In the context of global energy crises and growing environmental awareness, electric vehicles, particularly in China’s rapidly expanding EV market, have emerged as a pivotal direction for the automotive industry. However, challenges in enhancing driving controllability and comfort persist, especially in complex driving scenarios requiring precise steering control. Traditional steering systems in internal combustion engine vehicles rely on mechanical linkages, whereas electric vehicles utilize motor-driven systems that allow for more flexible wheel speed differential control. This fundamental difference necessitates comprehensive vehicle dynamics and motor kinetics analysis for effective differential steering in electric vehicles. Current research on electric vehicle steering control spans fuzzy logic, adaptive control, and PID-based strategies, yet issues like high algorithm complexity and poor real-time performance hinder practical applications. This paper proposes a differential steering automatic control method for the rear axle synchronous drive motor in electric vehicles, leveraging BP neural networks to optimize parameters and improve steering accuracy and responsiveness. By integrating permanent magnet synchronous motor (PMSM) dynamics and vehicle steering geometry, the control strategy aims to enhance the performance of China EV systems, ensuring better handling and comfort.
The core of the proposed system involves a rear axle drive configuration, where the front wheels handle steering independently of drive torque, thereby increasing agility. The overall control framework, as illustrated in the system diagram, processes inputs from the throttle pedal and steering wheel via CAN communication to determine vehicle speed and steering commands. A vehicle control chip analyzes these inputs to decide on differential steering activation based on the yaw angle. If the yaw angle is zero, the vehicle maintains straight-line motion; otherwise, differential steering is optimized using BP neural networks to adjust rear axle drive speeds. This approach addresses nonlinearities in electric vehicle dynamics, such as tire deformation and load transfer, which are influenced by speed and steering angle. The integration of BP neural networks allows for adaptive learning and nonlinear mapping, crucial for handling complex road conditions in China EV applications.
To establish a foundation for the control strategy, the dynamics of the permanent magnet synchronous motor (PMSM) are analyzed. PMSMs are preferred in electric vehicles for their high efficiency, rapid response, and compact size, which align with the needs of China EV designs for extended range and safety. The surface-mounted PMSM structure simplifies modeling by assuming equal inductances along the d and q axes, neglecting hysteresis and eddy current losses. The voltage equations in the d-q rotating frame, derived via Park transformation from the a-b-c stationary frame, are given by:
$$ u_d = R_s i_d + L_d \frac{di_d}{dt} – \omega_e L_q i_q $$
$$ u_q = R_s i_q + L_q \frac{di_q}{dt} + \omega_e (\psi_f + L_q i_q) $$
Here, \( u_d \) and \( u_q \) represent the stator voltages, \( i_d \) and \( i_q \) the stator currents, \( L_d \) and \( L_q \) the stator inductances, \( R_s \) the stator resistance, \( \omega_e \) the electrical angular velocity, and \( \psi_f \) the permanent magnet flux linkage. The flux linkage equations are:
$$ \psi_d = \psi_f + L_d i_d $$
$$ \psi_q = L_q i_q $$
The torque equation for the PMSM, considering the surface-mounted type where \( L_d = L_q \), and setting \( i_d = 0 \) for control simplicity, reduces to:
$$ T_e = \frac{3P_n \psi_f i_q}{2} $$
where \( T_e \) is the electromagnetic torque and \( P_n \) is the number of pole pairs. The motion equation of the motor is:
$$ J \frac{d\omega_m}{dt} = T_e – T_L – B \omega_m $$
with \( J \) as the moment of inertia, \( \omega_m \) the mechanical angular velocity, \( T_L \) the load torque, and \( B \) the damping coefficient. These equations form the basis for integrating PMSM dynamics into the differential steering control of electric vehicles.
Vehicle steering geometry is analyzed using the Ackermann model to determine wheel speeds during turns. The turning radii for each wheel are derived as follows:
$$ r_{fr} = \sqrt{d_g^2 + \left( \frac{d_g}{\tan \delta} – \frac{d_l}{2} \right)^2 } $$
$$ r_{br} = \frac{d_g}{\tan \delta} – \frac{d_l}{2} $$
$$ r_{fl} = \sqrt{d_g^2 + \left( \frac{d_g}{\tan \delta} + \frac{d_l}{2} \right)^2 } $$
$$ r_{bl} = \frac{d_g}{\tan \delta} + \frac{d_l}{2} $$
where \( r_{fr} \), \( r_{br} \), \( r_{fl} \), and \( r_{bl} \) are the turning radii for the front-right, rear-right, front-left, and rear-left wheels, respectively; \( d_g \) is the wheelbase, \( d_l \) is the track width, and \( \delta \) is the steering angle. The overall vehicle turning radius \( r \) is:
$$ r = \sqrt{ \left( \frac{d_g}{\tan \delta} \right)^2 + \left( \frac{d_g}{2} \right)^2 } $$
Assuming pure rolling motion around an instantaneous center, the wheel velocities are proportional to their distances from this center. Thus, the velocities are:
$$ v_{fr} = \frac{r_{fr} v}{r}, \quad v_{br} = \frac{r_{br} v}{r}, \quad v_{fl} = \frac{r_{fl} v}{r}, \quad v_{bl} = \frac{r_{bl} v}{r} $$
where \( v \) is the vehicle speed. This geometric analysis highlights that vehicle speed and steering angle are critical inputs for the differential steering control in electric vehicles, affecting factors like centrifugal force and tire slip.
The differential steering control strategy aims to generate a differential assist torque by varying the drive torques on the left and right wheels. The assist torque \( T_s \) is calculated as:
$$ T_s = \frac{d_\beta \cos \beta \Delta T}{r_a} $$
where \( d_\beta \) is the kingpin offset, \( \beta \) is the kingpin inclination angle, \( r_a \) is the effective wheel radius, and \( \Delta T \) is the torque difference between the steering wheels. The target differential assist torque \( T_Z \) is defined based on the steering wheel torque \( T_f \):
$$ T_Z = \begin{cases}
0, & 0 \leq T_f < T_{f0} \\
(0.7 e^{\mu T_f} – 1.3) k(v), & T_{f0} \leq T_f < T_{fmax} \\
T_{zmax}, & T_f \geq 6
\end{cases} $$
Here, \( T_{f0} \) is the initial steering wheel torque for assist intervention, \( T_{fmax} \) is the torque at maximum assist, \( T_{zmax} \) is the maximum assist torque, \( \mu \) is the assist coefficient, and \( k(v) \) is a speed-dependent factor. To ensure balanced torque distribution, the differential torque is averaged between wheels, promoting efficient and stable steering in electric vehicles.
BP neural networks are employed to optimize the control parameters, leveraging their nonlinear mapping and adaptive learning capabilities. The network structure consists of an input layer with two neurons (for normalized q-axis current and steering angle), a hidden layer with five neurons, and an output layer with two neurons (for adjusted rear axle drive speeds). Input features are normalized to a [0,1] range using min-max scaling:
$$ x_1 = \frac{i_q – i_{qmin}}{i_{qmax} – i_{qmin}}, \quad x_2 = \frac{\delta – \delta_{min}}{\delta_{max} – \delta_{min}}, \quad x_3 = \frac{\beta – \beta_{min}}{\beta_{max} – \beta_{min}} $$
During forward propagation, the hidden layer input \( x_p \) is computed as:
$$ x_p = \sum x_q w_{pq} $$
where \( w_{pq} \) are the weights between the input and hidden layers. The hidden layer output \( y_p \) uses the sigmoid activation function:
$$ y_p = s(x_p) = \frac{1}{1 + e^{-x_p}} $$
The output layer output \( y_s \) is:
$$ y_s = \sum y_p w_{ps} $$
where \( w_{ps} \) are the weights between the hidden and output layers. The error performance index \( E \) is minimized during training:
$$ E = \frac{\sum_{s=1}^N (y_{0s} – y_s)^2}{2} $$
Backpropagation adjusts the weights using gradient descent. The weight updates for the output-hidden layer connection include a momentum factor to avoid local minima:
$$ \Delta w_{ps} = \gamma y_p (y_{0s} – y_s) $$
$$ \Delta w_{pq} = \gamma \sum_{s=1}^N (y_{0s} – y_s) \frac{\partial y_s}{\partial w_{pq}} $$
where \( \gamma \) is the learning rate. This optimization enhances the control precision for China EV applications, ensuring robust differential steering automatic control.
Experimental validation was conducted to assess the performance of the proposed differential steering control system for electric vehicles. The BP neural network was configured with a 2-5-2 structure, initialized with random weights in [-1,1], and trained on 5,000 data points from NGSIM and EPA datasets. The error threshold was set to \( 5 \times 10^{-3} \). The electric vehicle parameters used in simulations are summarized in Table 1.
| Parameter | Specification |
|---|---|
| Vehicle Length | 5480 mm |
| Vehicle Height | 2100 mm |
| Vehicle Width | 2400 mm |
| Wheel Track | 1730 mm |
| Maximum Load Mass | 3000 kg |
| Minimum Ground Clearance | 170 mm |
| Maximum Gradeability | 12% |
Tests included returnability performance, steady-state rotation, and steering effort evaluations, adhering to GB/T 12354 and GB/T 12549 standards. The returnability test measured stabilization time and yaw rate variance under lateral accelerations of 3 m/s², 4 m/s², and 5 m/s² on a 15 m radius circular path. Stabilization time \( t \) and yaw rate total variance \( H_z \) were averaged over 10 trials:
$$ t = \frac{\sum_{i=1}^{10} t_i}{10}, \quad H_z = \frac{\sum_{i=1}^{10} H_{zi}}{10} $$
Comparative analysis with fuzzy PID control (Method 2) and active disturbance rejection servo control (Method 3) demonstrated the superiority of the proposed BP-based method (Method 1). For instance, in returnability tests, Method 1 achieved shorter stabilization times and lower yaw rate variances across all lateral accelerations, indicating improved control stability for electric vehicles.
Steady-state rotation tests evaluated the turning radius ratio and side-slip angle difference. The turning radius ratio is the ratio of the instantaneous turning radius \( R_i \) to the initial radius \( R_0 \), and the side-slip angle difference \( \delta_f – \delta_b \) is calculated as:
$$ \delta_f – \delta_b = \frac{360}{2\pi} d_g \left( \frac{1}{R_0} – \frac{1}{R_i} \right) $$
Results showed that Method 1 maintained a turning radius ratio close to 1.0 with minimal fluctuations, reflecting reduced understeer characteristics. The side-slip angle differences were smaller for Method 1, confirming better handling in China EV scenarios.
Steering effort tests on a standard double-lane change course measured average steering torque and force. As shown in Table 2, Method 1 yielded lower values, enhancing driver comfort in electric vehicles.
| Method | Average Friction Torque (N·m) | Average Friction Force (N) |
|---|---|---|
| Method 1 | 75 | 60 |
| Method 2 | 79 | 63 |
| Method 3 | 78 | 62 |
In conclusion, the integration of BP neural networks with PMSM dynamics and steering geometry enables effective differential steering automatic control for electric vehicles. Experimental results validate improvements in returnability, steady-state rotation, and steering effort, highlighting the system’s potential for enhancing China EV performance. Future work will focus on intelligent control integration and safety enhancements, such as redundancy designs, to further boost robustness and adaptability in electric vehicle applications.