## 3 Methodology
### Platoon Topology Control Based on Graph Theory and Safety Potential Field Theory
#### 3.1.1 Platoon Formation Control Model  
The formation of a platoon will definitely involve the lane changing process, and vehicles need to consider the complex road environment risk (e.g., surrounding vehicle information, road line information) to make motion decisions. It is necessary to ensure the absolute safety of the vehicles during the formation of the platoon. The safety potential field of a complex road environment can be divided into three categories: the lane marking potential field \(E_{L}\), the road boundary potential field \(E_{B}\) and the vehicle potential field \(E_{V}\)[30]. These three kinds of potential field can be expressed as  
\[ \begin{split} E_{v}&=M_{i}\lambda\frac{e^{-\beta_{ 1}\cos\theta}}{|K|^{5}}\cdot\frac{k^{\prime}}{|K|}\\ &=m_{i}\times\big{(}1.566\times 10^{-14}v^{6.687}+0.3345\big{)} \lambda\frac{e^{-\beta_{1}\cos\theta}}{\left\{\sqrt{\big{[}(x-x_{0})\frac{x}{ 2\pi}\big{]}^{2}+[(y-y_{0})\tau^{2}]}\right\}}\cdot\frac{k^{\prime}}{|K|}, \end{split}  \tag{8}\]  
\[ \begin{split} E_{L}=\sum\limits_{i,j=1}^{n}A_{i}e^{(-\frac{|U_{A }^{i}|}{2\sigma^{2}})}\cdot\frac{d_{A_{ij}}^{i}}{\left|d_{A_{ji}}^{i}\right|}, \\ d_{Aj}^{L}=y_{l,j}-y_{A},\end{split}  \tag{9}\]  
\[ \begin{split} E_{B}=\frac{1}{2}\eta\big{(}\frac{1}{\left|d_{ left left}^{B}\right|}\big{)}^{2}\cdot\frac{d_{left}^{B}}{\left|d_{left}^{B} \right|}+\frac{1}{2}\eta\big{(}\frac{1}{\left|d_{right}^{B}\right|}\big{)}^{2} \cdot\frac{d_{right}^{B}}{\left|d_{right}^{B}},\\ d_{left}^{B}=\left(y_{left}-y_{A}\right),d_{right}^{B}=\left(y_{ right}-y_{A}\right),\end{split}  \tag{10}\]  
where \(\lambda\), \(\beta_{1}\) and \(\zeta\) are determined coefficients, \(k^{\prime}\) is the pseudo-distance, \(M_{i}\) denotes the equivalent mass of object vehicle \(i\), \(\theta\) is the clockwise angle formed by any point around the object vehicle and the mass center of the object vehicle with the motion direction of the vehicle and \(a\) is the acceleration of the current motion state of the object vehicle. \(A_{i}\) represents the field intensity coefficient of different types of lane marking, which determines the maximum value of the lane marking potential field intensity. For example, the value of double amber lines \(A_{2}\) is much larger than that of lane-dividing lines \(A_{1}\). \(y_{A}\) is the ordinate value of point A in the road and \(y_{l,j}\) represents the position coordinates of the \(j\_th\) lane marking along the \(Y\)-axis; \(d_{Aj}^{L}\) represents the distance vector from point A to the \(j\_th\) lane marking; \(\sigma\) determines the velocity at which the potential field increases or decreases as the vehicle approaches or moves away from the lane marking. \(d_{left}^{B}\) and \(d_{right}^{B}\) represent the distance vector from point A to the left and right road boundaries, respectively. \(y_{left}\) and \(y_{right}\) are the position coordinates of the road boundary on the left and right sides, respectively. Finally, \(\eta\) is the coefficient of the road boundary potential field. Figure 2 displays the road boundary potential field (black dotted lines), the lane marking potential field (green dotted lines), the road vehicle potential field (red dotted lines) and the total potential field (blue solid line). The strength of the potential field at a certain point on the road represents the level of the potential traffic safety risk at that point. It can be found that the strength of the potential field reaches the maximum at the road boundary. The strength of the potential field is also relatively high at the position of the vehicle and the double yellow line. This characterization of the driving risk degree is completely consistent with the safety risks in the actual driving process of the vehicle. In  
our previous studies [29; 35], the vehicle potential field model was proven to accurately describe the driving risk of vehicles under different motion states in the CAVs environment.