In recent years, the rapid advancement of electric vehicle (EV) technology has positioned China as a global leader in the automotive industry, particularly in the development of distributed drive electric vehicles (DDEVs). These vehicles utilize in-wheel motors to independently control each wheel, offering enhanced maneuverability and energy efficiency. However, the integration of in-wheel motors introduces significant challenges, such as increased unsprung mass and electromechanical coupling effects, which adversely affect ride comfort and braking safety. This paper addresses these issues by proposing a cooperative control strategy that integrates active suspension and composite braking systems for DDEVs. The primary objective is to optimize multiple performance metrics, including vehicle smoothness, braking safety, and energy recovery efficiency, while mitigating the negative impacts of motor-induced vibrations. Through comprehensive modeling and simulation, we demonstrate the effectiveness of our approach in improving overall vehicle dynamics.
The foundation of our research lies in the development of a high-fidelity vehicle model that accounts for the electromechanical coupling effects in distributed hub-motor-driven electric vehicles. In-wheel motors, particularly permanent magnet synchronous motors (PMSMs), are susceptible to rotor eccentricity due to road excitations, leading to unbalanced electromagnetic forces. These forces directly influence the vertical dynamics of the vehicle, exacerbating vibrations and reducing tire-road contact performance. To capture these phenomena, we establish a detailed PMSM model that includes the effects of stator-rotor eccentricity. The electrical equations governing the motor are derived as follows:
$$ \frac{dI_A}{dt} = \frac{1}{L_s} \left( v_{AB} – v_{BC} \right) – R_s I_A + \frac{p \hat{\lambda}}{3} \left( 2e_A – e_B – e_C \right) \omega_r $$
$$ \frac{dI_B}{dt} = \frac{1}{L_s} \left( -v_{AB} + 2v_{BC} \right) – R_s I_B + \frac{p \hat{\lambda}}{3} \left( -e_A + 2e_B – e_C \right) \omega_r $$
$$ \frac{dI_C}{dt} = -\frac{dI_A}{dt} – \frac{dI_B}{dt} $$
where \( I_A, I_B, I_C \) are the phase currents, \( v_{AB}, v_{BC} \) are the phase voltages, \( R_s \) and \( L_s \) are the stator resistance and inductance, \( p \) is the number of pole pairs, \( \hat{\lambda} \) is the flux linkage amplitude, \( \omega_r \) is the rotor angular velocity, and \( e_A, e_B, e_C \) are the back EMFs. The electromagnetic torque \( T_e \) is calculated as:
$$ T_e = p \hat{\lambda} \left( e_A I_A + e_B I_B + e_C I_C \right) $$
When rotor eccentricity occurs, the air gap length varies, leading to unbalanced radial forces. The vertical component of these forces, \( F_g \), is expressed as:
$$ F_g = \frac{\hat{L} r}{\mu_0} \int_0^{2\pi} \left[ \left( B_{re}^2 – B_{te}^2 \right) \sin \theta + 2 B_{re} B_{te} \cos \theta \right] d\theta $$
where \( \hat{L} \) is the effective axial length, \( r \) is the radius, \( \mu_0 \) is the permeability of free space, \( B_{re} \) and \( B_{te} \) are the radial and tangential flux densities under eccentric conditions, and \( \theta \) is the angular position.
To model the vehicle’s vertical dynamics, we develop an active suspension system with eleven degrees of freedom, incorporating the effects of unbalanced electromagnetic forces. The equations of motion for the sprung mass, unsprung masses, and wheel assemblies are given by:
$$ m_s \ddot{x}_s = -\sum_{i=fl,fr,rl,rr} F_{si} $$
$$ I_y \ddot{\theta} = a \left( F_{sfl} + F_{sfr} \right) – b \left( F_{srl} + F_{srr} \right) + T_z $$
$$ I_x \ddot{\phi} = -d_l \left( F_{sfl} – F_{srl} \right) + d_r \left( F_{sfr} – F_{srr} \right) $$
$$ m_{usi} \ddot{x}_{usi} = F_{si} – k_{si} (x_{usi} – x_{ugi}) – F_{gi} $$
$$ m_{ugi} \ddot{x}_{ugi} = k_{mi} (x_{usi} – x_{ugi}) – k_{ti} (x_{ugi} – x_{gi}) $$
where \( m_s \) is the sprung mass, \( x_s \) is the vertical displacement of the sprung mass, \( \theta \) and \( \phi \) are the pitch and roll angles, \( I_x \) and \( I_y \) are the moments of inertia, \( F_{si} \) is the suspension force, \( k_{si} \) and \( k_{ti} \) are the suspension and tire stiffnesses, \( k_{mi} \) is the hub bearing stiffness, and \( F_{gi} \) is the vertical electromagnetic force. The suspension force \( F_{si} \) includes an active control component \( u_i \):
$$ F_{si} = k_{si} (x_s – x_{usi}) + C_{si} (\dot{x}_s – \dot{x}_{usi}) – u_i $$
For the longitudinal dynamics during braking, we establish a composite braking model that integrates hydraulic and regenerative braking systems. The vehicle longitudinal motion and wheel dynamics are described by:
$$ m a_b = -\sum F_{xi} $$
$$ J \dot{\omega}_i = F_{xi} R – T_{bfi} – T_{bmi} $$
where \( m \) is the total vehicle mass, \( a_b \) is the braking deceleration, \( F_{xi} \) is the longitudinal tire force, \( J \) is the wheel moment of inertia, \( \omega_i \) is the wheel angular velocity, \( R \) is the tire radius, \( T_{bfi} \) is the hydraulic braking torque, and \( T_{bmi} \) is the motor braking torque. The vertical tire load \( F_{zi} \) is influenced by suspension dynamics and is approximated as:
$$ F_{zfl} \approx \frac{m g b}{2L} + \frac{m g h_g}{2L} z – \frac{I_y \ddot{\theta}}{2L} + m_{usfl} \ddot{x}_{usfl} + m_{ugfl} \ddot{x}_{ugfl} $$
$$ F_{zfr} \approx \frac{m g b}{2L} + \frac{m g h_g}{2L} z – \frac{I_y \ddot{\theta}}{2L} + m_{usfr} \ddot{x}_{usfr} + m_{ugfr} \ddot{x}_{ugfr} $$
$$ F_{zrl} \approx \frac{m g a}{2L} – \frac{m g h_g}{2L} z + \frac{I_y \ddot{\theta}}{2L} + m_{usrl} \ddot{x}_{usrl} + m_{ugrl} \ddot{x}{ugrl} $$
$$ F_{zrr} \approx \frac{m g a}{2L} – \frac{m g h_g}{2L} z + \frac{I_y \ddot{\theta}}{2L} + m_{usrr} \ddot{x}_{usrr} + m_{ugrr} \ddot{x}_{ugrr} $$
where \( a \) and \( b \) are the distances from the center of gravity to the front and rear axles, \( L \) is the wheelbase, \( h_g \) is the center of gravity height, and \( z \) is the braking intensity.
The tire-road interaction is modeled using the Magic Formula, which expresses the longitudinal force \( F_x \) as a function of the longitudinal slip ratio \( \lambda \):
$$ F_x = D \sin \left[ C \arctan \left( B \lambda – E \left( B \lambda – \arctan(B \lambda) \right) \right) \right] $$
where \( B \), \( C \), \( D \), and \( E \) are parameters that depend on the tire vertical load and road conditions. The values of these parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| b1 | 0.004 |
| b2 | -0.598 |
| b3 | 49.6 |
| b4 | 226 |
| b5 | 0.069 |
| b6 | -0.006 |
| b7 | 0.056 |
| b8 | 0.486 |
To address the coupling between vertical and longitudinal dynamics, we design a cooperative control framework that integrates active suspension and composite braking systems. The control structure is hierarchical, consisting of a coordination layer and a subsystem decision layer. The coordination layer classifies braking scenarios based on slip ratio and braking intensity into non-emergency, low-intensity emergency, and high-intensity emergency conditions. The subsystem layer comprises an active suspension controller and a composite braking controller, which generate control signals for suspension actuation and brake torque distribution.
The active suspension controller is based on mixed H2/H∞ feedback control, which ensures robustness against road disturbances and motor-induced forces. The state-space representation of the suspension system is:
$$ \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B}_w \mathbf{w} + \mathbf{B}_u \mathbf{u} $$
$$ \mathbf{z}_\infty = \mathbf{C}_\infty \mathbf{x} + \mathbf{D}_{\infty w} \mathbf{w} + \mathbf{D}_{\infty u} \mathbf{u} $$
$$ \mathbf{z}_2 = \mathbf{C}_2 \mathbf{x} + \mathbf{D}_{2w} \mathbf{w} + \mathbf{D}_{2u} \mathbf{u} $$
where \( \mathbf{x} \) is the state vector, \( \mathbf{w} \) is the disturbance input (road profile and electromagnetic forces), \( \mathbf{u} \) is the control input (suspension active forces), \( \mathbf{z}_\infty \) and \( \mathbf{z}_2 \) are the controlled outputs for H∞ and H2 performance, respectively. The state feedback control law is \( \mathbf{u} = \mathbf{K} \mathbf{x} \), where \( \mathbf{K} \) is obtained by solving the linear matrix inequalities (LMIs):
$$ \begin{bmatrix} \mathbf{A} \mathbf{X} + \mathbf{B}_u \mathbf{W} + (\mathbf{A} \mathbf{X} + \mathbf{B}_u \mathbf{W})^T & \mathbf{B}_w & (\mathbf{C}_\infty \mathbf{X} + \mathbf{D}_{\infty u} \mathbf{W})^T \\ \mathbf{B}_w^T & -\gamma \mathbf{I} & \mathbf{D}_{\infty w}^T \\ \mathbf{C}_\infty \mathbf{X} + \mathbf{D}_{\infty u} \mathbf{W} & \mathbf{D}_{\infty w} & -\gamma \mathbf{I} \end{bmatrix} < 0 $$
$$ \begin{bmatrix} \mathbf{A} \mathbf{X} + \mathbf{B}_u \mathbf{W} + (\mathbf{A} \mathbf{X} + \mathbf{B}_u \mathbf{W})^T & \mathbf{B}_w \\ \mathbf{B}_w^T & -\mathbf{I} \end{bmatrix} < 0 $$
$$ \text{trace}(\mathbf{Z}) < \gamma_2^2 $$
with \( \mathbf{Z} > (\mathbf{C}_2 \mathbf{X} + \mathbf{D}_{2u} \mathbf{W})^T (\mathbf{C}_2 \mathbf{X} + \mathbf{D}_{2u} \mathbf{W}) \), and \( \mathbf{K} = \mathbf{W} \mathbf{X}^{-1} \).
The composite braking controller employs a model predictive control (MPC) approach for torque distribution and a PID-based anti-lock braking system (ABS) for slip regulation. The MPC optimizer minimizes a cost function that considers tire load utilization, energy recovery, and torque smoothness:
$$ \min J = w_1 J_1 + w_2 J_2 + w_3 J_3 $$
$$ J_1 = \sum_{k=1}^{h_p} \sum_{i=fl,fr,rl,rr} \left( \frac{T_{bi}(k) – T_{bmi}(k)}{F_{zi}(k) R} \right)^2 $$
$$ J_2 = \sum_{k=1}^{h_p} \sum_{i=fl,fr,rl,rr} P_{mi}(k) T_{bmi}(k) \eta_{mi}(k) $$
$$ J_3 = \sum_{k=1}^{h_p} \sum_{i=fl,fr,rl,rr} \left( \Delta u_i(k) \right)^2 $$
where \( h_p \) is the prediction horizon, \( T_{bi} \) is the total braking torque, \( T_{bmi} \) is the motor braking torque, \( P_{mi} \) is the motor power, \( \eta_{mi} \) is the motor efficiency, and \( \Delta u_i \) is the change in control input. The constraints include braking torque balance, motor torque limits, and front-rear torque distribution according to regulations.
The PID-based ABS adjusts the braking torque to maintain the slip ratio near an optimal value:
$$ \Delta T_i = K_P e_{\lambda i} + K_I \int e_{\lambda i} dt + K_D \frac{de_{\lambda i}}{dt} $$
where \( e_{\lambda i} = \lambda^* – \lambda_i \) is the slip error, and \( K_P, K_I, K_D \) are the PID gains.
The cooperative control strategy uses MPC to coordinate the active suspension and braking systems based on real-time conditions. For low-intensity emergency braking, the controller maximizes regenerative braking while ensuring stability; for high-intensity emergencies, it prioritizes braking performance. The coordination law is derived by solving an optimization problem that adjusts the suspension forces to modulate tire loads:
$$ \min \sum_{k=1}^{h_p} \sum_{i=fl,fr,rl,rr} \left( \frac{T_{bi}(k) – T_{bmi}(k)}{T_{bmi,max}} \right)^2 $$
$$ \text{subject to} \quad \Delta F_{zfl} = -\Delta F_{zrl}, \quad \Delta F_{zfr} = -\Delta F_{zrr}, \quad |\Delta F_{zi}| \leq 200 \, \text{N} $$
The additional suspension force \( \Delta F_{ai} \) is related to the vertical load change by \( \Delta F_{ai} = \eta \Delta F_{zi} \), with \( \eta = 5 \).
To validate the proposed control strategy, we conduct simulations under various driving conditions. The vehicle parameters used in the simulations are listed in Table 2.
| Parameter | Value |
|---|---|
| Sprung mass, \( m_s \) (kg) | 1421 |
| Unsprung fixed mass, \( m_{usi} \) (kg) | 42.5 |
| Unsprung rotating mass, \( m_{ugi} \) (kg) | 37.5 |
| Wheel radius, \( R \) (m) | 0.325 |
| Tire stiffness, \( k_{ti} \) (N/m) | 250,000 |
| Wheel inertia, \( J \) (kg·m²) | 1.9 |
| Battery voltage, \( E \) (V) | 320 |
| Battery capacity, \( C_m \) (Ah) | 120 |
| Distance to front axle, \( a \) (m) | 1.3 |
| Distance to rear axle, \( b \) (m) | 1.5 |
| Motor peak torque, \( T_{max} \) (N·m) | 335 |
| Motor peak power, \( P_{max} \) (kW) | 15 |
| Track width, \( d_l, d_r \) (m) | 0.8375 |
| Center of gravity height, \( h_g \) (m) | 0.6 |
| Roll inertia, \( I_x \) (kg·m²) | 440.6 |
| Pitch inertia, \( I_y \) (kg·m²) | 2426 |
| Suspension stiffness, \( k_{si} \) (N/m) | 23,500 |
| Suspension damping, \( C_{si} \) (N·s/m) | 1450 |
| Motor inductance, \( L_s \) (H) | 2.4e-3 |
| Stator resistance, \( R_s \) (Ω) | 0.5 |
| Hub bearing stiffness, \( k_{mi} \) (N/m) | 1.2e7 |
First, we evaluate the active suspension controller under braking conditions on a B-class road with an initial speed of 72 km/h. Comparing a controller that ignores electromagnetic forces (Controller 1) with our designed controller (Controller 2), the results show significant improvements in ride comfort and rotor eccentricity suppression. The root mean square (RMS) values of key performance metrics are summarized in Table 3.
| Performance Metric | Controller 1 (RMS) | Controller 2 (RMS) | Improvement |
|---|---|---|---|
| Sprung mass acceleration (m/s²) | 0.0966 | 0.0838 | 13.25% |
| Front suspension travel (m) | 0.0110 | 0.0109 | 0.91% |
| Rear suspension travel (m) | 0.0113 | 0.0112 | 0.88% |
| Front rotor eccentricity (m) | 1.0396e-4 | 6.4193e-5 | 38.25% |
| Rear rotor eccentricity (m) | 1.3559e-4 | 8.4035e-5 | 38.02% |
| Front tire dynamic load (N) | 338.98 | 339.12 | -0.04% |
| Rear tire dynamic load (N) | 383.56 | 383.57 | -0.001% |
Next, we test the cooperative control strategy under low-adhesion (μ=0.2) and low braking intensity (z=0.2) conditions. The initial speed is 72 km/h, and the battery state of charge (SOC) is 50%. Compared to independent control, the cooperative control reduces braking distance by 1.8%, increases recovered energy by 6.6%, and improves ride comfort metrics, as shown in Table 4.
| Performance Metric | Independent Control | Cooperative Control | Improvement |
|---|---|---|---|
| RMS braking deceleration (m/s²) | 1.920 | 1.945 | 1.3% |
| Maximum braking distance (m) | 103.76 | 101.93 | 1.8% |
| Recovered energy (J) | 334,000 | 356,000 | 6.6% |
| RMS sprung mass acceleration (m/s²) | 0.0958 | 0.0899 | 6.2% |
| RMS pitch acceleration (rad/s²) | 0.1015 | 0.0827 | 18.5% |
| RMS front suspension deflection (m) | 0.0114 | 0.0037 | 67.54% |
| RMS rear suspension deflection (m) | 0.0121 | 0.0037 | 69.42% |
| RMS front tire dynamic load (N) | 288.23 | 231.76 | 19.59% |
| RMS rear tire dynamic load (N) | 291.56 | 240.19 | 17.62% |
Under high-adhesion (μ=0.8) and high braking intensity (z=0.8) conditions at 120 km/h, the cooperative control prioritizes braking performance over ride comfort. The results, summarized in Table 5, indicate a 4.26% reduction in braking distance, albeit with some degradation in suspension metrics.
| Performance Metric | Independent Control | Cooperative Control | Change |
|---|---|---|---|
| RMS braking deceleration (m/s²) | 7.14 | 7.42 | 3.92% |
| Maximum braking distance (m) | 56.670 | 54.258 | 4.26% |
| RMS sprung mass acceleration (m/s²) | 0.093 | 0.110 | -18.28% |
| RMS pitch acceleration (rad/s²) | 0.321 | 0.360 | -12.15% |
| RMS front suspension travel (m) | 0.638 | 0.654 | -2.51% |
| RMS rear suspension travel (m) | 0.664 | 0.675 | -1.66% |
| RMS front tire dynamic load (N) | 0.246 | 0.262 | -6.5% |
| RMS rear tire dynamic load (N) | 0.279 | 0.250 | 10.39% |
In conclusion, this paper presents a novel cooperative control strategy for distributed drive electric vehicles that effectively integrates active suspension and composite braking systems. By accounting for electromechanical coupling effects and employing advanced control techniques, our approach enhances ride comfort, braking safety, and energy recovery across various driving scenarios. The simulation results validate the superiority of cooperative control over independent systems, highlighting its potential for real-world applications in the evolving landscape of China’s electric vehicle industry. Future work will focus on experimental validation and extension to more complex driving conditions.