In recent years, the rapid advancement of electric car technology, particularly in China EV markets, has driven significant innovations in autonomous driving systems. Trajectory tracking is a critical function for intelligent electric cars, ensuring safe and precise path following under various driving conditions. However, the dynamic nature of tire cornering stiffness poses a major challenge, as it varies with factors like vertical load and slip angle, leading to nonlinear behavior that affects control accuracy. Traditional control methods often assume constant cornering stiffness, which limits performance in extreme scenarios. This study addresses this issue by developing a robust trajectory tracking controller for distributed electric cars that incorporates real-time estimation of tire cornering stiffness. We focus on China EV advancements to highlight the global impact of electric car innovations.
Distributed electric cars, with their independent four-wheel drive capabilities, offer enhanced controllability and flexibility, making them ideal for testing advanced control algorithms. In China EV research, these vehicles are increasingly used to improve stability and efficiency. The core of our approach involves a seven-degree-of-freedom (DOF) vehicle dynamics model, which captures essential behaviors like longitudinal, lateral, and yaw motions. We analyze how vertical load and slip angle influence cornering stiffness, as these parameters are crucial for accurate modeling. For instance, under high lateral acceleration, the cornering stiffness can deviate significantly from linear assumptions, impacting the electric car’s trajectory tracking performance. To handle this, we design a cornering stiffness estimator using a radial basis function neural network (RBFNN), optimized with k-means clustering and least mean square (LMS) methods to enhance prediction accuracy and convergence speed. This estimator is integrated into a model predictive control (MPC) framework, which dynamically adjusts the control inputs while adhering to constraints like yaw rate and sideslip angle. Simulation results on both high and low adhesion roads demonstrate that our controller outperforms conventional methods, reducing tracking errors and improving stability. This work underscores the potential of China EV technologies in advancing autonomous driving systems for electric cars worldwide.
The vehicle dynamics model for the distributed electric car is based on a 7-DOF system, including longitudinal, lateral, yaw, and four wheel rotations. We assume equal wheelbases and tire characteristics for simplicity. The equations of motion in the vehicle coordinate system are derived as follows. The longitudinal dynamics account for forces from front and rear wheels, considering steering angles and tire slip. The lateral dynamics incorporate lateral forces and yaw moments, which are influenced by tire behavior. The yaw dynamics involve moments generated by tire forces and the vehicle’s inertia. Additionally, wheel dynamics describe the rotational motion of each wheel, driven by motor torques and braking forces. For an electric car, the independent wheel control allows precise torque distribution, enhancing stability in China EV applications.
The longitudinal dynamics equation is given by:
$$ m(\dot{v}_x – \omega v_y) = (F_{x,lf} + F_{x,rf}) \cos \delta_f – (F_{y,lf} + F_{y,rf}) \sin \delta_f + (F_{x,lr} + F_{x,rr}) $$
where \( m \) is the vehicle mass, \( v_x \) and \( v_y \) are longitudinal and lateral velocities, \( \omega \) is the yaw rate, \( \delta_f \) is the front steering angle, and \( F_{x,j} \) and \( F_{y,j} \) represent longitudinal and lateral tire forces for each wheel (e.g., lf for left front, rf for right front). This equation highlights how steering and tire forces affect the electric car’s acceleration.
The lateral dynamics are expressed as:
$$ m(\dot{v}_y + \omega v_x) = (F_{y,lf} + F_{y,rf}) \cos \delta_f + (F_{x,lf} + F_{x,rf}) \sin \delta_f + (F_{y,lr} + F_{y,rr}) $$
This shows the coupling between lateral motion and yaw, which is critical for trajectory tracking in electric cars.
The yaw dynamics equation is:
$$ I_z \dot{\omega} = [(F_{x,rf} – F_{x,lf}) \cos \delta_f + (F_{y,lf} – F_{y,rf}) \sin \delta_f] \frac{w_b}{2} + [(F_{x,lf} + F_{x,rf}) \sin \delta_f + (F_{y,lf} + F_{y,rf}) \cos \delta_f] l_f + (F_{x,rr} – F_{x,lr}) \frac{w_b}{2} – (F_{y,lr} + F_{y,rr}) l_r $$
where \( I_z \) is the yaw moment of inertia, \( w_b \) is the wheelbase, and \( l_f \) and \( l_r \) are distances from the center of gravity to the front and rear axles. This equation emphasizes the role of tire forces in generating yaw moments, which is vital for controlling the electric car’s orientation.
The wheel dynamics for each wheel are described by:
$$ J \dot{\omega}_i = T_{d,i} – T_{b,i} – F_{x,i} r_{e,i} $$
for \( i = 1 \) to \( 4 \), where \( J \) is the wheel moment of inertia, \( \omega_i \) is the wheel angular velocity, \( T_{d,i} \) is the drive torque, \( T_{b,i} \) is the brake torque, and \( r_{e,i} \) is the effective rolling radius. This independent wheel control is a key feature of distributed electric cars, enabling better traction and stability in China EV designs.
The tire model uses the Magic Formula to represent nonlinear tire behavior, which is essential for accurate cornering stiffness estimation. The general form is:
$$ y = D \sin[C \tan^{-1}(Bx – E(Bx – \tan^{-1} Bx))] $$
where \( y \) can be lateral force, longitudinal force, or aligning torque, and \( x \) is the slip angle or slip ratio. Parameters \( B \), \( C \), \( D \), and \( E \) are determined empirically and vary with road conditions. For electric cars, this model helps capture the complex interactions between tires and the road, especially in varying adhesion scenarios common in China EV testing.
The motor model for the hub motors in the distributed electric car is based on a permanent magnet synchronous motor. The simplified transfer function is:
$$ G(s) = \frac{T_r}{T_i} = \frac{1}{2\xi^2 s^2 + 2\xi s + 1} $$
where \( T_i \) is the input torque, \( T_r \) is the output torque, and \( \xi \) is a physical constant. The torque-speed characteristics are mapped to ensure efficient operation across different speeds, which is crucial for the performance of electric cars in dynamic environments.
To estimate the time-varying cornering stiffness, we develop an RBFNN-based estimator. The RBFNN uses Gaussian activation functions and is optimized with k-means clustering to initialize the centers and LMS to update the weights. The input to the network is the vertical load and slip angle, and the output is the estimated cornering stiffness. The Gaussian function for each hidden node is:
$$ u_i = \exp\left(-\frac{(x – c_i)^T (x – c_i)}{2\sigma_i^2}\right), \quad i = 1, 2, \ldots, M $$
where \( x \) is the input vector, \( c_i \) is the center vector, \( \sigma_i \) is the width parameter, and \( M \) is the number of hidden nodes. The k-means algorithm clusters the input data to determine optimal centers, improving convergence. The LMS update rule for weights is:
$$ \omega = \frac{e}{M c_{\text{max}}^2} (x_p – C_i) $$
where \( e \) is the error, \( c_{\text{max}} \) is the maximum center distance, and \( x_p \) is the input pattern. This approach enhances the estimator’s accuracy and speed, making it suitable for real-time applications in electric cars.
The trajectory tracking controller is designed using MPC, which predicts future states and optimizes control inputs. The state vector includes lateral velocity, longitudinal velocity, yaw angle, yaw rate, and position coordinates. The control input is the front steering angle. The discrete-time state-space model is linearized around the reference trajectory:
$$ \Delta \xi(k+1) = A_k \Delta \xi(k) + B_k \Delta u(k) $$
where \( \Delta \xi \) is the state deviation, \( \Delta u \) is the control increment, and \( A_k \) and \( B_k \) are matrices derived from linearization. The cornering stiffness estimates from the RBFNN are incorporated into these matrices to account for nonlinearities. For example, the matrix \( A_k \) includes terms like \( -\frac{2(C_f + C_r)}{m v_x} \), where \( C_f \) and \( C_r \) are the front and rear cornering stiffnesses, which are now time-varying. This integration allows the MPC to adapt to changing tire conditions, improving the electric car’s tracking performance.
The MPC objective function minimizes tracking errors and control efforts:
$$ J = \sum_{i=1}^{N_p} \| \eta(t+i|t) – \eta_{\text{ref}}(t+i|t) \|^2_Q + \sum_{i=1}^{N_c} \| \Delta u(t+i|t) \|^2_R + \rho \varepsilon^2 $$
where \( N_p \) is the prediction horizon, \( N_c \) is the control horizon, \( Q \) and \( R \) are weighting matrices, \( \rho \) is a slack variable weight, and \( \varepsilon \) is a relaxation factor. Constraints include limits on steering angle, steering rate, sideslip angle, and yaw rate to ensure stability. For instance, the yaw rate constraint is \( |\omega| \leq \frac{\mu g}{v_x} \), where \( \mu \) is the road adhesion coefficient and \( g \) is gravity. These constraints are vital for safe operation of electric cars, especially in adverse conditions encountered in China EV deployments.
Simulation results validate the controller’s performance on high and low adhesion roads. We compare our method (K_RBF_MPC) with a standard MPC without cornering stiffness estimation and an RBF-based MPC without optimization. The table below summarizes key parameters used in the simulations for the electric car model.
| Parameter | Value |
|---|---|
| Vehicle mass \( m \) | 1500 kg |
| Yaw inertia \( I_z \) | 2500 kg·m² |
| Wheelbase \( w_b \) | 2.8 m |
| Front axle distance \( l_f \) | 1.2 m |
| Rear axle distance \( l_r \) | 1.6 m |
| Prediction horizon \( N_p \) | 20 |
| Control horizon \( N_c \) | 5 |
| Sampling time \( T \) | 0.01 s |
On a high adhesion road (\( \mu = 0.85 \)) at 110 km/h, our controller reduces the maximum lateral tracking error to 0.115 m, compared to 0.152 m for RBF_MPC and 0.22 m for standard MPC. The steering angle and lateral acceleration show smoother responses, indicating better stability. The cornering stiffness estimates closely match reference values, with lower mean squared errors. For example, the front cornering stiffness error for K_RBF is \( 2.81 \times 10^6 \) N/rad, versus \( 5.95 \times 10^7 \) N/rad for RBF. This demonstrates the effectiveness of our estimator in capturing tire dynamics for electric cars.
On a low adhesion road (\( \mu = 0.3 \)) at the same speed, the improvements are even more pronounced. The maximum lateral error is 0.125 m for K_RBF_MPC, 0.172 m for RBF_MPC, and 0.249 m for MPC. The sideslip angle and yaw rate remain within safe limits, highlighting the controller’s robustness. The table below compares the mean squared errors for cornering stiffness and lateral forces in this scenario.
| Method | Front Cornering Stiffness Error (N/rad) | Rear Cornering Stiffness Error (N/rad) | Front Lateral Force Error (N) | Rear Lateral Force Error (N) |
|---|---|---|---|---|
| K_RBF | \( 2.69 \times 10^6 \) | \( 2.73 \times 10^6 \) | 355 | 180 |
| RBF | \( 5.63 \times 10^7 \) | \( 3.17 \times 10^7 \) | 7210 | 1510 |
These results confirm that our approach enhances trajectory tracking accuracy and stability for distributed electric cars, even under challenging conditions. The use of advanced estimation and control techniques aligns with the innovation trends in China EV development, where electric cars are expected to perform reliably in diverse environments.
In conclusion, this study presents a comprehensive trajectory tracking control strategy for distributed electric cars that dynamically accounts for tire cornering stiffness variations. By integrating an optimized RBFNN estimator with an MPC framework, we achieve significant improvements in tracking precision and vehicle stability. The methods are validated through simulations on both high and low adhesion roads, demonstrating superior performance compared to conventional controllers. This work contributes to the ongoing advancements in China EV technology, showcasing how intelligent control systems can enhance the safety and efficiency of electric cars. Future research could explore real-time implementation and integration with other vehicle systems to further optimize performance for autonomous electric cars.