vehicle steer by wire

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 4, APRIL 2016 2251
Robust Control of a Vehicle Steer-by-Wire
System Using Adaptive Sliding Mode
Zhe Sun, Jinchuan Zheng, Member, IEEE, Zhihong Man, Member, IEEE,
and Hai Wang, Member, IEEE
Abstract—This paper presents an adaptive sliding-mode
(ASM) control methodology for a vehicle steer-by-wire
(SbW) system. First, the SbW system is modeled as a
second-order system from the steering motor input volt-
age to the front-wheel steering angle. For simplicity, the
self-aligning torque and friction arising from the tire-to-
ground contact are regarded as external disturbance acting
on the SbW system. Next, an ASM controller is designed
for the SbW system, which can not only cope with the
parametric uncertainties in the plant model but also esti-
mate the coefficient of the self-aligning torque effectively.
The stability of the ASM control system is proved in the
sense of Lyapunov and the guidelines for selecting the
control parameters are given. Finally, experiments are car-
ried out for steering control to respectively follow a slalom
path and a circular path under various road conditions. It
is shown that the proposed ASM controller can achieve
stronger robustness against various road conditions lead-
ing to significantly smaller tracking errors in comparison
with a conventional sliding-mode controller and a linear
H∞ controller.
Index Terms—Adaptive sliding mode (ASM), robust
control, self-aligning torque, vehicle steer-by-wire (SbW)
system.
I. INTRODUCTION
A S a newly developed technology, drive-by-wire technol-
ogy has been playing a significant role in the automotive
industry. The main idea of drive-by-wire is to use electrome-
chanical actuators and electronic control systems to replace the
conventional mechanical linkages [1]. Different components
of a vehicle constitute different parts of the by-wire technol-
ogy, such as steer-by-wire (SbW) [2], [3], brake-by-wire [4],
[5], shift-by-wire [6], and throttle-by-wire [7] technologies. As
one part of the drive-by-wire systems, SbW system is an inno-
vative technology for automobile steering applications, which
possesses several remarkable advantages. First, the removal of
the steering column can eliminate the injury to a driver when
there is a front-end collision to a large extent. Second, in a
Manuscript received February 11, 2015; revised September 23, 2015;
accepted October 21, 2015. Date of publication November 10, 2015;
date of current version March 8, 2016.
Z. Sun, J. Zheng, and Z. Man are with the School of Software and
Electrical Engineering, Swinburne University of Technology, Melbourne,
Vic., 3122, Australia (e-mail: zsun@swin.edu.au; jzheng@swin.edu.au;
zman@swin.edu.au).
H. Wang is with the School of Electrical Engineering and Automation,
Hefei University of Technology, Hefei 230009, China (e-mail: haiwang_
1229@hfut.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2015.2499246
SbW system, the traditional mechanical linkages and hydraulic
systems between the steering wheel and front wheels are
replaced by electromechanical actuators and human–machine
interfaces, which not only reduce weight, noises, vibrations
and energy consumptions but also increase the freedom and
capability for a driver to tune the steering command, such that
the stability and maneuverability of a vehicle are improved
[8]. Finally, a feedback motor is typically coupled to the
steering wheel and provides the driver with mimic haptic feed-
backs of the interaction forces between front wheels and road
surface.
As the mechanical structure of a SbW system is different
from traditional steering systems, large amounts of studies have
been carried out on the mechanical modeling of SbW systems
in recent years. In [9], Gadda et al. used dc permanent mag-
net motors as the steering motors and reasonably modeled the
electrical properties as the series connection of an inductor,
a resistor, and a voltage source. In [10], Wang et al. utilized
two second-order differential equations to describe the effect of
tire forces in the front wheel and the dynamics in the steering
wheel, respectively. A simple second-order model is adopted in
[11] to represent the dynamics of SbW systems, where the self-
aligning torque was treated as an external disturbance resisting
the steering motion away from the straight-ahead position. In
addition, simplified second-order mathematical models were
used in [12] to describe the dynamics of the driver interface
subsystem and the direction control subsystem, respectively.
Apart from the modeling investigation, an appropriate con-
trol system is essential for the SbW system to regulate the
steering motor to generate the desired steering torques such that
the front wheels follow the steering wheel reference command
accurately for paths following. A variety of control method-
ologies for vehicle steering systems are reported in literature,
e.g., the sliding-mode-based learning control [13], robust H∞
control [14], iterative learning control [15], and neural network
control [16]. In particular, Yamaguchi and Murakami [17] pre-
sented an adaptive control system to estimate the tire cornering
stiffness, which is then used to realize the desired steering char-
acteristics. In [18], Falcone et al. presented a model predictive
control scheme for the steering system of an autonomous vehi-
cle, which computes the steering angle of front tires to follow a
known trajectory on slippery roads at possibly highest speed. In
[19], a haptic feedback control strategy with adjustable control
gains is provided for a vehicle steering system to increase the
drivers’ awareness of the roadway and driving conditions and to
enable the driver to avoid obstacles appearing in the road earlier
than that without feedback informations.
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2252 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 63, NO. 4, APRIL 2016
Sliding-mode control is a control methodology with strong
robustness to handle the problems of parametric uncertain-
ties existing in plant models and only the upper and lower
bounds of the parameters in the plant model are needed.
In [20], Wang et al. utilized a nominal feedback controller
to stabilize the nominal part of the plant model and aug-
mented a sliding-mode compensator to cope with the inﬂuence
of both the unknown system dynamics and uncertain road
conditions for a vehicle steering system. Adaptive control is
another control method which plays an important role in the
area of motion control owing to its effectiveness to deal with
time-varying parameters in a plant model. In [21], Ma and
Nejhad proposed an independent modal space adaptive con-
trol scheme and an adaptive fuzzy logic control strategy for
the manipulator vibration control and the active strut motion
control, respectively. In [22], an adaptive fuzzy controller
is proposed for an MEMS triaxial gyroscope, which elim-
inates the effect of system nonlinearities without the need
of accurate mathematical plant models. It is also quite com-
mon for researchers to combine sliding-mode control with
adaptive control to obtain more comprehensive advantages
[23]–[28]. For instance, in order to attain accurate positioning,
Sencer and Shamoto presented a robust adaptive sliding-mode
(ASM) control method to reject harmonic disturbances in servo
systems [29].
In this paper, we present an ASM control methodology for a
developed SbW system, where the adaptive control is utilized to
estimate the coefﬁcient of the self-aligning torque acting on the
steering system. Then, feedforward control inputs equivalent to
the estimated self-aligning torques are generated to compen-
sate for the effect of the self-aligning torques. Furthermore, a
feedback controller based on sliding mode is adopted to cope
with the system parametric uncertainties. The stability of the
ASM controller on the SbW system is proved in the sense of
Lyapunov. The guidelines for selecting the control parameters
are also given. Finally, experimental results are presented to
demonstrate the superior performance of the proposed ASM
controller in steering control for both a slalom path following
and a circular path following compared with a linear H∞ con-
troller and a conventional sliding-mode controller for the SbW
system.
The main contribution of this paper is that we combine
an adaptation method with the sliding-mode control, which is
theoretically proved to guarantee the tracking error to asymp-
totically converge to zero when applied to the SbW system.
The adaptation control law not only alleviates the chatter-
ing problem in the sliding-mode controller but also provides
explicit estimation of the self-aligning torque that can be
directly used by the steering wheel feedback motor to provide
the driver with road feels. Furthermore, experimental results
demonstrate that the developed controller is simple for practi-
cal implementation as well as effective to improve the tracking
accuracy.
This paper is organized as follows. Section II describes the
dynamic plant model of the SbW system. In Section III, the
ASM controller is designed for the SbW system. Section IV
presents and compares the experimental results of the ASM
Fig. 1. Experimental setup of an SbW system.
controller, the conventional sliding-mode controller, and the
H∞ controller. Section V concludes this paper.
II. PLANT MODELING
Fig. 1 shows our experimental setup for a SbW system,
where the conventional mechanical linkage between the steer-
ing wheel and the front wheels is removed and replaced by
a steering motor (Mitsubishi HF-SP102). The steering motor
is then controlled to provide appropriate torques to steer the
front wheels through a gear head, a pinion and rack gear box
and steering arms. The controller for the SbW system is imple-
mented on a HP personal computer (PC) by using MATLAB
Real-Time Workshop and an Advantech PCI multifunction card
to collect sensor signals and to generate control input signals in
real time. A motor servo driver is used to convert the control
input signals to current signals to drive the steering motor for
steering the front wheels. An angle sensor (MoTeC) is installed
on the pinion to measure the rotation angle of the pinion.
Multiplying the angle measurement by the transmission gain
from the pinion to the front wheels yields the steering angle of
the front wheels, which is the system output to be controlled in
our SbW system. In Fig. 1(b), an angle sensor is also installed
on the steering wheel to detect the angle of the steering wheel
handled by a driver. Similarly, multiplying this angle measure-
ment by a scaling factor from the steering wheel to the front
wheels yields the steering wheel reference angle, which is actu-
ally the reference command for the steering motor controller to
track.

SUN et al.: ROBUST CONTROL OF A VEHICLE SBW SYSTEM 2253
Based on a simplified bicycle model for vehicles [11], we
shall express the plant model of the SbW system as
Je ¨x + Be ˙x = κu − fc − τsel
κ = κ1 · κ2 · κ3 · κ4
fc = ξf sign( ˙x)
(1)
where x represents the steering angle of the front wheels, u is
the control input of the steering motor, Je and Be are the equiv-
alent moment of inertia and the viscous friction of the steering
system, respectively; fc is the Coulomb friction with ξf the
Coulomb friction constant; τsel is the self-aligning torque acting
on the steering system; κ1 is the scale factor accounting for the
conversion from the steering motor input voltage to the steering
motor output torque; κ2 is the gear ratio of the gear head; κ3 is
the gear ratio of the pinion and rack system; κ4 is the scale fac-
tor to account for the transmission from the linear motion of the
rack to the steering angle of front wheels; and sign(·) denotes
the standard signum function.
Note that in our experimental setup, the steering motor servo
driver has a much higher bandwidth than that of the mechan-
ical dynamics of the SbW system. Thus, we simply use the
scaling factor κ1 to approximate the model from the steering
motor input voltage to the steering motor output torque [10].
The values of the scaling factors contained in κ are given by
κ1 = 1.8
κ2 = 8.5
κ3 = 3.0
κ4 = 6.0.
(2)
Since the value of κ is fairly constant in time in our setup, it
is simply regarded as a constant during the control design as
will be addressed later. However, in this paper, we consider the
following parametric uncertainties with the bounds given by:
|ΔJe
| = |Je − Je0| ≤ ¯ΔJe
(Je0 = 85.5 kgm2
, ¯ΔJe
= 0.1Je0)
|ΔBe
| = |Be − Be0| ≤ ¯ΔBe
(Be0 = 218.8 Nms/rad, ¯ΔBe
= 0.1Be0)
|Δξf
| = |ξf − ξf0| ≤ ¯Δξf
(ξf0 = 4.2 Nm, ¯Δξf
= 0.1ξf0)
(3)
where Je0, Be0, and ξf0 denote the nominal parameters in our
plant model; ΔJe
, ΔBe
, and Δξf
are the parametric uncer-
tainties; ¯ΔJe
, ¯ΔBe
, and ¯Δξf
represent the bounds of the
corresponding parameters, respectively.
As shown in Fig. 1, limited by the experiment equipment,
our SbW platform cannot provide forward velocity for the front
wheels. Besides, the lack of vehicle weight distributed on the
front wheels results in the deficiency of the front-wheel cam-
ber angle [8]. Hence, during the experiments, the tires are not
exerted by actual self-aligning torques. However, under the
assumption of small slip angles of the tire, we can utilize a
hyperbolic tangent signal [10], [12] to mimic the self-aligning
torque, which can be expressed as
τsel = ρτ tanh(x) (4)
where ρτ is a time-varying coefficient with respect to various
road conditions, and tanh(·) represents the hyperbolic tangent
function
tanh(z) =
e2z
− 1
e2z + 1
. (5)
Now, it is easy to obtain the following voltage signal:
usel =
τsel
κ
+ ns =
ρτ
κ
tanh(x) + ns (6)
which is artificially generated during the experiments to mimic
the actual self-aligning torque acting on the SbW system. Note
that ns denotes a white noise signal added to account for
the unmodeled dynamics and uncertainties in the self-aligning
torque. In this way, we can evaluate the robustness of the
developed controller to some extent through experiments.
III. CONTROL DESIGN
Our control objective is to design a robust controller such that
the front-wheel steering angle can track a reference command
not only fast but also accurately in the presence of modeling
uncertainties and external disturbances. In order to improve
tracking accuracy and robustness, an ASM controller is devel-
oped to handle parametric uncertainties in the plant model and
estimate the coefficient of the self-aligning torque which is then
used to design a feedforward controller to compensate for the
self-aligning torque that is the primary disturbance. Finally, we
also present a conventional sliding-mode controller and an H∞
controller for comparison.
A. ASM Control for SbW Systems
Define the tracking error e as
e = xr − x (7)
where xr is the steering wheel reference command assumed to
be twice differentiable. Furthermore, a linear sliding variable s
is defined as
s = ˙e + λe (8)
where λ > 0 is to be designed.
Solving the sliding-mode dynamics as given by
˙s = λ˙e + (¨xr − ¨x) = 0 (9)
and neglecting the parametric uncertainties and the self-
aligning torque, namely, utilizing the nominal parameters to
replace the actual ones in (1) and supposing τsel = 0, we obtain
an expression of u0 which is also called the equivalent control
input [30] as follows:
u0 =
1
κ
[Je0λ˙e + Je0 ¨xr + Be0 ˙x + ξf0sign( ˙x)]. (10)
In order to guarantee the robustness of the controller against
the parametric uncertainties and the Coulomb friction fc, a
reaching control input u1 [31] is introduced as
u1 =
1
κ
[ s + Ksign(s)] (11)

where > 0 is to be designed, and K is given by
K = ¯ΔJe
λ|˙e| + ¯ΔJe
|¨xr| + ¯ΔBe
| ˙x| + ¯Δξf
. (12)
Lastly, a control input u2 is proposed to exclusively compensate
for the self-aligning torque acting on the SbW system
u2 =
ˆρτ
κ
tanh(x) (13)
where ˆρτ denotes the estimation of the actual coefficient ρτ ,
whose adaptation law is given by
˙ˆρτ = μstanh(x) (14)
with μ > 0 denotes the adaptation gain [29] to be designed and
s the sliding variable as defined in (8). The design of μ will be
detailed in Section III-B.
Lemma 1: Consider the SbW system (1) with the parametric
uncertainties in (3) and under the ASM control law
u = u0 + u1 + u2 (15)
with u0 defined in (10), u1 in (11), and u2 in (13), respectively.
Then, the tracking error (7) of the SbW closed-loop system
can asymptotically converge to zero for a given steering wheel
reference command.
Proof: Choose the Lyapunov function as
V =
1
2
s2
+
1
2μJe
(ρτ − ˆρτ )2
. (16)
Evaluating the first-order derivative of V along the system
trajectories yields
˙V = s ˙s +
1
μJe
(ρτ − ˆρτ )( ˙ρτ − ˙ˆρτ ). (17)
For a specific road condition, we assume ρτ is a constant
implying ˙ρτ = 0. Therefore, we have
˙V = s ˙s +
1
μJe
(ρτ − ˆρτ )(− ˙ˆρτ )
= s ˙s −
1
Je
(ρτ − ˆρτ )stanh(x)
= s(λ˙e + ¨xr − ¨x) −
1
Je
= s λ˙e + ¨xr +
Be
Je
˙x +
ξf
Je
sign( ˙x) +
ρτ
Je
tanh(x)
−
κu
Je
−
1
Je
(ρτ − ˆρτ )stanh(x). (18)
Substituting the control input (15) into (18) yields
˙V = s λ˙e + ¨xr +
Be
Je
˙x +
ξf
Je
sign( ˙x) +
ρτ
Je
tanh(x)
−
κ
Je
1
κ
[Je0λ˙e + Je0 ¨xr + Be0 ˙x
+ ξf0sign( ˙x) + s + ˆρτ tanh(x)
+ Ksign(s)]} −
1
Je
= s
ΔJe
Je
λ˙e +
ΔJe
Je
¨xr +
ΔBe
Je
˙x
+
Δξf
Je
sign( ˙x) −
K
Je
sign(s) −
Je
s
= −
Je
s2
+
Kms − K|s|
Je
(19)
with
Km = ΔJe
λ˙e + ΔJe
¨xr + ΔBe
˙x + Δξf
sign( ˙x). (20)
According to the following inequalities:
ΔJe
λ˙es ≤ ¯ΔJe
λ|˙e||s|
ΔJe
¨xrs ≤ ¯ΔJe
|¨xr||s|
ΔBe
˙xs ≤ ¯ΔBe
| ˙x||s|
Δξf
sign( ˙x)s ≤ ¯Δξf
|s|
(21)
we have
ΔJe
λ˙es + ΔJe
¨xrs + ΔBe
˙xs + Δξf
sign( ˙x)s
≤ ¯ΔJe
λ|˙e||s| + ¯ΔJe
|¨xr||s| + ¯ΔBe
| ˙x||s| + ¯Δξf
|s| (22)
namely,
Kms ≤ K|s|. (23)
Based on the above analysis, we can easily conclude that
˙V = −
Je
s2
+
Kms − K|s|
Je
< 0. (24)
The proof is thus completed.
Remark 1: In the ASM controller, we employ the adaptation
law to estimate the coefficient of the self-aligning torque ρτ
without the need of any prior information for ρτ . This is the
key benefit of adaptive estimation because the road conditions
are typically unknown to the vehicle system in reality. On the
other hand, for the uncertain parameters Je, Be, and ξf , their
nominal values and bounds can be accurately identified offline.
Thus, by using the sliding-mode control, the effects of these
uncertainties on the system performance can be compensated
fast and effectively.
Remark 2: The control input u1 (11) contains a discontinu-
ous term Ksign(s) which may induce undesired chattering to
the control signal. To alleviate this effect, the boundary layer
technique [32], [33] can be adopted. More specific, we use the
following saturation function to replace the signum function
in (11):
sat(z) =
z/ψ, if |z| < ψ
sign(z), if |z| ≥ ψ
(25)
where ψ denotes the boundary layer thickness.
B. Estimation of Coefficient of Self-Aligning Torque
As aforementioned, the self-aligning torque can be estimated
by adaptively estimating its coefficient ρτ as shown in (14). To

facilitate the adjustment for either good estimation accuracy or
fast estimation rate, we set the adaptation gain μ as a linear filter
[29] as follows:
μ = μ1 + μ2p (26)
where μ1 > 0, μ2 > 0, and p is the Laplace operator. Note
that the derivative of the sliding variable s is given in (19).
Substituting ˙s and (26) into the adaptation law described in (14)
yields
˙ˆρτ = μstanh(x)
= μ1stanh(x) + μ2 ˙stanh(x)
= μ1stanh(x) + μ2[−
s
Je
+
tanh(x)
Je
ρτ
−
tanh(x)
Je
ˆρτ +
Km − Ksign(s)
Je
]tanh(x)
= μ1stanh(x) −
μ2 stanh(x)
Je
+
μ2tanh2
(x)
Je
ρτ
−
μ2tanh2
(x)
Je
ˆρτ + w (27)
with
w =
μ2
Je
[Km − Ksign(s)]tanh(x). (28)
Rewrite (27) by using the Laplace transform, we have
μ2tanh2
(x)
Je
+ p ˆρτ =
μ2tanh2
(x)
Je
ρτ
+
(μ1Je − μ2 )stanh(x)
Je
+ w.
(29)
It can be seen that if there are no parametric uncertainties except
the self-aligning torque acting on the SbW system, selecting the
design parameter μ1 as
μ∗
1 = μ2
Je
(30)
leads to the term (μ1Je − μ2 )stanh(x)/Je = 0 as well as
w = 0. Thus, the adaptation dynamic equation (29) reduces to
ˆρ∗
τ =
μ2tanh2
(x)/Je
μ2tanh2
(x)/Je + p
ρτ (31)
where μ∗
1 and ˆρ∗
τ denote the parameters under the ideal condi-
tion without uncertainties. However, due to the existing para-
metric uncertainties in the plant model and also because the
equivalent moment of inertia Je is not completely known in
practice, we simply use the nominal parameter Je0 to replace
Je in (30), i.e.,
μ1 = μ2
Je0
. (32)
Substituting (32) into (29) yields
ˆρτ =
μ2tanh2
(x)/Je
μ2tanh2
(x)/Je + p
ρτ + da (33)
with da referred to as the adaptation perturbation, which is
given by
da =
μ2
Je
[Km − Ksign(s)]tanh(x)
+
ΔJe
Je0Je
μ2 stanh(x). (34)
It is noted from (34) that the adaptation perturbation da is
bounded in the presence of uncertainties and actually reduces
to zero without parametric uncertainties. This implies that ˆρτ
will converge to the actual ρτ in the case of no parametric
uncertainties or approach to a region close to the actual ρτ
when uncertainties arise. Moreover, we can see that the value
of μ2 affects the convergence rate of the estimation (i.e., the
adaptation bandwidth). The selection of μ2 will be discussed in
Section III-C.
C. Selection of Controller Parameters
To this end, we have presented the ASM control law and
the adaptation law for the self-aligning torque. It is clear that
the stability of the overall control system can be guaranteed
in the sense of Lyapunov. However, for practical implementa-
tion, the controller parameters should be carefully selected to
tradeoff among the tracking accuracy and robustness against the
measurement noises, the system uncertainties, and unmodeled
system dynamics.
1) Selection of λ : The parameter λ crucially determines
the tracking bandwidth of the sliding-mode function as given by
(8) and the decay rate of tracking errors on the sliding surface
[29], [31]. A larger λ leads to a faster response rate and higher
tracking accuracy, which, however, may bring excessive high-
frequency measurement noises to the system that deteriorate the
tracking accuracy inversely. To account for this tradeoff, we set
λ = 15 in our case.
2) Selection of : The term with the parameter in
(11) is introduced to enforce the tracking error onto the slid-
ing surface [29]. It is obvious that the control system band-
width becomes higher by increasing which leads to a faster
response to the reference command as well. Nevertheless, a
larger will similarly amplify the measurement noises from
the sliding variable s and meanwhile result in a larger adap-
tation perturbation da that degrades the estimation accuracy.
From the actual implementations, we find that = 45 is an
acceptable value.
3) Selection of μ2 : From (33), we can see that the param-
eter μ2 critically determines the adaptation bandwidth for the
estimation of the self-aligning torque. Apparently, a larger μ2
leads to a higher adaptation bandwidth implying a faster con-
vergence rate for the estimated coefficient ˆρτ to track the actual
one ρτ . However, we also note that the existence of μ2 reversely
increases the undesired adaptation perturbation da in (34).
Hence, a satisfactory μ2 = 2638 is chosen for implementation.
4) Selection of ψ : It is well known that a larger value of
the boundary layer thickness ψ (24) leads to less chattering but
at the cost of reducing tracking accuracy [32]. In our paper, the
disturbance of self-aligning torque is compensated by a sep-
arate control input (13), which means that a smaller value of

Fig. 2. Estimation of the coefficient of the self-aligning torque ˆρτ under
various road conditions in Case 1.
K in (11) would be sufficient to overcome the remaining uncer-
tainties and disturbances. This implies that the chattering effects
due to the sign function in (11) would be much less compared
to the conventional sliding-mode control. In actual implemen-
tation, we set ψ = 0.8 to obtain a satisfied balance between
control smoothness and tracking accuracy.
D. Two Controllers for Comparison
To compare the benefits of the proposed ASM controller,
a conventional sliding-mode controller and a linear H∞ con-
troller are also designed based on the methods employed in
[30] and [34], respectively. Here, we straightly give the design
results for simplicity.
1) Conventional Sliding-Mode Controller: The conven-
tional sliding-mode controller [30] is given by
ucsm =
1
κ
( ¯Jeλ|˙e| + ¯Je|¨xr| + ¯Be| ˙x| + ¯ξf + ¯τsel)sat(s) (35)
where sat(·) is the same saturation function defined in (25) with
the boundary layer thickness equal to 1.0; ¯Je, ¯Be, and ¯ξf denote
the upper bounds of the parameters Je, Be, and ξf , respectively;
and ¯τsel denotes the upper bound of the self-aligning torque
acting on the SbW system. For a fair comparison, the upper
bounds of the parameters Je, Be, and ξf of the conventional
sliding-mode controller are given by
¯Je = Je0 + ¯ΔJe
¯Be = Be0 + ¯ΔBe
¯ξf = ξf0 + ¯Δξf
.
(36)
The values of these parameters have been given in (3). We can
see that the main difference between the proposed ASM control
and conventional sliding-mode control lies in that the conven-
tional sliding-mode control requires a priori upper bound value
of the self-aligning torque, which may be difficult to predict in
reality due to the change of road conditions; while the ASM
controller eliminates this requirement and is able to adaptively
estimate the self-aligning torque for various road conditions and
subsequently cancel the resistant effect of self-aligning torque
through a feedforward path, which, on the other hand, alleviates
the control efforts of the sliding-mode controller.
Fig. 3. Control performances of the ASM controller in Case 1.
(a) Tracking profiles. (b) Tracking errors. (c) Control inputs.
2) H∞ Controller: The H∞ controller is of the following
form:
uh = 0.31¨xr + 20.66e + 9.06˙e + 0.79 ˙x (37)
which is designed based on a state-space approach and can
guarantee an optimal bounded tracking error in the presence
of the uncertainties [34].
IV. EXPERIMENTAL RESULTS
Experiments are carried out on the actual SbW setup to verify
the designed controllers with a sampling period of 1 ms. To
demonstrate the superiority of the ASM controller, we consider

Fig. 4. Control performances of the conventional sliding-mode controller
in Case 1. (a) Tracking profiles. (b) Tracking errors. (c) Control inputs.
two different road paths for the vehicle to follow, which thus
require different steering reference commands from the steering
wheel. Various road conditions are also considered to evaluate
the tracking performance.
A. Case 1: Steering for a Slalom Path Following
In this case, the steering wheel as shown in Fig. 1(b) is
maneuvered to generate an approximate sinusoidal waveform
which mimics the vehicle following a slalom path. Then, the
steering wheel angle sensor collects the corresponding steering
angle. By multiplying this angle sensor signal by the corre-
sponding scaling factor from the steering wheel to the front
wheels, a steering wheel reference angle is obtained and input
Fig. 5. Control performances of the H∞ controller in Case 1.
to the controllers under test. We also set the values of the
coefficient of the self-aligning torque ρτ in (6) as
ρτ =
⎧
⎪⎨
⎪⎩
155, 0 < t ≤ 20 s, Snowy road
585, 20 < t ≤ 40 s, Wet asphalt road
960, 40 < t ≤ 60 s, Dry asphalt road
(38)
to represent the change of the road conditions during the exper-
iments. The experimental results for this case are shown in
Figs. 2–5.
From Fig. 2, we can see that the designed adaptation law
can properly estimate various coefficients of the self-aligning
torque ρτ with respect to the change of road conditions. It is
seen that there exist oscillations in the curve of the estimated

Fig. 6. Estimation of the coefficient of the self-aligning torque ˆρτ in
Case 2.
coefficient ˆρτ . This is primarily caused by the high-frequency
measurement noises from the angle sensors on the steering
wheel and the front wheel. Though we have low-pass filtered
the high-frequency noises, the cutoff frequency of the filter can-
not be set too low as we also need to guarantee the fidelity of
the signals of the measured angles. It is also noted that when the
steering angle x is zero at the beginning of the experiment, the
estimated coefficient ˆρτ is zero as well. This is because we can
see from (33) that x = 0 implies ˆρτ = 0. Therefore, the adapta-
tion law requires the steering angles to be varying to guarantee
a good estimation of the coefficient. To solve this problem, we
actually can set the initial value of ˆρτ as the last estimated one
during the previous driving cycle.
The results in Figs. 3–5 obviously show that the tracking
performance of the ASM controller is superior to those of the
conventional sliding-mode controller and the H∞ controller.
From Fig. 3, the peak tracking errors of the ASM controller
under the three road conditions are 0.022, 0.025, and 0.028 rad,
respectively. The peak tracking errors occur at the beginning of
each change of the road conditions, at the time of which the
adaptation law starts to estimate the new value of ρτ . Once
the estimated coefficient ˆρτ converges to the actual one, the
peak tracking errors consistently converge into a small region
no matter how large the magnitudes of the self-aligning torque
are. In comparison, the peak tracking errors under the conven-
tional sliding-mode controller and the H∞ controller as shown
in Figs. 4 and 5 are much larger than that under the ASM con-
troller and also become larger with respect to the increase of the
self-aligning torque.
B. Case 2: Steering for a Circular Path Following
In this case, we consider a road path being straight and fol-
lowed by a circular curve which is more common in reality.
This is referred to as a circular path in this paper. Similarly,
we maneuver the steering wheel to generate the steering refer-
ence command for such a path following as shown in Fig. 7(a).
Experiments with a duration of 15 s are carried out and the coef-
ficient of the self-aligning torque ρτ is set as 700 to mimic the
condition on a wet asphalt road.
Fig. 6 shows the estimation of the coefficient of the self-
aligning torque under the ASM controller. We can see that the
Fig. 7. Control performances of the ASM controller in Case 2.
estimated ˆρτ deviates a bit from the actual ρτ at the steady
state, which may be primarily due to the adaptation perturba-
tion da in (33) being a constant in this case. At the time of 9 s,
the estimation curve contains a notched response which stems
from the change of the steering angle into a reverse direction.
The tracking profiles of the controllers are shown in Figs. 7–9.
It can be seen that the ASM controller is still superior to the
conventional sliding-mode controller and the H∞ controller in
this case. The peak tracking error under the ASM controller
is 0.055 rad, which is less than both the conventional sliding-
mode controller (0.059 rad) and the H∞ controller (0.07 rad).
More importantly, with online estimations of the coefficient
of the self-aligning torque ρτ , the tracking accuracy of the

Fig. 8. Control performances of the conventional sliding-mode controller
in Case 2. (a) Tracking profiles. (b) Tracking errors. (c) Control inputs.
ASM controller is improved to a large extent with the tracking
error almost converging to zeros eventually. Comparatively, the
tracking errors under the conventional sliding-mode controller
and the H∞ controller both contain significant steady-state
errors which are due to the insufficient capability to compensate
for the self-aligning torque.
C. Results of Steering Wheel Feedback Motor Control
In a SbW system, the feedback motor control for the steer-
ing wheel is also important because it provides haptic feed-
back of road conditions to the driver. For this purpose, we
Fig. 9. Control performances of the H∞ controller in Case 2.
employ a simple control scheme for the steering wheel feed-
back motor by making its control input proportional to the
estimated reaction force at the tire–road interface [35]. As
such, the driver is provided with road feels when counteracting
the adjustable feedback forces. More specifically, the dynamic
equation of the steering wheel in our setup as shown in Fig. 1 is
given by
Jsw
¨θ + Bsw
˙θ + Dswθ = τh − τf (39)
where θ is the steering wheel angle, τh is the driver input torque,
and τf is the reaction torque generated by the feedback motor.
Moreover, the model parameters equal to Jsw = 0.0791 kg ·
m2
, Dsw = 0.2 Nm/rad, and Bsw = 0.15 Nms/rad, respectively.

Fig. 10. Control performance of the steering wheel feedback motor.
(a) Steering wheel angle governed by driver and feedback motor.
(b) Feedback motor control torque.
By utilizing the estimated coefficient of self-aligning torque
ˆρτ in (14), we can design the control input of the feedback
motor as
uf =
1
ζ
ˆρτ tanh(x) (40)
where x is the measured front-wheel angle in (1) and ζ is a
positive scaling factor to adjust the level of the feedback force
according to the driver’s preference. In our setup, the feedback
motor power driver is configured to have a current feedback
servo loop. Hence, the resultant feedback motor control torque
approximately equals
τf ≈ κf uf (41)
where κf = 1.8 is the current-torque gain.
Experiments of the feedback motor control are carried out
and Fig. 10(a) shows the measured steering wheel angle output
which is governed by the driver input torque and the feedback
motor torque simultaneously. This steering wheel angle output
is then used as the reference command xr denoted in (7), which
is the front wheel’s control loop aimed to follow. Fig. 10(b)
presents the feedback motor control torque in response to var-
ious road conditions. During the experiments, the driver can
obviously feel the resistance when the steering wheel is steered
to deviate from its original position and also the variations
of feedback torques when the road condition changes. These
results demonstrate that the developed estimator for the self-
aligning torques is not only effective to improve the tracking
accuracy of the front wheel’s control but also useful for the
feedback motor control to provide the driver with road feels.
V. CONCLUSION
In this paper, we developed an ASM control scheme for the
SbW system which combines the sliding mode and adaptive
method such that the resultant control system is capable of
not only overcoming the plant parametric uncertainties but also
compensating for the dominant self-aligning torque disturbance
effectively. The stability of the ASM control-based SbW system
is proved in the sense of Lyapunov. Moreover, a practical adap-
tation law is designed for the estimation of the coefficient of
the self-aligning torque. The selection guidelines of the con-
troller parameters are given to account for the compromise
between the measurement noises and the desired system per-
formance in practice. Finally, experiment results are presented
to demonstrate the benefits of the ASM controller with better
tracking accuracy and stronger robustness against various road
conditions in comparison with the conventional sliding-mode
controller and the linear H∞ controller.
The self-aligning torque in this paper is simply modeled
by an approximate function based on the assumptions that the
vehicle mass, velocity, and other related tire parameters are con-
stant. In our future work, we will relax these assumptions and
investigate a complete self-aligning torque model with these
parameters being time-varying. Under this circumstance, we
will also develop advanced adaptive estimation and control
method to deal with the complexity of self-aligning torque. On
the other hand, we are currently developing a new experimen-
tal SbW system on a realistic vehicle and will subsequently
evaluate the performance of the proposed control scheme in
real driving environments. It is anticipated that the proposed
control scheme can enable the SbW system to track steering
command appropriately in practice. Based upon this perfor-
mance, we may further employ an effective method for altering
the vehicle handling dynamics [11] by adjusting the steering
command.
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Zhe Sun was born in Hefei, China, in 1989. He
received the B.E. degree in traffic and vehicle
engineering from Nanjing Agricultural University,
Nanjing, China, in 2011. He is currently working
toward the Ph.D. degree in science, engineer-
ing, and technology at Swinburne University of
Technology, Melbourne, Australia. His research
interests include vehicle dynamics and con-
trol, nonlinear control, adaptive control, and
mechatronics.
Jinchuan Zheng (M’13) received the B.Eng.
and M.Eng. degrees in mechatronics engi-
neering from Shanghai Jiao Tong University,
Shanghai, China, in 1999 and 2002, respec-
tively, and the Ph.D. degree in electrical
and electronic engineering from Nanyang
Technological University, Singapore, in 2006.
In 2005, he joined the Australian Research
Council (ARC), Centre of Excellence for
Complex Dynamic Systems and Control,
School of Electrical and Computer Engineering,
University of Newcastle, Callaghan, Australia, as a Research Academic.
From 2011 to 2012, he was a Staff Engineer with the Western Digital
Hard Disk Drive R/D Center, Singapore. Currently, he is serving as a
Lecturer at Swinburne University of Technology, Melbourne, Australia.
His research interests include mechanism design and control of
high-precision mechatronic systems, sensing and vibration analysis,
dual-stage actuation, and vision-based control.
Zhihong Man (M’94) received the B.E. degree
from Shanghai Jiao Tong University, Shanghai,
China, the M.Sc. degree from the Chinese
Academy of Sciences, Beijing, China, and the
Ph.D. degree from the University of Melbourne,
Melbourne, Australia, in 1982, 1987, and 1994,
respectively, all in electrical engineering.
From 1994 to 1996, he was a Lecturer
with the School of Engineering, Edith Cowan
University, Joondalup, Australia. From 1996 to
2001, he was a Lecturer and then a Senior
Lecturer with the School of Engineering, University of Tasmania, Hobart,
Australia. From 2002 to 2007, he was an Associate Professor of
Computer Engineering at Nanyang Technological University, Singapore.
From 2007 to 2008, he was a Professor and the Head of Electrical and
Computer Systems Engineering, Monash University, Sunway Campus,
Bandar Sunway, Malaysia. Since 2009, he has been with the Faculty
of Science, Engineering, and Technology, Swinburne University of
Technology, Melbourne, Australia, as a Professor. His research interests
include nonlinear control, signal processing, robotics, neural networks,
and vehicle dynamics and control.

Hai Wang (M’13) received the B.E. degree
from Hebei Polytechnic University, Tangshan,
China, in 2007, the M.E. degree from Guizhou
University, Guiyang, China, in 2010, and the
Ph.D. degree from Swinburne University of
Technology, Melbourne, Australia, in 2013,
respectively, all in electrical and electronic engi-
neering.
From 2014 to 2015, he was a Postdoctoral
Research Fellow with the Faculty of Science,
Engineering, and Technology, Swinburne
University of Technology. Since 2015, he has been with the School of
Electrical and Automation Engineering, Hefei University of Technology,
Hefei, China, as a Professor (Huangshan Young Scholar). His research
interests include sliding-mode control, adaptive control, robotics, neural
networks, nonlinear systems, and vehicle dynamics and control.

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