## 2 Optimal Lane Change Path Planning Based on the NSGA-II and TOPSIS Algorithms
### Optimization and Decision Result Analysis  
The proposed method is implemented and tested based on MATLAB. The computer is loaded with a Windows system with AMD Ryzen5 3.0 GHz processor and 16 GB memory.  
The weights of the three optimization objective functions are calculated by Equations (13)-(15) as follows: \(\omega_{1}=\)0.2940, \(\omega_{2}=\)0.2157, and \(\omega_{3}=\)0.4903. The top five solutions in the Pareto optimal solution set obtained through TOPSIS decision making are shown in Table 2.  
Therefore, the longitudinal displacement of lane changing is finally determined to be 78 m, and the lane changing time is 5.2 s. The optimal lane changing trajectory function can be obtained by substituting the two parameters into Equation (1) and combining the state quantities at the initial and termination moments of the ego vehicle lane changing. The lane changing trajectories of the ego vehicle and other surrounding vehicles are shown in Figure 8, where \(V_{e}\) represents the ego vehicle, \(V_{fc}\) represents the vehicle in front of the current line, \(V_{ft}\) represents the vehicle in front of the target line, and \(V_{rt}\) represents the vehicle in rear of the target line.  
In Equation (1), the lateral trajectory is a function of the longitudinal displacement \(x\). The lateral velocity and acceleration can be obtained by Equation (22).  
\[\left\{\begin{array}{l}\dot{y}(t)=\dot{y}(x)\dot{x}(t)\\ \ddot{y}(t)=\ddot{y}(x)\dot{x}^{2}(t)+\dot{y}(x)\ddot{x}(t)\end{array}\right. \tag{22}\]  
Then, the longitudinal and lateral displacement, velocity, acceleration, yaw angle, curvature, and other information of the vehicle are obtained, and the schematic diagram of these variables with time is drawn by Matlab as follows (Figure 9).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multicolumn{2}{c}{**The Decision Variables**} & \multicolumn{2}{c}{**Objective Optimization Function**} & \multirow{2}{*}{\(R_{m}\)} \\ \cline{2-2} \cline{4-6}  & **Longitudinal** & **Time** & & & & \\ \cline{2-2} \cline{4-6}  & **Displacement/m** & **/s** & & & & & \\ \hline
1 & 78 & 5.2 & 0.5947 & 0.0035 & 78.1286 & 0.9160 \\
2 & 80 & 5.2 & 0.5827 & 0.0034 & 80.1254 & 0.9154 \\
3 & 76 & 5.0 & 0.6137 & 0.0037 & 76.1320 & 0.9148 \\
4 & 82 & 5.4 & 0.5478 & 0.0032 & 82.1223 & 0.9136 \\
5 & 80 & 5.8 & 0.8051 & 0.0034 & 80.1254 & 0.9118 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The top 5 solutions in the Pareto optimal solution set.  
Figure 8: Lane changing trajectories of the ego vehicle and other surrounding vehicles.  
Figure 9: _Cont._  
It can be seen from the figure that the displacement, velocity, and curvature of the planned trajectory are smooth and continuous. The longitudinal velocity is gently accelerated from 13.89 m/s to 16.67 m/s, and the lateral velocity is gently accelerated from 0 to 1.35 m/s and then slowed down to 0. The lane changing process is relatively stable. The maximum value of longitudinal acceleration is about 0.95 m/s\({}^{2}\), the maximum value of lateral acceleration is about 0.91 m/s\({}^{2}\), and the acceleration value of the start and end of lane changing is 0. The optimal lane changing path designed in this study conforms to the constraints of lane changing trajectory and is suitable to be used as the actual lane changing reference trajectory.  
The path planning method in [20] is compared with the proposed method. Similar to the lane change scenario in [20], the maximum curvature of the lane change trajectory planned in this study is 0.0221/m, which is smaller than that in [20] (0.05729/m), and the lane changing velocity and acceleration curves in [20] are not smooth. The comparison shows that the method proposed in this study can ensure a more stable and comfortable lane changing process.