Corner-module architecture has emerged as one of the most promising chassis concepts for intelligent electric cars. In this architecture, each road wheel is enclosed in an independent module that integrates a hub motor, a steer-by-wire actuator and a brake-by-wire unit. Corner-module electric cars therefore provide four-wheel independent steering, four-wheel independent driving and four-wheel independent braking. This configuration removes most mechanical linkages of a conventional chassis and gives the vehicle extraordinary manoeuvrability, including in-situ rotation, lateral parking and oblique driving. However, the same redundancy also leads to strong coupling between longitudinal dynamics, lateral dynamics and roll dynamics. When electric cars are driven near the handling limit on slippery roads, the tyre forces can easily saturate, resulting in large vehicle sideslip, unintended drifting and even rollover. In this context, I have developed a novel “differential oblique steering” strategy for corner-module electric cars. The steering principle is to intentionally control the vehicle body to yaw into a drifting-like oblique posture during emergency cornering. By allocating a larger steering angle to the rear wheels and a smaller steering angle to the front wheels, all four tyres share the required lateral force in a more balanced way. The body posture simultaneously increases the effective wheelbase of the vehicle with respect to the turning centre, which significantly suppresses the roll angle. In the present work, I focus on the stability control of this differential oblique steering mode under extreme steering conditions, including modelling, steering-angle allocation, mode-switching control, model predictive path tracking and controller-in-the-loop validation.
From a broader viewpoint, high-speed lane-change and obstacle-avoidance manoeuvres are among the most dangerous driving scenarios for electric cars. When the steering wheel is turned sharply at high speed, the centrifugal force induces a large body roll moment. If the road surface is wet or icy, the usable lateral friction is greatly reduced. A conventional front-wheel-steered car often develops severe understeer, and when the driver continues to increase the steering input, the vehicle may leave the intended lane in an uncontrolled drift. Four-wheel steering can produce an additional lateral force from the rear axle, but in the extreme saturation region it is still difficult to keep the vehicle stable. The differential oblique steering method investigated in this paper therefore represents an unconventional yet practical alternative for enhancing the active safety of electric cars on harsh roads.
1. Introduction and Research Motivation
In recent years, electric cars have rapidly progressed toward higher levels of automation, electrification and networking. The corner-module architecture is especially attractive because it maximises freedom of motion for the vehicle body. Since each wheel module contains its own motor, steering gear and brake, the chassis can realise multiple driving modes that are not available in conventional vehicles. Some electric cars with this architecture can even perform crab steering, diagonal driving or zero-radius pivoting. Yet the full potential of corner-module electric cars is not confined to low-speed parking or manoeuvring. Under emergency avoidance at 80 km/h or above, the ability to control every wheel angle independently creates an opportunity to exploit controlled vehicle-body sideslip in a safe and systematic way. In a controlled drift-like turn, the rear wheels are steered actively in the same direction as the front wheels but with a larger angle, causing the rear axle to swing outward. The car turns into a corner while advancing with an oblique body attitude, a process which I refer to as differential oblique steering.
The motivation for studying differential oblique steering comes from practical limitations of conventional extreme-steering control. During a high-speed emergency lane change, a vehicle must generate a large lateral acceleration. If the attempt is made with ordinary steering, all four tyres may operate at their friction limits, and the body roll angle often exceeds the comfortable and even the safe threshold. By intentionally yawing the car into an oblique attitude, the component of the vehicle velocity perpendicular to its longitudinal axis changes the contact-patch slip conditions. In effect, the same curvature change can be obtained with less individual tyre slip. This reduces the possibility of side-slip saturation and rollover. Moreover, because the corner-module architecture allows each wheel to be commanded independently, the transition into the drifting-like oblique posture can be designed as a smooth, deterministic control procedure rather than an unpredictable instability phenomenon.
Many studies on electric cars and autonomous driving have investigated drift control or sliding-mode control at the limit. Those methods generally regard drifting as a nonlinear vehicle state to be stabilised. By contrast, the differential oblique steering developed in this paper deliberately maintains a positive body sideslip angle that may be larger than typical, while the four wheel angles are chosen so that the resultant vehicle travels along the desired reference path. A key point is that the total required lateral force is distributed over four tyres instead of mainly the front axle. This characteristic not only enlarges the achievable handling envelope but also keeps the roll angle at a lower level. From a systems point of view, the control scheme should be hierarchical: at the highest level, the desired body attitude and yaw rate are determined from path information; at the middle level, mode-switching selects between conventional four-wheel steering and differential oblique steering; at the lowest level, the model predictive controller computes the optimal front and rear steering commands. The present thesis thus includes: (1) development and validation of a full-vehicle dynamics model; (2) analysis of differential oblique steering mechanisms and wheel-angle allocation; (3) design of a roll-constrained switching path-tracking controller; and (4) controller-in-the-loop experiments that emulate realistic delays and sampling constraints.
2. Vehicle Dynamics Modelling for Corner-Module Electric Cars
To investigate the extreme steering behaviour of corner-module electric cars, I established a twelve-degree-of-freedom vehicle model that combines rigid-body motion, roll dynamics, wheel rotation and wheel steering. The degrees of freedom are the longitudinal velocity, lateral velocity, yaw rate, roll angle and the rotational/steering motions of the four wheels. Since the corner-module architecture uses an independent steering motor for each wheel, the model must consider wheel steering dynamics explicitly rather than treating the front wheel angle as a direct input.
2.1 Twelve-DOF Rigid-Body Model
The longitudinal, lateral and yaw equations are expressed in the vehicle-fixed coordinate system as
$$
\begin{aligned}
m(\dot{V}_x – V_y\gamma) & = \sum_{i=1}^{4} F_{xi}\cos\delta_i – \sum_{i=1}^{4}F_{yi}\sin\delta_i+F_{\mathrm{res}}, \\
m(\dot{V}_y + V_x\gamma) & = \sum_{i=1}^{4}F_{yi}\cos\delta_i + \sum_{i=1}^{4}F_{xi}\sin\delta_i, \\
I_z\dot{\gamma} & = \sum_{i \in \mathrm{front}}(F_{xi}\sin\delta_i + F_{yi}\cos\delta_i)a
– \sum_{i \in \mathrm{rear}}(F_{xi}\sin\delta_i + F_{yi}\cos\delta_i)b \\
& \quad + \frac{d}{2}\sum_{i \in \mathrm{left}}F_{xi\mathrm{sgn}} – \frac{d}{2}\sum_{i \in \mathrm{right}}F_{xi\mathrm{sgn}},
\end{aligned}
$$
where \(m\) is the total vehicle mass, \(V_x\) and \(V_y\) are the longitudinal and lateral velocity components, \(\gamma\) is the yaw rate, \(\delta_i\) is the steering angle of the \(i\)-th wheel, \(F_{xi}\) and \(F_{yi}\) are the longitudinal and lateral tyre forces, \(a\) and \(b\) are the distances from the centre of gravity to the front and rear axles, \(d\) is the half-track width, \(I_z\) is the yaw moment of inertia and \(F_{\mathrm{res}}\) is the aerodynamic drag. The sign convention is carefully selected to represent left/right wheel forces consistently.
The roll motion of the sprung mass is governed by
$$
I_{xs}\ddot{\rho} = m_s g h_s \sin\rho + m_s h_s(\dot{V}_y + V_x\gamma)\cos\rho – \frac{d}{2}\left(F_{s1}+F_{s3}-F_{s2}-F_{s4}\right),
$$
in which \(I_{xs}\) is the roll moment of inertia, \(\rho\) is the body roll angle, \(m_s\) is the sprung mass, \(h_s\) is the distance between the sprung-mass centre and the roll axis, and \(F_{si}\) denotes the suspension force at the \(i\)-th corner. The suspension forces are computed with stiffness and damping terms generated by the relative motion between the sprung mass and unsprung masses. Since this work is concerned primarily with roll stability during emergency steering, the suspension model is kept relatively simple but includes the geometric coupling caused by roll motion.
2.2 Tyre Model
The accuracy of the vehicle dynamics model depends strongly on tyre force reproduction. In the extreme situations discussed here, the tyre slip angle can enter the nonlinear region and even approach the saturation point; linear tyre models are therefore inadequate. I adopted the well-known Magic Formula tyre model, written in its generic form as
$$
F_x(x) = D\sin\!\big[C\arctan\!\big(Bx-E(Bx-\arctan Bx)\big)\big],
$$
with a similar expression for the lateral force as a function of slip angle. The stiffness, shape and curvature factors are functions of the vertical load, the camber angle and the friction coefficient. The parameter tables used for lateral force fitting are summarised in Table 1 and Table 2, and they were incorporated into the vehicle model in Simulink.
| Symbol | Value | Symbol | Value |
|---|---|---|---|
| \(a_0\) | 1.6 | \(a_4\) | 12.8 |
| \(a_1\) | −34 | \(a_6\) | −0.0053 |
| \(a_2\) | 1250 | \(a_7\) | 0.1925 |
| \(a_3\) | 2320 |
| Symbol | Value | Symbol | Value |
|---|---|---|---|
| \(b_0\) | 1.55 | \(b_5\) | 0.17 |
| \(b_1\) | 0 | \(b_6\) | 0 |
| \(b_2\) | 1000 | \(b_7\) | 0 |
| \(b_3\) | 60 | \(b_8\) | 0.2 |
| \(b_4\) | 300 |
In addition, the tyre vertical loads are affected by longitudinal and lateral load transfer, which is particularly important for roll stability. The vertical load on each wheel is calculated from the static load distribution, the longitudinal acceleration \(a_x\) and the lateral acceleration \(a_y\):
$$
\begin{aligned}
F_{z1} & = \frac{mgb}{2L}-\frac{m_s h_s a_x}{2L}-\frac{m_s h_s a_y b}{d L}, \\
F_{z2} & = \frac{mgb}{2L}-\frac{m_s h_s a_x}{2L}+\frac{m_s h_s a_y b}{d L}, \\
F_{z3} & = \frac{mga}{2L}+\frac{m_s h_s a_x}{2L}-\frac{m_s h_s a_y a}{d L}, \\
F_{z4} & = \frac{mga}{2L}+\frac{m_s h_s a_x}{2L}+\frac{m_s h_s a_y a}{d L}.
\end{aligned}
$$
These forces determine the friction-circle limits of each tyre and are indispensable for producing realistic responses in extreme cornering and in low-friction road tests.
2.3 Wheel-Steering Actuator Model
The corner-module architecture relies on a steer-by-wire system. The prototype corner module used as the object of this study integrates the suspension spring, damper, steering motor, reduction gear and wheel hub into one compact unit. The steering motor is a DC brush motor driven by a PWM voltage. The electrical equation is
$$
u V_{\mathrm{DC}} = L_a\frac{di_a}{dt}+R_a i_a + K_e\dot{\delta}_m,
$$
and the mechanical equation of the steering actuator is
$$
J_m\ddot{\delta}_m + B_m\dot{\delta}_m + T_{\mathrm{wm}} = T_{\mathrm{act}},
$$
where \(J_m\) is the motor inertia, \(B_m\) is the viscous friction coefficient, \(\delta_m\) is the motor angle, \(T_{\mathrm{act}}\) is the motor torque and \(T_{\mathrm{wm}}\) is the torque transferred from the wheel. With the reduction gear ratio \(k\), the wheel steering dynamics can be expressed as
$$
(J_w + k^2J_m)\ddot{\delta}_w + (B_w + k^2B_m)\dot{\delta}_w + T_e + T_F = kT_{\mathrm{act}},
$$
where \(J_w\) is the wheel inertia about the kingpin axis, \(T_e\) is the aligning torque and \(T_F\) is the Coulomb friction torque. The parameters of the steering system are presented in Table 3. In the subsequent control simulations, these actuator dynamics determine the practical rate and delay of steering-angle realisation.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Motor rotor inertia | \(J_m\) | 0.000828 | kg m² |
| Motor viscous friction | \(B_m\) | 0.008 | N m s/rad |
| Wheel inertia about kingpin | \(J_w\) | 1.829 | kg m² |
| Wheel viscous friction | \(B_w\) | 6 | N m s/rad |
| Reduction ratio | \(k\) | 60 | – |
| Motor voltage | \(V_{\mathrm{DC}}\) | 12 | V |
| Armature inductance | \(L_a\) | 0.16 | H |
| Armature resistance | \(R_a\) | 0.29 | Ω |
| Back-EMF coefficient | \(K_e\) | 0.021 | V s/rad |
| Torque coefficient | \(K_T\) | 0.081 | N m/A |
2.4 Hub-Motor Model and Model Validation
The hub motors of corner-module electric cars are permanent-magnet synchronous machines. Since the focus of this paper is the upper-level steering allocation rather than the inner-loop motor control, the hub motor torque output is represented by a second-order transfer function
$$
G(s) = \frac{T_{m}(s)}{T^*_{m}(s)} = \frac{1}{1 + 2\zeta s/\omega_n + s^2/\omega_n^2},
$$
where \(T_m\) is the actual torque, \(T_m^*\) is the target torque, \(\omega_n\) is the natural frequency and \(\zeta\) is the damping ratio. The motor output is bounded by the torque-speed envelope to respect physical limits.
Before employing the above model in controller design, I compared its response with the industry-standard vehicle model embedded in Carsim. In this validation, the vehicle travelled at 40 km/h and the steering wheel angle increased by 100° within 0.45 s at \(t=2\) s. The road friction coefficient was 0.85. The matching results of the yaw rate, sideslip angle, lateral velocity and roll angle were highly consistent, as listed in Table 4, thereby confirming that the formulated nonlinear vehicle model can be used for subsequent extreme-manoeuvre simulation.
| Response variable | Simulink model | Carsim model | Relative error |
|---|---|---|---|
| Yaw-rate peak | 23.6 deg/s | 23.1 deg/s | 2.2% |
| Sideslip-angle peak | 2.9 deg | 3.0 deg | 3.3% |
| Lateral speed peak | 1.9 m/s | 2.0 m/s | 5.0% |
| Roll-angle peak | 1.8 deg | 1.7 deg | 5.9% |
3. Wheel-Angle Allocation for Differential Oblique Steering
3.1 Comparison of Steering Modes
At low speed, a conventional front-wheel-steered vehicle already offers adequate controllability, but its turning radius is relatively large. Four-wheel steering with the rear wheels steered opposite to the front wheels can shift the instantaneous centre of rotation closer to the vehicle body, significantly reducing the required turning radius and improving low-speed manoeuvrability of electric cars. At high speed, rear wheels are steered in the same direction as the front wheels, producing a stabilising yaw moment and a faster transient response. Nevertheless, in an emergency lane change on wet or icy roads, even four-wheel-steered electric cars can be pushed beyond the linear tyre region. The differential oblique steering mode proposed here uses an opposite strategy: the front wheels turn only slightly, while the rear wheels turn more sharply in the same direction as the front wheels, but the vehicle’s body yaw angle is oriented to establish a large positive sideslip. The vehicle advances diagonally as it rotates, effectively “swinging” the tail outward around the corner.
Table 5 summarises the qualitative differences among front-wheel steering, ordinary four-wheel steering and differential oblique steering from the angles of wheel-angle distribution, body posture, stability mechanism and practical benefit for electric cars.
| Aspect | FWS | 4WS | Differential oblique steering |
|---|---|---|---|
| Front/rear angle relation | rear angle zero | rear is small and proportional | rear angle is larger than front |
| Body sideslip in high-speed turn | uncontrolled | small | large but controlled |
| Main roll moment lever | front-axle lateral force | distributed partly | distributed over four tyres |
| Effective wheelbase relative to roll centre | constant | constant | increased |
| Risk under low friction | high understeer/sideslip | moderate | low, because tyre-force margin is retained |
I have described the physical advantage intuitively: when the body adopts an oblique attitude and the tail swings outward, the instantaneous turning centre is closer to the vehicle front axle, but the longitudinal distance between the centre of gravity and the rear axle is effectively enlarged along the velocity direction. The roll moment is generated by lateral acceleration multiplied by sprung-mass height. At the same curvature and speed, the required equivalent lateral force is shared by all four wheels, so the body roll angle is smaller. This explains why differential oblique steering can improve both manoeuvrability and roll stability—two objectives that generally conflict in conventional chassis design.
3.2 Reference Model with Rear-Wheel Steering
In the design of the middle-layer and lower-layer controllers, I adopted a single-track model with rear-wheel steering. The single-track model avoids unnecessary complexity while preserving the primary lateral and yaw dynamics. With constant longitudinal speed and small-angle approximations limited to certain regions, the differential equations of lateral velocity and yaw rate are
$$
\begin{aligned}
m(\dot{V}_y + V_x\gamma) & = F_{yf}\cos\delta_f + F_{yr}\cos\delta_r, \\
I_z\dot{\gamma} & = aF_{yf}\cos\delta_f – bF_{yr}\cos\delta_r,
\end{aligned}
$$
where \(F_{yf}\) and \(F_{yr}\) are the resultant lateral tyre forces of the front and rear axles. If the tyre slip angles remain moderate, the forces are linearised as
$$
F_{yf}=-k_f\alpha_f, \qquad F_{yr}=-k_r\alpha_r,
$$
with cornering stiffnesses \(k_f\) and \(k_r\). The tyre slip angles are computed from the vehicle motion and steering angles:
$$
\alpha_f = \beta+\frac{a\gamma}{V_x}-\delta_f, \qquad \alpha_r=\beta-\frac{b\gamma}{V_x}-\delta_r.
$$
Substitution gives a two-degree-of-freedom model that is valid for moderate manoeuvres and forms the basis of the reference steering-angle design.
3.3 Rear-Wheel Follow-up Steering Allocation
In ordinary four-wheel steering, the rear-wheel angle is often formulated as a combination of three terms: a proportional term of the front-wheel angle, a proportional term of the yaw rate, and a disturbance-compensation term. The general form is
$$
\delta_r = G_\delta \delta_f + G_\omega \gamma + G_d.
$$
For steady-state cornering with zero sideslip angle, the coefficients \(G_\delta\), \(G_\omega\) and \(G_d\) can be derived by setting the desired sideslip angle to zero in the linearised model. The resulting coefficients are
$$
G_\delta = -\frac{k_f}{k_r}, \qquad
G_\omega = \frac{mV_x^2(a+b) – V_x(a k_f – b k_r)}{k_r V_x (a+b)},
$$
with \(G_d=0\). The relation indicates that, at high forward speed, the rear wheel angle remains in the same direction as the front wheel and gradually approaches a nearly proportional value. In the conventional highway driving region, this mode gives electric cars a quick and stable response while maintaining a small body sideslip angle.
3.4 Geometric Allocation of Differential Oblique Steering
Differential oblique steering is fundamentally different from ordinary four-wheel steering because the body sideslip angle is a controlled state rather than an undesired deviation. In the single-track geometry shown in Figure 3-9 of the original thesis, the turning centre is shifted so that the vehicle’s motion direction is not aligned with its longitudinal axis. Using the sine theorem in the velocity triangle, the relation among the front steering angle \(\delta_f\), rear steering angle \(\delta_r\), body sideslip angle \(\beta\), and the distance \(c\) from the turning centre to the front axle projection gives
$$
\frac{\sin\beta\cos\delta_f}{\sin(\delta_f-\beta)} = \frac{a}{R_0},
$$
where \(R_0\) is the effective turning radius of the centre of gravity. The rear-axle steering angle is simultaneously related to the body sideslip by
$$
\frac{\sin\beta\cos\delta_r}{\sin(\delta_r+\beta)} = \frac{b}{R_0}+c.
$$
After eliminating the intermediate distance variable, the fundamental relationship between front and rear steering angles and the body sideslip angle can be written as
$$
\frac{\tan\beta}{R_0} = \frac{\delta_f}{a\cos\delta_f} – \frac{\delta_r}{b\cos\delta_r}, \qquad \beta = \frac{a}{R_0} – \delta_f + \delta_r.
$$
For controller implementation, I use a simplified small-angle relation derived from the same geometry:
$$
\delta_f = \frac{a}{R_0}+\beta, \qquad \delta_r=\beta-\frac{b}{R_0}.
$$
Thus, for a given path curvature \(1/R_0\) and desired sideslip angle \(\beta\), the front and rear equivalent steering angles are determined independently. The four individual wheel angles are then obtained by applying an Ackermann-type distribution on each axle:
$$
\begin{aligned}
\delta_{fl} & = \frac{\tan\delta_f}{1 – \frac{B}{2L}\tan\delta_f}, &
\delta_{fr} & = \frac{\tan\delta_f}{1 + \frac{B}{2L}\tan\delta_f}, \\
\delta_{rl} & = \frac{\tan\delta_r}{1 – \frac{B}{2L}\tan\delta_r}, &
\delta_{rr} & = \frac{\tan\delta_r}{1 + \frac{B}{2L}\tan\delta_r},
\end{aligned}
$$
where \(B\) is the track width and \(L=a+b\) is the wheelbase. The resulting steering pattern guarantees that all four wheels rotate about one common instantaneous centre, even in the presence of a large body-slip posture.
The relationship between the front and rear wheels in the differential oblique mode corresponds to a different set of controller gains in the formula \(\delta_r = G_\delta\delta_f + G_\omega\gamma + G_d\). In ordinary four-wheel steering, \(G_\delta\) is negative or small positive depending on speed; in differential oblique steering, \(G_\delta>1\) in certain segments and the contribution of the body-sideslip term dominates. The switching process between the two modes is handled by a fuzzy supervisor discussed in the next section.
3.5 Model Verification of Wheel-Angle Allocation
I verified the ordinary four-wheel steering allocation first in a single-lane-change simulation at 36 km/h. The controller achieved accurate tracking of the centre line while keeping the body sideslip magnitude within ±0.5°, the wheel slip angles below 0.3° and the peak lateral deviation within 0.1 m. The rear wheel angle was generally opposite to the front wheel angle at this low speed, as expected.
For the differential oblique allocation, I prescribed a sinusoidal desired body-sideslip-angle input with amplitude relevant to high-speed turning. The reference model was evaluated at 80 km/h with a maximum steering-angle constraint of 40°. The front wheels and rear wheels followed the expected pattern: the rear angle amplitude was larger than the front angle amplitude, and the body sideslip angle resembled a biased sinusoidal waveform. Both the two-wheel equivalent model and the four-wheel Ackermann model produced smooth angles without violating the geometric constraint.
4. Lateral and Roll Stability Control under Extreme Steering
Having established the feedforward allocation method for differential oblique steering, I next designed a layered control system that can be executed in real time on embedded hardware. The complete framework consists of three functional layers: (i) steering-mode decision layer, (ii) roll-stability-based steering-angle constraint layer, and (iii) model predictive path-tracking control layer. This architecture separates the tasks of recognising the driving situation, protecting the vehicle against rollover, and optimising the tracking performance, which simplifies tuning and improves reliability.
4.1 Roll-Stability Steering-Angle Constraints
The first important design issue is to ensure that the commanded front and rear wheel angles do not force the tyres beyond their friction limits or cause excessive body roll. I derived two complementary constraints: a sideslip-angle constraint related to the road friction coefficient and a wheel-angle constraint related to the allowable body roll angle.
4.1.1 Sideslip-Angle Constraint Based on Road Friction
At the tyre level, the resultant force is limited by the friction ellipse. For an emergency manoeuvre on a road with friction coefficient \(\mu\), the total lateral force \(F_Y\) must satisfy
$$
F_{Y,\mathrm{req}} \leqslant \mu m g.
$$
Combining this condition with the linear tyre model and the relation between slip angle and vehicle state produces an allowable region of the body sideslip angle. Let the front and rear tyre cornering stiffnesses be \(k_f\) and \(k_r\). Replacing the axle lateral forces by their linearised expressions and imposing the friction-circle limit yields
$$
|\beta| \leqslant \frac{\mu m g + (k_r\delta_r+k_f\delta_f)+ (k_f a – k_r b)\gamma/V_x}{k_f + k_r},
$$
where the steering angles and the current yaw rate are measured or estimated. This bound is updated at every sampling instant. On a low-friction surface, the allowable sideslip region shrinks; however, because the differential oblique mode deliberately generates a positive \(\beta\), the constraint prevents the commanded \(\beta\) from exceeding the stabilisable region.
4.1.2 Four-Wheel Steering-Angle Constraint Based on Body Roll
To directly limit the roll angle, I modelled the roll dynamics of the sprung mass with the suspension forces expressed in terms of roll angle and roll rate. Starting from the roll equilibrium equation
$$
I_{xs}\ddot{\rho} + c_{\rho}\dot{\rho} + (k_{\rho} – m_sgh_s)\rho = m_s h_s a_y,
$$
I substitute the lateral acceleration expression containing the tyre forces at all four wheels. By further introducing the condition that the total tyre force at each corner must remain within the friction circle, the roll acceleration can be formulated as a state-space equation with the four wheel angles acting as inputs:
$$
\dot{\boldsymbol{x}}_{\rho} = \boldsymbol{A}_{\rho}\boldsymbol{x}_{\rho} + \boldsymbol{B}_{\rho}\boldsymbol{\delta} + \boldsymbol{E} W,
$$
where \(\boldsymbol{x}_{\rho}=[\rho,\dot{\rho}]^T\), \(\boldsymbol{\delta}=[\delta_1,\delta_2,\delta_3,\delta_4]^T\), and the matrices are
$$
\boldsymbol{A}_{\rho}=\begin{bmatrix}
0 & 1\\ -\frac{m_s g h_s}{I_{xs}} & -\frac{C_s}{I_{xs}}
\end{bmatrix},\qquad
\boldsymbol{B}_{\rho}=\begin{bmatrix}
0 & 0 & 0 & 0\\
\frac{m_sh_s}{I_{xs}m}\sqrt{F_{x1}^2+F_{y1}^2} & \cdots & \frac{m_sh_s}{I_{xs}m}\sqrt{F_{x4}^2+F_{y4}^2}
\end{bmatrix}.
$$
In practical terms, this equation predicts the roll angle one step ahead from the current steering command. Given a roll-angle threshold \(\rho_{\max}\), the allowed wheel-angle increment can be inverted. Different types of vehicles have different thresholds, as summarised in Table 6. For the corner-module electric car used in this study, I set an upper roll limit of 3.0°, which is conservative relative to the rollover boundary of approximately 35–45° for passenger electric cars, but still preserves comfort and controllability.
| Vehicle category | Roll angle threshold (deg) |
|---|---|
| Passenger car | 35–45 |
| SUV | 30–40 |
| Bus | 25–35 |
| Heavy truck | 20–30 |
| Sports electric car | 40–50 |
| ATV | 35–45 |
These two constraint sets are assembled before the model predictive controller performs the optimisation. The side-slip constraint is a hard inequality constraint on the state vector, while the roll-angle-derived constraint is expressed as an inequality on the control input sequence. The resulting feasible region is therefore based on both road state and vehicle attitude.
4.2 Fuzzy Steering-Mode Switching Strategy
Differential oblique steering is not necessary in all situations. When the tyre slip angles are small and the road curvature is mild, ordinary four-wheel steering provides sufficient stability with better passenger comfort. A switching logic is required to activate differential oblique steering only when the tyre forces approach saturation or when the path curvature demands a very high yaw rate. I constructed a fuzzy decision supervisor with two inputs:
- The maximum absolute tyre slip angle among all wheels, denoted \(\alpha_{\max}\); and
- The current reference path curvature \(\rho_{\mathrm{ref}}\).
The output is a continuous blending coefficient \(\lambda\in[0,1]\), where \(\lambda=0\) corresponds to ordinary four-wheel steering and \(\lambda=1\) corresponds to full differential oblique steering. Therefore, the effective steering coefficients are
$$
G_\delta = (1-\lambda)G_{\delta,\mathrm{4WS}}+\lambda G_{\delta,\mathrm{oblique}}, \quad
G_\omega = (1-\lambda)G_{\omega,\mathrm{4WS}}+\lambda G_{\omega,\mathrm{oblique}}.
$$
Table 7 Fuzzy rules for steering-mode switching
| \(\alpha_{\max}\) state / curvature state | Small curvature | Large curvature |
|---|---|---|
| Linear region | ordinary 4WS | oblique steering |
| Nonlinear region | ordinary 4WS | oblique steering |
| Saturation region | oblique steering | oblique steering |
According to the Magic Formula tyre curves, I classified the slip-angle state into three triangular membership functions: linear region from 0° to 2°, nonlinear transition from 1° to 4°, and saturation region beyond 3.5°. The curvature input was normalised in the range \(0-0.03\ \mathrm{m^{-1}}\). The fuzzy surface generated by the Mamdani inference engine is smooth, which avoids hard switching and reduces the bump in the steering command.
When the fuzzy controller decides that the vehicle is in a high-risk condition, the differential oblique mode is activated. Once activated, it remains active until the predicted roll angle falls below its threshold and the tyre slip angles return to the linear or nonlinear region. This hysteresis-like behaviour prevents frequent transitions when the path alternates between left and right bends, as in an S-shaped road.
4.3 Model Predictive Path-Tracking Controller
The lower-layer MPC controller tracks the reference path by solving a constrained finite-horizon optimal control problem at each time step. The prediction model is obtained from the linearised single-track vehicle model augmented with the path-tracking error dynamics. In the Frenet frame, the lateral deviation \(e_y\) and heading-error \(e_{\varphi}\) satisfy
$$
\dot{e}_y = V_x e_{\varphi}+V_y, \qquad \dot{e}_{\varphi}=\gamma – V_x\rho_{\mathrm{ref}}.
$$
The complete dynamic model for prediction is assembled as
$$
\dot{\boldsymbol{x}}_c = \boldsymbol{A}_c\boldsymbol{x}_c+\boldsymbol{B}_c u + \boldsymbol{d}_c,
$$
with \(\boldsymbol{x}_c=[\beta,\gamma,e_y,e_{\varphi}]^T\) and the front-wheel steering angle \(u=\delta_f\). The matrices are
$$
\boldsymbol{A}_c=\begin{bmatrix}
-\dfrac{k_f+k_r}{mV_x} & -\dfrac{k_f a-k_r b}{mV_x^2}-1 & 0 & 0\\
-\dfrac{k_f a-k_r b}{I_z} & -\dfrac{k_f a^2+k_r b^2}{I_z V_x} & 0 & 0\\
V_x & 0 & 0 & V_x\\
0 & 1 & 0 & 0
\end{bmatrix},
$$
$$
\boldsymbol{B}_c=\begin{bmatrix}
\dfrac{k_f}{mV_x}\\[2pt]
\dfrac{k_f a}{I_z}\\[2pt]
0\\[2pt]
0
\end{bmatrix},
\qquad
\boldsymbol{d}_c=\begin{bmatrix}
0\\[2pt]
0\\[2pt]
0\\[2pt]
-V_x\rho_{\mathrm{ref}}
\end{bmatrix}.
$$
The continuous system is discretised with a sampling period \(T_s=50\) ms. To embed the differential oblique mode into the path-tracking control, the rear-wheel angle is not an independent command but is obtained from the front-wheel angle through the mode-dependent geometric allocator discussed in Section 3. Therefore, the state-space model already reflects the influence of the desired sideslip posture: the body-sideslip state is not suppressed but guided by a reference value.
At each sampling time, I solve the following quadratic optimisation problem:
$$
\min_{\Delta U,\varepsilon}\sum_{i=1}^{N_p}\left\|\boldsymbol{\eta}_{t+i,t}-\boldsymbol{\eta}_{\mathrm{ref},t+i,t}\right\|^2_{\boldsymbol{Q}} + \sum_{i=0}^{N_c-1}\left\|\Delta U_{t+i,t}\right\|^2_{\boldsymbol{R}} + \rho_{\varepsilon}\varepsilon^2,
$$
subject to the following constraints:
$$
\begin{aligned}
& \Delta U_{\min} \le \Delta U \le \Delta U_{\max}, \\
& U_{\min} \le U \le U_{\max}, \\
& \boldsymbol{y}_{\min} – \boldsymbol{\varepsilon} \le \boldsymbol{Y} \le \boldsymbol{y}_{\max} + \boldsymbol{\varepsilon}, \\
& \varepsilon \ge 0,
\end{aligned}
$$
where \(\boldsymbol{\eta}=[e_y,e_{\varphi}]^T\), \(N_p\) is the prediction horizon, \(N_c\) is the control horizon, \(\boldsymbol{Q}\) and \(\boldsymbol{R}\) are weighting matrices, and \(\varepsilon\) is a slack variable used to soften the output constraints. In this paper, I set \(N_p=20\), \(N_c=5\), \(\boldsymbol{Q}=\operatorname{diag}(50,600)\), \(\boldsymbol{R}=5\times10^4\) and \(\rho_\varepsilon=1000\). The lower and upper limits of the wheel angle are defined by the steering actuator and by the roll-stability-angle constraint. The problem is converted to a standard quadratic programme and solved online with the active-set method.
At every control cycle, the first element of the optimal control sequence is applied to the vehicle:
$$
\delta_f(t)=\delta_f(t-T_s)+\Delta \delta_f^*(t).
$$
The rear-wheel commands are then generated by the selected allocation law. If \(\lambda=1\), the controller enters the differential oblique mode; if \(\lambda=0\), the ordinary four-wheel steering relation is used. Since the MPC objective already contains the path-tracking errors, the vehicle will automatically choose a larger body-sideslip reference when the path curvature is large, because a larger \(\beta\) reduces the lateral displacement error.
4.4 Simulation Results for Extreme Cornering of Electric Cars
4.4.1 High-Speed U-Shaped Bend on Medium-Friction Road
The first simulation scenario is a U-shaped bend with radius 100 m, road friction coefficient \(\mu=0.55\), and a constant longitudinal velocity of 80 km/h. I compared three steering modes: conventional front-wheel steering, ordinary four-wheel steering, and the proposed differential oblique steering mode. The simulation results are summarised in Table 8.
| Index | FWS | 4WS | Differential oblique |
|---|---|---|---|
| Maximum lateral tracking error | unstable | 2.90 m | 0.80 m |
| Maximum roll angle | – | 3.83 deg | 1.21 deg |
| Improvement in lateral error | – | baseline | 72.4% |
| Improvement in roll-angle peak | – | baseline | 68.3% |
In this situation, the front-wheel-steered vehicle could not track the reference path at all. Its rear axle lost lateral stability because the tyre slip angles exceeded the saturation value on the friction circle. The four-wheel-steered vehicle could complete the bend but exhibited an obvious overshoot, with the maximum lateral error of 2.9 m occurring during the transient phase. The differential oblique steering controller kept the lateral error below 0.8 m, which corresponds to a 72.4% improvement. More importantly, the body roll angle peak was reduced by 68.3% compared with ordinary four-wheel steering. The roll-angle reduction is exactly the expected result of the increased effective wheelbase in the oblique posture.
4.4.2 High-Speed S-Shaped Bend on Low-Friction Road
The second simulation scenario is a more demanding S-shaped lane-change path on a wet icy road with \(\mu=0.3\). The vehicle speed is still 80 km/h. The path consists of two consecutive bends in opposite directions, which poses a serious challenge to the switching logic. The performance comparison is shown in Table 9.
| Index | FWS | 4WS | Differential oblique |
|---|---|---|---|
| Maximum lateral tracking error | unstable | 7.21 m | 2.83 m |
| Maximum roll angle | – | 2.75 deg | 1.09 deg |
| Improvement in lateral error | – | baseline | 60.8% |
| Improvement in roll-angle peak | – | baseline | 60.4% |
On the low-friction road, the fuzzy mode-switching controller activated the oblique-steering mode during the sharp transitions. The sideslip angle reference was large and quickly changed sign; however, the roll angle peak remained at 1.09°, compared with 2.75° for ordinary four-wheel steering. The path-tracking error still reached 2.83 m, which is larger than in the U-shaped case, but this is reasonable because low friction fundamentally limits the maximum achievable lateral acceleration. In conclusion, the simulations confirm that the proposed controller provides better path-tracking accuracy and lower roll risk than both conventional steering strategies for high-speed electric cars on harsh roads.
5. Controller-in-the-Loop Verification
To assess the practical feasibility of the above controller, I built a controller-in-the-loop test platform. The main reason for using CIL rather than actual road testing is safety: emergency steering at 80 km/h on a low-friction surface can easily lead to a vehicle excursion or rollover if the control algorithm has any bug. In a CIL environment, the controller hardware runs the actual embedded code, while the virtual vehicle model runs on a real-time computer. This configuration prevents man-machine risk and allows repeated deterministic experiments.
5.1 Hardware and Real-Time Environment
The CIL platform consists of a dSPACE MicroAutoBox as the embedded controller, a PXI real-time system as the virtual vehicle host, a CAN communication bus and two host computers. A photograph and schematic of the platform are not reproduced here, but the data flow is described. The vehicle model compiled as a dynamic-link library is loaded into the PXI target. The dSPACE controller receives the vehicle states via a CAN database file, executes the fuzzy mode-switching and the MPC algorithm, and transmits the four wheel-angle commands back to the PXI vehicle model. To emulate the internal delays of a real chassis, CAN transmission delay is set to 10 ms, the controller sampling period is 1 ms, and the steering motor time constant is included in the wheel-steering actuator model.
All three CAN messages used in the experiment are listed in Table 10, Table 11 and Table 12. The first message carries the state-error vector, the second sends the longitudinal velocity input, and the third sends the four commanded wheel angles.
| Signal | Description | Unit | Resolution | Start bit | Length | Byte order |
|---|---|---|---|---|---|---|
| \(e_\beta\) | Sideslip-angle error | rad | 0.001 | 0 | 16 | Intel |
| \(e_\gamma\) | Yaw-rate error | rad/s | 0.0001 | 16 | 16 | Intel |
| \(e_y\) | Lateral displacement error | m | 0.1 | 32 | 16 | Intel |
| \(e_\varphi\) | Heading error | rad | 0.001 | 48 | 16 | Intel |
| Signal | Description | Unit | Resolution | Start bit | Length |
|---|---|---|---|---|---|
| \(V_x\) | Longitudinal speed | m/s | 0.001 | 0 | 16 |
| Signal | Description | Unit | Resolution | Start bit | Length |
|---|---|---|---|---|---|
| \(\delta_{fl}\) | Left-front wheel angle | rad | 0.001 | 0 | 16 |
| \(\delta_{fr}\) | Right-front wheel angle | rad | 0.001 | 16 | 16 |
| \(\delta_{rl}\) | Left-rear wheel angle | rad | 0.001 | 32 | 16 |
| \(\delta_{rr}\) | Right-rear wheel angle | rad | 0.001 | 48 | 16 |
5.2 CIL Results in the U-Shaped Bend Scenario
I repeated the U-shaped bend simulation on the CIL test bench with the same parameters as in Section 4.4.1. The measured maximum lateral error of differential oblique steering was 0.6 m, which is slightly smaller than the offline simulation value of 0.8 m; a small difference is expected because the discretisation and delay in the CIL environment alter the controller update phase. Ordinary four-wheel steering produced a maximum lateral error of 6.0 m. The body roll angle observed in the oblique mode was 1.24°, which is still considerably lower than that of ordinary four-wheel steering at 3.45°. The yaw rate and sideslip angle waveforms were consistent with the offline model, which demonstrates that the proposed algorithm is implementable on real-time embedded hardware.
5.3 CIL Results in the S-Shaped Bend Scenario.
In the second CIL test, the low-friction S-shaped bend was reproduced. The simulation time was 30 s, and the road friction coefficient was set to 0.3. The differential oblique steering controller again tracked the reference path better than the other two strategies. The maximum lateral error was 3.61 m, while ordinary four-wheel steering reached 10.3 m. The front-wheel-steered vehicle completely lost control and left the road after the first bend, so no reliable tracking-error value could be obtained. The roll angle of the oblique-steering car reached about 1.1° to 1.2° at the most critical transition, similar to the offline simulation.
The CIL tests validate that the proposed switching and differential oblique steering control strategy remains robust when realistic CAN delays and sampling-period constraints are present. Although the path-tracking performance is slightly degraded relative to the ideal offline model, the trend and final conclusions remain unchanged: differential oblique steering can significantly enhance both lateral tracking and roll stability for high-speed corner-module electric cars in emergency conditions.
6. Conclusions and Outlook
In this thesis, I have investigated the stability control of corner-module-architecture electric cars under extreme steering conditions and proposed a novel differential oblique steering approach. The main conclusions can be summarised as follows.
First, the twelve-degree-of-freedom vehicle model and the Magic Formula tyre model provide a reliable foundation for simulating the nonlinear behaviour of electric cars in emergency steering. The wheel-side steering system model and hub-motor model capture the actuator response characteristics and were validated against Carsim results with small discrepancies in the yaw rate, sideslip angle and roll angle.
Second, the wheel-angle allocation strategy for differential oblique steering has been established by geometric and dynamic analysis. In this mode, the rear wheels are steered with a larger angle than the front wheels; nevertheless, the steering directions of the four wheels remain consistent about the instantaneous turning centre. The vehicle actively adopts a large body-sideslip posture, which enlarges the effective wheelbase and reduces the body roll angle. This mechanism is fundamentally different from conventional four-wheel steering and gives electric cars a wider handling boundary.
Third, a layered control structure was designed for extreme steering on harsh roads. The fuzzy supervisor continuously evaluates the tyre slip state and the reference curvature and switches between ordinary four-wheel steering and differential oblique steering. The roll-stability layer based on road friction and body roll angle constrains the feasible sideslip and wheel-angle commands. The model predictive path-tracking controller tracks the reference path under rolling-horizon optimisation. In the simulated U-shaped and S-shaped emergency manoeuvres at 80 km/h, the maximum lateral error was improved by 60.8%–72.4% compared with ordinary four-wheel steering, while the peak body roll angle was reduced by 60.4%–68.3%. These results demonstrate that the proposed control scheme can effectively suppress the lateral-instability and rollover risk of electric cars in high-speed extreme cornering.
Finally, a controller-in-the-loop platform using dSPACE and PXI was established to emulate realistic CAN communication delays, sampling periods and actuator constraints. The CIL results for both the U-shaped and S-shaped scenarios show the same trends as offline simulation and confirm the practical reliability of the proposed strategy. The implementation details of the CAN messages and the timing characteristics provide useful guidance for subsequent hardware implementation.
There remain several limitations in this work which I intend to address in future studies. The desired body-sideslip angle used in the controller is currently chosen from a fixed reference law. A more complete rule for selecting the optimal sideslip reference according to visibility, passenger comfort and path curvature would extend the practical applicability. Moreover, the road friction coefficient and tyre cornering stiffness are fixed parameters in the current model; in real electric cars, these parameters vary with tyre wear, temperature, water-film thickness and other factors. Online estimation of time-varying friction and cornering stiffness should be integrated into the prediction model to make the differential oblique controller fully adaptive to complex road conditions. In addition, experimental implementation on a physical corner-module electric car prototype would provide final evidence of the benefits demonstrated in this study.
In summary, differential oblique steering is a promising unconventional steering mode for distributed-drive electric cars with corner-module architecture. It uses the redundancy of four-wheel independent steering not only to improve low-speed manoeuvrability but, more importantly, to stabilise high-speed extreme avoidances by actively shaping the vehicle body posture. Through simulations and controller-in-the-loop verification, I have shown that this method can improve the active-safety envelope of electric cars in severe driving conditions. The theoretical analysis and control framework presented here are expected to contribute to the development of future intelligent chassis systems for electric cars.