# Vehicle Lateral Control Based on Augmented Lagrangian DDPG Algorithm
## 5 Simulation Studies
### Simulation Platform and Model Verification
Training trajectories adopt the ISO 3888-1:1999 [26] emergency lane-change standard augmented with curvature perturbations (\(\Delta\kappa\sim\mathcal{U}(-0.2,0.2)\) m\({}^{-1}\)) and path length variations (\(\Delta s\sim\mathcal{N}(0,0.5\) m)), systematically expanding the operational envelope while preserving safety constraints. These explicitly defined parameters delineate the method's operational domain and enable reproducible validation under controlled autonomous driving conditions. The double lane change (DLC) maneuver, recognized as a representative scenario for evaluating vehicle stability during high-speed overtaking and obstacle avoidance, serves as a standard test for assessing vehicle handling and stability performance. The vehicle trajectory tracking control system is validated through co-simulation using the integrated Simulink and CarSim platform. To verify the simulation accuracy of both the 6-DOF vehicle dynamics model and CarSim S-Function module, a comparative validation was conducted under dual-lane change maneuvers at 60 km/h and 90 km/h operating conditions. The CarSim platform's built-in driver controller was employed to generate baseline vehicle responses, with both models receiving identical steering inputs and road signals. As shown in Figure 5, the comparative analysis of critical state parameters (including lateral velocity, yaw rate) demonstrates that the Simulink-implemented 6-DOF dynamics model achieves a high correlation with CarSim's reference outputs at 60 km/h and 90 km/h respectively. Quantitative comparisons between simulation platforms demonstrate exceptional agreement across operational scenarios. At 60 km/h, the lateral velocity (\(v_{y}\)) achieves an RMSE of \(1.990\times 10^{-4}\) m/s, while yaw rate (\(\omega\)) maintains \(3.376\times 10^{-5}\) rad/s precision. These metrics scale consistently with velocity, reaching \(4.228\times 10^{-4}\) m/s for \(v_{y}\) and \(7.023\times 10^{-4}\) rad/s for \(\omega\) at 90 km/h. The sub-millimeter/second level discrepancies validate the modeling fidelity of our Simulink implementation relative to the CarSim industry standard, confirming the vehicle dynamics replication accuracy essential for reliable reinforcement learning training. To verify the superiority of the proposed method, different models were used for training, including SAC (Soft Actor-Critic) [15], TD3 (Twin Delayed Deep Deterministic Policy Gradient) [16], DDPG [17], and DDPGALM algorithms in this paper, as shown in Table 1.  
Our methodology employs safety-constrained offline reinforcement learning to address the adaptation-safety dilemma. The training process utilizes the ISO 3888-1:1999 emergency lane-change trajectory as the baseline, introducing controlled perturbations during policy optimization to enhance generalization while preserving safety constraints. Highway geometry inherently bounds trajectory deviations through physical road boundaries, ensuring operational scenarios remain within the trained parameter space. Experimental validation in next section confirms the policy's robust adaptation to all tested trajectory variations without safety violations, demonstrating effective mitigation of offline training limitations through systematic environment-aware perturbation design.