# Vehicle Lateral Control Based on Augmented Lagrangian DDPG Algorithm
## 5 Simulation Studies
### Simulation Results and Analysis  
To comprehensively evaluate the performance of the proposed DDPGALM algorithm, comparative simulations were conducted against general deep reinforcement learning algorithms under two distinct speed conditions: 60 km/h (low speed) and 90 km/h (high speed). The selection of these speeds aims to validate the algorithm's adaptability across diverse driving scenarios. Low-speed scenario (60 km/h) reflect typical urban driving environments, where precise trajectory tracking and smooth control are essential. In contrast, high-speed scenario (90 km/h) impose greater challenges due to increased dynamic complexity, such as higher lateral acceleration and reduced reaction time, thereby testing the robustness and stability of the control strategy.  
Figure 6 shows the training process of different reinforcement learning algorithms in different scenarios. At 60 km/h (Figure 5(a),b), all algorithms demonstrate comparable training stability with similar episode termination frequencies and reward convergence patterns, attributable to reduced lane boundary constraint activations under lower dynamic complexity. The high-speed scenario (90 km/h, Figure 5(c),d) reveals critical differentiation DDPGALM significantly reduces early termination while sustaining stable reward progression, contrasting with general methods that exhibit frequent training interruptions. The off-road rate of DDPGALM, SAC, TD3, and DDPG algorithms is 24.8%, 51.4%, 64.4%, and 71.9% respectively at 60 km/h, and 14.8%, 35.7%, 75.5%, and 43.4% respectively at 90 km/h shown in (Figure 5(a),c). This performance divergence stems from DDPGALM's adaptive learning architecture, which dynamically adjusts exploration strategies and anticipates safety constraints under high-inertia conditions, thereby maintaining coherent policy evolution despite increased dynamic challenges.  
As shown in Figure 7, DDPGALM demonstrates remarkable trajectory tracking accuracy in a single episode of the simulation process. At 60 km/h (Figure 6(a),b), both algorithms achieve low lateral errors, but DDPGALM exhibits smoother trajectory adjustments, as evidenced by its reduced oscillation in Figure 6(b). At 90 km/h (Figure 6(c),d), the baseline DRL struggles with  
Figure 6: Reinforcement learning training process.  
accumulating trajectory deviations, particularly during sharp turns, leading to a maximum lateral error of 0.35 m. In contrast, DDPGALM limits the error to 0.15 m, highlighting its enhanced robustness under high-speed perturbations. Our rigorous lateral tracking standard (<0.05 m error threshold) yields 83.24% and 80.86% trajectory completion rates at 60 km/h and 90 km/h respectively, as quantitatively validated in Figure 7.  
Figure 8 analyzes steering angle and lateral acceleration. At 60 km/h (Figure 8a,b), the steering angle and lateral acceleration profiles of DDPGALM are smoother, ensuring passenger comfort. At 90 km/h (Figure 8c,d), the general DRL generates abrupt steering adjustments, resulting in unstable lateral acceleration peaks (up to 1.8 m/s\({}^{2}\)). DDPGALM, however, maintains controlled steering inputs, restricting lateral acceleration below 1.0 m/s\({}^{2}\).  
Figure 9 shows the reward values calculated by the reward function based on the vehicle's lateral tracking error, wheel steering angles, and lateral acceleration during the trajectory tracking process. The comparative simulations under low- and high-speed scenario demonstrate that DDPGALM outperforms conventional DRL in terms of training efficiency, trajectory tracking precision, and vehicle state stability. The high-speed scenario (90 km/h) particularly underscores the algorithm's advanced adaptability to dynamic challenges, a critical requirement for autonomous driving systems operating in complex environments. This dual-speed validation framework provides a comprehensive assessment of the algorithm's scalability and robustness, ensuring its applicability across diverse real-world scenarios.  
Figure 7: Simulation results of tracking trajectory.