1 Introduction
Active Front Steering (AFS) systems have been
introduced to improve handling stability under
adverse road conditions. In contrast to a
conventional steering system, the mechanical
linkage between the steering wheel and the front
wheels of an AFS system is complemented by an
extra angle augment motor. Therefore, a small
auxiliary front wheel angle, in addition to the
steering angle imposed by the driver, can be applied
to stabilize the vehicle besides improving vehicle
steering responses and avoiding critical handling
situations. Yet, the driver can still receive
information about road friction and vehicle stability
directly through the mechanical linkage without the
additional control as in the steer-by-wire (SBW)
systems.
Additionally to the enhanced dynamic behavior
of the steering system and vehicle stabilization, the
AFS system should also provide an improved
steering comfort by reducing steering effort [1].
Therefore, the torque assistance is required to limit
manual forces to a reasonable level. The existing
AFS system used the commercially available
hydraulic power assisted steering system with which
the oil volumetric flow should be adapted to the
output requirements. Then an additional flow
control valve was usually used resulting in a
complex control concept
This new developed AFS is based on a column
electric power steering system (C-EPS). Therefore,
the construction can vary the steering ratio by
superimposing steering angle and alleviate the
steering torque requirement to the driver by the EPS
actuator. The actuator can provide directional
control to the vehicle drive and reduce the engine
load in contrast to the hydraulic power assisted AFS
system. Furthermore, complex hydraulic system and
hydraulic delay can be eliminated. Therefore, two
motors are included in the AFS system. One
provides augment of the steering wheel angle, while
another in the EPS system provides the torque
assistance.
The goal of this paper is to develop an AFS
controller, which not only helps ensure the vehicle’s
response operability matching the driver’s sense, but
also helps prevent the vehicle from falling into an
unstable state. Most stability researches used only
the yaw rate to improve the vehicle handling
stability due to the difficulties associated with the
sideslip angle measurement [3, 4]. Theoretically, the
lateral motion of the vehicle is described by yaw
rate and sideslip angle. In addition, the sideslip
angle control can compensate the path deviation
occurring from the yaw rate control. Therefore, the
sideslip angle will be estimated and used to this
AFS system control together with the yaw rate.
To achieve better vehicle stability, the state
parameters of driving vehicle should be fed back.
The linear quadratic regulators (LQR) can be used
to find a suitable state feedback and optimize the
control. LQR has been widely used due to its simple
math disposal process and achieved optimal control
of the closed loop. However, LQR design provides
the optimal gain, which is a function of the system
matrices. Unfortunately, some parameter
uncertainties exist in the vehicle system, and result
in model inaccuracy. Therefore, although some
inherent robustness properties exist, the classical
LQR controller is not robust enough to system
uncertainties and cannot guarantee the stability of
the actual system [5].
Therefore, the classical LQR control should be
modified to overcome the limitations. The
uncertainties appearing in the problem under
consideration include unmodeled dynamics,
parameter perturbations and external disturbance [6].
As has been introduced in the literature of L.
Gianone, an active 4WS system was designed with
physical uncertainties. However, the system they
designed was only for the rear tyre stiffness
uncertainty that simply affected the state matrix A .
It is inadequate to the front wheel steering vehicle,
especially the AFS vehicle. Therefore, LQR
controller designed here proposed a matrix R with
parameter uncertainties and applied it to the AFS
control system.
The paper is organized as follows. Firstly, the
structure and modelling of the developed AFS
system is described. Then the control system
including the EPS actuator and AFS actuator is
designed based on the models in section 3. In
section 4, details of the AFS control has been
presented. To evaluate the designed controller, a
hardware-in-the-loop simulation (HILS) system is
described in section 5 and HILS tests are finally
conducted in section 6. Conclusions of this paper
are summarized in section 7.
2 Structure and modeling of an AFS system
The AFS system we developed is based on a column
electric power steering system (C-EPS). Therefore,
two motors are included, as illustrated in Fig.1: one
in the original EPS system provides the power
assistance to limit manual forces to a reasonable
level, whereas the permanent-magnet synchronous
motor (PMSM) provides augment of the steering
wheel angle. Therefore, the system can be divided
into two parts: EPS actuator and AFS actuator.
2.1 EPS actuator
A DC motor is used as EPS actuator to provide the
assistant torque for its maximum torque per given
current and controllable torque. The electric
equation for a DC motor is [7]:
dcdc dcE dc δ
& =+ uKiR (1)
where Rdc is the armature resistance, KE the
armature back emf constant, dc i the DC motor
current, udc the motor terminal voltage, δ dc the
angular position of the motor shaft.
The assistant torque of the EPS actuator is
applied to the steering system by a worm and worm
wheel. The dynamics of the DC motor then can be
given by:
dcdc dcdc cddcdc dcT δδδδ )( =−++ iKGKBJ &&& (2)
where dc J , Bdc , Kdc are the inertia moment, damping
coefficient, torsional stiffness of the DC column,
respectively; KT is the motor torque constant, δ c
the steering wheel angle, Gd the gear ratio of the
DC motor to the steering column.
2.2 AFS actuator
A PMSM is used as AFS actuator to provide the
assistant angle for its precise motor positioning
control as well as fast ratio change to the target one.
The PMSM model can be described with
simplification and Park’s transformation [8]:
)()()( 2
3 ′ q )( −= δλ mm q +− q tutRitNtiL & & (3)
where L′ is the stator inductance, N the number of
poles, qi the current of q -component, uq the stator
voltage of q -component, δ m the PMSM steering
angle, λ m the magnitude of the flux created by the
permanent magnets, R the armature resistance.
The torque produced by the PMSM can be
expressed with simplification as:
T (4)
The augmentation steering angle of the AFS
actuator is applied to the steering column through a
planetary gear mechanism, as shown in Fig.2. The
planetary gear mechanism makes it easy to realize
the variable steering ratio (VSR) and produce
superimposed steering angle.
where si is the gear ratio of the steering column to
sun wheel of the planetary gear set, mi the gear ratio
of PMSM to the ring of the planetary gear set, Gh
the gear ratio of worm-to-worm wheel, δ s the
superimposed angle of the steering column.
The PMSM dynamics can be described as [9]:
2
2
2
2
( )
() ) )((
m
p
mhm
pmh
sc
c
s
mhm
mh
sc
mm
mh
sc
mmmm
Tp
r
iGK
riG
iK
i
iGK
iG
iK K
iG
iK BJ
−−+ =
δδ ++ δ +++ δ &&&
(6)
where m J , Bm , K m are the inertia moment, damping
coefficient, torsional stiffness of PMSM column,
respectively; Kc the torsional stiffness of steering
column, p the displacement of the rack and tie rod,
p r the radius of the pinion.
2.3 Steering mechanics
Steering mechanics are also included in the system,
such as steering column, rack and tie rod. The
dynamic equations for the steering system besides
the EPS and AFS actuator can be expressed as [9]:
d
ps
mhm
p
sc
m
s
mhm
mh
sc
dcddccm
s
mh
dcdccccc
Tp ri
iGK
r
iK
i
iGK
iG
iK
GKK
i
iG KGKBJ
where c J , w J are the inertia moment of the steering
column, the front wheels, respectively; Kt , Kw the
torsional stiffness of the steering rack and tie rod,
the front wheels, respectively; Bc , Br , Bw the
damping coefficient of steering column, rack and tie
rod, front wheels; δ f the steering angle of the front
wheels, mr the mass of the rack and tie rod, k l the
length of the steering knuckle arm, Td the torque
provided by the driver, M z the resistance moment
of the front wheels.
Equation (7) describes the dynamic motion of the
steering column; Equation (8) represents the
dynamics of the rack and the tie rod. Equation (9)
describes the dynamics of the front wheels. In
Equation (9), the resistance moment of the front
wheels are depend on the longitudinal velocity.
When the vehicle is driven, with small sideslip
angle assumption of a linear vehicle model, the
resistance moment of the front wheels can be
simplified as:
( )f
x
afz
v
a dKM −+= δγβ (10)
where Kaf is the cornering coefficient of the front
wheels, d the pneumatic trail of the front wheels,
γ the yaw rate of vehicle, β the sideslip angle of
vehicle, x v the longitudinal velocity, a the distance
from gravity center to front axle.
When a vehicle moves slowly on dry asphalt and
changes direction, a large amount of steering torque
is required due to the road load on the tyres. The
tyres roll and change their directions simultaneously
[10]. The maximum resistance moment determining
the directional angle of front wheel can be expressed
as [11]:
P
G M z
3
1
3
μ= (11)
where μ is the friction coefficient between the tyre
and the road, G1 the load of the front wheels, P the
pressure of the tyres.
Express the AFS actuator model in state-space
3 Control system configurations
EPS control unit and AFS control unit are included
in the AFS system, as shown
The EPS control unit can realize the reduction of
steering torque exerted by a driver; while AFS control unit provides the steering angle augment for
vehicle safety and stability. Therefore, the augment
angle control and reference track control are
covered in the AFS control unit.
3.1 EPS control unit
The main functions of the EPS actuator are
reduction of steering torque and improvement of
return-to-center performance. These two functions
are not required to activate at the same time. A
proper amount of assist torque should be provided to
reduce the driver’s steering torque during cornering,
and to return the steering wheel to the original
position smoothly without overshoot and subsequent
oscillation of the vehicle right after reentering a
straight-line road [7].
The EPS control can optimize steering effort
characteristic for driver by varying a quantity of
assistant torque depending on various vehicletraveling situations, as shown in Fig.4. The target
current of the motor ri is determined based on the
driving conditions to reduce the steering torque
requirement. The actual current ai is generated
through the dynamics of the DC motor and the
steering column, and measured by the current
detecting unit. Then the controller calculates the
control signal to minimize the error te )( between ri
and ai .
Fi
3.2 AFS control unit
In order to control the AFS unit considering the
VSR (variable steering ratio) and vehicle stability, a
main-loop control and an inner-loop control are
included in the controller, as shown in Fig.5.
In the inner-loop control, a PI controller is design
to track the target angle of the front wheel δ fd with
a potentiometer measuring the displacement of the
steering knuckle arm. In addition, the target angle of
the front wheel is determined in the main-loop
control by both the feedforward and feedback
control. The feedforward control determines the
front wheel angle according the desired VSR and
the steering wheel angle. The stability parameters
are utilized in the feedback. Then the target angle
can be determined in the main-loop control by the
feedforward and feedback control.
4 Details of AFS main-loop control
In the main-loop control, different compensation
will be applied to the front wheels to achieve
desired state variables according to vehicle system
dynamics. Therefore, a reference model is then
established firstly.
where δ f is the steering angle controlled by the
driver steering command and the variable steering
ratio r , r cf = δδ / ; m the vehicle mass, z I the
vehicle moment of inertia; a , b are the distance
from the gravity center to the front and rear axle, respectively; Kaf , Kar the cornering coefficient of
the front and rear wheels, respectively.
This model represents the vehicle dynamic
behavior in the linear range. Suppose the vehicle
turns a constant radius circle of neutral steer, the
target responses can be obtained:
Then the feedback control can regulate the
compensating angle with reference values:
fb = Ku γ d + Kβ ββγγ d
）－（）－（ (16)
where fb u is the feedback compensation voltage of
the PMSM; KK βγ , are the feedback coefficient of
γ and β , respectively.
To implement the feedback control scheme,
accurate information on both sideslip angle and yaw
rate are required. The sideslip angle can control the
vehicle path deviation occurring from the
conventional yaw rate control. The yaw rate can be
measured with a gyroscope, while sideslip
measurement requires expensive sensors. Therefore,
estimation can serve as an option.
The sideslip angle estimation using lateral
acceleration signal, which has been tested by
Yoshifumi Aoki, was applied. According to the
2DOF model, the vehicle lateral acceleration y a can
be expressed as:
) 2)(2)(2(
)(
2 f
x
af
x
af ar
x
araf
x
xy
mv
K
mv
bKaK
mv
KK
v
va
β δγ
γβ
− − +
+ =
+= &
(17)
Then, with the measurable signals y a andγ , the
where 11 h , 12 h are the gains of estimation. A
particular h12 can be chosen to keep the observer
robust and estimate the sideslip angle exactly [14].
Thus, the sideslip angle can be fed back to the
AFS control. The feedback gain KK βγ , in Equation
(16) can be determined by linear quadratic regulator
(LQR).
4.2 LQR control
LQR can be used to find a control vector u to
minimize the cost function J for its good stability
and robustness properties [6]. The cost function J
of the AFS control takes the form:
J dt t
( ) 0
T
0 0
T = + uRueQe ∫ (19)
where ⎥
⎦
⎤ ⎢
⎣
⎡
−
−
⎥ =
⎦
⎤ ⎢
⎣
⎡ =
d
d
e
e
γγ
ββ
2
1 e , u is the control input,
Q0 and R0 are the weighting matrices of e and u .
Minimizing the cost function J , an optimized
feedback control input can be obtained:
(t) (t) 1 T
lqr 0 0 xPBRxKu* −
−=−= (20)
where P0 is the result of the Riccati equation
0 00
1 T
000
T
0 −+ =+ −
QPBBRPPAAP .
LQR provides the optimized feedback gain with
the function of system matrices A and B . And the
system uncertainties have not been dealt with in the
linear time-invariant nominal model. However,
many factors, such as inflation pressure, normal
load, nonlinearity, affect vehicle parameters [14].
As a result, the system will not perform optimally.
Although some inherent robustness properties exist;
the classical LQR controller is not robust enough to
system uncertainties and cannot guarantee the
stability of the actual system [15].Therefore, the
controller should be modified with regard to
parameter uncertainties of the vehicle.
The tire parameters represent the most important
source of uncertainties in the vehicles models [16].
The tire cornering stiffness uncertainties can be
represented by a linear function with a bounded
uncertainty [13].
⎪⎩
⎪
⎨
⎧
+= ≤
+= ≤
1)1(
1)1(
2 2
*
1 1
*
δδσ
δδσ
α α
α α
r r r
f f f
KK
KK
(21)
where σ f , σ r are the positive scaling factors
reflecting the magnitude of the deviation from the
normal values Kαf and Kαr .
Consider a general state equation including
external disturbance:
wBuBAxx 1 ++= 2
where u and w are the control input and external
lateral disturb forces, respectively. Then the
uncertain linear system can be described in the
following way with the parameter variations:
[(t)(t)][(t) (t)] (t) & += ΔAAx 1 ++ ΔBBx 1 + 2wBu(t) (23)
where 21 ,, BBA are normal matrices that describe
the system, ΔA(t) , (t) ΔB1 are perturbed matrices
representing time-varying parameter uncertainties.
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
− +
− + −
=
xz
af ar
z
af ar
x
af ar
x
af ar
vI
KbKa
I
bKaK
mv
bKaK
mv
KK
(2 (2) )
1 (2)(2 )
2 2
2
A ,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
− +
+ −
=
xz
faf rar
z
faf rar
x
faf rar
x
faf rar
vI
KbKa
I
aK bK
mv
aK bK
mv
KK
(2 (2) )
(2 (2) )
2
2
1
2
1 2
2
1 2 1 2
δσδσ δσδσ
δσδσ δσδσ
ΔA
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
z
af
x
af
I
aK
mv
K
2
2
B1 ,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
z
w
x
I
l
mv
1
B2 ,
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
=
z
af
x
af
f
z
faf
x
faf
I
aK
mv
K
I
aK
mv
K
2
2
2
2
1
1
1
1 δσ δσ
δσ
ΔB
Taking account of the uncertainties into the LQR
control design of the normal system, the Riccati
equation and cost function will be extended. The
Riccati equation takes the following form:
(1 δσ ) 0
A
0
T
1
1
01
2
1f
T
22
T T
+− =+
++++
−
QPBRPB
PBPB
φ
1 PAPA Δ ΔAP P 2
(24)
To keep the system robust, P should be
constrained as:
)1( 0 1 T
0
2
1
T
2 22
T T
++ ≥
−Δ−Δ−−−
−
PBPBR
PBPB
φ
1 PAAPPAPA
f δσ
(25)
Decompose (t)(t), ΔA ΔB1 and define the
matrices with , NL as follows:
T ΔA = LN (26)
Then the sufficient condition for (25) can be
expressed as:
0
5 HILS system
design
To evaluate the designed controller, some real-time
simulations are performed with the designed
steering equipment for the AFS system. The
hardware-in-the-loop simulation (HILS) integrates
the actual ECU and its peripheral hardware with the
virtual vehicle model, forming a closed loop to be
simulated in real time [17].
5.1 HILS system
Fig.7 shows the experimental equipment which is
made from an AFS unit with a reactive module. The
HILS system consists of three parts: the hardware
which includes steering system mechanics, ECU,
motor, data acquisition card, sensors, PCs; the
software part which includes the nonlinear vehicle
model in MATLAB, the AFS control logics, and
post-processing module; and interface part which
links the hardware and software parts.
In the HILS system, both the steering angles of
the driver and the assist motor are applied to move
the steering linkages, whose displacement is
measured by the sensor. With the sensor signal, the
vehicle model resides in the PC is computed, then
yaw rate and sideslip angle signals are sent back to
the ECU as the feedback. Thus, the desired assist
angle can be obtained to control the motor.
5.2 Nonlinear vehicle model
In the HILS system, a two-track nonlinear vehicle
model, which has been validated against test data in
our previous study, is presented to simulate and
evaluate the proposed AFS controller. The model is
derived through neglecting heave, roll and pitch
motions, as in Fig.8 [18]. The longitudinal direction
is ignored because only the lateral stability is of
interest in this study. However, the extra moment
produced by the unbalanced longitudinal forces is
covered in the model according with the crosswind.
The equations of lateral and yaw motions for the
vehicle model are derived:
where FFFF lrrlrllfrlfl ,,, ( Fli ) are the longitudinal
forces on the front left, front right, rear left, rear
right tyres, respectively; FFFF srrsrlsfrsfl ,,, ( Fsi ) the
lateral forces on the front left, front right, rear left,
rear right tyres; y v the lateral velocity; W the track
width, Fw the crosswind force, wl the distance from
the gravity centre to the action point of crosswind
force.
The cornering stiffness uncertainty is considered
using the nonlinear tyre model. As in the vehicle
model, the tyre force of each wheel can be
computed by the tyre model. In order to simulate the
performance of the tyres, the nonlinear Dugoff
model is employed [19]. The simplified model is
denoted as [20, 21]:
6 Simulation results
The performance of the designed AFS controller is
tested in the HILS system and compared with the
conventional vehicle without AFS control. The
lateral acceleration, which has been utilized to the
sideslip estimation, is used in the result comparison
instead of sideslip angle.
6.1 Torque assistant maneuver
In Fig.9, a sine wave steering is inputted at the
speed of 60 km/h where the required VSR is the
same as the ratio of mechanical system. (a) plots the
steering torque versus steering wheel angle of the
AFS controlled vehicle and the conventional vehicle
without AFS. (b), (c) are the lateral acceleration and
yaw rate responses, respectively.
It can be seen that, in order to track the similar
yaw rate and lateral acceleration, the steering torque
provided by the driver is reduced by the torque
compensation in active steering operation. Therefore,
the effectiveness of torque compensation by EPS
actuator is confirmed.
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