# Modeling of Lane Changing Behavior of Connected Driving Based on Vehicle Interaction Potential
## 3 Acceptable Clearance for Lane Changing
### Vehicle Interaction Potential Lane Change Model Considering Acceptable Lane Changing Clearance  
The vehicle potential field force is a short-range repulsive force, which is used to prevent other vehicles from approaching to ensure safe driving. In general, the greater the vehicle potential, the stronger the potential field force. In the process of lane changing, vehicles are affected by multiple vehicle potentials, so it is necessary to determine the potential field force generated by each vehicle on the lane changing vehicle, and then judge whether the vehicle meets the conditions of safe lane changing. The lane changing diagram of vehicles as shown in Figure 9 is drawn, and the action potential of vehicle \(TL\), vehicle \(TF\), and vehicle \(L\) on lane changing vehicle \(M\) is analyzed respectively. The resulting action potential of vehicle \(TL\) on vehicle \(M\):  
\[U^{M-TL}=\frac{\varepsilon}{v-u}\left[u\left(\frac{G_{a}^{M-TL}}{L_{M-TL}} \right)^{v}-v\left(\frac{G_{a}^{M-TL}}{L_{M-TL}}\right)^{u}\right]\cdot\exp(a _{M}\cos(\theta-\beta))\cdot\frac{d^{*}}{|d^{*}|} \tag{16}\]  
Figure 8: Lane changing process of vehicle \(M\) and vehicle \(L\). Figure 9: Force analysis of lane changing vehicle \(M\). To simplify the formula, let \(\lambda=\exp(a_{M}\cos(\theta-\beta))\cdot\frac{\theta}{|d^{*}|}\), and the potential field force generated by vehicle \(TL\) on vehicle \(M\):  
\[F^{M-TL}=\frac{\lambda uve}{v-u}\bigg{[}u\frac{\left(G_{a}^{M-TL}\right)^{u}}{L_ {M-TL}^{u+1}}-v\frac{\left(G_{a}^{M-TL}\right)^{v}}{L_{M-TL}^{v+1}}\bigg{]} \tag{17}\]  
With \(\eta=\frac{\lambda uve}{v-u}\), we get  
\[F_{x}^{M-TL}=\eta\bigg{[}u\frac{\left(G_{a}^{M-TL}\right)^{u}}{L_{M-TL}^{u+1}}- v\frac{\left(G_{a}^{M-TL}\right)^{v}}{L_{M-TL}^{v+1}}\bigg{]}\sin\beta \tag{18}\]  
\[F_{y}^{M-TL}=\eta\bigg{[}u\frac{\left(G_{a}^{M-TL}\right)^{u}}{L_{M-TL}^{u+1}} -v\frac{\left(G_{a}^{M-TL}\right)^{v}}{L_{M-TL}^{v+1}}\bigg{]}\cos\beta \tag{19}\]  
where: \(F_{x}^{M-TL}\) represents the potential field force generated by vehicle \(TL\) on vehicle \(M\) in the direction of \(x\); \(F_{y}^{M-TL}\) represents the potential field force generated by vehicle \(TL\) on vehicle \(M\) in direction \(y\); and \(\beta\) represents the angle between the centroid connection of vehicle \(TL\) and vehicle \(M\) and axis \(y\). Vehicle \(TF\) exerts force on vehicle \(M\):  
\[F^{M-TF}=\eta\left[u\frac{\left(G_{a}^{M-TF}\right)^{u}}{L_{M-TF}^{u+1}}-v\frac {\left(G_{a}^{M-TF}\right)^{v}}{L_{M-TF}^{v+1}}\right] \tag{20}\]  
\[F_{x}^{M-TF}=\eta\left[u\frac{\left(G_{a}^{M-TF}\right)^{u}}{L_{M-TF}^{u+1}}- v\frac{\left(G_{a}^{M-TF}\right)^{v}}{L_{M-TF}^{v+1}}\right]\sin\alpha \tag{21}\]  
\[F_{y}^{M-TF}=\eta\left[u\frac{\left(G_{a}^{M-TF}\right)^{u}}{L_{M-TF}^{u+1}}- v\frac{\left(G_{a}^{M-TF}\right)^{v}}{L_{M-TF}^{v+1}}\right]\cos\alpha \tag{22}\]  
where: \(F_{x}^{M-TF}\) represents the potential field force generated by vehicle \(T\)\(F\) on vehicle \(M\) in direction \(x\); \(F_{y}^{M-TF}\) represents the potential field force generated by vehicle \(T\)\(F\) on vehicle \(M\) in direction \(y\); and \(\alpha\) represents the angle between vehicle \(T\)\(F\) and vehicle \(M\)'s centroid connection and axis \(y\). In the same way, it can be analyzed that vehicle \(L\) exerts force on vehicle \(M\):  
\[F^{M-L}=\eta\bigg{[}u\frac{\left(G_{a}^{M-L}\right)^{u}}{L_{M-L}^{u+1}}-v\frac {\left(G_{a}^{M-L}\right)^{v}}{L_{M-L}^{v+1}}\bigg{]} \tag{23}\]  
Figure 8:0  
Figure 8:1  
where: \(F_{x}^{M-L}\) represents the potential field force generated by vehicle \(L\) on vehicle \(M\) in direction \(x\); \(F_{y}^{M-L}\) represents the potential field force generated by vehicle \(L\) on vehicle \(M\) in direction \(y\); and \(\zeta\) represents the angle between vehicle \(L\) and vehicle \(M\)'s centroid connection and axis \(y\). According to the vehicle interaction potential theory, when the workshop distance is less than the acceptable gap, the force generated by the interaction potential is a repulsive force.