# Vehicle Lateral Control Based on Augmented Lagrangian DDPG Algorithm
## 1 Introduction  
The rapid advancement of autonomous driving technology has imposed increasingly stringent requirements on vehicle trajectory tracking control. As a core component of intelligent driving systems for new energy vehicles, trajectory tracking control must balance tracking accuracy, safety, stability, and multi-objective optimization capabilities in complex dynamic environments. Traditional control methods such as model predictive control (MPC), and sliding mode control (SMC) demonstrate advantages including strong interpretability and straightforward implementation. Recent MPC methods achieve sub-cm tracking via adaptive parameter optimization (fuzzy particle swarm optimization) and hierarchical control, effectively managing over-actuated systems (four wheel steer and four wheel drive/agricultural machinery) through real-time quadratic programming in high-dimensional spaces [1, 2, 3]. Advanced SMC integrates neural disturbance estimation with adaptive feedback, demonstrating lateral error and disturbance rejection under coupled dynamic constraints across variable road conditions [4, 5, 6]. However, their model-dependent nature fundamentally limits effectiveness in addressing nonlinear system dynamics and multi-constrained optimization challenges. Deep Reinforcement Learning (DRL) has recently emerged as a transformative alternative through end-to-end learning mechanisms that bypass explicit model requirements. Recent advances in deep reinforcement learning [7; 8; 9; 10] have enhanced intelligent vehicle control through model-integrated end-to-end frameworks [11], expert-guided pre-training [12], and PID hybrid architectures [13]. However, two fundamental limitations persist: (1) Policy instability during early training phases due to random parameter initialization incurs high exploration costs, and (2) Frequent safety violations (boundary breaches/collisions) during exploration reveal critical constraint enforcement deficiencies. To resolve these challenges, we propose a constrained optimization framework modeling lateral control as a CMDP, integrating three key innovations: an augmented Lagrangian safety optimizer, replay-separated experience buffers, and dual-Critic value estimation to prevent overoptimistic Q-values while ensuring stable policy updates. Deep Reinforcement Learning (DRL) directly maps environmental states to control actions through deep neural networks, with its core advantages manifested in two aspects: (1) leveraging the feature extraction capabilities of deep learning to handle high-dimensional nonlinear systems, and (2) utilizing the trial-and-error mechanism of reinforcement learning to achieve multi-objective dynamic optimization. Its significance stems from the breakthrough solutions offered by Deep Q-Network (DQN) [14] and its derivative algorithms--such as Soft Actor-Critic (SAC) [15], Twin Delayed Deep Deterministic Policy Gradient (TD3) [16], Deep Deterministic Policy Gradient (DDPG) [17], Double Deep Q-Network (Double DQN) [18], and Asynchronous Advantage Actor-Critic (A3C) [19]--in addressing complex decision-making problems. Deep reinforcement learning in vehicle control evolves through two paradigms: model-free and model-based approaches. Model-free methods like DDPG [11] employ end-to-end frameworks to directly map states to control commands, enhancing tracking accuracy in complex scenarios though challenged by inefficient initial exploration. While expert-guided pretraining [12] narrows exploration spaces, it imposes strict requirements on demonstration data quality. Model-based approaches integrate six-DOF dynamics with DDPG for handling tire-road interactions, yet remain vulnerable to parametric uncertainties (e.g., friction coefficient variations) [20] in real-world deployments. Safe Deep Reinforcement Learning (Safe DRL) establishes a CMDP to transform physical safety boundaries into hard constraints for policy optimization [21]. By incorporating constraint optimization mechanisms, Safe DRL ensures safety during algorithm training.