## 4 Simulation Analysis
### Low Coefficient of Friction Road Test (\(\mu_{s}=0.4\))  
Figure 14 illustrates a comparison of the models on a low-friction road surface. It is evident that the model outlined in this paper demonstrates consistent control performance on skid-prone road surfaces, thereby improving vehicle control.  
Driving on low-friction surfaces reduces the available traction of the vehicle, posing a greater challenge to maintaining vehicle control than on level surfaces. Consequently, such conditions can result in reduced acceleration, longer braking distances, and increased sideslip.  
Figure 15 illustrates the impact of reduced road adhesion on vehicle tracking. The interference caused by low friction disrupts PID tracking, leading to errors and instability. Conversely, the LCP and QP solvers continue to guarantee precise tracking, with the fastness of the LCP ensuring accurate and stable tracking even under low adhesion conditions. Detailed error data can be found in Table 3 and Figure 16.  
Figure 17 shows the torque input to the vehicle on a low-friction road surface in Figure 16. The LCP approach shows a faster rate of torque change, which facilitates a quick response to disturbances and improves safety and tracking accuracy. Similarly, Figure 18 shows that the LCP achieves more stable front wheel angle changes.  
Figure 14: Model comparison (\(\mu_{s}=0.4\)).  
Figure 13: Runtime (\(\mu_{s}=1\)).  
Figure 16: Tracking error statistics (\(\mu_{s}=0.4\)).  
Figure 17: The torque (\(\mu_{s}=0.4\)).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & & \\ \cline{2-4}
**Method** & **Average** & **Max** & **Min** & **MSE** \\ \hline MPC (LCP) & 3.1\% & 0.387 & \(-\)0.242 & 0.153 \\ MPC (QP) & 4.4\% & 0.621 & \(-\)0.342 & 0.243 \\ PID & 6.5\% & 0.887 & \(-\)0.476 & 0.356 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tracking error (\(\mu_{s}=0.4\)).  
Figure 15: The path tracking effect (\(\mu_{s}=0.4\)).  
Figures 19 and 20 depict the variation in the vehicle's side inclination and yaw angle. The LCP method ensures that the variation in these two angles is consistent and controllable and is characterized by moderate amplitude variations. This will ensure smooth and safe vehicle operation.  
Figure 21 depicts the computation time of the solver for various parameters. As demonstrated in Figure 22, employing the LCP solution instead of the QP solution results in a significant reduction in controller computational time. This enhanced computational effi  
Figure 19: The sideslip angle (\(\mu_{s}=0.4\)).  
Figure 20: The yaw rate (\(\mu_{s}=0.4\)).  
ciency does not compromise tracking performance but, instead, accelerates the convergence of path tracking and enhances vehicle robustness.  
In conclusion, based on the analysis of the two road test cases discussed above, it is evident that employing LCP solvers to solve the MPC optimization problem offers clear advantages. This approach ensures computational efficiency and enhances real-time performance while preserving tracking accuracy. The algorithm's feasibility has been theoretically substantiated. Nevertheless, it is crucial to acknowledge that the simulation study inherently relies on ideal assumptions. The real-life operation of self-driving cars introduces a host of additional physical factors that may exert computational pressure on the controller. Therefore, further investigation into the practical feasibility of the controller in real-world scenarios is warranted for a comprehensive understanding of its capabilities.