In the rapidly evolving landscape of automotive technology, the advent of distributed drive battery electric vehicles represents a significant leap forward. Unlike conventional centralized powertrains, these vehicles utilize in-wheel or hub motors to independently control each wheel’s torque. This architecture offers unparalleled advantages in terms of precise torque vectoring, rapid response, and enhanced maneuverability. As a researcher deeply immersed in vehicle dynamics and control systems, I have focused on developing advanced stability control strategies for such battery electric vehicles. The core challenge lies in harnessing the independent torque control capability to improve vehicle stability and safety, particularly during extreme maneuvers, while also considering energy efficiency. This article presents our comprehensive work on a hierarchical direct yaw moment control (DYC) strategy based on model predictive control (MPC) for distributed drive battery electric vehicles. We aim to detail the mathematical modeling, controller design, and simulation validation, providing a robust framework for stability enhancement in modern electric mobility.
The motivation for this work stems from the inherent limitations of traditional stability control systems, such as the Electronic Stability Program (ESP), which often rely on differential braking. While effective, these methods can lead to significant power loss and speed reduction, which is particularly undesirable for battery electric vehicles where energy conservation is crucial. In contrast, distributed drive battery electric vehicles enable direct yaw moment control through coordinated drive and brake torque at each wheel, allowing for active stabilization without necessarily sacrificing longitudinal performance. Our goal is to design a control system that not only maintains vehicle stability but does so proactively by predicting and adjusting to dynamic states in real-time. This approach aligns with the broader vision of intelligent vehicle systems that prioritize safety, efficiency, and performance. Throughout this article, we will emphasize the application to battery electric vehicles, highlighting how their unique drivetrain characteristics can be leveraged for superior control.
To lay the foundation, we begin by establishing the vehicle dynamics models essential for control design. For stability control, the lateral and yaw motions are paramount. We adopt a two-degree-of-freedom (2-DOF) vehicle model that captures the essential dynamics while maintaining simplicity for real-time implementation. The model considers the vehicle as a single rigid body with states including sideslip angle and yaw rate. The equations of motion are derived from force and moment balances. Let \(m\) denote the vehicle mass, \(v_x\) and \(v_y\) the longitudinal and lateral velocities, \(\gamma\) the yaw rate, \(\delta\) the front steering angle, \(l_f\) and \(l_r\) the distances from the center of gravity to the front and rear axles, \(I_z\) the yaw moment of inertia, and \(\Delta M_z\) the additional yaw moment generated by the control system. The nonlinear equations are:
$$ m(\dot{v}_y + v_x \gamma) = F_{xf} \sin \delta + F_{yf} \cos \delta + F_{yr} $$
$$ I_z \dot{\gamma} = l_f (F_{xf} \sin \delta + F_{yf} \cos \delta) – l_r F_{yr} + \Delta M_z $$
Here, \(F_{yf}\) and \(F_{yr}\) represent the total lateral tire forces on the front and rear axles, respectively. For a battery electric vehicle with independent wheel control, these forces can be influenced by both steering and torque inputs, making the model suitable for DYC design.
The tire forces are highly nonlinear and depend on various factors such as vertical load, slip angle, and road conditions. To accurately represent this behavior, we employ the Magic Formula tire model, which is widely used in vehicle dynamics simulations. The lateral force \(F_{yi}\) for axle \(i\) (where \(i = f, r\)) is given by:
$$ F_{yi} = D \sin \left( C \arctan \left( B x – E (B x – \arctan(B x)) \right) \right) + \Delta S_v $$
$$ x = \alpha_i + \Delta S_h $$
In these equations, \(B\) is the stiffness factor, \(C\) the shape factor, \(D\) the peak factor, \(E\) the curvature factor, \(\Delta S_h\) the horizontal shift, \(\Delta S_v\) the vertical shift, and \(\alpha_i\) the tire slip angle. The parameters \(B, C, D, E, \Delta S_h, \Delta S_v\) are functions of the vertical load \(F_z\) and camber angle, but for simplicity, we focus on load dependency as it significantly affects force generation in a battery electric vehicle during dynamic maneuvers. The slip angles for the front and rear axles are calculated as:
$$ \alpha_f = \frac{l_f \gamma}{v_x} + \beta – \delta $$
$$ \alpha_r = \beta – \frac{l_r \gamma}{v_x} $$
where \(\beta\) is the vehicle sideslip angle. The vertical loads on each wheel vary due to load transfer during acceleration and braking, which is critical for a battery electric vehicle with high torque capabilities. The dynamic vertical loads \(F_{zij}\) for each wheel (where \(ij = fl, fr, rl, rr\) denote front-left, front-right, rear-left, rear-right) are expressed as:
$$ F_{zfl} = \frac{m g l_r}{2(l_f + l_r)} – \frac{m a_x h}{2(l_f + l_r)} – \frac{m a_y h l_r}{(l_f + l_r) B} $$
$$ F_{zfr} = \frac{m g l_r}{2(l_f + l_r)} – \frac{m a_x h}{2(l_f + l_r)} + \frac{m a_y h l_r}{(l_f + l_r) B} $$
$$ F_{zrl} = \frac{m g l_f}{2(l_f + l_r)} + \frac{m a_x h}{2(l_f + l_r)} – \frac{m a_y h l_f}{(l_f + l_r) B} $$
$$ F_{zrr} = \frac{m g l_f}{2(l_f + l_r)} + \frac{m a_x h}{2(l_f + l_r)} + \frac{m a_y h l_f}{(l_f + l_r) B} $$
Here, \(g\) is gravitational acceleration, \(h\) the center of gravity height, \(a_x\) and \(a_y\) the longitudinal and lateral accelerations, and \(B\) the track width. This detailed tire model ensures that our control strategy accounts for the nonlinearities and load variations inherent in battery electric vehicle operations.
With the dynamics model established, we proceed to design the hierarchical direct yaw moment control strategy. The overall architecture consists of an upper-level MPC controller that computes the required additional yaw moment to stabilize the vehicle, and a lower-level torque optimization allocator that distributes torque to the four wheels while minimizing tire adhesion utilization. This layered approach allows for modular design and effective handling of multiple constraints. The control system takes inputs such as vehicle speed, steering angle, and motion states, and outputs individual wheel torques for the battery electric vehicle’s in-wheel motors.
The upper-level MPC controller is designed based on the 2-DOF vehicle model. We formulate the control problem to track desired yaw rate and sideslip angle, which are derived from a reference model that represents ideal vehicle behavior. The state-space representation is discretized for digital implementation. Let the state vector be \(\mathbf{x} = [\beta, \gamma]^T\), the control input be \(u = \Delta M_z\), and the disturbance be the front steering angle \(\delta\). The continuous-time state equation is:
$$ \dot{\mathbf{x}} = A \mathbf{x} + B u + E \delta $$
where matrices \(A\), \(B\), and \(E\) are defined using linearized tire cornering stiffnesses \(C_f\) and \(C_r\):
$$ A = \begin{bmatrix} \frac{C_f + C_r}{m v_x} & \frac{l_f C_f – l_r C_r}{m v_x^2} – 1 \\ \frac{l_f C_f – l_r C_r}{I_z} & \frac{l_f^2 C_f + l_r^2 C_r}{I_z v_x} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ \frac{1}{I_z} \end{bmatrix}, \quad E = \begin{bmatrix} -\frac{C_f}{m v_x} \\ -\frac{l_f C_f}{I_z} \end{bmatrix} $$
Using Euler discretization with sampling time \(T_s = 0.02 \, \text{s}\), we obtain the discrete-time model:
$$ \mathbf{x}(k+1) = A_k \mathbf{x}(k) + B_k u(k) + E_k \delta(k) $$
where \(A_k = I + T_s A\), \(B_k = T_s B\), \(E_k = T_s E\), and \(I\) is the identity matrix. The MPC controller minimizes a cost function over a prediction horizon \(N_c\):
$$ \min J = \sum_{i=0}^{N_c-1} \left( \| \beta(k+i) – \beta_r(k+i) \|_{Q_1}^2 + \| \gamma(k+i) – \gamma_r(k+i) \|_{Q_2}^2 \right) + \sum_{i=0}^{N_c-1} \| u(k+i) \|_{R}^2 $$
Here, \(\beta_r\) and \(\gamma_r\) are the reference values, and \(Q_1\), \(Q_2\), and \(R\) are weighting matrices that balance tracking performance and control effort. Constraints are imposed on control input, its rate, and state variables to ensure physical feasibility and stability:
$$ u_{\min} \leq u(k) \leq u_{\max} $$
$$ \Delta u_{\min} \leq \Delta u(k) \leq \Delta u_{\max} $$
$$ \beta_{\min} \leq \beta(k) \leq \beta_{\max} $$
$$ \gamma_{\min} \leq \gamma(k) \leq \gamma_{\max} $$
By solving this quadratic programming problem in real-time, the MPC controller generates the optimal additional yaw moment \(\Delta M_z\) to correct deviations from desired vehicle behavior. This proactive control is especially beneficial for battery electric vehicles, as it can prevent instability before it occurs, leveraging the fast torque response of electric motors.
The lower-level torque optimization allocator takes the computed \(\Delta M_z\) and the desired longitudinal force \(F_{x,\text{des}}\) (from a separate speed controller) as inputs, and determines the individual wheel torques \(T_{xij}\). The objective is to minimize the total tire adhesion utilization, which ensures that tires operate within their linear region, maximizing stability and efficiency. For a battery electric vehicle, this is crucial to prevent tire saturation and maintain control authority. The adhesion utilization \(\lambda_{ij}\) for each wheel is defined as:
$$ \lambda_{ij} = \frac{\sqrt{F_{xij}^2 + F_{yij}^2}}{\mu F_{zij}} $$
where \(\mu\) is the road adhesion coefficient, and \(F_{xij}\) and \(F_{yij}\) are the longitudinal and lateral tire forces. Assuming small slip angles and using linear approximations, the lateral forces can be related to slip angles, but for optimization, we focus on distributing torques to minimize the sum of squares of utilization ratios. The optimization problem is formulated as:
$$ \min J_\lambda = \min \sum_{ij = fl, fr, rl, rr} c_{ij} \frac{T_{xij}^2}{(\mu F_{zij} r)^2} $$
where \(r\) is the wheel radius, and \(c_{ij}\) are weighting coefficients that can prioritize certain wheels. The constraints include the overall longitudinal force demand and yaw moment generation:
$$ (T_{xfl} + T_{xfr}) \cos \delta + T_{xrl} + T_{xrr} = \frac{T_{x,\text{des}}}{r} $$
$$ \frac{B}{2r} (T_{xfr} – T_{xfl}) \cos \delta + \frac{B}{2r} (T_{xrr} – T_{xrl}) + \frac{l_f (T_{xfl} + T_{xfr}) \sin \delta}{r} = \Delta M_z $$
Additionally, each wheel torque is limited by motor capability and road adhesion:
$$ |T_{xij}| \leq \min( \mu F_{zij} r, T_{\max} ) $$
where \(T_{\max}\) is the maximum motor torque. This constrained optimization is solved using sequential quadratic programming (SQP) methods, embedded in our simulation environment. For a battery electric vehicle, this allocation strategy ensures that torque is distributed efficiently, reducing energy consumption while maintaining stability.
To further enhance safety, we incorporate a slip ratio controller that prevents excessive wheel slip during aggressive maneuvers. The slip ratio \(s_{ij}\) for each wheel is calculated as:
$$ s_{ij} = \frac{\omega_{ij} r – u_{ij}}{\max(\omega_{ij} r, u_{ij})} $$
where \(\omega_{ij}\) is the wheel angular velocity and \(u_{ij}\) is the wheel center velocity. A PID controller adjusts the torque if the slip ratio exceeds a optimal threshold \(s_{\text{opt}} = 0.15\), which is typical for various road surfaces. The correction torque \(\Delta T_{ij}\) is:
$$ \Delta T_{ij} = K_P (s_{\text{opt}} – s_{ij}) + K_I \int (s_{\text{opt}} – s_{ij}) \, dt + K_D \frac{d(s_{\text{opt}} – s_{ij})}{dt} $$
The final wheel torque command is \(T_{ij}^* = T_{xij} + \Delta T_{ij}\). This layered control strategy, from high-level MPC to low-level allocation and slip control, provides a comprehensive solution for direct yaw moment control in battery electric vehicles.
To validate our control strategy, we developed a co-simulation platform using CarSim and MATLAB/Simulink. CarSim provides high-fidelity vehicle dynamics, while Simulink implements our control algorithms. The test vehicle is a battery electric vehicle with four in-wheel motors. Key parameters are summarized in the table below:
| Vehicle Parameter | Symbol | Value |
|---|---|---|
| Vehicle Mass | \(m\) | 1120 kg |
| Distance from CG to Front Axle | \(l_f\) | 1.16 m |
| Distance from CG to Rear Axle | \(l_r\) | 1.16 m |
| CG Height | \(h\) | 0.375 m |
| Yaw Moment of Inertia | \(I_z\) | 1020 kg·m² |
| Track Width | \(B\) | 1.46 m |
| Wheel Radius | \(r\) | 0.3 m (assumed) |
We conducted simulations under two critical scenarios: double lane change and serpentine maneuvers. These test the vehicle’s stability during transient and continuous steering inputs, which are common in emergency avoidance situations for battery electric vehicles.
In the double lane change test, the initial speed is set to 120 km/h on a road with adhesion coefficient \(\mu = 0.6\). Without stability control, the vehicle exhibits significant overshoot in yaw rate and sideslip angle, potentially leading to loss of control. With our MPC-based DYC activated, the vehicle closely tracks the reference values. The results are quantified in the table below, showing errors compared to the reference model:
| Parameter | Error Metric | Without Control | With MPC Control |
|---|---|---|---|
| Sideslip Angle \(\beta\) | Maximum Error | 0.939° | 0.135° |
| Average Error | 0.101° | 0.015° | |
| Root Mean Square Error | 0.320° | 0.089° | |
| Yaw Rate \(\gamma\) | Maximum Error | 5.106°/s | 2.423°/s |
| Average Error | 0.565°/s | 0.201°/s | |
| Root Mean Square Error | 1.532°/s | 0.873°/s |
The reduction in errors demonstrates the effectiveness of our controller in maintaining stability for the battery electric vehicle under high-speed evasion maneuvers.
For the serpentine test, which involves continuous steering around cones, the speed is 70 km/h on a road with \(\mu = 0.55\). Without control, the vehicle quickly loses stability and deviates from the path after the fifth cone. With MPC control, the vehicle successfully follows the trajectory. The error analysis is presented below:
| Parameter | Error Metric | Without Control | With MPC Control |
|---|---|---|---|
| Sideslip Angle \(\beta\) | Maximum Error | 12.598° | 0.356° |
| Average Error | 4.339° | 0.025° | |
| Root Mean Square Error | 24.783° | 0.112° | |
| Yaw Rate \(\gamma\) | Maximum Error | 30.823°/s | 2.025°/s |
| Average Error | 10.526°/s | 0.125°/s | |
| Root Mean Square Error | 13.952°/s | 0.618°/s |
These results confirm that our control strategy significantly enhances stability and path-following capability for battery electric vehicles in challenging driving conditions. The hierarchical approach allows for real-time computation while respecting physical constraints, making it suitable for implementation in actual battery electric vehicle systems.
In conclusion, we have developed and validated a comprehensive direct yaw moment control strategy for distributed drive battery electric vehicles. By integrating model predictive control at the upper level and torque optimization at the lower level, we achieve proactive stability control that leverages the fast and precise torque response of in-wheel motors. The simulations under double lane change and serpentine maneuvers demonstrate substantial improvements in tracking desired yaw rate and sideslip angle, reducing errors and preventing loss of control. This work contributes to the advancement of intelligent control systems for battery electric vehicles, offering a balance between safety, performance, and energy efficiency. Future research will focus on hardware-in-the-loop testing and real-world validation, as well as extending the strategy to include adaptive parameters for varying road conditions and vehicle configurations. The potential for such systems in next-generation battery electric vehicles is immense, paving the way for safer and more efficient electric mobility.