investigation of robust roll motion control considering varying speed and actuator dynamics

Mechatronics 14 (2004) 35–54

Investigation of robust roll motioncontrol considering varying speed

and actuator dynamics

Hyo-Jun Kim a,*, Young-Pil Park b

a Department of Mechanical Engineering, Samchok National University, 253 KyoDong Samchok,

KangwonDo 245-711, South Koreab Department of Mechanical Engineering, Yonsei University, South Korea

Received 29 August 2001; received in revised form 27 May 2002; accepted 23 September 2002

Abstract

This paper presents the design of an active roll controller for a vehicle and an experimental

study using the electrically actuating roll control system. Firstly, parameter sensitivity analysis

is performed based on the 3DOF linear vehicle model. The controller is designed in the

framework of lateral acceleration control and gain-scheduled H1 control scheme considering

the varying parameters induced by laden and running vehicle condition. In order to investigate

the feasibility of an active roll control system, experimental work is performed using a

hardware-in-the-loop (Hil) setup which has been constructed using the devised electrically

actuating system and a full vehicle model with tire characteristics. The performance is eval-

uated by experiment using the devised Hil setup under the conditions of steering maneuvers

and parameter variations. Finally, in order to enhance the control performance in the transient

region, a hybrid control strategy is proposed.

� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

Ground vehicle design typically represents a trade-off between performance and

safety. Design parameters affecting lateral dynamics can influence maneuvering

ability, but also have some influence on dynamic stability including spinout and

rollover. In steering maneuvers, vertical loads on tires at the outer track increase and

* Corresponding author. Tel.: +82-33-570-6322; fax: +82-33-574-2993.

E-mail address: [email protected] (H.-J. Kim).

0957-4158/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0957-4158(02)00094-6

mail to: [email protected]

36 H.-J. Kim, Y.-P. Park / Mechatronics 14 (2004) 35–54

those on the inner track decrease, which is called lateral load transfer. When moment

equilibrium is broken at some conditions, a vehicle loses roll stability. Geometric

dimensions, suspension characteristics as well as maneuvering conditions influence

the dynamic roll behavior of a car. This is a problem for automobiles [15], but alsoon railroad vehicles with a high center of gravity. To improve the roll characteristics

of a car, the customary approach is to increase the roll stiffness using a stabilizer bar.

Unfortunately, this method affects the ride comfort with respect to high frequency

isolation induced by road excitation.

In order to enhance vehicle performance in an active manner, advanced suspen-

sion systems, such as active suspension, have been widely analyzed in the literature

over the years. At present one of the most efficient methods for further development

is the roll control option combined with a semi-active suspension system [1]. Thissystem performs the following functions: (1) isolation of the driver from uneven

roadway noise, road holding on irregular road surfaces using a variable damper

system, (2) safe turning through steering using the active roll control (ARC) system

[6]. This provides many of the benefits of a fully active system with much reduced

cost and power consumption. Unfortunately, most studies have focused on the active

or semi-active suspension system and there have been few studies of the ARC system.

Lin et al. [2] performed a theoretical study of active roll reduction in heavy ve-

hicles. Their system used an anti-roll bar equipped with a hydraulic linear actuatorwhich provided the necessary torque to counteract the roll moment of the car body.

The lateral acceleration feedback and LQR control scheme were used, based on the

linear vehicle model. Ross-martin et al. [3] and Sharp et al. [4] performed a simu-

lation study of the ARC system for passenger cars with a hydraulic rotary actuator

using lateral acceleration feedback control.

As is the case with any vehicle system, an actual car is expected to operate in a

highly variable environment. For instance, parameter variations resulting from

loading pattern and driving condition will influence vehicle dynamics. The influenceof time-variable parameters such as forward speed, which is assumed to be constant

in previous works, on dynamic characteristics will also be considered. This raises

questions about the robustness of the control system which mean that the controller

must cope with these uncertainties successfully. For a system with time-variable

parameters, a customary approach to the design of a controller is to switch the gain

according to a pre-fixed value or to handle it as uncertainties with a varying range. In

these cases, it gives rise to delicate stability question in the switching zone and it

cannot guarantee satisfactory performance and robustness across a range of varia-tions. Among the control schemes, it has been recognized that the gain-scheduled H1control scheme guarantees robust stability and disturbance rejection in the presence

of time-variable parameters. Packard and Gahinet have made an important con-

tribution toward this approach. Packard proposed a control structure in which the

controller adjusts to variations in the plant dynamics in order to maintain stability

and high performance along all trajectories, under the assumption of real-time

measurement of parameters [8]. Apkarian and Gahinet described the design and

synthesis of a gain-scheduled controller with guaranteed H1 performance based onthe linear matrix inequality (LMI) approach [7,9].

H.-J. Kim, Y.-P. Park / Mechatronics 14 (2004) 35–54 37

In this paper, parameter sensitivity analysis is performed based on a three DOF

vehicle model including lateral/roll dynamics related to steering maneuvers. After

investigation of dynamic characteristics, the active roll controller is designed in theframework of a gain-scheduled H1 control scheme considering parameter variations

resulting from loading conditions and operating speed. In order to evaluate the

control performance, the prototype passenger car electric actuation system, com-

prised of an active anti-roll bar with electric motor and ball screw actuator, is

constructed. That system can improve the high frequency isolation with high com-

pliance characteristics during inoperative periods. Using the hardware-in-the-loop

(Hil) setup including the prototype roll control system and full vehicle model with

tire characteristics, the control performance of lateral acceleration control and gain-scheduled H1 control including actuator dynamics is investigated. Finally, in order

to enhance the control performance in the transient region, a hybrid control scheme

is proposed and evaluated.

2. Mathematical vehicle model

In developing the active controller, it is not desirable to use the complex vehicle

model because of sampling time and implementation of the control system. In this

paper, the linear vehicle model is used for the design of a controller. Fig. 1 shows a

Fig. 1. Handling characteristics model: (a) top view; (b) front view.


three DOF model including the yaw and roll dynamics of a car, related to driver

steering maneuvers, traveling on a road surface at a constant speed V with the tire

steering angle di ði ¼ 1; . . . ; 4Þ. The coordinates xyz are a vehicle-fixed frame and XYZare an earth fixed coordinates system. The vehicle is assumed to be symmetrical inthe x–z plane, the tire characteristics and road conditions for the left and right tires

are the same. In a conventional front steer vehicle, the tire has a steering angle df atthe front only, that is, df ¼ d1 ¼ d2, dr ¼ 0 ¼ d3 ¼ d4.

In a yaw plane representation (Fig. 1(a)), the side slip angle b, at the center of

gravity in a car body, is expressed in Eq. (1) when a vehicle is rotating at some

angular velocity r relative to the inertial frame:

b ¼ tan�1 vyvx

� �V

�¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2x þ v2y

q �ð1Þ

where vx, vy are velocity components in x, y directions, respectively.

Where the slip angle b is assumed to be small, jbj � 1, each velocity component

can be written as vx ¼ V cos b � V , vy ¼ V sin b � V b. The tire slip angles bi ði ¼1; . . . ; 4Þ at each tire can be expressed as follows:

b1 ¼ tan�1 V bþ lfrV � tfr=2

� �� df � bþ lfr

V� df ð2Þ

b2 ¼ tan�1 V bþ lfrV þ tfr=2

� �� df � bþ lfr

V� df ð3Þ

b3 ¼ tan�1 V b� lrrV � trr=2

� �� b� lrr

Vð4Þ

b4 ¼ tan�1 V b� lrrV þ trr=2

� �� b� lrr

Vð5Þ

In Eqs. (2)–(5), under assumption that the magnitudes of jtfr=2j, jtrr=2j are small in

the first term, the equations have been simplified by neglecting second order terms.

Tire slip angles at front and rear, expressed as bf ¼ b1 ¼ b2 and br ¼ b3 ¼ b4,

have the following matrix form if roll angle effects are considered:

bf

br

� �¼

1lfV

1 � lrV

264

375 b

c

� �� df

0

� �� af

ar

� �/ ð6Þ

where af , ar are the fixed coefficients of roll effects.

There are many models that describe tire characteristics. In this paper, under the

assumption that the lateral tire forces Yf , Yr are linear functions with slip angles bf ,

br, Eq. (7) is used:

Yf ¼ �2Kfbf ; Yr ¼ �2Krbr ð7Þ
where Kf , Kr are the cornering stiffness for the front and rear tires, respectively.
In a roll plane representation like Fig. 1(b), the car body has a roll motion with

roll angle / and roll angular velocity p relative to the roll center.


Considering the previous forces and moments, equations of motion including

lateral, yaw and roll motion are written as

MV ð _bbþ rÞ þMshs _pp ¼ 2Kf df

�þ af/� b� lf

Vr�þ 2Kr ar/

�� bþ lr

Vr�

ð8Þ

Iz _rr ¼ 2Kf df

�þ af/� b� lf

Vr�lf � 2Kr ar/

�� bþ lr

Vr�lr ð9Þ

Ix _pp þMshsV ð _bbþ rÞ ¼ �K//� C/p ð10Þ

where K/ is roll stiffness and C/ is roll damping.

From Eqs. (8)–(10), the state-space representation can be expressed as Eq. (11).

Where u is an active roll moment Md to reduce the roll response resulting from steer

disturbance df . Considering that one inclinometer about the roll axis is the available

sensor, the measured output variable is the car body roll angle. The measured output

equation is written as Eq. (12).

E _xx ¼ A0xþ B0uþ L0df ð11Þ

y ¼ Cpxþ Dpu ð12Þ

where

xp ¼ ½ v r / p �T

A0 ¼

�ðKf þ KrÞV

�MV � KflfV

þ KrlrV

� �ðKfaf þ KrarÞ 0

�ðKflf � KrlrÞV

�ðKfl2f þ Krl2r ÞV

ðKfaflf � KrarlrÞ 0

0 0 0 1

0 �MshsV �K/ �C/

26666664

37777775

B0 ¼ ½ 0 0 0 1 �T; L0 ¼ ½Kf Kflf 0 0 �T;Cp ¼ ½ 0 0 1 0 �; Dp ¼ ½0�

E ¼

M 0 0 Mshs0 Iz 0 0

0 0 1 0

Mshs 0 0 Ix

2664

3775

3. Parameter sensitivity analysis

3.1. Eigenvalue sensitivity analysis with speed variation

In this study, the effects of changes in vehicle parameters were examined. For each

variation of parameters, the eigen-problem [10] was solved and dynamic behavior

was investigated.


In order to analyze the eigenvalue sensitivity of the linear system associated with

vehicle parameters, the characteristic equation can be rewritten from Eq. (11) as

_xx ¼ Apx ðAp ¼ E�1A0Þ ðAp ¼ ½akl�; k ¼ 1; . . . n; l ¼ 1; . . . ; nÞ ð13Þ

The partial differentiation form of the characteristic equation with respect to anelement akl of Ap gives:

oAp

oakl

� �� ui þ Ap �

ouioakl

� �¼ oki

oakl

� �ui þ ki

ouioakl

� �ð14Þ

where ki is eigenvalue and ui is eigenvector with i ¼ 1; . . . ; n:Using orthogonality of eigenvector like:

uTi � vj ¼ vTj � ui ¼ dij

where dij is Kronecker delta and vi is left eigenvector with j ¼ 1; . . . ; n.Premultiplying by vTi to Eq. (14) and reordering the terms gives

okioakl

� �¼ vTi �

oAp

oakl

� �� ui ð15Þ

The eigenvalue sensitivity ki;r with respect to vehicle parameter �zzr ðr ¼ 1; . . . ; qÞfrom the numerical calculation of ðoakl=o�zzrÞ can be calculated as

okio�zzr

� �¼ ki;r ¼

Xn

k;l¼1

okioakl

� �� oakl

o�zzr

� �ð16Þ

ki;r can be expressed as:

ki;r ¼ ai;r þ ibi;r ð17Þ

where ai;r ¼ Rebki;rc, bi;r ¼ Imbki;rc.The vehicle parameters, �zzr ðr ¼ 1; . . . ; 6Þ in Eq. (16), are selected as mass of car

body (sprungmass), cornering stiffness, roll moment of inertia, yaw moment of in-

ertia, roll stiffness and roll damping. Each of the eigenvalue sensitivity related to �zzr iscalculated with respect to the forward speed range of 30–120 km/h. Fig. 2(a) and (b)

illustrate the complex eigenvalue sensitivity k3;r divided into a real part and an

imaginary part resulting from the calculation of Eq. (17). These figures show that the

behavioral characteristics of the vehicle are affected by vehicle parameter variations,

but also by variations in forward speed. In case of r ¼ 2 and 4, they are influentialwith respect to damping with an increase of speed above 30 km/h, while the other

parameters mainly affect the frequency shift.

3.2. Parameter sensitivity analysis

In order to investigate the influence of parameter variations induced by loading

patterns, roll dynamic analysis is performed. Fig. 3(a) and (b) illustrate the influence

of variations in sprungmass Ms, roll moment of inertia Ix on roll angle response at a

constant speed 50 km/h using a step steer maneuver like:

Fig. 2. Eigenvalue sensitivity of k3;r with varying forward speed: (a) real part; (b) imaginary part.


df ¼

h0 t < t0

h0 þ ðh1 � h0Þðt � t0Þðt1 � t0Þ

� �2

3� 2ðt � t0Þðt1 � t0Þ

� �� t0 < t < t1

h1 t > t1

8>><>>:

ð18Þ

with parameters varying in the range of �20% relative to nominal values in Table 1,subjected to steer input with h0 ¼ 0 deg, h1 ¼ 3:5 deg, t0 ¼ 0:0 s, t1 ¼ t0 þ 0:2 s.

The results clearly show that the variations in car body mass affect the steady-

state response as well as transient response and Ix mainly affect the transient response

of the vehicle. The influence of forward speed V on roll transfer function is also

illustrated in Fig. 4 when the speed is varied from 20 to 120 km/h with nominal

0 1 2 3 4 5-0.05

-0.04

-0.03

-0.02

-0.01

Rol

lAng

.(ra

d)

-20 %-10 %0 %+10 %+20 %

(a) 0 1 2 3 4 5

-0.05

-0.04

-0.03

-0.02

-0.01

Rol

lAng

.(ra

d)

-20 %-10 %0 %+10 %+20 %

(b) Time(sec) Time(sec)

0.00 0.00

Fig. 3. Influence of parameter variations on roll angle gain. Influence of (a) sprungmass and (b) roll

moment of inertia.

1.0E-001 1.0E+000 1.0E+001 1.0E+002Frequency (rad/s)

-80

-60

-40

-20

0

Mag

(dB

)

V = 20 km/hV = 40 km/hV = 60 km/hV = 80 km/hV = 100 km/hV = 120 km/h

20

Fig. 4. Influence of speed variations on roll angular velocity.

Table 1

Vehicle parameters

Parameter Nominal vehicle Perturbed vehicle

Sprungmass (MS, kg) 1011 1215

Roll moment of inertia (Ix, kgm2) 440 520

Yaw moment of inertia (Iz, kgm2) 2400 2800


vehicle parameters. Fig. 4 shows that the gain and damping characteristics are

affected by the variation in V .The results show that the influence of uncertainties associated with loading

conditions and operating speed should be considered in the design of the controller.


4. Design of controller

4.1. Lateral acceleration control

A lateral acceleration controller is the conventional control method, which is

simple and easy to implement. A lateral acceleration signal from a body mounted

transducer provides the main source for the roll control. In order to improve the

transient roll behavior, a roll angular velocity feedback term is added. This con-

troller computes the desired anti-roll moment like:

MdðtÞ ¼ KaayðtÞ þ Kd_//ðtÞ ð19Þ

Each constant gain Ka, Kd is determined by experiment using the Hil-setup based onfull vehicle dynamics in Section 5.

In a previous work [3], performance of this type of ARC system was shown to be

adversely affected by the operating speed. In order to improve this problem, con-

troller gain was switched based on a pre-fixed value with respect to the speed range.

In this paper, the above method is not adopted because this is a partial solution to be

carefully investigated in a wide operating range, especially, in the gain switching

region.

4.2. Gain-scheduled H1 control

4.2.1. Control system formulation

This section presents the design of an active roll controller based on a gain-

scheduled H1 control scheme, which can be automatically gain-scheduled along

varying parameter trajectories. As parameter sensitivity analysis, the roll control

problem has a large operating range of forward speed as well as the presence of

uncertainties with respect to loading patterns. Assuming real-time measurement ofvarying forward speed, it can be fed to the controller to optimize the performance

and robustness of the closed-loop system in the framework of the gain-scheduled H1control scheme [7].

Fig. 5 shows the configuration of the control structure including exogeneous input

wd, control input u to controlled outputs z, measured output y, varying parameter

vector g and time invariant weighting functions WaðsÞ, WsðsÞ, WnðsÞ.The plant transfer function Puðs; gÞ and disturbance transfer function Pwðs; gÞ are:

Puðs; gÞ ¼ CpðsI � ApðgÞÞ�1Bp ð20Þ

Pwðs; gÞ ¼ CpðsI � ApðgÞÞ�1Lp ð21Þ
where Ap ¼ E�1A0, Bp ¼ E�1B0 and Lp ¼ E�1L0 are expressed from Eq. (4).
Considering the input–output relationships in Fig. 5 yields the state-space rep-

resentation like:

_XX ¼ AðgÞX þ B1ðgÞwþ B2ðgÞu ð22Þz ¼ C1ðgÞX þ D11ðgÞwþ D12ðgÞu ð23Þ

Fig. 5. Configuration of feedback control system.


y ¼ C2ðgÞX þ D21ðgÞwþ D22ðgÞu ð24Þ
where state vector X ¼ ½xTa xTs xTp �
Twith plant state vector xp and weight function

WsðsÞ, WaðsÞ state vector xs, xa. WnðsÞ is fixed weighting function with level of Nw and

w ¼ ½wdn�T with steer disturbance wd, measurement noise n and z ¼ ½z1 z2�T respec-

tively. B2ðgÞ, C2ðgÞ, D12ðgÞ, D21ðgÞ of Eqs. (22)–(24) are parameter-independent and

each matrix is defined as:

AðgÞ ¼Aa 0 0

0 As BsCp

0 0 ApðgÞ

24

35; B1ðgÞ ¼

0 0

L 0

� �; B2ðgÞ ¼

Ba

BsDp

Bp

24

35;

C1ðgÞ ¼Ca 0 0

0 Cs DsCp

� �; D11ðgÞ ¼ ½0�; D12ðgÞ ¼

Da

DsDp

� �;

C2ðgÞ ¼ 0 0 Cp½ �; D21ðgÞ ¼ 0 Nw½ �; D22ðgÞ ¼ ½Dp�
When the time-varying vector g of q real parameters varies in a polytope U of
vertices ni ði ¼ 1; . . . ; rÞ:
g 2 U :¼ C0fn1; n2; . . . ; nrg ð25Þ
where r ¼ 2q and C0 means convex hull.

The system of Eqs. (22)–(24) also ranges in a polytope like:

AðgÞ B1ðgÞ B2ðgÞC1ðgÞ D11ðgÞ D12ðgÞC2ðgÞ D21ðgÞ D22ðgÞ

24

35 2 C0

Ai B1i B2i

C1i D11i D12i

C2i D21i D22i

24

35; i

8<: ¼ 1; . . . ; r

9=; ð26Þ

where Ai;B1i; . . . denote the values of AðgÞ;B1ðgÞ; . . . at the vertices g ¼ ni of the para-meter polytope.

Under this condition, the controller has the form of

_xxK ¼ AKðgÞxK þ BKðgÞy ð27Þu ¼ CKðgÞxK þ DKðgÞy ð28Þ


that guarantees quadratic H1 performance for the closed loop system of Fig. 5 for all

parameter trajectories.

A unified framework for the controller, of which dynamics can be updated invarying parameters, has been proposed by Apkarian and Gahinet [7] in which the

practical synthesis problem of the parameter-dependent controller is reduced to

solving a system of LMIs [11,13].

The trajectories of varying parameters g can be expressed as

gðtÞ ¼Xr

i¼1

aiðtÞni ð29Þ

where ai is a convex coordinates with 06 ai 6 1.The controller XðgÞ is defined as an interpolant of the vertex controller Xi based

on the position of g in the polytope like:

AK BK

CK DK

� �:¼

Xr

i¼1

aiXi ¼Xr

i¼1

aiAKi BKi

CKi DKi

� �ð30Þ

where AK , BK , CK , DK are the state-space matrices of the controller.

The resulting controller XðgÞ enforces stability and performance over the entireparameter polytope and for arbitrary parameter variations (proof: see Ref. [7]).

4.2.2. Selection of weighting functions and synthesis

The selection of weighting functions follows the same method as classical H1synthesis based on frozen varying parameters [14]. The magnitude and shape of the

weighting functions WaðsÞ, WsðsÞ in Fig. 5 have an influence on the characteristics of

the controller KðsÞ. It is desirable to select proper weighting functions. In order to

guarantee the robust stability relative to additive modelling error resulting fromvehicle parameter variations induced by loading conditions, the weighting function

WaðsÞ should be properly chosen to match the following condition:

rmaxfDaðjwÞg6 jWaðjwÞj; 8w ð31Þ
In this paper, WaðsÞ is selected as a second order function like Eq. (32) to match Eq.
(31) where additive modeling error DaðsÞ is calculated by considering the parameters

in Table 1 at forward speed V ¼ 50 km/h. It also has effective magnitude in the highfrequency range considering the unmodelled higher frequency dynamics.

When the bandwidth of the weighting function WsðsÞ is too wide, the control

performance declines. Thus WsðsÞ is chosen as a fourth order function in Eq. (33), to

have sufficient magnitude in the low frequency range including the roll mode fre-

quency of the vehicle. The frequency responses of the adopted weighting functions

WsðsÞ, WaðsÞ, WnðsÞ are presented in Fig. 6.

WaðsÞ ¼0:0003ðs2 þ 2sþ 1Þ

s2 þ 1:3sþ 36ð32Þ

WsðsÞ ¼ð75� 104Þðsþ 1Þ

s4 þ 38:4s3 þ 1828:5s2 þ 19;814sþ 331;776ð33Þ

1.0E-001 1.0E+000 1.0E+001 1.0E+002Frequency (Hz)

-150

-100

-50

0M

ag(d

B)

Ws(s)

Wa(s)

Wn(s)

50

Fig. 6. Weighting functions.


The time-variable parameters g1ðtÞ, g2ðtÞ are defined as V and 1=V with ranges of

10–180 km/h of V . The vertices ni ði ¼ 1; . . . ; 4Þ are also defined as:

n1 ¼ g1min g2min½ �; n2 ¼ g1max g2min½ �; n3 ¼ g1min g2max½ �;n4 ¼ g1max g2max½ �

ð34Þ

From the approach of solving LMIs, the controller XðgÞ can be adopted as Eq.

(35) with convex coordinates ai ði ¼ 1; . . . ; 4Þ of Eq. (36).

AKðgÞ BKðgÞCKðgÞ DKðgÞ

� �¼

X4

i¼1

aiðtÞAKi BKi

CKi DKi

� �ð35Þ

a1 ¼ xy; a2 ¼ ð1� xÞy; a3 ¼ xð1� yÞ; a4 ¼ ð1� xÞð1� yÞ ð36Þ
where � �
x ¼ g1max � g1ðtÞg1max � g1min

; y ¼ g2ðtÞ � g2min

g2max � g2min

; g ¼ g1ðtÞg2ðtÞ

The final structure of the roll controller with the measurement of roll angle and

forward speed can be obtained as the form of Eqs. (27) and (28).

5. Experimental Work

5.1. Configuration of experimental setup

In order to evaluate the performance of the ARC system, a series of experimental

works were performed using a Hil setup. The Hil simulation technique is an efficientway to realistically test dynamic vehicle behavior in a laboratory. A schematic dia-

gram of the experiment is shown in Fig. 7. The prototype active roll-bar system,

including the electrically actuated system and anti-roll bar fixed with a bush, is

implemented as a hardware part [16]. Not considering the real-time roll moment

Fig. 7. Schematic diagram of Hil setup.


distribution control between front and rear axle, the actuating system is considered

as a single unit. The actuator has a maximum force of 7200 N and maximum velocity

of 320 mm/s. The stiffness of anti-roll bar is 5100 Nm/rad. In order to make the

system respond faster, the stiffness must be increased. If there is too much additional

roll stiffness, it will spoil the ride comfort because of the high roll excitation which is

induced when a bump is met by wheels on one side of the car only. To consider this

problem, Darling [17] devised a hydraulic actuating system with a P-port closedproportional valve, which allows the actuator to free wheel during straight line

driving. Moreover, in the case of the hydraulically actuated system, the system is

very complex because of the large number of hydraulic elements including pump,

relief valve, accumulator, control valve, oil tank, pipes, and so on. Contamination by

leakage during operation has to be considered and maintenance and changing of

parts are complicated. In this paper, to overcome the aforementioned problem, an

electrically actuating roll control system is devised, which would effectively remove

the additional roll stiffness introduced by the active anti-roll bar and improve thehigh frequency isolation of the car body during periods when the system is inoper-

ative. In the computer, 10 DOF vehicle dynamics as well as control logic are sim-

ulated in a real time for interfacing with the ARC system. The vehicle model, based

on Ref. [5], includes bounce, yaw, pitch, roll dynamics and longitudinal, lateral

motions. The wheel-axle (unsprung mass) dynamics were described in the vertical

plane while the suspension characteristics were modelled as linear components with

stiffness kf ¼ 10,947 N/m, kr ¼ 14,559 N/m and damping cf ¼ 526 N s/m, cr ¼ 925

N s/m. Tire forces were calculated using a nonlinear tire model presented by Pacejka[12]. The vehicle model had dimensions with tf ¼ 1:51 m, tr ¼ 1:48 m, lf ¼ 1:13 m,


lr ¼ 1:44 m, roll steer af ¼ 0:2, ar ¼ �0:2 and the parameters in Table 1, where

subscript f is the front and r is the rear. The overall processing loop, with a sampling

time of 2 ms, is constructed as follows; the dynamic behavior and tire force in a

vehicle induced by a driver steering maneuver are simulated in the computer. Theactive roll moment is calculated using the feedback signal such as the car body roll

angle and forward speed. The control signal corresponding to the desired active

moment is transmitted to the motor driver through an interfacing board with a D/A

converter. The actual moment, generated by the prototype active roll-bar system, is

measured by a force transducer and delivered to the computer through an interfacing

board with an A/D converter. The actual active roll moment is exerted on the vehicle

handling dynamic model.

5.2. Lateral acceleration feedback control

Using the Hil setup illustrated in Fig. 7, lateral acceleration control with Eq. (19)

was performed in order to investigate the performance of a prototype ARC system.

In J-turn maneuver with h0 ¼ 0 deg, h1 ¼ 3:5 deg, t0 ¼ 0:0 s, t1 ¼ 0:2 s in Eq. (7), the

experimental results at constant speed 50 km/h are presented in Fig. 8. In case of

controller gain Ka ¼ 200, Kd ¼ �1000, the resulting roll angle of the car body is

reduced about 13% relative to passive vehicle conditions in the steady-state region.

When gains are Ka ¼ 250, Kd ¼ �1000, the car body may be flat in a steady-state

response. In the transient region, the response is affected by actuating system dy-namics. Fig. 9 illustrates the comparison between the experiment and simulation

results with a bandwidth of 3.3 Hz. As in the previous work [2], the use of a low

bandwidth actuating system causes some performance degradation in the transient

region. To improve the transient response, a high bandwidth actuating system can be

used but from a practical point of view, this is more costly. In this paper, a hybrid

control strategy is presented in Section 5.4 in order to enhance the response in the

transient region. This is a combined control method using a continuously variable

damper subjected to vibration control.

0 1 2 3Time (sec)

-0.06

-0.04

-0.02

0.00

0.02

Ang

le (r

ad)

Uncontrolled case

Ka = 200, Kd = -1000

Ka = 250, Kd = -1000

Fig. 8. Roll angle responses in step steer maneuver with some feedback gain.

0 1 2Time (sec)

-0.15

-0.10

-0.05

0.00

0.05

Rol

lrate

(rad

/s) Experiment

Simulation

0.10

Fig. 9. Roll angular velocity responses for limited bandwidth actuating system (Ka ¼ 200, Kd ¼ �1000).


5.3. Gain-scheduled H1 control

Fig. 10 illustrates the experimental results at various speeds in J-turn maneuver

with h0 ¼ 0 deg, t0 ¼ 0:0 s, t1 ¼ t0 þ 0:2 s in Eq. (7). At constant forward speed 50

km/h with h1 ¼ 3:5 deg, the car body can be kept flat in the steady-state region by

active control. In case of a constant forward speed 100 km/h with h1 ¼ 2 deg, the rollangle is reduced 16% relative to passive vehicle conditions. Fig. 11 illustrates the

active roll moment generated by the prototype ARC system. In order to investigate

the control performance relative to vehicle parameters and speed variations, these

factors are considered. Fig. 12 presents the control result subjected to single sine

steer maneuver with df max ¼ 2 deg under the conditions of an initial forward speed

of 100 km/h and a speed decay rate of 10 km/h/s after 0.2 s with perturbed vehicle

parameters in Table 1. This maneuver represents an abrupt driver operation to avoid

some obstacle by operating the brake pedal. In the previous work [3], gain-selected

0 1 2 3Time (sec)

-0.08

-0.06

-0.04

-0.02

0.00

Rol

lAng

.(ra

d) Controlled (V=50km/h)

Controlled (V=100km/h)

Uncontrolled (V=50km/h)

Uncontrolled (V=100km/h)

0.02

Fig. 10. Comparison of roll angle responses of J-turn maneuver at constant speed in nominal vehicle

condition.

0 1 2 3-0.08

-0.04

0.00

0.04

Rol

lAng

.(ra

d) Uncontrolled case

Controlled case

Time (sec)

0.08

Fig. 12. Comparison of roll angle responses of single sine steer maneuver at varying forward speed in

laden vehicle condition.

0 1 2 3-400

0

400

800

1200

Mom

ent(

Nm

)

1600

Time (sec)

Fig. 11. Actuating roll moment at J-turn steering maneuver.


lateral acceleration control, not applied in this paper, was introduced. The gain was

pre-fixed in a certain speed range and switched by the current speed. If the range of

the forward speed were wide, the response in the switching region had to be con-sidered because of the discontinuity of controller gain. The gain-scheduled H1controller in Section 3 is constructed by interpolating the vertex controller, thus

enabling smoother gain change. Fig. 12 shows that the resulting roll angle of the car

body can be reduced and Fig. 13 shows the comparison of roll behavior in a roll

mode phase plot.

5.4. Hybrid control

In the previous results of Figs. 8 and 10, control responses in the transient region

are affected by the bandwidth of the actuating system [18]. In order to improve the

transient responses, the effects of the variable damping system on ARC are inves-

-0.08 -0.04 0.00 0.04 0.08Angle (rad)

-0.4

-0.2

0.0

0.2

0.4

0.6

Rat

e (r

ad/s

)

Uncontrolled case

Controlled case

Fig. 13. Comparison of roll mode phase plot in laden condition at varying forward speed.


tigated. In general, the variable damper is used for vibration control induced byuneven road input, such as orifice adjusting oil-filled damper and magnetorheo-

logical damper [19].

The damping force of the variable damper is determined by a controllable dam-

ping coefficient as in the following rule:

Cd ¼fd

ð _xxs � _xxuÞ½fdð _xxs � _xxuÞ > 0�

Cmin ½fdð _xxs � _xxuÞ6 0�

8<: ð37Þ

If Cd > Cmax, Cd ¼ Cmax, where Cd is desired damping coefficient, Cmax and Cmin are

the maximum, minimum damping coefficient of the variable damper. fd is the desiredforce for suppression of motion, ð _xxs � _xxuÞ is relative velocity between the car body

and wheel at a corner.

Considering the roll motion in a steering maneuver, the desired damping force ateach damper on roll control can be calculated by the following relationships:

Md ¼X2

j¼1

Mj ð38Þ

Mi ¼ Fliti2� Fri

ti2

ðFli þ Fri ¼ 0Þ ði ¼ 1; 2Þ ð39Þ

In the above equations, Md is the total desired roll moment, Mj is the moment

component of subscript j with the front and rear axle, Fli and Fri are the desired

forces of left and right side of axle, ti is the track width with subscript i of left andright. The desired damping coefficient of each variable damper can be determinedby Eq. (37).

Fig. 14(a) and (b) illustrate the roll angle responses in J-turn maneuver at a

constant speed 50 km/h with the nominal vehicle parameters in Table 1. The results

of simulation using the variable dampers to roll control of Fig. 14(a), clearly show

that the roll angle response in the transient region can be smooth although there is

Fig. 15. Schematic diagram of hybrid control scheme.

0 1 2 3Time (sec)

-0.06

-0.04

-0.02

0.00

0.02

Rol

lAng

.(ra

d)

Uncontrolled case

Semi-active only

(a)0 1 2 3

Time (sec)

-0.06

-0.04

-0.02

0.00

0.02

Rol

lAng

.(ra

d) ARC only

Hybrid control

Uncontrolled case

(b)

Fig. 14. Comparison of roll angle responses in J-turn maneuver at 50 km/h and nominal vehicle condition.

(a) Control using variable damper, (b) hybrid control.


no improvement in the steady-state region. This resulted from the operating char-

acteristics of a semi-active actuator which cannot generate active force.

From the above investigation, each of variable dampers can be controlled with an

anti-roll actuating system simultaneously based on Eqs. (37)–(39). This hybrid

control strategy is presented in Fig. 15. Fig. 14(b) illustrates the resulting roll re-

sponse using the hybrid control with respect to passive and ARC conditions. In the

figure, a hybrid control with a variable damper can give a smoother response in the

transient region and improve steady-state response with respect to a passive vehicle.

6. Conclusions

In this work, an ARC system represented by an electric actuating system andvariable damper was proposed for robust roll motion control of a vehicle. In order to


have an high compliance to external excitation during inoperative periods and a

compact configuration, an electric actuating roll control system was devised, which

could improve the undesirable dynamic response induced by additional roll stiffness.Following the investigation of parameter sensitivity analysis based on a linear vehicle

model, a gain-scheduled H1 controller has been formulated by considering the time-

varying forward speed and parameter uncertainties with respect to loading condi-

tions. By experimental work using a hardware-in-the-loop setup and full vehicle

dynamics with nonlinear tire characteristics, which was a useful method for inves-

tigating the characteristics of a prototype hardware system, the control performance

of lateral acceleration control and gain-scheduled H1 control have been investigated.

Moreover, a hybrid control system with a variable damper has been proposed and ithas been shown that the proposed control strategy is very effective for the im-

provement of both steady-state and transient responses in spite of using a limited

bandwidth actuating system.

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