With the sustainable growth of the automotive industry and the rapid deployment of intelligent mobility, the electric vehicle has become the dominant carrier for future transportation. In my research, I focus on a distributed-drive electric vehicle architecture, in which each wheel is driven by an independent in-wheel motor. This configuration offers remarkable advantages for an electric vehicle because it removes the conventional driveshaft, shortens the powertrain path, shortens the control latency and, more importantly, enables flexible torque generation at each wheel. Those properties make the distributed-drive electric vehicle an ideal platform for implementing advanced chassis-domain control algorithms. In particular, accurate perception of the tyre–road interaction and robust yaw stability control are two coupled problems that determine whether the full potential of an electric vehicle can be released in safety-critical manoeuvres.
The main objective of my work is to construct a comprehensive chassis-domain control framework for a distributed-drive electric vehicle. The framework consists of three closely connected layers. First, a nonlinear seven-degree-of-freedom electric-vehicle dynamics model is established and validated against a high-fidelity software environment. Second, a road adhesion coefficient estimator is developed on the basis of interacting multiple models and adaptive unscented Kalman filtering. The estimator supplies real-time information about the available friction between the tyre and the road. Third, a yaw stability controller is designed for an electric vehicle with four-wheel independent steering and four-wheel independent driving. The controller uses the estimated road friction as a dynamic boundary condition, and it coordinates the front/rear steering angles and the four-wheel drive torques to prevent the vehicle from entering an unstable lateral region.
After this introductory discussion, I will present the dynamic modelling and the platform used for an electric vehicle in my study. Then I will describe the proposed road-friction estimation method in detail, including the identification of the tyre-model parameters. Following that, the yaw stability control strategy based on a nonlinear three-step method is explained. Finally, I provide simulation results to compare the proposed control architecture with baseline algorithms.
1. Dynamic Modelling of the Distributed-Drive Electric Vehicle
Accurate mathematical models are essential for both state observation and controller synthesis. In this section, I introduce a seven-degree-of-freedom model of an electric vehicle, together with the tyre model and the in-wheel motor model. The model considers the longitudinal motion, the lateral motion, the yaw motion and the rotational dynamics of four wheels. The vertical dynamics are neglected because the road is assumed to be flat and the vehicle body is assumed to be rigid over the operating range considered here.
I use a coordinate system fixed to the electric vehicle, where the x-axis points in the forward longitudinal direction and the y-axis points to the left side of the vehicle. The longitudinal force balance is written as
$$
\begin{aligned}
m a_x &= (F_{xfl}+F_{xfr})\cos\delta_f
-(F_{yfl}+F_{yfr})\sin\delta_f \\
&\quad +F_{xrl}+F_{xrr},
\end{aligned}
\tag{1}
$$
and the lateral force balance is written as
$$
\begin{aligned}
m a_y
&=(F_{xfl}+F_{xfr})\sin\delta_f
+(F_{yfl}+F_{yfr})\cos\delta_f \\
&\quad+F_{yrl}+F_{yrr},
\end{aligned}
\tag{2}
$$
where \(m\) denotes the vehicle mass, \(a_x\) and \(a_y\) are the longitudinal and lateral accelerations, \(\delta_f\) is the front-wheel steering angle, and \(F_{xij}\) and \(F_{yij}\) are the longitudinal and lateral forces of the individual tyres. The subscripts \(fl\), \(fr\), \(rl\) and \(rr\) denote the left-front, right-front, left-rear and right-rear wheels respectively.
The yaw moment balance with respect to the centre of gravity is expressed as
$$
\begin{aligned}
I_z\dot{\gamma}
&=l_f\left[(F_{xfl}+F_{xfr})\sin\delta_f
+(F_{yfl}+F_{yfr})\cos\delta_f\right] \\
&\quad-\frac{B_f}{2}(F_{xfl}-F_{xfr})\cos\delta_f
+\frac{B_f}{2}(F_{yfl}-F_{yfr})\sin\delta_f \\
&\quad-l_r(F_{yrl}+F_{yrr})
-\frac{B_r}{2}(F_{xrl}-F_{xrr}),
\end{aligned}
\tag{3}
$$
where \(I_z\) is the yaw moment of inertia, \(\gamma\) is the yaw rate, \(l_f\) and \(l_r\) are the distances from the centre of gravity to the front and rear axles, and \(B_f\) and \(B_r\) are the front and rear track widths.
In addition to the planar motion, each wheel has a rotational degree of freedom. The wheel rotational dynamics are represented by
$$
I_w\dot{\omega}_{ij}=T_{d,ij}-T_{b,ij}-R F_{xij},
\tag{4}
$$
where \(I_w\) is the wheel moment of inertia, \(\omega_{ij}\) is the wheel angular velocity, \(T_{d,ij}\) and \(T_{b,ij}\) are the driving and braking torques, and \(R\) is the effective rolling radius.
The vertical forces on the tyres are affected by longitudinal and lateral load transfer. I compute them from the static load and the dynamic load transfer terms:
$$
\begin{aligned}
F_{zfl}&=\frac{m g l_r}{2l}-\frac{m a_x h}{2l}
-\frac{m a_y h l_r}{B l},\\
F_{zfr}&=\frac{m g l_r}{2l}-\frac{m a_x h}{2l}
+\frac{m a_y h l_r}{B l},\\
F_{zrl}&=\frac{m g l_f}{2l}+\frac{m a_x h}{2l}
-\frac{m a_y h l_f}{B l},\\
F_{zrr}&=\frac{m g l_f}{2l}+\frac{m a_x h}{2l}
+\frac{m a_y h l_f}{B l},
\end{aligned}
\tag{5}
$$
where \(l=l_f+l_r\) is the wheelbase, \(h\) is the centre-of-gravity height, and \(B\) is the average track width used in the simulation vehicle.
For an electric vehicle, the tyre model must be accurate not only in the linear region but also near the friction limit. I compared the Dugoff tyre model with the Pacejka Magic Formula tyre model. The Magic Formula has a unified expression:
$$
\begin{aligned}
y(x)
&=D\sin\left\{C\arctan\left[B x
-E\left(B x-\arctan(B x)\right)\right]\right\}\\
&\quad+S_V,
\end{aligned}
\tag{6}
$$
where \(y\) can represent either the longitudinal tyre force or the lateral tyre force, and \(x\) is the corresponding slip variable, such as the slip ratio or the slip angle. The parameters \(B\), \(C\), \(D\) and \(E\) are the stiffness factor, the shape factor, the peak factor and the curvature factor, respectively, while \(S_V\) is the vertical offset.
For the pure longitudinal slip condition, the tyre model is expressed as
$$
F_{x0}=D_x\sin\left\{C_x\arctan\left[B_x \lambda
-E_x\left(B_x \lambda-\arctan(B_x\lambda)\right)\right]\right\},
\tag{7}
$$
where \(\lambda\) is the longitudinal slip ratio. For the pure lateral slip condition, the lateral tyre force is written as
$$
F_{y0}=D_y\sin\left\{C_y\arctan\left[B_y \alpha
-E_y\left(B_y \alpha-\arctan(B_y\alpha)\right)\right]\right\},
\tag{8}
$$
where \(\alpha\) is the tyre slip angle.
The in-wheel motor is modelled as a first-order inertial system because the motor’s electromagnetic response is much faster than the vehicle dynamics. The dynamic relation between the command torque and the actual output torque is given by
$$
\frac{T_{out}(s)}{T_{cmd}(s)}=\frac{1}{\tau s+1},
\tag{9}
$$
where \(\tau\) is the time constant of the motor-drive system. The output torque is also constrained by the motor speed and the available power. The maximum torque \(T_{\max}\) remains constant in the constant-torque region and decreases hyperbolically when the motor speed exceeds the rated speed.
The vehicle parameters used in my simulation model are summarized in the following table.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Total vehicle mass | \(m\) | 1412 | kg |
| Sprung mass | \(m_s\) | 1270 | kg |
| Distance, CG to front axle | \(l_f\) | 1.015 | m |
| Distance, CG to rear axle | \(l_r\) | 1.895 | m |
| Centre-of-gravity height | \(h\) | 0.54 | m |
| Track width | \(B\) | 1.675 | m |
| Yaw moment of inertia | \(I_z\) | 1536.7 | kg·m² |
| Tyre rolling radius | \(R\) | 0.325 | m |
In order to validate the seven-degree-of-freedom model, I built a co-simulation platform that connects the Simulink model with a high-fidelity vehicle dynamics software package. Several manoeuvres were simulated, including a high-speed step-steering input and a single-lane-change input on both high-friction and low-friction roads. The dynamic responses obtained from my model and from the high-fidelity software agree well in terms of the lateral acceleration, the yaw rate and the sideslip angle. This verified model is subsequently used as the basis for the state estimator and the stability controller of the electric vehicle.
2. Road Adhesion Coefficient Estimation for an Electric Vehicle
The road adhesion coefficient \(\mu\) is a critical parameter for an electric vehicle because it determines the maximum tyre force that can be generated under a given vertical load. A reliable estimation of \(\mu\) allows the chassis controller of an electric vehicle to adapt its intervention threshold and to prevent the tyres from exceeding the friction limit.
In my research, I formulate the road adhesion coefficient estimation problem as a nonlinear state-estimation problem. The four wheel-road friction coefficients are treated as slowly varying states,
$$
\mathbf{x}=[\mu_{fl},\ \mu_{fr},\ \mu_{rl},\ \mu_{rr}]^{T}.
\tag{10}
$$
The state transition is modelled as a random-walk process because the road friction coefficient changes only when the tyre enters a different road surface:
$$
\mathbf{x}(k+1)=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}
\mathbf{x}(k)+\boldsymbol{\xi}(k),
\tag{11}
$$
where \(\boldsymbol{\xi}(k)\) is the process noise with covariance \(\mathbf{Q}\).
The measurement model was designed for two distinct driving conditions. Under purely longitudinal driving, the tyre lateral forces are small, so the lateral acceleration and the yaw rate do not provide sufficient excitation for estimating \(\mu\). In this condition, I augment the measurement vector with the four wheel angular accelerations, leading to
$$
\mathbf{z}_x=[a_x,\ \dot{\omega}_{fl},\ \dot{\omega}_{fr},\ \dot{\omega}_{rl},\ \dot{\omega}_{rr}]^{T}.
\tag{12}
$$
Under combined longitudinal-lateral driving, the lateral acceleration and the yaw rate carry important information. Therefore the measurement vector is selected as
$$
\mathbf{z}_{xy}=[a_x,\ a_y,\ \gamma,\ \mu_{fl}-\mu_{fr},\ \mu_{rl}-\mu_{rr}]^{T}.
\tag{13}
$$
The last two virtual measurements constrain the estimated difference between the left-side and right-side road friction coefficients, which improves the observability when the road surface is not uniform.
2.1 Adaptive Unscented Kalman Filter
The conventional unscented Kalman filter, abbreviated as UKF, uses the unscented transformation to propagate a set of deterministic sigma points through the nonlinear vehicle model. However, the performance of the UKF depends strongly on the prior statistics of the process noise and the measurement noise. When the driving condition of an electric vehicle changes abruptly, the fixed noise covariance matrices become inaccurate, causing a slow convergence or even divergence of the estimator.
To solve that problem, I introduce the Sage-Husa adaptive algorithm into the UKF framework. The resulting adaptive unscented Kalman filter, abbreviated as AUKF, updates the measurement noise covariance in real time. The innovation sequence is written as
$$
\tilde{\mathbf{z}}_{k+1}
=\mathbf{z}_{k+1}-\hat{\mathbf{z}}_{k+1|k},
\tag{14}
$$
where \(\hat{\mathbf{z}}_{k+1|k}\) is the predicted measurement. The weighting coefficient is
$$
d_{k+1}=\frac{1-b}{1-b^{k+1}},
\tag{15}
$$
where \(b\) is a forgetting factor normally selected between 0.95 and 0.995. The estimated covariance matrix of the measurement noise is then updated by
$$
\hat{\mathbf{R}}_{k+1}
=(1-d_{k+1})\hat{\mathbf{R}}_{k}
+d_{k+1}
\left(
\tilde{\mathbf{z}}_{k+1}\tilde{\mathbf{z}}_{k+1}^{T}
-\mathbf{P}_{zz,k+1}
\right),
\tag{16}
$$
where \(\mathbf{P}_{zz,k+1}\) is the innovation covariance computed by the unscented transformation. To guarantee the positive definiteness of the noise covariance, I enforce a symmetric positive-definite projection after each update.
2.2 Tyre Model Parameter Identification with GA-RLS
The accuracy of a road-friction estimator depends strongly on the accuracy of the tyre force calculation. In many previous studies, the Magic Formula tyre model is fitted with a set of parameters identified under a single loading condition, which is insufficient for a distributed-drive electric vehicle that frequently operates under changing vertical loads and combined slip conditions.
I therefore proposed a hybrid genetic-algorithm and recursive-least-squares method, abbreviated as GA-RLS, to identify the Magic Formula tyre parameters. The identification problem is formulated as a constrained nonlinear optimization problem:
$$
J(\boldsymbol{\theta})
=\sum_{j=1}^{m}\sum_{i=1}^{n_j}
\left[
y_{ji}^{exp}-y_{ji}^{model}(\boldsymbol{\theta})
\right]^2,
\tag{17}
$$
where \(\boldsymbol{\theta}\) is the parameter vector, \(y_{ji}^{exp}\) is the measured tyre force of the \(j\)-th loading condition and the \(i\)-th sample, and \(y_{ji}^{model}\) is the corresponding model output. The search domain is bounded by
$$
\boldsymbol{\theta}_{lb}\le\boldsymbol{\theta}\le\boldsymbol{\theta}_{ub}.
\tag{18}
$$
The GA phase performs a global search in the feasible parameter space. Each individual in the population is represented by a real-coded vector
$$
\mathbf{x}_k=[x_{k,1},x_{k,2},\dots,x_{k,p}]^T,
\tag{19}
$$
where \(p\) is the number of unknown parameters. The initial population is generated randomly, and the fitness of the \(k\)-th individual is defined as
$$
f(\mathbf{x}_k)=\frac{1}{1+J(\mathbf{x}_k)}.
\tag{20}
$$
The genetic operators include roulette-wheel selection, simulated binary crossover and polynomial mutation. The optimal solution obtained by the GA is then supplied as the initial value to the second identification stage. In this stage, a recursive least-squares or a quadratic-programming algorithm refines the parameter vector to satisfy the Karush-Kuhn-Tucker optimality conditions.
I divided the Magic Formula tyre model into four groups: the pure longitudinal force model, the pure lateral force model, the combined-slip longitudinal force model and the combined-slip lateral force model. The number of unknown parameters for each model group is listed in the table below.
| Tyre model group | Number of parameters | Typical input variables |
|---|---|---|
| Pure longitudinal model | 11 | \(\lambda,\ F_z\) |
| Pure lateral model | 22 | \(\alpha,\ F_z\) |
| Combined longitudinal model | 6 | \(\lambda,\ \alpha,\ F_z\) |
| Combined lateral model | 14 | \(\alpha,\ \lambda,\ F_z\) |
To show the identification quality, I compare the experimental tyre force data with the predicted force from the identified Magic Formula parameters. The root-mean-square error between the measured data and the model output remains small for all four model groups. The identified tyre model is then embedded into the state estimator so that the tyre longitudinal and lateral forces can be accurately calculated under multiple driving conditions.
2.3 Interacting Multiple Model AUKF
A single AUKF estimator is still not sufficient when an electric vehicle suddenly changes its operating regime, for example from a purely longitudinal manoeuvre to a combined cornering and braking manoeuvre. To handle the transition smoothly, I use an interacting multiple model framework. The structure consists of two parallel AUKFs: one tailored to the longitudinal driving condition and the other tailored to the combined driving condition.
The mixing probability between model \(i\) and model \(j\) is computed as
$$
\mu_{i|j}(k-1)
=\frac{p_{ij}\,\mu_i(k-1)}{c_j(k-1)},
\tag{21}
$$
where \(p_{ij}\) is the model transition probability and
$$
c_j(k-1)=\sum_{i=1}^{r}p_{ij}\mu_i(k-1)
\tag{22}
$$
is the predicted model probability. The mixed state estimate and covariance for model \(j\) are obtained by
$$
\hat{\mathbf{X}}^{0j}_{k-1|k-1}
=\sum_{i=1}^{r}
\hat{\mathbf{X}}^{i}_{k-1|k-1}\,\mu_{i|j}(k-1),
\tag{23}
$$
$$
\begin{aligned}
\mathbf{P}^{0j}_{k-1|k-1}
&=\sum_{i=1}^{r}\mu_{i|j}(k-1)
\Big\{
\mathbf{P}^{i}_{k-1|k-1}\\
&\quad+
\left[
\hat{\mathbf{X}}^{i}_{k-1|k-1}
-\hat{\mathbf{X}}^{0j}_{k-1|k-1}
\right]
\left[
\hat{\mathbf{X}}^{i}_{k-1|k-1}
-\hat{\mathbf{X}}^{0j}_{k-1|k-1}
\right]^T
\Big\}.
\end{aligned}
\tag{24}
$$
After each AUKF runs independently, the likelihood of model \(j\) is calculated from the innovation vector \(\tilde{\mathbf{z}}_j(k)\) and its covariance \(\mathbf{S}_j(k)\):
$$
\Lambda_j(k)
=
\frac{1}{\sqrt{\left|2\pi\mathbf{S}_j(k)\right|}}
\exp\left\{
-\frac{1}{2}
\tilde{\mathbf{z}}_j^T(k)\mathbf{S}_j^{-1}(k)
\tilde{\mathbf{z}}_j(k)
\right\},
\tag{25}
$$
and the model probability is updated as
$$
\mu_j(k)
=
\frac{\Lambda_j(k)\,c_j(k-1)}
{\sum_{i=1}^{r}\Lambda_i(k)\,c_i(k-1)}.
\tag{26}
$$
Finally, the fused state estimate is
$$
\hat{\mathbf{X}}(k|k)
=
\sum_{j=1}^{r}
\mu_j(k)\hat{\mathbf{X}}^j(k|k),
\tag{27}
$$
and the fused covariance is
$$
\begin{aligned}
\mathbf{P}(k|k)
=
\sum_{j=1}^{r}
\mu_j(k)
\Big\{
&\mathbf{P}^j(k|k)\\
&+
\left[
\hat{\mathbf{X}}^j(k|k)-\hat{\mathbf{X}}(k|k)
\right]
\left[
\hat{\mathbf{X}}^j(k|k)-\hat{\mathbf{X}}(k|k)
\right]^{T}
\Big\}.
\end{aligned}
\tag{28}
$$
In contrast to a conventional UKF, which uses a fixed process model and fixed noise statistics, the proposed IMM-AUKF method provides more robustness when an electric vehicle travels on a road with a varying friction coefficient. The interaction between the two filters smooths the transition and prevents the discontinuous jumps that are often observed when the estimator switches between models.
3. Simulation and Experimental Validation of the Friction Estimator
I evaluated the proposed IMM-AUKF road-friction estimator under several typical operating conditions of an electric vehicle. The conditions include double lane-change manoeuvres on split-friction roads, constant-speed driving on roads with different friction levels, and pure longitudinal acceleration-braking manoeuvres.
I first simulated a double lane-change manoeuvre at a vehicle speed of 60 km/h. The road friction coefficient was set to different values on the left and right sides to test the estimator under asymmetric excitation. The conventional UKF exhibited a clear convergence delay at the beginning of the simulation, and its estimates contained large oscillations when the steering wheel was quickly reversed. In contrast, the proposed IMM-AUKF converged almost immediately and maintained an estimate close to the true road friction value throughout the manoeuvre.
I also simulated an abrupt friction change from a high-friction road to a low-friction road. In that case, the conventional UKF reacted slowly to the change and produced a large temporary overshoot. The proposed IMM-AUKF tracked the step change quickly and smoothly, which indicates that the interacting multiple model structure improves the dynamic response of the estimator.
In the pure longitudinal condition, I simulated an electric vehicle accelerating from standstill and then braking to standstill on both high-friction and low-friction roads. The conventional UKF was again slower in the early acceleration phase and showed a significant undershoot on the high-friction road. The proposed IMM-AUKF reduced the initial transient and provided a more accurate estimate over the entire operating range.
The estimator was also tested with experimental data from a distributed-drive electric vehicle platform. The vehicle was driven on dry asphalt and on wet gravel. In the high-friction coupled manoeuvre, the proposed estimator remained close to a stable value even during rapid turning and hard braking, while the conventional UKF fluctuated strongly. In the low-friction straight-line manoeuvre, the proposed method also showed a faster convergence and a higher immunity to the noise excited by acceleration and braking.
The improvement of the proposed estimator over the conventional UKF is summarized in the table below.
| Validation scenario | Conventional UKF behaviour | Proposed IMM-AUKF behaviour |
|---|---|---|
| Split-friction double lane change | Slow initial convergence, visible oscillation | Fast convergence, small oscillation |
| Friction step change | Noticeable lag and overshoot | Smooth and rapid tracking |
| High-friction longitudinal manoeuvre | Undershoot in the acceleration phase | Accurate estimate during the whole manoeuvre |
| Low-friction straight-line braking | Large disturbance-induced fluctuations | Robust and stable estimate |
4. Yaw Stability Control of the Distributed-Drive Electric Vehicle
The second main contribution of my work is a yaw stability controller that fully uses the four-wheel independent steering and four-wheel independent driving capabilities of an electric vehicle. The control objective is to keep the yaw rate close to a desired reference while simultaneously preventing the vehicle sideslip angle from becoming excessively large.
4.1 Stability Indicators and Phase-Plane Stability Boundary
Two variables are widely used to evaluate the lateral stability of an electric vehicle. The first variable is the yaw rate \(\gamma\), which characterizes the rotational response of the vehicle. The second variable is the body sideslip angle \(\beta\), which characterizes the difference between the vehicle heading direction and the actual velocity direction. When \(\beta\) becomes too large, the tyre force can no longer increase with the steering angle, and the electric vehicle may enter an irreversible spin.
The desired yaw rate of the controlled electric vehicle is calculated from the driver’s steering input and the current road friction. The nominal steady-state yaw rate is given by
$$
\gamma_{ss}
=
\frac{v_x}{l(1+K v_x^2)}\delta_f,
\tag{29}
$$
where
$$
K=\frac{m}{l^2}
\left(
\frac{l_f}{K_r}-\frac{l_r}{K_f}
\right)
\tag{30}
$$
is the stability factor of the electric vehicle. Since the lateral acceleration cannot exceed the friction limit, the actual reference yaw rate is bounded by
$$
\gamma^{*}
=
\min\left\{
\left|\gamma_{ss}\right|,\
0.85\frac{\mu g}{v_x}
\right\}
\operatorname{sgn}(\delta_f),
\tag{31}
$$
where \(\mu\) is the estimated road friction coefficient and the factor 0.85 provides a safety margin. The ideal body sideslip angle is set to zero because it ensures that the vehicle heading is aligned with the vehicle velocity direction.
To determine whether an electric vehicle is approaching instability, I use the \(\beta-\dot{\beta}\) phase plane. The stable region is defined by the double-linear-boundary method:
$$
B_1\le A\beta+\dot{\beta}\le B_2,
\tag{32}
$$
where \(A\), \(B_1\) and \(B_2\) are parameters that depend on the vehicle speed \(v_x\), the steering angle \(\delta_f\) and the road friction coefficient \(\mu\). Because computing the phase plane online is computationally expensive, I pre-compute the stability boundaries over a grid of operating conditions and store them in a three-dimensional lookup table. During real-time control, the phase-plane boundary is obtained by interpolation according to the current vehicle states.
4.2 Four-Wheel-Steering Model and Reference Dynamics
For an electric vehicle equipped with four-wheel steering, the driver’s steering command is sent to the front axle, while the controller determines an additional steering command for the rear axle. I use a two-degree-of-freedom bicycle model as the basis for the control design. The equations of motion are
$$
m v_x(\dot{\beta}+\gamma)
=
F_{yf}\cos\delta_f+F_{yr}\cos\delta_r,
\tag{33}
$$
$$
I_z\dot{\gamma}
=
l_f F_{yf}\cos\delta_f-l_r F_{yr}\cos\delta_r+\Delta M,
\tag{34}
$$
where \(F_{yf}\) and \(F_{yr}\) are the front and rear axle lateral forces, and \(\Delta M\) is the additional yaw moment generated by torque vectoring. In the linear region, the lateral forces are expressed as
$$
F_{yf}
=
K_f\alpha_f,\qquad
F_{yr}
=
K_r\alpha_r,
\tag{35}
$$
with the tyre slip angles defined by
$$
\alpha_f
=
\beta+\frac{l_f\gamma}{v_x}-\delta_f,
\tag{36}
$$
$$
\alpha_r
=
\beta-\frac{l_r\gamma}{v_x}-\delta_r.
\tag{37}
$$
In the controller, I use a front-axle-feedforward rear-axle steering strategy. Under the zero-sideslip condition, the rear steering angle is proportional to the front steering angle with a speed-dependent gain:
$$
\delta_r=K_{vw}(v_x)\delta_f.
\tag{38}
$$
The function \(K_{vw}(v_x)\) is derived from the steady-state solution of the two-degree-of-freedom model under the constraint \(\beta=0\). At low speeds, the gain is negative, which produces an opposite-phase steering mode and reduces the turning radius. At high speeds, the gain becomes positive, which produces a same-phase steering mode and enhances the directional stability of the electric vehicle.
4.3 Nonlinear Three-Step Controller Design
After defining the desired dynamics, I designed a nonlinear yaw stability controller based on the three-step control method. The three-step method decomposes the control law into a quasi-steady-state component, a reference dynamic feedforward component and an error-feedback component.
In the first step, I assume that the electric vehicle has reached a steady state. The derivatives of the states are set to zero, and the control inputs are calculated from the algebraic force and moment balance equations. This produces the quasi-steady-state control part.
In the second step, I consider the dynamic change of the reference signal. To make the vehicle state track a time-varying reference trajectory, I add a feedforward term that depends on the derivative of the desired yaw rate or the desired sideslip angle. This feedforward term improves the transient response of the electric vehicle when the driver changes the steering direction quickly.
In the third step, I introduce a feedback term to reject modelling errors and external disturbances. The closed-loop error dynamics are designed to behave as a stable linear system:
$$
\dot{e}_\beta+K_{p\beta}e_\beta+K_{i\beta}\int e_\beta dt=0,
\tag{39}
$$
$$
\dot{e}_\gamma+K_{p\gamma}e_\gamma+K_{i\gamma}\int e_\gamma dt=0,
\tag{40}
$$
where
$$
e_\beta=\beta-\beta^{*},\qquad
e_\gamma=\gamma-\gamma^{*}.
\tag{41}
$$
The controller gains \(K_{p\beta}\), \(K_{i\beta}\), \(K_{p\gamma}\) and \(K_{i\gamma}\) are selected to guarantee a satisfactory convergence speed without causing overshoot.
By solving the algebraic equations produced by the three steps, the target front axle lateral force and the additional yaw moment are obtained. Because the lateral force cannot be directly commanded on the vehicle, I transform the target lateral force into an additional front steering angle through the inverse tyre model:
$$
\begin{aligned}
\delta_{f,ts}
&=
\delta_f
+
\arctan
\left(
\frac{F_{yf}-K_f\alpha_{f,base}}{K_f}
\right),\\
\delta_{r,ts}
&=
\arctan
\left(
\frac{F_{yr}-K_r\alpha_{r,base}}{K_r}
\right),
\end{aligned}
\tag{42}
$$
where \(\alpha_{f,base}\) and \(\alpha_{r,base}\) are the baseline slip angles calculated from the current vehicle states. The final control structure is illustrated by the following signal flow:
Driver steering and vehicle states → reference model → three-step yaw controller → target lateral forces and yaw moment → inverse tyre model → additional front and rear steering angles → torque allocation layer → in-wheel motor commands.
4.4 Torque Allocation Optimization
For the distributed-drive electric vehicle, the additional yaw moment is realized by distributing different longitudinal torques to the four wheels. To ensure that no tyre is excessively loaded, I formulate the torque allocation problem as an optimization problem. The objective is to minimize the sum of the squared tyre workload:
$$
J
=
\min
\sum_{i=1}^{4}
c_i
\frac{T_{xi}^{2}}
{\left(\mu F_{zi}R\right)^{2}},
\tag{43}
$$
where \(c_i\) is a weighting coefficient for the \(i\)-th wheel and \(T_{xi}\) is its longitudinal torque. The equality constraints are the total driving torque demand and the additional yaw moment demand:
$$
T_{x1}+T_{x2}+T_{x3}+T_{x4}=T_{req},
\tag{44}
$$
$$
\frac{B\cos\delta_f}{2R}
(-T_{x1}+T_{x2})
+
\frac{B\cos\delta_r}{2R}
(-T_{x3}+T_{x4})
=
\Delta M,
\tag{45}
$$
where \(T_{req}\) is the total propulsion torque requested by the driver. The individual torques are also bounded by the tyre-road friction and the motor output capability:
$$
|T_{xi}|
\le
\min\left(\mu F_{zi}R,\ T_{motor,max}\right).
\tag{46}
$$
The above optimization problem can be solved analytically after eliminating two of the four torque variables using the equality constraints. The resulting torque commands always satisfy the total traction demand while generating the required yaw moment with minimum tyre workload utilization.
4.5 Simulation Verification
I verified the proposed controller on an electric vehicle model in several challenging scenarios. I compared three configurations: the uncontrolled electric vehicle, an electric vehicle with conventional sliding-mode active-front-steering and torque-vectoring control, and the electric vehicle with the proposed three-step control strategy. The scenarios include double lane-change manoeuvres on high-friction and low-friction roads, a step-steering manoeuvre on a low-friction road, and a double lane-change manoeuvre at high speed on a low-friction road.
On a high-friction road at 72 km/h, the uncontrolled electric vehicle already exhibited visible oscillation in the lateral acceleration and yaw rate after the first lane change. The sliding-mode controller suppressed part of this oscillation, but the proposed three-step controller achieved the smallest peak values and the fastest convergence. In the phase plane, the trajectory controlled by the three-step method stayed closest to the origin.
When the road friction coefficient was reduced to 0.5, the advantage of the proposed controller became more evident. The uncontrolled electric vehicle displayed a large body side-slip angle and a clear tendency toward spin in the double-lane-change manoeuvre. The sliding-mode controller kept the vehicle stable but still showed noticeable chattering. The proposed three-step controller maintained both the yaw rate and the side-slip angle within narrow bounds throughout the entire manoeuvre.
In the high-speed step-steering test at 108 km/h on a low-friction road, the uncontrolled electric vehicle suffered from a sustained oscillation. The sliding-mode controller reduced this oscillation, but the proposed controller eliminated it almost completely. The controlled electric vehicle reached the steady-state yaw rate without overshoot and preserved the driver’s intended path.
Finally, in the limit-condition test at 108 km/h on a low-friction road, the proposed three-step controller maintained a very small body side-slip angle and limited the yaw rate to the friction-bounded reference. The phase-plane trajectory of the controlled vehicle was tightly confined around the stable region, which demonstrates that the controller expands the safe operating envelope of the electric vehicle.
The simulation results are summarized in the following table.
| Scenario | Speed | Friction | Key observation |
|---|---|---|---|
| Double lane change | 72 km/h | 0.8 | Three-step control has the smallest yaw-rate peak |
| Double lane change | 72 km/h | 0.5 | Three-step control suppresses body side slip |
| Step steering | 108 km/h | 0.5 | No overshoot and rapid steady-state convergence |
| Double lane change | 108 km/h | 0.5 | Vehicle states remain inside the stable phase-plane region |
5. Conclusion
In this research, I developed a systematic estimation and control framework for a distributed-drive electric vehicle. A seven-degree-of-freedom model of an electric vehicle was first established and validated. A road adhesion coefficient estimator was then designed using an interacting multiple model adaptive unscented Kalman filter. The proposed estimator relied on a Magic Formula tyre model whose parameters were identified by a hybrid GA-RLS approach. Simulation and experimental results demonstrated that the proposed estimator converges faster and behaves more robustly than a conventional UKF when an electric vehicle experiences strong longitudinal and lateral excitation.
Based on the estimated road adhesion coefficient, a yaw stability controller was designed for an electric vehicle with four-wheel independent steering and four-wheel independent driving. The controller was derived from the nonlinear three-step method and consisted of a quasi-steady-state part, a reference dynamic feedforward part and an error-feedback part. An optimization-based torque allocation algorithm was further developed to distribute the required yaw moment among the four wheels. The simulation results show that the proposed controller achieves better yaw-rate tracking and a smaller body side-slip angle than a conventional sliding-mode controller, especially on low-friction roads and in high-speed limit manoeuvres.
The contribution of my work can be viewed from both the state-estimation perspective and the control perspective. The proposed friction estimator provides an electric vehicle with reliable information about the environment, while the proposed yaw stability controller makes full use of the over-actuated chassis of a distributed-drive electric vehicle. In future research, I plan to extend the control framework toward a more integrated chassis-domain strategy that simultaneously coordinates longitudinal speed regulation, active suspension damping and four-wheel steering. I also intend to investigate data-driven disturbance observers to further improve the robustness of the electric vehicle under uncertain road and tyre conditions.