J. Cent. South Univ. (2015) 22: 1378−1388
DOI: 10.1007/s11771-015-2655-y
Vehicle path tracking by integrated chassis control
Saman Salehpour1, Yaghoub Pourasad2, Seyyed Hadi Taheri3
1. Department of Mechanical Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran;
2. Faculty of Electrical Engineering, Urmia university of Technology, Urmia, Iran;
3. Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran
© Central South University Press and Springer-Verlag Berlin Heidelberg 2015
Abstract: The control problem of trajectory based path following for passenger vehicles is studied. Comprehensive nonlinear vehicle
model is utilized for simulation vehicle response during various maneuvers in MATLAB/Simulink. In order to follow desired path, a
driver model is developed to enhance closed loop driver/vehicle model. Then, linear quadratic regulator (LQR) controller is
developed which regulates direct yaw moment and corrective steering angle on wheels. Particle swam optimization (PSO) method is
utilized to optimize the LQR controller for various dynamic conditions. Simulation results indicate that, over various maneuvers, side
slip angle and lateral acceleration can be reduced by 10% and 15%, respectively, which sustain the vehicle stable. Also, anti-lock
brake system is designed for longitudinal dynamics of vehicle to achieve desired slip during braking and accelerating. Proposed
comprehensive controller demonstrates that vehicle steerability can increase by about 15% during severe braking by preventing
wheel from locking and reducing stopping distance.
Key words: vehicle dynamics; active control system; optimal controller; electronic stability program (ESP); particle swam
optimization (PSO)
1 Introduction
Improving vehicle safety and handling properties in
recent years is an indispensable issue which, can
efficiently reduce accident possibility and protect the
crews. So, increasing the stability and maneuverability of
vehicle by active control system is a critical subject in
automotive industry.
Active control systems development and their
effects on vehicle performance were investigated in
previous studies [1−5]. Improving stability and
steerability of vehicle stopping in the shortest possible
distance with maintaining control over a vehicle and
desired path following is probably the most important
active safety requirement of any vehicle that can reduce
traffic jam and accident risk.
HAMERSMA and SCHALKELS [6] developed an
integrated ABS and semi active suspension system for
improving vehicle directional control and braking
condition in a rough roads. WANG et al [7] designed
controllers in the electronic stability program (ESP)
system according to fuzzy logic and PID control theory
to improve control performance. Then, a joint simulation
model was established in a MATLAB/Simulink
environment, in which the ESP control module was
embedded into the vehicle dynamic model achieved from
ADAMS/Car. Two yaw motion control systems
improving a vehicle lateral stability were proposed based
on braking and steering controllers [8]. A 15
degree-of-freedom (DOF) vehicle model, simplified
steering system model, and driver model were used to
evaluate the proposed controllers. Also, a robust ABS
(anti-skid brake system) controller was designed and
developed. LI et al [9] developed an optimal steering
angle and torque distribution for omni-directional vehicle
which was a vehicle that has an in-wheel steering motor
and in-wheel driving motor installed with each wheel.
Optimal distribution of longitudinal tire force and lateral
tire force were achieved by slip angle estimation in
various directions. The coordinated control system was
proposed to improve vehicle handling and stability by
coordinating control of ESP and AFS [10]. The
fuzzy-PID controller was developed to calculate the
target yawing moment required to keep the vehicle stable.
Simulation results were compared the performance of the
integrated system with other situations such as only AFS
control, only ESP control and no control. The results
showed that the integrated controller was able to improve
the driving dynamics and steering stability of the vehicle
effectively. KECECI and TAO [11] proposed a yaw
moment controller of which parameters are dependent on
the vehicle forward velocity for improving vehicle
handling and stability. Robustness of the controller was
Received date: 2014−03−17; Accepted date: 2015−01−27
Corresponding author: Seyyed Hadi Taheri; Tel: +98−9363062187; E-mail: [email protected]
J. Cent. South Univ. (2015) 22: 1378−1388
enhanced with considering the parameter uncertainties
arising from tyre cornering stiffness and the actuator
saturation limitations.
Adaptive vehicle skid control, for stability and
tracking of a vehicle during slippage of its wheels
without braking, was addressed in Ref. [12]. Two
adaptive control algorithms were developed for unknown
road condition and certain road type. Their results show
that the vehicle control system with an adaptive control
law keeps the speed of the vehicle as desired by applying
more power to the drive wheels where the additional
driving force at the non-skidding wheel will compensate
for the loss of the driving force at the skidding wheel,
and also arranges the direction of the vehicle motion by
changing the steering angle of the two front steering
wheels.
WANG et al [12] designed a lateral control law and
developed a strategy to determine the given speed of
autonomous vehicles. They proposed an improved
method for calculating the lateral offset and heading
angle error to reduce the impact of reference path data
noise. Multiple fuzzy inference engines are used to
design the steering controller and determine the given
driving speed, including the forward and backward
directions. For vehicle path tracking, an unexpected
sliding effects were taken into consideration [13]. Robust
anti-sliding controllers were designed for the observation
and suppression of such sliding effects. TALVALA et al
[14] designed a look-ahead controller coupled with
longitudinal control based on the path position and the
wheel slip to create an autonomous race car. ZAKARIA
et al [15] proposed controller based on steering wheel
angle and yaw rate of the vehicle for vehicle path
tracking in autonomous vehicles. The controller
consisted of relationship between future point lateral
error, linear velocity, heading error and reference yaw
rate.
MASHADI et al [16] developed the optimal path
following controller based on genetic algorithm for
lateral dynamics of vehicle. Simulation results
demonstrated that the proposed controller was able to
effectively keep the vehicle path appropriately close to
the desired path even in the presence of the driver
commands. The main objective of this work is designing
optimal controller of integrated lateral/longitudinal
vehicle dynamics based on controlling combined slips to
improve vehicle handling properties and tracking desired
path. PEDRO et al [17] used PSO for optimization of
active suspension system for vehicle ride and road
holding. They compared the performance of this
controller with the performance of passive, PID and
non-optimized intelligent controllers. Several methods
for controlling vehicle stability has been presented in
previous works [18−19]. For instance, MOKHIAMAR
1379
and ABE [18] used a direct yaw moment control (DYC)
and showed that change in chassis control from 4WS to
DYC inevitably improves handling performance and
active safety in vehicle motion with larger slip angles
and/or higher lateral accelerations. Also, NAGAI et al
[19] presented an integrated robust control system of
active rear wheel steering (ARS) and DYC for improving
vehicle handling and stability.
In this work, for a suitable model for improving
vehicle path following and handling properties,
combined lateral and longitudinal vehicledynamics are
developed to be tracked by integrated ABS/ESP control
system. Firstly, an optimal LQR and PID controller are
designed for improving stability, maneuverability and
path following of comprehensive vehicle model. Then,
PSO algorithm is utilized to optimize the controller
parameters in various driving condition, based on closed
loop driver/vehicle model. Lateral dynamic controlled by
considering yaw rate, side-slip angle and lateral
acceleration, whilst longitudinal dynamics of the vehicle
can be controlled with the throttle/brake pedal and
longitudinal slip.
2 Vehicle modelling
In this work, to model the vehicle during the path
following at various maneuvers, a non-linear 4-DOF
vehicle model that includes both lateral and longitudinal
dynamics is used. The input of this model is front wheel
steer angle while the output variables to be controlled are
vehicle side slip and yaw rate. A schematic of typical
front wheel steering passenger car is illustrated in Fig. 1.
The DOFs associated with this model are the
longitudinal velocity u, the lateral velocity vy, the yaw
rate r, and the roll rate. Sources of nonlinearity include
nonlinear behavior of tires, nonlinearities existing in the
longitudinal direction and the lateral tire normal load
transfers, the roll steer effects, and the camber angle
changes due to the vehicle roll.
The governing dynamic equations for the
longitudinal, lateral, and yaw motions of the vehicle
body are described as follows.
Fig. 1 4-DOF vehicle model [20]
J. Cent. South Univ. (2015) 22: 1378−1388
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Longitudinal motion:
m(u  rv)   Fx
(1)
Lateral motion:
4
  F
m(v  ru )  ms hsΦ
 y
(2)
i 1
Yaw motion:
I zz r  I xzΦ   m z
(3)
Roll motion:
I xxΦ  I xz r   m x
(4)
where m and ms are the total mass and the rolling mass,
respectively; Izz and Ixx are mass moments of inertia
about z-axis and x-axis, respectively; Ixz is the product of
inertia with respect to x and z axes; hs is the height of
sprung mass CG to roll axis; the terms  Fx and  Fy
are external forces along the x and y directions;
 M x and  M z are the sums of the moments acting
around the roll and yaw axes of the vehicle-fixed
coordinate system and can be evaluated by
 Fx  Fxfl  Fxfr  Fxrl  Fxrr
(5)
 Fy  Fyfl  Fyrl  Fyrl  Fyrr
(6)
the tire. It is commonly called Fiala’s theory and is
related to the tire cornering characteristics which takes
into account the interactions between longitudinal and
side forces as
u Fz sgn( ),    cr
Fy  
3
u Fz sgn( )(1  H ),    cr
(13)
u  u max  (u max  u min ) S s
(14)
S s  S s2  tan 2 
(15)
 cr  tan 1 (
H  1
3u Fz
C
)
C tan 
(17)
3u Fz
where α, Ss and Ssα are side slip angle, longitudinal slip
ratio and the comprehensive slip ratio, respectively.
Lateral side slip angle for front (αf) and rear tires (αr) are
given as
  ar
)
 f    tan 1 (
(18)
  br
)
 r  (
(19)
u
u
 M z  a( Fyfl  Fyfr )  b( Fyrl  Fyrr )  0.5T [( Fxfl 
4
Fxrl )  ( Fxfr  Fxrr )]   M zi
(7)
 M x  ms h(v  ru )  (ms gh  kΦ )Φ  CΦΦ
(8)
i 1
In the above equations, a and b are respectively the
distances measured from the CG to the front and the rear
axles; T is the track width; Kf and Cf are the overall roll
stiffness and the damping coefficient, respectively.
External forces  Fx and  Fy , can be related to
the tractive and the lateral tire forces through below
equations:
Fxi  Fli cos  i  Fli sin  i , i  fl, fr, rl and rr
(9)
Fyi  Fli cos  i  Fri sin  i , i  fl, fr, rl and rr
(10)
Given that the vehicle is the front wheel steering,
steering angle (δi) can be considered
 fl   fr   f   fl  K rsf Φ
(11)
 rl = rr  K rsr
(12)
where Krsf and Krsr are steered by roll coefficient for the
front and rear wheels which depend on suspension
geometry.
The mathematical model proposed by FIALA [21]
is used to the analysis of lateral force due to side slip of
(16)
According to integrated lateral/longitudinal
dynamics of vehicle, vertical tire loads on wheels can be
expressed as
Fz1 
mg
2
Fz 2 
mg
2
a ys h
 b ax h

m h
( )  KR [
( )  ( s )( s ) sin  ]
 
g l
g T
m T
l

(20)
a ys h
 a ax h
( )  (1  K R )[
( )
 
g l
g T
l
m h

( s )( s ) sin  
m T

Fz 3 
mg
2
Fz 4 
mg
2
(21)
a ys h
 b ax h

m h
( )  KR [
( )  ( s )( s ) sin  ]
 
l
g
l
g
T
m
T


(22)
a ys h
 a ax h
( )  (1  K R )[
( )
 
g l
g T
l
m h

( s )( s ) sin  ]
m T

(23)
where ax and ay are longitudinal and lateral accelerations,
respectively; KR is the ratio of the front roll stiffness to
the total roll stiffness which determines the front/rear
distribution of total lateral load transfer; and h denotes
the height of CG relative to the ground.
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3 Controller design
The purpose of control system proposed in this
work is controlling the vehicle to follow a desired path,
whereas maintains the vehicle actual motions, yaw rate
and slip angles, close to their desired responses with a
minimum external yaw moment, for improving vehicle
stability and handling condition. To achieve this aim,
ABS controller and an optimal LQR and PID approaches
should be applied for longitudinal dynamics and lateral
dynamics, respectively.
At the first step in the design of driver/vehicle
controller, the simplest form of planar motion should
develop an integrated lateral/longitudinal vehicle
dynamic model:
Lateral motion:
m(Vy  Ur )  Fyr cos  r  Fxr sin  r  Fyf cos  f 
Fxf sin  f
(24)
 bFyr
sin  r
 bFyr
cos  r
(25)
State space equation form of bicycle model is
represented as
X  AX  BU
(26)
where
a
v
X   , A   11
 
a 21
a11  2
a21  2
b1  2
dVx
  FN
dt
(27)
J   RFN  Tb
(28)
m
Slip ratio in relation to longitudinal speed and tire
rotation is defined as

Yaw motions:
I z r  aFyf cos  f  aFxf sin  f
Fig. 2 Quarter car model for ABS design
a12 
b 
, B   1 , U   f

a 22 
b2 
bC  Cf
Cf  Cr
, a12  2 r
U ,
MU
MU
a 2 Cf  b 2 Cr
bCr  Cf
, a 22  2
,
I zU
I zU
V x  R
Vx
(29)
Controllability (stabilization) of vehicle and braking
performance of vehicle is related to lateral and
longitudinal forces which act on tire respectively. As
shown in Fig. 3, these forces depend on coefficient of
friction between the tire and the road which changes with
wheel slip ratio. This value differs according to the road
type. From Fig. 3 it is clear that for almost all road
surfaces the frictional coefficient value is optimum when
the wheel slip ratio is approximately 0.2 and the worst
when the wheel slip ratio is 1, in other words, when
wheel is locked. So, objective of ABS controller is to
regulate the wheel slip ratio to target value of 0.2 to
maximize the frictional coefficient (μ) for any given road
surface.
aC
Cf
, b2  2 f .
Iz
M
3.1 ABS (longitudinal) controller
In order to develop ABS controller for longitudinal
dynamic model of vehicle and slip regulation, as shown
in Fig. 2, a quarter car vibration model is considered.
The model consists of a single wheel attached to a
mass m. As the wheel rotates, driven by inertia of the
mass m in the direction of the velocity Vx, a tyre reaction
force Fx is generated by the friction between the tyre
surface and the road surface. The tyre reaction force will
generate a torque that initiates a rolling motion of the
wheel causing an angular velocity ω. A brake torque
applied to the wheel will act against the spinning of the
wheel causing a negative angular acceleration. The
equations of motion of the quarter car are
Fig. 3 Lateral and longitudinal forces versus slip ratio
3.2 ESP (Lateral) controller
In order to path following, in addition to
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longitudinal controller which is conducted by ABS,
lateral motion of vehicle should be controlled by
considering side slip angle, lateral acceleration and yaw
rate. So, designing a sufficient controller for lateral
motion of vehicle should be performed by considering
the path parameters including lateral deviation from
desired path and heading angle error.
Despite to control the vehicle in desired path,
relationship between the vehicle and the intended path is
identified by expressing in terms of a lateral position
error, ye (the lateral distance between the vehicle and the
intended path), and an orientation angle error (ψv−ψr). So,
the parameters of road geometry are combined by
vehicle model as follows:
y e  u sin  ν cosψ
(30)
ψ  ψ v  ψ r
(31)
where subscripts v and r are representatives of vehicle
and road, respectively; and  r is defined as road
curvature rate which equals longitudinal velocity divided
by road curvature, u/R.
Combining the vehicle model equations with
external yaw moment and Eqs. (30) and (31) can be
described by the following state space equations based
on small heading angle error and constant longitudinal
vehicle speed assumptions:
X  AX  E  BM z  K (1 / R )
(32)
0 
 ye 
0 1 u 0 
e 
 
0 a


1
11 0 a12 
X   , A  
, E   ,
 
0 
0 0 0 1 
 
 


0 a21 0 a 22 
r 
 e2 
0 
0 
0 
0 
B   , K   
0 
 u 
 
 
b
0 
a21  2
e1 =2
Cαf +Cαr
Mu
, a12  2
bCαr  aCαf
Mu
(33)
U,
bCαr  aCαf
a 2 Cαf +b 2 Cαr
, a22  2
,
Izu
Izu
Cαf
M
, e2 =  2
aCαf
Iz
, b
3.3 Optimal LQR controller
To control a vehicle to track a driver intended path
with constant longitudinal velocity, there by LQR theory,
the performance index that penalizes the tracking errors
and control expenditure is formulated as
tf
J  1 / 2 [ w1 ( ye  yed ) 2  w2 (  d ) 2  w3 (v  vd ) 2 
t0
w4 (r  rd )+w5 (   d ) 2 +w6 M z2 ]dt
1
Iz
For the vehicle model, the lateral velocity ν and the
yaw rate r are considered the two state variables while
the yaw moment Mz is the control input, which must be
calculated through the control law. Furthermore, the
vehicle steering angle δ is considered the external
disturbance controlled by driver model that should be
(34)
where Mz denotes external yaw moment. The subscript d
denotes the desired response of each variable. The first
and second terms in the performance index are lateral
deviation and heading error, respectively, which are
representations of vehicle path following. The third and
fourth terms in performance index denote handling and
stability property of vehicle, respectively, and the fifth
term is vehicle steering angle which is regulated by
driver. w1−w6 are weighting factors which indicate the
relative importance of the corresponding terms.
The typically defined optimal control consists of the
state variable feedback signal and the disturbance feed
forward signal that is related to the road specification,
expressed as
M z =K υ +K r r +K ψ +K δ +K δ δ  K ye ye  K R R
where
a11  2
added to vehicle model states through driver model.
In order to develop the yaw moment control law for
vehicle path following two different control strategies are
developed: LQR and PID. Then, their parameters are
optimized by PSO for various conditions.
(35)
where K υ , K r , K ψ , K δ , K δ and K ye are known as the
state gains of lateral velocity, yaw rate, heading angle,
steering angle, steering angle rate, and lateral
displacement, respectively, which act on the vehicle
states and kR is the preview gain, acting on the previewed
path information.
Consequently, Eq. (34) can be rewritten in the
standard form of the optimal control as
J (u ) 
1  T
[U RU  ( X d  X ) T Q( X d  X ) 2 ]dt
2 0
(36)
where Xd is the desired values that the vehicle states
should track. Since Q is positive semi-definite,
XT(t)QX(t)≥0, which represents the penalty incurred at
time t for statetrajectories which deviate from 0.
Similarly, since R is positive definite, UT(t)RU(t)>0
unless U(t)=0. It represents the control effort at time t in
trying to regulate X(t) to 0. The entire cost criterion
reflects the cumulative penalty incurred over the infinite
horizon. Values of thestate gainsaresetup based on time
variation, in such a way that the optimal controller is able
to control the vehicle’s track and stability. It results in the
J. Cent. South Univ. (2015) 22: 1378−1388
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minimum lateral deviation, obtaining vehicle stability
and maneuverability over various paths.
3.4 PID controller
The PID controller is a three-mode controller
(Fig. 4). That is, its activity and performance are based
on the values chosen for three tuning parameters, one
each nominally associated with the proportional, integral
and derivative terms.
Fig. 4 PID controller acting on closed loop vehicle
A proportional controller (Kp) will have the effect of
reducing the rise time and will reduce but never
eliminate the steady-state error. An integral control (Ki)
will have the effect of eliminating the steady-state error
for a constant or step input, but it may make the transient
response slower. A derivative control (Kd) will have the
effect of increasing the stability of the system, reducing
the overshoot, and improving the transient response. The
effects of each of controller parameters, Kp, Kd, and Ki on
a closed-loop system are summarized in Table 1.
exploration directions (their flights) using the following
speed equation:
vi , j  w  vi , j  c1  r1  ( Pb, j  xi , j )  c2  r2  ( Pg , j  xi , j )
(37)
where w is inertia factor influencing the local and global
abilities of the algorithms; vi,j is the velocity of the
particle i in the j-th dimension; c1 and c2 are the weights
affecting the cognitive and social factors, respectively; r1
and r2 are a generator of random numbers uniformly
distributed in (0, 1) range; Pb stands for the best value
found by particle i; Pg denotes the global best found by
the entire swarm. When the velocity is updated, the new
position i in its j-th dimension is calculated. This process
is repeated for every dimension and for all the particles
in the swarm. As shown in Fig. 5, the structure of PSO is
initialized with a population of random solutions and
searches for optima by updating generations.
Table 1 Effects of PID controller parameters on closed loop
system response
Closed loop Rise
Settling
Steady state
Overshoot
response
time
time
error
Kp
Decrease Increase Small change
Decrease
Ki
Decrease Increase
Small
Decrease
change
Increase
Eliminate
Decrease
No change
Kd
3.5 Controller optimization
In this work, PSO algorithm is utilized for
optimization of controller’s parameters. Q and R
matrixes in LQR controller and PID parameters are
optimized by using PSO algorithm.
The PSO algorithm is a kind of heuristic global
optimization technology which is based on the behavior
of communities having both social and individual
conducts [22]. This algorithm which was developed by
KENNEDY and EBERHART [23] is a population-based
algorithm and each individual (particle) represents a
solution in the n-dimensional space. Each particle also
has knowledge of the previous best experience and
knows the global best experience (solution) which is
found by the entire swarm. Particles also update their
Fig. 5 Evolution procedure of PSO Algorithms
Unlike GA, PSO has no evolution operators such as
cross over and mutation. In PSO, the potential solutions,
called particles, fly through the problem space by
following the current optimum particles. PSO is easy to
implement and there are few parameters to adjust. PSO
has been successfully applied in many areas: function
optimization, artificial neural network training, fuzzy
system control. Therefore, fitness function, which should
be minimized by PSO, can be written with regard to the
closed-loop system response as [24−25]
I  tr0.2  ts0.5  Ess5  M p2
(38)
The exponents of Eq. (22) are adjusted in such a
way that the transient response properties, namely, rise
time (tr), settling time (ts), steady-state error (Ess) and
overshoot all attain lower values. The parameters used in
1384
PSO algorithm are as follows: population size is 100;
acceleration constants c1=1.5, c2=1.5, and iteration
number is 40.
4 Simulation results
J. Cent. South Univ. (2015) 22: 1378−1388
allows the driver to control the vehicle. However,
without ABS controller after the slip ratio becomes 100%,
wheel locks and losing steerability of vehicle loses.
Also, to evaluate the performance of ABS controller
on braking status, stopping distance and braking torque
on tire are illustrated in Fig. 7.
Simulation is performed for vehicle comprehensive
model with control, optimized control by PSO and
without controller modes. Simulations are run in
Matlab/Simulink software and effects of ABS and lateral
controller (ESP) for these cases are compared.
4.1 ABS controller results
In order to investigate the effects of ABS controller
on vehicle longitudinal dynamic, slip ratio, vehicle speed
and wheel rotation speeds are compared for no control,
control and optimized controller with ABS in Fig. 6,
respectively.
Fig. 7 Effects of ABS controller and optimization on vehicle
longitudinal dynamic performance: (a) Stopping distance; (b)
Torque on tire
Results indicate that torques acting on tire in
uncontrolled mode, which is proportional to the
longitudinal tire slip, are less. Therefore, road holding
and steerability of vehicle reduce in this mode. Whereas,
in controlled modes, tire slip was limited in sufficient
range, which results in improving maneuverability and
reducing stooping distance(time). Also, optimized
controller with PSO has the minimum stopping distance
and best performance.
Fig. 6 Effects of ABS controller (optimized by PSO) on
braking and steerability: (a) Vehicle and wheel rotation speed;
(b) Slip ratio
Results reveal that optimized ABS controller by
PSO can sustain the slip ratio in optimum rate (0.2) and
prevent wheel from locking during barking, which
4.2 ESP controller
In order to investigate the effects of ESP controller
on vehicle lateral dynamics, simulation results are
performed over different paths, including double-lane
change and j-turn maneuvers forno control, control, and
optimized controller by PSO (PSO-control).
J. Cent. South Univ. (2015) 22: 1378−1388
4.2.1 j-turn maneuver
Vehicle path following, lateral deviation from
desired path, handling properties (lateral velocity and
acceleration) and control efforts (external momentum on
tire and steering angle) for standard j-turn maneuver in
no control, control, and optimized controller by PSO
modes are illustrated in Fig. 8.
As shown in Fig. 8, vehicle without controller
cannot follow desired path properly and reaches the
maximum lateral acceleration and speed, which makes
vehicle unstable. Whilst, addition of optimized controller
improves stability and vehicle can track desired path
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with a minimal deviation. Also, in contrast to
uncontrolled one, with optimal ESP controller,
fluctuation of steering angle and yaw moment for closed
loop vehicle model in steady state condition are
minimized.
4.2.2 Double lane change maneuver
In this stage, vehicle state variation during standard
double-lane change maneuver is illustrated in Fig. 9, for
path, lateral error, lateral velocity, acceleration, external
yaw moment and steering angle.
Results comparison indicates that optimized
controller improves vehicle handling conditions. Lateral
Fig. 8 Simulation results of j-turn maneuver in no control, control and PSO-control modes: (a) Vehicle path following; (b) Lateral
deviation during path following; (c) Lateral velocity; (d) Lateral acceleration; (e) Steering angle; (f) External yaw momentum
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J. Cent. South Univ. (2015) 22: 1378−1388
Fig. 9 Simulation results of double-lane change maneuver in no control, control and PSO-control modes: (a) Vehicle path following;
(b) Lateral deviation during path following; (c) Lateral velocity; (d) Lateral acceleration; (e) Steering angle; (f) External yaw
momentum
velocity and acceleration always remain below the
critical margins (1 m/s and 0.8 g) [26−27]. So, vehicle
can track desired path with a minimum deviation and
maintains stability and steerability. In order to analyze
the controller performance, control efforts are compared
for proposed controllers in Figs. 8(e) and (f), respectively.
It is obvious that optimized controller by PSO needsless
control efforts. Also, results depict that designed
controllers are adaptive to various maneuvers, whereas
optimal controller can track the desired path by
considering vehicle handling properties.
4.3 Mass analysis
Finally, the effectiveness of optimal controller is
evaluated for various vehicle weights. As shown in
Figs. 10, vehicle weights vary in range of passenger
vehicle cars (1150−2000 kg) with particular center of
gravity for all cases. It shows that vehicle with optimized
controller with regard to handling properties and
maneuverability, can follow desired path with a
J. Cent. South Univ. (2015) 22: 1378−1388
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Fig. 10 Evaluation of mass effects on controller for vehicle path following: (a) Lane change path; (b) Lateral deviation; (c) Lateral
velocity; (d) Lateral acceleration
minimum deviation for various vehicle weights.
in comparison with the other cases.
5 Conclusions
References
1) Nonlinear vehicle model for longitudinal and
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3) The effectiveness of integrated ESP and steering
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evaluate in the closed-loop driver/vehicle system, for
vehicle path following.
4) Lateral controller applies corrective steering
angle and torques on tires to maintaining vehicle stability
and improving handling properties.
5) Performance of proposed controllers and their
parameters are optimized by PSO.
6) Sensitivity analysis for various vehicle masses
demonstrates the robustness of optimized controller.
7) Optimized ESP controller represents a significant
improvement in the vehicle stability and path following
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(Edited by DENG Lü-xiang)