Abstract
Introduction:
Connected and Automated Vehicles face challenges of fixed parameters, poor dynamic adaptability, and the lack of longitudinal and lateral coordination, which affect safe and stable vehicle operations. To solve these problems, this study aims to develop an advanced coordinated control system for intelligent vehicles.
Methods:
This study proposes a dynamics modeling technique based on online calibration of connected parameters. This technique integrates the real-time data update patterns of vehicle-infrastructure cooperation to construct an accurate motion model, which combines dynamic parameter inputs to achieve precise evaluation of vehicle driving states. In addition, this study adopts a control technique based on Adaptive Fuzzy Sliding Mode (AFSM) and fuzzy Reinforcement Learning (RL) for coordinated vehicle management and control. This technique takes dynamic states as inputs, enhances the suppression of chattering interference by introducing a fuzzy inference layer, and calibrates the final control results through a dual strategy integrating spatial constraints and adaptive mechanisms.
Results:
Experiments are conducted based on a commercial bus. In lateral target tracking control, the model in this study reaches a displacement of 3.78 m at 20 s, and the lateral tracking error drops to −0.02 m, outperforming similar models. In real-vehicle extreme lane-changing control experiments, the root-mean-square lateral error of the model in this study is 0.035 m, while the maximum control chattering rate is only 2.4%. Finally, in longitudinal and lateral coordinated performance analysis, the maximum speed error of the model in this study is 0.25 km/h, and the jerk is 0.12 m/s3, both of which are superior to similar models.
Discussion:
The proposed technique demonstrates good application effects in addressing parameter ambiguity and dynamic control imbalance. This study provides technical support for trajectory planning and coordinated vehicle control of intelligent vehicles, contributing to high-precision obstacle avoidance and multi-objective coordinated control of connected and automated vehicles.
1 Background
1.1 Research motivation
The longitudinal and lateral coordinated control of Connected and Automated Vehicles (CAVs) plays a critical role in fields such as autonomous driving trajectory planning, active safety collision avoidance, and traffic efficiency improvement (). However, traditional vehicle dynamics control models are mostly constructed based on offline fixed parameters. Facing complex and changing road conditions, these models encounter challenges such as poor dynamic adaptability, weak parameter robustness, and the lack of coordination between longitudinal and lateral motions (). With the rapid development of artificial intelligence and vehicle-infrastructure cooperation technologies, an intelligent coordinated control system that integrates multidimensional fuzzy logic and deep Reinforcement Learning (RL) algorithms provides technical support for precise obstacle avoidance and high-stability driving of CAVs under complex environments ().
1.2 Related research and highlights
In the era of CAVs, vehicles achieve obstacle avoidance trajectory planning through longitudinal and lateral control, and relevant scholars conducted extensive research on this topic. Zhang C et al. addressed the problems of complex longitudinal and lateral coupling and parameter uncertainty in autonomous driving, and proposed a robust model predictive control strategy based on tubes. This method utilized T-S fuzzy technology to construct an integrated model and converted the friction circle approximation into convex constraints. Experiments indicated that this method outperformed traditional linear time-varying methods in tracking performance for both longitudinal and lateral control (). To solve the problem of vehicle obstacle avoidance trajectory tracking deviation caused by road condition fluctuations, Liu Y et al. investigated a robust predictive control and longitudinal-lateral control decoupling algorithm. Based on the octagon approximation of the friction circle, that study designed a model controller based on linear and fuzzy tubes to coordinate vehicle motion. From the simulation results, it was observed that this approach played a vital role in improving the safe tracking operation in both longitudinal and lateral planes (). The group of researchers led by Zadeh A Y et al. attempted to solve the issue that both energy efficiency optimization and safety of driving cannot be balanced simultaneously in autonomous vehicles, thus introduced an optimized intelligent system using longitudinal and lateral control mechanisms ().
Fuzzy control algorithms are widely applied in obstacle avoidance and tracking control of CAVs, ensuring the stability and safety of vehicle driving. To address the lack of coordination in hierarchical collision avoidance during path tracking, Han J et al. proposed a unified safe tracking strategy based on barrier functions. This strategy simplified trajectory tracking into a yaw angle problem and combined an adaptive sliding mode controller to handle input saturation. Experiments demonstrated that this method possessed excellent obstacle avoidance performance (). Mousaei A et al. aimed at the problems of fuzzy parameters and limited computational budget in electric vehicles, and designed a robust optimal fuzzy controller based on the Takagi-Sugeno model. This method obtained various feedback gains and upper bounds by solving linear matrix inequalities offline. Simulations showed that this technique possessed excellent control performance (). In addressing the issue of inadequate adaptability due to the use of fixed weight in distributed electric vehicles, Kong X et al. have designed a hierarchical yaw moment control strategy. From testing, it has been found that such a strategy enhanced driving stability (). As a solution to the issue of tracking errors due to poor trajectory prediction in high-speed obstacle avoidance situations, Fu X et al. have proposed an obstacle avoidance control strategy using long short-term memory networks. The control strategy involved the utilization of several controllers such as predictive modeling and speed tracking. Such a system enhanced interactive trajectory prediction and minimized positional and sideslip angle errors ().
In summary, longitudinal and lateral coordinated control of vehicles is widely applied in autonomous driving trajectory planning and active safety fields (). Existing methods are mostly constructed based on fixed parameters, facing problems such as poor adaptability to complex non-steady-state road conditions, insufficient path tracking accuracy in curves, and unsmooth switching between driving and braking (). To solve the problems of parameter ambiguity and dynamic control imbalance, this study proposed a longitudinal and lateral coordinated control method based on multidimensional fuzzy algorithms to address the low driving robustness of vehicles. This study has two innovations. First, this study constructed an Adaptive Fuzzy Sliding Mode (AFSM) lateral control system to dynamically adjust reaching parameters through fuzzy inference to suppress chattering, achieving high-precision path tracking. Second, this study introduced a Fuzzy Twin Delayed Deep Deterministic Policy Gradient Proportional Integral Derivative (FTD3-PID) longitudinal control system to adaptively optimize speed and smoothly switch between driving and braking. This study provides technical support for high-precision obstacle avoidance and multi-objective coordinated control of CAVs.
2 Methodology
2.1 Vehicle dynamics model construction for longitudinal and lateral coordinated control
In the era of connected and automated vehicles, Vehicle to Everything (V2X) technology enables real-time information interaction between vehicles and roads, surrounding vehicles, and the cloud, which provides data support for longitudinal and lateral coordinated control of vehicles. However, traditional vehicle dynamics models are mostly constructed based on offline fixed parameters, making them unable to adapt to dynamically changing conditions such as road adhesion and slope in a connected environment. This limitation leads to insufficient control accuracy and robustness of subsequent longitudinal and lateral coordinated controllers. To solve this problem, this study constructs a vehicle dynamics model that integrates online calibration of connected parameters. The technical process is shown in Figure 1.
FIGURE 1
As shown in Figure 1, the multi-source data fusion module integrates real-time data from vehicle-mounted sensors and the V2X unit. The vehicle dynamics model outputs the current motion state of the vehicle. The longitudinal and lateral coordinated controllers generate path tracking and speed tracking control variables, respectively, and perform bidirectional information interaction. The coordinated control command fusion module performs global optimization on the control variables. Finally, the optimized commands are sent to the steering, driving, and braking actuators to achieve stable coordinated control of the vehicle.
2.1.1 Longitudinal dynamics model construction
To accurately describe the force and motion characteristics of longitudinal vehicle movement, a longitudinal dynamics equation considering connected road slope information is established, as shown in Equation 1 ().
In Equation 1, represents the curb weight of the vehicle, and represent the rotational inertia of the front and rear wheels, respectively, represents the effective rolling radius of the wheel, represents the longitudinal acceleration of the vehicle, represents the output torque of the driving wheels, represents the gravitational acceleration, represents the road rolling resistance coefficient, represents the road slope angle, represents the air density under standard atmospheric pressure, represents the air resistance coefficient, and represents the windward area of the vehicle.
2.1.2 Lateral dynamics model construction
To meet the requirements of vehicle lateral path tracking control, a simplified two-degree-of-freedom vehicle dynamics model is established. First, the lateral force balance equation is derived, as shown in Equation 2 ().
In Equation 2, represents the longitudinal driving speed of the vehicle, represents the vehicle sideslip angle at the center of gravity, represents the rate of change of the sideslip angle at the center of gravity, represents the vehicle yaw rate, and and represent the lateral forces experienced by the front and rear axle tires, respectively. On this basis, the vehicle yaw moment balance equation is derived, as shown in Equation 3 ().
In Equation 3, represents the rotational inertia of the vehicle around the center of gravity, represents the yaw acceleration, and and represent the distances from the front and rear axles to the vehicle center of gravity, respectively, which determine the axle load distribution and basic handling stability of the vehicle. The front and rear tire slip angles are related to the vehicle states by and , where is the steering angle, is the sideslip angle, is the yaw rate, and is the longitudinal speed. These slip angles then determine the lateral forces through and under the linear tire assumption. This completes the physical link among the steering input, vehicle motion states, and tire force generation.
2.1.3 Online calibration mechanism of connected parameters
Based on the assumption that the tire sideslip angle lies in the small-angle linear range, the linear relationship between the lateral force of the front tire and sideslip angle can be expressed as indicated by Equation 4.
In Equation 4, represents the lateral stiffness of the front tire, and represents the front tire sideslip angle. Similarly, a linear relationship between the rear tire lateral force and the sideslip angle is established, as shown in Equation 5 ().
In Equation 5, represents the lateral stiffness of the rear tire, and represents the rear tire sideslip angle. This study obtains real-time dynamic parameters such as road slope and road surface adhesion coefficient through V2X technology. The linear tire model is valid under small slip angle conditions, which covers most normal driving scenarios. The cornering stiffness serves as the local slope of the lateral force-slip angle curve within this region, and the RLS estimator tracks its variations online. To ensure validity under large slip or low-friction conditions, the adhesion coefficient estimated from normalized tire force is used as an indicator. When the slip angle exceeds a preset threshold, the estimator update is paused to prevent erroneous parameter correction from nonlinear tire data.
Key parameters in the model, such as the rolling resistance coefficient and the lateral stiffness of the front and rear wheels, are updated online. The online calibration is implemented via a recursive least squares (RLS) estimator with forgetting factor, as shown in Equation 6.
In Equation 6, is the estimated parameter vector, is the regressor vector, is the measured output, is the Kalman gain, is the covariance matrix, and is the forgetting factor.
For road slope estimation, the longitudinal dynamics is reformulated as , where is identified by RLS using IMU acceleration and wheel speed. For rolling resistance, the force balance is used to estimate . For cornering stiffness, the linear tire model is adopted to estimate and from lateral acceleration, yaw rate, and steering angle. For adhesion coefficient, the normalized tire force is evaluated, with V2X-provided road surface type as the initial prior.
This update allows the dynamics model to adapt to different road driving conditions, which provides accurate state inputs for subsequent longitudinal and lateral coordinated control. The forgetting factor controls the correction rate, balancing adaptation speed and noise sensitivity. The Kalman gain adjusts the correction magnitude based on the covariance matrix . In the experiments, is set to 0.98 for slope and rolling resistance, and 0.95 for cornering stiffness and adhesion coefficient. The initial covariance is , with a 5 s initialization period.
2.2 Design of AFSM lateral control system with longitudinal and lateral coordination
Based on the vehicle dynamics model, traditional lateral controllers do not consider the coordinated effects of longitudinal motion states and road conditions. They feature fixed preview distances and poor parameter robustness, which leads to insufficient path tracking accuracy under large-curvature curves. Therefore, this study proposes an AFSM lateral control system. The technical process is shown in Figure 2.
FIGURE 2
In Figure 2, this process takes the output vehicle dynamics state information as the core input, and sequentially completes four core steps: preview road error calculation, longitudinal and lateral coordinated preview distance adjustment, adaptive fuzzy sliding mode control law derivation, and front wheel steering angle command output. The preview error module calculates the lateral and heading deviations between the vehicle and the desired path. The preview distance adjustment module dynamically optimizes preview parameters based on longitudinal speed, longitudinal acceleration, and road curvature obtained via V2X. The AFSM controller generates the optimal front wheel steering angle control variable. Finally, the command is sent to the steering actuator.
2.2.1 Preview road error model with longitudinal and lateral coordination
Based on preview follow theory, a road error differential equation is established to describe the relative motion relationship between the vehicle and the path, as shown in Equation 7 ().
In Equation 7, represents the lateral error between the vehicle and the preview point, represents the heading error between the vehicle and the preview point, represents the longitudinal error, and represents the heading angular velocity of the preview point. The preview distance dynamically adjusts according to the current longitudinal speed and road curvature, achieving preview coordination of longitudinal and lateral motions.
2.2.2 Design of AFSM controller
To balance control robustness and smoothness, an integral sliding surface is designed as the foundation of the controller, as shown in Equation 8 ().
In Equation 8, represents the sliding surface function, and represents the positive sliding surface coefficient, which regulates the error convergence speed. This integral sliding surface corresponds to the core calculation unit in the sliding mode control main loop of the AFSM structure shown in Figure 3, and provides the basis for subsequent reaching law calculation and fuzzy parameter adjustment. An improved double-power reaching law is adopted to ensure fast convergence of the system while reserving parameter space for subsequent fuzzy adjustments, as shown in Equation 9.
FIGURE 3
In Equation 9, and represent the reaching law coefficients, while and represent the power coefficients. A dual-input dual-output fuzzy controller is designed, which takes the sliding surface and its rate of change as inputs to dynamically adjust the reaching law coefficients, as shown in Equation 10.
In Equation 10, and represent the initial values of the parameters, and and represent the parameter adjustment amounts output by the fuzzy controller. The dual-input dual-output fuzzy controller is configured with explicit parameters. Inputs are the sliding surface s and its time derivative , both with a universe of discourse of [-0.2, 0.2]. Outputs are the reaching law adjustment values and with universes of [-0.05, 0.05] and [-0.03, 0.03] respectively. All variables adopt triangular membership functions divided into five fuzzy sets: NB, NS, ZE, PS, PB. The 25 inference rules follow the chattering suppression principle: large increases the reaching coefficient for fast convergence, while small reduces the coefficient to smooth control output. The overall structure of the AFSM controller is shown in Figure 3.
As shown in Figure 3, this controller consists of a sliding mode control main loop, a fuzzy parameter adaptive adjustment loop, and a longitudinal state feedback module. The main loop takes the lateral error as input to generate the basic control variable. The fuzzy loop dynamically adjusts the reaching law parameters through fuzzy inference, which suppresses the chattering phenomenon of traditional sliding mode control. The longitudinal state feedback module corrects the sliding surface coefficients based on longitudinal acceleration, achieving parameter coordination for longitudinal and lateral control. Combining the sliding surface and the reaching law, the final front wheel steering angle control law is derived, as shown in Equation 11.
In Equation 11, represents the front wheel steering angle control command, and represents the lateral control gain, which is jointly determined by vehicle dynamics parameters and driving conditions. Stability of the closed-loop lateral system is proven via the Lyapunov criterion. A Lyapunov function is defined as , and its time derivative satisfies ()) under the designed reaching law, ensuring finite-time convergence of the sliding surface. The controller remains stable within the lateral error range of [-0.5 m, 0.5 m] and the longitudinal speed range of [10 km/h, 80 km/h].
2.3 Design of FTD3-PID longitudinal control system
2.3.1 Overall system structure
On the basis of longitudinal and lateral coordinated control, traditional longitudinal controllers do not coordinate with the lateral path tracking state. This lack of coordination results in unsmooth driving-braking switching and an inability to dynamically adjust speed according to curve curvature, which compromises vehicle driving stability. Therefore, this study proposes an FTD3-PID longitudinal control system with longitudinal and lateral coordination. The system structure is shown in Figure 4.
FIGURE 4
As shown in Figure 4, this system takes the vehicle state output by the dynamics model, the reference speed planned cooperatively via V2X, the feedback lateral error, and the road curvature as inputs. It sequentially completes four core stages: coordinated desired speed generation, FTD3-PID parameter adaptive adjustment, inverse longitudinal dynamics calculation, and driving-braking logic switching. The coordinated desired speed module dynamically corrects the reference speed based on lateral road curvature and lateral error. The acceleration command is generated through the FTD3-PID controller, and the acceleration information is analyzed and converted into regulations such as brake pressure. The driving-braking switching module achieves a smooth transition between the two operating conditions.
2.3.2 Design of FTD3-PID controller
The speed tracking error is defined as the difference between the coordinated desired speed and the actual speed, as shown in Equation 12.
In Equation 12, represents the longitudinal speed tracking error, represents the desired driving speed generated through longitudinal and lateral coordination, and represents the actual driving speed of the vehicle. Based on PID control theory, a basic speed control law is constructed, as shown in Equation 13 ().
In Equation 13, represents the desired acceleration output by the controller. A fuzzy inference module is designed to constrain the action space of the TD3 algorithm, which improves the exploration efficiency of the algorithm. A Gaussian membership function is used to describe the fuzzy characteristics of the input variables, as shown in Equation 14 ().
In Equation 14, represents the membership degree of the -th input variable corresponding to the -th fuzzy set, represents the speed error and its rate of change, represents the mean value of the Gaussian function, and represents the standard deviation of the Gaussian function. The internal structure of the FTD3 module is shown in Figure 5.
FIGURE 5
As shown in Figure 5, this module consists of a fuzzy preprocessing unit, a twin critic network, an actor network, and a target network. The fuzzy preprocessing unit fuzzifies the input error to constrain the action output range of TD3. The twin critic network evaluates action values, the actor network outputs PID parameter adjustment amounts, and the target network utilizes a delayed update mechanism to improve algorithm stability. The TD3 algorithm updates parameters via a loss function, as shown in Equation 15.
In Equation 15, represents the loss function of the critic network, is the target Q-value, is the output value of the critic network, is the system state, and is the action output by the actor network. The actor network updates parameters by maximizing the Q-value to generate PID parameter adjustment amounts, as shown in Equation 16.
In Equation 16, represents the objective function of the actor network, and represents the PID parameter adjustment amounts output by the actor network. Finally, the adaptively updated PID control parameters are obtained, as shown in Equation 17.
In Equation 17, , , and represent the initial values of the PID parameters, and , , and represent the parameter adjustment amounts output by the FTD3 module. Bounded-input bounded-output stability of the longitudinal control system is guaranteed by the PID baseline structure with constrained parameter adjustment ranges. The adjustment outputs of the FTD3 module are limited to ±30% of the initial parameters, which prevents control divergence and maintains stable operation under the speed range of 0–100 km/h and road adhesion coefficient above 0.2.
3 Results and discussion
3.1 Experimental environment setup
To verify the effectiveness of the proposed technology in vehicle longitudinal and lateral control, this study carried out corresponding lateral and longitudinal scenario experiments. Table 1 lists the experimental hardware and software.
TABLE 1
| Project | Content |
|---|---|
| Processor | Intel i7-10700K |
| Random access memory | 32 GB |
| Hard disk | 1 TB |
| GPU | NVIDIA RTX 3060 |
| System | Ubuntu 20.04 LTS |
| Deep learning framework | PyTorch 1.10 |
| Predictive control tools | Simulink/TruckSim |
Experimental environment setup configuration.
As shown in Table 1, the experiment relied on the Ubuntu 20.04 LTS operating system and integrated the Simulink/TruckSim co-simulation platform to build a vehicle dynamics verification model. Key parameters of the experimental vehicle dynamics model and controller: vehicle mass set to 2,408 kg, wheelbase set to 2.675 m, yaw moment of inertia set to 3,231 kg m2, and front/rear cornering stiffness to 88,500 N/rad and 118,200 N/rad, respectively. The road adhesion coefficient is set to 0.85 for dry asphalt and 0.35 for wet roads. V2X communication adopts an update rate of 10 Hz with a packet loss rate of up to 5%. The controller sampling time is 0.02 s. The TD3 reward function is designed as , where and are speed and acceleration errors, is control effort, and is control variation. Training is conducted for 1,000 episodes with an actor learning rate of , critic learning rate of discount factor of 0.99, batch size of 64, and exploration noise of 0.2. It is worth noting that the controller is designed based on a general vehicle dynamics framework. When simulating and verifying the algorithm logic, a passenger car level model weighing 2,408 kg was used; At the same time, an online parameter calibration mechanism is adopted to adapt to the parameter characteristics of large commercial vehicles in real vehicle testing, ensuring that the control algorithm has cross vehicle applicability.
In addition, this study selected an 8-m-long drive-by-wire chassis experimental vehicle as the test object in real scenarios. The vehicle platform integrated a HiRain ControlBase_S rapid prototyping controller and carried an RTK-GPS high-precision differential positioning antenna to obtain real-time vehicle speed and heading information, achieving bottom-level steering and throttle interactions via the vehicle CAN bus. A consistent control architecture is applied to the real vehicle, with core parameters calibrated online according to the 8-m commercial bus to match its dynamic characteristics.
3.2 AFSM lateral control experiment
To verify the superiority of the proposed AFSM, the experiment introduced three baseline methods for comparison: Radial Basis Function Neural Network Proportional Integral Derivative (RBF-PID), Model Predictive Control with Linear Quadratic Regulator (MPC-LQR), and Fuzzy Adaptive Terminal Sliding Mode Control (FATSMC).
To verify the dynamic response performance of AFSM under sudden path changes, the experiment set up multiple step lateral target displacements. The test aimed to evaluate the transient tracking capability and steady-state error of the controllers, covering a time range from 10 s to 150 s, as shown in Figure 6.
FIGURE 6
As shown in Figure 6a, during the initial phase of the step change (within 20 s), the target displacement was 3.80 m, while AFSM reacted quickly, achieving 3.78 m and FATSMC attained 3.65 m. From a relatively comparative perspective, the performance of RBF-PID was not up to the mark during abrupt changes; within 100 s–110 s, where the target displacement reduced suddenly from 5.80 m to 3.50 m, the tracking displacement was limited to 4.20 m. Figure 6b shows the lateral tracking displacement error results. The error curve of AFSM stabilized rapidly, dropping to −0.02 m at 20 s, and the maximum error did not exceed 0.15 m in subsequent step changes. In contrast, the error value of MPC-LQR rebounded drastically from −0.12 m to 0.70 m during the 100 s–110 s period, which presented obvious abnormal fluctuations. This phenomenon reflected that linear models struggle to achieve global smooth convergence under strongly nonlinear operating conditions.
To verify the robustness of the controller during continuous smooth lane changes, the experiment designed a desired displacement trajectory that grew linearly over time. The test aimed to examine the anti-lag capability and steady-state following effect, as shown in Figure 7.
FIGURE 7
Figure 7a display the tracking displacements of lane changing. The good result of AFSM is obtained since its tracking value is 7.98 m after 60 s when the target displacement is 8.00 m, which means that there exists almost no difference between the two kinds of curves. However, when RBF-PID and MPC-LQR were used to track strongly coupled features, their performance suffered a decline; for instance, the tracking value of RBF-PID was only 7.25 m at 60 s. Figure 7b illustrates the lane-changing tracking displacement errors. The FATSMC method generated prominent abnormal fluctuations between 40 s and 60 s, where the error expanded rapidly from 0.18 m to 0.65 m. The error of MPC-LQR also exhibited noticeable decay, reaching 0.52 m at 80 s. Conversely, the error of AFSM remained stable within the [-0.05 m, 0.08 m] interval. This indicates that traditional algorithms without adaptive sliding mode screening mechanisms easily generate serious accumulated tracking deviations during continuous ramp changes.
To verify the comprehensive anti-noise ability and real-time response capability of the lateral controller in real physical environments, this study selected 12 operating conditions with different road friction and crosswind disturbances for comprehensive validation using the RTK-GPS system and the micro rapid prototyping controller. The 12 real-vehicle operating conditions were as follows: Conditions one to three represented straight driving on dry asphalt at 20, 40, and 60 km/h, respectively; Conditions four to six represented straight driving on wet roads at 20 km/h with light breeze, 40 km/h with strong crosswind, and 60 km/h with wind gusts, respectively; Conditions seven to nine represented dry curve driving with large curvature at 20 km/h, medium curvature at 40 km/h, and small curvature at 60 km/h, respectively; Condition 10 involved driving on an ice-snow low-adhesion road at 30 km/h; Condition 11 represented a dry curve at 40 km/h with strong crosswind; and Condition 12 represented comprehensive extreme lane changing on a double-lane-change wet road at 50 km/h. The extreme double-lane-change test procedure was designed with reference to ISO 15037-2:2002 (general conditions for heavy vehicle dynamics testing), ISO 14793:2011 (lateral transient response test methods for heavy commercial vehicles and buses), and the ISO 3888-2 double lane change maneuver protocol. The test was conducted on an 8-m commercial bus (heavy vehicle category, >5 t) on wet asphalt with a friction coefficient of 0.35. The entry speed was set to 50 km/h with a tolerance of ±2 km/h. The vehicle was loaded to curb weight plus 50% payload, representing a typical operating condition. The pass/fail criteria required that the lane change be completed as one continuous movement within 10 s, with maximum lateral acceleration not exceeding 1 m/s2 and 0.5 s moving average lateral jerk not exceeding 5 m/s3, while avoiding lane markings or cones. The results are shown in Figure 8.
FIGURE 8
Figures 8a,b show the Root-Mean-Square (RMS) performance of lateral and heading errors. Under the extreme test of Condition 12, AFSM-LC maintained the lateral and heading error RMS values at extremely low levels of 0.035 m and 0.85°, respectively. Meanwhile, the lateral error of RBF-PID rose to 0.152 m under Condition 12, which indicated insufficient anti-noise capability. Figure 8c displays the steering wheel angle control chattering rate results. Benefiting from its adaptive smoothing mechanism, AFSM generated a maximum chattering rate of only 2.4%, meeting the comfort requirements for vehicle-mounted real-time monitoring, whereas the chattering rate of FATSMC increased abnormally to 9.5%. Figure 8d presents the average computational time of the algorithms. The longest execution time of AFSM was only 4.12 ms, satisfying high-frequency computing power requirements. However, the execution time of MPC-LQR generated an abnormal peak of 18.50 ms under Condition 10, which reduced the safety and reliability of the underlying drive-by-wire hardware.
3.3 FTD3-PID longitudinal control experiment
To verify the control advantages of the proposed FTD3-PID longitudinal control system with longitudinal and lateral coordination, the experiment introduced three baseline algorithms for comparison: Deep Deterministic Policy Gradient (DDPG), Adaptive Fuzzy Proportional Integral Derivative (AF-PID), and Genetic Algorithm optimized Model Predictive Control (GA-MPC).
In order to achieve comprehensive validation on control performance, this research conducted two simulations with complicated traffic conditions using actual roads. One is the low to medium speed sharp curve condition with 20 km/h speed for checking how the controller could compensate power longitudinally with respect to sharp steering resistance. The other is the medium to high speed continuous curve condition with 40 km/h speed for investigating the global stability with coordinated longitudinal and lateral speed control.
The experiment first tested the low-to-medium speed sharp curve condition. To verify the suppression effect of longitudinal and lateral coordination on lateral errors and heading deviations at 20 km/h, this study selected 12 driving path nodes for validation, as shown in Figure 9.
FIGURE 9
The results for lateral errors can be seen from Figure 9a. FTD3-PID was effective in spatial temporal feature learning. Within the first 100 m of the path, its lateral error had reached an error convergence of 0.015 m before stabilizing at 0.012 m at 600 m. The next best algorithm was AF-PID with a final error value of 0.038 m. Compared to others, DDPG failed in dealing with continuous curve pathing. During the range of 250 m–300 m, there was an error of drastic drop from 0.045 m to 0.085 m. Figure 9b presents the heading error results. The heading error curve of FTD3-PID decreased smoothly, dropping to 0.21° at 200 m and finally converging to an extremely low error of 0.15°. Meanwhile, the heading error value of GA-MPC rebounded from 0.52° to 0.95° during the 300 m–350 m period. This reflected that traditional metaheuristic optimization networks experience gradient fluctuations in complex curvature mapping, making it difficult to achieve global smooth convergence.
In addition to studying the accurate assignment of speed longitudinally and the smooth regulation of longitudinal acceleration at the speed of 40 km/h, 12 path points were also selected for comparison, as illustrated in Figure 10.
FIGURE 10
In Figure 10a, the FTD3-PID controller showed outstanding results and successfully recognized the target velocity with outstanding stability, where the largest speed error was found to be just 0.25 km/h. The performance of both DDPG and AF-PID degraded due to extraction of longitudinal-lateral coupling features; for example, DDPG increased the speed error to 1.85 km/h in the 300 m curve. Figure 10b illustrates the longitudinal acceleration fluctuations. The acceleration values of GA-MPC exhibited clear attenuation anomalies, undergoing severe oscillations with a fluctuation rate as high as 0.65 m/s2 when handling sudden curvature changes (400 m–450 m), and AF-PID also generated a fluctuation of 0.58 m/s2 at this location. This indicates that traditional optimization methods without deep action screening mechanisms can easily cause control confusion with similar dynamic curvatures.
Finally, to verify the effectiveness of the algorithm in practical scenarios, this study selected 10 road sections in a specific campus for testing: Section 1 was a conventional straight road; Sections 2 and 3 were single and continuous speed bump segments, respectively; Section 4 was a large-curvature curve entry; Section 5 was a continuous right-angle turn; Section 6 was an S-shaped continuous shuttle curve; Section 7 was a bumpy cobblestone road; Section 8 was an undulating slope; Section 9 was a partially sprinkled wet and slippery road; and Section 10 was an emergency braking avoidance zone before the finish line. The experiment extracted performance indicators of these 10 feature road sections using AutoBox, as shown in Figure 11.
FIGURE 11
Figures 11a,b present the maximum lateral offset and speed RMS performance. Under the extreme condition of Section 8, FTD3-PID maintained the maximum offset at 0.042 m and the speed error at 0.15 km/h, whereas the offset of DDPG rose to 0.185 m, indicating insufficient resistance to lateral disturbances. Figure 11c illustrates real-time memory usage. Benefiting from network compression, FTD3-PID reached a maximum usage of only 45.2 MB, satisfying the computing requirements of vehicle-mounted hardware, while the usage of GA-MPC spiked to 125.6 MB. Figure 11d shows the ride comfort jerk results. The jerk of FTD3-PID on Section 5 was only 0.12 m/s3. Conversely, AF-PID generated a high jerk of 0.85 m/s3 on Section 8, which would frequently cause psychological anxiety for occupants in practical industrial applications, significantly reducing the longitudinal psychological comfort and driving safety reliability of autonomous vehicles.
To quantify the contribution of the online-calibrated rolling resistance coefficient to longitudinal control performance, an ablation study was conducted under the medium-to-high speed continuous curve condition (40 km/h). Two configurations were compared: the full model with V2X-based online calibration of the rolling resistance coefficient, and the fixed model with the coefficient held constant at 0.015. The rolling resistance coefficient in the full model was updated online via a forgetting-factor recursive least squares estimator based on longitudinal dynamics. Five evaluation metrics were recorded over 10 independent trials, and the results are presented in Table 2.
TABLE 2
| Metric | Full model (online calibration) | Fixed model (f = 0.015) | p-value |
|---|---|---|---|
| Maximum speed error (km/h) | 0.25 ± 0.03 | 0.82 ± 0.07 | <0.01 |
| Mean absolute speed error (km/h) | 0.08 ± 0.01 | 0.34 ± 0.04 | <0.01 |
| Longitudinal acceleration fluctuation (m/s2) | 0.12 ± 0.02 | 0.43 ± 0.06 | <0.01 |
| Speed tracking settling time (s) | 1.8 ± 0.2 | 4.3 ± 0.5 | <0.01 |
| Control effort variation (%) | 6.2 ± 0.8 | 15.7 ± 1.9 | <0.01 |
Ablation study results of rolling resistance coefficient online calibration (mean ± std, n = 10).
From Table 2, the full model with online calibration achieves significantly lower speed errors and acceleration fluctuations across all metrics (p < 0.01, paired t-test). The maximum speed error is reduced from 0.82 km/h to 0.25 km/h, and the acceleration fluctuation decreases from 0.43 m/s2 to 0.12 m/s2. The settling time is also shortened by 58.1%, indicating faster convergence. These results confirm that online calibration of the rolling resistance coefficient plays a critical role in maintaining accurate speed tracking and ride comfort under varying road conditions, validating its necessity as a key calibrated parameter in the proposed coordinated control system.
4 Conclusions and perspectives
Longitudinal and lateral control of intelligent vehicles is critical for safe driving. At present, traditional models face problems such as weak adaptability to dynamic road conditions, disconnected longitudinal and lateral coupling, and a tendency to cause system chattering. To address these issues, this study proposed a vehicle lateral control technique that combines V2X dynamic calibration with AFSM. This technique utilizes V2X to update dynamics parameters and adjusts the sliding mode reaching law through fuzzy inference to suppress chattering and achieve precise path tracking. In addition, this study proposed a longitudinal coordinated technique based on FTD3-PID. It takes states and errors as inputs, constrains the action space of TD3 through fuzzy inference, and utilizes twin networks to adaptively update PID parameters, which achieves precise speed tracking and smooth switching. In ramp lane-changing tracking tests, the error of AFSM stabilized within the range of −0.05 m–0.08 m, which was significantly superior to the 0.52 m error of MPC-LQR. In real-vehicle extreme lane changes, the lateral error RMS of AFSM was 0.035 m, and the control chattering rate was only 2.4%, outperforming the 0.152 m of RBF-PID and the 9.5% of FATSMC. In curve control, FTD3-PID yielded a maximum speed error of 0.25 km/h, and the jerk was as low as 0.12 m/s3, which was superior to the 1.85 km/h of DDPG and the 0.85 m/s3 of AF-PID. Therefore, the proposed technology demonstrated successful application in coordinated vehicle control. However, this study has some limitations. AFSM relies heavily on V2X communication, making parameter calibration prone to mismatch during communication packet loss. In addition, the fuzzy rules of FTD3-PID depend on expert prior experience, which leads to insufficient generalization ability under extreme operating conditions. Future work will involve applying a delay compensation technique for increased robustness in cases where there is a poor link in the network and use unsupervised learning algorithms to produce fuzzy rules on the fly.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
YL: Writing – original draft, Methodology, Data curation, Conceptualization. AN: Formal Analysis, Investigation, Project administration, Writing – review and editing.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Summary
Keywords
AFSM, Cav, coordinated control, RL, vehicle dynamics model
Citation
Luo Y and Niu A (2026) Integrated longitudinal and lateral coordinated control system for connected and automated vehicles driven by fuzzy control algorithm. Front. Mech. Eng. 12:1932727. doi: 10.3389/fmech.2026.1932727
Received
09 July 2026
Revised
07 August 2026
Accepted
12 August 2026
Published
07 September 2026
Volume
12 - 2026
Edited by
Adel Razek, UMR8507 Laboratoire Génie électrique et électronique de Paris (GeePs), France
Reviewed by
Beomjoon Pyun, Korea Automotive Technology Institute, Republic of Korea
James Pickering, Harper Adams University Engineering Subject Area, United Kingdom
Mohamad Ezral Bin Baharudin, Universiti Malaysia Perlis, Malaysia
Updates
Copyright
© 2026 Luo and Niu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Yingzhe Luo, luoyingzhe425@163.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.