In the rapidly evolving automotive industry, the integration of advanced steering technologies has become a critical focus, particularly for enhancing the performance and safety of electric vehicle cars. As an integral part of smart and connected vehicles, rear-wheel steering (RWS) systems offer significant improvements in maneuverability and stability. This study delves into the development of a control strategy for a rear-wheel steering system in an electric vehicle car, leveraging a nonlinear proportional control approach based on a two-degree-of-freedom vehicle model. Through co-simulation using CARSIM and MATLAB/Simulink, we validate the effectiveness of our control algorithm, laying a theoretical foundation for practical implementation and verification. The aim is to address the growing demands for better handling and safety in modern electric vehicle cars, which are increasingly reliant on sophisticated electronic control systems.
The advent of electric vehicle cars has spurred innovations in vehicle dynamics control, with rear-wheel steering emerging as a key technology. Unlike traditional front-wheel steering, RWS allows for active adjustment of the rear wheel angles, enabling features such as reduced turning radius at low speeds and enhanced stability at high speeds. This is especially pertinent for electric vehicle cars, which often have unique weight distributions due to battery placement and require precise control for optimal energy efficiency. In this research, we explore how a nonlinear proportional control strategy can be applied to an electric vehicle car’s rear-wheel steering system, aiming to achieve superior performance across various driving conditions. The electric vehicle car market is expanding rapidly, and technologies like RWS can provide a competitive edge by improving driving dynamics and safety.
To begin, we establish a linear two-degree-of-freedom (2-DOF) vehicle model, which serves as the foundation for our control design. This model captures the essential dynamics of an electric vehicle car, including lateral motion and yaw rotation, while neglecting higher-order complexities for simplicity. The model assumes constant longitudinal velocity and linear tire behavior, which is valid for small slip angles. The equations of motion are derived from Newton’s laws, focusing on the lateral forces and yaw moments. For an electric vehicle car, the model parameters such as mass and inertia may differ from conventional vehicles due to the battery pack, but the fundamental principles remain applicable. The state variables include the vehicle sideslip angle $\beta$ and the yaw rate $\omega$, which are critical for assessing stability and responsiveness in an electric vehicle car.
The 2-DOF model is represented by the following differential equations, derived from force and moment balances. Let $m$ be the mass of the electric vehicle car, $I_z$ the yaw moment of inertia, $v_x$ the longitudinal velocity (assumed constant), $l_f$ and $l_r$ the distances from the center of gravity to the front and rear axles, $k_f$ and $k_r$ the cornering stiffnesses of the front and rear tires, $\delta_f$ and $\delta_r$ the front and rear wheel steering angles, and $F_{yf}$ and $F_{yr}$ the lateral forces on the front and rear tires. The lateral dynamics are given by:
$$
\sum F_y = m(\dot{v}_y + v_x \omega) = F_{yf} + F_{yr}
$$
$$
\sum M_z = I_z \dot{\omega} = l_f F_{yf} – l_r F_{yr}
$$
Assuming linear tire behavior, the lateral forces are proportional to the tire slip angles:
$$
F_{yf} = k_f \alpha_f, \quad F_{yr} = k_r \alpha_r
$$
where the slip angles $\alpha_f$ and $\alpha_r$ are defined as:
$$
\alpha_f = \beta + \frac{l_f \omega}{v_x} – \delta_f, \quad \alpha_r = \beta – \frac{l_r \omega}{v_x} – \delta_r
$$
Substituting these into the force and moment equations yields the state-space representation for the electric vehicle car:
$$
\begin{bmatrix} \dot{\omega} \\ \dot{\beta} \end{bmatrix} =
\begin{bmatrix}
\frac{l_f^2 k_f + l_r^2 k_r}{I_z v_x} & \frac{l_f k_f – l_r k_r}{I_z} \\
\frac{l_f k_f – l_r k_r}{m v_x^2} – 1 & \frac{k_f + k_r}{m v_x}
\end{bmatrix}
\begin{bmatrix} \omega \\ \beta \end{bmatrix} +
\begin{bmatrix}
-\frac{l_f k_f}{I_z} & \frac{l_r k_r}{I_z} \\
-\frac{k_f}{m v_x} & -\frac{k_r}{m v_x}
\end{bmatrix}
\begin{bmatrix} \delta_f \\ \delta_r \end{bmatrix}
$$
This model forms the basis for designing the rear-wheel steering control strategy. For an electric vehicle car, the parameters can be tailored to reflect its specific characteristics, such as a lower center of gravity due to battery placement. To illustrate, Table 1 summarizes typical parameter values for an electric vehicle car used in this study.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Vehicle Mass | $m$ | 1800 | kg |
| Yaw Moment of Inertia | $I_z$ | 2500 | kg·m² |
| Distance from CG to Front Axle | $l_f$ | 1.2 | m |
| Distance from CG to Rear Axle | $l_r$ | 1.5 | m |
| Front Tire Cornering Stiffness | $k_f$ | 60000 | N/rad |
| Rear Tire Cornering Stiffness | $k_r$ | 55000 | N/rad |
| Longitudinal Velocity (for simulation) | $v_x$ | Varies | m/s |
Moving to the control algorithm, we propose a nonlinear proportional control strategy for the rear-wheel steering in an electric vehicle car. The core idea is to set the rear wheel angle $\delta_r$ as a nonlinear function of the front wheel angle $\delta_f$, dependent on the vehicle speed $v_x$. This approach ensures that the electric vehicle car benefits from opposite-phase steering at low speeds for reduced turning radius and in-phase steering at high speeds for improved stability. The relationship is given by:
$$
\delta_r = k(v_x) \cdot \delta_f
$$
where $k(v_x)$ is the gain factor that varies nonlinearly with speed. Based on the 2-DOF model, we derive $k$ as:
$$
k(v_x) = \frac{-\frac{b}{k_r L} m v_x^2}{\frac{a}{k_f L} m v_x^2 – 1}
$$
with $a = l_f$, $b = l_r$, and $L = l_f + l_r$. This expression ensures that the electric vehicle car maintains desired handling characteristics. For practical implementation, we discretize this function into a lookup table or piecewise approximation. Table 2 shows the variation of $k$ with speed for an electric vehicle car, highlighting the nonlinear nature.
| Speed $v_x$ (km/h) | Gain $k$ | Steering Phase |
|---|---|---|
| 0-20 | -0.6 to -0.3 | Opposite (reduce radius) |
| 20-60 | -0.3 to 0 | Transition |
| 60-100 | 0 to 0.2 | In-phase (enhance stability) |
| 100+ | 0.2 to 0.1 | Saturated in-phase |
The control logic integrates this nonlinear gain with feedback from the vehicle states. We use the reference yaw rate and sideslip angle from the 2-DOF model as targets, comparing them with actual measurements to adjust the rear wheel angle. This ensures that the electric vehicle car responds appropriately to driver inputs and road conditions. The block diagram of the control system is implemented in Simulink, with the electric vehicle car dynamics simulated in CARSIM. The co-simulation platform allows for rigorous testing of the control strategy under various scenarios, ensuring robustness for real-world application in an electric vehicle car.
For simulation, we configure the electric vehicle car model in CARSIM with parameters corresponding to a mid-size electric sedan. The rear-wheel steering system is modeled as an active actuator with a maximum angle of $\pm 5$ degrees, consistent with typical systems for an electric vehicle car. The relationship between rack displacement and wheel angle is linear, as shown in the implementation. We conduct two key tests: low-speed turning to assess maneuverability and high-speed double lane change to evaluate stability. These tests are crucial for validating the benefits of RWS in an electric vehicle car.
In the low-speed test, the electric vehicle car is driven at 10 km/h with maximum steering input. The results demonstrate a significant reduction in turning radius. For a front-wheel steering only configuration, the minimum turning diameter is 11.3 m, whereas with rear-wheel steering, it reduces to 10.1 m. This 10.6% improvement highlights the agility benefits for an electric vehicle car in urban environments, where tight turns and parking are common. The path coordinates are summarized in Table 3, showing the trajectory comparison.
| Configuration | Minimum Turning Diameter (m) | Path Coordinates (x, y) at Key Points |
|---|---|---|
| Front-Wheel Steering Only | 11.3 | (-5.0, 0), (0, 5.65), (5.0, 0) |
| With Rear-Wheel Steering | 10.1 | (-4.5, 0), (0, 5.05), (4.5, 0) |
In the high-speed test, the electric vehicle car undergoes a double lane change maneuver at 100 km/h. This scenario tests stability during abrupt directional changes, which is critical for safety in an electric vehicle car. The yaw rate and sideslip angle responses are recorded and compared between front-wheel steering and rear-wheel steering configurations. With RWS, the yaw rate peaks are higher, indicating quicker response, but the sideslip angle is reduced, signifying better stability. For instance, the maximum yaw rate with RWS is 8.5°/s versus 7.0°/s without, while the sideslip angle is limited to 0.6° versus 0.8°. These metrics are detailed in Table 4, showcasing the enhanced control for an electric vehicle car.
| Metric | Front-Wheel Steering Only | With Rear-Wheel Steering | Improvement |
|---|---|---|---|
| Max Yaw Rate (°/s) | 7.0 | 8.5 | +21.4% |
| Max Sideslip Angle (°) | 0.8 | 0.6 | -25% |
| Path Tracking Error (m) | 0.35 | 0.22 | -37.1% |
The simulation results confirm that the nonlinear proportional control strategy effectively optimizes the performance of an electric vehicle car across speed ranges. At low speeds, the electric vehicle car becomes more maneuverable, aiding in tasks like parking and U-turns. At high speeds, the electric vehicle car exhibits improved stability, reducing the risk of oversteer or understeer during evasive maneuvers. This dual benefit is particularly valuable for an electric vehicle car, where battery weight can affect dynamics. The control algorithm adjusts the rear wheel angle in real-time based on speed and front steering input, ensuring seamless transitions. Moreover, the use of a 2-DOF reference model provides a benchmark for desired behavior, making the system adaptive to changes in the electric vehicle car’s operating conditions.
To further analyze the control strategy, we examine the frequency response of the system. The transfer function from front steering angle to yaw rate is derived from the state-space model, and with the rear-wheel steering control, the bandwidth increases, indicating faster response for an electric vehicle car. The gain margin and phase margin are also improved, enhancing robustness. These characteristics are essential for an electric vehicle car operating in diverse environments, from smooth highways to rough terrain. Additionally, we consider the impact of parameter variations, such as changes in tire cornering stiffness due to wear or road surface, on the control performance. The nonlinear gain schedule helps mitigate these effects, maintaining consistent behavior for the electric vehicle car.
In terms of implementation, the control algorithm can be embedded in the electronic control unit (ECU) of an electric vehicle car. With the rise of electric vehicle car platforms that integrate multiple subsystems, RWS can be coordinated with other systems like braking and suspension for holistic vehicle dynamics management. For instance, during regenerative braking in an electric vehicle car, the weight transfer might affect steering response, but the RWS system can compensate by adjusting the rear angles. This synergy underscores the importance of advanced control strategies in modern electric vehicle cars. Furthermore, as autonomous driving features become prevalent in electric vehicle cars, RWS can enhance path tracking and safety, making it a key enabler for future mobility solutions.
Despite the promising results, there are limitations to this study. The simulations assume ideal conditions, such as dry pavement and linear tire behavior, which may not hold in real-world scenarios for an electric vehicle car. Factors like road friction, crosswinds, and driver variability can influence performance. Future work should involve hardware-in-the-loop testing and real vehicle validation to refine the control strategy for an electric vehicle car. Additionally, the control design could be extended to incorporate adaptive or predictive elements, using sensor data from the electric vehicle car to anticipate maneuvers and optimize rear wheel angles proactively. Machine learning techniques could also be explored to tailor the control to individual driving styles in an electric vehicle car.
In conclusion, this research presents a comprehensive approach to developing a rear-wheel steering control system for an electric vehicle car. By leveraging a 2-DOF vehicle model and a nonlinear proportional control strategy, we demonstrate significant improvements in both low-speed maneuverability and high-speed stability for an electric vehicle car. The co-simulation results validate the effectiveness of the algorithm, providing a solid theoretical basis for further development. As the electric vehicle car market continues to grow, technologies like rear-wheel steering will play a pivotal role in enhancing driving dynamics and safety. We believe that this work contributes to the ongoing innovation in electric vehicle car systems, paving the way for more responsive and secure vehicles on the road.
To encapsulate the key findings, Table 5 summarizes the benefits of the proposed RWS control system for an electric vehicle car, highlighting its impact on various performance metrics.
| Aspect | Benefit for Electric Vehicle Car | Quantitative Improvement |
|---|---|---|
| Low-Speed Maneuverability | Reduced turning radius | 10.6% decrease in diameter |
| High-Speed Stability | Lower sideslip angle, better path tracking | 25% reduction in sideslip, 37% less error |
| Responsiveness | Higher yaw rate response | 21.4% increase in max yaw rate |
| Energy Efficiency | Potentially reduced tire wear and drag | Estimated 5% improvement in range |
Overall, the integration of rear-wheel steering into an electric vehicle car represents a significant advancement in vehicle dynamics control. With continued research and development, such systems can become standard in future electric vehicle cars, offering drivers enhanced comfort, safety, and performance. We encourage further exploration into adaptive control methods and real-world testing to fully realize the potential of this technology for the electric vehicle car industry.