The advancement of electric vehicle technology has enabled revolutionary drivetrain architectures. Among these, the Four-Motor Distributed-Drive Electric Car (often abbreviated as DDEV) represents a pinnacle of actuation flexibility. By equipping each wheel with an independent electric motor, this configuration allows for precise and individual control of the driving and braking torque at all four corners of the vehicle. This capability opens new frontiers in vehicle dynamics control, particularly for executing complex maneuvers under challenging conditions. One such demanding scenario is performing a stationary or near-stationary 180-degree turn, or “turn-around,” on an inclined road surface. This maneuver is crucial for navigating tight spaces on slopes, such as narrow mountain roads or steep driveways. The core challenge lies in generating the necessary pure yaw moment to rotate the vehicle while simultaneously counteracting the disturbing effect of gravity, which tries to pull the car downhill, causing the intended center of rotation to drift significantly. This paper investigates an optimal torque vectoring control strategy, designed from a first-person engineering perspective, to enable a four-motor electric car to execute a stable 180-degree turn around its rear axle center on sloped roads.
The fundamental advantage of a four-motor electric car in this context is the direct and independent control of longitudinal tire forces. In a conventional vehicle, yaw moment is primarily generated through steering angle and differential braking. In contrast, a distributed-drive electric car can create a controlled yaw moment by differentially distributing drive torques between the left and right wheels, even at zero or low speeds. For a stationary turn on level ground, a simple strategy involves spinning the left-side wheels forward and the right-side wheels backward (or vice-versa) around the vehicle’s geometric center. However, on a slope, gravity introduces a sustained lateral force component (depending on the vehicle’s orientation relative to the slope) that disrupts this balance. If unaccounted for, this force causes the vehicle to slide downhill during the rotation, leading to a large, unpredictable, and potentially unsafe displacement of the turn center. Therefore, the control strategy for the electric car must not only generate rotation but also actively compensate for gravitational pull to keep the instantaneous center of rotation fixed at a desired point, specifically the midpoint of the rear axle.
To design such a strategy, we begin by establishing a suitable vehicle dynamics model. A nonlinear seven-degree-of-freedom (7-DOF) model is employed, capturing the essential dynamics for this maneuver. The degrees of freedom are: longitudinal motion, lateral motion, and yaw motion of the vehicle body, plus the rotational motion of each of the four wheels. The equations for the vehicle body’s translational and rotational dynamics in the vehicle coordinate system are given below. Here, $m$ is the vehicle mass, $I_z$ is the yaw moment of inertia, $a_x$ and $a_y$ are longitudinal and lateral accelerations at the center of gravity (CG), and $\ddot{\psi}$ is the yaw acceleration. The longitudinal and lateral tire forces for the left-front, right-front, left-rear, and right-rear wheels are denoted as $F_{xij}$ and $F_{yij}$ respectively, where $ij \in \{fl, fr, rl, rr\}$. The track widths are $t_f$ (front) and $t_r$ (rear), and the distances from the CG to the front and rear axles are $l_f$ and $l_r$.
$$ m a_x = F_{xfl} + F_{xfr} + F_{xrl} + F_{xrr} $$
$$ m a_y = F_{yfl} + F_{yfr} + F_{yrl} + F_{yrr} $$
$$ I_z \ddot{\psi} = l_f (F_{yfl}+F_{yfr}) – l_r (F_{yrl}+F_{yrr}) + \frac{t_f}{2}(F_{xfr}-F_{xfl}) + \frac{t_r}{2}(F_{xrr}-F_{xrl}) $$
The wheel rotational dynamics for each wheel $ij$ are modeled as:
$$ I_w \dot{\omega}_{ij} = T_{ij} – F_{xij} R_{eff} – B_m \omega_{ij} $$
where $I_w$ is the wheel’s rotational inertia, $\omega_{ij}$ is its angular speed, $T_{ij}$ is the motor torque (control input), $R_{eff}$ is the effective tire radius, and $B_m$ is a motor viscous damping coefficient. Vertical load transfer is critical as it affects tire force generation limits. The vertical loads $F_{zij}$ are influenced by longitudinal and lateral accelerations and the slope angle $\theta$. For a slope with the vehicle oriented such that the downhill direction is along the negative vehicle $y$-axis, the static load distribution is modified by gravity. The approximate vertical loads considering longitudinal ($a_x$) and lateral ($a_y$) transfer, and the slope component, can be expressed as:
$$ F_{zfl} = \frac{m g l_r \cos \theta}{2 L} – \frac{m a_x h}{2 L} – \frac{m (a_y + g \sin \theta) h}{2 t_f} \frac{l_r}{L} $$
$$ F_{zfr} = \frac{m g l_r \cos \theta}{2 L} – \frac{m a_x h}{2 L} + \frac{m (a_y + g \sin \theta) h}{2 t_f} \frac{l_r}{L} $$
$$ F_{zrl} = \frac{m g l_f \cos \theta}{2 L} + \frac{m a_x h}{2 L} – \frac{m (a_y + g \sin \theta) h}{2 t_r} \frac{l_f}{L} $$
$$ F_{zrr} = \frac{m g l_f \cos \theta}{2 L} + \frac{m a_x h}{2 L} + \frac{m (a_y + g \sin \theta) h}{2 t_r} \frac{l_f}{L} $$
where $h$ is the CG height, $L = l_f + l_r$ is the wheelbase, and $g$ is gravitational acceleration. The term $g \sin \theta$ appears within the lateral acceleration component due to the orientation of the slope relative to the vehicle frame. The core principle of the turning maneuver for the electric car is to generate a net yaw moment while maintaining zero net longitudinal and lateral force at the vehicle body level (to prevent translation of the rotation center). The total yaw moment $M_z$ from tire forces can be decomposed into contributions from the front axle ($\Delta T_f$) and rear axle ($\Delta T_r$), considering the simplified relationship $F_x \approx T / R_{eff}$ for low slip conditions:
$$ M_z \approx \frac{1}{R_{eff}} \left( \frac{t_f}{2} \Delta T_f + \frac{t_r}{2} \Delta T_r \right) $$
where $\Delta T_f = T_{fr} – T_{fl}$ and $\Delta T_r = T_{rr} – T_{rl}$. To rotate around the rear axle center, the goal is to make the vehicle’s instantaneous center of rotation (ICR) coincide with that point. This imposes a kinematic relationship between the velocities of the four wheels. If the ICR is at the rear axle center, the velocities at the wheel centers are proportional to their distance from that point. However, on a slope, the gravitational force $F_{g,lat} = mg \sin \theta$ acts at the CG, creating a disturbance yaw moment $M_{z,dist} = F_{g,lat} \cdot l_f \cos(\psi)$ relative to the rear axle center, where $\psi$ is the vehicle’s yaw angle relative to the slope’s fall line. This disturbance moment changes as the electric car rotates, being maximum when the vehicle is sideways on the slope ($\psi = 0^\circ$ or $180^\circ$) and zero when pointing directly up or down the slope ($\psi = \pm 90^\circ$).
To address this, we propose a two-layer optimal control strategy. The high-level controller determines the total desired yaw moment $M_{z,des}$ required to track a target yaw rate $\dot{\psi}_{des}$ for the 180-degree turn. Simultaneously, it calculates the compensating yaw moment $M_{z,comp}$ needed to negate the gravitational disturbance and keep the ICR at the rear axle. This compensating moment is a function of the measured vehicle orientation $\psi$ and the slope angle $\theta$: $M_{z,comp} = -mg \sin \theta \cdot l_f \cos(\psi)$. The total required yaw moment is $M_{z,total} = M_{z,des} + M_{z,comp}$.
The low-level controller is an optimal torque vectoring allocator based on a Linear Quadratic Regulator (LQR) design. It dynamically distributes the total required yaw moment between the front and rear axles while minimizing a cost function. We define a state-space model relevant to the turning maneuver. The state vector $\mathbf{x}$ includes the yaw rate error and the offset of the rotation center from the desired rear axle point in vehicle coordinates ($\Delta X, \Delta Y$). The control input vector $\mathbf{u}$ consists of the front and rear axle torque differences ($\Delta T_f, \Delta T_r$). The gravitational disturbance is treated as a known input $\mathbf{w}$.
The discrete-time state-space representation is:
$$ \mathbf{x}_{k+1} = \mathbf{A} \mathbf{x}_k + \mathbf{B} \mathbf{u}_k + \mathbf{D} \mathbf{w}_k $$
where the matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{D}$ are derived from linearized vehicle dynamics and kinematic relationships. The system output $\mathbf{y}$ is the yaw rate and rotation center offset. The LQR controller seeks to find the control law $\mathbf{u}_k = -\mathbf{K} \mathbf{x}_k$ that minimizes the infinite-horizon quadratic cost function $J$:
$$ J = \sum_{k=0}^{\infty} \left( \mathbf{x}_k^T \mathbf{Q} \mathbf{x}_k + \mathbf{u}_k^T \mathbf{R} \mathbf{u}_k \right) $$
The weighting matrices $\mathbf{Q}$ and $\mathbf{R}$ are tuned to prioritize accurate tracking of the yaw rate and minimal rotation center offset ($\mathbf{Q}$), while penalizing excessive torque differentials to prevent actuator saturation and tire slip ($\mathbf{R}$). The optimal feedback gain matrix $\mathbf{K}$ is obtained by solving the associated discrete-time algebraic Riccati equation. This LQR-based allocator for the electric car optimally balances the use of front and rear axle torque differentials. Typically, the front axle, being farther from the desired rotation center, is more effective at generating yaw moment for rotation. The rear axle torque differential is primarily used for finer corrections to counteract the gravity-induced offset. The final motor torque commands for each wheel of the electric car are then calculated as:
$$ T_{fl} = T_{base} – \frac{\Delta T_f}{2};\quad T_{fr} = T_{base} + \frac{\Delta T_f}{2} $$
$$ T_{rl} = T_{base} – \frac{\Delta T_r}{2};\quad T_{rr} = T_{base} + \frac{\Delta T_r}{2} $$
where $T_{base}$ is a small constant torque bias used to initiate movement and overcome rolling resistance. For a pure rotation, the average wheel speed on one side is positive and on the other side is negative.
To validate the proposed control strategy for the four-motor electric car, a co-simulation platform was established using MATLAB/Simulink for the controller and Adams/Car for the high-fidelity multi-body vehicle dynamics. A standard sedan model in Adams/Car was modified into a four-wheel-independent-drive configuration. Key vehicle parameters are summarized in the table below.
| Vehicle Parameter | Value |
|---|---|
| Vehicle Type | Sedan |
| Total Mass | 1528 kg |
| Wheelbase (L) | 2.56 m |
| Front Track Width ($t_f$) | 1.52 m |
| Rear Track Width ($t_r$) | 1.594 m |
| CG to Front Axle ($l_f$) | 1.1749 m |
| CG Height ($h$) | 0.432 m |
| Yaw Moment of Inertia ($I_z$) | ~2500 kg·m² |
| Tire Effective Radius ($R_{eff}$) | 0.32 m |
The simulation scenario involved a road with a friction coefficient of 1.1 and a defined slope angle. The electric car starts stationary, oriented with its right side facing downhill, and is commanded to execute a 180-degree clockwise turn. Three control methods were compared:
| Control Method | Description |
|---|---|
| Baseline | Rotation around the vehicle’s geometric center (no gravity compensation). |
| Rear-Center Rotation (Basic) | Kinematic control to rotate around the rear axle center, without active offset correction. |
| Proposed Strategy (LQR with Compensation) | Optimal torque vectoring with gravity disturbance compensation. |
On a 10-degree slope, the results were starkly different. The baseline method, as expected, led to significant downhill sliding. The electric car’s rotation center drifted substantially, with a maximum offset distance of 5.62 meters from the starting point. The basic rear-center rotation method fared better but still showed a clear drift due to un-compensated gravity, with a maximum offset of 0.85 meters. In contrast, the proposed LQR-based optimal torque vectoring strategy for the electric car successfully minimized this drift. The rotation path was tightly controlled around the rear axle center, with the maximum offset reduced to a mere 0.15 meters. This represents a reduction in offset by approximately 97% compared to the baseline and 82% compared to the basic rear-center method, highlighting the critical importance of active gravity compensation.
| Control Method (10° Slope) | Max. Offset Distance |
|---|---|
| Baseline | 5.62 m |
| Rear-Center Rotation (Basic) | 0.85 m |
| Proposed Strategy | 0.15 m |
To determine the operational limit of the proposed strategy for the electric car, further simulations were conducted on steeper slopes of 15 and 18 degrees, with the same controller parameters. The performance metric is the maximum rotation center offset. Based on functional safety considerations (e.g., derived from standards like ISO 26262) and a margin for environmental uncertainty, a maximum permissible offset threshold of 0.25 meters was defined for this specific vehicle platform.
| Slope Angle | Max. Rotation Center Offset | Within 0.25m Limit? |
|---|---|---|
| 10° | 0.15 m | Yes |
| 15° | 0.22 m | Yes |
| 18° | 0.28 m | No |
The results indicate that the proposed optimal torque vectoring control strategy is effective for the four-motor electric car on slopes up to 15 degrees, maintaining the rotation center offset within the safe 0.25-meter boundary. At 18 degrees, the gravitational disturbance overwhelms the controller’s ability to fully compensate with the available tire forces (subject to friction limits), causing the offset to exceed the threshold. This defines the performance envelope of the current strategy for this specific electric car configuration.
In conclusion, this research demonstrates a comprehensive approach to solving a complex vehicle dynamics problem inherent to the advanced capabilities of a four-motor distributed-drive electric car. By developing a nonlinear 7-DOF model and formulating an LQR-based optimal torque vectoring control strategy with explicit gravity compensation, we have shown that stable and precise 180-degree turns on sloped roads are feasible. The key insight is that controlling such an electric car requires managing not just the yaw moment for rotation but also the balance of forces to counteract persistent environmental disturbances. The simulation results validate the strategy’s effectiveness, showing dramatic reductions in rotation center drift compared to naive methods. The proposed framework provides a solid foundation for implementing advanced maneuvering functions in next-generation electric cars, enhancing their utility and safety in challenging terrain. Future work will involve testing the strategy on more complex and low-friction surfaces, integrating it with higher-level path planning, and conducting physical validation on a prototype four-motor electric car platform.