## 4 Critical Safety Distance in Different Critical Conditions
### The Safety Distance Threshold of Avoiding Side Crashes  
As shown in Figure 2g, side crashes may occur between the \(LV\) and \(MV\), the \(MV\) and \(MV\), and the \(MV\) and \(RV\). For the convenience of expression, this study expresses the \(MV\) or \(RV\) of the vehicle that intends to change lanes as the \"ego vehicle,\" and the \(LV\) or \(MV\) of the vehicle in front of the target lane is expressed as the \"front vehicle.\" In other words, the ego vehicle may cause side crashes with the front vehicle in the target lane when changing lanes. This study calculated the IPF distribution of the two vehicles in the lane-changing scene according to the motion state of the two vehicles. As shown in Figure 6, there were two critical cases. In Figure 6a, the inner circles of the field value distributions of the two vehicles are tangent to each other, and in Figure 6b, the outer circles of the distributions of the real field values are tangent to each other. In the scene where the inner circles in Figure 6a are tangent to each other, in order to ensure that the ego vehicle avoids colliding with the front vehicle during and after  
Figure 6: The critical situation for side collisions: (**a**) the inner circles of the fields are tangent to each other and (**b**) the outer circles of the fields are tangent to each other. changing lanes, the minimum headway between the ego vehicle and the front vehicle should be greater than \(D_{min}(\theta)\); the relevant formulas are as follows:  
\[D_{min}(\theta)=D_{min}^{\text{IPF}}-\frac{L_{\text{e}}}{2}\cdot cos\theta+\frac{ L_{f}}{2} \tag{20}\]  
\[D_{min}^{\text{APF}}(\theta)=L_{e}^{\text{IPF}}\cdot cos\theta+L_{f}^{\text{IPF}} \tag{21}\]  
\[L_{e}^{\text{IPF}}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{e}\ cos\theta_{e}+ ax_{e}}\cdot\left(-\sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{ \text{p}}}\right) \tag{22}\]  
\[L_{f}^{\text{IPF}}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{f}+ax_{f}} \cdot\left(-\sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{\text{p}}}\right) \tag{23}\]  
where \(D_{min}(\theta)\) is the minimum headway; \(L_{\text{e}}\) and \(L_{f}\) are the vehicle lengths of the ego vehicle and front vehicle, respectively; \(D_{min}^{\text{APF}}\) is the minimum longitudinal distance between the vehicle centroids; and \(D_{min}^{\text{IPF}}(\theta)\) is the sum of \(L_{e}^{\text{IPF}}\) and \(L_{f}^{\text{IPF}}\). \(L_{e}^{\text{IPF}}\) and \(L_{f}^{\text{IPF}}\) 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 are derived from the potential field (Equation (1)). In Figure 6b, 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 \(D_{max}(\theta)\); this is shown in the following formulas:  
\[D_{max}(\theta)=D_{max}^{\text{IPF}}-\frac{L_{\text{e}}}{2}\cdot cos\theta+ \frac{L_{f}}{2} \tag{24}\]  
\[D_{max}^{\text{IPF}}(\theta)=L_{e}^{\text{IPF}}\cdot cos\theta+L_{f}^{\text{ IPF}} \tag{25}\]  
\[L_{e}^{\text{IPF}}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{e}\ cos\theta_{e}+ ax_{e}}\cdot\left(\sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{\text{p}}}\right) \tag{26}\]  
\[L_{f}^{\text{IPF}}=\frac{1}{2}M_{i}\gamma\tau\cdot e^{-\beta a_{f}+ax_{f}} \cdot\left(\sqrt{{E_{\text{e}}}^{2}-4{M_{i}}^{2}\lambda\gamma}+{E_{\text{p}}}\right) \tag{27}\]  
where \(D_{max}(\theta)\) is the maximum headway; \(L_{\text{e}}\) and \(L_{f}\) are the vehicle lengths of the ego vehicle and front vehicle, respectively; \(D_{max}^{\text{IPF}}\) is the maximum longitudinal distance between the vehicle centroids; and \(D_{max}^{\text{IPF}}(\theta)\) is the sum of \(L_{e}^{\text{IPF}}\) and \(L_{f}^{\text{IPF}}\).