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Linear Quadratic Optimal Regulator for Steady State Drifting
Paper:
Linear Quadratic Optimal Regulator for Steady State Drifting of
Rear Wheel Drive Vehicle
Ronnapee Chaichaowarat and Witaya Wannasuphoprasit†
Department of Mechanical Engineering, Chulalongkorn University
254 Phayathai Road, Wangmai, Pathumwan, Bangkok 10330, Thailand
E-mail: witaya.w@chula.ac.th
†Corresponding author
[Received April 15, 2014; accepted January 28, 2015]
Drifting is a large sideslip cornering technique with
counter steering, which is advantageous in some driv-
ing conditions where vehicle-handling capability over
linear tire slip-friction characteristics is imperative. In
this paper, the dynamics of a rear-wheel-drive (RWD)
vehicle cornering at steady states was simplified us-
ing a single-track vehicle model. In addition, tire fric-
tions in any slip conditions were estimated from the
combination of the Pacejka’s magic formula and the
modified Nicolas-Comstock tire model. A computer
program was developed, on the basis of the equations
of motion (EOMs) derived via the body-fixed coordi-
nate so that the suitable cornering speed and its corre-
sponding steady-state driving control inputs (the steer-
ing angle and rear wheel slip ratio) could be calcu-
lated automatically for any given radius of curvature
and vehicle sideslip. The other set of EOMs was de-
rived via the normal-tangential coordinate and then
linearized so that the state space description could be
constructed. Eventually, the linear quadratic optimal
regulator was designed and simulated via MATLAB
for various regulation problems where the initial con-
dition of each individual state deviated from its de-
sired steady-state value. According to the simulation
results, the physical explanation of the control inputs
can be used as guidance for adjusting vehicle behavior
in manual driving.
Keywords: vehicle dynamics, automotive control, drift-
ing, nonlinear control systems, quadratics optimal regula-
tors
1. Introduction
Nowadays, a variety of active safety systems have
been developed for general passenger vehicles so that
drivers can control their vehicles even in emergency con-
ditions [1]. Most of the existing systems – e.g., differ-
ential braking, active steering, and independent wheel
torque control systems in modern electric or hybrid ve-
hicles – are developed on the basis of a simplified vehi-
cle model of steady-state cornering with small sideslip, as
in [2]. In addition, the elementary active safety systems
in all current automobiles – an anti-lock braking system
(ABS) and a traction control system (TCS) – are primar-
ily intended to prevent extreme sliding so that all tires
operate within the linear region of the tire slip-friction
characteristic, allowing ordinary drivers to maintain con-
trol. It is envisioned that rather than restricting the tire
slip limit within the linear region, both the full handling
performance of vehicle and the expert skills of the driver
in extreme slip conditions can be advantageous in terms
of accident avoidance.
Drifting is a cornering technique with large sideslip as
a result of the rear tires experiencing extreme slip in both
the lateral and longitudinal directions. Although drifting
is not convenient for a normal driver, in addition it re-
quires expert skills to control the suitable slip of each of
the tires to maintain the equilibrium of forces and the mo-
ment so that the vehicle travels along a desired trajectory.
This technique is popular and is frequently used by expert
rally drivers when turning sharply on dirt tracks to ensure
that their vehicles are not understeered.
That the number of studies on drifting has increased
significantly over the last decade reflects the increase in
researcher awareness of how important an understanding
of drifting is. According to the computational simulation
in [3], though the minimum time trajectory cannot be ac-
quired, drifting provides an advantage to return to a con-
trollable straight-line driving state faster than small-slip
driving. As a result, the driver has enough time to respond
to emergencies or unexpected changes in the driving envi-
ronment. Some basic drifting techniques – e.g., trail brak-
ing and pendulum turning – have been studied and simu-
lated using experimental data recorded from drift testing
by an expert rally driver in [4, 5]. The test vehicle had an
inertial measurement unit, a GPS module, and other nec-
essary equipment to measure the vehicle state. In addi-
tion, neural networks were used in [6] so that the vehicle
sideslip could be simply related with the data measured
from an accelerometer array. Although some controversy
about the drifting dynamics had begun to be revealed by
a number of works, very few studies have been conducted
on the development of a controller for automatic drifting.
As previously mentioned, drifting with extreme slip
may be advantageous in some cornering conditions. With
Journal of Robotics and Mechatronics Vol.27 No.3, 2015 225
Chaichaowarat, R. and Wannasuphoprasit, W.
this vision in mind, steady-state drifting, which is the sim-
plest case, was studied first. The optimal controller for
steady-state drifting of an all-wheel-drive (AWD) vehicle
was initiated in [7], where computational simulation was
conducted via a single-track vehicle model along with the
combination of the Bakker-Nyborg-Pacejka (BNP) magic
formula [8] and the friction circle model, which assumes
symmetrical tire characteristics in both the lateral and lon-
gitudinal directions. The linear quadratic optimal regula-
tor (LQR) controller was designed corresponding to the
state space model constructed from the linearization of
the vehicle model and the tire model about the desired
steady state. However, the steering angle was fixed so that
the open-loop driving control inputs – the front- and rear-
wheel slip ratios – could be directly computed withoutany
iteration. As a result, it was difficult to generate most of
the calculated control inputs in real-world situations. For
example, independent torque distribution among the front
and rear drive shafts was needed in case a positive slip ra-
tio was required in both the front and rear wheels. In ad-
dition, the left-foot-braking technique may be required in
case a negative slip ratio was required at the front wheels
and a positive slip ratio was required at the rear wheels.
Hence, it would be more appropriate to derive con-
trol inputs that could be controlled conveniently. With-
out making the steering angle constant, the algorithms to
calculate the control inputs for steady-state cornering of
an RWD vehicle were developed in [9–13] by neglecting
rolling resistance and assuming that no brake torque was
applied to the front wheels. The optimal controller for
steady-state drifting of an RWD vehicle was initiated and
simulated computationally in [9, 10], where the compre-
hensive full-car vehicle model, along with the tire fric-
tion estimating approach as used in [7], was used to de-
velop a computer program for iteration of the driving con-
trol inputs: the steering angle and the rear drive torque.
However, details of the iteration algorithm were not pro-
vided. In addition, the tire friction estimation procedure
used in [7, 9, 10] gave an estimation with discrepancies at
a high degree of combined slip. Soon afterward, the com-
puter program for automatic iteration of the steady-state
control inputs – the steering angle and rear wheel slip
ratio – was developed in [11] using the single-track ve-
hicle model and the semi-empirical combined-slip Pace-
jka’s magic formula tire model [14]. However, the tire
model for simulation in [14] was too complicated for con-
trol purposes.
To estimate the tire frictions accurately despite ex-
tremely combined slip, the BNP-MNC tire friction model
used in [12, 13] was developed from the combination of
the BNP magic formula [8] and the modified Nicolas-
Comstock (MNC) tire model [15, 16]. Building on the
framework of [11], the computer program for calcula-
tion of a suitable cornering speed and its corresponding
steady-state driving control inputs – the steering angle
and rear wheel slip ratio – for any given radius of curva-
ture and vehicle sideslip was developed in [12,13] based
on the single-track vehicle model and the BNP-MNC tire
model.
Although the objective of finding control inputs for
steady-state cornering in [12], as well in [11], was not
for the control application but for the determination of the
maximum cornering speed among different sideslips in-
stead, those could be sufficiently fundamental for [12].
The feasible steady state – i.e., the cornering speed cor-
responding to a given radius of curvature and vehicle
sideslip – acquired from the algorithm proposed in [12]
were used as a reference for the desired steady state
in [13] where the LQR stabilizing controller had been
designed and simulated via MATLAB. In the proposed
control scheme, stabilization is achieved by regulating the
steering angle and rear slip ratio.
In this paper, the stabilizing controller developed
in [13] is tested via computational simulation with a va-
riety of regulation problems from which the initial con-
dition of each individual state deviates from its desired
steady-state value. The following content will be orga-
nized in the same way as in [13]. The single-track vehicle
model will be introduced first. Then, the application of the
BNP-MNC tire friction model will be briefly described.
Afterward, the algorithm of computation for steady-state
driving control inputs will be explained via the program
flowchart. In addition, the equations of motion (EOMs)
in terms of the state space description will be proposed
for the control scheme. Next, the design procedure of the
LQR stabilizing controller will be proposed. Finally, the
simulation results will be presented and discussed from a
physical perspective.
2. Vehicle Model
In this study, the dynamics of steady-state cornering of
an RWD vehicle along a constant radius curve on a hor-
izontal plane with a constant cornering speed and vehi-
cle sideslip was simplified using a two-dimensional dy-
namic model – namely, the single-track model or the bi-
cycle model. With the application of this model, the com-
plexity of the actual automobile dynamics can be reduced
with the consideration of a two-wheeled vehicle in which
the dynamics of suspension and the effects of lateral load
transfer are neglected.
2.1. Single Track Vehicle Model
To define the symbols used in the single-track vehicle
model and their sign conventions, the translation of the
vehicle drifting along the arc from position A to B is il-
lustrated in Fig. 1.
In this figure, the x-ycoordinate is the moving frame
affixed to the vehicle body at its center of gravity (CG),
where the x-axis is oriented in the longitudinal direction
directed toward the vehicle heading, and the y-axis is ori-
ented in the lateral direction. The other moving frame
located at the CG is the n-tcoordinate, the direction of
which always changes depending on the direction of the
vehicle velocity. Longitudinal distances Lfand Lrare
measured from the CG to the front and rear axles, respec-
226 Journal of Robotics and Mechatronics Vol.27 No.3, 2015
Linear Quadratic Optimal Regulator for Steady State Drifting
ܮ
ܮ
Fig. 1. Symbols used in the single-track vehicle model.
tively. The vehicle velocity
Vmeasured at the CG is per-
pendicular to the direction of the curved radius
ρ
,which
is measured from the instantaneous center zero velocity
(ICZV). The angle between the actual velocity and the in-
stant heading direction is given by the vehicle sideslip
β
.
Likewise, the direction of local velocities measured at the
front axle
Vfand rear axle
Vrare perpendicular to the di-
rections of the curved radii measured from the ICZV to
the front and rear axles, respectively. The sideslip
β
,the
steering angle
δ
, the front-wheel slip angle
α
f,andthe
rear-wheel slip angle
α
rare positive in the clockwise di-
rection, whereas the yaw angle
ψ
, the yaw rate ˙
ψ
,andthe
yaw acceleration ¨
ψ
are measured in the counterclockwise
direction. Finally, the positive directions and the points of
applications of all tire forces are indicated. In this paper,
drifting is defined by the counter steering behavior – i.e.,
the positive steering angle (
δ
>0)in the counterclockwise
cornering.
According to the body-fixed x-ycoordinate, the set of
equations of motion (EOMs) is obtained. The equilibri-
ums of the forces in the longitudinal and lateral directions
are shown in Eqs. (1) and (2), respectively. In addition,
the equilibrium of the moment about the vertical axis is
shown in Eq. (3).
Fxw f cos
δ
−Fyw f sin
δ
+Fxr =max..... (1)
Fxw f sin
δ
+Fyw f cos
δ
+Fyr =may..... (2)
−Fxw f sin
δ
+Fyw f cos
δ
Lf+Fyr Lr=ICG ¨
ψ
(3)
The lateral and longitudinal accelerations can be derived
from certain normal and tangential accelerations via the
coordinate transformation between the x-yand n-tcoordi-
nates, as shown in Eqs. (4) and (5).
ax=ansin
β
+atcos
β
......... (4)
ay=−ancos
β
+atsin
β
......... (5)
In the special case of steady-state drifting, tangential ac-
celeration is always zero, as expressed in Eq. (6). In ad-
dition, normal acceleration can be simply calculated from
Eq. (7).
at=˙
ρ
˙
ψ
+
ρ
¨
ψ
=0 ........... (6)
an=
ρ
˙
ψ
2=V2
ss
ρ
ss
............ (7)
According to the assumptions that no rolling resistance
and no braking and driving torques are applied to the front
wheels, the EOMs describing steady-state drifting of an
RWD vehicle can be modified as given in Eqs. (8)–(10).
−Fyw f sin
δ
+Fxr =mV 2
ss
ρ
ss
sin
β
ss ...... (8)
Fyw f cos
δ
+Fyr =−mV 2
ss
ρ
ss
cos
β
ss ...... (9)
−Fyw f cos
δ
Lf+Fyr Lr=0 ........(10)
The obtained EOMs were rearranged to a usable form as
follows:
Fyw f =mayLr
Lf+Lrcos
δ
........ (11)
Fxr =max+Fywf sin
δ
........ (12)
Fyr =may−Fywf cos
δ
........ (13)
This set of EOMs was used to construct the computer pro-
gram for calculation of the suitable cornering speed and
corresponding open-loop driving control inputs for any
desired steady state.
By using the single-track vehicle model, lateral load
transfer can be neglected. However, longitudinal load
transfer is still significant. The calculations of vertical
loads at the front and rear axles are shown in Eqs. (14)
and (15), respectively, where his the vertical distance of
the CG from the ground.
Fzf =m
Lf+Lr
(gLr−axh)........(14)
Fzr =m
Lf+LrgLf+axh........(15)
For a given set of vehicle states – the radius of curvature
(
ρ
), vehicle sideslip (
β
), and cornering speed (V),the
vehicle yaw rate (˙
ψ
)can be computed directly, as shown
in Eq. (16). In addition, the local velocities at the CG in
each component are given in Eqs. (17) and (18).
˙
ψ
=V
ρ
................(16)
Vx=Vcos
β
..............(17)
Vy=Vsin
β
..............(18)
From the kinematic relation, the components of the front
axle velocity along the longitudinal and lateral directions
of the vehicle are given in Eqs. (19) and (20), respectively.
In addition, the components parallel and perpendicular to
Journal of Robotics and Mechatronics Vol.27 No.3, 2015 227
Chaichaowarat, R. and Wannasuphoprasit, W.
the wheel heading direction, which are required for the
calculation of the slip angle and slip ratio, are given in
Eqs. (21) and (22), respectively.
Vxf =Vx.............. (19)
Vyf =Vy−˙
ψ
Lf........... (20)
Vxwf =Vxf cos
δ
+Vyf sin
δ
....... (21)
Vywf =−Vxf sin
δ
+Vyf cos
δ
...... (22)
Likewise, components parallel and perpendicular to the
wheel heading direction of the rear axle velocity are given
in Eqs. (23) and (24), respectively.
Vxr =Vx...............(23)
Vyr =Vy+˙
ψ
Lr.............(24)
For the front tire radius of rfand the rotational speed of
ω
f, the slip angle can be calculated by using Eq. (25) or
(26), whereas Eq. (27) is for slip ratio calculation. For the
rear tire radius of rrand the rotational speed of
ω
r, the slip
angle and slip ratio can be calculated by using Eqs. (28)
and (29), respectively.
α
f=tan−1Vyw f
Vxwf .......... (25)
α
f=tan−1Vyf
Vxf −
δ
......... (26)
kf=
ω
frf−Vxwf
ω
frf
.......... (27)
α
r=tan−1Vyr
Vxr .......... (28)
kr=
ω
rrr−Vxr
ω
rrr
........... (29)
The slip angle and slip ratio obtained in this section are
the important information directly affecting tire friction,
as will be described in the following section.
2.2. BNP-MNC Tire Friction Model
While drifting, extremely combined slip at rear tires
is inevitable. Thus, the appropriate tire friction model
must be deliberately selected. In this study, the modi-
fied Nicolas-Comstock (MNC) tire model for combined
slip [15] was used to estimate the tire frictions in both the
longitudinal and lateral directions, as shown in Eqs. (30)
and (31), respectively. The MNC tire model allows non-
isotropic tire characteristics in both directions; ordinary
tire slip-friction functions for pure slip in the longitudi-
nal direction Fx(k)and the lateral direction Fy(
α
)can be
selected individually. The traction stiffness coefficient Ck
and the cornering stiffness coefficient C
α
are given by the
initial slope of Fx(k)and Fy(
α
), respectively. The first
quotients of Eqs. (30) and (31) are the original Nicolas-
Comstock model, as given in [16], and the second quo-
tients are the correction factors that affect the shape of the
friction ellipses.
Table 1. Magic formula parameters of a P225/60R16 tire.
Parameter B C D E K Fz[N]
Fx(k)0.12 1.48 3308 0.01 100 3101
Fy(
α
)0.08 1.44 6004 −1.84 100 6145
Longitudinal friction coefficient
Fig. 2. Longitudinal friction coefficient versus slip.
Fx(
α
,k)= Fx(k)Fy(
α
)k
k2F2
y(
α
)+F2
x(k)tan2
α
×k2C2
α
+(1−|k|)2cos2
α
F2
x(k)
kC
α
. (30)
Fy(
α
,k)= Fx(k)Fy(
α
)tan
α
k2F2
y(
α
)+F2
x(k)tan2
α
×(1−|k|)2cos2
α
F2
y(
α
)+sin2
α
C2
k
Cksin
α
(31)
In this study, the tire slip-friction functions for pure slip
Fx(k)and Fy(
α
)were formulated in terms of the Bakker-
Nyborg-Pacejka (BNP) magic formula [8] as given in
Eqs. (32) and (33).
Fi(si)=Disin Citan−1(Bi/0i)......(32)
/0i=(1−Ei)Kisi+Ei
Bitan−1(BiKisi)...(33)
where the subscript i=xy indicates the direction of inter-
est, whether it is the longitudinal or the lateral direction of
the friction and the slip. The longitudinal slip (sx)refers
to the slip ratio (k), whereas the lateral slip (sy)refers to
the slip angle (
α
). The other symbols are the magic for-
mula parameters.
By using the BNP-MNC tire friction model with the
magic formula parameters gathered from the experiment
of a P225/60R16 tire in [14], as shown in Ta bl e 1 ,the
estimation results of the longitudinal tire friction coeffi-
cient
μ
x=FxFzcan be plotted with varied slip angles
and slip ratios as shown in Fig. 2. In a similar manner,
the estimation results of the lateral tire friction coefficient
228 Journal of Robotics and Mechatronics Vol.27 No.3, 2015
Linear Quadratic Optimal Regulator for Steady State Drifting
li i
Lateral friction coefficient
Fig. 3. Lateral friction coefficient versus slip.
μ
y=FyFzare plotted with varied slip angles and slip
ratios as shown in Fig. 3.
According to Fig. 2, at any given slip ratio, maximum
magnitude of longitudinal friction occurs at the zero slip
angle when the tire does not slip in the lateral direction.
At the zero slip angle, the longitudinal friction increases
linearly with the increasing slip ratio and then reaches the
maximum value. Finally, it slightly decreases to satura-
tion until the slip ratio reaches unity. According to Fig. 3,
at any given slip angle, maximum magnitude of lateral
friction occurs at the zero slip ratio when the tire does not
slip in the longitudinal direction. At the zero slip ratio,
the lateral friction increases linearly with the increasing
slip angle and then reaches the maximum value. Finally, it
slightly decreases to saturation until the slip angle reaches
90◦.
3. Computation of Feasible Steady-State Drift-
ing and Control Inputs
As previously mentioned, without making the steer-
ing angle constant, computational iteration is necessary
for the derivation of steady-state control inputs for RWD
vehicle drifting. The vehicle model and the tire friction
model described in the previous section were used to con-
struct the computer program for the calculation of a suit-
able cornering speed for a given radius of curvature and
vehicle sideslip. In addition, the corresponding driving
control inputs for the steering angle and the rear-wheel
slip ratio were calculated. The algorithm of the devel-
oped computation will be briefly described via the pro-
gram flowchart in Fig. 4.
First, the vehicle parameters of vehicle mass (m);mo-
ment of inertia (ICG); wheelbase (L); longitudinal dis-
tance from the CG to the front axle (Lf)and to the
rear axle (Lr); vertical distance of the CG from the
ground (h); and the BNP magic formula parameters
(B,C,D,E,and K)were defined. Then, the radius of cur-
vature (
ρ
), vehicle sideslip (
β
), and positive initial esti-
mation of the cornering speed (V)had to be entered into
the program. After that, the components of the accel-
eration in the n-tand x-ydirections were computed by
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Fig. 4. Algorithm to compute steady-state control inputs.
Eqs. (4)–(7). The vertical loads at the front and rear axles
were also computed using Eqs. (14) and (15). In addition,
the vehicle yaw rate and local velocities at the CG, also at
the front and rear axles, were calculated using Eqs. (16)–
(24). Finally, the rear slip angle was directly calculated
by Eq. (28).
For the ensuing iteration, the initial value of the steer-
ing angle must be set as any arbitrary constant – e.g., zero
degrees (
δ
=0◦)– so that the first iteration of the front
lateral tire force F∗
yw f (ay,
δ
)based on the modified vehi-
cle model in Eq. (11) can be computed. In addition, the
front slip angle, which will be increased by 0.01◦at ev-
ery loop of iteration, was primarily set at zero degrees
(
α
f=0◦). The front tire lateral friction, varying with the
slip angle at the zero slip ratio Fywf (
α
f,kf=0,Fzf),was
estimated by the BNP-MNC tire model. The lateral force
Journal of Robotics and Mechatronics Vol.27 No.3, 2015 229
Chaichaowarat, R. and Wannasuphoprasit, W.
Table 2. Vehicle parameters used in the simulation.
Parameters Va l u e Parameters Va l u e
m[kg] 1250 L[m] 2.52
ICG [kg·m2]2500 h[m] 0.28
Lf[m] 1.13 rf[m] 0.3
Lr[m] 1.39 rr[m] 0.3
computed from the vehicle model (F∗
yw f )was compared
with the maximum of lateral friction estimated by the tire
model (Fyw f )to identify whether the front tire was still
in equilibrium. The front tire will slide out in the lateral
direction if the vehicle model-based tire force exceeds the
maximum limit of friction generated from the tire model.
If the front tire can sustain equilibrium, the corresponding
front slip angle will be gained by iteration when the error
deviation between the vehicle model-based tire force and
the tire model-based tire force (Ferr
yw f )is less than the tol-
erance limit (Ftol
yw f ). In addition, the steering angle can be
directly computed by Eq. (26).
The obtained steering angle was used to compute the
rear-axle longitudinal force F∗
xr(ax,F∗
yw f ,
δ
)and lateral
force F∗
yr(ay,F∗
yw f ,
δ
)using the modified vehicle model in
Eqs. (12) and (13), respectively. In addition, both the lon-
gitudinal friction Fxr (
α
r=cons,kr,Fzr)and lateral friction
Fyr (
α
r=cons,kr,Fzr)of the rear tire, varying with the slip
ratio (kr)from 0 to 1, were estimated by using the BNP-
MNC tire model at a previously calculated constant rear
slip angle.
The vehicle model-based tire forces (F∗
xr,F∗
yr)in both
the longitudinal and lateral directions were separately
compared with the tire model-based frictions (Fxr ,Fyr)
to identify whether the rear tire was still in equilibrium.
The rear tire will slide out in some direction if the vehi-
cle model-based tire force exceeds the maximum limit of
the friction generated from the tire model in that direc-
tion. If the rear tire could sustain equilibrium in both di-
rections, the longitudinal friction-based rear tire slip ratio
(krx), which entails equality between the tire model-based
friction (Fxr )and vehicle model-based tire force (F∗
xr),
was specified. Likewise, the lateral friction-based rear
tire slip ratio (kry), which entails equality among the tire
model-based friction (Fyr )and vehicle model-based tire
force (F∗
yr), was specified simultaneously. The existence
of the tangible slip ratio was at a suitable cornering speed
when both rear slip ratios were equivalent (krx =kry).
In this study, the vehicle parameters in Table 2 were
used for simulation. The results of the demo computation
using the developed program are shown in Fig. 5.The
attentive radius of curvature, the vehicle sideslip, and the
arbitrary positive initial estimation of the cornering speed
must be entered at the beginning. Then, the magnitude of
the suitable steady-state velocity and the corresponding
driving control inputs – the steering angle and rear wheel
slip ratio, along with other important data, such as front-
and rear-wheel slip angles and the tangible wheel rota-
ڭ
Fig. 5. Demo results of the developed program.
tional speeds at the front and rear axles – were computed
and displayed, respectively. From the given set of the en-
tered data, only the simulation results corresponding to
the feasible minimum and maximum cornering speeds are
exhibited, respectively, in Fig. 5. In this paper, drifting
is defined by the positive steering angle, which indicates
counter-steering behavior.
4. Design of Stabilizing Controller for Steady-
State Drifting
4.1. State Space Description
To design the linear quadratic optimal regulator (LQR)
stabilizing controller, the new set of EOMs must be de-
rived into the expression of the vehicle states and driving
control inputs and then linearized into the state space de-
scription.
The equilibrium equations corresponding to the direc-
tion of absolute velocity at the CG were taken into con-
sideration; therefore, they could then be derived in terms
of tangential acceleration (at), normal acceleration (an),
and yaw acceleration (¨
ψ
)as Eqs. (34)–(36), respectively.
d
dtV=1
mFxw f cos (
δ
−
β
)−Fyw f sin (
δ
−
β
)
+Fxr cos
β
+Fyr sin
β
]..... (34)
d
dt
β
=1
mV Fxw f sin (
δ
−
β
)+Fyw f cos (
δ
−
β
)
−Fxr sin
β
+Fyr cos
β
+mV ˙
ψ
]. (35)
d
dt ˙
ψ
=1
ICG −Fxw f sin
δ
+Fyw f cos
δ
a+Fyr b(36)
According to the kinematic relation Eq. (37), the EOMs of
an RWD vehicle drifting on a plane could be expressed by
the implicit functions of the state variables and the driv-
ing control inputs as Eqs. (38)–(40) by neglecting rolling
230 Journal of Robotics and Mechatronics Vol.27 No.3, 2015
Linear Quadratic Optimal Regulator for Steady State Drifting
resistance and assuming that no brake torque was applied
to the front wheel.
d
dt
ρ
=˙
ψ
d
dtV−Vd
dt ˙
ψ
˙
ψ
2.........(37)
d
dt
ρ
=f1(
ρ
,
β
,V,
δ
,kr)
=
ρ
mV −Fyw f sin (
δ
−
β
)+Fxr cos
β
+Fyr sin
β
+
ρ
2
ICGVFyw f cos
δ
a−Fyr b.... (38)
d
dt
β
=f2(
ρ
,
β
,V,
δ
,kr)
=1
mV Fyw f cos (
δ
−
β
)−Fxr sin
β
+Fyr cos
β
+mV2
ρ
..... (39)
d
dtV=f3(
ρ
,
β
,V,
δ
,kr)
=1
m−Fyw f sin (
δ
−
β
)+Fxr cos
β
+Fyr sin
β
]......... (40)
The standard form of the state space description as
Eqs. (41) and (42) could then be established by lin-
earization of Eqs. (38)–(40) about the desired equilibrium
(
ρ
ss,
β
ss,Vss).
˙
˜
X
X
X=A
A
As
s
ss
s
s˜
X
X
X+B
B
Bs
s
ss
s
s˜
U
U
U...........(41)
˜
Y
Y
Y=C
C
C˜
X
X
X...............(42)
The Jacobian matrices A
A
As
s
ss
s
sand B
B
Bs
s
ss
s
swere constructed by
performing the partial derivative of Eqs. (38)–(40) by each
of the state variables and each of the driving control inputs
about the desired equilibrium, as given in Eqs. (43) and
(44), respectively. Furthermore, the output matrix C
C
Cwas
the identity matrix with a dimension of 3 ×3.
A
A
As
s
ss
s
s=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∂
f1
∂ρ
ss
∂
f1
∂β
ss
∂
f1
∂
V
ss
∂
f2
∂ρ
ss
∂
f2
∂β
ss
∂
f2
∂
V
ss
∂
f3
∂ρ
ss
∂
f3
∂β
ss
∂
f3
∂
V
ss
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
....(43)
B
B
Bs
s
ss
s
s=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
∂
f1
∂δ
ss
∂
f1
∂
kr
ss
∂
f2
∂δ
ss
∂
f2
∂
kr
ss
∂
f3
∂δ
ss
∂
f3
∂
kr
ss
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.......(44)
The state vector ˜
X
X
Xrepresented the deviation of the cur-
rent states from the desired states, as given in Eq. (45).
In addition, the input vector ˜
U
U
Urepresented the deviation
of the current control inputs from the steady-state control
inputs, as shown in Eq. (46).
˜
X
X
X=⎡
⎣
ρ
−
ρ
ss
β
−
β
ss
V−Vss
⎤
⎦...........(45)
˜
U
U
U=
δ
−
δ
ss
kr−krss ...........(46)
4.2. LQR Stabilizing Controller
In this study, the state variable feedback controller gain
for steady-state drifting stabilization was designed to min-
imize the quadratic performance index in Eq. (47).
J=1
2T
t0˜
X
X
XT
T
TQ
Q
Q˜
X
X
X+˜
U
U
UT
T
TR
R
R˜
U
U
Udt ......(47)
Appropriate choices for the composition of the weighting
matrices – i.e., the semi-positive definite matrix (Q
Q
Q≥0
0
0)
and the positive definite matrix (R
R
R>0
0
0)– were suggested
as Eqs. (48) and (49), respectively.
1
Qii
=tf−t0×Max acceptable value of [xi(t)]2(48)
1
Rii
=tf−t0×Max acceptable value of [ui(t)]2(49)
The subscript iof the elements in the weighing matrices
indicates the considered state or the control input. For ex-
ample, Q11 is used for the gain consideration of the first
state variable. The final time (tf)is determined by the
desired setting time for state regulation. Deviation of the
considered state varyingwith time from the desired steady
state is denoted by (xi(t)). Likewise, deviation of the con-
sidered control input varying with time from the steady
state reference is denoted by (ui(t)).
5. Simulation Results
The designed LQR stabilizing controller was simulated
via MATLAB for various regulation problems, in which
the initial condition of each individual state deviated from
its desired steady state – the 22 m radius of curvature(
ρ
ss),
15◦sideslip (
β
ss), and 50.23 km/h cornering speed (Vss).
In this section, the simulation results for the regulation of
the initial error in the radius of curvature, the initial error
in the vehicle sideslip and the initial error in the cornering
speed will be discussed from a physical perspective. The
resulting plots of each case will exhibit the error of the
radius of curvature, the error of the vehicle sideslip, the
error of the cornering speed, the steering angle control
input, and the rear-wheel slip ratio control input, changing
with time, respectively.
5.1. Regulation of Initial Error in Curve Radius
The simulation results of the steady-state drifting stabi-
lization when the initial radius of curvature deviated 1 m
Journal of Robotics and Mechatronics Vol.27 No.3, 2015 231
Chaichaowarat, R. and Wannasuphoprasit, W.
Fig. 6. Resulting plots for regulation of initial error in radius
of curvature.
from its desired steady state (
ρ
0=23 m)are shown in
Fig. 6. According to the plots of state errors, the error of
the radius suddenly decreases to zero, resulting in an in-
crease in the sideslip and speed error. However, all state
errors were eventually eliminated by stabilization.
Based on the plots of the control inputs, a sharp drop
in the steering angle below its steady reference (
δ
ss =
4.326◦)was observed at the beginning of the simulation.
This implies that the steering angle tends to be conven-
tional steering such that the radius of curvature is reduced
and then increases to its steady reference with a small
overshoot. The plots of the state errors depicts that the
sharp drop in the steering angle at the beginning affected
the decrease of the sideslip and, later, the increase of the
speed. In addition, the sharp rise of the rear slip ratio
above its steady reference (kss =0.169)at the beginning
of simulation affects the reduction of the lateral friction
at the rear tire so that the decrease of the vehicle sideslip
slowed down and returned to its desired value.
Furthermore, the trajectory plot of the RWD vehicle
drifting on the X-Yplane is shown in Fig. 7. The plot
depicts that the controlled vehicle could drift at a steady
state with a constant radius of curvature, sideslip, and cor-
nering speed.
5.2. Regulation of Initial Error in Vehicle Sideslip
The simulation results of the steady-state drifting stabi-
lization when the initial sideslip of the vehicle deviated 2◦
from its desired steady state (
β
0=17◦)is shown in Fig. 8.
According to the plots of the state errors, the error of the
Fig. 7. Trajectory of the RWD vehicle drifting on the X-Yplane.
Fig. 8. Resulting plots for regulation of initial error in vehi-
cle sideslip.
sideslip slowly decreased to zero, resulting in a sudden
increase of the radius and speed error. However, all state
errors were eventually eliminated by stabilization.
Based on the plots of control inputs, an extremely sharp
drop in the steering angle below its steady-state value at
the beginning of simulation could be observed. This im-
plies that the steering angle was regulated to be hard con-
ventional steering to reduce the exceeding sideslip. This
232 Journal of Robotics and Mechatronics Vol.27 No.3, 2015
Linear Quadratic Optimal Regulator for Steady State Drifting
Fig. 9. Resulting plots for regulation of initial error in cor-
nering speed.
response of the steering angle also directly affects the
decrease of the radius of curvature. The steering angle
increased to a positive value and eventually reached its
steady reference. At the beginning of the simulation, the
rear slip ratio increased sharply to cancel the effect of the
sharply increased steering angle. At the same time that
the steering angle became positive, the rear slip ratio de-
creased to its steady reference so that the rear tire could
generate greater lateral friction to reduce the sideslip er-
ror.
5.3. Regulation of Initial Error in Cornering Speed
The simulation results of the steady-state drifting stabi-
lization when the initial cornering speed deviated 2 km/h
from its desired steady state (V0=52.23 km/h)are shown
in Fig. 9. According to the plots of the state errors, the
error of speed suddenly decreased to zero, resulting in an
increase of the radius and sideslip error. However, all state
errors were eventually eliminated by stabilization.
Based on the plots of the control inputs, a negative rear
slip ratio at the beginning of the simulation to reduce the
instant speed could be observed. This response of the rear
slip ratio also directly affects the sudden decrease of the
sideslip. Furthermore, the steering angle was regulated
to be hard conventional steering at the beginning of the
simulation to prevent too much negative deviation of the
vehicle sideslip. This response of the steering angle also
directly affects the negative deviation of the radius of cur-
vature.
6. Conclusions
In this study, the single-track vehicle model and the
BNP-MNC tire friction model were used to simplify the
dynamics of RWD vehicle cornering at a steady state.
The computer program for the calculation of a suitable
cornering speed and its corresponding steady-state con-
trol inputs for any given radius of curvature and vehicle
sideslip was developed on the basis of the EOMs derived
via the body-fixed coordinate. The LQR stabilizing con-
troller was designed from the state space description lin-
earized from the other set of EOMs derived via the n-t
coordinate; in addition, it was simulated by MATLAB,
and the initial condition of each individual state deviated
from its desired steady-state value. According to the sim-
ulation results, all state errors could be regulated. The
discussion on the response of the control inputs may be
used as guidance for adjusting vehicle behavior in manual
driving. The steady-state drifting controller can be im-
proved from other nonlinear system control schemes [17].
The application of an automatic drifting assistant system
in general passenger vehicles may be feasible in the fu-
ture.
Acknowledgements
This research was partly funded by the Junior Science Talent
Project (JSTP) and the Department of Mechanical Engineering at
Chulalongkorn University.
References:
[1] N. Wada, A. Takahashi, M. Saeki, and M. Nishimura, “Vehicle yaw
control using an active front steering system with measurements of
lateral tire force,” J. of Robotics and Mechatronics, Vol.23, No.1,
pp. 83-93, 2011.
[2] T. D. Gillespie, “Fundamentals of Vehicle Dynamics,” SAE Int.,
PA, USA, 1992.
[3] E. Velenis and P. Tsiotras, “Minimum time vs maximum exit veloc-
ity path optimization during cornering,” the IEEE Int. Symposium
on Industrial Electronics, 2005.
[4] E. Velenis, P. Tsiotras, and J. Lu, “Modeling aggressive maneuvers
on loose surfaces: data analysis and input parameterization,” the
Mediterranean Conf. on Control and Automation, 2007.
[5] E. Velenis, P. Tsiotras, and J. Lu, “Modeling aggressive maneuvers
on loose surfaces: the cases of trail-braking and pendulum-turn,”
the European Control Conf., 2007.
[6] M. Abdulrahim, “On the dynamics of automobile drifting,” SAE
Technical Paper, No.2006-01-1019, 2006.
[7] E. Velenis, E. Frazzoli, and P. Tsiotras, “On steady-state cornering
equilibria for wheeled vehicles with drift,” the 48th IEEE Conf. on
Decision and Control, pp. 3545-3550, 2009.
[8] E. Bakker, L. Nyborg, and H. B. Pacejka, “Tyre modeling for use in
vehicle dynamics studies,” SAE Technical Paper, No.870421, 1987.
[9] E. Velenis, D. Katzourakis, E. Frazolli, P. Tsiotras, and R. Happee,
“Stabilization of steady-state drifting for a RWD vehicle,” the 10th
Int. Symposium on Advanced Vehicle Control, 2010.
[10] E. Velenis, D. Katzourakis, E. Frazolli, P. Tsiotras, and R. Happee,
“Steady-state drifting stabilization of RWD vehicles,” Control En-
gineering Practice, Vol.10, No.11, pp. 1363-1376, 2011.
[11] R. Chaichaowarat and W. Wannasuphoprasit, “Two dimensional dy-
namic model of drifting vehicle,” the 7th TSAE Int. Conf. on Auto-
motive Engineering, 2011.
[12] R. Chaichaowarat and W. Wannasuphoprasit, “Dynamics and simu-
lation of RWD vehicle drifting at steady state using BNP-MNC tire
model,” SAE Technical Paper, No.2013-01-0001, 2013.
[13] R. Chaichaowarat and W. Wannasuphoprasit, “Optimal control for
steady state drifting of RWD vehicle,” Proc. of the 7th IFAC Sym-
posium of Advances in Automotive Control, pp. 824-830, 2013.
Journal of Robotics and Mechatronics Vol.27 No.3, 2015 233
Chaichaowarat, R. and Wannasuphoprasit, W.
[14] H. T. Szostak, W. R. Allen, and T. J. Rosenthal, “Analytical mod-
eling of driver response in crash avoidance maneuvering. volume
II: an interactive model for driver/vehicle simulation,” U.S. Depart-
ment of Transportation Report, NHTSA DOT HS-807-271, 1988.
[15] R. Branch and M. Branch, “The tire-force ellipse (friction ellipse)
and tire characteristics,” SAE Technical Paper, No.2011-01-0094,
2011.
[16] V. T. Nicolas and T. R. Comstock, “Predicting directional behav-
ior of tractor semitrailers when wheel anti-skid brake systems are
used,” ASME Paper, No.72 - WA/AUT-16, 1972.
[17] M. Tanaka, H. Ohtake, and K. Tanaka, “A simple, natural and ef-
fective framework of nonlinear systems control and its application
to aerial robot,” J. of Robotics and Mechatronics, Vol.26, No.2,
pp. 140-147, 2014.
Name:
Ronnapee Chaichaowarat
Affiliation:
Department of Mechanical Engineering, Chula-
longkorn University
Address:
254 Phayathai Road, Wangmai, Pathumwan, Bangkok 10330, Thailand
Brief Biographical History:
2012 Received Bachelor degree in Mechanical Engineering from
Chulalongkorn University (1st Class Honors)
2013 Received Master degree in Mechanical Engineering from
Chulalongkorn University
2013- Ph.D. Student, Department of Mechanical Engineering,
Chulalongkorn University
Main Works:
•“Tire Test for Drifting Dynamics of a Scaled Vehicle,” TSME J. of
Research and Applications in Mechanical Engineering, Vol.1, No.3,
pp. 33-39, 2013.
•“Dynamics and Simulation of RWD Vehicle Drifting at Steady State
using BNP-MNC Tire Model,” SAE Int. J. of Transportation Safety, Vol.1,
No.1, pp. 134-144, 2013.
•“Optimal Control for Steady State Drifting of RWD Vehicle,” IFAC
Proc. Volumes, Vol.7, pp. 824-830, 2013.
Membership in Academic Societies:
•Junior Science Talent Project (JSTP)
•Society of Automotive Engineers Thailand (TSAE)
Name:
Witaya Wannasuphoprasit
Affiliation:
Department of Mechanical Engineering, Chula-
longkorn University
Address:
254 Phayathai Road, Wangmai, Pathumwan, Bangkok 10330, Thailand
Brief Biographical History:
1990 Received Bachelor degree in ME (Honor) from King Mongkut’s
Institute of Technology Ladkrabang
1993 Received Master degree in ME from Northwestern University
1999 Received Ph.D. in ME from Northwestern University
2000 Post-Doctoral Fellow, Northwestern University
Main Works:
•Cobots: Collaborative Robots, Haptic Interface, Intelligent Assisted
Devices
•US Patents: 6928336, 6907317, 6813542, 6241462
•Awards: Best Paper Awards: IEEE ICRA 1996, ASME IMECE 1998
(MHED)
Membership in Academic Societies:
•Thai Society of Mechanical Engineering (TSME)
•Society of Automotive Engineers Thailand (TSAE)
234 Journal of Robotics and Mechatronics Vol.27 No.3, 2015
