Fluctuations around mean walking behaviors in diluted pedestrian flows

Understanding and modeling the dynamics of pedestrian crowds can help with designing and increasing the safety of civil facilities. A key feature of a crowd is its intrinsic stochasticity, appearing even under very diluted conditions, due to the variability in individual behaviors. Individual stochasticity becomes even more important under densely crowded conditions, since it can be nonlinearly magnified and may lead to potentially dangerous collective behaviors. To understand quantitatively crowd stochasticity, we study the real-life dynamics of a large ensemble of pedestrians walking undisturbed, and we perform a statistical analysis of the fully resolved pedestrian trajectories obtained by a yearlong high-resolution measurement campaign. Our measurements have been carried out in a corridor of the Eindhoven University of Technology via a combination of Microsoft Kinect 3D range sensor and automatic head-tracking algorithms. The temporal homogeneity of our large database of trajectories allows us to robustly define and separate average walking behaviors from fluctuations parallel and orthogonal with respect to the average walking path. Fluctuations include rare events when individuals suddenly change their minds and invert their walking directions. Such tendency to invert direction has been poorly studied so far, even if it may have important implications on the functioning and safety of facilities. We propose a model for the dynamics of undisturbed pedestrians, based on stochastic differential equations, that provides a good agreement with our field observations, including the occurrence of rare events.

Figure 1

(a) Sketch of the measurement site (staircase landing) with dimensions. Pedestrians walk from region L to region R or vice versa. From the individual trajectories (cf. thin lines, only seven reported for the sake of readability) we can define an average path $\overline{\mathrm{\Gamma }}$ (cf. thick dashed line) around which the ensemble of pedestrians fluctuates during their walk. (b) Color picture of a pedestrian walking from region L to region R (left to right). The picture has been taken via the color camera embedded in the Kinect sensor [12]. The framed region is larger than what was reliably captured by the Kinect depth sensor (cf. Fig. 3) and eventually considered in the analyses. This picture was shot in January 2017, i.e., more than 2 yr after the end of the observations. Notably, during the measurement period the dark banner at the right-hand side of the image was absent.

Figure 2

Probability distribution function of the number of pedestrians, $N_i$, passing in the corridor between two trajectory-inversion events (i.e., the number of consecutive crossings of the corridor). Comparison of measurements (red dots), simulation data from Eqs. (1) and (2) (black open circles), and a Poisson process with expectation $\mathbf{E}\left[N_i\right]=N_0=450$ pedestrians (blue dotted line).

Figure 3

Examples of Kinect depth maps of a single pedestrian walking undisturbed in our measurement site reaching the opposite side [cf. Fig. 1 and similar condition in Fig. 1]. (a) Three backgroundless depth maps from three instants close in time. The reconstructed trajectory of the pedestrian head is superimposed as a solid line. The grayscale colorization follows the depth levels: Darker pixels are closer to the camera plane; thus, heads, which are local extrema of the depth field, are darkest. The background, immutable in time, has been subtracted. (b) Example of a raw depth map for the middle frame in (a). Pixels whose depth could not be assessed reliably by the sensor are in black. These typically include far background pixels or shaded regions.

Figure 4

Transversal dynamics: comparison between measurements and model. We model the transversal motion as a harmonically bounded Langevin motion [cf. $y$ and $v$ dynamics in Eqs. (3) and (4)]. In (a) we report the time-correlation function of the transversal displacement $y$. The analytic solution [proportional to a cosine function with exponential decay; cf. Eq. (F5)] is reported as a blue dotted line. Measurements (red dots) and simulations (open dots) in a domain of equal size are in good agreement with the analytic solution. (b),(c) Probability distribution function of, respectively, transversal positions $y$ and transversal velocities $v$. In both cases the analytic solution is a Gaussian distribution (blue dotted line), which is in good agreement with the measurements (red dots). In the case of transversal positions $y$ we observe rare deviations from the Gaussian behavior at $\left|y\right|>0.4$. These are due to stopping events [cf. peak at $u=0$ in Fig. 5]. We refer the reader to Appendixes pp4, pp5, pp6 for further details on the calculations. Simulations included the same number of trajectories $N$ as in the observations $\left(N=72376\right)$.

Figure 5

Longitudinal dynamics: comparison between measurements and model. We model the longitudinal motion as a Langevin dynamics in a double-well velocity potential [cf. $x$ and $u$ dynamics in Eqs. (1) and (2)]. In (a) we compare the potential obtained from field measurements [after symmetrization of the velocities—cf. Eq. (8)—red dots] with the rescaled potential $R\varphi \left(u\right)=R{(u^2-u_p^2)}^2$ (dotted line). (b) Time correlation of the longitudinal velocity $u$. The analytic exponential decay of the linearized dynamics $\mathrm{[}exp(-8\alpha u_p^{2}t)$—cf. Eq. (9)—blue dotted line] is compared with measurements (red dots) and simulations of Eqs. (1) and (2) (in a virtual corridor with dimensions similar to those of our observations; open dots). The finite size of the corridor is responsible for a deviation from an exponential decay: From simulations, we expect the correlation to decay exponentially for small times only $\left(\tau <1.5\mathrm{s}\right)$. The measured time correlation (cf. Appendix pp4 for detailed formulas) decays around the expected exponential trend with larger discrepancies after $\tau >0.75\mathrm{s}$. Following the exponential decay at small times, we fit the correlation time $\left[{\left(8\alpha u_p^2\right)}^{-1}\right]$, i.e., $\alpha$. (c) Probability distribution function of longitudinal velocity $u$: comparison between measurements (red dots) and model (open dots). The simulated dynamics captures the entity of the fluctuation as well as the negative velocity tail within the considered approximation (neglected high-velocity behavior and stops).

Figure 6

(a),(b) Probability density function of pedestrian positions, respectively for pedestrians going left to right (a) and right to left (b). Average paths $\overline{\mathrm{\Gamma }}$ from (C1) are reported as a solid line. These are calculated after a binning of the measurements in 40 equal intervals within $[-1.0,0.8]$ in dependence on the $x$ coordinate. The figure axes report Cartesian coordinates aligned with the main directions of the corridor, respectively longitudinal, $x$, and transversal, $y^{\prime }$ [cf. Eqs. (C1) and (C2)]. (c),(d) Average velocity fields for pedestrians going left to right (a) and right to left (b). The fields are computed after binning the measurements in a $40\times 40$ grid on the region $[-1.0,0.8]\times [-0.6,0.6]$ and averaging velocity measurements bin by bin (fields are downsampled for readability). The evaluation of the transversal walking fluctuation $y$ and the longitudinal and transversal components of the walking velocity $u$ and $v$ [cf. Eqs. (1) to (4)] employ the references set in this way. The fluctuation $y$ is the distance [parallel to the $y^{\prime }$ axis, i.e., Eq. (C2)] from the average path [Eq. (C1)]. Longitudinal and transversal velocity components are computed after a projection of the measured velocity on the (normalized) velocity fields. Components are calculated independently for the two classes of pedestrians, then the contributions are merged to obtain the probability distribution functions in Figs. 4 and 5.

Figure 7

Error in the evaluation of the average path [Eq. (C1)] for pedestrians going left to right and vice versa. In both cases we split evenly and randomly the measurement sets in two. We report the absolute error on Eq. (C1) between the sets, which remains within the millimeter.