# Behavioral Dynamics of Pedestrians Crossing between Two Moving Vehicles
## Appendix A Analysis of Two-Step Crossings  
In order to model two-step crossings in which there are two acceleration events (Figure 3b), we extend the model in the following way: We first take the acceleration equation  
\[\ddot{y}=\frac{\dot{y}}{\tau}\left(1-\frac{\dot{y}}{v_{\text{max}}}\right), \tag{1}\]which is equivalent to Equations 1 and 2 with appropriate initial conditions. In the two-step crossing, after acceleration (i.e., at time \(t>t_{a}+2\tau\)), the pedestrian will decelerate at a point \(y_{s}\) and stop until she/he accelerates again at time \(t_{s}\). This behavior may be described by adding two terms in Equation (A1), which leads to  
\[\dot{y}= \frac{\dot{y}}{\tau}\left(1-\frac{\dot{y}}{v_{\max}}\right)-r_{s} \dot{y}\exp\left[-\frac{(y-y_{s})^{2}}{\sigma_{s}^{2}}\right]\theta(t_{s}-t)\] \[+v_{s}\delta(t-t_{s}),\] (A2)  
where the second and the third terms of the right-hand side represent repulsion and impulse force, respectively. The repulsion is centered at position \(y_{s}\) with range \(\sigma_{s}\); \(y_{s}\) may be interpreted as the point beyond which the pedestrian perceives to be unsafe, due to the incoming traffic. Accordingly, \(y_{s}\) is the position of the flat region of the curve in Figure 3b, i.e., \(y_{s}=-2.3\,\mathrm{m}\) in this example. The Heaviside step function \(\theta(t_{s}-t)\) effectively \"turns off\" the repulsion force at time \(t_{s}\), thus removing the potential for collision after the vehicle has passed. The parameter \(r_{s}\) adjusts the overall strength of the repulsion. The impulse term is necessary for the model to undergo sharp acceleration from rest, so that the pedestrian starts to walk again at time \(t_{s}\). In the example of Figure 3b, the time at which the subject begins the second acceleration is given by \(t_{s}=2.0\,\mathrm{s}\). The magnitude \(v_{s}\) of the impulse is a fraction of \(v_{\max}\), and determines how quickly the model regains the maximum velocity.  
We fit Equation (A2) to the data for the case \(y_{0}=-3.5\,\mathrm{m}\) and \(l_{g}=25\,\mathrm{m}\), yielding \(y_{s}=-2.29\pm 0.22\,\mathrm{m}\). This implies that participants who performed two-step crossings walked forward about \(1.2\,\mathrm{m}\) before stopping, which amounts to about \(1.5\,\mathrm{m}\) from the path of the vehicles. We also have the impulse magnitude \(v_{s}/v_{\max}=0.67\pm 0.22\) and time \(t_{s}=2.41\pm 0.26\,\mathrm{s}\), which corresponds to the time for the pedestrian to start walking again. This range includes the time (about \(2.5\,\mathrm{s}\)) at which the leading vehicle passes the interception point. Other parameters, repulsion strength \(r_{s}\) and range \(\sigma_{s}\), which suffer from large fluctuations due to the very limited sample number, are obtained as \(r_{s}=520\pm 480\,\mathrm{s}^{-1}\) and \(\sigma_{s}=0.26\pm 0.24\,\mathrm{m}\). However, the sample size of two-step crossings was insufficient to derive statistically meaningful results. A possible behavioral interpretation of this type of crossing is as exchanging the additional energy required for deceleration and acceleration in favor of more safety or security.