Yaw Stability Improvement for Four-Wheel Active Steering Vehicle

using Sliding Mode Control

Norhazimi Hamzah Faculty of Electrical Engineering

Universiti Teknologi MARA Pulau Pinang, Malaysia

norhazimi880@ppinang.uitm.edu.my

Yahaya Md Sam, Hazlina Selamat Faculty of Electrical Engineering

Universiti Teknologi Malaysia Johor, Malaysia

yahaya@fke.utm.my, hazlina@fke.utm.my

M Khairi Aripin Control, Instrumentation & Automation Department.

Faculty of Electrical Engineering, UTeM Melaka, Malaysia

khairiaripin@utem.edu.my

Muhamad Fahezal Ismail Industrial Automation Section

UniklMfi Selangor, Malaysia

fahezal@mfi.unikl.edu.my

Abstract Active steering control is one of the approach that can be used to improve the vehicles lateral and yaw stability. By combining active front steering and active rear steering control, the vehicles handling and stability can be improved via four wheel active steering (4WAS) control. In this paper, a robust control algorithm of sliding mode control is designed for 4WAS vehicle. Single track 2 d.o.f linear model is utilized for controller design and simulation purpose. Simulation for 4WAS and front steering (AFS) is carried out in Simulink for step steer and double lane change maneuver to verify the effectiveness of the proposed control system. The result shows that the 4WAS perform better than the AFS in tracking the desired response trajectory.

Keywords- Active Steering Control, Four Wheel Active Steering, Yaw Stability Control, Sliding Mode Control

I. INTRODUCTION Vehicle stability control (VSC) is one of the important

topics in vehicle dynamics where ongoing research is actively conducted. The main objective of vehicle stability control is to maintain the vehicle keep on the road or desired track/path. Lateral force that exists in vehicle dynamic motion has great influence to the vehicle stability. According to [1], yaw stability control system that purely based on kinematic and dynamics motion of vehicle is one of vehicle lateral control system that have been developed by researchers. As reported [2], there are three main control objectives of vehicle yaw stability control system which are yaw rate control, sideslip control and combination of both yaw rate and sideslip. The purpose of yaw stability control system is to keep the vehicles yaw rate as closer as possible to the nominal motion expected by the driver. Conventionally, the lateral force of vehicle is being controlled by steering system that directly commanded by the driver.

Nowadays, vehicle dynamics studies is focusing on active steering control to improve the yaw stability control system. In general, there are three techniques for active steering control have been developed that are active front steering (AFS), rear wheel active steering (ARS) and four wheel active steering (4WAS). In AFS, front wheel steer angle is a sum of steer angle commanded by the driver and a corrective steer angle that generated by the designed controller. AFS is used to improve the handling and stability performance. The performance is good when vehicle is driving or handling at steady state condition or tires forces is in linear region but it is less effective when vehicle dynamics become nonlinear or tires forces approaches their adhesion limit. The ARS on the other hand is used to improve the vehicle transient response for low speed cornering manoeuvres. In order to enhance the manoeuvrability at low speed and the handling stability at high speed, combination of active front steering and active rear steering or so called four wheel active steering (4WAS) has been proposed in [3], [4], and [5]. By implementing 4WAS control, the lateral and yaw motion can be controlled simultaneously using two independent control inputs via active front steering and active rear steering where it is impossible to control lateral and yaw motion using active front or active rear steering only.

In previous research works, 4WAS control has been designed based on various control algorithms. In [3], a decentralized nonlinear P and PI active steering control is designed for active front and active rear steering control but robustness issue is not discusses intensively. Optimal control theory is implemented in [4] for optimal model following control design that consists of feedforward and feedback control but simulation and comparison analysis with other controller is not carried out. In [5], sliding mode control is designed for uncertainties parameters in 4WAS vehicle.

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However, comparison between controlled 4WAS and uncontrolled AFS is insufficient.

In this paper, a prominent sliding mode control algorithm is utilized for both 4WAS and AFS vehicle due to the robustness properties against parameters uncertainties and disturbances. This paper is organized in five sections as overview briefs in Section I, vehicle dynamics modelling for controller design and simulation in Section II, robust control design of sliding mode control in Section III. The simulations and analysis is carried out in Section V and this research paper concludes in Section V.

II. VEHICLE DYNAMICS MODEL

A. 4WAS Single Track Model

Single track model or also called bicycle model in [4] as shown in Figure 1 is utilized to describe the dynamic model of 4WAS vehicle. This model used to design the controller for yaw rate and lateral motion.

Figure 1. Single track model 2 d.o.f

In simplest form of planar motion, 2 d.o.f linear model for yaw and lateral dynamic motion (roll motion is neglected) are describe in the following equations:

Lateral motion;

yryfxx FFrmVmV +=+ (1) Yaw rate;

yryfz bFaFrI = (2) where m is vehicle mass, xV is vehicle forward speed, is vehicle body sideslip, r is yaw rate, a/b is distance from front/rear axle to center of gravitiy (CG), f and r are front and rear steering angle, zI is yaw moment of inertia and

yryf FF , are lateral force of front and rear tires. With assumptions linear relationship between tire lateral force and tire slip angle;

ffyf CF = (3)

rryr CF = (4)

where fC and rC are front and rear cornering stiffness respectively. Slip angles of front and rear tires, f and r can be obtained by using the following equations;

xff V

ar= (5)

xrr V

br= (6)

By substitutes and rearrange equations (3), (4), (5) & (6) into (1) & (2), state space model can be obtained as follows;

BuAxx += (7)

+

=

r

f

bbbb

raaaa

r

2221

1211

2221

1211

(8)

where

212111,

x

rf

x

rf

mV

bCaCa

mVCC

a

=

+= (9)

xz

rf

z

rf

VICbCa

aI

bCaCa

22

2221 ,+

=

= (10)

x

r

x

f

mVC

bmVC

b == 1211 , (11)

z

r

z

f

IbC

bI

aCb == 1221 , (12)

where

In equation (5), noted that front wheel steering angle, f is sum of additional/corrective steer angle by controller, c and steering wheel angle demanded/commanded by the driver, d ;

cdf += (13)

III. CONTROLLER DESIGN FOR 4WAS VEHICLE The purpose of the control system is to maintain the vehicle

actual response close to the desired response determined by the reference model. To ensure robustness of the system against parameter uncertainties and external disturbance [6], sliding mode controller is chosen as the steering controller.

A. Vehicle Reference Model The desired vehicle handling performance is expressed as

a reference model which has zero vehicle sideslip angle at the center of gravity and the desired yaw rate is computed based on the steering input and the vehicle speed.

r

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The reference model [5] is represented by the following: ddddd uBxAx += (14)

*

//

/100/1

frrd

d

rd

d

kk

rr

+

=

(15)

where

)1(,0 2

x

xrdd KVL

Vkk+

== and

=

rf Ca

Cb

LmK2

(16)

B. Sliding Mode Controller Design The first step in designing the sliding mode controller is to

choose a suitable sliding surface. Then, the control input is design to drive the system trajectories to the sliding surface so that the closed loop dynamics are completely governed by the equation that define the surface.

The difference between the actual response and the desired response define the tracking error.

==

d

dd rr

xxe

(17)

Then, the sliding surface, 0= is defined as follows,

Ce= (18) where C is a full rank constant matrix. The matrix C is chosen such that CB is nonsingular. Taking derivative on equ. 18,

[ ] [ ][ ][ ]dddd

dddddd

ddddd

uBBuxAAAeCuBBuxAAxAxAxC

uBxABuAxCxxCeC

++=

++=

++===

)(

)( (19)

Based on constant reaching law method [7],

)sgn( Q= (20) Thus,

[ ])sgn()(

)()(1

1

QCB

uBxAAAeCCBu dddd

+= (21)

The reaching law approach will establish the reaching condition which is the condition where the state will move toward and reach the sliding surface as well as specifies the dynamic characteristic of the system during the reaching phase.

C. Stability Analysis The stability of the closed loop system is determined using

Lyapunov stability theory. Let the positive definite function ;

TV 5.0= (22)

be a Lyapunov function candidate. The time derivative of V along the system trajectories is calculated as

[ ]ddddTT uBBuxAAAeCV ++== )( (23) Replacing equation (21) into (23)

[ ]

QQ

uBQCB

uBxAAAeCCBB

xAAAe

CV

Tdd

dddd

dd

T

+

++

=

)sgn(

)sgn()(

)()(

)(

1

1

(24)

where )sgn( T=

For 0>Q , then 0

0 1 2 3 4 5 6-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

t(sec)

veh

icle

sides

lip a

ngle

(deg)

desired4WASAFS

1 1.2 1.4-0.8-0.6-0.4-0.2

00.20.4

Figure 2. Vehicle sideslip angle response with J-turn maneuver

0 1 2 3 4 5 6-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t(sec)

yaw

ra

te(ra

d/s)

desired4WASAFS

1.2 1.4 1.60.25

0.3

0.35

Figure 3. Yaw rate response with J-turn maneuver

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0 1 2 3 4 5 6-5

-4

-3

-2

-1

0

1

2

3

4

5

t(sec)

veh

icle

sid

eslip

ang

le(de

g)

desired4WASAFS

Figure 4. Vehicle sideslip angle response with lane change maneuver

0 1 2 3 4 5 6-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

t(sec)

yaw ra

te(ra

d/s)

desired4WASAFS

Figure 5. Yaw rate response with lane change maneuver

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V. CONCLUSION A sliding mode controller for 4WAS vehicle is presented in

this paper. The application of SMC enable the lateral and yaw motion of 4WAS to both simultaneously trace the response of the reference model exactly, which is better than AFS.

ACKNOWLEDGMENT The authors would like to thank UTM (Vot number

Q.J130000.2623.04J48), UiTM, UTeM and MoHE for supporting the present work.

REFERENCE [1] R. Rajamani, "Lateral Vehicle DynamicsVehicle Dynamics and

Control," Springer US, 2006, pp. 15-49.

[2] W. J. Manning and D. A. Crolla, "A review of yaw rate and sideslip controllers for passenger vehicles," Transactions of the Institute of Measurement and Control, vol. 29, pp. 117-135, 2007.

[3] R. Marino, S. Scalzi, and F. Cinili, "Nonlinear PI front and rear steering control in four wheel steering vehicles," Vehicle System Dynamics, vol. 45, pp. 1149-1168, 2007.

[4] B. Li and F. Yu, "Optimal model following control of four-wheel active steering vehicle," in 2009 IEEE International Conference on Information and Automation, ICIA 2009, 2009, pp. 881-886.

[5] F. Du, J. S. Li, L. Li, and D. H. Si, "Robust control study for four-wheel active steering vehicle," in Proceedings - International Conference on Electrical and Control Engineering, ICECE 2010, 2010, pp. 1830-1833.

[6] N. Hamzah, Sam, Y.M. and Shuib, N.M., "Longitudinal Tire Slip Control Utilizing Sliding Mode Control," in The Second International Conference on Control, Instrumentation and Mechatronic Engineering (CIM09), Malacca, Malaysia, 2009, pp. 62-65.

[7] J. Y. Hung, W. Gao, and J. C. Hung, "Variable structure control. A survey," IEEE Transactions on Industrial Electronics, vol. 40, pp. 2-22, 1993.

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