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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

ߚ

ߩܸൌͲ

ߙ

݇

ൌͲ ߜൌͲιǡ

ܨ௬௪ሺߙǡ݇ୀǡܨ௭ሻ

ܨ௬௪

כሺܽ௬ǡߜሻ

ܨ௬௪

כ൏ሺܨ

௬௪ሻ ߙൌߙͲǤͲͳι

ܨ௬௪ሺߙǡ݂݇ൌͲǡܨ௭ሻ, ߜ

ߙൌͲι

ܨ௬௪

൏ܨ

௬௪

௧

ߜǡߙǡߙ

ܨ௫

כሺܽ௫ǡܨ௬௪

כǡߜሻ

ܨ௬

כሺܽ௬ǡܨ௬௪

כǡߜሻ

ܨ௫ሺߙୀ௦ǡ݇ǡܨ௭ሻ

ܨ௬ሺߙୀ௦ǡ݇ǡܨ௭ሻ

ܨ௫

כ൏ሺܨ

௫ሻ

ܨ௬

כ൏ሺܨ

௬ሻ

݇௫

݇௬

݇௫ ൌൌ݇௬

ܸൌܸͲǤͲͲͷ

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.

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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