Dynamic analysis of stepping behavior of pedestrian social groups on stairs

A video camera located on the fifth floor of the teaching building was adopted to record pedestrian movement on stairs. The frame rate and resolution of the video camera were 25 fps and 1920 × 1080 pixels, respectively. In order to quantify pedestrian groups' stepping behavior, we need to obtain pedestrians' movement trajectories by detecting and tracking them from the video recordings. Here, the optimal flow method [40, 44–46] was used to automatically extract each pedestrian's trajectory in image space, i.e., X and Y coordinates of each pedestrian's head at each frame. Then, the direct linear transformation method [47] was employed to correct and transform the trajectories in image space to those in real space, namely in the elevated perspective plane parallel with the incline direction of the stairway (see the blue dashed outline in figure 1(a)). An example of two groups' trajectories in real space is depicted in figure 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the purpose of analyzing stepping behavior of pedestrian social groups on stairs, such as step length, step width and speed, it is necessary to first detect stepping locomotion. From figure 2, it is evident that group members' trajectories show the feature of lateral swaying due to human biped walking, i.e. the mass center of a pedestrian alternates from one leg to the other during walking. Specifically, pedestrians' trajectories are composed of curves which oscillate continuously. Locations of pedestrians' feet can be reflected by peaks and valleys of the curves (see figure 3). In fact, the curvature change of the curves is similar to the trend of corresponding trajectories [20, 24, 25]. Thus, locations of steps can be obtained by calculating local maximum and minimum curvatures. The detailed calculation processes are in the following.

• (a)  
  Calculating curvature of trajectories (C(t)) at time t by equations (1) and (2).

  $$\begin{equation}C\left(t\right)=\frac{{x}^{\prime }\left(t\right){y}^{{\prime\prime}}\left(t\right)-{y}^{\prime }\left(t\right){x}^{{\prime\prime}}\left(t\right)}{{\left({x}^{\prime }{\left(t\right)}^{2}+{y}^{\prime }{\left(t\right)}^{2}\right)}^{\frac{3}{2}}},\end{equation} \tag{ 1 }$$

  $$\begin{equation}\begin{cases}{x}^{\prime }\left(t\right)=\frac{x\left(t+{\Delta}t\right)-x\left(t\right)}{{\Delta}t}\hfill \\ {y}^{\prime }\left(t\right)=\frac{y\left(t+{\Delta}t\right)-y\left(t\right)}{{\Delta}t}\hfill \\ {x}^{{\prime\prime}}\left(t\right)=\frac{{x}^{\prime }\left(t+{\Delta}t\right)-{x}^{\prime }\left(t\right)}{{\Delta}t}\hfill \\ {y}^{{\prime\prime}}\left(t\right)=\frac{{y}^{\prime }\left(t+{\Delta}t\right)-{y}^{\prime }\left(t\right)}{{\Delta}t}\hfill \end{cases}\end{equation} \tag{ 2 }$$

  where Δt = 0.4 s [20], and (x, y) represents the coordinate points of trajectories.
• (b)  
  Smoothing the curvature. As an example in figure 4, the time development of curvature (represented by magenta points) fluctuates periodically, but displays some noise. Thus, it is difficult to directly gain local peaks and valleys. Here, the Savitzky–Golay filter [24, 25, 48], which is able to maintain the original shape of a signal, is employed to smooth the time series of the curvature. The final smoothed result is depicted by the black line in figure 4.
• (c)  
  Obtaining all peaks and valleys of the smoothed curvature. In figure 4, it is easy to find all peaks and valleys (namely local maximums and minimums respectively represented by green full triangles and blue full stars) from the smoothed curve. Moreover, their corresponding time can be also identified. Thus, the stepping points of left and right feet can be obtained from the original trajectories. The precision of this method has been compared with manual identification in references [20, 24, 25]. The largest difference in step length was ±0.05 m with a relative error margin of ±9.8%.
• (d)  
  Calculating step width, step length and speed. After gaining stepping points of left and right feet from trajectories above, the step width (W(t)) and length (L(t)) between two successive stepping points (s(t₁) and s(t₂), or s(t₂) and s(t₃) in figure 3) can be calculated as follows.

  $$\begin{equation}W\left({t}_{2}\right)=\frac{\left\vert x\left({t}_{1}\right)y\left({t}_{2}\right)+x\left({t}_{2}\right)y\left({t}_{3}\right)+x\left({t}_{3}\right)y\left({t}_{1}\right)-x\left({t}_{1}\right)y\left({t}_{3}\right)-x\left({t}_{2}\right)y\left({t}_{1}\right)-x\left({t}_{3}\right)y\left({t}_{2}\right)\right\vert }{\sqrt{{\left(x\left({t}_{1}\right)-x\left({t}_{3}\right)\right)}^{2}+{\left(y\left({t}_{1}\right)-y\left({t}_{3}\right)\right)}^{2}}},\end{equation} \tag{ 3 }$$

  $$\begin{equation}L\left({t}_{2}\right)=\sqrt{{\left(x\left({t}_{1}\right)-x\left({t}_{2}\right)\right)}^{2}+{\left(y\left({t}_{1}\right)-y\left({t}_{2}\right)\right)}^{2}-W{\left({t}_{2}\right)}^{2}}\end{equation} \tag{ 4 }$$

  where (x, y) denotes the coordinate points of trajectories. t₁, t₂ and t₃ represent time corresponding to three successive stepping points (i.e., s(t₁), s(t₂) and s(t₃) in figure 3).

Zoom In Zoom Out Reset image size

Pedestrians' instantaneous velocity can be decomposed into two components parallel to step length (V_L) and width (V_w). Thus, V_L and V_w respectively denote the speeds of forward movement and body-sway motion, and can be obtained in the following.

$$\begin{equation}{V}_{L}\left({t}_{2}\right)=\frac{L\left({t}_{2}\right)}{{t}_{2}-{t}_{1}},\end{equation} \tag{ 5 }$$

$$\begin{equation}{V}_{W}\left({t}_{2}\right)=\frac{W\left({t}_{2}\right)}{{t}_{2}-{t}_{1}}\end{equation} \tag{ 6 }$$

Figure 5 displays the relationship between V_L (i.e., forward movement speed) and V_w (body-sway motion speed) for all singles and group members during ascending and descending movement. It is evident in figure 5(a) that when V_L is smaller than approximately 0.9 m s⁻¹ during ascending movement, increasing V_L will increase the value of V_w. When V_L is larger than approximately 0.9 m s⁻¹ during ascending movement, increasing V_L will decrease the value of V_w, i.e. the lateral swaying of body will be reduced, and pedestrians focus more on forward movement. However, these phenomena above are not very noticeable during descending movement in figure 5(b). The average value of V_w for all singles and group members in descending processes is 0.11 (±0.05) m s⁻¹, which is a little higher than that in ascending processes (i.e., 0.07 (±0.03) m s⁻¹). Both average values of V_w are a little smaller than those on flat surface in reference [24] where controlled single-file experiments were performed under different densities.

Zoom In Zoom Out Reset image size

Moreover, for pedestrians with the same value of V_L in figures 5(a) and (b), their values of V_w disperse over a certain range. The range is larger when V_L is higher. This is due to that pedestrians are in different groups with different sizes, and the speed of body-sway motion may be affected by group structure and sizes. It can be also seen that the maximums of different ranges during descending movement are relatively larger than those during ascending movement, as pedestrians' average descending speed is higher than average ascending speed.

Figure 6 illustrates speed–step length and speed–step time relations for all singles and groups during ascending movement. It is apparent that the step length increases with the increasing value of V_L. This trend is consistent with that on flat surface [24]. When V_L is smaller than around 0.9 m s⁻¹, the increase rate of step length is not evident (see figure 6(a)), as pedestrians' speeds are not large, and always walk step by step. They prefer to decrease step time in order to increase V_L (see figure 6(b)). When V_L is higher than 0.9 m s⁻¹, the increasing rate of step length is noticeable, due to that pedestrians (especially singles and dyads) may ascend two steps one time. Accordingly, their step time begins to slightly increase.

Zoom In Zoom Out Reset image size

For all singles and groups during descending movement, their step length slowly increases with the increasing value of V_L (see figure 7(a)), i.e. they always descend a step when V_L < 1.5 m s⁻¹, with the exception of two steps at large speed (V_L > 1.5 m s⁻¹). Correspondingly, their step time decreases with a decreasing rate, when speed increases but is smaller than around 1.5 m s⁻¹ (see figure 7(b)). Their step time starts to slightly increase if V_L > 1.5 m s⁻¹.

Zoom In Zoom Out Reset image size

We further compare the effects of group size and movement processes on step length and step time in tables 2 and 3. The statistical test, i.e., analysis of variance ((ANOVA)) [18, 49] is used to discuss the impact. From table 2, it is observed that the average step lengths of singles, dyads, triads and four-person groups during ascending movement are respectively 0.476 (±0.132), 0.438 (±0.094), 0.446 (±0.093) and 0.418 (±0.033) m. The average step lengths for different group types during ascending movement are all a little larger than those during descending processes, as the number of pedestrians who ascent two steps one time is larger than that of pedestrians descending two steps one time (see figures 6(a) and 7(a)). However, all the average step lengths during ascending and descending movement are a little larger than the inclined length of a step (i.e., 0.382 m), due to that some pedestrians' movement directions are not parallel with the incline direction of the stairway. It is noteworthy that differences in average step length induced by group size are not statistically significant both during ascending and descending movement, as shown by corresponding low ANOVA F and high p values (p > 0.05) in table 2.

Considering the influence of group size on step time in table 3, it is clear that difference in average step time induced by group size is not statistically significant during ascending movement, as shown by corresponding low ANOVA F and high p values (p > 0.05). Nevertheless, the difference is statistically significant during descending movement. The ANOVA F value is large, and the ANOVA p value is small (p = 0.006 < 0.05). Here, with the increasing group size, the step time increases during descending movement. This may be due to that with the increasing number of group members, they spend more time in communicating with each other and maintaining group structure, especially in the case of descending movement.

Given the effect of movement processes on step time in table 3, it is observed that for the same group size, the average step time during ascending movement is higher than that during descending movement. This is due to that pedestrians' average ascending speed is smaller than average descending speed (see figure 5), and they need to spend more time in maintaining larger step length (see table 2).

Figure 8 depicts average ascent and descent step widths of each single or group member. A serial number is given to each single or group. The mean in each subgraph is shown in table 4. It is evident that each pedestrian's average step width fluctuates around the mean in figure 8. The step width varies in the ranges of [0.01, 0.11] and [0.01, 0.14] during ascending and descending processes, respectively. The ranges are close to those obtained through controlled experiments on stairs by Chen et al [23]. Moreover, the maximum step width on stairs in this paper is similar to that on flat ground in references [22, 24]. This demonstrates that pedestrians may use the same gait to maintain motion balance, as test subjects in references [22, 24] and this paper were all Chinese college students. Nevertheless, the minimum step width on stairs in this paper is lower than that on flat ground. This may be due to different experimental setups and scenarios.

Zoom In Zoom Out Reset image size

Figure 9 illustrates the detailed distribution of step width for different group sizes during ascending and descending movement. It can be found that difference in average step width resulting from group size is not statistically significant during ascending movement, as reflected from corresponding low ANOVA F and high p values (p > 0.05) in table 4. However, the difference is statistically significant during descending movement, as shown by corresponding high ANOVA F and low p values (p < 0.05) in table 4. More specifically, the average step width increases with the increasing number of group members during descending movement.

Zoom In Zoom Out Reset image size

Considering the influence of movement processes on step width, it is observed that for the same group size, the average step width during ascending processes is a little smaller than that during descending processes (see table 4). This is because that pedestrians' average step length during ascending movement is larger than that during descending movement (see table 2), i.e. pedestrians focus more on forward movement during ascending movement.