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#  Traction and Stability Control using Fuzzy-based Controller Integration for
Electric Vehicles

Nimantha Dasanayake, and Shehara Perera Nimantha Dasanayake is with the
Department of Mechanical Engineering, Faculty of Engineering, University of
Moratuwa, Moratuwa, Sri Lanka<EMAIL_ADDRESS>Perera is with the
Dyson School of Design Engineering, Imperial College London, United Kingdom,
<EMAIL_ADDRESS>0000-0001-6221-1610 0000-0001-6695-4307

###### Abstract

Adverse road conditions can cause vehicle yaw instability and loss of
traction. To compensate for the instability under such conditions, corrective
actions must be taken. In comparison to a mechanical differential, an
electronic differential can independently control the two drive wheels and
provide means of generating more effective corrective actions. As a solution
for traction and stability issues in automobiles, this study has developed a
controller for a vehicle electronic differential consisting of two program-
controlled rear motors. The control algorithm adjusts to changing road
conditions. Traction was controlled using a motor reaction torque observer-
based slip ratio estimation, and yaw stability was achieved by tracking a
reference yaw rate calculated using estimated tyre cornering stiffnesses. A
recursive least squares algorithm was used to estimate cornering stiffness.
The yaw rate of the vehicle, as well as its longitudinal and lateral
accelerations, were measured, and the body slip angle was estimated using an
observer. A fuzzy inference system was used to integrate the independently
developed traction control and yaw control schemes. The fuzzy inference system
modifies the commanded voltage generated by the driver’s input to account for
the traction and yaw stability controller outputs. A vehicle simulator was
used to numerically simulate the integrated controller. For a racetrack
simulation, the peak slip ratio was reduced by 42.31% and the RMS yaw rate
error was reduced by 88.17%. For the double lane change test at 40 km.h^-1 and
100 km.h^-1, the RMS yaw rate error was reduced by 86.96% and 92.34%,
respectively.

###### Index Terms:

Electronic differential, Slip control, Traction control, Stability control,
Fuzzy inference system

##  I Introduction

The study of controlling vehicle dynamics to produce dependable and safer
rides has recently evolved dramatically. More emphasis has been placed on
creating Electronic Differentials with Traction and Stability Control
(EDTSCs). These technologies are utilised to manage the longitudinal and
lateral dynamics of the vehicle, making Electric Vehicles (EVs) safer and more
robust in operation. Because the driver’s input and the control signals from
the Traction and Stability Control (TSC) system must be carefully combined to
have stable control over the drive train, the requirements for an EDTSC system
for an EV are more demanding and sophisticated than those for internal
combustion engine vehicles. This complexity, however, opens up new options for
improving control over the vehicle’s nonlinear dynamics.

Commercial EV manufacturers have used technologies such as torque vectoring,
EDTSCs, and TSCs to improve traction and stability during critical driving
situations and ensure optimal driving performance [1, 2, 3].

Figure 1: Simplified representation of the proposed control architecture for
the EDTSC with Fuzzy Inference System (FIS) and yaw and traction controllers.

##  II Related Work

A commonly used method for vehicle yaw stability control in electronic
differentials is direct yaw-moment management [4, 5, 6, 7]. The system
independently adjusts the driving and braking forces on each wheel to generate
an inverse yaw moment to counteract the yaw moments that cause instability in
crucial situations under direct yaw-moment control. To prevent wheel spin or
skid caused by saturation of longitudinal tyre force, direct torque control of
motors has been developed singly or together with direct yaw-moment
management.

TABLE I: Traction and yaw moment control algorithms

Control Objectives |  State Measurements/ Estimations |  Control Algorithm |  Disadvantages
---|---|---|---
Yaw rate |  Yaw rate, (In addition: Slip angle observer, Cornering stiffness estimators) |  LQR [4], FIS [8], Adaptive Neuro-FIS [9], SMC [10], Adaptive-SMC [11], Feedback and adaptive feedforward control [6, 7], MPC [12] |  No traction control incorporated
Slip ratio |  Wheel speed, (In addition: Slip ratio, Friction coefficient estimators) |  SMC [13, 14], Disturbance observer based current control [15] |  No yaw control incorporated
Yaw rate and Slip ratio |  Yaw rate, Slip angle observer, Cornering stiffness estimator, Wheel speed (In addition: Slip ratio estimator) |  Feedback control [5] |  No yaw and traction control integration

LQR: Linear Quadratic Regulator, FIS: Fuzzy Inference System, SMC: Sliding
Mode Control, MPC: Model Predictive Control

###  II-A State Estimation

To assess the amount of stability and hence conduct corrective actions, it is
critical to perceive the dynamical states of the vehicle such as vehicle
speed, accelerations, yaw rate, and wheel speeds. Direct torque and speed
calculations of electric motors are one of the advantages of EVs over IC
engine-driven vehicles in terms of control. A prominent control tool utilised
by many researchers for states that are not immediately quantifiable is state
estimator. Some examples are body slip angle observer [16], cornering
stiffness and road friction coefficient estimator [17] and, yaw-moment
observer [5]. Along with the above estimators, techniques like the Kalman
filter [6] and disturbance observer [15] have been used to reduce the noise
and uncertainty of the estimated values.

Calculation of slip ratio using the vehicle velocity incorporates the long
time integral errors since the only way to estimate the velocity is to
integrate the measured longitudinal acceleration. Therefore, in this study
instead of using vehicle speed to directly calculate slip ratio an indirect
way of estimating slip ratio that is based on reaction torque observer was
used. The reaction torque observer is a disturbance observer which helps to
achieve a robust estimation of slip ratio. The disturbance observer is widely
used for estimating plant uncertainties and external disturbances [18]. A
brief study of the disturbance observation will be offered herein to elucidate
the theory behind the reaction torque observer.

###  II-B Disturbance Observer

For robust control against system disturbances, disturbance observer-based
controllers are used in applications such as servo systems and robot
manipulation. A deliberately designed disturbance observer deduces robust
control, which compensates for the disturbance by appropriate feedback.
According to [19], a disturbance observer was used to measure the reaction
torque of joint actuators on a robotic manipulator. To improve system dynamics
and stability by reducing response time and torque ripples against load
disturbances, a torque disturbance observer has been implemented for model-
predictive control of electric drives [20].

Although most recent work has used a disturbance observer to compensate for
the effects of uncertainties on closed-loop control, the proposed method has
only used it to realise the reaction torque under the assumption that model
coefficient variations from nominal values are negligible.

###  II-C Control Strategies

Moving beyond the standard approaches to dealing with the problem of vehicle
dynamics control, the methodologies listed below have been added to the
control architectures.

#### 1) Model Matching Control (MMC)

A simple algorithm for developing a desired model of a dynamic system and
matching the real states of the system with the desired model. The most
typical states used by this type of algorithm are slip ratio, body slip angle,
and yaw rate [21].

#### 2) Optimal Control

Used to determine the optimal values for control signals. The most common
approach is to use the optimal control method to determine the control yaw
moment by minimising a performance index or cost function composed of errors
in body slip angle and yaw rate relative to desired/reference values [5].

#### 3) Model Predictive Control (MPC)

As an extension of model matching control, the optimal path to match the
dynamical states to the reference states is predicted and the control signal
is updated accordingly. In terms of definition, MPC is a strategy that employs
the theory of optimal control [22, 6].

#### 4) Sliding Mode Control (SMC)

Limits the dynamic system state trajectory to a sliding surface with the
desired characteristics and ensures that the system converges to the surface
in a finite time [13]. This approach is widely used in vehicle traction and/or
stability control algorithms to deal with dynamical uncertainty [14, 23, 15].
One drawback of SMC is that the control state trajectory chatters when it
approaches closer to the controlled parameter’s set point. This difficulty is
overcome by utilising a solution that uses a conditional integrator to transit
from SMC to PI controller when the parameter is close to the set point [24].

#### 5) Fuzzy Logic and Neural Network Control

Have been utilised to improve the robustness of control algorithms in traction
control, stability control, or both. Neural networks have been utilised as
sub-algorithms within the main algorithm, which is in charge of a localised
control [25, 26]. Fuzzy logic control has been adapted in car traction control
in a variety of ways [27, 8, 10, 9, 11]. Adaptive fuzzy systems has also been
used to integrate two different vehicle models with different cornering
stiffnesses to obtain a yaw moment controller which is more robust to
variations of the road surface [9].

Table I offers a summary of the control algorithms used in previous studies,
along with their disadvantages. The primary drawback observed by the authors
is the absence of either traction control, yaw moment control, or their
coalition in action. It is challenging to integrate an EDTSC’s traction and
yaw stability control counterparts together. The essential issue is how to
ensure that longitudinal dynamics are not compromised by efforts to control
lateral dynamics. In previous work, fuzzy systems hasn’t been employed to
address the issue of controller integration for a combined traction and yaw
moment control. This work offers a controller integration method that uses a
Fuzzy Inference System (FIS) to create corrective yaw moments utilising the
slip ratio and error in yaw rate compared to the desired yaw rate as inputs.
Furthermore, a motor reaction torque observer-based slip ratio estimation is
proposed while employing the Recursive Least Square (RLS) algorithm for
cornering stiffness estimation (see Fig. 1).

More precisely and compactly, the contributions of this article are as
follows:

  1. 1.

Estimation of the slip ratio indirectly using the disturbance observer applied
for reaction torque observance.

  2. 2.

Using a fuzzy inference system for traction and yaw rate controller
integration.

The rest of the article is structured as follows. Section III presents the
vehicle dynamics model used to develop the EDTSC. The Section IV discusses the
mathematical formulation of the proposed EDTSC. Section V shows the simulation
model and experiments. Section VI presents results and Section VII concludes
the findings and presents the future work.

##  III Vehicle Dynamics Model

The vehicle status variables must be determined before the EDTSC system can be
implemented. The yaw rate can be measured directly with an Inertial
Measurement Unit (IMU) sensor, but the side slip angle cannot be measured and
must be approximated using a vehicle mathematical model. The yaw stability
controller is based on the vehicle’s lateral motion dynamics. Because of its
simplicity, a two-degrees-of-freedom (2-DoF) linear vehicle model is used to
calculate the vehicle lateral motion state variables.

As shown in Figure 2, the vehicle was modelled as the equivalent half-vehicle.
In the figure, $\beta$ is the body slip angle and $\gamma$ is the yaw rate.
$u$ and $v$ are the vehicle’s lateral and longitudinal velocities
respectively. $\delta$ is the steering angle and $\alpha_{f}$ and $\alpha_{r}$
are the front and rear tire side slip angles. $Y_{f}$ and $Y_{r}$ are the
lateral tire forces on the front and rear wheels, respectively. $l_{f}$ and
$l_{r}$ are the distances between the center of mass and front and rear axles
respectively.

6.5 2.5 atan2(,) 30 45 25 37 0.2

(Origin) at (0,0); [-latex,thick] (Origin)–++(0,+1.5) coordinate (yaxis);
[-latex,thick] (Origin)–++(+1.5,0) coordinate (xaxis); [dashed]
(Origin)–++(:8.5cm) coordinate (AngleEnd); [dotted,thick] (,0) node [below]
$y$ –++(0,) coordinate (Yf); [dotted,thick] (0,) node [left] $x$ –++(,0);
pic[”$\psi$”, draw=black, text=black, -latex, angle eccentricity=1.25, angle
radius=0.8cm] angle=xaxis–Origin–Yf;

(Yr) at ($(Origin)+(\Angle:2.5cm)$); at (Yr) [rotate=,draw,thick,rounded
corners=1mm,minimum width=1cm, minimum height=0.4cm] ; [red,-latex,thick]
(Yr)–++(+:1.3cm) coordinate (RedArrowOne); pic[”$\alpha_{r}$”, draw=black,
text=red, -latex, angle eccentricity=1.45, angle radius=0.8cm]
angle=Yf–Yr–RedArrowOne; [latex-,thick,black] (Yr)–++(-90:0.75cm) node
[rotate=,right] $Y_{r}$;

[thick] (Yr)–(Yf);

at (Yf) [rotate=+,draw,thick,rounded corners=1mm,minimum width=1cm, minimum
height=0.4cm] ; [red,-latex,thick] (Yf)–++(+:1.3cm) coordinate (RedArrowTwo);
[dashed] (Yf)–++(+:1.3cm) coordinate (DeltaAngleEnd); pic[”$\delta$”,
draw=black, text=black, -latex, angle eccentricity=1.35, angle radius=0.8cm]
angle=AngleEnd–Yf–DeltaAngleEnd; pic[”$\alpha_{f}$”, draw=black, text=red,
-latex, angle eccentricity=1.45, angle radius=1cm]
angle=DeltaAngleEnd–Yf–RedArrowTwo; [latex-,thick,black] (Yf)–++(-90:0.75cm)
node [rotate=,right] $Y_{f}$;

(COM) at ($(Origin)+(\Angle:4.7cm)$); [rotate=] [radius=] (COM) – ++(,0) arc
[start angle=0,end angle=90] – ++(0,-2*) arc [start angle=270, end angle=180];
[thick,radius=] (COM) circle; [-latex,thick,olive] (COM)–++(+90:0.8cm) node
[left,rotate=] $u$; [-latex,thick,olive] (COM)–++(:1cm) node [below,rotate=]
$v$; pic[”$\gamma$”, draw=red, text=red, -latex, angle eccentricity=1.4, angle
radius=0.5cm]angle=Yr–COM–Yf; [blue,-latex,thick] (COM)–++(+:1.3cm) coordinate
(BlueArrowBeta); pic[”$\beta$”, draw=black, text=blue, -latex, angle
eccentricity=1.45, angle radius=1.0cm] angle=Yf–COM–BlueArrowBeta;

(LrLabel) at ($(Yr)+(\Angle+90:1.25cm)$); (COMLabel) at
($(COM)+(\Angle+90:1.25cm)$); (LfLabel) at ($(Yf)+(\Angle+90:1.25cm)$);
(LlrLabel) at ($(Yr)+(\Angle+90:1.75cm)$); (LlfLabel) at
($(Yf)+(\Angle+90:1.75cm)$);

[Barlatex-latexBar] (LrLabel)–(COMLabel) node [midway,sloped,fill=white]
$l_{r}$; [Barlatex-latexBar] (COMLabel)–(LfLabel) node
[midway,sloped,fill=white] $l_{f}$; [Barlatex-latexBar] (LlrLabel)–(LlfLabel)
node [midway,sloped,fill=white] $L$;

Figure 2: Geometric representation of the bicycle model

The dynamics of the lateral motion of a vehicle can be represented as,

$Mv\left(\frac{\mathrm{d\beta}}{\mathrm{dt}}+\gamma\right)=2(Y_{f}+Y_{r})$ (1)

$I\frac{\mathrm{d\gamma}}{\mathrm{dt}}=N_{z}-N_{t}$ (2)

where $I$ is the vehicle’s inertia around the vertical axis. ${N_{z}}$ is the
control moment which is generated by the torque differences between the left
and right in-wheel motors and ${N_{t}}$ is the moment which is generated by
the lateral forces. Other terms are the same as the way described in the
preceding sections. Furthermore, $N_{t}$ can be represented as,

$N_{t}=2C_{f}\left(\beta+\frac{l_{f}}{v}\gamma-\delta\right)l_{f}-2C_{r}\left(\beta-\frac{l_{v}}{v}\gamma\right)l_{r}$
(3)

where $C_{f}$ and $C_{r}$ are the cornering stiffnesses of the front and rear
tyres, which are defined as the gradients of the lateral tyres force against
the slip angle. For a typical tyre, cornering stiffness increases linearly
with slip angle for a particular range before peaking, causing tyre forces to
be saturated. Cornering stiffnesses are treated as constants in linearized
dynamics, assuming that the vehicle is only operated inside the linear region
of the slip angle.

The yaw moment controller’s objective is to generate a corrective yaw moment
on the vehicle using differential motor torques in oversteer and understeer
situations. The corrective yaw moment is generated to maintain the desired yaw
rate. The required yaw rate is determined by the vehicle’s longitudinal
velocity and steering angle. The yaw rate of the vehicle should satisfy the
following equation during steady-state cornering:

$\gamma_{des}=\frac{v}{(L+Kv^{2})}\delta$ (4)

where L is the wheelbase and K is the stability factor given by,

$K=\frac{M}{2L}\left(\frac{l_{r}}{C_{f}}-\frac{l_{f}}{C_{r}}\right)$ (5)

If $K$ is negative, the car will oversteer, while if $K$ is positive, the
vehicle will understeer. As the equation (5) shows, $C_{f}$ and $C_{r}$ are
required to calculate the desired yaw rate required by the yaw moment
controller. The section IV discusses its estimation.

##  IV EDTSC Model Architecture

Figure 3:  Schematic diagram for motor plant with the reaction torque
observer.

The EDTSC proposed in this article is an adaptive model for controlling both
traction and yaw stability. First, a disturbance observer-based reaction
torque observer for an electric motor with current and rotor speed
measurements is established. This facilitates an accurate estimation of the
driving force acting on the wheels, resulting in more accurate direct torque
control. The reaction torque observer formulation will be discussed next,
followed by the direct torque control system, direct yaw moment control
system, differential action, and FIS for integrating the traction and yaw
stability control signals.

###  IV-A Reaction Torque Observer

The equations of the motor model can be expressed as follows:

$\mathrm{J\dot{\omega_{m}}}={K_{t}}I-{K_{f}}{\omega_{m}}-{T_{L}}$ (6)

$\mathrm{L\dot{I}}=-RI-{K_{b}}{\omega_{m}}+{V_{cmd}}$ (7)

Here, $\mathrm{J}={J_{m}}+{J_{w}}/G^{2}$ is the combined inertia of the rotor
($J_{m}$) and the wheel ($J_{w}$) and $G$ is the gear ratio between the motor
and the drive shaft. In equations (6) and (7), $\omega_{m}$ is the motor
speed, $I$ is the motor current, $T_{L}$ is the motor load torque, $V_{cmd}$
is the commanded voltage and, $R$ and $L$ are the resistance and the
inductance of the rotor winding. $K_{t}$, $K_{f}$, $K_{b}$ are the torque,
friction and back-EMF constants. The reaction torque ($T_{R}$) and the Coulomb
friction ($F_{c}$) can be used to calculate motor load torque as
${T_{L}}={F_{c}}+{T_{R}}$ .

The nominal motor model for velocity dynamics can be expressed as follows.

$\mathrm{{J_{n}}\dot{\omega}_{m}}={K_{tn}I+{K_{fn}}\omega_{m}-{T_{D}}}$ (8)

The nominal combined inertia is $J_{n}$, the nominal torque and friction
constants are $K_{tn}$ and $K_{fn}$, and the disturbance torque is $T_{D}$. It
is assumed that current dynamics disturbances (equation (7)) are
insignificant. As a result, for the nominal plant transfer function
($P_{n}(s)$) that will be utilised to observe the disturbance, only the
velocity dynamics are considered.

Figure 4: Schematic diagram of the direct torque controller.

$\mathrm{{P_{n}}(s)}=\frac{\omega_{m}(s)}{I(s)}=\frac{{K_{tn}}}{{J_{n}}s+{K_{fn}}}$
(9)

The inertia of the motion system will change depending on its mechanical
arrangement. Because of the irregular distribution of magnetic flux on the
rotor’s surface, the torque coefficient varies with rotor position in an
electric motor. As a result, the disturbance torque is a combination of load
torque and fluctuations in the constants $J_{n}$, $K_{tn}$, and $K_{fn}$. The
constants with uncertainty can be written as $J={J_{n}}+\Delta J$,
${K_{t}}={K_{tn}}+\Delta{K_{t}}$ and ${K_{f}}={K_{fn}}+\Delta{K_{f}}$. By
substituting these constants to equation (6), and comparing with the equation
(8), equation for disturbance torque can be derived as:

$\mathrm{{T_{D}}}=\Delta
J\dot{\omega_{m}}-\Delta{K_{t}}I-\Delta{K_{f}}\omega_{m}+{T_{L}}$ (10)

From the equation (8), $T_{D}$ can be expressed as,

$\mathrm{{T_{D}}}={K_{tn}}I-{K_{fn}}\omega_{m}-{J_{n}}{\dot{\omega}_{m}}$ (11)

For several reasons, the equation (11) cannot be used directly to calculate
$T_{D}$. To begin, when the calculation is implemented in a real system, the
inverse of the nominal plant’s transfer function (${P_{n}}^{-1}(s)$), which is
not proper, must be implemented to obtain the result of the equation (11).
However, improper transfer functions are not physically implementable [28].
Second, the equation incorporates possibly noisy measurements $I$ and
$\omega_{m}$. Third, the disturbance’s direct feedback results in an algebraic
loop. Furthermore, the time derivative of $\omega_{m}$ may increase the level
of noise.

A first-order low-pass filter was used to overcome these concerns with a
direct calculation of $T_{D}$ [28]. The so-called Q-filter must be a low-pass
filter with a relative degree greater than that of $P_{n}(s)$ and a wide
bandwidth, thus a high cut-off frequency. A first-order filter with a cut-off
frequency of $\omega_{c}$ is used in this work.

$\mathrm{Q(s)}=\frac{{\omega_{c}}}{s+{\omega_{c}}}$ (12)

The Q-filter is then used to get an estimate of ${\hat{T}_{D}}$ as follows:

$\mathrm{{\hat{T}_{D}}}={\frac{{\omega_{c}}}{s+{\omega_{c}}}}.{({K_{tn}I-{K_{fn}\omega(s)-s{J_{n}}\omega(s))}}}$
(13)

Overall motor dynamics with the disturbance torque observer is depicted in
Figure 3.

Figure 5: Controller schematics: (a) yaw stability controller, (b) traction
and stability controller with fuzzy inference system (FIS) for controller
integration.

The motor constants, namely, $J$, $K_{t}$, $K_{f}$, and $F_{c}$ can be
determined using system identification and it was assumed that their variation
during the operation is negligible. Therefore, from the equation (10) and
motor load torque expression, the estimated reaction torque can be derived as:

$\mathrm{\hat{T}_{D}}\simeq{T_{L}}={F_{c}}+{\hat{T}_{R}}$ (14)

$\mathrm{\hat{T}_{R}}\simeq{\hat{T}_{D}}-{F_{c}}$ (15)

###  IV-B Direct Torque Control

The wheel slip ratio is a significant parameter used in traction control to
avoid wheel skid and spin. To compute the slip ratio value of each wheel, the
vehicle speed and wheel speed must be known. The vehicle speed is measured in
conventional cars by the speeds of the non-driven wheels. Non-driven wheels
slip in various conditions, such as braking and cornering over tight turns,
and are thus unsuitable for calculating vehicle speed. An accelerometer cannot
avoid the problem of speed offset caused by the long-time integral calculation
[5]. Based on the method provided in [5], a slip-ratio estimator was
constructed to acquire an accurate wheel slip ratio value without detecting
wheel speeds. However, unlike the original method, the slip ratio is
calculated based on the reaction torque that was calculated by the reaction
torque observer described in the previous section.

The longitudinal motion equation of the vehicle can be expressed as:

$M\dot{v}=F_{d}-F_{dr}$ (16)

Where, $F_{d}$ is the driving force, $F_{dr}$ is the driving resistance, and
$M$ is the vehicle mass. The relationship between the driving force and the
reaction torque of the motor can be written using the wheel radius $r$ as
$F_{dr}=T_{R}/r$. The wheel speed ($\omega$) can be obtained using the gear
ratio of the gearbox ($G$) and motor speed $\omega_{m}$ as
$G\omega_{m}=\omega$.

Slip ratio ($\lambda$) is a value that represents the difference between the
vehicle speed and the tangential speed of the tire surface (${V_{w}}$) and is
defined as:

$\lambda=(V_{w}-V)/V_{w}$ (17)

where, ${V_{w}}=r\omega$. By differentiating the equation (17) and
substituting from above expressions, state equation for $\lambda$ is derived
as:

$\dot{\lambda}=\frac{\dot{\omega}_{m}}{\omega_{m}}\lambda+\frac{\dot{\omega}_{m}}{\omega_{m}}-\frac{\hat{T}_{R}}{r^{2}MG\omega_{m}}+\frac{F_{dr}}{rMG\omega_{m}}$
(18)

The motor torque is regulated based on the reference slip ratio values to
achieve the highest traction force at the tyre contact patch of each wheel. A
PID controller can be used to manage motor torque by adjusting the commanded
voltage by slip ratio error, as shown in Figure 4. However, a problem arises
when the suggested direct torque controller is used in conjunction with the
direct yaw moment controller. The control signals provided to the motor must
respect both controllers; therefore, achieving a model-based optimal control
strategy is challenging. To address this issue, FIS is proposed to generate
corrective factors for motor voltage commands that account for yaw rate error
as well as slip ratio, as stated in the following section.

###  IV-C Direct Yaw Moment Control

Since tyre cornering stiffnesses vary greatly depending on road conditions and
vehicle load transfer, it is desirable to devise a method for estimating them.
In previous research, a cornering stiffness estimator based on the Recursive
Least Square (RLS) algorithm was utilised. The RLS algorithm presented in [6]
calculates the yaw moment created by lateral forces ($N_{t}$) using the
measured yaw rate and estimated body slip angle, whereas the equations of
front and rear lateral forces ($Y_{f}$, $Y_{r}$) are employed in [29]. The
equations used in [29] were employed in this work, but the consideration of a
parabolic relationship between slip and cornering stiffness was eliminated
since the vehicle in this analysis is assumed to be operating in the linear
region of slip angle.

Front and rear lateral forces ($Y_{f}$, $Y_{r}$) can be derived as:

$Y_{f}=M_{y}a_{yf}=M\frac{l_{r}}{L}(v(\hat{\dot{\beta}}+\gamma)+l_{f}\dot{\gamma})\cos(\delta)$
(19)

$Y_{r}=M_{y}a_{yr}=M\frac{l_{f}}{L}(v(\hat{\dot{\beta}}+\gamma)-l_{r}\dot{\gamma})$
(20)

Here, $\hat{\dot{\beta}}$ is the estimated body slip angle and its derivation
will be discussed shortly. Now the regression equations can be constructed as
follows:

$\bm{Y}=\bm{\xi}\bm{\theta}$ (21)

where,

$\mathbf{Y}^{T}=\begin{bmatrix}\hat{Y_{f}}&\hat{Y_{r}}\end{bmatrix}\hskip
10.0pt\bm{\theta}^{T}=\begin{bmatrix}C_{f}&C_{r}\end{bmatrix}$ (22)

are the estimates of lateral forces and cornering stiffnesses respectively.
The regressor $\bm{\xi}$ was derived as:

$\bm{\xi}(s)=F(s)\begin{bmatrix}-2(\beta+\mfrac{l_{f}}{v}\gamma-\delta)&0\\\
0&-2(\beta-\mfrac{l_{r}}{v}\gamma)\end{bmatrix}$ (23)

where, $F(s)=\omega_{c}/(s+\omega_{c})$ is a low pass filter used to filter
out the noise of the time derivative terms by converting the transfer function
into a proper one. The estimation update is obtained from the following
equation.

$\bm{\hat{\theta}}(k)=\bm{\hat{\theta}}(k-1)+\bm{K}(\bm{Y}-\bm{\xi}(k)\bm{\hat{\theta}}(k-1))$
(24)

The optimum RLS gain was calculated such that the squared error through time
is minimized.

$\bm{K}=\bm{\Gamma}(k-1)\bm{\xi}^{T}(k)(\bm{R}+\bm{\xi}(k)\bm{\Gamma}(k-1)\bm{\xi}^{T}(k))^{-1}$
(25)

where, $\bm{\Gamma}$ is the estimation error covariance matrix and $\bm{R}$ is
the measurement error covariance. An update of $\bm{\Gamma}$, is obtained as
follows.

$\bm{\Gamma}(k)=(I-\bm{K}\bm{\xi}(k))\bm{\Gamma}(k-1)$ (26)

This regressor $\bm{\xi}$ consists of the body slip angle $\beta$. As a
result, an adaptive observer was used to estimate the body slip angle
($\hat{\beta}$). It is referred to as an adaptive observer because the
coefficients of the state equations presented below (equations (27) and (28))
are dependent on cornering stiffnesses and are updated at each time step by
the values provided by the cornering stiffness estimator (equation (24)).

$\bm{\hat{\dot{x}}=A(\hat{\theta})\hat{x}+B(\hat{\theta})(y-\hat{y})}$ (27)

$\bm{\hat{y}=C(\hat{\theta})\hat{x}+D(\hat{\theta})u}$ (28)

Here, $\bm{\hat{x}}$ and $\bm{\hat{y}}$ are the estimated state vector and
estimated output vector. $\bm{u}$ is the input vector. These three vectors are
defined as follows.

$\bm{x}=\begin{bmatrix}\beta\\\ \gamma\end{bmatrix}\hskip
5.0pt\bm{y}=\begin{bmatrix}\gamma\\\ a_{y}\end{bmatrix}\hskip
5.0pt\bm{u}=\begin{bmatrix}\delta\\\ N_{z}\end{bmatrix}$ (29)

The state equations of the observer are based on the Eqs. (1), (2) and (3).
$\bm{A}$ is the state transition matrix, $\bm{B}$ is the input matrix,
$\bm{C}$ is the output matrix and $\bm{D}$ is the direct transition matrix
which is defined as,

$\bm{A}(\theta)=\begin{bmatrix}-2\left(\mfrac{C_{f}+C_{r}}{Mv}\right)&-1-2\left(\mfrac{l_{f}C_{f}-l_{r}C_{r}}{Mv^{2}}\right)\\\\[10.00002pt]
-2\left(\mfrac{l_{f}C_{f}-l_{r}C_{r}}{I}\right)&-2\left(\mfrac{l_{f}^{2}C_{f}-l_{r}^{2}C_{r}}{Iv}\right)\end{bmatrix}$
(30)

$\bm{B}(\theta)=\begin{bmatrix}\mfrac{2C_{f}}{Mv}&0\\\\[10.00002pt]
\mfrac{2l_{f}C_{f}}{I}&\mfrac{1}{I}\end{bmatrix}$ (31)

$\bm{C}(\theta)=\begin{bmatrix}0&1\\\ va_{11}&v(a_{12}+1)\end{bmatrix}$ (32)

$\bm{D}(\theta)=\begin{bmatrix}0&0\\\ vb_{11}&0\end{bmatrix}$ (33)

where $a_{ij}$ and $b_{ij}$ are the elements of $A$ and $B$ matrices.

As shown in Figure 5(a), the estimated cornering stiffnesses are used to
calculate the yaw moment due to lateral forces using equation (3) and
compensate for it when the corrective yaw moment is generated. Furthermore, a
PI controller has been attached to generate a control yaw signal to track the
desired yaw rate.

The estimation error can be represented as $e=\beta-\hat{\beta}$ and
$\dot{e}=(\bm{A}-\bm{K}\bm{C})e$. For robust performance, the gain matrix
should be selected properly. All eigenvalues of $\bm{(A-KC)}$ must be negative
to ensure the stability of the observer. Based on the pole assignment and
robustness, the gain matrix $\bm{K}(\theta)$ was derived as,

$\bm{K}(\theta)=\begin{bmatrix}\mfrac{\lambda_{1}\lambda_{2}(l_{f}C_{f}-l_{r}C_{r})I}{(2C_{f}C_{r}(l_{f}+l_{r})^{2})}-1&\mfrac{1}{v}\\\
-(\lambda_{1}+\lambda_{2})&\mfrac{M(l_{f}^{2}C_{f}+l_{r}^{2}C_{r})}{I(l_{f}C_{f}+l_{r}C_{r})}\end{bmatrix}$
(34)

where, $\lambda_{1}$ and $\lambda_{2}$ are the assigned poles of the observer.

###  IV-D Differential Action

During cornering, the wheels rotate at different speeds because each wheel has
its own turning radius. The turning radius of the vehicle can be obtained
using the steering angle ($\delta$) and yaw rate ($\gamma$) as follows,

$R=L/tan(\delta)$ (35)

The radii of rear left and rear right wheels ($R_{l}$ and $R_{r}$) can be
derived as ${R_{l}}=R+t/2$ and ${R_{r}}=R-t/2$, where $t$ is the track width.
The commanded voltage on the motors ($V_{cmd}$) was considered as a function
of the torque demand of the driver ($T_{dem}$) that is sensed by the
accelerator pedal position sensor. Then $V_{cmd}$ was modified based on the
turning radii of each wheel to generate the individual commanded voltages for
left and right wheel motors ($V_{cmd,l}$,$V_{cmd,r}$ respectively).

${V_{cmd,l}}=\mfrac{R_{l}}{R}{V_{cmd}}\hskip
10.0pt{V_{cmd,r}}=\mfrac{R_{r}}{R}{V_{cmd}}$ (36)

It should be noted that the turning radii of the two wheels are not the same
as the values given by $R_{l}$ and $R_{r}$ expressions when the vehicle is
running. This is due to non-zero slip angles generated by each wheel due to
lateral tire forces. Thus, the radii must be modified depending on the
cornering stiffnesses. But in this context, it is assumed that the error of
cornering radii is small enough since the yaw rate is controlled in such a way
that slip angles hardly exceed the $None$ range.

###  IV-E Controller Integration using Fuzzy Inference System (FIS)

TABLE II: Fuzzy rules for controller integration

$V_{corr,l}$ | Yaw Rate Error ($\gamma_{err}$)
---|---
NL | NS | Z | PS | PL | NL | NS | Z | PS | PL
Left Motor | Right Motor
Slip
Ratio | VS | PL | PL | Z | NL | NL | NL | NL | Z | PL | PL
S | PL | PS | N | NS | NL | NL | NS | N | PS | PL
M | PS | Z | NS | NL | NL | NL | NL | NS | Z | PS
L | PS | Z | NL | NS | NL | NL | NS | NL | Z | PS
VL | Z | Z | NL | NL | NL | NL | NL | NS | Z | Z

Normalised corrective voltage ($V_{corr}$) and yaw rate error ($\gamma_{err}$)
fuzzy sets: Negative Large: NL, Negative Small: NS, Zero: 0, Positive Small:
PS, Positive Large: PL. Normalised slip ratio ($\lambda$) fuzzy sets: VS: Very
Small, S: Small, M: Medium, L: Large, and VL: Very Large.

Figure 6: Input and output membership functions of the fuzzy inference system
(FIS): (a) Normalised yaw rate error ($\gamma_{err}$), (b) Normalised slip
ratio ($\lambda$) and (c) Normalised corrective voltage ($V_{corr}$).

The FIS generates modifications needed for the commanded voltages using
correction factors so that the traction and yaw rate are controlled as
expected by the individual controllers. The inputs to the FIS are slip ratio
($\lambda$) and yaw rate error ($\gamma_{err}$) which is defined as follows.

$\gamma_{err}=\gamma-\gamma_{des}$ (37)

The voltage correction factors for left and right wheel motors are
$V_{corr,l}$ and $V_{corr,r}$ respectively. All variables except for slip
ratio were normalized and fuzzified into five fuzzy sets namely, NL - negative
large, NS - negative small, Z - zero, PS - positive small, and PL - positive
large. Since only positive and zero slip ratio values were considered (e.i
non-braking situations) a separate set of five fuzzy sets were used: VS - Very
Small, S - Small, M - Medium, L - Large, VL - Very Large. Triangular
membership functions were used for all four variables as shown in Figure 6.
Min and Max functions were employed in fuzzy rules for AND and OR operations.
A Mamdani-type fuzzy inference system was employed with the Min implication
and Max aggregation methods. The centroids of the inferred membership
functions were used for defuzzification. Table II shows the rule base for the
right and left motors. The corrective yaw moment is generated by the
difference between the correction factors of the two wheels. When a non-
trivial yaw rate error is identified, the correction factor for one wheel
increases while the other decreases. However, when a high slip ratio is
detected, both correction factors are shifted more towards the negative values
to make sure that the magnitude of the positive correction factor is as small
as possible, but the torque difference is enough to correct the yaw moment at
the same time.

##  V Model Implementation and Experiments

TABLE III: Simulation model parameters ((a)) Dimensional parameters of the
vehicle used in the simulation model.

M (kg) | Iz (kgm2) | lf (m) | lr (m) | w (m) | Jw (kgm2) | d (m)
---|---|---|---|---|---|---
260 | 60 | 0.83 | 0.7 | 1.2 | 0.23 | 0.46

((b)) Motor parameters used in the simulation model.

Jm (kgm2) |  Kt (NmA-1) |  Kf (Nmrad-1s) |  KE (Vrad-1s) |  R ($\Omega$) |  L (H) |  Pole pairs
---|---|---|---|---|---|---
1.26e-2 | 0.5 | 0.01 | 0.04 | 7.0e-3 | 7.6e-5 | 10

Figure 7: FSG Hockenheimring race track: Segment marked in red was used for
the simulation.

This section describes the simulation model that was used to implement the
EDTSC and the experimental method followed to assess the performance of EDTSC.

###  V-A Simulation Model

The EDTSC was integrated to a twin-track non-linear transient-state vehicle
model with non-linear tire dynamics. Aerodynamic forces were added to the
model to improve the accuracy of the simulation. Parameters of a formula-type
electric racecar was used for the simulation. The parameters are given in
table III(b). A permanent magnet synchronous motor model was used as the
power-plant for the vehicle model and its parameters are given in table
III(b). A simplified version of a pacejka tire model [30] was used to simulate
tire dynamics.

The 6-DOF model was implemented MATLAB and solved numerically using the MATLAB
ode23 ordinary differential equation solver. Furthermore, a constant time step
of 5ms was used. All simulations were performed on a laptop with an i7-10750H
processor (2.60 GHz) and 8GB of RAM running on Windows 10.

###  V-B Experiments

A predefined track using an adaptive driver model and double lane change by
following a reference path was used to validate the proposed EDTSC in a
simulated environment. In each scenario, the initial phase of testing was done
to confirm that the cornering stiffness estimator, reaction force observer and
slip ratio estimator are closely estimating the parameters on which the EDTSC
rely on. For track simulation, the Formula Student Germany (FSG)
Hockenheimring endurance track was used [31]. For simplicity, only the results
of a segment (represented by red in Figure 7) are discussed in the next
section VI. The driver model was fed with a series of curvature-distance data,
and its task was to keep the path as close to the prescribed curvature as
possible. The initial vehicle velocity was set to zero.

The ISO 3888-2:2011 standard [32] was used to define the double lane change
path. The path was tracked by the driver model in the same way as in the
previous case, with the exception that the driver let off the throttle at the
start of the path after reaching the test velocity. The controller was put
through its paces at two different longitudinal speeds: 40 km.h^-1 and 100
km.h^-1.

##  VI Results

This section presents the numerical results obtained when the EDTSC
architecture presented in section IV is applied to the simulation model in
section V.

###  VI-A Race Track Simulation

Figure 8: Variation of estimated front and rear cornering stiffnesses ($C_{f}$
and $C_{r}$) with time for the race track simulation. Figure 9: Reaction force
(top) and Slip ratio (bottom) estimates against time. Figure 10: Race track
simulation results against time: (i) - Variation of slip ratio ($\lambda$),
(ii) - Variation of yaw rate error ($\gamma_{err}$), with and without fuzzy
inference system (FIS), (iii) - Variation of normalized corrective voltage
factors ($V_{corr}$) generated by the FIS.

The accuracy with which the cornering stiffness estimator estimates the actual
cornering stiffnesses are shown in Figure 8. It is evident that the estimator
is capable of reaching the actual cornering stiffness values faster.

The top graph of Figure 9 depicts actual reaction force ($F_{R}$) and observed
reaction force variations against time. It can be seen that for positive
reaction force values, the reaction force observer tracks the actual force.
However, a significant deviation can be observed at negative values because
the reaction force observer does not account for braking situations. This is
not a problem because the proposed FIS copes with non-braking situations only.
In accelerating situations, the observer can be seen tracking the actual value
accurately with a correlation coefficient $>0.85$. The slip ratio ($\lambda$)
estimation is accurate, and the error is negligible (with a correlation
coefficient $>0.99$) as shown in the bottom graph of Figure 9. This suggests
that slip ratio can be safely used in the FIS.

TABLE IV: Simulation Results

| Simulation
---
type
Parameter |  | RMS value
---
reduction (%)
| Peak value
---
reduction (%)
Race track | $\lambda$ | 96.14 | 42.31
$\gamma_{err}$ | 88.17 | 71.93
Double lane change |  $\gamma_{err}$ (40 km.h^-1) | 86.96 | 97.93
$\gamma_{err}$ (100 km.h^-1) | 92.34 | 120.24

$\lambda$: Slip ratio, $\gamma_{err}$: Yaw rate error

Figure 10 depicts the resulting slip ratio ($\lambda$), yaw rate error
($\gamma_{err}$), and FIS output ($V_{corr}$) variations. Figure 10-(i) shows
the slip ratio improvement with the FIS. The FIS has flattened out the peaks
caused by the sudden increase in wheel velocities. Figure 10-(ii) depicts the
compensation for yaw rate error, and it shows how a lower error could be
achieved throughout the run time compared to the case without the FIS. The
percentage reductions in slip ratio and yaw rate error for track simulation
are shown in Table IV. Another significant finding is that the total time
required by the vehicle to complete a track segment with FIS ($18.845$ s) is
less than that required without it ($18.85$ s). This indicates that the
implemented EDTSC system does not degrade the vehicle’s overall performance in
exchange for improved stability, but rather the opposite. This $0.3\%$ time
improvement is significant in terms of racecar lap times where racecars have
to complete multiple laps.

As shown in Figure 10-(iii), the values of the corrective voltage factors are
biased towards positive values due to the nature of the membership functions.
This ensures that the motor control saturation caused by the generation of
negative voltage commands is limited, thereby increasing the range of the
corrective yaw moment.

###  VI-B Double Lane Change Simulation

Figure 11: Double lane change manoeuvre simulation results with with and
witout Fuzzy Inference System (FIS): A(i) - A(iv) for 40 km.h^-1 and B(i) -
B(iv) for 100 km.h^-1.

The paths traced by the vehicle with and without the FIS are shown in 11 A(i)
and B(i). It is clear that the addition of the FIS has not compromised path
tracking, and it will be demonstrated in the following that stability has
improved. Although the driver’s throttle input is zero during the double lane
change manoeuvre, the FIS is permitted to generate non-zero motor voltage
commands to ensure that the yaw rate is as close to the desired value as
possible. This explains why the curves in 11 A(ii) and B(ii) show non-zero
reaction forces. The correlation between the observed reaction force and
actual reaction force is $>0.8$ for 40 km.h^-1 and $>0.75$ for 100 km.h^-1. As
a result of the FIS’s behaviour, the slip ratio becomes positive at each
location where the vehicle attempts to negotiate a curve, as illustrated in 11
A(iii) and B(iii). However, it is clear that when the vehicle is run without
the FIS, the slip ratio is negative and less than $2\times 10^{-4}$, whereas
it increases at the beginning of curved parts of the path when the FIS is
used. The root mean square slip ratios for 40 km.h^-1 test is $<0.001$ and for
100 km.h^-1 test it is $<0.002$. Because the peak slip ratio is within the
linear region of the tyre longitudinal force ($\lambda<0.04$ for 40 km.h^-1,
$\lambda<0.08$ for 100 km.h^-1), this sacrifice in low slip ratio is
acceptable.

From Figure 11 A(iv) and B(iv) it can be concluded that the FIS performs well
in reducing the yaw rate error relative to the desired yaw rate. The
reductions in root mean square and peak values of yaw rate error are given in
Table IV.

It was also observed that increase in FIS gain could lead to oscillatory
behaviour of yaw rate thus the gain value must be carefully selected otherwise
it would result in instability. However, the selection of FIS gain considering
the stability of the controller is out of the scope of this research. The
oscillatory behaviour origins from the corrective voltage factors that are
generated by the FIS and is evident from 11 B(v). As shown in Figure 11 A(v)
and B(v), the corrective voltage factors suddenly goes up at high yaw rate
error points (during cornering) to generate the corrective yaw moment. As in
the track simulation case, both factors are biased towards positive to avoid
quick zero saturation of motor torque.

##  VII Conclusion

This article proposes a novel controller for electronic differentials with an
FIS to integrate the direct torque and direct yaw moment controllers. The
controller was integrated with a 6-DOF vehicle model and simulated for two
test cases. The slip ratio and cornering stiffness values were estimated
successfully to generate EDTSC reference parameters.To construct the cornering
stiffness estimator’s regressor in the RLS algorithm, a slip angle observer
based on the bicycle model was used. The slip ratio was estimated closely to
the actual value using the reaction torque observer that is based on a
disturbance observer.

To reduce dynamical instabilities, the FIS could handle the trade-off between
slip control and yaw rate control by sending appropriate control signals to
the traction motors. Accurate enough slip reaction force and slip ratio
estimations could be achieved in various drive conditions. Furthermore, the
FIS could secure a substantial reduction in yaw rate error relative to the
desired yaw rate and mitigate the peak slip ratios resulted from sudden change
in driving maneouver. The compromise in slip ratio during double lane change
test was negligible. Future work could focus on implementing the EDTSC in a
test vehicle to evaluate the performance. In addition, four-wheel electric
drive vehicles could be considered when developing the control architecture to
improve traction and stability.

## Acknowledgment

The authors thank the Falcon E Racing, the electric formula student team of
the University of Moratuwa for the insightful discussions. Also, the authors
acknowledge the technical and financial sponsors of the formula student
project.

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