## 3 Car-Following IDM Considering PS  
PS is variable and regional. Currently, the quantitative calculation method for the shape and distance of PS has not been determined. [PERSON] et al. [17] proved through the simple stopping distance method that the boundary of PS is approximately circular. Furthermore, the results of anisotropic testing showed that the human anterior thoracic area has a larger demand for space than the back area [18]. This paper focuses on the forward and lateral distances of the vehicle. An egg-shaped boundary is defined based on the above elliptical boundary. The variability is determined by the velocity of the vehicle and individual driver differences. The regionality is determined by the PS boundary formula expressed in Equation (1) below.  
\[S_{i}(t)=\left\{\begin{array}{l}\frac{(x-x_{i}(t))^{2}}{A_{i}(t)^{2}}+\frac {(y-y_{i}(t))^{2}}{B_{i}(t)^{2}}\leq 1,x\geq x_{i}(t)\\ \frac{(x-x_{i}(t))^{2}+(y-y_{i}(t))^{2}}{B_{i}(t)^{2}}\leq 1,x<x_{i}(t)\end{array}\right. \tag{1}\]  
where \(S_{i}(t)\) is the PS boundary of vehicle \(i\) at time \(t\); \(x_{i}(t)\) and \(y_{i}(t)\) are the coordinates of vehicle \(i\) at time \(t\); \(x\) and \(y\) are points on the PS boundary; \(A_{i}\) is the forward distance of vehicle \(i\); and \(B_{i}\) is the lateral distance of vehicle \(i\). The specific shapes are shown in Figure 3.  
Figure 2: Relationship of vehicle velocity to forward and lateral distances.  
Based on the intelligent driver model (IDM) [8], as shown in Equation (2), and by considering the influence of PS on car-following behavior, a PS-IDM analytical model of car-following behavior is established, as shown in Equation (3).  
\[\left\{\begin{matrix}\dot{v}=a\left[1-\left(\frac{v_{0}}{v_{f}}\right)^{4}-\left( \frac{s_{1}}{h-1}\right)^{2}\right]\\ s_{1}=s_{0}+v_{0}T_{s}+\frac{v_{0}\Delta v}{2\sqrt{ab}}\end{matrix}\right. \tag{2}\]  
where \(\dot{v}\) is the vehicle acceleration; \(v_{0}\) is the current velocity of the vehicle; \(a\) is the maximum acceleration; \(v_{f}\) is the free-flow velocity; \(s_{1}\) is the desired vehicle distance; \(h\) is the current space headway of the vehicle; \(l\) is the vehicle length; \(s_{0}\) is the minimum parking distance; \(T_{s}\) is the safe time headway; \(\Delta v\) is the velocity difference from the vehicle ahead; and \(b\) is the comfortable deceleration.  
Based on literature review and test analysis, PS has excellent coupling with car-following model. Considering the effects of \(A_{i}\) and \(B_{i}\) in Equation (2), the following equation for the car-following behavior was obtained.  
\[\left\{\begin{matrix}\dot{v}=a\left[1-\left(\frac{v_{0}}{v_{f}}\right)^{4}- \left(\frac{s_{1}}{h-1}\right)^{2}\right]+\lambda c_{i}\\ s_{1}=s_{0}+v_{0}T+\frac{v_{0}\Delta v}{2\sqrt{ab}}\end{matrix}\right. \tag{3}\]  
In Equation (3), the time headway is an intuitive representation of the longitudinal dimension of the driver's difference, i.e., \(A_{i}\) denotes the corrected time headway \(T=T_{s}+\Delta T\), and \(\Delta T\) is the time headway correction value. \(B_{i}\) denotes the lateral distance factor term \(\lambda c_{i}\), \(\lambda\) is the lateral distance utility parameter, and \(c_{i}\) is the lateral intrusion distance of vehicle \(i\). As shown in Figure 4, the intrusion distance is related to the vehicle position and the current velocity.  
Figure 4: Schematic diagram of intrusion rate.  
Figure 3: PS boundary of vehicle.