## 4 Critical Safety Distance in Different Critical Conditions
### The Safety Distance Threshold for Avoiding a Rear-End Crash  
As shown in Figure 2f, rear-end crashes may occur between \(LV\) and \(MV\), \(MV\) and \(MV\), and \(MV\) and \(RV\). For the convenience of expression, an \(MV\) or \(RV\) that intends to change lanes is expressed as the \"ego vehicle,\" and an \(LV\) or \(MV\) of the vehicle in front of the target lane is expressed as the \"front vehicle.\" In other words, the ego vehicle may collide with the front vehicle in the target lane when changing lanes. According to the motion state of the two vehicles, the IPF distribution of the two vehicles in the lane-changing scene is shown in Figure 5, where there are two critical cases. In Figure 5a, the inner circles of the field value distribution of the two vehicles are tangent to each other, and the outer circles of the distribution of the real field values are tangent to each other in Figure 5b. In the scene where the inner circles in Figure 5a are tangent to each other, it can be understood that in order to ensure that the ego vehicle avoids colliding with the front vehicle during and after lane changing, the minimum headway between the ego vehicle and the front vehicle should be greater than \(D_{min}\). Its formulas were as follows:  
\[D_{min}=D_{min}^{IPF}-\frac{L_{e}}{2}+\frac{L_{f}}{2} \tag{11}\]  
\[D_{min}^{IPF}=L_{e}^{IPF}+L_{f}^{IPF} \tag{12}\]\[L_{e}^{IPF}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{e}\cos\theta_{e}+\alpha v _{e}}\cdot\left(-\sqrt{E_{v}{}^{2}-4M_{i}{}^{2}\lambda\gamma}+E_{v}\right) \tag{13}\]  
\[L_{f}^{IPF}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{f}+\alpha v_{f}}\cdot \left(-\sqrt{E_{v}{}^{2}-4M_{i}{}^{2}\lambda\gamma}+E_{v}\right) \tag{14}\]  
where \(D_{min}\) is the minimum headway; \(L_{e}\) and \(L_{f}\) are the vehicle lengths of the ego vehicle and front vehicle, respectively; \(D_{min}^{IPF}\) is the minimum longitudinal distance between vehicle centroids; and \(D_{min}^{IPF}\) is the sum of \(L_{e}^{IPF}\) and \(L_{f}^{IPF}\). \(L_{e}^{IPF}\) and \(L_{f}^{IPF}\) are the distances from the tangent point to the mass center of the ego vehicle and front vehicle, respectively, under the critical potential field value, which can be derived from the potential field equation (Equation (1)). Figure 4: Flow chart of the platoon lane-changing process. In Figure 5b, where the outer circles are tangent to each other, it can be understood that in order to ensure that the distance between the two vehicles is not too large, which would affect the stability of the platoon, the maximum distance between the ego vehicle and the front vehicle in the lane changing process should be less than \(D_{max}\); this is shown in the following formulas:  
\[D_{max}=D_{max}^{IPF}-\frac{L_{\text{e}}}{2}+\frac{L_{f}}{2} \tag{15}\]  
\[D_{max}^{IPF}=L_{e}^{IPF}+L_{f}^{IPF} \tag{16}\]  
\[L_{e}^{IPF}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{e}\ cos\theta_{e}+ax_{ e}}\cdot\left(\sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{\text{e}}}\right) \tag{17}\]  
\[L_{f}^{IPF}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{f}+ax_{f}}\cdot\left( \sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{\text{e}}}\right) \tag{18}\]  
where \(D_{min}\) is the maximum headway; \(L_{e}\) and \(L_{f}\) are the vehicle lengths of the ego vehicle and front vehicle, respectively; \(D_{max}^{IPF}\) is the maximum longitudinal distance between the vehicle centroids; and \(D_{max}^{IPF}\) is the sum of \(L_{e}^{IPF}\) and \(L_{f}^{IPF}\). \(L_{e}^{IPF}\) and \(L_{f}^{IPF}\) are the distances from the tangent point to the mass center of the ego vehicle and the front vehicle, respectively, under the critical potential field value. They are derived from the potential field (Equation (1)) and the formulas are shown above. During the process of lane changing, it is assumed that the motion state of the surrounding vehicles remains unchanged.