. (a) Experimental site. (b) Staircase side view. (c) Staircase top view. (d) Evacuation map of experiment participants.

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Considering the number of steps in the experimental staircase, we conducted evacuation experiments under three different densities (9, 14, 20) and three different speed gradients (\({v}_{1},{v}_{2},{v}_{3}\)). To collect sufficient data, each experiment lasted 2 to 3 min, resulting in a total of 9 experiments. Each experiment involved evacuation movements under the three speed gradients, with one staff member providing acceleration prompts according to the pedestrians’ evacuation process, leading to a total of 27 sets of experimental data. During each session, participants were initially lined up evenly at the bottom of the staircase. They were then instructed to move naturally along the staircase in a loop, as they would in real life, without overtaking the person in front. Additionally, the order of participants in the corridor was determined randomly to ensure variation in each participant’s adjacent predecessors and followers.

To track head movement, each participant was required to wear a red hat, as is common in most previous studies. To monitor the movement of the left and right feet, each participant was instructed to wear a pink shoe cover on their left foot and a blue shoe cover on their right foot, alternating with those in front of them (Fig. 1d). The different colors of shoe covers were used to clearly distinguish the feet of consecutive pedestrians. To reduce the impact of the shoe covers’ friction on participants’ foot movements, the covers were made of thick non-woven fabric, which has a friction level similar to that of regular shoe soles and helps prevent slipping on the tiles. These colored markers facilitate the simultaneous tracking of both head movements and the movement of pedestrians’ feet when extracting data.

To simultaneously capture the movement of both feet and the head of pedestrians, a camera (with a resolution of 1920 × 1080 and a frame rate of 30 fps) was installed in the middle of the horizontal slope on the side of the staircase (Fig. 1c). After the experiment, the recorded video clips were transferred to a computer and analyzed using TRACKER software. This software is widely used for extracting data on the trajectory of object movement and can extract preliminary trajectory data of evacuees from each experiment. TRACKER includes color recognition and tracking capabilities, allowing it to automatically identify and lock onto the coordinates and head positions of evacuees in each frame. However, in the process of automatic recognition, there will be errors and partial occlusion during the staggered movement of the feet, so in the process of data extraction, the software manual calibration method is used to obtain the coordinate position of each frame of each evacuator.

The camera was positioned in the middle of the left side of the horizontal slope during filming to simultaneously capture the position coordinates of pedestrians’ left foot, right foot, and head. Using multiple cameras for simultaneous recording would complicate the task of capturing all three positions—head, left foot, and right foot—at the same moment. Even if it were possible, time discrepancies among the three coordinates would complicate the analysis of pedestrians’ synchronization mechanisms. When extracting coordinates using TRACKER, some errors can arise due to the imaging principle not being perfectly horizontal or vertical. However, since the number of footprint samples collected during each experiment is consistent, regardless of the camera’s placement, the number of samples remains the same. When studying the effect of pedestrian speed on synchronization rate, the analysis is based on the number of footprint samples divided by the experiment duration. This approach mitigates issues related to coordinate errors from the non-horizontal imaging principle and ensures that head, left foot, and right foot coordinates are extracted simultaneously in each frame, eliminating the time discrepancies that would occur with multiple cameras.

Experiment Introduction

The experiment primarily focuses on analyzing two variables, density and velocity, to investigate the synchronous patterns of footsteps between leading and trailing individuals during crowd evacuation under various density and velocity gradient conditions. The study aims to calculate the characteristic density values that are most likely to trigger synchronization mechanisms and to analyze whether changes in velocity impact the occurrence of synchronization.

Experimental step

Step 1: Randomly assign numbers to 60 volunteers.

Step 2: Have the volunteers begin the experiment according to the sequence outlined in Table 1.

Table 1 Experimental statistical table.
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Step 3: Before starting the experiment, arrange the participants in a single-file line at a specific density. Upon receiving the signal from the staff, the participants begin the experiment.

Step 4: Start recording when the evacuation speed of the participants stabilizes. Initially, the pedestrians walk at a comfortable pace suited to their physical condition. After a period of movement, the \({v}_{1}\) time period ends when the last pedestrian completes one lap. Next, the staff issues the first acceleration command, prompting all pedestrians to increase their speed compared to their initial walking pace, although overtaking is not permitted. The \({v}_{2}\) time period ends when the last pedestrian completes one lap. Following this, the staff issues a second acceleration command, allowing pedestrians to move at a faster pace, nearly approaching a running speed. After a period of movement, the \({v}_{3}\) time period ends when the last pedestrian completes one lap. This completes one set of experiments. The experimental flow chart is shown in Fig. 2.

Fig.2
figure 2

Experimental procedures.

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During the experiment, the staff provided two acceleration prompts. When processing the data with TRACKER software, video segments for the \({v}_{1}\), \({v}_{2}\), and \({v}_{3}\) time periods were extracted separately using the software’s editing features. Each set of experiments yields data for three different speed gradients, resulting in a total of 27 sets of experimental data from 9 experiments.

Analysis and discussion

Data collection

Figure 3 (a) and (b) depict velocity change schematics, where the blue line represents the left foot, the red line represents the right foot, and v0 = 0.1 m/s denotes the velocity threshold. When the velocity is less than 0.1 m/s, the foot is considered in a stationary state. Figure 3 (c) and (d) illustrate the movement trends of the front and rear pedestrians, respectively. Figure 3 (e) and (f) present scatter plots of the left and right feet of pedestrians, with color intensity indicating the magnitude of movement speed—lighter colors correspond to higher speeds. From these figures, it is evident that pedestrians ascend stairs more slowly than they descend, which is consistent with previous findings 36. Additionally, within a single gait cycle, pedestrians take more time to ascend stairs compared to descending them. The motion trajectory during stair ascent forms a semi-circular arc, while during stair descent, it follows a straight line. Consequently, the speed variation trend during stair ascent is smoother, whereas during stair descent, the speed variation exhibits a unimodal pattern.

Fig. 3
figure 3

Schematic diagram of footprint sample extraction. (a) Velocity diagram of the front pedestrian. (b) Velocity variation diagram of the rear pedestrian. (c) Space–time diagram of the front and back pedestrians in X direction. (d) Space–time diagram of the front and back pedestrians in Y direction. (e) Pedestrian’s left foot track diagram. (f) Pedestrian’s right foot track diagram.

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By extracting the start and end times of each step for each pedestrian and calculating the longitudinal instantaneous velocities of each pedestrian’s left and right feet in each frame, the experiment ultimately collected a total of 17,464 complete footprint samples from pedestrian interactions, as shown in Fig. 3.

Synchronization trigger mechanism

In real-life scenarios, evacuation on stairs exhibits two trends: ascending and descending movements 59. When moving upward, pedestrians perform negative work, while moving downward involves positive work. Consequently, pedestrians display different phenomena when ascending and descending stairs. Figure 4 is a schematic illustration of pedestrian evacuation on stairs, both ascending and descending. Figure 4 (a) and (b) are ascending and descending stairs, while Fig. 4 (c), (d), (e), and (f) are velocity profiles. Specifically, Fig. 4 (c) and (d) display velocity changes during ascending and descending stairs, while Fig. 4 (e) and (f) focus on velocity changes during descending stairs. Figure 4 (g) and (h) illustrate the synchronized motion trends of pedestrians ascending stairs, while Fig. 4 (i) and (j) depict the synchronized motion trends of pedestrians descending stairs. The figures reveal that the trajectory of pedestrians going upstairs resembles a semi-circular arc, while the trajectory during downstairs evacuation trends towards a straight line. The depth of color in the figure represents the magnitude of speed. Analyzing the velocity change plots shows that speed variations in the first two plots are relatively smoother than in the latter two, indicating a more stable motion trend. Pedestrians experience rapid changes in speed while descending stairs, with velocity variation exhibiting a unimodal pattern. This phenomenon occurs because the trajectory of pedestrians going up stairs is a semicircular arc, while the trajectory when going down is a straight line.

Fig. 4
figure 4

Synchronous diagram of ascending and descending stairs. (a) Stair path. (b) Descent path. (c) Left foot up the stairs synchronization diagram. (d) Right foot up the stairs synchronization diagram. (e) Left foot down the stairs synchronization diagram. (f) Right foot down the stairs synchronization diagram. (g) Left foot going up the stairs in sync. (h) Right foot going up the stairs in sync. (i) Left foot going down the stairs in sync. (j) Right foot going down the stairs in sync.

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In the process of crowd evacuation, there are two forms of coordinated movement between the two feet: either synchronous or asynchronous. Synchronization, a self-organized coordination mechanism among individuals, relies on intrinsic cooperation to achieve alignment. By detecting the start and end times of each step for pedestrians, we find that when a leading pedestrian and a closely following pedestrian reach the same frequency and phase during one step cycle, synchronization occurs. Figure 5 (a) and (b) depict the schematic diagram of synchronization principles in the X-direction. In Fig. 5 (a), it illustrates a pedestrian ascending stairs, while Fig. 5 (b) represents a pedestrian’s descending stairs. In both figures, thick lines denote the leading pedestrian, and thin lines represent the trailing pedestrian. The variable \({\Delta t}_{k}^{i}\) represents the duration of any walking cycle for pedestrian \(i\). \({\Delta t}_{l,k}^{i}\) (\({\Delta t}_{r,k}^{i}\)) denotes the duration of the left (right) foot for pedestrian i during any walking cycle, and \({t}_{l,k}^{i,s}\) ((\({t}_{r,k}^{i,s}\)) signifies the initiation time of the left (right) foot for pedestrian i during any walking cycle. \({t}_{l,k}^{i,e}\) (\({t}_{r,k}^{i,e}\)) indicates the termination time of the left (right) foot for pedestrian \(i\) during any walking cycle. Figure 5 (c) and (d) illustrate the spatiotemporal diagram of the instantaneous velocity of the left (right) foot for the two pedestrians. The coincidence of the two lines indicates that the footstep frequencies of the two pedestrians are in the same frequency and phase state, i.e., frequency difference \({D}_{f}^{i}\cong 0\), phase difference \({D}_{p}^{i}\cong 0\). At this point, the two pedestrians are in a synchronized state. The calculation formula as in Eq. (2), \({\Delta t}_{f}^{r}\) and \({\Delta t}_{i}^{r}\) (where \({t}_{f,e}^{r}\) and \({t}_{i,e}^{r}\)) represent the step duration (end time) of the right foot for the leading pedestrian and their first follower. \({\Delta t}_{f}^{l}\) and \({\Delta t}_{i}^{l}\)​ (where \({t}_{f,e}^{l}\)​ and \({t}_{i,e}^{l}\)) represent the step duration (end time) of the left foot for the leading pedestrian and their first follower. Pedestrian step movement is similar to a sine function. When the frequencies and phases of two sine functions are the same, the two sine functions will coincide. If the frequencies and phases of the leading and following pedestrians are close, \({D}_{f}^{i}\cong 0\) and \({D}_{p}^{i}\cong 0\), indicating that the leading and following pedestrians are almost perfectly synchronized. Specifically, for two pedestrians to achieve synchronization, two conditions need to be satisfied. (1) start time conformity. and (2) duration conformity, as specified in Eq. (1). In the equations, \({t}_{f}^{s}\) and \({t}_{i}^{s}\) (\({\Delta t}_{f}\) and \({\Delta t}_{i}\)) represent the single-step start times (durations) of the follower and leader, and \({K}_{1}\) (\({K}_{2}\)) is the permissible error threshold for the difference in start times (durations), set to 0.1 s 60. The values of \({K}_{1}\) and \({K}_{2}\) are crucial here, as extreme values for \({K}_{1}\) and \({K}_{2}\) can impact statistical accuracy. In this case, their values are derived from empirical values reported in the literature53. As shown in Fig. 5 (a) and (b), synchronization is considered when, within one step cycle, the left and right feet of the front pedestrian and their follower synchronize sequentially, and the instantaneous velocity change graphs of the left (right) feet of the front and rear pedestrians tend to overlap, as illustrated in Fig. 5 (c) and (d). The two 2/T represent one synchronization cycle, where the left and right feet of the front and rear pedestrians synchronize sequentially. According to this method, a total of 2,467 pairs of synchronized samples were detected.

Fig. 5
figure 5

Schematic diagram of synchronization formation. (a) A synchronized space–time diagram of the stairs. (b) A synchronized space–time diagram of going down the stairs. (c) Left foot speed sync diagram. (d) Right foot speed sync diagram.

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$$\left\{\begin{array}{c}\left|{t}_{f}^{s}-{t}_{i}^{s}\right|\le {K}_{1}\\ \left|{\Delta t}_{f}-{\Delta t}_{i}\right|\le {K}_{2}\end{array}\right.$$
(1)

Analysis reveals that once synchronization is established, if pedestrians maintain a constant speed and continue moving in a straight line, this synchronization will persist throughout the period of crowd movement, as shown in Fig. 6. Figure 6 (a) is a spatiotemporal diagram, where color depth indicates the magnitude of pedestrian velocities. It can be observed that, due to pedestrians performing negative work while ascending stairs, their speed is relatively slower during this phase. The interval before time Ta represents the stair ascent phase for both leading and trailing pedestrians. The \(\Delta \text{T}1\) period indicates the transition from ascending to descending stairs for the leading and trailing pedestrians, while the \(\text{Tb}-\text{Tc}\) period represents the stair descent phase. The \(\Delta \text{T}2\) period is the transition from descending to ascending stairs. Figure 6 (b) and 6 (c) depict time-momentum diagrams, where momentum is defined as \(p=mv\). Here, \(m\) represents mass, and in the diagram, the motion trend of a particular point is used for detection, so \(m=1\) and \(p=v\). In the ∆T1 phase, both the direction and velocity of the particles change, disrupting the synchronization mechanism established before this phase. After this disruption, due to inherent coordination among leading and trailing pedestrians, the synchronization mechanism is re-triggered, initiating synchronized movement during the Tb-Tc period. In the ∆T2 phase, the change in direction disrupts synchronization once more. In the ∆T2 phase, the change in direction disrupts synchronization once more. Thus, pedestrian motion follows an infinite loop of order → disorder → order (synchronization → asynchronization → synchronization). Once pedestrians activate the synchronization mechanism and maintain nearly constant linear motion with consistent velocities, synchronization persists.

Fig. 6
figure 6

Synchronous – asynchronous- synchronous cycle diagram. (a) Spatiotemporal diagram. (b) Synchronous-asynchronous velocity variation diagram of the left foot. (c) Synchronous-asynchronous velocity variation diagram of the right foot. (d) Spatiotemporal diagram. (e) Asynchronous—synchronous velocity variation diagram of the left foot. (f) Asynchronous—synchronous velocity variation diagram of the right foot.

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$$\left\{\begin{array}{c}{D}_{f}^{i}=\left|\frac{1}{{\Delta t}_{f}^{r}}-\frac{1}{{\Delta t}_{i}^{r}}\right|+\left|\frac{1}{{\Delta t}_{f}^{l}}-\frac{1}{{\Delta t}_{i}^{l}}\right|\\ {D}_{p}^{i}=\left|{t}_{f,e}^{r}-{t}_{i,e}^{r}\right|+\left|{t}_{f,e}^{l}-{t}_{i,e}^{l}\right|\end{array}\right.$$
(2)

The impact of velocity changes on synchronization rate

When studying the impact of velocity on pedestrian evacuation efficiency, most current research primarily focuses on pedestrians’ instantaneous velocity61. Instantaneous velocity provides information only about speed at a specific moment, while overall speed over a period of time is typically represented by average velocity. To quantitatively analyze the relationship between speed and the occurrence of synchronization, a model using the original speed formula (v = ∆s/∆t) is applied, and an average speed formula for a foot is defined. The following section will primarily analyze the relationship between foot average speed and synchronization.

Tables (2), (3), and (4) present the relationship between pedestrian’s foot speed and synchronization rates. Figure 6 provides a visual analysis of a portion of the data from these tables. It can be observed from Fig. 7 (a), (b), and (c) that changes in velocity do not exhibit a direct correlation with synchronization rates. To further validate the relationship between velocity and synchronization rates, a one-way analysis of variance (ANOVA) was employed, yielding F = 0.9 and P = 0.455 (greater than 0.05). Therefore, it can be concluded that there is no significant difference between these variables, indicating that changes in velocity do not have a direct correlation with synchronization.

Fig. 7
figure 7

Speed and synchronization rate graph.

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Tables 2, 3, and 4 provide numerical values for three experiments, including the sample size of pedestrian footprints, synchronous samples, and synchronization rates. In the tables, \({\rho }_{i}\) represents pedestrian density, \({v}_{g}\) represents velocity gradient, \({v}_{\rho }^{i}\left(c/s\right)\) denotes the magnitude of foot speed, \({\overline{v} }_{0}(c/s)\) represents the optimal speed value, \({F}_{s}\left(c\right)\) indicates footprint samples, \({S}_{s}\left(r\right)\) represents synchronous samples, and \({S}_{o}^{r}\left(\%\right)\) denotes the synchronization rate. The synchronization rate is defined by Eqs. (3) and (4), representing the synchronous occurrence rate of two pedestrians over a certain period and the total synchronous occurrence rate of all pedestrians under specific pedestrian density during a time interval. In these equations, N denotes the total number of pedestrian footprints, \({n}_{i}\) represents the individual footprint count, and \({S}_{n}\)) indicates the number of synchronous occurrences. Foot speed magnitude and average foot speed are calculated using Eqs. (5) and (6), where \(\rho\) represents pedestrian density, \(i\) indicates the velocity gradient, N is the number of footprint samples during this time, \({n}_{i}\) is the number of individuals, and \({n}_{\rho }\) is the count of speed magnitudes. The speed of pedestrians is influenced by both the surrounding environment and individual psychological tolerance, causing it to vary. To quantitatively analyze the impact of speed on synchronization occurrence, we use the definition of speed from the original physical formula (v = ∆s/∆t) as a prototype. A formula for calculating the speed of a foot when a pedestrian disperses vertically has been defined, as shown in Formula (5). The calculated pedestrian motion speeds \({v}_{9}^{1}\), \({v}_{14}^{1}\), and \({v}_{20}^{1}\) were used to define the error values \(\kappa\), which are the differences between them. When the error value is less than 1, its average can be considered an optimal value for quantitative analysis. The choice of 1 as the critical point for κ < 1 is based on the definition of foot speed for footprint samples, where the number of samples is considered the distance in the speed formula. If this value is less than 1, it will not form an independent foot cycle, making it unsuitable for synchronization rate analysis; hence, 1 is chosen as the critical value.

Table 2 Record of footprint sample data for experiment 1.
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Table 3 Record of footprint sample data for experiment 2.
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Table 4 Record of footprint sample data for experiment 3.
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$${S}_{r}=\frac{{S}_{n}}{\left({n}_{i}+{n}_{i+1}\right)/2} {P}_{{S}_{n}}<0.001$$
(3)
$${S}_{o}^{r}=\frac{{S}_{n}}{\sum_{i=1}^{n}{n}_{i}/2}=\frac{{S}_{n}}{\left({n}_{1}+{n}_{2}+\cdots +{n}_{i}\right)/2}$$
(4)
$${v}_{\rho }^{i}=\frac{N/n}{\Delta t} {P}_{{v}_{\rho }^{i}}<0.001$$
(5)
$${\overline{v} }_{0}=\frac{\sum_{i=1}^{n}{v}_{\rho }^{i}}{{n}_{\rho }}$$
(6)

In summary, under typical circumstances, the movement pattern of human footsteps follows an infinite cycle of disorder- order–disorder (asynchronous-synchronous-asynchronous). From experimental data and statistical analysis, it can be concluded that variations in speed do not significantly impact the occurrence of synchronization. This finding contradicts common intuition, as people generally believe that an increase in speed would disrupt the inherent coordinated movement mechanism between individuals, leading to a decrease in synchronization. However, this overlooks the fact that prior to the increase in speed, individuals have already engaged in a certain duration of evacuation movement. During this time, continuous mutual coordination occurs among individuals, establishing an intrinsic coordination mechanism (similar to the everyday concept of unspoken understanding). Even with changes in speed, this intrinsic coordination mechanism continuously adjusts to external factors, maintaining a balance in movement trends. This is why the impact of increased speed on the synchronization rate is not substantial.

Characteristic density to induce synchronization mechanism

The preceding analysis quantitatively examined the relationship between the average speed of footsteps and synchronization under three velocity gradients for small groups at different densities. Some researchers have indicated the existence of a characteristic density during horizontal evacuation 62, at which pedestrians are most likely to trigger synchronization mechanisms. In comparison to horizontal evacuation, during vertical evacuation, pedestrians exhibit both longitudinal and lateral displacements. However, pedestrian movement still follows the alternating cyclic motion of two feet. Therefore, during vertical evacuation, there exists a characteristic density at which the likelihood of triggering pedestrian synchronization mechanisms is maximized. The subsequent focus will be on computing this characteristic density value.

Studies have indicated the existence of a characteristic density at which synchronization is most likely to occur, with an increase in population density being a primary triggering factor for synchronization53. This section primarily analyzes the relationship between density and synchronization rate. Firstly, the Pearson correlation coefficient is calculated to assess the strength of the relationship between density and synchronization rate. The calculated Pearson coefficient is -0.95, as per Eq. (7), indicating a strong correlation between density and synchronization rate. Secondly, a visual analysis of the density and synchronization rate from the data in Tables 2, 3, and 4 is conducted. The trend of density and synchronization rate follows a normal distribution, as shown in Fig. 8(a), and the effect of density on synchronization rate is statistically significant, as indicated in Fig. 8(b) and (c) (Fv1 = 290.67023, Pv1 < 0.001; Fv3 = 21.981, Pv3 = 0.0161 < 0.05). From statistical and visual analysis, it can be concluded that there is a significant difference between the two, and there may be a characteristic density at which pedestrian density reaches a maximum synchronization occurrence rate.

Fig. 8
figure 8

Conceptual Illustration of the Relationship Between Density and Synchronization Rate. (a) Effect of density on synchronization rate. (b) Significance analysis at v1. (c) Significance analysis at v3.

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$${r}_{\rho v}=\frac{\sum {\rho }_{i}{v}_{i}-n\overline{\rho }\overline{v} }{\left(n-1\right){s}_{\rho }{s}_{v}}=\frac{n\sum {\rho }_{i}{v}_{i}-\sum {\rho }_{i}\sum {v}_{i}}{\sqrt{n\sum {\rho }_{i}^{2}-{\left(\sum {\rho }_{i}\right)}^{2}}\sqrt{n\sum {v}_{i}^{2}-{\left(\sum {v}_{i}\right)}^{2}}}$$
(7)

From the above analysis, it can be concluded that there is a characteristic density at which the synchronization occurrence rate of pedestrians is highest. Synchronization is only meaningful when the density between individuals is relatively high, as illustrated in Fig. 9. At low densities, the distance between people increases, minimizing the risk of collisions and reducing the likelihood of injuries or fatalities. However, at higher densities, individuals are closer together. When the synchronization mechanism is triggered, it enhances movement efficiency and helps prevent incidents like stampedes and congestion, thereby reducing casualties and economic losses. When the distance between two pedestrians is small enough to accommodate the length of one foot of the follower (specifically one step vertically, as shown in Fig. 9), two scenarios can arise when synchronization is not maintained: 1) The follower, trying to avoid a collision with the leader, may wait for the leader to step first. This inaction can prevent the follower from signaling those behind, potentially causing chaos if an accident occurs, as individuals might engage in irrational evacuation behaviors due to fear 63. 2) The follower may collide with the leader, leading to various forms of congestion and stampede accidents. In either case, synchronization is crucial for accident prevention.

Fig. 9
figure 9

Foot-to-Foot distance diagram under synchronization conditions.

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According to the analysis above, the synchronization mechanism proves highly beneficial in reducing accidents, and a characteristic density value exists. When the population density reaches this value, the synchronization mechanism is most likely to be triggered. Figure 10 (a) and (b) depict the head-to-head distance of two pedestrians at a specific moment, representing the distance between their heads at the beginning and end of a synchronization cycle (30 frames). By following this method, a total of 2,467 head distance diagrams for synchronized pedestrian pairs were extracted. Subsequently, using Eqs. (8) to (11), We computed the characteristic density value most likely to trigger the synchronization mechanism, as shown in Fig. 10 (c) and (d). Figure 10 (c) displays the probability density function (PDF), resulting in parameter values of \(\mu =2.48704\) and \(\sigma =0.48828\). Figure 10 (c) displays the probability density function (PDF), resulting in parameter values of μ = 2.48704 and σ = 0.48828, while Fig. 10 (d) corresponds to the cumulative distribution function (CDF). From Fig. 10, it can be seen that at the maximum of the approximate PDF, the characteristic density value most likely to trigger the synchronization mechanism is \(2.5{m}^{-1}\). In the formula, \({r}_{i}^{s}\)(\({r}_{f}^{s}\))denotes the spatial position of the leading pedestrian (follower) at the beginning of a synchronization cycle, \({r}_{i}^{e}\)(\({r}_{f}^{e}\))denotes the spatial position of the leading pedestrian (follower) at the end of a synchronization cycle, \({\Delta r}_{s}\) represents the spatial distance between two pedestrians before and after the start of a synchronization cycle, \({\Delta r}_{e}\) denotes the spatial distance between the two pedestrians at the end of a synchronization cycle, \({\Delta \overline{r} }_{h}\) indicates the distance between the two pedestrians within a synchronization cycle, \(\rho\) is the characteristic density of pedestrians triggering the synchronization mechanism.

Fig. 10
figure 10

Fitted graph of characteristic density. (a) Initial moment. (b) End moment. (c) Probability density function. (d) Cumulative distribution function.

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$${\Delta r}_{s}=\left|{r}_{f}^{s}-{r}_{i}^{s}\right| { P}_{{r}_{f}^{s}}=0$$
(8)
$${\Delta r}_{e}=\left|{r}_{f}^{e}-{r}_{i}^{e}\right|\quad{P}_{{r}_{f}^{e}}=0$$
(9)
$${\Delta \overline{r} }_{h}=\frac{{\Delta r}_{s}+{\Delta r}_{e}}{2}$$
(10)
$$\rho =\frac{1}{{\Delta \overline{r} }_{h}}\quad{P}_{\rho }=0$$
(11)

It has been calculated that the synchronization mechanism is most likely triggered at \(\rho ={2.5}^{-m}\). Next, a fitting plot of head distance versus pedestrian density for the extracted 2,467 pairs of synchronization footprint samples is shown in Fig. 11 (\(SSE=7.56*{e}^{-29}\), \(MSE=4.13*{e}^{-16}\), \({R}^{2}=95\%\)). The power-law characteristics depicted in Fig. 11 indicate a monotonic relationship between distance and density, demonstrating that the characteristic density value triggering the synchronization mechanism is unique. The synchronization mechanism is most likely triggered between pedestrians in front and behind only when the pedestrian density reaches this specific value during vertical evacuation. This also implies that, at this density, pedestrian evacuation efficiency is at its peak, maximizing safety benefits.

Fig. 11
figure 11

Fitted graph of density and distance.

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Utilization of single file evacuation in real-life scenarios

Currently, single-file evacuation is widely applied in emergency evacuations in public places, fire situations, and other emergencies, as well as in emergency drills for schools and organizations, large event management, and traffic management. For instance, in crowded places like shopping malls, sports venues, and cinemas, single-file evacuation helps people leave in an orderly manner, reducing chaos and congestion while enhancing safety. Schools and other institutions often conduct fire drills using single-file evacuation strategies to ensure that individuals can exit quickly and orderly. In daily life, pedestrians are often required to engage in single-file evacuation, such as when dining in restaurants, conducting business in banks, or going through security checks at train stations and airports. In these cases, individuals may either be asked to line up in single file, or the layout is designed to facilitate single-file movement. For example, at airport security checkpoints, the space from the entrance to the exit is designed in an "S" shape, which allows pedestrians to pass through in a single-file manner. This design helps alleviate evacuation pressure during peak times, thereby improving evacuation efficiency, as illustrated in Fig. 12.

Fig.12
figure 12

Application of single file evacuation in real life.

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Conclusion and discussion

The article primarily investigates the inherent coordination mechanism of groups during stair evacuation—specifically, "synchronous behavior." We quantitatively analyzed the relationship between the average speed of footsteps and the occurrence of synchronization, calculating the characteristic density value that most easily triggers the synchronization mechanism. This research provides a theoretical foundation for improving the efficiency and safety of stair evacuations and suggests potential directions for future research and practice. By enhancing our understanding of crowd evacuation behavior, we can effectively raise public awareness of evacuation safety, thereby protecting lives in emergencies. The main research findings include:

(1) Once pedestrians trigger the synchronization mechanism and maintain a near-uniform linear motion, synchronous behavior will continue until the crowd movement stops, provided their speed remains unchanged. In actual evacuation scenarios, pedestrian movement is not merely mechanical; emotions can also influence behavior. Negative emotions may spread among the crowd during an incident, leading to a higher likelihood of stampedes and congestion compared to normal circumstances. “Synchronous behavior” can enhance crowd evacuation efficiency and reduce the occurrence of congestion and stampede accidents. (2) Using one-way ANOVA and experimental data visualization, we found that the average speed of footsteps does not have a direct relationship with the probability of synchronization occurring. (3) The characteristic density value we calculated for triggering the synchronization mechanism most easily is \({2.5m}^{-1}\), which holds significant implications for evacuation guidance during incidents. The calculated safe distance for triggering the synchronization mechanism is 0.4 m, which is the distance between the heads of the person in front and the follower, roughly equivalent to the reach of a person extending their arm to the back of the person ahead.

This article calculates the characteristic density value at which pedestrians are most likely to trigger synchronization mechanisms during staircase evacuation. It finds that when single-file pedestrians maintain a constant-like linear motion, once the synchronization mechanism is triggered, it is difficult to disrupt, and this movement mechanism significantly impacts evacuation efficiency. Additionally, the researchers have calculated the characteristic density value for triggering synchronization mechanisms during horizontal plane evacuations4. Despite the differences in evacuation mechanisms between stairs and horizontal planes, both synchronization behaviors are crucial for enhancing evacuation efficiency. Future research could further explore synchronization mechanisms under different environments and formations to gain a more comprehensive understanding of the dynamic characteristics of pedestrian crowd evacuation. Currently, experimental studies on pedestrian crowd evacuation primarily focus on single-file evacuations on horizontal planes13,54,55,56, while recent years have seen a gradual increase in research on stairs, including evacuations of individuals of different age groups on stairs, escalator evacuation models, and firefighters’ evacuations on stairs. It has been found that factors such as gender, age, and load weight can all influence evacuation efficiency on stairs21,41,42,43,44,45,46. However, studies on the synchronization behavior of pedestrians on stairs are relatively scarce, and existing literature only mentions that synchronization behavior can enhance evacuation efficiency, noting that stair conditions tend to lead pedestrians to form a trend of linear movement during evacuation47,64. Furthermore, research on self-organizing behavior, synchronization mechanisms, step speed, step length, and step timing characteristics of pedestrians on stairs remains insufficient, with current studies mainly concentrated on horizontal areas48,60,65,66. These characteristics provide theoretical support for optimizing stair space structures, offering effective evacuation guidance strategies, and reducing the occurrence of stampede accidents. Therefore, future research should emphasize the study of movement characteristics of pedestrians on stairs. Given the widespread use of escalators in many large public places and their higher evacuation efficiency, in-depth research on the evacuation mechanisms of pedestrians on escalators is also particularly important43,67,68,69.

The limitations of this study are as follows: Our experiments primarily focused on the youth population, which does not fully reflect the behaviors of individuals of different age groups during evacuation. Therefore, future research should include participants of varying ages, physical conditions, and psychological characteristics to more comprehensively simulate real evacuation scenarios. Moreover, actual evacuation processes often involve complex dynamics with multiple queues, while this study was mainly based on single-queue experiments. Future research could expand from one-dimensional experiments to multi-dimensional models through numerical simulations, allowing for a more accurate analysis of crowd evacuation behavior under different conditions. Lastly, the lack of real data from natural scenarios limits our comprehensive understanding of crowd behavior in actual emergencies. Obtaining data from natural settings would facilitate a deeper exploration of which factors are most likely to trigger synchronization mechanisms, predict potential bottlenecks, optimize evacuation route designs, and better understand pedestrian behavior patterns and psychological responses. This would provide more practical references for emergency management and architectural design.

In summary, this study has deepened our understanding of crowd dynamics and their behavioral characteristics in emergencies, revealing the potential of synchronized behavior to enhance evacuation efficiency. These findings not only enrich the theoretical content of crowd dynamics research but also offer practical guidance for evacuations during natural disasters such as earthquakes and fires. Future research could further explore synchronization mechanisms in different contexts to enhance our understanding and predictive capabilities regarding crowd behavior.