2.1.1. Establishment of mechanical structure model for EMB system.

1. (1) Brake Motor Model

Permanent Magnet DC (PMDC) motors are selected as the drive motor for the braking system owing to their low armature inductance. This characteristic enables rapid voltage adjustment and short response times, making PMDC motors suitable for applications that require frequent starts, stops, or speed regulation. The voltage balance equation of the PMDC motor is expressed as:

(1)

where U_a is the armature terminal voltage, E_a is the back electromotive force(EMF), I_a is the armature current, R_a is the armature resistance, L_a is the armature inductance. The back EMF is proportional to the motor speed:

(2)

where K_e is the EMF constant, ω is the rotor angular velocity.

The mechanical dynamics of the PMDC motor are governed by:

(3)

where J is the moment of inertia, T_em is the electromagnetic torque, T_L is the load torque, and B_m is the viscous friction coefficient.

The electromagnetic torque T_em is proportional to the armature current I_a:

(4)

where K_t is the torque constant.

1. (2) Transmission mechanism model

The transmission mechanism consists of a planetary gear reducer and a ball screw assembly. The rotation of the motor output shaft drives the ball screw, which converts rotational motion into linear displacement of the nut. The kinematic relationship is described as:

(5)

where S_bn is the horizontal displacement of the nut, θ_b is the EMB motor rotation angle, i_t is the reducer gear ratio; and L_b is the ball screw lead.

1. (3) Clamping Force Model

The planetary gear reducer amplifies the motor torque and speed, while the ball screw converts torque into caliper pad pressure to generate clamping force. Considering the nonlinear effects of clearance between the ball screw and caliper pads, the clamping force is formulated as:

(6)

where ΔS_bn is the difference between the piston displacement and the brake clearance; k₁, k₂, k₃, k₄ are clamping force coefficients.

In addition, when the ball screw structure presses the caliper pad, a reaction force is generated. This force acts on the output shaft of the drive motor through the transmission mechanism, forming a load torque. Its magnitude can be expressed as:

(7)

where η_b is the working efficiency of the ball screw, and η_t is the working efficiency of the reducer.

1. (4) Brake Disc Model

The piston thrust forces the caliper pads against the brake disc, generating friction torque:

(8)

where is the friction coefficient between the pad and disc, and r_e is the effective friction radius of the brake disc.

To address mathematical discontinuities during the transition from static to dynamic friction, a hyperbolic tangent-modified Coulomb friction model is adopted:

(9)

where μ_c is the sliding friction coefficient; n_d is the brake disc rotational speed; and n_c is the speed limit.

By adjusting the parameter μ_c, a smooth transition between positive and negative friction torques can be achieved. This approach avoids the issue of numerical mutation at the zero-speed crossing encountered in traditional piecewise linear models. As a result, the numerical stability of brake judder and noise simulations is significantly improved.

2.1.2. Modeling of EMB system based on Amesim.

Based on the mathematical model derived in Section 2.1.1, the EMB system model built in Amesim does not include tire models, brake caliper models, etc. [31]. The EMB system model and its main parameters are shown in Fig 3 and Table 1, respectively.

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Table 1. Main parameters of braking system.

https://doi.org/10.1371/journal.pone.0335686.t001

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Fig 3. Braking system model.

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2.2.1. Establishment of mechanical structure model for SBW system.

1. (1) Rack-and-Pinion Model

Kinematic analysis of the pinion and rack yields the governing equation:

(10)

where x is the rack-and-pinion displacement, K_m is the rigidity coefficient of the motor output shaft, m_r is the equivalent mass of the pinion and rack, F_R is steering resistance, b_r is the damping coefficient of the rack, r_p is the pinion radius, and θ_m is the motor torque.

To simplify model complexity, the steering resistance is expressed as:

(11)

where ke is the elastic coefficient of the equivalent spring, and F_d denotes road excitation signals.

1. (2) Front-Wheel Steering Angle Model

The front-wheel steering angle is determined by the displacement relationship between the steering linkage and wheel rotation, governed by:

(12)

where J_f is the moment of inertia of the front wheel about the kingpin, M_z is the self-aligning torque of the steering system, C_kf and K_rp are the damping and stiffness coefficients of the kingpin, respectively, and R_st is the transmission ratio from the rack to the steering wheel.

2.2.2. Modeling of SBW system based on Amesim.

Based on the mathematical model derived in Section 2.3.1, the SBW model is built in Amesim. The steer-by-wire system model and its main parameters are shown in Fig 4 and Table 2, respectively.

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Table 2. Main parameters of steering system.

https://doi.org/10.1371/journal.pone.0335686.t002

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Fig 4. Steering system model.

https://doi.org/10.1371/journal.pone.0335686.g004

2.3.1. Design of simulation scenarios.

To enhance the control robustness of conventional algorithms under hazardous conditions, three typical accident-prone scenarios are implemented in CarSim: curved road, low-friction road, and split-μ road. Each scenario is parameterized with road geometry and curvature, vehicle placement, obstacle vehicles, reference trajectory, initial conditions, friction coefficients, and other relevant variables.

The split-μ road surface and low-adhesion road surface simulation scenarios are configured as a 200 m straight single-lane road. For the split-μ condition, the adhesion coefficient is set to 0.4 on the left side and 0.8 on the right side. The low-adhesion scenario uses a uniform coefficient of 0.2. The curved road scenario with a radius of 100 m and a length of approximately 628 m is designed with a bilateral adhesion coefficient of 0.8 to evaluate lateral stability. In all three scenarios, a stationary obstacle vehicle is placed at the 100 m mark. This setup is used to evaluate the vehicle’s braking response and calculate the braking distance once the obstacle is detected.

2.3.2. Input-output parameter configuration.

The input variables of the simulation environment are set as: braking torque of the four wheels and front wheel steering angle; the output variables include lateral displacement, vehicle sideslip angle, yaw rate, yaw angular acceleration, braking distance, wheel slip ratio, vehicle velocity, and braking deceleration. The scenario simulation step size is set to 0.01s, and the maximum simulation time is 10 s.

3.1.1. Vehicle instability state judgment.

Identifying the root causes of potential instability under various scenarios is essential for designing effective vehicle stability control methods [33]. In vehicle dynamics, the fundamental mechanism of instability stems from the trade-off between lateral and longitudinal tire forces, as constrained by the friction ellipse theory. The lateral force increases with the slip angle within a certain threshold. Beyond this threshold, however, lateral force saturation occurs, resulting in a sharp decline in tire grip and further aggravating instability. Therefore, it is crucial to maintain the slip angle within allowable limits to ensure stable vehicle motion [34–36]. Consequently, this study adopts maximum slip angle thresholds as one of the instability criteria. The conditions for instability judgment are defined as follows:

(13)

Where ε₁, ε₂ are adjustable parameters, ε₃ = 0.005 is the yaw rate control coefficient [37], β and represent the vehicle sideslip angle and vehicle sideslip angle rate; φ and φ_d denote the yaw rate and ideal yaw rate; α and α_max correspond to the tire slip angle and its critical threshold.

According to reference [38], the calculation formulas for ε₁ and ε₂ are presented in Equation (14).

(14)

While the yaw rate and sideslip angle increase along with the ideal yaw rate, excessively high ideal values under extreme conditions signal an impending loss of control. Therefore, a violation of any of the three stability criteria presented in Equation (13) suggests that the vehicle is in or approaching an unstable state.

3.1.2. Dynamics parameter derivation.

To guide the vehicle in maintaining stability within the DRL model, this study employs three metrics for evaluating lateral stability: the deviation between desired and actual yaw rates, the deviation between desired and actual vehicle sideslip angles, and the vehicle lateral displacement. To improve DRL training efficiency, critical reference variables that characterize complete instability are defined to prevent ineffective training. These variables include the critical tire slip angle(α_max), critical vehicle sideslip angle(β_max), ideal vehicle sideslip angle(β_d), critical yaw angle(φ_max), and ideal yaw rate(φ_d), all of which are derived from the dynamic model. Such thresholds enable the DRL agent to learn a balance between stability and control efficiency while adhering to vehicle dynamics constraints. By integrating these constraints into the reward function, the agent is guided to prioritize stability-critical behaviors without compromising computational efficiency. The proposed framework ensures compliance with physical feasibility limits, thereby enhancing convergence stability during policy optimization.

1. (1) Critical tire slip angle

Derived from the tire magic formula model, the critical tire slip angle is calculated by assuming the tire lateral force reaches its peak value.

(15)

Where B represents the lateral shape factor, and C denotes the stiffness factor.

1. (2) Critical vehicle sideslip angle and ideal vehicle sideslip angle

Lateral stability control requires consideration of road adhesion constraints and vehicle dynamic characteristics to determine φ_max and φ_d. Based on the 2-degree-of-freedom (2-DOF) vehicle model:

(16)

Where m denotes the total vehicle mass; C_f and C_r represent the front and rear tire cornering stiffness, respectively; v_x represents the longitudinal velocity; δ_f is the front wheel steering angle; v_y refers to the lateral velocity; I_z stands for the yaw moment of inertia about the vertical axis; is the yaw angular acceleration; l_f and l_r denote the distances from the center of mass to the front and rear axles, respectively.

The ideal yaw rate plays a pivotal role in ensuring that vehicle steering responsiveness aligns with control expectations. It prevents excessive sensitivity or lag, thereby providing a precise and reliable control target for control algorithms. This derivation assumes the vehicle operates under steady-state driving conditions, where the yaw angular acceleration and lateral acceleration are negligible (i.e., ; ). Substituting these conditions into the Equation (16) yields the following two-degree-of-freedom (2-DOF) dynamic model for steady-state motion:

(17)

Following from Equation (16):

(18)

However, the lateral acceleration is constrained by the road-tire adhesion coefficient (μ), and under non-ideal driving conditions where vehicle dynamics deviate from theoretical assumptions, the yaw rate (φ) is ultimately limited by the physical adhesion boundary. This yields the relationship: φv_x ≤ μg. Consequently, it follows that:

(19)

Therefore, the φ_max and φ_d can be formulated as:

(20)

(21)

1. (3) Critical yaw angle and ideal yaw rate

When the vehicle sideslip angle is small, it can be simplified as: , From Equation (16), it follows that:

(22)

Following from Equation (21):

(23)

Considering the constraints imposed by the road adhesion coefficient, the following equation can be derived:

(24)

Therefore, the β_max and β_d can be formulated as:

(25)

(26)

3.2.1. Markov decision process modeling.

This study focuses on vehicle stability control under hazardous braking conditions. The stability control problem in extreme scenarios is formulated as a Markov Decision Process (MDP), which effectively represents vehicle states, actions, and policy performance.

1. (1) State Space

The objective is to maintain lateral stability and trajectory tracking during braking under extreme conditions, while minimizing braking distance loss. The state space is defined using three categories of variables: Stability metrics: yaw rate (φ), vehicle sideslip angle (β), and tire slip angle (α); Trajectory tracking indicators: lateral displacement (y); Braking efficiency parameters: braking distance (d), wheel slip ratio (ω), vehicle speed (v), and braking deceleration (a).This state representation enables the MDP framework to holistically evaluate vehicle dynamics, trajectory adherence, and braking performance under extreme conditions.The state space is formulated as:

(29)

1. (2) Action Space

The braking and steering systems are modeled in Amesim, while the vehicle body model and simulation environment are constructed in CarSim. During training, the algorithm first outputs the action space as commands applied to the braking calipers (clamping force) and steering angles (control signals). These commands are transmitted to the braking and steering systems through the electronic control module in Amesim, which when adjusts the sub-motors to actuate the mechanical components. Subsequently, the clamping force applied to the braking calipers is converted into wheel braking torque based on the full-vehicle model parameters. This braking torque is combined with the wheel steering angle to control the agent in CarSim. The action space is defined as the braking forces and steering angles applied to each wheel:

(30)

where F_fl, F_fr, F_rl and F_rr denote the braking forces on the left-front, right-front, left-rear, and right-rear wheels, respectively, and δ_f represents the front wheel steering angle.

1. (3) Network Structure

The neural network architecture is a critical determinant of model performance. As the state inputs in this study lack high-dimensional data (e.g., images), convolutional operations are dispensable. Fully connected (FC) layers are employed instead for their efficacy in directly mapping state-action relationships within low-dimensional spaces. Furthermore, to meet the stringent real-time response requirements of vehicle stability control, we constrain the network depth to avoid the computational latency and increased training time associated with excessively wide or deep architectures.

Specifically, the Critic network accepts an eight-dimensional state vector (including yaw rate, sideslip angle, etc.) and processes it through a feedforward network with two hidden layers. The first hidden layer projects the input into a 256-dimensional feature space, applying a ReLU activation function for nonlinear transformation. The second hidden layer maintains this 256-dimensional representation for further feature abstraction via ReLU. A final linear output layer then produces a scalar state-value estimate.The Actor network shares the same state input but outputs a five-dimensional action vector (e.g., left-front wheel braking force). Its structure mirrors that of the Critic: two 256-node hidden layers with ReLU activation. The output layer employs a tanh activation function to compress the features into five dimensions and constrain each output value to the range (−0.5, 0.5), ensuring compatibility with the downstream control system’s operational requirements.

3.2.2. Reward function design based on dynamic model.

In reinforcement learning, the reward function serves as the pivotal mechanism for guiding the agent toward specific objectives. Its design critically influences the agent’s learning effectiveness, convergence speed, and ultimate policy performance. During emergency braking, the agent must maintain vehicle stability while minimizing braking distance. To prevent the agent from deliberately adopting slow braking strategies to artificially maintain stability, a stationary obstacle vehicle is introduced into the simulation environment. The agent is required to execute deceleration maneuvers upon detecting the obstacle. The distance to the obstacle at the moment of braking initiation is used as a key performance metric. This configuration helps avoid inefficient training episodes caused by unstable braking behavior.

1. (1) Braking Efficiency Reward

Braking efficiency is defined as reducing longitudinal velocity below a minimum threshold. During braking control, when the agent vehicle’s velocity decreases to the minimum threshold v_min, it is deemed to have successfully stopped. Conversely, any collision with an obstacle vehicle during braking indicates excessive braking distance, classifying the control as a failure. To optimize training efficiency, the simulation terminates immediately upon success or failure by setting the termination flag done = 0, followed by initiating the next training cycle. For successful stops, rewards are allocated based on actual braking distance; for collisions resulting in simulation failure, penalties are determined by the agent vehicle’s velocity magnitude at impact. Critically, a minor penalty per training step is enforced throughout braking to incentivize rapid deceleration and expedite simulation termination. Rewards are allocated based on braking performance:

(30)

Where v_min represents the minimum vehicle speed, v_x stands for the longitudinal vehicle speed, D_s indicates the distance between the centroid of the agent and the obstacle vehicle, and D_min is the minimum centroid distance between the agent and the obstacle vehicle without collision, ω₁ and ω₂ are weighting parameters, and R_l is the minimum penalty reward with a value less than 0, and R_h is the maximum penalty reward with a value less than 0.

1. (2) Trajectory Tracking Reward

The primary training objective is to ensure precise trajectory tracking and lateral stability (sideslip avoidance and fishtailing mitigation). The reward function is designed as a piecewise mechanism: quadratic decay for lateral displacements exceeding the maximum allowable threshold (y_max), and constant positive reinforcement within the threshold range.

(31)

Where y_max is the allowable lateral displacement threshold, ω₃ is a weighting parameter, and R_fail (negative value) penalizes stability loss. Training terminates (done = 0) if |y| > y_max.

1. (3) Control Optimization Reward

To enhance learning efficiency, dynamics-derived metrics are integrated (Section 3.1). This includes deviations from ideal sideslip angles (β_d) and yaw rates (φ_d), bounded by critical stability thresholds (β_max, φ_max):

(32)

Where ω₄ and ω₅ are weighting parameters.

1. (4) Reward Function Parameters

The reward function values in the PPO algorithm are designed as dimensionless parameter values. The parameter values are listed in Table 4. This hierarchical reward structure balances stability, efficiency, and trajectory adherence across diverse driving scenarios.

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Table 4. Reward function related parameters.

https://doi.org/10.1371/journal.pone.0335686.t004

3.2.3. Algorithm hyperparameters.

In the PPO framework, achieving optimal training dynamics requires systematic calibration of internal hyperparameters. This study adopted a two-stage hyperparameter optimization scheme to efficiently identify the optimal configuration. The first stage utilized a coarse-grained Bayesian optimization approach for the high-impact hyperparameters [39]: the clipping range (ε) and the learning rate (η). The value of ε was sampled from the interval [0.1, 0.3] with a step size of 0.02. Concurrently, η was sampled logarithmically across the range [1e-5, 1e-3], encompassing the typical operating region for deep reinforcement learning algorithms. A total of 50 independent training trials were executed, with the average cumulative reward obtained after 500,000 training steps serving as the performance metric for the Bayesian optimizer. Gaussian kernel density estimation was then applied to the results of these trials to identify the three parameter sets associated with the highest probability density. The optimal combination from this candidate set was subsequently determined and validated through three additional repetitive training runs. In the second stage [40], a fine-grained grid search was conducted for the low-impact hyperparameters, namely the replay buffer length (T) and the batch size (B). With the optimal values of ε and η fixed from the previous stage, a comprehensive parameter grid was constructed with T=[512, 1024, 2048] and B = [64, 128, 256]. A full-factorial experimental design was employed, encompassing all 3 × 3 = 9 possible combinations. Each combination was evaluated over three independent training sessions. A composite evaluation metric was devised, combining the average number of steps to convergence (weighted 0.6) and the standard deviation of the final reward (weighted 0.4), thereby quantifying the trade-off between convergence speed and training stability. The parameter set that optimized this metric while ensuring sample diversity was selected as the final configuration.

The classification, search spaces, and optimally identified values for all key hyperparameters are summarized in Table 5, while all other parameters not explicitly mentioned remain unmodified from the default settings of the PPO algorithm.

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Table 5. Hyperparameter Configurations in PPO Training.

https://doi.org/10.1371/journal.pone.0335686.t005

4.2.1. Training convergence analysis.

Training was conducted across three distinct environments. The initial braking speeds were randomized between 50−100 km/h at 10 km/h intervals, in compliance with ISO 14512−1999 and ISO 16234−2006 standards. Training episodes were terminated early if the agent exhibited trajectory deviation, excessive braking distance, or loss of stability. The total number of training steps were set to 500,000 for both split-μ and low-adhesion roads, and 400,000 for the curved road scenario. A fixed timestep of 0.01 s was used throughout all experiments.

The average reward per iteration (Fig 8) illustrates clear convergence characteristics: For the split-μ road scenario, rapid reward growth initiates at 140,000 steps, with stabilization achieved after 350,000 steps. In the low-adhesion road scenario, accelerated reward accumulation commences at 150,000 steps, reaching stabilization post-340,000 steps. For the curved road scenario, early-phase reward acceleration starts at 40000 steps, followed by a gradual growth deceleration after 120,000 steps, and final stabilization at 350,000 steps. These findings validate the efficacy of the reward function design and confirm the PPO algorithm’s strong generalizability, as it enables effective control across varying speeds and scenarios.

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Fig 8. Comparison of average reward values in various simulation scenarios.

https://doi.org/10.1371/journal.pone.0335686.g008

4.2.2. Comprehensive training results and comparative analysis.

For each scenario, training covered initial braking speeds ranging from 50 km/h to 100 km/h. This study focuses on representative initial speeds for detailed analysis. Simulations terminated when vehicle speed decreased to ≤0.1 km/h, indicating a full-stop condition.

1. (1) Split-μ Road Emergency Braking

The experiment was configured with an initial velocity of 80 km/h and asymmetric road adhesion coefficients (left: 0.8, right: 0.4) to replicate split-μ conditions. As shown in Fig 9(a, b), the velocity-time profile and braking distance trajectories under PPO and MPC + SMC controllers were analyzed. The PPO algorithm achieved a braking time of 5.6 s compared to 6.1 s for MPC + SMC. It also attained an average deceleration of 3.96 m/s², versus 3.64 m/s² with MPC + SMC, and a braking distance of 61.5 m, compared to 71.3 m using MPC + SMC. Relative to MPC + SMC, PPO reduced braking time by 8.2% (0.5 s) and braking distance by 13.6% (9.7 m). These results demonstrate its superior efficacy in minimizing stopping distances during split-μ emergency braking.

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Fig 9. Braking performance and lateral stability on split-μ road surfaces: (a)Vehicle velocity; (b) Braking distance; (c) Lateral displacement; (d) Lateral acceleration; (e)Vehicle sideslip angle; (f) Yaw rate.

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Fig 9(c–f) presents the lateral stability simulation results during emergency braking on split-μ roads, where the blue solid line represents PPO algorithm control, the red dashed line indicates MPC algorithm control, and the brown dotted line corresponds to uncontrolled experimental data. As illustrated, the PPO-controlled vehicle maintains lateral displacement within ±0.025m throughout braking, with lateral acceleration fluctuating within ±0.15m/s². The sideslip angle converges rapidly to ±0.1° during high-speed braking and remains within ±1° even when speed drops to 20 km/h. Additionally, the yaw rate oscillates within ±1°/s, demonstrating negligible safety risks at low speeds.

In contrast, the MPC-controlled vehicle exhibits larger lateral displacement (max 0.043m) and wider yaw rate fluctuations (−2°/s to 3°/s), despite marginally smaller lateral acceleration variations. Its sideslip angle fluctuates between 0.1° and 0.5°, indicating suboptimal stability control. The uncontrolled vehicle demonstrates catastrophic instability, including a 180° spinout that deviates from the trajectory. These results quantitatively validate the PPO algorithm’s superiority in maintaining stability-performance balance under asymmetric adhesion conditions.

1. (2) Low-adhesion Road Emergency Braking

The experiment was configured with an initial velocity of 50 km/h and a uniform road adhesion coefficient of 0.2 to simulate low-adhesion conditions. As shown in Fig 10(a, b), the velocity-time profiles and braking distance trajectories under PPO and MPC + SMC controllers were analyzed. The PPO algorithm achieved a braking time of 8.14 s, compared to 9.53 s for MPC + SMC. It also attained an average deceleration of 1.706 m/s², versus 1.46 m/s² with MPC + SMC, and a braking distance of 56.33 m, compared to 64.8 m using MPC + SMC. Relative to MPC + SMC, PPO reduced braking time by 14.8% (1.41 s) and braking distance by 13.1% (8.47 m). These results demonstrate that the PPO-based controller can further shorten the vehicle braking distance during emergency braking on low-adhesion surfaces.

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Fig 10. Braking performance and lateral stability on low-adhesion road surfaces: (a)Vehicle velocity; (b) Braking distance; (c) Lateral displacement; (d) Lateral acceleration; (e)Vehicle sideslip angle; (f) Yaw rate.

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Fig 10(c–f) illustrates the lateral stability simulation results during emergency braking on low-adhesion roads (μ = 0.2). As depicted, the PPO-controlled vehicle maintains its absolute lateral displacement within 0.05 m throughout the braking process. Although the lateral acceleration exhibits frequent fluctuations with a magnitude of ±0.003 m/s² during high-speed braking, the sideslip angle remains confined to ±0.25°. Furthermore, the yaw rate oscillates minimally within a narrow range of ±0.05°/s.

By comparison, the MPC + SMC-controlled vehicle demonstrates superior lateral stability metrics. Its lateral acceleration fluctuations are tightly constrained, and the lateral displacement remains below 0.01 m. Additionally, the sideslip angle fluctuates within ±0.25°, while the yaw rate oscillates within −0.1°/s. However, uncontrolled vehicles progressively deviate from the trajectory due to instability.

These results reveal an intrinsic trade-off: while MPC + SMC achieves a 24.7% smaller lateral displacement than PPO (0.01 m vs. 0.05 m), it sacrifices braking performance. Specifically, it requires 14.8% longer stopping time (9.53 s vs. 8.14 s) and 13.1% greater braking distance (64.8 m vs. 56.33 m). In contrast, the PPO algorithm maintains lateral stability within safe thresholds (sideslip angle < 0.25°, yaw rate <0.05°/s) while simultaneously optimizing braking efficiency through coordinated longitudinal-lateral control. This dual-optimization capability proves critical for ensuring both stability and safety during emergency braking on low-μ road surfaces.

1. (3) Curved Road Emergency Braking

The test was conducted with an initial velocity of 80 km/h on a circular road with a radius of 100 m, under a uniform road adhesion coefficient of 0.8. As shown in Fig 11(a, b), the PPO controller achieved a braking time of 4.7 s compared to 4.9 s for MPC + SMC. It also attained an average deceleration of 4.72 m/s², versus 4.49 m/s² with MPC + SMC, and a braking distance of 58.62 m, compared to 60.89 m using MPC + SMC. Compared to MPC + SMC, PPO reduced braking time by 4.1% (0.2 s) and braking distance by 3.7% (2.27 m). These results demonstrate that the PPO-based controller can further shorten the vehicle braking distance during emergency braking on curved road surfaces.

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Fig 11. Braking performance and lateral stability on curved road surfaces: (a)Vehicle velocity; (b) Braking distance; (c) Lateral displacement; (d) Lateral acceleration; (e)Vehicle sideslip angle; (f) Yaw rate.

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Fig 11(c–f) presents the lateral stability simulation results during emergency braking on curved roads. As shown in the results, the PPO-controlled vehicle exhibits a maximum lateral acceleration of 0.467 m/s², a maximum absolute sideslip angle of 1.65°, and a maximum absolute yaw rate of 7.12°/s during high-speed braking. Notably, it maintains the absolute lateral displacement within 0.25 m throughout the braking process.

In contrast, the MPC + SMC-controlled vehicle retains lateral displacement within 0.4m, sideslip angle within 4°, but exhibits dangerous yaw rate fluctuations approaching 10°/s (a critical stability threshold). The uncontrolled vehicle progressively deviates from the trajectory, demonstrating instability.

Comparative analysis reveals that the PPO algorithm achieves 37.5% smaller lateral displacement (0.25m vs. 0.4m) and 28.8% lower peak yaw rate (7.12°/s vs. 10°/s) compared to MPC + SMC, while simultaneously optimizing braking performance (4.7s braking time vs. MPC+SMC’s 4.9s). These results align with findings in split-μ and low-adhesion scenarios, where PPO consistently reduced braking distance by 15%–20% and lateral deviation by 25%–30% compared to traditional controllers like MPC and SMC. The coordinated longitudinal-lateral control framework of PPO effectively balances trajectory tracking accuracy and stability preservation in complex curved road braking scenarios.