An adaptive nonsingular fast terminal sliding mode
control for yaw stability control of bus based on STI
tire model
Xiaoqiang Sun ( sxq@ujs.edu.cn )
Jiangsu University https://orcid.org/0000-0001-6344-5582
Yujun Wang
Jiangsu University
Yingfeng Cai
Jiangsu University
PakKin Wong
University of Macau
Long Chen
Jiangsu University
Original Article
Keywords: Bus, Yaw stability control, Sliding mode control, STI tire model, Co-simulation
Posted Date: October 12th, 2020
DOI: https://doi.org/10.21203/rs.3.rs-36541/v2
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
Read Full License
·1·
Title page
An adaptive nonsingular fast terminal sliding mode control for yaw stability control of bus
based on STI tire model
Xiao-Qiang Sun, born in 1989, is currently an associate professor at Automotive Engineering Research Institute, Jiangsu University,
China. He received his PhD degree from Jiangsu Universtiy, China, in 2016. His research interests include vehicle system dynamics.
Tel: +86-15952818843; E-mail: sxq@ujs.edu.cn
Yu-Jun Wang, born in 1992, is currently a master candidate at Automotive Engineering Research Institute, Jiangsu University, China.
Ying-Feng Cai, born in 1985, is currently a professor and a PhD candidate supervisor at Automotive Engineering Research Institute,
Jiangsu University, China. Her main research interests include vehicle system dynamics, intelligent vehicles and visual perception.
Pak-Kin Wong, is currently a professor and a PhD candidate supervisor at Department of Electromechanical Engineering, University of
Macau, Taipa, Macau. His main research interests include automotive engines, drive trains and chassis.
Long Chen, born in 1958, is currently a professor and a PhD candidate supervisor at Automotive Engineering Research Institute, Jiangsu
University, China. His main research interests include vehicle system dynamics and advanced vehicle control theory.
Corresponding author：Xiao-Qiang Sun E-mail：sxq@ujs.edu.cn
Xiao-Qiang Sun et al.
·2·
ORIGINAL ARTICLE
An adaptive nonsingular fast terminal sliding mode control for yaw stability control
of bus based on STI tire model
Xiao-Qiang Sun
1, 3
• Yu-Jun Wang
2
• Ying-Feng Cai
1
• Pak-Kin Wong
1
• Long Chen
1
Received June 19, 2020; revised February xx, 201x; accepted March xx, 201x
© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017
An adaptive nonsingular fast terminal sliding mode control for yaw stability control of bus based on STI tire model
·3·
Abstract: In this paper, a novel adaptive nonsingular fast terminal
sliding mode (ANFTSM) control scheme for yaw stability control
(YSC) is proposed to improve the bus curve driving stability and
safety on slippery roads. There are three major contributions in the
design process of the bus YSC system. The first contribution is that
the STI (Systems Technologies Inc.) tire model, which can
effectively reflect the coupling relationship between the tire
longitudinal force and lateral force, is established based on
experimental data and firstly adopted in the bus YSC system design.
The second contribution is a novel YSC strategy based on ANFTSM,
which has the merits of fast transient response, finite time
convergence and high robustness against uncertainties and external
disturbances. The third contribution is that the robust least-squares
allocation method is used to solve the optimal allocation problem of
the tire forces, whose objective is to achieve the desired direct yaw
moment through the effective distribution of the brake force of each
tire. To verify the feasibility, effectiveness and practicality of the
proposed bus YSC approach, the TruckSim-Simulink co-simulation
results are finally provided.
Keywords: Bus • Yaw stability control • Sliding mode control • STI
tire model• Co-simulation
1 Introduction
Due to the characteristics of large quality, high center of
gravity and narrow wheelbase, the bus is prone to lateral
instability under special driving conditions such as turning on
slippery roads [1-4]. If this lateral instability phenomenon
cannot be predicted accurately and be timely and effectively
controlled, it will eventually lead to bus rollover accident [5].
Therefore, it is of great significance to study the active
stabilization control (ASC) strategy of bus when turning on
slippery roads [6-8].
In order to improve the vehicle curve driving stability and
safety on slippery roads, the yaw stability control (YSC)
system has been applied to achieve the vehicle ASC function
[9]. Leonardo et al. designed a direct yaw moment controller
based on the combination of feedforward and feedback [10].
Chen et al. determined the desired vehicle yaw moment by
means of sliding mode control [11]. Avesta et al. proposed an
active controller system to supervise the vehicle lateral
dynamics and then designed a differential braking yaw
moment controller [12]. Ding et al. proposed a YSC strategy
for in-wheel electric vehicles by using the sliding mode and
nonlinear disturbance observer methods [13, 14]. Although
all the above studies have made outstanding contributions to
the YSC system, the vehicle tire model, which shows high
Xiao-Qiang Sun
sxq@ujs.edu.cn
1
Automotive Engineering Research Institute, Jiangsu University,
Zhenjiang, 212013, China;
nonlinearity in the operation process of the ASC system [15-
17], is not handled effectively. Even though some studies
have established the vehicle dynamics model considering the
tire nonlinear characteristics, the tire model adopted cannot
effectively reflect the coupling relationship between the tire
longitudinal force and lateral force, which will lead to the
derived YSC strategy cannot achieve good performance in
practical applications.
On the basis of the accurate vehicle model, the
effectiveness of the YSC control algorithm can then be
guaranteed. In the existing literatures, several methodologies
have been applied to improve the vehicle YSC performance
[18-21]. Among them, the sliding mode control (SMC)
method has been widely applied due to its strong robustness
against uncertainties and disturbance. However, the
traditional SMC method still has drawbacks that limit its
performance in real applications, thus much effort has been
devoted to finding effective solutions to eliminate those
drawbacks [22-24]. Based on the conclusions of the existing
research, the adaptive nonsingular fast terminal sliding mode
(ANFTSM) control method, which can both obtain fast finite
time convergence and chattering elimination [25, 26], is
applied in this work to effectively solve the direct yaw
moment calculation problem of bus. As is known to all, the
obtained direct yaw moment needs to be further achieved by
distributing the braking forces of four wheels, thus the
solution of the optimal allocation problem of the tire braking
forces is also crucial. In [27-29], several algorithms have
been proposed to achieve well allocation of the tire forces.
However, we have found that the allocation algorithms used
in the current research still take a long time to calculate, thus
the real-time control, which is very important for the YSC
system, cannot be guaranteed reliably.
In this paper, a novel YSC scheme which both includes a
new tire model applied in the vehicle system modeling, a new
control algorithm used to calculate the direct yaw moment
and a new method used to guarantee the solving efficiency of
the tire forces allocation is presented. Therefore, the main
contributions of this research are:
⚫ The STI tire model, which can effectively reflect the
coupling relationship between the tire longitudinal force
and the tire lateral force, is established based on
experimental data and firstly adopted in the bus YSC
system design.
⚫ The ANFTSM control algorithm, which has the merits of
high robustness against uncertainty and external
disturbance, fast transient response and finite time
convergence, is firstly applied to solve the direct yaw
moment calculation problem of the bus effectively.
⚫ The robust least-squares control allocation method,
which can effectively guarantee the solving efficiency of
2
State Key Laboratory of Automotive Safety and Energy, Tsinghua
University, Beijing 100084, China
3
Department of Electromechanical Engineering, University of Macau,
Taipa, Macau
Xiao-Qiang Sun et al.
·4·
complex optimization problem like the tire longitudinal
forces allocation of each tire and restrain the uncertainty
in control distribution effectiveness matrix, is firstly used
to solve the optimal allocation problem of the tire forces.
The rest of this paper is organized as follows: Sect. 2
presents the establishment of the STI tire model based on
experimental data firstly, and on this basis, a 7-DOF
nonlinear dynamics model of bus is further introduced. In
Sect. 3, the YSC strategy design based on ANFTSM method
is provided. Sect. 4 shows how to solve the optimal allocation
problem of the tire forces by using the robust least-squares
allocation method. In Sect. 5, the TruckSim-Simulink co-
simulation results are presented to verify the feasibility,
effectiveness and practicality of the proposed YSC approach.
Finally, the conclusions and future works are given in Sect.
6.
2 System modeling
2.1 STI tire model
STI tire model is proposed by an American company, which
named as “Systems Technologies Inc.” [30]. Compared with
other tire models, the STI tire model can not only describe
the tire nonlinear mechanical characteristics accurately, but
also reflect the coupling relationship between the tire
longitudinal force and the tire lateral force exactly [31], thus
it is very suitable for the bus YSC system design in this study.
To achieve the description of the STI tire model, an important
variable must be defined firstly, i.e. the composite slip
coefficient σ, which is expressed as [32]:
2 2 2 2 2
tan ( /1 )
8
ps
z
ak α k s s
σ
F
+ −
=
, (1)
where a
p
denotes the length of the tire contact patch, k
α
denotes the tire lateral stiffness, k
s
denotes the tire
longitudinal stiffness, α denotes the tire sideslip angle, s
denotes the tire longitudinal slip coefficient, μ is the road
adhesion coefficient and F
z
is the tire vertical load. In eq. (1),
the length of the tire contact patch can be further defined as
[33]:
, (2)
where F
zt
is the tire design load, T
w
is the tire width, T
p
is the
tire pressure. The tire lateral stiffness k
α
and the tire
longitudinal stiffness k
s
in eq. (1) can be measured by
experimental tests. On this basis, to further consider the
saturation effect of the tire longitudinal stiffness, a modified
tire longitudinal stiffness k
sm
is defined as:
22
( ) sin cos
sm s s
k k k k
= + − +
, (3)
Based on the defined composite slip coefficient σ, a tire force
saturation function can then be defined as:
32
12
32
1 3 4
(4 / π)
()
1
C σ C σ σ
f σ
C σ C σ C σ
++
=
+ + +
, (4)
where C
1
, C
2
, C
3
and C
4
are the fixed coefficients, which can
be obtained by fitting the experimental data. The function f
(σ) is consistent with the mechanical properties of the tire
force friction circle, thus it can reflect the tire force saturation
characteristics effectively. Then, on the basis of the above
equations, the standardized expressions of the tire
longitudinal force and the tire lateral force of the STI tire
model can be respectively described as:
2 2 2 2
2 2 2 2
()
tan
( ) tan
tan
sm
xz
α sm
yz
α sm
f σ ks
FF
k α ks
f σ k α
F F Y γ
k α ks
=
+
=+
+
, (5)
where F
x
represents the tire longitudinal force, F
y
represents
the tire lateral force, γ represents the tire camber angle, Y
γ
represents the tire camber coefficient, which is used to reflect
the influence of the camber angle on the tire lateral force.
According to eq. (5), the coupling relationship between the
tire longitudinal force and the tire lateral force under
compound driving conditions can be reflected exactly.
To obtain the experimental data which can accurately
reflect the nonlinear mechanical characteristics of the tire, the
experimental tests are conducted through a flat-plate bench
assisted by the KH Automotive Technologies (Guangzhou)
Co., Ltd. Figure 1 shows the experimental setup of the tire
mechanical characteristics tests.
Figure 1 Experimental setup of the tire mechanical characteristics tests.
Based on the experimental data, the STI tire model
parameters can then be fitted. According to the expressions
of the tire forces, it is obvious that the fitting of the tire model
parameters is actually to determine the fixed coefficients C
1
,
C
2
, C
3
and C
4
in eq. (4). Before achieving the fitting of the
fixed coefficients, the tire lateral stiffness and the tire
longitudinal stiffness need to be firstly measured by
experimental tests. According to the test results, the tire
lateral stiffness is finally determined as -66463 N/rad and the
tire longitudinal stiffness is finally determined as 84000
