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Robust Design of Body Slip Angle Observer with
Cornering Power Identification at Each Tire for
Vehicle Motion Stabilization
Yoshifumi Aoki, Zhao Li, Yoichi Hori
Department of Electrical Engineering
The University of Tokyo
731, Hongo, Bunkyoku, Tokyo 1138656,Japan
yaoki@horilab.iis.utokyo.ac.jp
http://mizugaki.iis.utokyo.ac.jp/
Abstract—Body slip angle is important for vehicle’s safety.
However as sensors to measure β are very expensive, we need to
estimate β from only variables to be measurable.
In this paper,we propose a novel method based on γ and side
acceleration ay. To make this observer more robust, we design
the observer’s gain matrix for robustness and propose how to
identify cornering power at each tire.
Next, we proposed new control methods for 2Dimension
control. We control β by yaw moment with PID controller.
This method is known as DYC (Direct Yaw momnet Control) in
Internal Combustion engine Vehicles (ICVs). However, the torque
difference can be generated directly with inwheel motors.
We performed experiments by UOT MarchII. The experimen
tal results proved that our proposed method was good.
I. INTRODUCTION
Electric Vehicles (EVs) are environmentfriendly and ex
pected to be a promising solution for solving today’s energy
problems. Thanks to the dramatic improvement of motors’ and
batteries’ performances, EVs will become more popular in the
near future. It is predicted that before pure EVs, Hybrid EVs
(HEVs) will be widely used in the next 10 years.
However, it is not well recognized that EVs have other
advantages over Internal Combustion engine Vehicles (ICVs)
[1]. Those advantages can be summarized in three aspects.
First, motor’s torque generation is fast and accurate. Electric
motor’s torque response is only several milliseconds, which is
10100 times as fast as combustion engine’s. This advantage
can enable us to realize high performance control of EVs.
Second, motor torque can be known precisely. Therefore
we can easily estimate driving and braking forces between
tire and road surface in realtime. This advantage can be used
to realize novel control based on road condition.
Third, inwheel motors can be installed in EVs’ each rear
and front tires. We can control each torques of the four motors
so that it is easier to control EVs’ slip angle β and yaw rate
γ than ICVs’. In order to make full use of EVs’ advantages,
it is essentially important to research on β and γ control and
β observer.
In order to estimate β, we utilize full order linear observer
because nonlinear observer is too complex to control β and γ.
Fig. 1. UOT MarchII
We design observer’s gain matrix and propose a new method
based on γ and side acceleration ay. Additionally to make the
novel observer more robust against cornering power CP, we
proposed a novel method of cornering power identification.
We did experiments by using UOT March II. Experiments
show the proposed observer can estimate β accurately and the
observer is robust against parameter variation.
Next, we design controller to control only β. If β is too
big, EVs cannot be controlled and spin off. β should be
controlled. We utilize Model Following Control (MFC) and
PID controller. Experiments by UOT March II demonstrated
the effectiveness of our control method.
II. MODELING OF EVS
We use twowheel model [2] for twodimensional movement
of EVs as shown in Fig. 2. Generally, in order to describe
vehicle’s two dimension movement exactly, fourwheel model
is needed. However because four wheeled model is nonlinear
model, it cannot be used for linear observer design. Where P
is the center of gravity, lf is the distance from P to the front
wheel, lr is the distance from P to the rear wheel, αf is the
front wheel slip angle, αris the rear wheel slip angle and δf
is actual steering angle at tire.
Usually, we express state equations with β, γ, and vehicle
speed v. Motion equations are expressed in Equs. (1).
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γ
V
y
x
Pαr
β
δf
αf
lr
lf
2Fr
2Ff
l
Fig. 2. Twowheel model of vehicle motion
From Equs. (1) we can get the state equation:
˙ x = Ax + Bu
(1)
A
=
?
?
−(Cfl+Cfr+Crr+Crl)
mv
−(lf(Cfl+Cfr)+lr(Crl+Crr)
I
Cfl+Cfr
mv
lf(Cfl+Cfr)
I
−lf(Cfl+Cfr)+lr(Crl+Crr)
mv2
−l2
?
−1
f(Cfl+Cfr)−l2
?
r(Crl+Crr)
Iv
?
B
=
0
1
I
?
, x =
β
γ
, u =
?
δf
N
?
Cfl∼ Crrare cornering powers at each tire, which is defined
as Equs. (2) and (3).
Cf=∂Ff
∂αf
αf=0
(2)
Cr=∂Fr
∂αr
αr=0
(3)
Input N is yawmoment by driving force distribution, which
is expressed by Equs. (4).
N =d
2(−Fx fl+Fx fr−Fx rl+Fx rr)
Because N cannot be measured, we must estimate N.
(4)
A. yaw moment estimation
To estimate N, It is necessary to estimate driving forces
Fd. We proposed this yawmoment estimation method, which
is defined as Equs. (6) and expressed by Fig. (5).
Fm
Vw
Fd
?
?
Vehicle
? ?
Fx ?
Mws
Fig. 3.driving force estimator
ˆFx
=
Fm− MwdVw
d
2(−ˆFx fl+ˆFx fr−ˆFx rl+ˆFx rr)
dt
(5)
ˆ N
=
(6)
III. DESIGN OF PROPOSED LINEAR OBSERVER
A. Restructuring of output equation by using side acceleration
ay
Various methods for estimating β were proposed previously.
For example, direct integral method [3] estimates β based on
Equ. (7). In this method, estimated β contains steady state
error, therefore it can’t estimate β exactly. Nonlinear observers
[4] [5] [6] [7] aim to design an accurate model based on
actual vehicle’s dynamics and to estimate β. These methods
are suitable for simulation. But due to the complexity of the
models, these methods are difficult for β estimation in real
time.
?
The advantage of conventional linear observers is it’s simple
structure. However they are not robust enough against model
error. Moreover it cannot estimate β exactly in nonlinear
region.
In order to overcome these disadvantages, we propose
a novel linear observer in this paper. Unlike conventional
observers using only γ as measurable signal, we utilize ay
together with γ to construct the linear observer [8], which can
estimate β in nonlinear region.
To design the observer, it is necessary to restructure output
equation by measurable parameters. Using Equs. (1) and (7),
aycan be restructured as:
v(˙β + γ) = ay⇔ β =(ay
v
− γ)dt
(7)
ay= v(a11β + a12γ + b1δ + γ)
(8)
The output equation is:
y =Cx + Du
?
(9)
?
C =
01
va11v(a12+ 1)
?
, D =
?
0 0
vb110
?
, y =
?γ
ay
B. Full order linear observer
We use full order observer, which is defined by the follow
ing equations.
˙ˆ x = Aˆ x + Bu − K(ˆ y − y)
ˆ y = Cˆ x + Du
(10)
(11)
where K is observer matrix gain. ˆ x s estimated x. The esti
mation error equation e =ˆβ − β should satisfy the following
error equation:
˙ e = (A − KC)e
Full order observer’s characteristic is decided by value of
matrix gain K.
(12)
C. Design of gain matrix for robustness
If the selected matrix gain is inadequate, the linear observer
will have a poor robust performance against model error and
sometimes cannot estimate β exactly. To decide matrix gain,
we must consider two important factors. First, the observer
must be designed robust against model error. Since twowheel
model is used in our design, some model error exists more or
less. Especially, cornering power Cf and Cr depend on road
condition and loads on each tires. Therefore, their values are
changing and cannot be measured. Second, all eigenvalues
of A − KC must be located in stable region. A − KC is
the state transition matrix of Equ. (11). The positions of
A−KC eigenvalues will affect control system’s time response
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performances, such as overshoot, rising time and settling time.
To make the observer robust, we referred [9]. By calculate
Equs. (7) and (8), we can getˆβ:
˙ˆβ = a11ˆβ+a12ˆ γ+b11δf−k11(ˆ γ−γ)−k12( ˆ ay−ay)(13)
State equation of β is expressed as:
˙β = ´
a11β + ´
b11δf
a12and´
b11are the real values. Any model error is not
contained in this equation.
By Equs. (13) and (14), the state equation forˆβ−β is given
by following equation.
˙ˆβ −˙β =a11(1 − k12v)(ˆβ − β)
+[a12− k12v(a12+ 1) − k11](ˆ γ − γ)
−(1 − k12v)( ´
−(1 − k12v)( ´
−(1 − k12v)(´
The best condition for robustness in Equ. (15) is:
1 − k12v = 0 ⇔ k12=1
v
Based on consideration of pole assignment and robustness
against cornering power, K is decided as:
?[ lf(Cfr+Cfl)−lr(Crr+Crl)] λ1λ2I
−λ1− λ2
λ1and λ2are the assigned poles of the observer.
Fig. 4 is simulation result of the designed observer. This
result shows us that the novel proposed observer can estimate
β well.
a12γ +´
(14)
´
a11, ´
a11− a11)β
a12− a12)γ
b11− b11)δf
(15)
(16)
K =
(Cfr+Cfl)(Crr+Crl)(lf−lr)2 −1
1
v
m((Cfr+Cfl)lf2+(Crr+Crl)lr2)
[Cfr+Cfl)lf−(Crr+Crl)lr]I
?
0 20406080 100
0
20
40
60
80
time
velocity(km/h)
0 2040
time
6080 100
0
0.5
1
1.5
2
acceleration(m/s
2)
0 2040
time
6080 100
5
0
5
body slip angle β(deg)
0 20 40
time
6080 100
5
0
5
10
15
20
25
30
yaw rate β (deg/s)
Vx
Vy
Vw
ax
ay
simulated γ
estimstited γ
simulated β
estimatited β
[s]
[s]
[s]
[s]
(a) vehicle and tire speeds
(b) acceleration
(d) simulated and estimated γ
Vy
Vx
Vw
ay
ax
(c) simulated and estimated β
Fig. 4. Simulation result
IV. PROPOSED METHOD OF CORNERING POWER
IDENTIFICATION
A. lateral force estimator for friction circle estmation and
cornering power identification
By driving force estimator shown as Equ. 5, we can know
longitudinal forces at tire. If lateral force can be estimated,
we can know friction circle of tire. Lateral force is needed to
estimate.
Additionally, cornering power CP can be identified from
estimated side forces by using Equ. 17 and fixed trace method.
Fy= Cfα
(17)
Lateral forces are estimated by Equs.19, which is derived
from linear equation of vehicle motion shown as Equ. 18.
I ˙ γ =2Fyflfcosδf− 2Fyrlr+ N
may=2Fyfcosδf+ 2Fyr
(18)
?
ˆFyf=I ˙ γ + mlray− N
2(lf+ lrcosδf)
ˆFyr=−I ˙ γ + maylf+ Ncosδf
2(lrcosδf+ lf)
Moreover, we proposed the method of lateral force estimation
at each tire from Equs. 19 and 20.
Lateral force Fyis also defined by Equ. 20.
(19)
Fy= µ(λy)Fz
(20)
where Fz f and Fz r which are defined by Equ. 21, are
average normal components of reaction at front and rear tires.
µ(λy) is friction coefficient of lateral direction.
Fz f=m(lrg − hax)
2(lf+ lr)
Fz r=m(lrg + hax)
2(lf+ lr)
Equ. 20 shows us that lateral force be proportional to the
product of µy and normal component of reaction. Left and
right tires’ µ are nearly equal. So lateral force is directly
proportional to Fz. Fzcan be calculated by Equ. 22.
Fz fr=m(lrg − hax)
2(lf+ lr)
Fz fl=m(lrg − hax)
2(lf+ lr)
Fz rr=m(lrg + hax)
2(lf+ lr)
Fz rl=m(lrg + hax)
2(lf+ lr)
where h is the distance from P to the ground, ayis lateral
acceleration and axis longitudinal acceleration.
From Equs. 20 ∼ 22, we can get lateral forces at each tire.
(21)
+hmay
d
−hmay
+hmay
d
(22)
d
−hmay
d
ˆFy fr=µ(λy f)Fz fr= µ(λy f)Fz fFz fr
Fz f
=ˆFyf(1 +2(lf+ lr)hay
d(lrg − hax))
ˆFy fl=µ(λy f)Fz fr= µ(λy f)Fz fFz fr
Fz f
=ˆFyf(1 −2(lf+ lr)hay
ˆFy rr=µ(λy f)Fz fr= µ(λy f)Fz fFz fr
d(lrg − hax))
(23)
Fz f
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=ˆFyr(1 +2(lf+ lr)hay
d(lrg + hax))
ˆFy rl= µ(λy f)Fz fr= µ(λy f)Fz fFz fr
Fz f
=ˆFyr(1 −2(lf+ lr)hay
d(lrg + hax))
Fig. 5 is simulation result. This result shows that the
poposed lateral force estimator can estimate lateral forces at
each tire.
05 1015
500
0
500
1000
1500
2000
2500
3000
time
Fyfl
Fyfl est
05 10 15
500
0
500
1000
1500
2000
2500
3000
time
Fyfr
Fyfr est
05 1015
500
0
500
1000
1500
2000
2500
3000
time
Fyrl
Fyrl est
05 1015
500
0
500
1000
1500
2000
2500
3000
time
Fyrr
Fyrr est
Lateral force [N]
Lateral force [N]
Lateral force [N]
Lateral force [N]
Fig. 5. Simulation result;lateral force estimation
N estmator
body slip angle
observer
Tr
?f
N
^
Electric
Vehicle
lateral force
estimator
ay
?
β^
N
^
CP estimator
Fy
^
CP
^
Fig. 6. conrnering power estimator
TABLE I
SENSORS OF UOT MARCHII
PC to controlPentium MMX 223[MHz]
AMD K6233[MHz]
Slackware Linux 3.5
RTLinux rel. 9K
3600[ppr]
OS
encoder pulse
number
acceleration sensor
Yaw rate sensor
ANALOG DEVICES ADXL202
HITACHI OPTICAL FIBER
GYROSCOPE HOFGCLI(A)
CORREVIT S400Noncontact
Optical sensor
V. CORNERING POWER IDENTIFICATION
As shown in the preceding chapter, lateral force can be
estimated by the proposed method. We apply Equ. 17 to
adaptive identification (Equ. 24).
y[k]=ϕT[k]ˆθ[k] + e[k]
ˆθ[k]=ˆθ[k−1] +P[k−1]ϕ[k](y[k] − ϕT[k]ˆθ[k−1])
κ2+ ϕT[k]P[k−1]ϕ[k]
P[k]=1
(24)
κ1(P[k − 1] −P[k − 1]ϕ[k]ϕT[k]P[k − 1]
κ2+ ϕT[k]P[k − 1]ϕ[k]
We utilize fixed trace method. κ1and κ2are expressed as:
)
κ1=κ,κ2= 1
?P[k − 1]ϕ[k]?2
1 + ϕTP[k − 1]ϕ[k]
(25)
κ =1 −
1
ξ
(26)
where ξ is trace gain defined by following equation in case of
fixed trace method:
ξ = trP[k] = const.
(27)
In this case, y is lateral foce, ϕ is tire slip angle α and θ is
cornering power.
VI. STRUCTURE OF ESTIMATING BODY SLIP ANGLE
We estimate body slip angle β by the proposed observer
based on γ and ay with cornering power identification. This
system is shown in Fig. 6.
In this system, body slip angle β and cornering power
estimate each other.
VII. EXPERIMENTAL DEMONSTRATION FOR OBSERVER BY
UOT MARCH II
A. Experiment setup for the proposed observer
UOT MarchII is our experimental EV built to prove EVs’
advantages. We made this EV by ourselves, which is remod
eling of Nissan March. The EV equips acceleration sensor,
gyro sensor and noncontact speed meter which enable us to
measure β. Table. I explains specification.
We did various experiments and some experimental results
were shown you. Vehicle velocity, driver’s input steering wheel
angle δ and road type were changed to test the effectiveness
and robustness of the proposed observer and controller. While
the experiment was done, road type was changed from dry
road to wet road. But observer’s and controller’s parameters
are kept unchanged. Steering angle was changed freely by the
test driver.
We recorded β, γ, δ, v, Vw, Fmand ayin hard disk drive
by the sampling time of 1 [ms] and calculated by Matlab.
B. Experimental results for the proposed observer
Figs. 7 and 8 show two experiments’ results under the
difference conditions.
Fig. 7 shows experimental results in linear region. Fig. 7 (a)
is measured and estimated β. Fig. 7 (b) is identified cornering
power. Fig. 7 (c) and (d) are measured and output ayand γ.
In Fig. 7 (a), EV is in linear region because steering angle
δ is small. Fig. 7 shows us that the novel proposed observer
can estimate β well in linear region. Even if we cannot know
cornering power, we can know precise β by cornering power
identification.
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2626.5 2727.5 2828.5 29
time(s)
29.5 3030.5 31
1
0.5
0
0.5
1
1.5
2
2.5
3
β obtained by the proposed observer
β measured by noncontact optical sensor
^
^
β
β
body slip angle β(deg)
(a)measured value and
estimation of β
2626.5 2727.5 28 28.5 29
time [s]
29.5 3030.5 31
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4
CPfl
CPfr
CPll
CPlr
cornering power [N/rad]
(b)identified value of cornering
power
8
20 2530354045
2
1
0
1
2
3
4
5
6
β obtained by the proposed observer
β measured by noncontact optical sensor
^
time(s)
body slip angle β(deg)
^
β
β
(a)measured value and
estimation of β
2025 30 354045
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4
CPfl
CPfr
CPll
CPlr
cornering power [N/rad]
time [s]
(b)identified value of cornering
power
26
26.5 27 27.5 2828.5 29
time(s)
29.5 30 30.5 31
5
0
5
10
15
20
25
30
35
40
yaw rate g (deg/s)
g obtained by the proposed observer
g measured by gyro sensor
^
(c)measured value and
observer’s output of γ
26 26.5 2727.5 2828.5 29
time(s)
29.5 30 30.5 31
2
1
0
1
2
3
4
5
6
7
ay obtained by the proposed observer
ay measured by acceleration sensor
^
ay(m/s2)
lateral acceleration
(d)measured value and
observer’s output of ay
Fig. 7. estimation results; steering
angle δ is 90[deg]
20253035 4045
5
0
5
10
15
20
25
30
35
g obtained by the proposed observer
g measured by gyro sensor
^
time(s)
yaw rate g (deg/s)
(c)measured value and
observer’s output of γ
20
25 30 35 40 45
2
1
0
1
2
3
4
5
6
7
8
time(s)
ay(m/s2)
lateral acceleration
ay obtained by the proposed observer
^
ay measured by acceleration sensor
(d)measured value and
observer’s output of ay
Fig. 8. estimation results; steering
angle δ is 180[deg]
Fig. 8 is experimental results in nonlinear region. Because δ
is larger than δ in Fig. 7, β becomes larger and EV enters non
linear region. Fig. 8 demonstrates that the novel observer can
estimate β even in nonlinear region. In Fig. 8 (b), estimated
Conering powers is smaller than in linear region.
VIII. DESIGN OF β CONTROLLER
Body slip angle β must be controlled for driver’s safety.
We proposed this method based on DYC (Direct Yaw Moment
control), which makes full use of EVs’ advantages.
Proposed method is based on PID controller and MFC
(Model Following Control).
We did experiments by UOT March II to demonstrate the
effectiveness of our control method.
A. Model Following Control (MFC)
We utilize Model Following Control (MFC) in order to gen
erate the reference of the vehicle dynamics, or the ”desired”
response to driver’s input [2]. In this paper, MFC calculates
the desired value βref.
We use linear state equations expressed by Equ. (1) as the
desired model. Using Equ. (1), βref is:
βref=
(s − a22)b11+ a12b21
(s − a11)(s − a22) − a12a21δf
where matrix A and B are defined in Equ. (1)
If we can keep β following the desired value βref freely,
EVs’ movements and safety will be improved dramatically.
(28)
C
F12
EV
controller
N
β ref
β
+

desired
model
δ
Fig. 9. the proposed control diagram
B. Design of body slip controller
The proposed control diagram is Fig. 9. We control body
slip angle β by yaw rate N. We use linear two wheel
(Equ. 1)model for PID controller to make EV’s advantage,
that is ”motor’s torque generation is fast and accurate”.
F12is transfer function from input yaw moment N to body
slip angle β.
F12=
a12b22
(s − a11)(s − a22) − a12a21
Transfer function from βref to β in Fig. 9 is:
(29)
β
βref
=
CF12
1 + CF12
(30)
To decide poles of the transfer function in Equ. (30) freely,
we use PID gain (Equ. (31)).
C = K1s + K2+K3
s
(31)
By Equs. (29), (31) and (30), we can get following equation.
β
βref
=
(K1s2+K2s+K3)a12b22
s3−s2(a11a22−a12a21)+(K1s2+K2s+K3)a12b22(32)