Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 643548 15 pageshttpdxdoiorg1011552013643548

Research ArticleStudy on Integrated Control of Vehicle Yaw andRollover Stability Using Nonlinear Prediction Model

Jianyong Cao12 Lixin Jing3 Konghui Guo1 and Fan Yu1

1 School of Mechanical Engineering Shanghai Jiao Tong University Shanghai 200240 China2 Shanghai Motor Vehicle Inspection Center Shanghai 201805 China3 China Automotive Technology amp Research Center Tianjin 300300 China

Correspondence should be addressed to Fan Yu fanyusjtueducn

Received 4 February 2013 Revised 24 April 2013 Accepted 10 June 2013

Academic Editor Pedro Ribeiro

Copyright copy 2013 Jianyong Cao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes the integrated controller of the yaw and rollover stability controls based on the prediction model A nonlinear3-DoF vehiclemodel with a piecewise linearization tiremodel is built up as the rollover predictivemodel and its accuracy is verifiedby vehicle tests A yaw stability controller and a rollover stability controller are proposed respectively Then coordinated controlstrategy is investigated for the integration of vehicle yaw and roll stability controlsThe additional yaw torque and braking torque ofeach wheel are calculatedThe unified command of valves is sent combined with ABS control algorithm Virtual tests in CarSim arecarried out including slalom condition and double-lane change condition Results indicate that the coordinated control algorithmimproves vehicle yaw and roll stability effectively

1 Introduction

Yaw and roll motions are two key lateral dynamic charac-teristics of the ground vehicles The safety of driving vehicleis mainly dependent on yaw and rollover stability especiallyin emergency conditions Yaw stability is the ability to keepvehicle turning and avoid the motion of skidding and rollstability is to avoid vehicle turnover under certain lateralacceleration InUSA the percentage of rollover occurrence inall crashes was about 27 in 2007 However the percentageof rollover occurrence in fatal crashes was about 215 andwas significantly higher than that in injury and property-damage-only crashes Untripped rollovers account for lessthan 5 of rolled passenger vehicles in single-vehicle crasheswhich often occur during high-speed collision avoidancemanoeuvres [1] Furthermore rollovers account for nearly33 of all deaths from sport utility vehicle (SUV) whichbeing popular in recent years is considered to be easier toencounter rollover accident than other kinds of passengervehicles for its higher gravity center and a lower rolloverthreshold [2ndash4] Hence it is necessary to design a rolloverprevention controller

Approaches proposed by researchers to preventuntripped rollover can be classified into three types Usingroller warning system the first type controls the vehicle rollmotions by the driver using roller warning systems indirectlyRollover warning is one common method to prevent fromrollover [5] Rollover warning system predicts whether thevehicle has a rollover risk in the next time period based onthe current driving state of the vehicle Preston-Thomas andWoodrooffe developed one stability control and warningsystem on heavy vehicles in 1990 They used the lateral loadtransfer ratio (LTR) to indicate rollover threat [6] Ervin etal proposed a rollover stability advisor (RSA) system in 1998This system determined the rollover acceleration thresholdbased on real-time measurements of the status of the vehicle[7] Chen proposed a time-to-rollover (TTR) metric whichpredicted the time to an impending rollover incident Asuper real-time vehicle model is used in the metric andfuture vehicle states can be predicted in real-time [8] Thesecond type controls vehicle roll motions using active antirollbar [9] continuous damping control (CDC) [10] or activesuspension control [11] to enhance the rollover stabilitythrough influencing the lateral load transfer and longitudinal

2 Mathematical Problems in Engineering

speed control [12 13] It can prevent rollover by increasingthe rollover thresholds The third type utilizes integratedchassis control (ICC) which consists of active front steeringyaw stability control (YSC) and activesemiactive suspensionactive antiroll bar (AARB) and so on Some studies haveshown that integration of individual modular chassis controlsystems is amore efficientmethod to improve vehicle stabilityand handling [14ndash16] Recently Chou and DrsquoAndrea-Novelproposed a global vehicle control system using differentialbraking torques and active suspension forces [17] Yoon etal designed an integrated system by using YSC and activeantiroll bar [12] Li et al developed a coordinated systemby using direct yaw moment active steering and activestabilizer [18] By integrating electronic stability controlactive front steering and continuous damping control Yoonet al proposed unified chassis control strategy to preventrollover and improve lateral stability [19]

Among these three types the third one would have thebetter performance than the other two However multipleactuators are needed and complexity would be increasedAmong those ICC the YSC contributes to maintain vehicleyaw stability by reducing the lateral acceleration through con-trolling the longitudinal speed [20ndash22] and the differentialbraking is only used to reduce the lateral acceleration It isunable to obtain the roll motions information of the vehicleas rollover warning systemThe roll motions may deterioratethe maneuverability of the vehicle and vice versa So in thispaper a novel coordinated control system is proposed whichcontrols the vehicle yaw and rollover stability only using thedifferential braking based on the devices of YSC In detail anonlinear prediction model-based vehicle TTR estimator isdesigned to predict an impending rollover incident for theYSC system Then the coordinate system of RSC and YSC isproposed However the difficult task for the coordinate sys-tem is to obtain TTR properly To overcome this problem a 3-DoF vehicle model considering tire nonlinear characteristicsis proposed to calculate the load transfer ratio online whichcan estimate the rollover characters of the vehicle At last thecontroller of the coordinate system is designed by using PIand P control method

Taking an SUV as the research object whose parameterswere given inTable 1 the coordinated control of RSC andYSCis studied on the basis of in-depth study of the yaw and rollstability controls The presentation of this paper is organizedas follows Section 2 provides the derivation detail of a non-linear 3-DoF vehicle model with a piecewise linearization tiremodel whose parameters can be identified and verified InSection 3 the proposed controller is described In Section 4simulations are performed on CarSim and the results arediscussed Section 5 gives the conclusion of this paper

2 Vehicle Dynamics Model

This section describes a three-degree-of-freedom roll predic-tion model which is shown in Figure 1 taking ISO vehiclecoordinate system A roll degree-of-freedom is consideredon the basis of two-degree-of-freedom vehicle nonlinearprediction model including a lateral movement along the

Table 1 Parameter list of vehicle model

Element ValueVehicle masskg 1910Sprung mass of vehiclekg 1664Distance between the center of sprung mass and thevehicle roll axism 0316

CG heightm 055CG to rear axism 1165Inertia around the z-axisKglowastm2 30892Inertia around the z-axis and x-axisKglowastm2 332Inertia around the z-axis of sprung massKglowastm2 25497Inertia around the z-axis and x-axis of sprungmasskglowastm2 321

CG to front axism 1411

119910-axis a yaw motion around the 119911-axis and a roll motionaround the 119909-axis [23] Some other assumptions are madein order to simplify the motion equations The principalhypotheses are as follows

(i) Small perturbations from straight running at constantforward speed on the flat and level road are neglected

(ii) The frontwheel steering angle is taken asmodel input

(iii) The vertical motion and pitching motion areneglected when the vehicle is on the flat and levelroad

(iv) Aerodynamic effects are neglected

(v) Change of tire characteristic and the aligning torqueare neglected

(vi) Axle lateral stiffness considers steering system stiff-ness suspension stiffness and tire lateral elasticity

(vii) The location of the center of gravity is fixed

Based on the above assumptions the balance equations ofthe force along the 119910-axis and torques around the 119911-axis and119909-axis can be expressed as

sum119865119910 = 119898 sdot (119903 sdot 119881119909 +

119881119910) minus 119898119904 sdot 119890 sdot

sum119872119911 = 119868119911 sdot119903 + 119868119909119911 sdot

sum119872119909 = 119868119909119904 sdot minus 119898119904 sdot 119890 sdot (119903 sdot 119881119909 +

119881119910) + 119868119909119911119904 sdot119903

(1)

External forces can be expressed as

sum119865119910 = 119865119910119891 + 119865119910119903 = 119896119891 sdot 120572119891 + 119896119903 sdot 120572119903

sum119872119911 = 119886 sdot 119865119910119891 + 119887 sdot 119865119910119903 = 119886 sdot 119896119891 sdot 120572119891 minus 119887 sdot 119896119903 sdot 120572119903

sum119872119909 = 119898119904 sdot 119892 sdot 119890 sdot 120593 minus 119870120593 sdot 120593 minus 119862120593 sdot

(2)

As a roll prediction model vehicle model with threedegrees-of-freedom should represent the real vehicle When

Mathematical Problems in Engineering 3

120572r120572r

120575f120575f

120575w120575w

u

L

a

Y

120573r

b

Vy

VxX

t

Fyr

2

Fyr

2

Fyf

2

Fyf

2 z

K120593e

o

C120593

tFzi Fzo

hR

msay

120593

y

msg

Figure 1 3-DoF vehicle nonlinear prediction model schematic diagram

120583

120572f1205721205831 1205721205832

kf middot 1205721205831

Fyf

120583 middot Fz

Figure 2 Diagram of simplified front wheel sideslip characteristics

rollover occurs the lateral acceleration of vehicle will gener-ally reach a larger value and then the lateral forces of tires arein a nonlinear area In this case the limit of tirersquos lateral forcedue to road adhesion limit should be considered In order toprovide vehicle yaw stability and avoid oversteer caused bylateral force saturation of rearwheel the saturation restrictionis used for front wheel only The cornering property offront wheel is nonlinear and that of the rear wheel is linear

0 2 4 6 8 10

0

1

2

3

Time (s)

Test dataSimulation

Late

ral a

ccel

erat

ion

(ms2)

Figure 3 Steering wheel angle step input test

0 10 20 30 40

0

1

2

Time (s)

minus1

minus2Late

ral a

ccel

erat

ion

(ms2)

Test dataSimulation

Figure 4 Slalom test (30 kmh)

4 Mathematical Problems in Engineering

RSC braking torque

Vehicle status calculation

HCU

TTR

Valve command calculation

ABS

Rollover predictive model

YSC yaw torque

Unified valve command

Valve

Reference speed

Steering angle

Wheel speed

Lateral acceleration

Master cylinder pressure

Yaw rate

Reference speed

Road adhesion

coefficient

Braking torque and yaw torque distribution

CCYR braking torque and yaw torque

rlowast rlowast minus r2-DoF vehicle model

120573

command 2Valve

command 1

Wheel cylinder Pw

Figure 5 Overall structure of coordinated control algorithm of vehicle yaw and rollover stability

respectively The cornering characteristic of front tire ispiecewise and the lateral force is obtained as follows119865119910119891

=

119896119891 sdot 119886119891 119886119891 le 1198861205831

119896119891 sdot 1198861205831 +

120583119865119911119891 minus 119896119891 sdot 1198861205831

1198861205832 minus 1198861205831

(119886119891 minus 1198861205831) 1198861205831 le 119886119891 le 1198861205832

120583 sdot 119865119911119891 119886119891 ge 119886119911119891

(3)where 1198861205831 and 1198861205832 are sideslip angles respectively whichcorrespond to the turning point of cornering characteristicswhen the road friction coefficient is 120583 sdot 119886119891 is front wheelsideslip angle 119865119911119891 is front wheel vertical load and thediagram is shown in Figure 2

A 3-DOF vehicle model is derived from (1)ndash(3) and theparameters (119896119891 119896119903 119870120593 119862120593) of the model can be identified

through the inputoutput data of the vehicle test data tableA typical signal for example step test is chosen as driversteering wheel input to obtain the vehicle lateral accelerationas the output Meanwhile the same signal is also as the inputof 3-DOF vehicle model in simulation Using the errors of 119886119910and 119886119910 an objective function in (4) is employed to optimizethe vehicle parameters

119869 = int

1199050

0

(119886119910 minus 119886119910)2119889119905 (4)

The 3-DoF vehicle dynamics model is verified by vehiclehandling and stability experiment data The main contentsof the experimental verification include vehicle steady-stateroll characteristics and steady-state cornering performancevehicle transient steering characteristics and steer on-center characteristics and returnability The constant-speedvariable-steer tests are carried out on a uniform dry level

Mathematical Problems in Engineering 5

120575f

120572f

Fyf

lf

Vy

Vx120574

Vy + lf middot 120574

120573

Vy minus lr middot 120574

Fyr

120572r

lr

Figure 6 Bicycle model

and hard road surface A step input of steering wheel angleis applied when the vehicle runs at a constant-speed and theslalom test was carried out at a speed of 30 kmh Vehicle dataare recorded at sampling rate of 200HzThe results show thatthe responses of the vehicle model coincide with the actualvehicle measurements (see Figures 3 and 4)

3 Coordinated Control Algorithm ofVehicle Yaw and Rollover Stability

The objectives of YSC and RSC are to follow the driverintention and limit the maximum of the vehicle lateralacceleration respectively In order to maximize the vehicleyaw and roll stability YSC and RSC will be integrated inthis section Figure 5 indicates the cascade structure of thecontrol algorithm which consists of three levels First a3-DoF rollover prediction model and a 2-DoF single-trackmodel are implemented to obtain the roll track error and theyaw track error respectively Then an RSC braking torquecontroller and a YSC yaw moment controller are establishedto generate a desired torque control input based on the errorsobtained previously and the coordinated control of yaw androllover (CCYR) stability controllers is applied to coordinatethe demand of these two controllers based on the TTR Inthe underlying structure the braking torque and the yawmoment are distributed to each wheel and achieved by the

hydraulic control unit (HCU) valve controller and the ABScontroller is also used to prevent wheel lock In additionto the cascaded control logic a vehicle status observer isimplemented to obtain the status used in the controller whichincludes reference longitudinal speed and lateral speed roadadhesion coefficient vehicle sideslip angle wheel rotationspeed and wheel cylinder pressure

31 Design of YSC YSC is designed to enhance vehiclemaneuverability by tracking a reference yaw rate generated bya driverrsquos steering input The yaw moment control is adoptedto generate a desired yaw moment in order to reduce the yawrate error between the reference and actual yaw rate A linearbicycle model is used to compute the reference yaw rate andFigure 1 shows the 3-DoF model including the direct yawmoment

In order to obtain the reference yaw rate and vehiclesideslip angle a single-track model is proposed As is shownin Figure 6 yaw and lateral motion in the bicycle model canbe obtained as follows

119898119881119909 (

120573 + 120574) = 119865119910119891 + 119865119910119903

119868119911120574 = 119865119910119891119897119891 minus 119865119910119903119897119903

(5)

The wheel lateral force is considered to be linear with the tireslip angle

119865119910119891 = 119896119891120572119891

119865119910119903 = 119896119903120572119903

120572119891 = 120575119891 minus

119881119910 + 119897119891120574

119881119909

120572119903 =

119881119910 minus 119897119903120574

119881119909

(6)

Therefore the transfer function of yaw rate and sideslip anglecan be obtained

120574

120575

(119904) = 119866119903

1205911119904 + 1

11987911199042+ 1198792119904 + 1

120573

120575

(119904) = 119866120573

1205911015840

1119904 + 1

11987911199042+ 1198792119904 + 1

(7)

where

119866119903 =

119881119909 tan 120575119891119897119894119871 [1 + (119881119909119881ch)

2]

119866120573 =

119897119903 minus 1198981198971198911198812

119909119896119903119897

119897 [1 + (119881119909119881ch)2]

1205911 =

119898119881119909119897119891

119897119896120574

1205911015840

1=

119868119911

119887119896120574119881119909 minus 119898119881119909119897119891

1198791 =

1198981198812

119909119868119911

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

speed control [12 13] It can prevent rollover by increasingthe rollover thresholds The third type utilizes integratedchassis control (ICC) which consists of active front steeringyaw stability control (YSC) and activesemiactive suspensionactive antiroll bar (AARB) and so on Some studies haveshown that integration of individual modular chassis controlsystems is amore efficientmethod to improve vehicle stabilityand handling [14ndash16] Recently Chou and DrsquoAndrea-Novelproposed a global vehicle control system using differentialbraking torques and active suspension forces [17] Yoon etal designed an integrated system by using YSC and activeantiroll bar [12] Li et al developed a coordinated systemby using direct yaw moment active steering and activestabilizer [18] By integrating electronic stability controlactive front steering and continuous damping control Yoonet al proposed unified chassis control strategy to preventrollover and improve lateral stability [19]

Among these three types the third one would have thebetter performance than the other two However multipleactuators are needed and complexity would be increasedAmong those ICC the YSC contributes to maintain vehicleyaw stability by reducing the lateral acceleration through con-trolling the longitudinal speed [20ndash22] and the differentialbraking is only used to reduce the lateral acceleration It isunable to obtain the roll motions information of the vehicleas rollover warning systemThe roll motions may deterioratethe maneuverability of the vehicle and vice versa So in thispaper a novel coordinated control system is proposed whichcontrols the vehicle yaw and rollover stability only using thedifferential braking based on the devices of YSC In detail anonlinear prediction model-based vehicle TTR estimator isdesigned to predict an impending rollover incident for theYSC system Then the coordinate system of RSC and YSC isproposed However the difficult task for the coordinate sys-tem is to obtain TTR properly To overcome this problem a 3-DoF vehicle model considering tire nonlinear characteristicsis proposed to calculate the load transfer ratio online whichcan estimate the rollover characters of the vehicle At last thecontroller of the coordinate system is designed by using PIand P control method

Taking an SUV as the research object whose parameterswere given inTable 1 the coordinated control of RSC andYSCis studied on the basis of in-depth study of the yaw and rollstability controls The presentation of this paper is organizedas follows Section 2 provides the derivation detail of a non-linear 3-DoF vehicle model with a piecewise linearization tiremodel whose parameters can be identified and verified InSection 3 the proposed controller is described In Section 4simulations are performed on CarSim and the results arediscussed Section 5 gives the conclusion of this paper

2 Vehicle Dynamics Model

This section describes a three-degree-of-freedom roll predic-tion model which is shown in Figure 1 taking ISO vehiclecoordinate system A roll degree-of-freedom is consideredon the basis of two-degree-of-freedom vehicle nonlinearprediction model including a lateral movement along the

Table 1 Parameter list of vehicle model

Element ValueVehicle masskg 1910Sprung mass of vehiclekg 1664Distance between the center of sprung mass and thevehicle roll axism 0316

CG heightm 055CG to rear axism 1165Inertia around the z-axisKglowastm2 30892Inertia around the z-axis and x-axisKglowastm2 332Inertia around the z-axis of sprung massKglowastm2 25497Inertia around the z-axis and x-axis of sprungmasskglowastm2 321

CG to front axism 1411

119910-axis a yaw motion around the 119911-axis and a roll motionaround the 119909-axis [23] Some other assumptions are madein order to simplify the motion equations The principalhypotheses are as follows

(i) Small perturbations from straight running at constantforward speed on the flat and level road are neglected

(ii) The frontwheel steering angle is taken asmodel input

(iii) The vertical motion and pitching motion areneglected when the vehicle is on the flat and levelroad

(iv) Aerodynamic effects are neglected

(v) Change of tire characteristic and the aligning torqueare neglected

(vi) Axle lateral stiffness considers steering system stiff-ness suspension stiffness and tire lateral elasticity

(vii) The location of the center of gravity is fixed

Based on the above assumptions the balance equations ofthe force along the 119910-axis and torques around the 119911-axis and119909-axis can be expressed as

sum119865119910 = 119898 sdot (119903 sdot 119881119909 +

119881119910) minus 119898119904 sdot 119890 sdot

sum119872119911 = 119868119911 sdot119903 + 119868119909119911 sdot

sum119872119909 = 119868119909119904 sdot minus 119898119904 sdot 119890 sdot (119903 sdot 119881119909 +

119881119910) + 119868119909119911119904 sdot119903

(1)

External forces can be expressed as

sum119865119910 = 119865119910119891 + 119865119910119903 = 119896119891 sdot 120572119891 + 119896119903 sdot 120572119903

sum119872119911 = 119886 sdot 119865119910119891 + 119887 sdot 119865119910119903 = 119886 sdot 119896119891 sdot 120572119891 minus 119887 sdot 119896119903 sdot 120572119903

sum119872119909 = 119898119904 sdot 119892 sdot 119890 sdot 120593 minus 119870120593 sdot 120593 minus 119862120593 sdot

(2)

As a roll prediction model vehicle model with threedegrees-of-freedom should represent the real vehicle When

Mathematical Problems in Engineering 3

120572r120572r

120575f120575f

120575w120575w

u

L

a

Y

120573r

b

Vy

VxX

t

Fyr

2

Fyr

2

Fyf

2

Fyf

2 z

K120593e

o

C120593

tFzi Fzo

hR

msay

120593

y

msg

Figure 1 3-DoF vehicle nonlinear prediction model schematic diagram

120583

120572f1205721205831 1205721205832

kf middot 1205721205831

Fyf

120583 middot Fz

Figure 2 Diagram of simplified front wheel sideslip characteristics

rollover occurs the lateral acceleration of vehicle will gener-ally reach a larger value and then the lateral forces of tires arein a nonlinear area In this case the limit of tirersquos lateral forcedue to road adhesion limit should be considered In order toprovide vehicle yaw stability and avoid oversteer caused bylateral force saturation of rearwheel the saturation restrictionis used for front wheel only The cornering property offront wheel is nonlinear and that of the rear wheel is linear

0 2 4 6 8 10

0

1

2

3

Time (s)

Test dataSimulation

Late

ral a

ccel

erat

ion

(ms2)

Figure 3 Steering wheel angle step input test

0 10 20 30 40

0

1

2

Time (s)

minus1

minus2Late

ral a

ccel

erat

ion

(ms2)

Test dataSimulation

Figure 4 Slalom test (30 kmh)

4 Mathematical Problems in Engineering

RSC braking torque

Vehicle status calculation

HCU

TTR

Valve command calculation

ABS

Rollover predictive model

YSC yaw torque

Unified valve command

Valve

Reference speed

Steering angle

Wheel speed

Lateral acceleration

Master cylinder pressure

Yaw rate

Reference speed

Road adhesion

coefficient

Braking torque and yaw torque distribution

CCYR braking torque and yaw torque

rlowast rlowast minus r2-DoF vehicle model

120573

command 2Valve

command 1

Wheel cylinder Pw

Figure 5 Overall structure of coordinated control algorithm of vehicle yaw and rollover stability

respectively The cornering characteristic of front tire ispiecewise and the lateral force is obtained as follows119865119910119891

=

119896119891 sdot 119886119891 119886119891 le 1198861205831

119896119891 sdot 1198861205831 +

120583119865119911119891 minus 119896119891 sdot 1198861205831

1198861205832 minus 1198861205831

(119886119891 minus 1198861205831) 1198861205831 le 119886119891 le 1198861205832

120583 sdot 119865119911119891 119886119891 ge 119886119911119891

(3)where 1198861205831 and 1198861205832 are sideslip angles respectively whichcorrespond to the turning point of cornering characteristicswhen the road friction coefficient is 120583 sdot 119886119891 is front wheelsideslip angle 119865119911119891 is front wheel vertical load and thediagram is shown in Figure 2

A 3-DOF vehicle model is derived from (1)ndash(3) and theparameters (119896119891 119896119903 119870120593 119862120593) of the model can be identified

through the inputoutput data of the vehicle test data tableA typical signal for example step test is chosen as driversteering wheel input to obtain the vehicle lateral accelerationas the output Meanwhile the same signal is also as the inputof 3-DOF vehicle model in simulation Using the errors of 119886119910and 119886119910 an objective function in (4) is employed to optimizethe vehicle parameters

119869 = int

1199050

0

(119886119910 minus 119886119910)2119889119905 (4)

The 3-DoF vehicle dynamics model is verified by vehiclehandling and stability experiment data The main contentsof the experimental verification include vehicle steady-stateroll characteristics and steady-state cornering performancevehicle transient steering characteristics and steer on-center characteristics and returnability The constant-speedvariable-steer tests are carried out on a uniform dry level

Mathematical Problems in Engineering 5

120575f

120572f

Fyf

lf

Vy

Vx120574

Vy + lf middot 120574

120573

Vy minus lr middot 120574

Fyr

120572r

lr

Figure 6 Bicycle model

and hard road surface A step input of steering wheel angleis applied when the vehicle runs at a constant-speed and theslalom test was carried out at a speed of 30 kmh Vehicle dataare recorded at sampling rate of 200HzThe results show thatthe responses of the vehicle model coincide with the actualvehicle measurements (see Figures 3 and 4)

3 Coordinated Control Algorithm ofVehicle Yaw and Rollover Stability

The objectives of YSC and RSC are to follow the driverintention and limit the maximum of the vehicle lateralacceleration respectively In order to maximize the vehicleyaw and roll stability YSC and RSC will be integrated inthis section Figure 5 indicates the cascade structure of thecontrol algorithm which consists of three levels First a3-DoF rollover prediction model and a 2-DoF single-trackmodel are implemented to obtain the roll track error and theyaw track error respectively Then an RSC braking torquecontroller and a YSC yaw moment controller are establishedto generate a desired torque control input based on the errorsobtained previously and the coordinated control of yaw androllover (CCYR) stability controllers is applied to coordinatethe demand of these two controllers based on the TTR Inthe underlying structure the braking torque and the yawmoment are distributed to each wheel and achieved by the

hydraulic control unit (HCU) valve controller and the ABScontroller is also used to prevent wheel lock In additionto the cascaded control logic a vehicle status observer isimplemented to obtain the status used in the controller whichincludes reference longitudinal speed and lateral speed roadadhesion coefficient vehicle sideslip angle wheel rotationspeed and wheel cylinder pressure

31 Design of YSC YSC is designed to enhance vehiclemaneuverability by tracking a reference yaw rate generated bya driverrsquos steering input The yaw moment control is adoptedto generate a desired yaw moment in order to reduce the yawrate error between the reference and actual yaw rate A linearbicycle model is used to compute the reference yaw rate andFigure 1 shows the 3-DoF model including the direct yawmoment

In order to obtain the reference yaw rate and vehiclesideslip angle a single-track model is proposed As is shownin Figure 6 yaw and lateral motion in the bicycle model canbe obtained as follows

119898119881119909 (

120573 + 120574) = 119865119910119891 + 119865119910119903

119868119911120574 = 119865119910119891119897119891 minus 119865119910119903119897119903

(5)

The wheel lateral force is considered to be linear with the tireslip angle

119865119910119891 = 119896119891120572119891

119865119910119903 = 119896119903120572119903

120572119891 = 120575119891 minus

119881119910 + 119897119891120574

119881119909

120572119903 =

119881119910 minus 119897119903120574

119881119909

(6)

Therefore the transfer function of yaw rate and sideslip anglecan be obtained

120574

120575

(119904) = 119866119903

1205911119904 + 1

11987911199042+ 1198792119904 + 1

120573

120575

(119904) = 119866120573

1205911015840

1119904 + 1

11987911199042+ 1198792119904 + 1

(7)

where

119866119903 =

119881119909 tan 120575119891119897119894119871 [1 + (119881119909119881ch)

2]

119866120573 =

119897119903 minus 1198981198971198911198812

119909119896119903119897

119897 [1 + (119881119909119881ch)2]

1205911 =

119898119881119909119897119891

119897119896120574

1205911015840

1=

119868119911

119887119896120574119881119909 minus 119898119881119909119897119891

1198791 =

1198981198812

119909119868119911

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

120572r120572r

120575f120575f

120575w120575w

u

L

a

Y

120573r

b

Vy

VxX

t

Fyr

2

Fyr

2

Fyf

2

Fyf

2 z

K120593e

o

C120593

tFzi Fzo

hR

msay

120593

y

msg

Figure 1 3-DoF vehicle nonlinear prediction model schematic diagram

120583

120572f1205721205831 1205721205832

kf middot 1205721205831

Fyf

120583 middot Fz

Figure 2 Diagram of simplified front wheel sideslip characteristics

rollover occurs the lateral acceleration of vehicle will gener-ally reach a larger value and then the lateral forces of tires arein a nonlinear area In this case the limit of tirersquos lateral forcedue to road adhesion limit should be considered In order toprovide vehicle yaw stability and avoid oversteer caused bylateral force saturation of rearwheel the saturation restrictionis used for front wheel only The cornering property offront wheel is nonlinear and that of the rear wheel is linear

0 2 4 6 8 10

0

1

2

3

Time (s)

Test dataSimulation

Late

ral a

ccel

erat

ion

(ms2)

Figure 3 Steering wheel angle step input test

0 10 20 30 40

0

1

2

Time (s)

minus1

minus2Late

ral a

ccel

erat

ion

(ms2)

Test dataSimulation

Figure 4 Slalom test (30 kmh)

4 Mathematical Problems in Engineering

RSC braking torque

Vehicle status calculation

HCU

TTR

Valve command calculation

ABS

Rollover predictive model

YSC yaw torque

Unified valve command

Valve

Reference speed

Steering angle

Wheel speed

Lateral acceleration

Master cylinder pressure

Yaw rate

Reference speed

Road adhesion

coefficient

Braking torque and yaw torque distribution

CCYR braking torque and yaw torque

rlowast rlowast minus r2-DoF vehicle model

120573

command 2Valve

command 1

Wheel cylinder Pw

Figure 5 Overall structure of coordinated control algorithm of vehicle yaw and rollover stability

respectively The cornering characteristic of front tire ispiecewise and the lateral force is obtained as follows119865119910119891

=

119896119891 sdot 119886119891 119886119891 le 1198861205831

119896119891 sdot 1198861205831 +

120583119865119911119891 minus 119896119891 sdot 1198861205831

1198861205832 minus 1198861205831

(119886119891 minus 1198861205831) 1198861205831 le 119886119891 le 1198861205832

120583 sdot 119865119911119891 119886119891 ge 119886119911119891

(3)where 1198861205831 and 1198861205832 are sideslip angles respectively whichcorrespond to the turning point of cornering characteristicswhen the road friction coefficient is 120583 sdot 119886119891 is front wheelsideslip angle 119865119911119891 is front wheel vertical load and thediagram is shown in Figure 2

A 3-DOF vehicle model is derived from (1)ndash(3) and theparameters (119896119891 119896119903 119870120593 119862120593) of the model can be identified

through the inputoutput data of the vehicle test data tableA typical signal for example step test is chosen as driversteering wheel input to obtain the vehicle lateral accelerationas the output Meanwhile the same signal is also as the inputof 3-DOF vehicle model in simulation Using the errors of 119886119910and 119886119910 an objective function in (4) is employed to optimizethe vehicle parameters

119869 = int

1199050

0

(119886119910 minus 119886119910)2119889119905 (4)

The 3-DoF vehicle dynamics model is verified by vehiclehandling and stability experiment data The main contentsof the experimental verification include vehicle steady-stateroll characteristics and steady-state cornering performancevehicle transient steering characteristics and steer on-center characteristics and returnability The constant-speedvariable-steer tests are carried out on a uniform dry level

Mathematical Problems in Engineering 5

120575f

120572f

Fyf

lf

Vy

Vx120574

Vy + lf middot 120574

120573

Vy minus lr middot 120574

Fyr

120572r

lr

Figure 6 Bicycle model

and hard road surface A step input of steering wheel angleis applied when the vehicle runs at a constant-speed and theslalom test was carried out at a speed of 30 kmh Vehicle dataare recorded at sampling rate of 200HzThe results show thatthe responses of the vehicle model coincide with the actualvehicle measurements (see Figures 3 and 4)

3 Coordinated Control Algorithm ofVehicle Yaw and Rollover Stability

The objectives of YSC and RSC are to follow the driverintention and limit the maximum of the vehicle lateralacceleration respectively In order to maximize the vehicleyaw and roll stability YSC and RSC will be integrated inthis section Figure 5 indicates the cascade structure of thecontrol algorithm which consists of three levels First a3-DoF rollover prediction model and a 2-DoF single-trackmodel are implemented to obtain the roll track error and theyaw track error respectively Then an RSC braking torquecontroller and a YSC yaw moment controller are establishedto generate a desired torque control input based on the errorsobtained previously and the coordinated control of yaw androllover (CCYR) stability controllers is applied to coordinatethe demand of these two controllers based on the TTR Inthe underlying structure the braking torque and the yawmoment are distributed to each wheel and achieved by the

hydraulic control unit (HCU) valve controller and the ABScontroller is also used to prevent wheel lock In additionto the cascaded control logic a vehicle status observer isimplemented to obtain the status used in the controller whichincludes reference longitudinal speed and lateral speed roadadhesion coefficient vehicle sideslip angle wheel rotationspeed and wheel cylinder pressure

31 Design of YSC YSC is designed to enhance vehiclemaneuverability by tracking a reference yaw rate generated bya driverrsquos steering input The yaw moment control is adoptedto generate a desired yaw moment in order to reduce the yawrate error between the reference and actual yaw rate A linearbicycle model is used to compute the reference yaw rate andFigure 1 shows the 3-DoF model including the direct yawmoment

In order to obtain the reference yaw rate and vehiclesideslip angle a single-track model is proposed As is shownin Figure 6 yaw and lateral motion in the bicycle model canbe obtained as follows

119898119881119909 (

120573 + 120574) = 119865119910119891 + 119865119910119903

119868119911120574 = 119865119910119891119897119891 minus 119865119910119903119897119903

(5)

The wheel lateral force is considered to be linear with the tireslip angle

119865119910119891 = 119896119891120572119891

119865119910119903 = 119896119903120572119903

120572119891 = 120575119891 minus

119881119910 + 119897119891120574

119881119909

120572119903 =

119881119910 minus 119897119903120574

119881119909

(6)

Therefore the transfer function of yaw rate and sideslip anglecan be obtained

120574

120575

(119904) = 119866119903

1205911119904 + 1

11987911199042+ 1198792119904 + 1

120573

120575

(119904) = 119866120573

1205911015840

1119904 + 1

11987911199042+ 1198792119904 + 1

(7)

where

119866119903 =

119881119909 tan 120575119891119897119894119871 [1 + (119881119909119881ch)

2]

119866120573 =

119897119903 minus 1198981198971198911198812

119909119896119903119897

119897 [1 + (119881119909119881ch)2]

1205911 =

119898119881119909119897119891

119897119896120574

1205911015840

1=

119868119911

119887119896120574119881119909 minus 119898119881119909119897119891

1198791 =

1198981198812

119909119868119911

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

RSC braking torque

Vehicle status calculation

HCU

TTR

Valve command calculation

ABS

Rollover predictive model

YSC yaw torque

Unified valve command

Valve

Reference speed

Steering angle

Wheel speed

Lateral acceleration

Master cylinder pressure

Yaw rate

Reference speed

Road adhesion

coefficient

Braking torque and yaw torque distribution

CCYR braking torque and yaw torque

rlowast rlowast minus r2-DoF vehicle model

120573

command 2Valve

command 1

Wheel cylinder Pw

Figure 5 Overall structure of coordinated control algorithm of vehicle yaw and rollover stability

respectively The cornering characteristic of front tire ispiecewise and the lateral force is obtained as follows119865119910119891

=

119896119891 sdot 119886119891 119886119891 le 1198861205831

119896119891 sdot 1198861205831 +

120583119865119911119891 minus 119896119891 sdot 1198861205831

1198861205832 minus 1198861205831

(119886119891 minus 1198861205831) 1198861205831 le 119886119891 le 1198861205832

120583 sdot 119865119911119891 119886119891 ge 119886119911119891

(3)where 1198861205831 and 1198861205832 are sideslip angles respectively whichcorrespond to the turning point of cornering characteristicswhen the road friction coefficient is 120583 sdot 119886119891 is front wheelsideslip angle 119865119911119891 is front wheel vertical load and thediagram is shown in Figure 2

A 3-DOF vehicle model is derived from (1)ndash(3) and theparameters (119896119891 119896119903 119870120593 119862120593) of the model can be identified

through the inputoutput data of the vehicle test data tableA typical signal for example step test is chosen as driversteering wheel input to obtain the vehicle lateral accelerationas the output Meanwhile the same signal is also as the inputof 3-DOF vehicle model in simulation Using the errors of 119886119910and 119886119910 an objective function in (4) is employed to optimizethe vehicle parameters

119869 = int

1199050

0

(119886119910 minus 119886119910)2119889119905 (4)

The 3-DoF vehicle dynamics model is verified by vehiclehandling and stability experiment data The main contentsof the experimental verification include vehicle steady-stateroll characteristics and steady-state cornering performancevehicle transient steering characteristics and steer on-center characteristics and returnability The constant-speedvariable-steer tests are carried out on a uniform dry level

Mathematical Problems in Engineering 5

120575f

120572f

Fyf

lf

Vy

Vx120574

Vy + lf middot 120574

120573

Vy minus lr middot 120574

Fyr

120572r

lr

Figure 6 Bicycle model

and hard road surface A step input of steering wheel angleis applied when the vehicle runs at a constant-speed and theslalom test was carried out at a speed of 30 kmh Vehicle dataare recorded at sampling rate of 200HzThe results show thatthe responses of the vehicle model coincide with the actualvehicle measurements (see Figures 3 and 4)

3 Coordinated Control Algorithm ofVehicle Yaw and Rollover Stability

The objectives of YSC and RSC are to follow the driverintention and limit the maximum of the vehicle lateralacceleration respectively In order to maximize the vehicleyaw and roll stability YSC and RSC will be integrated inthis section Figure 5 indicates the cascade structure of thecontrol algorithm which consists of three levels First a3-DoF rollover prediction model and a 2-DoF single-trackmodel are implemented to obtain the roll track error and theyaw track error respectively Then an RSC braking torquecontroller and a YSC yaw moment controller are establishedto generate a desired torque control input based on the errorsobtained previously and the coordinated control of yaw androllover (CCYR) stability controllers is applied to coordinatethe demand of these two controllers based on the TTR Inthe underlying structure the braking torque and the yawmoment are distributed to each wheel and achieved by the

hydraulic control unit (HCU) valve controller and the ABScontroller is also used to prevent wheel lock In additionto the cascaded control logic a vehicle status observer isimplemented to obtain the status used in the controller whichincludes reference longitudinal speed and lateral speed roadadhesion coefficient vehicle sideslip angle wheel rotationspeed and wheel cylinder pressure

31 Design of YSC YSC is designed to enhance vehiclemaneuverability by tracking a reference yaw rate generated bya driverrsquos steering input The yaw moment control is adoptedto generate a desired yaw moment in order to reduce the yawrate error between the reference and actual yaw rate A linearbicycle model is used to compute the reference yaw rate andFigure 1 shows the 3-DoF model including the direct yawmoment

In order to obtain the reference yaw rate and vehiclesideslip angle a single-track model is proposed As is shownin Figure 6 yaw and lateral motion in the bicycle model canbe obtained as follows

119898119881119909 (

120573 + 120574) = 119865119910119891 + 119865119910119903

119868119911120574 = 119865119910119891119897119891 minus 119865119910119903119897119903

(5)

The wheel lateral force is considered to be linear with the tireslip angle

119865119910119891 = 119896119891120572119891

119865119910119903 = 119896119903120572119903

120572119891 = 120575119891 minus

119881119910 + 119897119891120574

119881119909

120572119903 =

119881119910 minus 119897119903120574

119881119909

(6)

Therefore the transfer function of yaw rate and sideslip anglecan be obtained

120574

120575

(119904) = 119866119903

1205911119904 + 1

11987911199042+ 1198792119904 + 1

120573

120575

(119904) = 119866120573

1205911015840

1119904 + 1

11987911199042+ 1198792119904 + 1

(7)

where

119866119903 =

119881119909 tan 120575119891119897119894119871 [1 + (119881119909119881ch)

2]

119866120573 =

119897119903 minus 1198981198971198911198812

119909119896119903119897

119897 [1 + (119881119909119881ch)2]

1205911 =

119898119881119909119897119891

119897119896120574

1205911015840

1=

119868119911

119887119896120574119881119909 minus 119898119881119909119897119891

1198791 =

1198981198812

119909119868119911

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

120575f

120572f

Fyf

lf

Vy

Vx120574

Vy + lf middot 120574

120573

Vy minus lr middot 120574

Fyr

120572r

lr

Figure 6 Bicycle model

and hard road surface A step input of steering wheel angleis applied when the vehicle runs at a constant-speed and theslalom test was carried out at a speed of 30 kmh Vehicle dataare recorded at sampling rate of 200HzThe results show thatthe responses of the vehicle model coincide with the actualvehicle measurements (see Figures 3 and 4)

3 Coordinated Control Algorithm ofVehicle Yaw and Rollover Stability

The objectives of YSC and RSC are to follow the driverintention and limit the maximum of the vehicle lateralacceleration respectively In order to maximize the vehicleyaw and roll stability YSC and RSC will be integrated inthis section Figure 5 indicates the cascade structure of thecontrol algorithm which consists of three levels First a3-DoF rollover prediction model and a 2-DoF single-trackmodel are implemented to obtain the roll track error and theyaw track error respectively Then an RSC braking torquecontroller and a YSC yaw moment controller are establishedto generate a desired torque control input based on the errorsobtained previously and the coordinated control of yaw androllover (CCYR) stability controllers is applied to coordinatethe demand of these two controllers based on the TTR Inthe underlying structure the braking torque and the yawmoment are distributed to each wheel and achieved by the

hydraulic control unit (HCU) valve controller and the ABScontroller is also used to prevent wheel lock In additionto the cascaded control logic a vehicle status observer isimplemented to obtain the status used in the controller whichincludes reference longitudinal speed and lateral speed roadadhesion coefficient vehicle sideslip angle wheel rotationspeed and wheel cylinder pressure

31 Design of YSC YSC is designed to enhance vehiclemaneuverability by tracking a reference yaw rate generated bya driverrsquos steering input The yaw moment control is adoptedto generate a desired yaw moment in order to reduce the yawrate error between the reference and actual yaw rate A linearbicycle model is used to compute the reference yaw rate andFigure 1 shows the 3-DoF model including the direct yawmoment

In order to obtain the reference yaw rate and vehiclesideslip angle a single-track model is proposed As is shownin Figure 6 yaw and lateral motion in the bicycle model canbe obtained as follows

119898119881119909 (

120573 + 120574) = 119865119910119891 + 119865119910119903

119868119911120574 = 119865119910119891119897119891 minus 119865119910119903119897119903

(5)

The wheel lateral force is considered to be linear with the tireslip angle

119865119910119891 = 119896119891120572119891

119865119910119903 = 119896119903120572119903

120572119891 = 120575119891 minus

119881119910 + 119897119891120574

119881119909

120572119903 =

119881119910 minus 119897119903120574

119881119909

(6)

Therefore the transfer function of yaw rate and sideslip anglecan be obtained

120574

120575

(119904) = 119866119903

1205911119904 + 1

11987911199042+ 1198792119904 + 1

120573

120575

(119904) = 119866120573

1205911015840

1119904 + 1

11987911199042+ 1198792119904 + 1

(7)

where

119866119903 =

119881119909 tan 120575119891119897119894119871 [1 + (119881119909119881ch)

2]

119866120573 =

119897119903 minus 1198981198971198911198812

119909119896119903119897

119897 [1 + (119881119909119881ch)2]

1205911 =

119898119881119909119897119891

119897119896120574

1205911015840

1=

119868119911

119887119896120574119881119909 minus 119898119881119909119897119891

1198791 =

1198981198812

119909119868119911

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

PI controller

Vehicle

Rollover predictive model

TTR

Effective value calculation

Effective error calculation

PI controller

P controller

rlowastr

120573 e120573 ΔM

120573

ΔMb_RSCminus

minus

+

ΔMz

ΔMb

ΔMr

TTRE

1205821

1205822

Δr er

ΔMYSC

TTRth

Figure 7 Diagram of the additional yaw moment and brake torque

0 1 2 3 4 5 6 7 8 90

20

40

60Vehicle reference speed

Wheel speed

(km

h)

FL FR

RL RRVref

0 1 2 3 4 5 6 7 8 9

02040

Wheel accelerationFL FR

RL RR

(ms2)

minus40minus60

minus20

0 1 2 3 4 5 6 7 8 90

2

4

6

(MPa

)

Time (s)

Time (s)

Time (s)

RL RRFL FR

Brake pressure

Figure 8 Braking in low adhesion road

1198792 =

119881119909 [119868119911 (119896119891 + 119896119903) + 119898 (1198972

119891119896119891 + 119897

2

119903119896119903)]

1198972119896119891119896119903 (1 + 119881

21199091198812

ch)

1198812

ch =1198961198911198961199031198972

119898(119896119903119897119903 minus 119896119891119897119891)

(8)

Hence the steady-state response of yaw rate and sideslip anglecan be obtained as follows

120574state = 119866120574120575119891 120573state = 119866120573120575119891 (9)

In addition the tire-road adhesion limitation is taken intoaccount

120574max =120583119892

119881119909

(10)

Therefore the nominal yaw rate can be obtained as

1205741198730 = min1003816100381610038161003816120574state

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

120583119892

119881119909

10038161003816100381610038161003816100381610038161003816

sign (120575119891) (11)

Similarly the nominal vehicle sideslip angle could also beobtained by the steady-state response and the tire-roadfriction limitation

1205731198730 = min 1003816100381610038161003816120573119873

10038161003816100381610038161003816100381610038161003816arctan (002120583119892)100381610038161003816

1003816 sign (120575119908) (12)

The strategy of yaw rate control method takes the devia-tion 119890 between the actual yaw rate and the reference yaw rateas a control variableWhen the absolute value of the deviation119890 exceeds the threshold set by control algorithm the vehicleshould be in an instability state and needs to intervene yawmoment [24] The additional yaw moment Δ119872 demandedby YSC can be expressed denoting the error between the

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

00 05 10 15 20 25 30 3502468

10

(MPa

)

Time (s)

Time (s)

Time (s)

Brake pressureFL FR

RL RR

00 05 10 15 20 25 30 350

204060

(km

h)

Vehicle reference speedWheel speed

RL RR FL FR

Vref

00 05 10 15 20 25 30 35

03060

Wheel accelerationFL FR RL RR(m

s2)

minus30minus60

Figure 9 Braking in high adhesion road

0 50 100 150 200 250 300 350 400

024

0 2 4 6 8 10 12 14 16 18 20

0300600

0306090

Y co

ordi

nate

(m)

X coordinate (m)

Stee

ring

angl

e (de

g)

Time (s)

0 2 4 6 8 10 12 14 16 18 20Time (s)

Vehi

cle sp

eed

(km

h)

minus600

minus300

minus2

minus4

With CCYROnly with YSCOnly with RSC

Figure 10 Driving track steering wheel angle and vehicle speed

nominal yaw motion and sideslip angle and the actual valueof them respectively as follows

119890120574 (119899) = 1205741198730 (119899) minus 120574 (119899)

119890120573 (119899) = 1205731198730 (119899) minus 120573 (119899)

(13)

And the desired yaw moment can be obtained from theyaw error and vehicle sideslip angle error

Δ119872120574 = 119870120574119901Δ119890120574 (119899) + 119870120574119894119890120574 (119899)

Δ119872120573 = 119870120573119901Δ119890120573 (119899) + 119870120573119894119890120573 (119899)

(14)

where119870120574119901 119870120574119894 119870120573119901 and 119870120573119894 are controller parameters

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

10

20

30

40

50Ya

w ra

te (d

egs

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus10

minus20

minus30

minus40

minus50

(a)

0

2

4

6

8

Roll

angl

e (de

g)

0 2 4 6 8 10 12 14 16 18 20Time (s)

minus2

minus4

minus6

minus8

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

minus2

minus4

minus6

minus8

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

Vehi

cle sl

ip an

gle (

deg)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2

4

6

8

10

minus2

minus4

minus6

minus8

With CCYROnly with YSCOnly with RSC

(d)

Figure 11 Yaw rate roll angle lateral acceleration and sideslip angle

A weight is used to obtain the desiredmoment of the YSCas follows

Δ119872YSC = (1 minus 120572) Δ119872120574 + 120572Δ119872120573 (15)

where 120572 is a positive weight coefficient up to 1 and 120572 increasesas the vehicle sideslip angle increases

32 Design of RSC RSC is designed to generate a targetbraking force to reduce the lateral acceleration based on the

LTR while the brake intervention is decided on the TTRthreshold Both TTR and LTR are calculated based on the3-DoF vehicle nonlinear prediction model described in thesecond section and with denoting 119909 = [120573 120574 120593] thevehicle model can be transferred to the state-space model asfollows

= 119860119909 + 119861120575 (16)

where is the roll rate of vehicle body

Consider

119860 = 119872minus1119873 119861 = 119872

minus1119875

119872 =

[

[

[

[

119868119911 0 119868119909119911119904 0

0 119898119881119909 minus119898119904119890 0

119868119909119911119904 minus119898119904119890119881119909 119868119909119904 0

0 0 0 1

]

]

]

]

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

L(N

)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

Fz_F

R(N

)

(b)

0 2 4 6 8 10 12 14 16 18 20Time (s)

0

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

With CCYROnly with YSCOnly with RSC

(c)

0

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

0 2 4 6 8 10 12 14 16 18 20Time (s)

With CCYROnly with YSCOnly with RSC

(d)

Figure 12 Wheel vertical force

119873 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

minus2(

1198961198911198862+ 1198961199031198872

119881119909

) 2 (minus119886119896119891 + 119887119896119903 + 119873119891 + 119873119903) 0 2 (119886119864119891119896119891 minus 119887119864119903119896119903)

2(

minus119886119896119891 + 119887119896119903

119881119909

) minus 119898119881119909 minus2 (119896119891 + 119896119903) 0 2 (119864119891119896119891 + 119864119903119896119903)

119898119904119890119881119909 0 minus119862120601 119898119904119892119890 minus 119870120601

0 0 1 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

119875 = [minus2 (119886119896119891 minus 119873119891) 2119896119891 0 0]

119879

(17)

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 50 100 150 200 250 300

012345

Y co

ordi

nate

(m)

X coordinate (m)

minus1minus2

0 1 2 3 4 5 6 7 8 9 10

0200400

Stee

ring

angl

e (de

g)

Time (s)

minus600minus400minus200

0 1 2 3 4 5 6 7 8 9 100

306090

120150

Vehi

cle sp

eed

(km

h)

Time (s)With CCYROnly with YSCOnly with RSC

Figure 13 Driving track steering wheel angle and vehicle speed

LTR can be defined as

LTR =

1003816100381610038161003816sum119899

119894=1(FL119894 minus FR119894)

1003816100381610038161003816

sum119899

119894=1(FL119894 minus FR119894)

(18)

The roll dynamic can be described by

minusFR 1199052

+ FL 1199052

minus 119870120593120593 minus 119862120593 = 0 (19)

Hence the LTR can be obtained from the previousequation

LTRestamit = minus

2 (119862120593 + 119870120593120593)

119898119892119905

(20)

So LTR can be obtained from the sate space model asfollows

LTRestamit = 119862LTR119909 (21)

where

119862LTR = [0 0 minus

2119862120593

119898119892119905

minus

2119870120593

119898119892119905

] (22)

TTRestamit is calculated according to the LTRestamit at thecurrent state with the proposed 3-DoF vehicle model onlineA TTR threshold TTR1198730 is used to control the brakingintervention of RSC

119890TTR (119899) = TTR1198730 (119899) minus TTRestamit (119899) (23)

The error between the nominal TTR and the estimatedTTR is used to calculate the braking yaw moment of the RSCwith a proportional controller where119870120593119901 is the parameter

Δ119872119887 RSC = 119870120593119901119890TTR sign (120575119891) (24)

33 Design of CCYR Stability Controller The safety of drivingvehicle is mainly dependent on yaw and rollover stabilityUnstability of yaw takes place on all roads while rollovermainly happens on the high adhesion road So YSC mainlyworks on the low adhesion road alone while YSC and RSCmay be simultaneously or separately triggered on the highadhesion road There are two conditions when YSC and RSCsimultaneously trigger One when vehicle exists the danger ofoversteer and rollover at the same time YSC and RSC makevehicle tend to understeer The other when the rollover takesplace RSC gradually intervenes the control system whichleads to larger understeer of the vehicle Then the understeerof vehicle triggers the YSC which makes the vehicle tend tooversteer

Overall there are three primary relationships betweenYSC and RSC

(1) When separately triggered YSC and RSC are inde-pendent and maximize achievement of the controltarget separately

(2) When simultaneously triggered in line with controltrend YSC and RSC control the system at the sametime with coupling interaction

(3) When simultaneously triggered in contrast to controltrend there is contradiction between YSC and RSC

Hence the control targets are to keep yaw and rolloverstability and reduce contradiction betweenYSC andRSC andcoordinate those two controllers working as an integration toensure vehicle stability according to the danger level of yawand roll

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60Ya

w ra

te (d

egs

)

Time (s)

minus20

minus40

minus60

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus8

minus6

minus4

minus2

(b)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Time (s)

minus8

minus6

minus4

minus2

Ay (m

s2)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus8

minus10

minus12

minus6

minus4

minus2

With CCYROnly with YSCOnly with RSC

(d)

Figure 14 Yaw rate roll angle lateral acceleration and sideslip angle

Coordinated control of yaw and rollover stability con-trollers is designed to coordinate the YSC and RSC shownin Figure 7

The additional yawmomentΔ119872 is generated by applyingbraking forces In the research the braking torqueses areapplied to the diagonal wheels based on detection of theundersteer and oversteer during driving situation When thevehicle has characteristic of understeer the braking force isapplied on the inside front wheel Similarly the braking forceis applied on the outside rear wheel when the vehicle hascharacteristic of oversteer The control algorithm calculatesthe braking force of each wheel based on the states of vehicleand the value of the additional yaw moment Δ119872

Δ119872 = 1205821Δ119872YSC + 1205822Δ119872119887 RSC (25)

where 1205821 and 1205822 are chosen carefully according to the vehiclestate and 1205821 and 1205822 emphasize the importance of yaw stabilityand rollover stability respectively In order to obtain 1205821 and1205822 the normalization coefficients 1198861 and 1198862 are introduced

1198861 =

1003816100381610038161003816Δ119872YSC

1003816100381610038161003816

Δ119872YSC 0 1198862 = 1 minus

TTRTTRth

(26)

where Δ119872YSC 0 is the nominal value of additional yawmoment is Δ119872YSC and TTRth is the nominal value ofTTR 1205821 and 1205822 can be calculated through (27) for coordi-nation of the Δ119872YSC and Δ119872119887 RSC when YSC and RSC are

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

L(N

)

Time (s)

(a)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_F

R(N

)

Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

L(N

)

Time (s)

With CCYROnly with YSCOnly with RSC

(c)

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000

12000

Fz_R

R(N

)

Time (s)With CCYROnly with YSCOnly with RSC

(d)

Figure 15 Wheel vertical force

simultaneously triggered 1205822 increases as the TTR decreasesand 1205821 + 1205822 = 1

1205821 =

1198861

1198861 + 1198862

1205822 =

1198862

1198861 + 1198862

(27)

34 Design of ABS Controller In order to prevent the wheelfrom being locked in braking and provide greater lateralforce ABS control is added to the coordinated control ofyaw and roll stability ABS control algorithm mainly usingthe wheel acceleration and deceleration threshold methodis supplemented by reference slip rate threshold controlThe comparisons between wheel acceleration and slip rate

are carried out between vehicle data and correspondingthreshold values According to decision logic the controlcommand is sent to ABS hydraulic modulator to keep sliprate at the optimal value As a result the longitudinal orlateral force can obtain fully utilization which will causesmall braking distance and good yaw stability [25]

Figures 8 and 9 are the simulations of the ABS controlalgorithm The simulation is straight line braking in uni-form pavement (road surface with low adhesion and highadhesion) and the initial speed is 60 kmh The adhesioncoefficient of low adhesion surface is 02 and the adhesioncoefficient of high adhesion road is 08

The results show that the ABS control algorithm designedin this paper can realize good control effect in the generalcondition and it is helpful for supplying a good foundationfor the coordinated control of yaw and roll stability

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

4 Simulation Results of Coordinated Control

The proposed CCYR stability controller is implementedunder the Simulink platform and simulated with the Car-Sim software package To evaluate the control strategy thealgorithms of YSC and RSC are also established respectivelyAnd all the algorithms are evaluated through the vehiclesimulation

For each system three simulations including slalomtest under high adhesion double-lane change test underhigh adhesion and sine dwell test under high adhesion areconducted to test the rollover prevention capabilities Thesame driver model is used for each rollover controller andeach simulation In these simulations the antilock brakingsystem (ABS) is used to prevent the lock of wheels

41 Slalom Test under High Adhesion The adhesion coeffi-cient of slalom test is 1 and the initial speed is 80 kmh

As is shown in Figure 10 RSC vehicle has not completedthe given trajectory when the driver tries to operate thesteering wheel When RSCmainly controls front wheel brak-ing the lateral force of the outside front wheel reduces andsaturates so that the yawmoment reduces greatly resulting ina greater dynamic vehicle understeer In addition the speeddecreases greatly because the driving force reduces due tolateral force saturation YSC vehicle follows the intent of thedriver well and completes the given trajectory and the speedchanges a little CCYR vehicle completes the given trajectoryand the path is basically similar to that of YSC vehicle

In Figures 11 and 12 the decrease of the sideslip angleand the roll angle is achieved by the RSC which gives thesmaller vehicle response than the RSC because of rapidtire load change However the erratic fluctuation of tireload under larger lateral acceleration increases the responsetime of controller The delay of response time leads toinstability of vehicle The additional control moment is madebecause of the lager sideslip angle and the roll angle at thebeginning of simulation test whichmake the vehicle stabilizein the subsequent responds of vehicle The CCYR stabilitycontroller combines the advantages of the YSC and RSC Yawrate body roll angle and sideslip angle of CCYR vehicle arerelatively small and consequently the vehicle has good yawstability

42 Double-Lane Change Test under High Adhesion Theadhesion coefficient of double-lane change test is 1 and theinitial speed is 130 kmh

In Figure 13 RSC vehicle turns to understeer due tooutside front wheel braking and has not completed thegiven trajectory and the driverrsquos steering wheel angle isgreater YSC vehicle follows the driverrsquos intentions well andcompleted the given trajectory and has small changes inspeed CCYR vehicle completed the given trajectory and thepath is basically similar to that of YSC vehicle

In Figures 14 and 15 the changes of response signalsincluding tire load yaw rate body roll angle and sideslipangle are the same under larger lateral acceleration betweenthe YSC and RSC and the YSC or RSC is triggered separately

0 1 2 3 4

0

100

200

300

Stee

ring

angl

e (de

g)

Time (s)

minus100

minus200

minus300

Figure 16 Steering angle

and simultaneously to maximize achievement of the controltargetTheCCYR stability controller coordinates theYSC andRSC to control the system at the same time with couplinginteraction The results of simulations show that this controlmethod has good control effectiveness

43 Sine Dwell Test under High Adhesion The adhesioncoefficient of sine dwell test is 1 The initial speed is 80 kmhThe input of steering wheel angle is a 07Hz sine with a dwellmaneuver The maximum amplitude of steering wheel angleis 280 degrees shown in Figure 16 The tests were carried outin CarSim

In Figure 17 the YSC and the RSC demonstrate theireffectiveness in achieving enhanced vehicle stability by theirperformance in the NHTSA sine-with-dwell tests Comparedto the RSC the values of response signals including yaw ratebody roll angle and sideslip angle are larger for the YSCThesmall fluctuations of response signals make the stable time ofvehicle longer with the RSC Figure 17 shows that the max-imum values of yaw rate and sideslip angle of the proposedCCRY are always smaller than those of the uncoordinatedsystem which demonstrate that the new control algorithmhas certain enhancement in the comfortable and stability ofvehicle

5 Conclusion

In this paper the coordinated control strategy of yaw androllover stability is developed A nonlinear 3-DoF vehicleprediction model is built and tested The YSC traces the yawmotion and the sideslip error with a 3-DoF vehicle modelvia differential braking The RSC applies a braking forceaccording to the TTR threshold and TTR estimation witha 3-DoF nonlinear prediction vehicle model of the vehicleSimulation is implemented on CarSim electronic stabilitycontrol platform by the comparison among CCYR YSC andRSC The results can be summarized as follows

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

0 1 2 3 4

0

10

20

30

40

50Ya

w ra

te (d

egs

)

Time (s)

minus40

minus30

minus20

minus10

(a)

0 1 2 3 4

0

2

4

6

8

10

Roll

angl

e (de

g)

Time (s)

minus10

minus8

minus6

minus2

minus4

(b)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Time (s)

minus10

minus8

minus6

minus2

minus4

Ay (m

s2)

(c)

With CCYROnly with YSCOnly with RSC

0 1 2 3 4

0

2

4

6

8

10

Vehi

cle sl

ip an

gle (

deg)

Time (s)

minus10

minus8

minus6

minus2

minus4

(d)

Figure 17 Yaw rate roll angle lateral acceleration and sideslip angle

(1) The nonlinear 3-DoF vehicle model is built up andtested and the results show that the model has veryhigh simulation accuracy

(2) The coordinated control algorithm has better stabilityof yaw and roll than YSC and RSC

(3) As the rollover predictive model the proposed non-linear 3-DoF vehicle model is effective for the coordi-nated control algorithm

Nomenclature

119886 Distance from center of gravity (CG) to frontaxle

119887 Distance from CG to rear axle119905 Track width119896119891 Front tire cornering stiffness119896119903 Rear tire cornering stiffness119898 Total mass of vehicle119898119904 Sprung mass of vehicle

119868119911 Inertia around the 119911-axis119868119909119911 Inertia around the 119911-axis and 119909-axis119868119911119904 Inertia around the 119911-axis of sprung mass119868119909119911119904 Inertia around the 119911-axis and 119909-axis of

sprung mass119892 Gravity acceleration119890 Distance between the center of sprung mass

and the vehicle roll axis119862120593 Roll damping119865119910119891 Front tire lateral force119865119910119903 Rear tire lateral forceFL Left side tire vertical forceFR Right side tire vertical force119870120593 Roll stiffness119872119909 Roll moment119872119911 Yaw moment119881119909 Longitudinal vehicle speed119881119910 Lateral vehicle speed

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

119881119910 Lateral vehicle acceleration120575119891 Front tire steer angle120572119891 Front tire slip angle120572119903 Rear tire slip angle120573 Vehicle sideslip angle120574 Yaw angle119903 Yaw rate119903 Yaw angular acceleration120593 Roll angle of vehicle body Roll angular acceleration of vehicle body120583 Tire-road friction coefficient

Acknowledgments

The authors thank the ldquoresearch team of test technologyon the active safety of automobilerdquo The work is sponsoredby Shanghai Automobile Incorporation Company IndustrialTechnology Development Fund (SAIC Grant 1007)

References

[1] P L Boyd ldquoNHTSArsquos NCAP rollover resistance rating systemrdquoin Proceedings of the 19th International Technical Conference onthe Enhanced Safety of Vehicles 2005

[2] Motor vehicle traffic crash injury and fatality estimatesNational Highway Traffic Safety Administration 2003

[3] V Cherian R Shenoy A Stothert J Shriver J Ghidella andT D Gillespie ldquoModel-Based Design of a SUV anti-rollovercontrol systemrdquo SAE paper 2008-01 2008

[4] J S Tang ldquoActive roll and stability controlrdquo SAE InternationalJournal of Passenger Cars vol 1 no 1 pp 1060ndash1070 2009

[5] D Hyun ldquoPredictive modeling and active control of rollover inheavy vehiclesrdquo Texas AampM University 2001

[6] J Preston-Thomas and J Woodrooffe ldquoA feasibility study of arollover warning device for heavy trucksrdquo Transport CanadaPublication TP 10610E 1990

[7] R D Ervin C B Winkler P S Fancher et al ldquoCooperativeagreement to foster the deployment of a heavy vehicle intel-ligent dynamic stability enhancement systemrdquo Interim ReportNHTSA-US DOT Contract DTNH22-95-H-07002 1998

[8] B-C Chen Warning and control for vehicle rollover prevention[PhD thesis] Department of Mechanical Engineering andApplied Mechanics University of Michigan 2001

[9] A Y Lee ldquoCoordinated control of steering and anti-roll bars toalter vehicle rollover tendenciesrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 124no 1 pp 127ndash132 2002

[10] S Yoon J Jung and B Koo ldquoDevelopment of rollover pre-vention system using unified chassis control of ESP and CDCsystemsrdquo SAE 2006-01-1276 2006

[11] M Kamal ldquoDevelopment of Active Suspension Control forCombined Handling and Rollover Propensity EnhancementrdquoSAE SP 2007-01-0826 2007

[12] J Yoon D Kim and K Yi ldquoDesign of a rollover index-basedvehicle stability control schemerdquo Vehicle System Dynamics vol45 no 5 pp 459ndash475 2007

[13] J Yoon W Cho B Koo and K Yi ldquoUnified chassis control forrollover prevention and lateral stabilityrdquo IEEE Transactions onVehicular Technology vol 58 no 2 pp 596ndash609 2009

[14] B Guvenc T Acarman and L Guvenc ldquoCoordination ofsteering and individual wheel braking actuated vehicle yawstability controlrdquo in Proceedings of the IEEE Intelligent VehiclesSymposium pp 288ndash293 IEEE 2003

[15] M A Selby Intelligent vehicle motion control [PhD disserta-tion] University of Leeds 2003

[16] Y Kou H Peng and D Jung ldquoDevelopment of an integratedchassis control system for worst-case studiesrdquo in Proceedings ofAVEC pp 47ndash52 2006

[17] H Chou and B DrsquoAndrea-Novel ldquoGlobal vehicle control usingdifferential braking torques and active suspension forcesrdquoVehicle System Dynamics vol 43 no 4 pp 261ndash284 2005

[18] D Li S Du and F Yu ldquoIntegrated vehicle chassis control basedon direct yaw moment active steering and active stabiliserrdquoVehicle System Dynamics vol 46 no 1 pp 341ndash351 2008

[19] J Yoon S YimW Cho B Koo and K Yi ldquoDesign of an unifiedchassis controller for rollover prevention manoeuvrability andlateral stabilityrdquo Vehicle System Dynamics vol 48 no 11 pp1247ndash1268 2010

[20] B-C Chen C-C Yu W-F Hsu and M-F Lo ldquoDesign ofelectronic stability control for rollover prevention using slidingmode controlrdquo International Journal of Vehicle Design vol 56no 1ndash4 pp 224ndash245 2011

[21] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe Germany1999

[22] B-C Chen and H Peng ldquoDifferential-braking-based rolloverprevention for sport utility vehicles with human-in-the-loopevaluationsrdquo Vehicle System Dynamics vol 36 no 4-5 pp 359ndash389 2001

[23] A Sorniotti and N DrsquoAlfio ldquoVehicle dynamics simulation todevelop an active roll control systemrdquo SAEPapers 01-0828 2007

[24] Q Z Yan J M Williams J Li Q Z Yan-Daimler ChryslerJ M Williams-Daimler Chrysler and J Li-Daimler ChryslerldquoChassis control system development using simulation soft-ware in the loop rapid prototyping and hardware in the looprdquoin Proceedings of the SAE Conference pp 1ndash12 2002

[25] H Ding K Guo J Zhang H Fu and J Lv ldquoDevelopmentand application of the hardware and driver-in-the-loop testrig for automotive electronic stability programs automotiveengineeringrdquo 2006-04 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of