The global transformation toward electric and intelligent mobility has placed unprecedented demands on vehicle dynamic control. Distributed-drive electric vehicles, in which each wheel is driven by an in-wheel or wheel-side motor, provide a natural platform for precise tyre force control. When four-wheel independent steering is combined with this distributed architecture, the vehicle acquires redundant actuation in both longitudinal tyres and lateral wheel angles, enabling features such as in-situ steering, diagonal driving, and high-speed yaw stabilisation. However, the same features introduce strong coupling among longitudinal, lateral, and yaw motions. In this work, I focus on the integrated stability control of a distributed-drive electric vehicle with four-wheel independent steering. The central problem considered is how to judge the vehicle stability boundary accurately under varying road and driving conditions, and how to coordinate the independent steering angles and drive torques so that both trajectory tracking and yaw stability are preserved.
1. Introduction and Background
Electric vehicles have become a global mainstream direction due to concerns about energy security and carbon emissions. The Chinese “dual-carbon” strategy and the long-term plan for the new-energy-vehicle industry make it clear that full electrification and intelligent chassis control should be developed in parallel. Compared with conventional centralised powertrains, a distributed-drive system can independently control each wheel’s driving torque. Therefore, the response delay is smaller, the energy recovery efficiency can be higher, and the chassis layout freedom is enlarged. In particular, the four-motor four-wheel distributed architecture is regarded as a promising solution for future high-performance electric vehicles.
On the basis of distributed drives, four-wheel independent steering further increases the controllable degree of freedom. A conventional vehicle with front-wheel steering has difficulty in maintaining yaw stability near the tyre adhesion limit. Four-wheel independent steering can generate an additional yaw moment through the lateral tyre forces of the rear wheels. At low speed, the four wheels can be steered in opposite directions to turn around the vehicle centre with a near-zero radius, which greatly improves manoeuvrability in narrow spaces. At medium and high speed, the rear wheels can steer in phase with the front wheels, reducing the vehicle sideslip angle and enhancing lateral stability. At any speed, the independent front and rear wheel angles can be used to generate a lateral force margin that supports direct yaw-moment correction. For this reason, I have selected the four-wheel-independent-steering distributed electric vehicle as the principal research target.
Many researchers have investigated distributed-drive electric vehicles and four-wheel independent steering. Early prototypes such as the Lohner-Porsche electric vehicle used wheel-hub motors more than a century ago. Modern concepts introduced by several motor companies have demonstrated in-situ steering, transverse parking, and crab-like motion. In addition, numerous papers examine vehicle stability control using active front steering, direct yaw-moment control, torque vectoring, and integrated chassis control. Nevertheless, there remain two key issues that are not fully solved.
The first issue is the vehicle stability judgement method. The conventional fixed-threshold method uses constant upper and lower limits of the yaw-rate error or sideslip-angle error. It is simple but cannot adapt to changing road friction, vehicle speed, or steering-frequency. The phase-plane method represents the stable region with a fixed boundary in the \(\beta\)–\(\dot{\beta}\) or \(\beta\)–\(\omega_r\) plane, but the shape of the boundary changes with road surface and velocity. Therefore, I propose a dynamic-threshold stability judgement method. The threshold values of the yaw-rate error and the sideslip-rate error are tuned online using fuzzy control based on road adhesion, steering-angle change rate, vehicle speed, and lateral acceleration.
The second issue is the coordination of longitudinal and lateral tyre forces in the lower control-allocation layer. Many existing methods minimise the longitudinal tyre workload only or treat the longitudinal and lateral tyre forces separately. However, because of the tyre friction ellipse, the longitudinal force and lateral force on every wheel are strongly coupled. Neglecting this coupling may generate an allocation result that exceeds the physical tyre adhesion limit, especially on low-friction roads. In this thesis, I formulate a multi-constraint allocation problem in which the objective is to minimise the sum of squared tyre workload factors, while considering the tyre-friction ellipse, motor torque limits, and the required longitudinal force, lateral force, and yaw moment. The lateral force demanded from each tyre is then transformed into a wheel steering angle through an inverse tyre model.
The main contributions of this thesis are summarised in the following points. First, an accurate seven-degree-of-freedom vehicle model is developed and verified with a commercial simulation platform. Second, a dynamic-threshold stability judgement method is designed using fuzzy logic. Third, a three-layer hierarchical control architecture is constructed, including a stability-analysis and judgement layer, a control-variable-calculation layer, and an execution-allocation layer. Fourth, the proposed scheme is systematically evaluated under in-situ steering, diagonal driving, high-speed double lane-change, and low-adhesion double lane-change conditions.
To position my work, I summarise several representative existing stability-judgement methods and their meaningful limitations in Table 1.
| Approach | Typical criterion | Advantage | Limitation |
|---|---|---|---|
| Fixed-threshold | \(|\omega_r-\omega_{r,d}| \le \Delta\omega\) | Simple, low-cost | One threshold cannot match all road and speed conditions |
| Phase plane | \(\beta\)–\(\dot{\beta}\) envelope | Graphically intuitive | Boundary is quasi-static; difficult to update in real time |
| Dynamic-threshold | \(|\omega_r-\omega_{r,d}| \le \Delta\omega(v_x,\mu,\cdots)\) | Adaptable to operation condition | Need reliable decision rule |
2. Vehicle Dynamics Modelling
An accurate vehicle model is the foundation for control design and validation. In this section, I describe a seven-degree-of-freedom (7-DOF) vehicle model that includes longitudinal, lateral, and yaw motion of the sprung mass plus rotational motion of the four wheels. The model is intentionally simplified by ignoring suspension vibration, roll, pitch, and vertical motion. The input quantities to the model are the four wheel steering angles and the four wheel torques. Figure 2 of the thesis shows a vehicle plane model; the corresponding body equations are given below.
2.1 Body and wheel dynamics
Let \(v_x\) and \(v_y\) be the longitudinal and lateral velocities of the centre of mass, and let \(\omega_r\) be the yaw rate. The tyre forces are decomposed along the body axes. The longitudinal and lateral equilibrium equations are
$$
m(\dot{v}_x – \omega_r v_y) = F_{xfl}+F_{xfr}+F_{xrl}+F_{xrr},
$$
$$
m(\dot{v}_y + \omega_r v_x) = F_{yfl}+F_{yfr}+F_{yrl}+F_{yrr}.
$$
The yaw-moment equilibrium around the vertical axis is written as
$$
I_z \dot{\omega}_r =
(F_{xfr}+F_{xrr}-F_{xfl}-F_{xrl})b
+ l_f(F_{yfl}+F_{yfr})
– l_r(F_{yrl}+F_{yrr}),
$$
where \(I_z\) is the yaw moment of inertia, \(l_f\) and \(l_r\) are the distances from the centre of mass to the front and rear axles, \(b\) is half of the track width, and \(F_{xij}\), \(F_{yij}\) are the body-axis tyre forces for the \(ij\) wheel, with \(ij \in \{fl, fr, rl, rr\}\).
The wheel rotation dynamics are governed by
$$
I_w \dot{\omega}_{ij} = T_{ij} – r F_{xij},
$$
where \(I_w\) is the wheel rotational inertia, \(T_{ij}\) is the net wheel torque, \(r\) is the effective rolling radius, and \(\omega_{ij}\) is the wheel rotational speed.
The body-axis forces are obtained from the tyre-axis forces through the wheel steering angle \(\delta_{ij}\):
$$
F_{xij}=F_{txij}\cos \delta_{ij}-F_{tyij}\sin \delta_{ij},
$$
$$
F_{yij}=F_{txij}\sin \delta_{ij}+F_{tyij}\cos \delta_{ij}.
$$
The vertical tyre load is an important input to the tyre force model. Considering longitudinal and lateral load transfer, I use the following steady-state vertical-load expressions:
$$
F_{zfl}=\frac{m g l_r}{l_f+l_r}
-\frac{m a_x h}{2(l_f+l_r)}
-\frac{m a_y h l_r}{2b(l_f+l_r)},
$$
$$
F_{zfr}=\frac{m g l_r}{l_f+l_r}
-\frac{m a_x h}{2(l_f+l_r)}
+\frac{m a_y h l_r}{2b(l_f+l_r)},
$$
$$
F_{zrl}=\frac{m g l_f}{l_f+l_r}
+\frac{m a_x h}{2(l_f+l_r)}
-\frac{m a_y h l_f}{2b(l_f+l_r)},
$$
$$
F_{zrr}=\frac{m g l_f}{l_f+l_r}
+\frac{m a_x h}{2(l_f+l_r)}
+\frac{m a_y h l_f}{2b(l_f+l_r)}.
$$
Here \(h\) is the centre-of-gravity height and \(a_x\), \(a_y\) are the longitudinal and lateral accelerations.
2.2 Tyre model
I employ the well-known Magic Formula tyre model to reproduce nonlinear tyre characteristics. The general formula can be written as
$$
y(x)=D \sin\left\{C \arctan\left[Bx-E\left(Bx-\arctan(Bx)\right)\right]\right\},
$$
where \(D\) is the peak factor, \(C\) is the shape factor, \(B\) is the stiffness factor, and \(E\) is the curvature factor. For pure longitudinal slip, the driving/braking force is obtained by substituting the slip ratio \(\lambda_{ij}\) into the formula together with the coefficients \(a_0\)–\(a_8\). For pure lateral slip, the cornering force is obtained from the slip angle \(\alpha_{ij}\) using coefficients \(b_0\)–\(b_8\). The lateral slip angle of each wheel is computed as
$$
\alpha_{fl} = \delta_{fl}-\tan^{-1}\left(\frac{v_y+l_f \omega_r}{v_x-b \omega_r}\right),
$$
$$
\alpha_{fr} = \delta_{fr}-\tan^{-1}\left(\frac{v_y+l_f \omega_r}{v_x+b \omega_r}\right),
$$
$$
\alpha_{rl} = \delta_{rl}-\tan^{-1}\left(\frac{v_y-l_r \omega_r}{v_x-b \omega_r}\right),
$$
$$
\alpha_{rr} = \delta_{rr}-\tan^{-1}\left(\frac{v_y-l_r \omega_r}{v_x+b \omega_r}\right).
$$
At combined slip, when a tyre is both slipping and sideslipping, I use the total slip approach to limit the resultant force inside the adhesion boundary. The vehicle parameters used in the simulation model are summarised in Table 2.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Vehicle mass \(m\) | 1413 kg | CG height \(h\) | 0.54 m |
| Front axle distance \(l_f\) | 1.015 m | Wheel radius \(r\) | 0.325 m |
| Rear axle distance \(l_r\) | 1.895 m | Track width \(t_w\) | 1.675 m |
| Wheelbase \(L\) | 2.91 m | Yaw inertia \(I_z\) | 1537.6 kg m² |
2.3 Drive system model
The driven wheels are equipped with permanent-magnet synchronous motors. Because the motor current-control loop is much faster than the vehicle dynamics, I approximate the dynamic relation between the demanded torque \(T_m^*\) and the actual torque \(T_m\) with a second-order response of the form
$$
G(s)=\frac{T_m(s)}{T_m^*(s)}
=\frac{1}{\tau_m^2 s^2+2\zeta_m \tau_m s+1},
$$
where \(\tau_m\) is the motor time constant and \(\zeta_m\) is the damping coefficient. To follow the desired longitudinal speed, I design a PID speed controller as shown in the thesis structure. The PID output represents the acceleration command or the total longitudinal force demand required by the upper-level controller.
2.4 Model verification
Before the control system is developed, the 7-DOF vehicle model must be compared with a benchmark vehicle model. I therefore create a co-simulation environment consisting of MATLAB/Simulink and the commercial software CarSim. The benchmark CarSim model is a high-fidelity full-vehicle model, whereas my 4WIS model inherits the same gross parameters. In the verification manoeuvre, the vehicle speed is 36 km/h and the front-wheel steering angle is a sine wave with amplitude 0.05 rad and period \(\pi\) s. The simulation lasts 10 s.
The comparison result is shown in Fig. 4 of the thesis but not repeated numerically here. The simulation outputs of speed, sideslip angle, and yaw rate are highly consistent between the 7-DOF model and the CarSim model. Small discrepancies are caused by the neglected suspension kinematics and roll dynamics. Therefore, the 7-DOF model is considered accurate enough for control synthesis and evaluation.
3. Integrated Control Strategy Design
This section describes the complete controller architecture. The control system consists of three layers. The first layer is the stability-analysis and judgement layer. It uses a linear two-degree-of-freedom vehicle model as an ideal reference, calculates the ideal yaw rate and ideal sideslip rate, and compares the actual values with the ideal values. The comparison errors are then checked against fuzzy dynamic thresholds. According to the result, the controller selects either trajectory-tracking control or active stability control.
When the vehicle is judged to be stable, a model predictive controller is used for lateral trajectory tracking while the PID controller tracks the longitudinal speed. When the vehicle is judged to be unstable, a second-order sliding-mode controller is activated to calculate the additional yaw moment. The lower allocation layer converts the required forces into the optimal longitudinal and lateral forces of the four tyres; these forces are then converted to drive torques and steering angles using the inverse tyre model.
3.1 Dynamic-threshold stability judgement
I choose two state variables to indicate instability: the yaw-rate error and the sideslip-rate error. The actual yaw rate is \(\omega_r\); its desired value is \(\omega_{r,ref}\). The actual sideslip rate is \(\dot{\beta}\), while its desired value is \(\dot{\beta}_{ref}\). Hence, the error variables are
$$
e_{\omega_r} = \omega_r-\omega_{r,ref},
$$
$$
e_{\dot{\beta}} = \dot{\beta}-\dot{\beta}_{ref}.
$$
The vehicle is considered stable when both errors satisfy
$$
|e_{\omega_r}| \le \Delta\omega_{r,th},
$$
$$
|e_{\dot{\beta}}| \le \Delta\dot{\beta}_{th}.
$$
If one of the inequalities is violated, the stability control mode is activated. The thresholds \(\Delta\omega_{r,th}\) and \(\Delta\dot{\beta}_{th}\) are not fixed. They are adjusted according to the instantaneous driving condition.
The desired yaw rate is obtained from the ideal linear bicycle model with a steady-state steering response. For a front steering angle \(\delta_f\), the desired yaw rate is bounded by the road adhesion limit:
$$
\omega_{r,ref} = \min\left\{\left|\frac{v_x}{(l_f+l_r)(1+K v_x^2)}\delta_f\right|,\;
\left|\frac{0.85\mu g}{v_x}\right|\right\}
\operatorname{sgn}(\delta_f),
$$
where \(K\) is the stability factor and \(\mu\) is the road friction coefficient. Similarly, the desired sideslip angle is bounded by
$$
\beta_{ref} = \min\left\{\left|\beta_d\right|,\;\left|\arctan(0.02\mu g)\right|\right\}
\operatorname{sgn}(\beta_d).
$$
The desired sideslip rate \(\dot{\beta}_{ref}\) is obtained by differentiating \(\beta_{ref}\). This reference model is used not only for stability judgement but also for the sliding-mode controller.
Since the mapping from driving conditions to stability thresholds is highly nonlinear and difficult to model analytically, I design two fuzzy controllers. The first fuzzy controller determines \(\Delta\omega_{r,th}\). Its inputs are the road adhesion coefficient \(\mu\) and the absolute change rate of the steering angle \(|\dot{\delta}_f|\). The second fuzzy controller determines \(\Delta\dot{\beta}_{th}\). Its inputs are the vehicle speed \(v_x\) and the lateral acceleration \(a_y\). All inputs and outputs are normalised to seven fuzzy sets.
Table 3 summarises the key variables of the fuzzy threshold generators.
| Threshold | Inputs | Output universe | Fuzzy sets |
|---|---|---|---|
| \(\Delta\omega_{r,th}\) | \(\mu\), \(|\dot{\delta}_f|\) | \([0,8]\) | NB,NM,NS,ZE,PS,PM,PB |
| \(\Delta\dot{\beta}_{th}\) | \(v_x\), \(a_y\) | \([0,25]\) | NB,NM,NS,ZE,PS,PM,PB |
The fuzzy rules are formulated according to the following physical logic. On a low-friction road, a smaller threshold should be used because a small yaw-rate error is already dangerous at high speed. On a high-friction road and low speed, the threshold can be relaxed, allowing the trajectory-tracking controller to be used for a larger region. Tables 4 and 5 present examples of the fuzzy rule tables for \( \Delta\omega_{r,th} \) and \( \Delta\dot{\beta}_{th} \).
| \(\mu\) / \(|\dot{\delta}_f|\) | NB | NM | NS | ZE | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | P1 | P1 | P1 | P2 | P2 | P3 | P3 |
| NM | P1 | P1 | P2 | P2 | P3 | P3 | P4 |
| NS | P1 | P2 | P2 | P3 | P3 | P4 | P4 |
| ZE | P2 | P2 | P3 | P3 | P4 | P4 | P5 |
| PS | P2 | P3 | P3 | P4 | P4 | P5 | P5 |
| PM | P3 | P3 | P4 | P4 | P5 | P5 | P6 |
| PB | P3 | P4 | P4 | P5 | P5 | P6 | P7 |
| \(a_y\) / \(v_x\) | NB | NM | NS | ZE | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | P7 | P7 | P6 | P6 | P5 | P4 | P3 |
| NM | P5 | P4 | P4 | P3 | P3 | P2 | P2 |
| NS | P4 | P4 | P3 | P3 | P2 | P2 | P2 |
| ZE | P4 | P3 | P3 | P2 | P2 | P1 | P1 |
| PS | P3 | P2 | P2 | P1 | P1 | P1 | P1 |
| PM | P2 | P2 | P2 | P1 | P1 | P1 | P1 |
| PB | P1 | P1 | P1 | P1 | P1 | P1 | P1 |
In these tables, the linguistic output values P1 to P7 correspond to monotonically increasing magnitudes of the threshold, from the smallest threshold P1 to the largest threshold P7. In the second frame, the desired yaw rate is filtered through a non-linear condition to avoid sign errors; centre-gravity-centre sideslip rate is used as an extra stability indicator. The fuzzy inference process uses a centre-of-gravity defuzzification method.
3.2 MPC trajectory tracking controller
When the dynamic-threshold judgement says that the vehicle is still inside the stable region, I apply a model predictive control algorithm for trajectory tracking. The main advantage of MPC is its ability to handle multiple constraints on steering angle, steering increment, and system output. In this study, the nonlinear 4WIS vehicle model is linearised along the operating trajectory and then discretised.
The state vector and the control vector are defined as
$$
\xi = [v_y,\; \omega_r,\; \varphi,\; Y]^T,
\qquad
u = [\delta_f,\; \delta_r]^T,
$$
where \(\varphi\) is the yaw angle and \(Y\) is the lateral position. A linear time-varying state equation is obtained by linearisation:
$$
\dot{\xi}(t)=A(t)\xi(t)+B(t)u(t).
$$
After discretisation with sampling period \(T\), the system becomes
$$
\xi(k+1)=A(k)\xi(k)+B(k)u(k).
$$
To limit control increments, I extend the state vector with the previous control input and take the control increment \(\Delta u\) as the new control variable. The objective function is designed as
$$
J(k)=
\sum_{i=1}^{N_p}\left\|\eta(k+i|k)-\eta_{ref}(k+i|k)\right\|_Q^2
+
\sum_{i=0}^{N_c-1}\left\|\Delta u(k+i|k)\right\|_R^2
+\rho \varepsilon^2,
$$
where \(N_p\) is the prediction horizon, \(N_c\) is the control horizon, \(Q\) is the state-error weighting matrix, \(R\) is the control-increment weighting matrix, \(\rho\) is the penalty weight, and \(\varepsilon\) is a slack variable introduced for constraint softening. The trajectory-tracking problem is then converted into the following constrained quadratic-programming problem:
$$
\min_{\Delta u,\varepsilon} \; J(k)
$$
subject to the steering-angle bound \(u_{\min} \le u \le u_{\max}\), increment bound \(\Delta u_{\min} \le \Delta u \le \Delta u_{\max}\), output bound \(Y_{\min}-\tau \le Y \le Y_{\max}+\tau\), and the linearised vehicle dynamic constraints. The solution is computed online with the MATLAB quadprog solver. Only the first control increment of the optimal sequence is applied to the vehicle; the optimisation is repeated at the next sampling instant.
3.3 Sliding-mode direct yaw moment controller
If the vehicle state lies outside the stable region, the yaw-rate reference and sideslip-angle reference can no longer be achieved with steering alone. The controller therefore computes an additional yaw moment \(\Delta M_z\) to force the vehicle state back toward the stable region. I choose a sliding-mode controller because it is robust against model uncertainty, parameter variation, and external disturbances.
The yaw-rate tracking error is denoted by \(e_{\omega_r} = \omega_r-\omega_{r,ref}\), and the sideslip-rate tracking error by \(e_{\dot\beta}=\dot\beta-\dot\beta_{ref}\). A second-order sliding surface is defined as
$$
s = \lambda e_{\omega_r} + (1-\lambda) e_{\dot\beta},
$$
where \(\lambda \in (0,1]\) is a weighting parameter. The equivalent control necessary to maintain \(s=0\) is derived from the nominal vehicle model, while a switching term is added for robustness. In practice, the discontinuous sign function is replaced by the saturation function \(\operatorname{sat}(s/\Delta)\) to attenuate chattering:
$$
\operatorname{sat}\left(\frac{s}{\Delta}\right)=
\begin{cases}
1 & s>\Delta,\\
s/\Delta & |s|\le\Delta,\\
-1 & s<-\Delta,
\end{cases}
$$
where \(\Delta>0\) is the boundary-layer thickness. The total additional yaw moment is expressed as
$$
\Delta M_z = \Delta M_{z,eq} – K_v \, I_z \operatorname{sat}\left(\frac{s}{\Delta}\right),
$$
where \(K_v>0\) is the switching gain. The equivalent part compensates the known vehicle yaw dynamics obtained from the 2DOF linear reference model, while the switching part ensures convergence. To prove the closed-loop stability, I define the Lyapunov function \(V=s^2/2\). Its derivative satisfies
$$
\dot{V}=s\dot{s}=-K_v s \operatorname{sat}\left(\frac{s}{\Delta}\right)\le 0,
$$
which confirms the reachability condition of the sliding surface. Hence, the direct yaw moment controller is asymptotically stable.
3.4 Lower-level force allocation
The lower layer receives the desired longitudinal force determined from the speed controller and the additional yaw moment from the sliding-mode controller. It must allocate the required forces to the four tyres. In this thesis, I formulate an optimisation problem with the objective of minimising the tyre utilisation. The tyre workload is defined as
$$
\eta_{ij}=\frac{\sqrt{F_{xij}^2+F_{yij}^2}}{\mu_{ij}F_{zij}}.
$$
The optimisation objective is therefore
$$
\min \;
J_{alloc}=\sum_{ij}\frac{F_{xij}^2+F_{yij}^2}{(\mu_{ij}F_{zij})^2},
$$
where \(\mu_{ij}\) is the road friction coefficient of that corner and \(F_{zij}\) is the vertical load. Two equality constraints must be satisfied. The first is the total longitudinal force demand \(F_{xd}\):
$$
F_{xfl}+F_{xfr}+F_{xrl}+F_{xrr}=F_{xd}.
$$
The second is the yaw-moment balance, which must include both the longitudinal tyre-force difference and the lateral tyre-force arm:
$$
b(F_{xfr}+F_{xrr}-F_{xfl}-F_{xrl})
+l_f(F_{yfl}+F_{yfr})
-l_r(F_{yrl}+F_{yrr})
=M_{z,d}+\Delta M_z.
$$
The inequality constraints are essential for practical feasibility. First, each tyre force vector must remain inside the friction ellipse:
$$
F_{xij}^2+F_{yij}^2 \le (\mu_{ij}F_{zij})^2.
$$
Second, the drive-torque limit is transferred into a longitudinal-force limit:
$$
-\frac{T_{\max,ij}}{r} \le F_{xij} \le \frac{T_{\max,ij}}{r}.
$$
Third, the tyre lateral force should be bounded so that the inverse tyre model remains in a reasonable operating range:
$$
F_{yij,\min} \le F_{yij} \le F_{yij,\max}.
$$
Because the objective function is quadratic and the constraints are linear or convex, the allocation problem can be solved by the quadratic-programming method. I transform the equations into the standard QP form in MATLAB and solve them with the quadprog solver.
3.5 Inverse tyre model
The outputs of the allocation layer are one longitudinal force and one lateral force for every wheel. The longitudinal force can be tracked by the in-wheel motor torque directly. The lateral force, however, cannot be applied directly. I therefore derive an inverse tyre model that maps the required lateral force \(F_{yij}\) and vertical load \(F_{zij}\) to a tyre slip angle \(\alpha_{ij}\). The simplified forward tyre model is written as
$$
F_y = \frac{\mu}{k} C_{\alpha} G_x \tan\left(\frac{k}{\mu}\alpha\right),
$$
where \(C_{\alpha}\) is the cornering stiffness, \(k\) is a tyre parameter, and \(G_x\) is a combined-slip correction factor:
$$
G_x = \sqrt{1-\left(\frac{F_x}{\mu F_z}\right)^2}.
$$
Solving the above equation for \(\alpha\) gives the inverse expression:
$$
\alpha_{ij}
=
\frac{\mu}{k}
\arctan
\left(
\frac{k F_{yij}}
{\mu C_{\alpha} G_x}
\right).
$$
Finally, the wheel steering angle demanded by the controller is determined from the kinematic slip relation:
$$
\delta_{ij} = \alpha_{ij}
+ \arctan\left(\frac{v_y \pm l_{f/r}\omega_r}{v_x \mp b \omega_r}\right),
$$
where the sign depends on the wheel position. The tyre inverse model was verified under two operating conditions: one on a road with \(\mu=0.8\) and a vertical load of \(4000\,\mathrm{N}\), and the other on a road with \(\mu=0.2\) and a vertical load of \(5000\,\mathrm{N}\). The computed slip angles were compared with the values from the CarSim tyre model. The curves agree closely in both conditions, confirming the inverse model is suitable for the four-wheel independent steering system.
4. Simulation Results and Analysis
The final control architecture was evaluated with MATLAB/Simulink and CarSim co-simulation. Four representative driving scenarios are selected to assess the controller over a wide operation envelope: in-situ steering, diagonal driving, high-speed double lane change, and low-adhesion double lane change. The first two scenarios mainly test the low-speed flexibility and special functions of four-wheel independent steering. The last two scenarios evaluate trajectory-tracking accuracy and yaw stability in extreme manoeuvres.
For the in-situ steering test, I compare pure-drive mode with a “steering plus drive” mode that is produced by the proposed controller. For the other tests, the proposed four-wheel-independent-steering control is compared against two baseline controllers: front-wheel steering with equal torque distribution, and conventional four-wheel steering with a fixed proportional rear-angle relation. Table 6 summarises the basic simulation settings of the four scenarios.
| Scenario | Speed | Friction \(\mu\) | Manoeuvre | Major evaluation indexes |
|---|---|---|---|---|
| In-situ steering | 0 km/h start | 0.9 | rotate 180° about CG | heading angle, lateral drift, tyre workload |
| Diagonal driving | 36 km/h | 0.9 | single lane-change lateral shift | trajectory error, yaw rate, heading angle |
| High-speed double lane change | 72 km/h | 0.9 | double lane-change obstacle avoidance | trajectory error, yaw rate, sideslip rate |
| Low-adhesion double lane change | 36 km/h | 0.3 | double lane-change | trajectory error, sideslip angle, stability thresholds |
4.1 In-situ steering
In-situ steering is an important function enabled by the four-wheel independent steering architecture. In the simulation, the initial vehicle speed is zero. The desired final heading angle is 180 degrees. The proposed controller is allowed to use both the drive torques and the four wheel steering angles.
In pure-drive mode, the vehicle cannot complete the 180-degree rotation quickly; large heading-angle errors and large lateral drift appear. In contrast, the “steering plus drive” mode uses the front and rear wheels in opposite directions to create a pure yaw moment around the vehicle centre. The heading angle reaches 180 degrees in approximately 13.67 s, whereas pure-drive mode requires more than 25 s. Therefore, the completion time is reduced by 45.31%.
Moreover, the maximum lateral displacement of the front-axle centre is reduced by 1.72%, the peak yaw-rate error is reduced by 91.38%, and the peak sideslip-rate error is reduced by 62.78% relative to pure-drive mode. The tyre workload remains low and balanced because the proposed allocation strategy distributes the force according to vertical load and adhesion margin rather than sliding around the contact patch. Table 7 provides the numerical results of this comparison.
| Metric | Pure drive | Steering + drive | Improvement |
|---|---|---|---|
| Completion time (s) | 25 | 13.6725 | 45.31% faster |
| Peak front-axle lateral drift (m) | 0.9677 | 0.9511 | 1.72% lower |
| Peak sideslip-rate error (rad/s) | 0.1690 | 0.0629 | 62.78% lower |
| Peak yaw-rate error (rad/s) | 0.1415 | 0.0122 | 91.38% lower |
It is interesting to observe that the yaw-rate error of the pure-drive mode exceeds the fuzzy dynamic threshold quickly after the command is applied. As a result, the pure-drive mode is classified as unstable. The proposed combined steering-and-drive mode, on the other hand, keeps both the yaw-rate error and the sideslip-rate error within the threshold boundary during the whole rotation process. The vehicle rotates around a stable vertical axis and only a very small translational drift is observed. This confirms that the dynamic-threshold stability judgement and the lower allocation law work together properly.
4.2 Diagonal driving
Diagonal driving, or crab-like motion, is another special function of four-wheel independent steering. The vehicle should move along a straight line inclined to the vehicle longitudinal axis while maintaining a nearly zero heading angle. This is useful for parallel parking and avoiding obstacles in narrow road spaces. The simulation speed is 36 km/h and the road friction coefficient is 0.9.
The proposed controller follows a target path with a lateral offset. The results are compared with conventional front-wheel steering and conventional four-wheel steering. The maximum trajectory error is only 0.0937 m under the proposed scheme, whereas the front-wheel-steering vehicle has a maximum error of 0.1676 m and the conventional four-wheel-steering vehicle has a maximum error of 0.1325 m. Thus, the trajectory-tracking accuracy is improved by at least 44.09% relative to the conventional four-wheel steering and by an even larger margin against front-wheel steering.
The yaw-rate peak under the proposed scheme is 0.0244 rad/s, which is 79.39% lower than that of the conventional four-wheel-steering vehicle. The sideslip-angle peak under the proposed scheme is 0.0032 rad, 79.75% lower than front-wheel steering. More importantly, the heading-angle deviation is kept below about 1.62 degrees during the diagonal move. In the baseline schemes, however, the vehicle body rotates by roughly 10 degrees because the tyre lateral forces are not coordinated. Table 8 compares these results.
| Metric | Front-wheel steering | Conventional 4WS | Proposed 4WIS | Maximum improvement |
|---|---|---|---|---|
| Peak trajectory error (m) | 0.1676 | 0.1325 | 0.0937 | 44.09% lower |
| Peak yaw rate (rad/s) | 0.1117 | 0.1184 | 0.0244 | 79.39% lower |
| Peak sideslip angle (rad) | 0.0158 | 0.0099 | 0.0032 | 79.75% lower |
| Peak heading-angle error (deg) | 9.9639 | 10.0755 | 1.6223 | 83.90% lower |
The fuzzy threshold plots in this scenario reveal that the baseline front-wheel-steering and conventional-four-wheel-steering schemes repeatedly violate the dynamic thresholds, indicating temporary yaw instability. The proposed scheme remains below the thresholds during the whole simulation. The tyre workload of the four tyres is also well balanced. Therefore, diagonal driving is achieved with negligible body rotation and excellent lateral stability.
4.3 High-speed double lane change
High-speed double lane change is a classical test for vehicle handling stability. The vehicle speed is set to 72 km/h and the road friction is 0.9. The desired path represents an emergency obstacle avoidance manoeuvre followed by a return to the original lane.
Compared with the baseline front-wheel-steering vehicle, the proposed controller reduces the maximum trajectory error from 0.1428 m to 0.0914 m, a 35.99% improvement. The peak yaw rate is decreased by 16.75% relative to front-wheel steering and by 3.24% relative to conventional four-wheel steering. The peak sideslip angle is reduced by 54.73% relative to conventional four-wheel steering. The improvement of motion stability is more obvious because the yaw-rate-error peak is reduced by 50.16% and the sideslip-rate-error peak by 63.79% with respect to the conventional four-wheel-steering baseline.
The wheel steering angles produced by the proposed low-level controller are smooth and free from severe oscillation. Therefore, the vehicle follows the lane-change path with a more natural body attitude. The tyre workload peak of the proposed vehicle is around 65.31%, which means that the tyres still have a reserve of grip under this high-speed manoeuvre. Table 9 lists the numeric comparison.
| Metric | Front-wheel steering | Conventional 4WS | Proposed 4WIS | Maximum improvement |
|---|---|---|---|---|
| Peak trajectory error (m) | 0.1428 | 0.1108 | 0.0914 | 35.99% lower |
| Peak yaw rate (rad/s) | 0.2830 | 0.2435 | 0.2356 | 16.75% lower |
| Peak sideslip angle (rad) | 0.0101 | 0.0284 | 0.0129 | 54.73% lower |
| Peak yaw-rate error (rad/s) | 0.1651 | 0.2530 | 0.1261 | 50.16% lower |
| Peak sideslip-rate error (rad/s) | 0.0667 | 0.1149 | 0.0416 | 63.79% lower |
Close inspection of the threshold plots shows that both baselines exceed the dynamic threshold during the first and second lane-change transitions. The front-wheel-steering vehicle suffers from a delayed yaw response; the conventional four-wheel-steering vehicle has a large sideslip-angle overshoot because its rear steering angle follows the front angle by a fixed ratio without considering the actual state. The proposed independent-steering controller is able to assign a different rear steering angle on each side, and therefore it generates a corrective yaw moment that is matched to the current vehicle states. This is the key reason for the improved stability.
4.4 Low-adhesion double lane change
Low-friction roads are the most challenging condition for vehicle stability control. A small tyre-force error can cause a large sideslip angle or even spin. In this scenario, the road friction coefficient is 0.3 and the vehicle speed is 36 km/h. The desired path is the same double lane-change path used in the high-speed scenario.
The proposed controller achieves an extremely small trajectory error of 0.0144 m. The front-wheel-steering vehicle has an error of 0.2268 m, and the conventional-four-wheel-steering vehicle has an error of 0.1767 m. Therefore, the proposed scheme improves trajectory-tracking accuracy by 93.65% compared with front-wheel steering and by 91.85% compared with conventional four-wheel steering. The peak yaw rate is reduced by 25.92% compared with conventional four-wheel steering, and the peak sideslip angle is reduced by 39.06% compared with front-wheel steering.
From a stability-judgement viewpoint, the proposed control law maintains the yaw-rate error and sideslip-rate error within the dynamic threshold for the entire manoeuvre. It is interesting that the conventional-four-wheel-steering baseline has a large yaw-rate-error peak of 0.0474 rad/s, which is much larger than the front-wheel steering baseline. This is because the fixed-ratio rear steering without allocation can degrade yaw damping on a low-friction road. The independent control in the proposed method determines the rear-wheel steering angle and drive torque on each side so that the active yaw moment provides a damping effect. The tyre workload under the proposed scheme is also balanced, with a maximum value near 30.06%, preventing any single wheel from exceeding the friction ellipse.
A summary of the low-adhesion results is given in Table 10.
| Metric | Front-wheel steering | Conventional 4WS | Proposed 4WIS | Maximum improvement |
|---|---|---|---|---|
| Peak trajectory error (m) | 0.2268 | 0.1767 | 0.0144 | 93.65% lower |
| Peak yaw rate (rad/s) | 0.1093 | 0.1655 | 0.1226 | 25.92% lower |
| Peak sideslip angle (rad) | 0.0128 | 0.0101 | 0.0078 | 39.06% lower |
| Peak yaw-rate error (rad/s) | 0.0329 | 0.0474 | 0.0256 | 45.99% lower |
| Peak sideslip-rate error (rad/s) | 0.0075 | 0.0064 | 0.0040 | 46.67% lower |
This scenario clearly demonstrates the benefit of the dynamic stability judge and the low-level constrained allocation. When the road friction coefficient is low, the dynamic threshold automatically becomes smaller. Hence, the sliding-mode controller intervenes earlier and prevents a large sideslip angle from building up. The trajectory-tracking controller is still used when the errors are sufficiently small, but it is not allowed to excite the vehicle with aggressive steering increments. This seamless mode switching is possible because of the first-layer fuzzy threshold generator.
5. Conclusion and Outlook
In this thesis, I have investigated the integrated stability and trajectory-tracking control of distributed-drive electric vehicles with four-wheel independent steering. The main conclusions can be summarised as follows.
First, a seven-degree-of-freedom vehicle model was constructed, which contains longitudinal, lateral, yaw, and four-wheel rotational dynamics. The nonlinear tyre forces were modelled with the Magic Formula. The model was supplied with a second-order electro-mechanical motor model and verified against a high-fidelity CarSim vehicle model under a sinusoidal steering manoeuvre. The model responses agree well with the benchmark model, so the model can provide a reliable test platform for controller evaluation.
Second, I proposed a fuzzy dynamic threshold method to judge the vehicle stability boundary. Instead of setting one fixed yaw-rate error threshold and one fixed sideslip-rate threshold, the thresholds are updated online from the road adhesion coefficient, steering-angle change rate, vehicle speed, and lateral acceleration. This method can detect instability earlier on low-friction roads and avoid excessive interventions on high-friction roads. Compared with the common fixed-threshold method, the dynamic threshold method is more consistent with the real physical stability boundary of the electric vehicle.
Third, a three-layer hierarchical control architecture was designed. The stability-analysis layer decides whether the vehicle is stable. If the state is stable, the MPC controller generates smooth front- and rear-axle steering commands to track the desired path. If the state is not stable, a second-order sliding-mode controller calculates the required additional yaw moment. The proposed control does not simply add the yaw moment to every wheel; instead, it uses a constrained optimisation to distribute the longitudinal and lateral forces among the four tyres. The objective function is the sum of squared tyre workloads, and the constraints include the friction ellipse, motor torque limits, and generated lateral-force limits. The use of the inverse tyre model makes it possible to convert the required lateral force into the wheel steering angle.
Fourth, extensive simulations were performed in four representative scenarios. In-situ steering shows that the steering-plus-drive mode can rotate the vehicle about its centre with much higher efficiency and much less yaw-rate error than pure-drive mode. Diagonal driving shows that the proposed 4WIS strategy establishes a direct straight path without significant heading rotation. The high-speed double lane-change manoeuvre indicates that the proposed scheme reduces the trajectory error by 35.99% and significantly lowers the yaw-rate/sideslip-rate peaks. Finally, the low-adhesion double lane-change test confirms that the dynamic-threshold stability judge along with the lower allocation is especially valuable on low-friction roads; the trajectory error is reduced by more than 90% compared with baselines, and the vehicle remains stable throughout the avoidance manoeuvre.
Despite these encouraging results, there are still several limitations in this study. The proposed controller is evaluated only by simulation; real-vehicle testing is necessary to evaluate actuator delay, parameter uncertainty, and communication faults. The vehicle model ignores suspension roll and vertical load dynamics, which may become important in highly dynamic manoeuvres. Future research should extend the model to a full multi-body version and augment the control architecture with fault-tolerant allocation for actuator failure. In addition, the dynamic threshold method can be combined with learning-based estimation of the road friction coefficient to make the stability boundary even more adaptive.
In summary, the distributed-drive electric vehicle with four-wheel independent steering offers a highly redundant actuation architecture for integrated vehicle control. With the proposed fuzzy dynamic threshold stability judgement, model predictive trajectory tracking, sliding-mode yaw stabilisation, and optimal tyre-force allocation, the electric vehicle can achieve both agile low-speed special manoeuvres and robust high-speed stability in challenging driving conditions.