# Path Tracking Control Strategy Based on Adaptive MPC for Intelligent Vehicles
## 5 Simulation Verification  
To validate the efficacy of the model for path tracking under varying curvature conditions, a continuous variable curvature curve condition is configured within CarSim, and a simulation test is conducted to compare and assess the control performance of the condition-adaptive controller against the traditional MPC and the HOSMC. As illustrated  
Figure 8: Overall structure of working-condition-adaptive path tracking controller. Figure 7: Time domain parameter adaptive MPC algorithm flow. in Figure 9, the reference path comprises segments featuring serpentine curves, a left-turn maneuver and line-shifting conditions. In this study, the standard MPC algorithm serves as a baseline for comparison with the proposed adaptive MPC method. The parameters for the standard MPC were adopted from established references [10,11,15], ensuring a fair evaluation. Specifically, the prediction horizon was set to 20 steps (1.0 s at Ts = 0.05 s), the control horizon to 5 steps and the weight matrices to Q = diag [200, 100, 100] and R = 5 \(\times\) 10\({}^{4}\). These settings align with widely accepted practices in vehicle dynamics control. The standard MPC, with its fixed parameters, is included to demonstrate the limitations of traditional methods in handling variable curvature paths, thereby underscoring the need for adaptive control strategies. The parameters for the standard MPC were tuned iteratively through simulation tests under typical conditions, following methodologies outlined in [11,15]. The chosen values prioritize tracking accuracy while maintaining computational feasibility. Set the road adhesion coefficient as 0.8 and the initial vehicle speed as 60 km/h. The simulation results are shown in Figure 10.  
Figure 10: Simulation results of \(\mu=0.8\) variable curvature. (**a**) Path tracking trajectories; (**b**) Lateral tracking deviation; (**c**) Yaw angle stability; (**d**) Yaw rate constraints. Figure 9: Reference path under variable curvature. In the path tracking trajectory (Figure 10a), the MPC controller exhibits a maximum lateral deviation of approximately 0.35 m in sharp curves, with the trajectory showing noticeable \"curve-cutting\" behavior. The HOSMC controller reduces the maximum deviation to 0.18 m, adhering more closely to the reference path, though slight oscillations persist in high-curvature sections. In contrast, the conditional adaptive controller achieves the smallest deviation (<0.1 m) with a smooth, oscillation-free trajectory, demonstrating superior performance in variable curvature regions (e.g., S-shaped curves). For lateral tracking deviation (Figure 10b), the MPC shows a peak deviation of 0.32 m at curve entry and a steady-state error of 0.05 m. The HOSMC reduces the peak deviation to 0.15 m with near-zero steady-state error, but its response time is slower (-1.2 s). The conditional adaptive controller achieves the lowest peak deviation (0.08 m), faster response (0.8 s) and zero steady-state error. Regarding yaw angle stability (Figure 10c), the MPC exhibits fluctuations of up to \(\pm 4^{\circ}\), with overshoot during abrupt curvature changes. The HOSMC reduces the fluctuation amplitude to \(\pm 2.5^{\circ}\) but shows high-frequency jitter. The conditional adaptive controller maintains minimal fluctuations (\(\pm 1.5^{\circ}\)), smooth dynamic response and no overshoot. For the yaw rate constraint (Figure 10d), the MPC exceeds the safety threshold (>15\({}^{\circ}\)/s) twice, lasting 0.5 s; the HOSMC remains within the threshold but approaches the upper limit (12\({}^{\circ}\)/s); meanwhile, the conditional adaptive controller consistently keeps the yaw rate below 10\({}^{\circ}\)/s with gradual variations. Under high-adhesion road surface and medium-to-high vehicle speed driving conditions, the conditional adaptive controller reduces tracking errors by over 50% compared to the MPC and improves yaw stability by 60%. Its key advantage lies in parameter adaptation, enabling real-time optimization of prediction and control horizons based on curvature.