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Predictive safety filter enhanced curriculum learning control for efficient vehicle dynamics controller
Baocong Zhang, Siliang Lu, Chenyang Li
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Abstract
Abstract:Recent advances in learning-based control have enabled impressive achievements in solving complex control problems in various domains. However, since learning-based control may not be able to realize safety-guaranties, it is of great importance to enhance safety and robustness while maintaining good performances. Take vehicle motion \& dynamics control as an example, in order to overcome the pain points of traditional methods such as heavy parameter calibration effort and learning-based control to bring better performance and efficiency in stability \& agility over prior work for state-based vehicle control tasks, in this work, our method aims to develop a curriculum learning controller enhanced with physics-based predictive safety filter. The validation is conducted with the Python-CarSim platform, demonstrating better improvements and scalability under various maneuvers.
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- Source: https://arxiv.org/abs/2608.09653v1
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Predictive safety filter enhanced curriculum learning control for efficient vehicle dynamics controller Baocong Zhang∗1, Siliang Lu∗2,Chenyang Li2 *Siliang Lu and Baocong Zhang contributed equally to this work, ordered alphabetically. Siliang Lu is the corresponding author (email: siliang.lu@cn.bosch.com).2Bosch Corporate Research, Shanghai, China1Shanghai Jiaotong University, Shanghai, China Abstract Recent advances in learning-based control have enabled impressive achievements in solving complex control problems in various domains. However, since learning-based control may not be able to realize safety-guaranties, it is of great importance to enhance safety and robustness while maintaining good performances. Take vehicle motion & dynamics control as an example, in order to overcome the pain points of traditional methods such as heavy parameter calibration effort and learning-based control to bring better performance and efficiency in stability & agility over prior work for state-based vehicle control tasks, in this work, our method aims to develop a curriculum learning controller enhanced with physics-based predictive safety filter. The validation is conducted with the Python-CarSim platform, demonstrating better improvements and scalability under various maneuvers. I INTRODUCTION Modern vehicle motion & dynamics control systems are typically built on a model-based feedforward plus feedback architecture where a nominal vehicle model is used to generate the targets (feedforward) while a feedback loop (often PID or gain-scheduled control) compensates for modeling errors and disturbances to achieve stability and tracking performances. Even if this framework is mature, efficient, and widely deployed, its performance and robustness depend heavily on extensive manual calibration of vehicle types, tire configurations, load conditions, road friction levels, and maneuver sets, thus making the design and validation process time-consuming and engineering intensive. In addition, due to lack of a centralized control design to adapt to different test conditions, existing solutions are often decoupled according to different functions such as the brake controller, the steering controller, and the suspension controller. However, with the rapid development of X-by-wire technologies in electric vehicles, learning-based control has great potential in the vehicle integrated controller (integrating braking, steering and suspension) for various scenarios, especially those safe-critical ones such as Federal Motor Vehicle Safety Standard (FMVSS) 126, J-turn & Double Lane change (DLC) with actuators as many as possible. In terms of learning-based control, representative approaches include (i) imitation learning, where an AI controller learns to reproduce reference control trajectories generated by existing classical dynamics controller, and (i) reinforcement learning, where a policy is trained through interaction with a high-fidelity simulator (e.g., CarSim) to maximize stability and performance objectives. These methods can learn complex control behaviors and potentially generalize over a broader set of operating conditions than hand-tuned controllers. Nevertheless, AI controllers are commonly treated as black-box policies and often lack explicit guaranties of physical consistency and satisfaction with the safety constraint. In practical development, generalization gaps may cause an AI policy to output incorrect or counterproductive actions under rare or out-of-distribution conditions (e.g., producing a yaw moment in the wrong direction when correcting understeer), which is unacceptable for safety-relevant dynamics control functions. To mitigate risks of unsafe or unreliable actions, the broader field has investigated predictive safety filter methods that update a nominal control command to enforce safety constraints. Conventional predictive filtering typically relies on (a) an accurate and computationally efficient prediction model, (b) sufficient reference information over the horizon, and (c) an online solver capable of meeting real-time constraints. In practice, these requirements create challenges: high-fidelity simulators, such as CarSim, are accurate but are not designed for large-scale parallel training or tight-loop online optimization; meanwhile, simplified models may not be reliable enough for closed-loop predictive correction. In addition, many existing approaches are not fully integrated with a data-driven model calibration pipeline, which limits their scalability when deployed across vehicle variants and maneuvers. I-A Contributions To mitigate these gaps above, this paper makes the following contributions: • Data-driven auto-calibration of a structured vehicle dynamics model A differentiable, physics-structured (white-box) vehicle model is aligned to high-fidelity simulation trajectories in classic maneuvers, substantially reducing reliance on time-consuming manual calibration and enabling faster iteration across variants. • Physics-consistent safety filtering on top of AI to mitigate generalization failures A model-based safety filter supervises the nominal AI brake command and corrects unsafe or physically inconsistent actions, providing an interpretable and verifiable safeguard against rare but critical failure cases (e.g., wrong yaw-moment direction under understeer). • Minimal intervention predictive correction with real-time feasibility Beyond kinematics control such as path tracking, the predictive safety filter is formulated to stay close to the nominal AI command while enforcing feasibility/safety requirements for the lower-level such as dynamics control to support high-frequency action (i.e. 200Hz) correction suitable for real control loops. • End-to-end, learning-based pipeline from data to deployment The work establishes a repeatable workflow connecting trajectory generation, model alignment, predictive safety filtering, and closed-loop validation, and provides a fast, parallel white-box simulation environment that supports large-scale training/evaluation beyond what traditional high-fidelity tools can efficiently provide. I-B Related Work This section reviews the existing control methods for vehicle motion and dynamics control, from model-free (proportional-integral-derivative, PID), model-based (model predictive control, MPC) to learning-based control (Reinforcement learning). In addition, an overview about safety filter is presented. • PID: Rodic and Vukobratovic [3] proposed a synthesized control strategy for an autopilot system. The approach aims to track pre-defined trajectories by coordinating four-wheel steering, active damping, and independent wheel torque distribution. The low-level controller employs a PID scheme informed by vehicle dynamics principles. The method has been evaluated in simulation under low lateral acceleration conditions, demonstrating accurate trajectory tracking and robustness to disturbances such as friction variations and crosswind effects. • Model Predictive Control: In [4][5], a linear model predictive control (MPC) framework is developed for lateral stability control of a steer-by-wire vehicle. The prediction model relies on a linear tire formulation, while axle slip angle limits are derived offline from a brush tire model and imposed as constraints. The controller is designed to intervene only when slip angle bounds are exceeded and explicitly accounts for actuator limitations, thereby ensuring predictable vehicle behavior under aggressive maneuvers such as double step steering. Linear MPC has also been applied to integrated path tracking and yaw stability control in autonomous vehicles [6][7]. In these works, vehicles equipped with active front and rear steering as well as direct yaw moment control are considered. Compared with a linear quadratic controller, MPC demonstrates improved tracking and stability performance across different actuator configurations and maneuvers, although with increased actuator effort in most cases. In addition, since the highest model accuracy can be achieved by employing the non-linear system dynamics in the prediction model, which leads to non-linear MPC, there are also works on non-linear MPC [9] [10] [11]. • Reinforcement learning Reinforcement Learning has emerged as a promising solution to address the limited flexibility and scalability of traditional integrated chassis control frameworks, which are not well suited for distributed in-wheel motor-driven electric vehicles. To overcome these limitations, an intelligent chassis dynamic control framework based on a multi-agent architecture is proposed for electric vehicles independently driven by four-wheels [1]. The simulation results demonstrate that the proposed controller agent achieves the desired dynamic control objectives and significantly improves lateral handling stability, thus validating the effectiveness of the proposed control framework and providing a solid foundation for further development. • Safety filter I Preliminary I-A Double track model The 7-DOF model is implemented as a composition of differentiable sub-models [2], each corresponding to a physical subsystem, as shown in Fig. 1. Figure 1: Structure of the 7DOF Double-Track Model At each time step, the modules are executed in a consistent data flow to produce a complete state update: • Steering system: The steering subsystem maps the driver steering input to each wheel angle δi _i. In addition, the calibration parameters include steering gain/ratio coefficients and geometric conversion factors. • Wheel and brake dynamics: The handling model adopts a 7-DOF formulation, consisting of three vehicle-body DOFs (longitudinal motion, lateral motion, and yaw motion) and four rotational DOFs corresponding to wheel angular speeds ωi _i. The vehicle is assumed to operate on a flat road surface[11]. By summing forces along the longitudinal (x) and lateral (y) directions, the vehicle accelerations are obtained according to Newton’s second law. The wheel rotational dynamics are governed by Jwω˙i=Ta,i−Tb,i−RFx,i−Cfωi,J_w ω_i=T_a,i-T_b,i-RF_x,i-C_f _i, where JwJ_w denotes the wheel inertia, Ta,iT_a,i is the applied drive torque, Tb,iT_b,i is the brake torque, R is the effective tire radius, Fx,iF_x,i is the longitudinal tire force, and CfC_f is the viscous damping coefficient. In addition, the calibration parameters include JwJ_w, brake gain (pressure-to-torque coefficients), and viscous damping terms. • Vehicle body dynamics: The dynamics of the planar vehicle describes the longitudinal velocity vxv_x, the lateral velocity vyv_y, and the yaw rate ψ˙ ψ. The equations of motion are given by m(v˙x−vyψ˙)=∑Fx,m( v_x-v_y ψ)=Σ F_x, m(v˙y+vxψ˙)=∑Fy,m( v_y+v_x ψ)=Σ F_y, Jzψ¨=∑Mz,J_z ψ=Σ M_z, where m is the vehicle mass and JzJ_z is the yaw moment of inertia. The yaw motion depends on tire forces and self-aligning moments. The vehicle-level states (xI,yI,v,β,ψ,ψ˙)(x^I,y^I,v,β,ψ, ψ) are updated from the summed tire forces/moments and aerodynamic effects. The calibration parameters include mass m, yaw inertia JzJ_z, geometric parameters (lfl_f, lrl_r, track widths wfw_f, wrw_r), and CG height h. • Load transfer model: Dynamic load transfer describes the redistribution of normal forces among wheels due to longitudinal and lateral accelerations. The total wheel load is composed of static load distribution and quasi-static dynamic transfer effects. The normal load per-wheel Fz,iF_z,i is calculated based on the accelerations of the vehicle and the geometry of the chassis. Calibrated parameters primarily include shared geometric quantities such as Center Gravity (CG) height and track width. • Aerodynamics model: The aerodynamic subsystem computes drag force, lateral force, and yaw moment as functions of vehicle states (i.e., speed and side slip angle). In addition, the calibration parameters include the reference area, characteristic length, air density, and fitted coefficients that describe the force/moment dependence on the side slip angle. • Tire model (Magic Formula): The tire model is a critical component of the vehicle handling model, as it directly influences vehicle performance in acceleration, braking, and cornering conditions, it characterizes both lateral and longitudinal tire forces. In this study, the well-known Pacejka Magic Formula (MF) tire model is adopted to describe tire behavior. The model provides the self-alignment moment, the lateral force FyF_y and the longitudinal force FxF_x as functions of the normal load of the wheel FzF_z, side-slip angle α and the longitudinal slip ratio κ. Mathematically, the per-wheel tire forces are computed as (Fx,Fy)=MF(Fz,α,κ;θMF),(F_x,F_y)=MF(F_z,α,κ; _MF), where θMF _MF denotes the set of calibrated Magic Formula coefficients and scaling factors. This formulation is fully differentiable, enabling gradient-based optimization for control and calibration purposes.In addition, the model parameters are organized into a unified container that exposes learnable chassis parameters, steering parameters, wheel/brake parameters, tire MF parameters, and aerodynamic parameters. This design supports selective freezing/unfreezing of subsets during calibration. I Methodology The core of the paper are a coupled framework that (i) constructs a differentiable, physics-structured (white-box) vehicle dynamics model and aligns it to high-fidelity trajectories through data-driven calibration, and (i) curriculum learning (i) uses the calibrated model as a predictive backbone to perform online, physics-informed safety correction of a nominal control command for a nonlinear real-time dynamics system. I-A Curriculum learning with CarSim This section describes the specific application setting and control architecture, as illustrated in Fig. 2. In general, the AI controller proposes per-wheel brake-pressure commands ptAIp_t^AI, using the yaw-rate target ψ˙ttar ψ_t^tar as a reference, along with other observations such as the steering input δt _t of a human driver or the steering system. The AI controller ultimately outputs the brake-pressure commands applied to the plant. Figure 2: Customized gym environment for CarSim-python co-simulation To facilitate stepwise controller training, a curriculum learning scheme is adopted, consisting of three stages, as shown in Fig. 3. In all stages, the dynamic inputs are augmented with noise to improve generalization. For CarSim inputs, both control variables (i.e. torque or pressure) and random initial conditions (i.e. steering angles, friction coefficients, initial speeds) are used to simulate vehicle behaviors with mathematical vehicle models inside CarSim. As a result, multiple outputs can be sent to Simulink model, including vehicle states (i.e. driving velocity, wheel velocities, accelerations, yaw rate, pitch rate, roll rate, slip angle & ratio, etc.) so that Simulink model can calculate target value (i.e. target yaw rate), which is one of the inputs for reward functions and some of the key states are used as observations in the reinforcement learning framework. Figure 3: Inductance of oscillation winding on amorphous magnetic core versus DC bias magnetic field For stage 1, easy maneuvers related to FMVSS 126 are used, which is defined as the sine-with-dwell test with initial driving speeds over 80 kph commonly used for the evaluation of electronic stability control, as shown in Fig. 4 Figure 4: FMVSS126 For stage 2, in terms of medium-level tasks, for vehicle stability tests, J-turn maneuvers are used, which tests vehicle stability during a sudden steering input over 90 degrees with initial driving speeds over 80 kph, as shown in Fig. 5. Figure 5: J-turn Last but not least, the most difficult tasks are depicted as double lane change (DLC) with fixed trajectory tracking with high driving speeds, rapid target yaw rate changes but no restrictions about the steering input. One of the example maneuvers is shown in Fig. 6 Figure 6: Double lane change In addition, the curriculum learning design for training and inference is shown below. • Actions: Front-left (FL) wheel pressure, rear-left (RL) wheel pressure, rear-right (R) wheel pressure & front-right (FR) wheel pressure. • State space: FL, FR, RL, R wheel speeds; FL, FR, RL, and R wheel pressures at last time step; Lateral-velocity / acceleration, longitudinal velocity / acceleration, steering wheel angle, yaw rate, target yaw rate • Reward function: The overall reward is defined as a weighted sum of penalties for yaw-rate and wheel slip: left_penalty =∑i∈0,2ai⋅(ψ˙>ψ˙tar), = _i∈\0,2\a_i·1 ( ψ> ψ^tar ), (1) right_penalty =∑i∈1,3ai⋅(ψ˙<−ψ˙tar), = _i∈\1,3\a_i·1 ( ψ<- ψ^tar ), (2) slip ratio_penalty =(κfront,outer>0.8)+(κfront,inner>0.2)+(κrear>0.05), = aligned &1 ( _front,outer>0.8 )\\ &+1 ( _front,inner>0.2 )\\ &+1 ( _rear>0.05 ) aligned, (3) Reward =−w1left_penalty−w2right_penalty−w3slip_penalty. = aligned &-w_1\,left\_penalty\\ &-w_2\,right\_penalty\\ &-w_3\,slip\_penalty\\ aligned. (4) where aia_i denotes the wheel action (brake or torque) on wheel i, ψ˙ ψ is the current yaw rate, ψ˙tar ψ^tar is the target yaw rate and κ denotes the slip ratio of the corresponding wheel. The weights w1,w2,w3w_1,w_2,w_3 determine the relative importance of each penalty. In terms of learning-based controller, various reinforcement learning algorithms, including PPO and SAC, can be applied with multi-layer perceptron (MLP) for the policy network and efficient deployment on the vehicle. I-B Data-Driven Auto-Calibration Pipeline for Aligning the 7DOF Model to CarSim In terms of the double track vehicle model, its parameters are automatically calibrated using high-fidelity simulation data from target maneuvers in CarSim by minimizing prediction errors via gradient-based optimization. This approach replaces labor-intensive manual tuning with an automated calibration pipeline, enables systematic alignment of the model to a reference simulator, and produces a prediction model that is physically interpretable and more reliable for downstream control and verification. The process is illustrated in Fig.7. Figure 7: Pseudo algorithm of data-driven auto-calibration Training data: CarSim provides a set of simulated trajectories =T(i)D=\T^(i)\. Each trajectory consists of time stamps and synchronized sequences of control inputs and observed states (tk,uk,xk)(t_k,u_k,x_k). Dataset construction: For each sample, a fixed-length sliding window (length L, stride 1) is applied along the time dimension to create subsequences, without crossing the trajectory boundaries. All windows are then aggregated into a single training data set W for mini-batch training. Initialization of vehicle setup: The parameter vector θ is initialized using the vehicle configuration used in CarSim (mass/inertia, geometry, wheel radius, nominal tire/steering / aero settings). This provides a physically meaningful starting point and reduces the calibration search space. augmented state initialization (InitState): Unlike a simple assignment of measured signals, the model requires a complete initialization of the augmented states z at the beginning of each window. An initialization routine InitState(us,xs)InitState(u_s,x_s) computes consistent internal variables (e.g., road wheel angles δ, initial slip angles α, slip ratios κ, normal loads FzF_z, and initial force estimates) so that the subsequent roll-out is physically valid. Differentiable roll-out and loss: For each window, the model rolls out (^,^)( x, z) over the horizon using (^t+1,^t+1)=fθ(^t,^t,ut;Δt).( x_t+1, z_t+1)=f_θ( x_t, z_t,u_t; t). The calibration objective minimizes the mean squared error between the predicted observable states x and the CarSim states x over the entire window. Parameters are updated using gradient-based optimization (e.g., Adam) with backpropagation through time. Validation / testing: After training, the calibrated parameters θ are evaluated on disjoint CarSim trajectories not used during training. For testing, the model rolls out the entire trajectory from a single initial condition without applying sliding windows, and reports trajectory-level error metrics. I-C Predictive Safety filter control flow The paper introduces an online safety-correction paradigm for nonlinear systems in which the safety filter computes. At each control step, a corrected control input by solving for a descent direction of a safety-aware objective using gradients through the calibrated prediction model. This realizes an MPC-style runtime loop that combines multi-step prediction with online optimization, while maintaining efficient computation. As a result, unsafe or physically inconsistent actions produced by a nominal controller (e.g., a black-box AI policy) can be corrected in a physics-informed manner, providing deployable, efficient online action correction without relying on slow high-fidelity simulators for tight-loop optimization, as shown in Fig. 8 Figure 8: Predictive safety filter structure Moreover, the definitions of the state and input are shown as follows: Observable state x typically includes measurable vehicle-level signals and wheel states, such as longitudinal/lateral accelerations (ax,ay)(a_x,a_y), body-frame velocities (vx,vy)(v_x,v_y), wheel angular speeds ω, steering/road wheel angle proxy, and yaw rate ψ˙ ψ. Augmented state z includes internal variables required for a physically consistent roll-out but not directly measured or not always available as signals, such as tire slip angles α, slip ratios κ, normal loads FzF_z, tire forces (Fx,Fy)(F_x,F_y), aerodynamic forces/moments, and intermediate subsystem states. Input u includes driver/actuator commands, in particular steering and per-wheel brake pressures p (pfl,pfr,prl,prr)(p_fl,p_fr,p_rl,p_r). At a high level, the safety filter is formulated as a constrained MPC problem with a decision variable P, representing a sequence of brake-pressure per-wheel on the prediction horizon. The formulation enforces vehicle dynamics using a calibrated differentiable 7DOF double-track prediction model with augmented internal states, and enforces actuator and safety limits (i.e. limits on brake pressure, limits on slip-related quantities, and comfort/feasibility limits such as restricting the magnitude of longitudinal acceleration |ax||a_x| to avoid overly aggressive braking). For online efficiency, the constrained objective and constraints are combined into a single loss J(P)J(P) by introducing soft penalty terms (e.g., hinge/rectifier penalties for constraint violations). This converts the constrained MPC into an unconstrained optimization problem that can be solved by gradient-based methods for millisecond-level online action correction on commodity CPU platforms. The optimized command is applied in a receding-horizon manner. Here is a representative implementation of the MPC-based safety filter (single control step), consistent with Fig. 9. At each control step t, the filter receives the measured vehicle state tx_t, the steering command δt _t, the yaw-rate target ψ˙ttar ψ_t^tar, and the AI-proposed per-wheel brake pressure tAIp_t^AI. Figure 9: Pseudo algorithm of predictive safety filter with double track model Horizonal construction from one-step inputs: Because only one-step values are available externally, the MPC internally extends the steering and yaw-rate target to horizon-length sequences using hold/extrapolation operators (ZOH/FOH-like) with a short history buffer. This produces signals such as δt:t+T−1 _t:t+T-1 and ψ˙t:t+T−1tar ψ_t:t+T-1^tar. Initialization of the brake pressure sequence: Let P denote the decision variable (i.e., per-wheel brake pressure or torque over the horizon). A warm-start sequence is obtained by shifting the previous solution forward by one step. Two initialization modes are supported by setting the first step of the initialized sequence to the AI proposal tAIp_t^AI while keeping the remaining steps from the shifted warm start. Augmented-state initialization: The predictive model maintains augmented internal states z in addition to measured states x. The filter initializes the prediction model via InitState(⋅)InitState(·), which takes the measured (t,t)(u_t,x_t) and estimates a consistent augmented state tz_t. In the present implementation, InitState(⋅)InitState(·) leverages the available CarSim-like measurements embedded in tx_t (i.e., accelerations, velocities, yaw rate, wheel angular speeds), together with the current steering/brake command, to compute internal variables required by the 7DOF model modules (e.g., tire slip ratios, tire slip angles, and other intermediate subsystem states). MPC optimization by gradient descent: Given an initialized P, the filter evaluates the loss at the horizon-level J(P)J(P) (computed over the entire predicted horizon) and updates P using gradient descent with step size α. Typical loss terms include yaw-rate tracking against ψ˙tar ψ^tar, closeness-to-AI (penalizing deviation from AIp^AI), input smoothness, soft safety penalties (hinge penalties for violating slip-related limits), and a comfort/feasibility term limiting |ax||a_x| to avoid overly aggressive braking. After each gradient update, P is projected/clipped to satisfy the actuator limits. Receding-horizon execution and warm-start update: After a fixed number of gradient steps, the optimized sequence P∗P^* is used to apply only the first a few brake-pressure commands (execution length) to the plant. The remaining steps are stored and shifted to warm-start the next control step, providing continuity and supporting online millisecond-level execution. At each time step t, the safety filter solves an MPC problem over an horizon with the decision variable P, using inputs tx_t, δt _t, ψ˙ttar ψ_t^tar, and tAIp_t^AI. Because the model maintains augmented states z, the roll-out starts from an initialized augmented state provided by InitState(⋅)InitState(·). IV Results IV-A Training Setup The RL agent is trained and inferenced on a server equipped with 12th Gen Intel(R) Core i5-12600K CPU (3.7GHz). The vehicle model is obtained from the CarSim 2021 simulator. Both the timesteps of the controller and the simulation are 0.001s. In addition, each training episode lasts 9s9~s and the RL agent is trained for a maximum of N=3000N=3000 episodes, with an adaptive learning rate by KL divergence and a discount factor of 0.99. In terms of the double track model with auto-calibration, it has been trained with 101 epochs when it converges to loss under 0.3 on a server with an NVIDIA A100 GPU. IV-B Double track model validation To evaluate the generalization performance of double track model with the auto-calibration algorithm, the parameters are identified with CarSim traces from 12 different maneuvers, including 5 FMVSS126 traces with different magnitudes, 1 Jturn trace, 6 DLC traces with different steering angles where each trace has 9000 samples, and validated with unseen maneuvers. The validation performances are shown in Fig. 10 and Fig. 11: Figure 10: Validation between double track model & CarSim in FMVSS126 Figure 11: Validation between double track model & CarSim in Jturn IV-C Learning-based correction of model inaccuracies To evaluate the improvements about the proposed controller compared to baseline controller, which is a feed-forward (model-based) and feed-backward controller (PID), Table I below shows the RMSE between the target yaw rate and the actual yaw rate with or without the predictive safety filter for FMVSS126 maneuver. TABLE I: Performance Comparison of Different Methods Method RMSE Average Time per step(ms) baseline 2.72 10 CL w/ safety filter 1.44 2.72 CL w/o safety filter 2.41 2 In addition, the benchmark performances about the yaw rate tracking and the deceleration of vehicle velocity with or without safety filter are shown in Fig. 12 and Fig. 13: Figure 12: Curriculum learning controller performance w/o safety filter Figure 13: Curriculum learning controller performance w/ safety filter IV-D Safety Filters vs Baseline (FMVSS126) IV-E Computational time The RL controller training takes around 12 hours. Although the predictive safety filter increases the inference time, the proposed controller outperforms the baseline controller. For time-critical scenarios such as on-road applications, the proposed controller could realize less than 6 ms on average for each time step with CPU, which is accepted for all test maneuvers. IV-F Generalization To evaluate generalization, the proposed controllers were tested and compared with the baseline,on two unseen maneuvers with RMSE, one is the J-turn and the other is the free-style driving of a human driver, as shown in Table I. Although both controllers exhibit performance degradation under these new maneuvers compared to maneuvers seen such as FMVSS126, the proposed controller achieves better results in Jturn while a bit worse than the baseline in free-style driving due to unpredictable steering inputs. Nevertheless, this still demonstrates its potential for improved robustness to environmental variations and its potential for real-world deployment. TABLE I: CL performance w/ safety filter vs Baseline in generalization Method Jturn Human dirver baseline 2.44 2.09 CL w/ safety filer 2.17 5.67 Figure 14: Benchmark between baseline and proposed controller with maneuver by human driver (upper:baseline) Figure 15: Benchmark between baseline and proposed controller with maneuver with Jturn (upper:baseline) V Discussions Plant variations: The controlled plant may be the differentiable 7DOF model, a high-fidelity simulator (e.g., CarSim), a hardware-in-the-loop environment, or a real vehicle. The same safety-filtering principle applies as long as the plant can accept steering and brake-pressure commands and provide the measured states needed by the filter. Model variations: The predictive model can be extended to higher degrees of freedom (e.g., 14DOF) and can incorporate additional subsystems (e.g., suspension, driveline, road/tire effects) while preserving differentiability for gradient-based optimization. Subsets of parameters may be calibrated depending on the target vehicle and available data. Deployment of on-road vehicle tests: Below shows the brief workflow of the real vehicle tests to be used: 1. The trained policy model is deployed onto a PC with sufficient computational capability and CAN communication functionality, and the CAN communication interfaces are configured accordingly. 2. CAN communication interfaces are configured to receive or send sensor signals between the vehicle and the PC tot can decode the command information transmitted to the vehicle. 3. In addition, the inference frequency of the policy model running on the PC is adapted to the CAN communication frequency to ensure consistent timestamps of the sensor signals transmitted via the CAN and to keep the inference rate within an acceptable range for on-road control. VI Conclusions In order to overcome those pain points to bring better performance, in this work, our method aims to develop a curriculum learning controller enhanced with physics-based predictive safety filter to reduce the huge tuning effort and bring superior performance in stability & agility over previous work for state-based vehicle control tasks. Our simulation results demonstrate that the proposed controller outperforms the state-of-the-art controller, achieving a reduction of up to 47% in RMSE between the target yaw rate and the actual yaw rate while still satisfying computational time requirements. Moving forward, the goal is to extend this approach to on-road tests, targeting real-world, safety-critical scenarios with stringent computational constraints. References [1] Guodong Yin, Tong Zhu, Zuping Ren et al. Intelligent Control system framework for multi-agent based electric vehicle chassises. China Mechanical Engineering, 2018, p. 1796-1801. [2] M. A. Subari, K. Hudha, N. 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