A Yaw Rate Tracking Control of Active Front Steering System 233
control technique. In Section 2, the dynamics of a nonlinear vehicle model and a lin- ear single track model are discussed. The theory of CNF control and design proce-
dures are explained in Section 3. The simulation results and discussion are presented in Section 4. Finally, a conclusion and future works is presented in Section 5.
2 Vehicle Dynamic Models
In this section, dynamic equations of a nonlinear two track vehicle model and a linear single track model are presented and discussed. These two models are constructed for
vehicle plant and controller design respectively, whose performance is analyzed using a computer simulation tool.
2.1 Nonlinear Two Track Model
A nonlinear two track model as shown in Figure 1 is used as the actual vehicle plant for controller evaluation. The nonlinear dynamics for lateral and yaw motion are de-
scribe as in equations 1 and 2 respectively;
r F
F mv
y y
−
−
=
β β
β
sin cos
1
1
[ ]
z y
y r
f x
f x
f y
f y
f z
M F
F l
F F
F F
l I
r +
+ −
+ +
+ =
sin sin
cos cos
1
4 3
2 1
2 1
δ δ
δ δ
2 where the sum of longitudinal forces
y
F
, sum of lateral forces
x
F
and yaw mo- ment
z
M in the above equations are given as follows;
sin cos
2 1
2 1
y y
f x
x f
y
F F
F F
F +
− +
=
δ δ
3 cos
sin
2 1
2 1
y y
f x
x f
y
F F
F F
F +
+ +
=
δ δ
4
4 3
2 1
2 1
sin sin
cos cos
2
x x
f y
f y
f x
f x
z
F F
F F
F F
d M
− +
+ −
− =
δ δ
δ δ
5 The vehicle parameters involved in equations 1 - 5 above are vehicle speed v ,
vehicle mass m , vehicle width track d, distance of front axle to center of gravity CG
f
l and distance of rear axle to CG
r
l . The front wheel steer angle
f
δ
is the input to the system while the nonlinear longitudinal tire forces
xi
F and lateral tire forces
yi
F can be described using Pacejka tire model. Notice that the vehicle speed v
is always assumed constant when no braking and accelerating are involved.
234 M.K. Aripin et al.
d
2 x
F
2 y
F
1 y
F
1 x
F
4 x
F
3 x
F
4 y
F
3 y
F
x
v
v
r
d
β
f
l
r
l
y
v
z
M
f
δ
f
δ
Fig. 1. Two track model 2.2
Tire Model
In the equations 1 - 5 above, the longitudinal tire force
xi
F and lateral tire force
yi
F may exhibit nonlinear characteristics. The pure longitudinal and lateral tire forces
during pure side slip can be described using the Pacejka tire model as described in the following equations 6 and 7 respectively
[ ]
. tan
. .
tan sin
1 1
i xi
i xi
xi i
xi xi
xi xi
B B
E B
C D
F λ
λ λ
− −
− −
= 6
[ ]
. tan
. .
tan sin
1 1
f yi
f yi
yi f
yi yi
yi yi
B B
E B
C D
F α
α α
− −
− −
= 7
where the parameters
xi yi
xi yi
xi yi
xi
E B
B C
C D
D ,
, ,
, ,
, and
yi
E are known as tire model
parameters that depending on tire characteristics, road surface and vehicle conditions while
i
λ
and
i
α
are longitudinal wheel slip and tire sides slip angle respectively that given by the following equations
i i
i i
V V
R −
=
ω λ
8
+
− =
=
−
v r
l
f f
. 1
2 1
tan β
δ α
α 9
+ −
= =
−
v r
l
r
β α
α
1 4
3
tan
10 where in equation 8, R is wheel radius,
i
ω
is wheel angular velocity and
i
V is ground contact speed for each tire.
A Yaw Rate Tracking Control of Active Front Steering System 235
2.3 Linear Single Track Model
