In recent years, the global shift towards sustainable transportation has accelerated the development of electric vehicles (EVs), with China EV markets leading in adoption and innovation. As a researcher in automotive dynamics, I have focused on enhancing the safety and performance of electric cars through advanced control systems. One critical aspect is acceleration slip regulation (ASR), which prevents wheel spin during acceleration on varying road surfaces, ensuring optimal traction and stability. For dual-motor electric cars, where independent motors drive the front and rear axles, ASR becomes even more vital due to the instantaneous torque delivery of electric motors. In this paper, we explore a control strategy that combines fuzzy logic for road identification and sliding mode control for torque adjustment, tailored for China EV applications. This approach aims to maximize longitudinal force while maintaining lateral stability, addressing common challenges in electric car operations.
The dynamics of a wheel in an electric car are fundamental to understanding ASR. When a driving torque $T_d$ is applied, the wheel rotates with angular velocity $\omega$, and the ground reaction force $F_d$ propels the vehicle forward at longitudinal velocity $v_x$. The wheel’s moment of inertia $I_\omega$ leads to the equation: $$\frac{d\omega}{dt} = \frac{T_d – F_d \cdot r}{I_\omega}$$ where $r$ is the wheel radius. The utilization adhesion coefficient $\mu$ is defined as $\mu = \frac{F_d}{F_z}$, with $F_z$ being the vertical load. For a dual-motor electric car, the vertical loads on each wheel must account for dynamic weight transfer, which is crucial for accurate road identification. The loads for the left front ($F_{z,F_l}$), right front ($F_{z,F_r}$), left rear ($F_{z,R_l}$), and right rear ($F_{z,R_r}$) wheels are given by: $$F_{z,F_l} = \frac{mgb}{L} – \frac{m a_x h}{L} – \frac{m a_y h b}{d_1 L}, \quad F_{z,F_r} = \frac{mgb}{L} – \frac{m a_x h}{L} + \frac{m a_y h b}{d_1 L}$$ $$F_{z,R_l} = \frac{mga}{L} + \frac{m a_x h}{L} – \frac{m a_y h a}{d_2 L}, \quad F_{z,R_r} = \frac{mga}{L} + \frac{m a_x h}{L} + \frac{m a_y h a}{d_2 L}$$ where $m$ is the vehicle mass, $g$ is gravity, $a$ and $b$ are distances from the center of gravity to the front and rear axles, $L$ is the wheelbase, $h$ is the height of the center of gravity, $d_1$ and $d_2$ are track widths, and $a_x$ and $a_y$ are longitudinal and lateral accelerations. These equations highlight the complexity of managing electric car dynamics, especially in China EV environments where road conditions can vary widely.
The slip rate $s$ of a wheel is a key parameter in ASR, defined as: $$s = \frac{r \omega – v_x}{r \omega} \times 100\%$$ This relationship determines the tire’s adhesion characteristics; as $s$ increases, the longitudinal adhesion coefficient $\mu$ initially rises to a peak $\mu_p$ at the optimal slip rate $s_b$, then declines. Beyond $s_b$, the lateral adhesion coefficient decreases, compromising steering stability. For electric cars, maintaining $s$ near $s_b$ ensures maximum traction and control. The adhesion coefficient-slip rate curve varies with road conditions, such as dry asphalt or ice, making real-time road identification essential. In China EV applications, where urban and rural roads mix, this adaptability is crucial for safety.
To achieve effective ASR, we employ a fuzzy control system for road identification. This method estimates the maximum utilization adhesion coefficient by monitoring the derivative $\frac{d\mu}{dt}$ during acceleration; when it transitions from positive to negative, the peak adhesion coefficient is identified. We use the Burckhardt model to fit standard road curves: $$\mu(s) = c_1 \cdot (1 – e^{-c_2 s}) – c_3 s$$ where $c_1$, $c_2$, and $c_3$ are road-specific parameters. The optimal slip rate $s_b$ and peak adhesion coefficient $\mu_p$ are derived as: $$s_b = \frac{1}{c_2} \ln\left(\frac{c_1 c_2}{c_3}\right), \quad \mu_p = c_1 \left(1 – \frac{c_3}{c_1 c_2} \left(1 + \ln\left(\frac{c_1 c_2}{c_3}\right)\right)\right)$$ We define six standard roads with their parameters, optimal slip rates, and peak adhesion coefficients in the table below, which aids in fuzzy inference for China EV scenarios.
| Road Surface | $c_1$ | $c_2$ | $c_3$ | $s_b$ | $\mu_p$ |
|---|---|---|---|---|---|
| Dry Asphalt | 1.28 | 23.99 | 0.52 | 0.17 | 1.17 |
| Dry Cement | 1.19 | 25.16 | 0.537 | 0.16 | 1.09 |
| Wet Asphalt | 0.85 | 33.82 | 0.347 | 0.13 | 0.801 |
| Cobblestone | 0.40 | 33.70 | 0.12 | 0.14 | 0.34 |
| Snow | 0.19 | 94.12 | 0.064 | 0.06 | 0.19 |
| Ice | 0.05 | 306.3 | 0.001 | 0.03 | 0.05 |
The fuzzy controller processes inputs such as the identified maximum adhesion coefficient and actual slip rate, using membership functions and rules to compute similarity weights for each road type. For instance, if the adhesion coefficient is low and the slip rate is small, the system might identify the road as ice with high similarity. The output road parameters are weighted averages: $$\mu_p = \frac{\sum_{i=1}^6 \mu_{p,i} x_i}{\sum_{i=1}^6 x_i}, \quad s_b = \frac{\sum_{i=1}^6 s_{b,i} x_i}{\sum_{i=1}^6 x_i}$$ where $x_i$ are the similarity coefficients. This method ensures robust identification for electric cars, even in dynamic China EV driving conditions.
For ASR control, we implement a sliding mode approach to adjust motor torque. The system intervenes when the wheel slip rate exceeds the identified $s_b$, replacing the driver’s torque request with a controlled value. The sliding surface is defined as $e = s – s_b$, and we use an exponential reaching law with a saturation function to minimize chattering: $$\frac{de}{dt} = -\epsilon \cdot \text{sat}(e) – k e$$ where $\epsilon$ and $k$ are gains, and $\text{sat}(e)$ is: $$\text{sat}(e) = \begin{cases} 1 & \text{if } e > \Delta \\ \frac{e}{\Delta} & \text{if } |e| \leq \Delta \\ -1 & \text{if } e < -\Delta \end{cases}$$ The controlled torque $T_{\text{SMC}}$ is derived as: $$T_{\text{SMC}} = \frac{I_\omega r}{v_x} \left( \dot{s}_b + \frac{v_x}{r} \right) + r F_d – \frac{I_\omega r^2}{v_x} \left( \epsilon \cdot \text{sat}(e) + k e \right)$$ This ensures that the slip rate quickly converges to $s_b$, enhancing the electric car’s performance. The control strategy is summarized in a flowchart, illustrating how torque decisions are made based on slip rate comparisons.
We validated this ASR strategy through co-simulation using Simulink and Carsim, focusing on low-speed scenarios common in urban China EV operations. For straight-line acceleration on low-adhesion roads (e.g., $\mu = 0.2$), with full throttle application, the system accurately identified the optimal slip rate with less than 7% error. The wheels’ slip rates were controlled near $s_b$, and the electric car achieved higher speeds and longer distances compared to uncontrolled cases—specifically, a 9% increase in speed and 7.4% in distance over the same time. This demonstrates the effectiveness of ASR in boosting the动力 of electric cars. Additionally, in scenarios with crosswinds, the controlled vehicle maintained minimal lateral displacement (0.05 m) and steering correction (8°), whereas the uncontrolled vehicle nearly lost stability, underscoring the importance of ASR for safety in China EV applications.
On split-mu roads, where left and right sides have different adhesion coefficients, the ASR system adopted a low-select principle, targeting the optimal slip rate of the low-adhesion side. This resulted in reduced motor torque on the high-adhesion side, minimizing lateral deviation. The maximum lateral displacement and steering wheel angle were 40% and 15% of those in uncontrolled cases, respectively. During lane-change maneuvers on low-adhesion surfaces, the controlled electric car closely followed the desired path with a maximum deviation of 0.29 m and steering angle of 112°, while the uncontrolled vehicle experienced tailspin. These results highlight the strategy’s ability to maintain lateral stability in electric cars, a critical factor for China EV safety standards.
In conclusion, the integration of fuzzy road identification and sliding mode control provides a robust ASR solution for dual-motor electric cars. This approach ensures optimal slip rate tracking, enhancing both longitudinal performance and lateral stability across diverse road conditions. For the China EV market, where reliability and safety are paramount, such systems can significantly improve vehicle dynamics. Future work could explore adaptive gains for varying speeds and integration with other stability controls, further advancing electric car technologies.