Research ArticleVehicle Velocity and Roll Angle Estimation withRoad and Friction Adaptation for Four-Wheel IndependentDrive Electric Vehicle

Linhui Zhao and Zhiyuan Liu

Department of Control Science and Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Linhui Zhao zhaolinhuihiteducn

Received 30 April 2014 Accepted 7 October 2014 Published 22 October 2014

Academic Editor Bo Shen

Copyright copy 2014 L Zhao and Z Liu 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

Vehicle velocity and roll angle are important information for active safety control systems of four-wheel independent drive electricvehicle In order to obtain robustness estimation of vehicle velocity and roll angle a novel method is proposed based on vehicledynamics and the measurement information provided by the sensors equipped in modern cars The method is robust with respectto different road and friction conditions Firstly the dynamic characteristics of four-wheel independent drive electric vehicleare analyzed and a four-degree-of-freedom nonlinear dynamic model of vehicle and a tire longitudinal dynamic equation areestablished The relationship between the longitudinal and lateral friction forces is derived based on Dugoff tire model Theunknown input reconstruction technique of sliding mode observer is used to achieve longitudinal tire friction force estimationA simple observer is designed for the estimation of the roll angle of the vehicle And then using the relationship the estimatedlongitudinal friction forces and roll angle a sliding mode observer for vehicle velocity estimation is provided which does not needto know the tire-road friction coefficient and road angles Finally the proposed method is evaluated experimentally under a varietyof maneuvers and road conditions

1 Introduction

Vehicle active safety control systems such as yaw stabilitycontrol system and roll stability control system can signifi-cantly reduce the number of road accidents [1ndash3] Howeverthese systems usually depend upon information about vehiclevelocity yaw rate and roll angle Generally the yaw rate ismeasurable but the vehicle velocity and roll angle cannotbe measured directly in modern cars due to the cost andreliability issues As a consequence they must be estimatedand the accurate and reliable vehicle velocity and roll angleinformation is very important for the vehicle active safetycontrol systems [4 5] Since the wheel torques of four-wheelindependent drive electric vehicle can be obtained easily thehigher accuracy of vehicle velocity estimation compared toconventional vehicles can be achieved which should be usedto improve the performance of vehicle active safety controlsystems

In [6] an extended Kalman filter is used to estimatevehicle velocity and friction forces Since a random walk is

chosen to model each friction force the estimation accuracymay be degraded when the friction forces are time-varyingduring braking and driving In addition as the vehicle modelis significantly nonlinear model errors related to currentlyestimated state may be introduced by linearization A non-linear observer for vehicle velocity estimation with stabilityguarantees based on acceleration and yaw ratemeasurementsin addition towheel angular velocity and steeringwheel anglemeasurements is investigated in [7] But the wheel angularvelocity measurements are transformed to measurementsof longitudinal velocity and used in the feedback term ofthe observer This is based on an assumption of zero tirelongitudinal and lateral slips So the measurements will con-tain large errors and deteriorate the estimate of longitudinalvelocity when the tire slips are high Unfortunately the tireslips usually are high during braking and steering especiallyon low friction surfaces Moreover the longitudinal velocityobserver gains must be determined at each sample timeThis has an adverse effect on the real-time performance ofthe observer In [8] a sliding mode observer is suggested to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 801628 11 pageshttpdxdoiorg1011552014801628

2 Mathematical Problems in Engineering

estimate vehicle sideslip angle while a linear lateral frictionmodel is used and longitudinal friction forces are treatedas inputs of the observer However the linear relationshiprepresenting lateral friction force is not sufficiently accurateanymore for lateral acceleration above 4ms2 and large tiresideslip angle and longitudinal friction force is not alwaysavailable in modern cars [9] However the researches aboveassume that the road grade angle and bank angle are zeroor are known for accurate estimation of vehicle velocity Theroad angles are nonzero if the road is nonflat and then havesignificant effect in both vehicle dynamics and accelerationmeasurements [10] Amethod for identifying road bank angleand vehicle roll angle separately using measurements fromglobal positioning system and inertial navigation systemsensors is developed in [11] In [12] the lateral tire forcesensors are used to estimate the vehicle sideslip angle androll angle for four-wheel independent drive electric vehicleand high estimation accuracy is achieved But the sensorsmentioned above are expensive for modern cars In [13] theestimation of the road grade and bank angles is discussedbased on vehicle longitudinal and lateral dynamics and theassumption that the angles vary slowly enough compared tothe dynamics of the system the effect of the roll angle has notbeen considered in this paper

In this paper using the measurements from the existingsensors equipped in the four-wheel independent drive elec-tric vehicle including the wheel angular velocities longitudi-nal and lateral accelerations yaw rate roll rate wheel steeringangles and wheel torques a method for vehicle velocity androll angle estimation with road and friction adaptation is pro-posed based on the nonlinear vehicle dynamics and DugofftiremodelThe proposedmethod is evaluated experimentallyunder a variety of maneuvers and road conditions

2 Vehicle Model

As shown in Figures 1 and 2 a vehicle maneuvering androll model is used to design the vehicle velocity and rollangle observers This model has four degrees of freedomfor longitudinal motion lateral motion yaw motion androll motion of the vehicle During handling maneuvers onsmooth roads the vehicle roll motion is primarily inducedby lateral acceleration [14] Based on this assumption a body-fixed coordinate systemwith the origin at the vehicle center ofgravity (COG) is used to set up the model and the kinematicrelationships among the vehicle velocity yaw rate roll angleand acceleration are as follows

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin (120601V + 120601119903)

119869

119909

120601V + 119862roll

120601V + 119870roll120601V = 119898119904119886119910ℎroll

119903 =

119872

119911

119869

119911

(1)

where V119909and V

119910are the longitudinal and lateral velocities

of vehicle COG 119903 is the yaw rate 120579119903and 120601

119903denote the

road grade and bank angles The effect between the road

bR2

bF2

bF

bR

Fx1 Fx2

Fy1x lF

r

y COG

120575

lRFx3 Fx4

Fy3 Fy4

Fy2

Figure 1 Vehicle maneuvering model

grade and bank angle is not considered in this paper 119892 isthe gravitational constant 120601V is the roll angle as a result ofvehicle lateral motion by steering maneuvers 119886

119909and 119886

119910are

the longitudinal and lateral accelerationsmeasurements fromthe sensors attached to the vehicle body and aligned with thebody-fixed coordinate system and they will be different fromthe actual accelerations of the COG when there are nonzeroroad grade angle and bank angle 119869

119909is the vehicle moment

of inertia about longitudinal axis 119862roll is the roll stiffnessand 119870roll is the roll damping 119898

119904is the sprung mass of the

vehicle and ℎroll is the height of the roll center 119869119911 is the vehiclemoment of inertia about vertical axis119872

119911is the torque about

vertical axisDue to the presence of measurement errors such as

biases and noise from the sensors some feedback should beused to make the estimated results converge and then thevehicle dynamics is introduced into the vehicle model (1)According to Figure 1 the force balances in the direction ofthe longitudinal and lateral axis as well as the torque balanceabout the vertical axis are given by

119886

119909=

119865

119909

119898

119886

119910=

119865

119910

119898

119872

119911= 119897

119865((119865

1199091+ 119865

1199092) sin 120575 + (119865

1199101+ 119865

1199102) cos 120575)

minus 119897

119877(119865

1199103+ 119865

1199104)

+

119887

119865((119865

1199101minus 119865

1199102) sin 120575 minus (119865

1199091minus 119865

1199092) cos 120575)

2

minus

119887

119877(119865

1199093minus 119865

1199094)

2

(2)

Mathematical Problems in Engineering 3

X

Z

COG

120579r

(a) Force induced by road grade angle

Y

Z

COG

120601

120601r

(b) Vehicle roll model

Figure 2 Force induced by road grade angle and vehicle roll model

with

119865

119909= (119865

1199091+ 119865

1199092) cos 120575

minus (119865

1199101+ 119865

1199102) sin 120575 + 119865

1199093+ 119865

1199094minus 119862

119909V2119909

119865

119910= (119865

1199091+ 119865

1199092) sin 120575

+ (119865

1199101+ 119865

1199102) cos 120575 + 119865

1199103+ 119865

1199104

(3)

where the effect of wheel rolling resistance forces is ignored119898 denotes the total mass of the vehicle 120575 is the frontwheel steering angle and can be obtained from the measuredsteering wheel angle directly 119862

119909represents the aerodynamic

resistances coefficient 119897119865and 119897119877are the distances from the

vehicle COG to the front and rear axle and 119887119865and 119887119877are the

front and rear track width 119865119909119894and 119865

119910119894are the longitudinal

and lateral friction forces of the 119894th wheel and 119894 is 1 2 3 and4 and represents four wheels respectively

The nonlinearity of vehicle tires will become a criticalfactor during emergency maneuvers in which the linear tiremodel is not sufficiently accurate anymore To account for thenonlinearities of the tire friction force Dugoff tire model [15]is used in this paper and the longitudinal friction force 119865

119909119894

and the lateral friction force 119865119910119894

of the 119894th wheel in (3) aregiven as

119865

119909119894=

119862

120590120590

119894

1 + 120590

119894

119891 (120582

119894)

119865

119910119894=

119862

120572tan120572119894

1 + 120590

119894

119891 (120582

119894)

(4)

where 119862120590and 119862

120572are the longitudinal and cornering stiffness

of the tire 120590119894and 120572

119894denote the tire longitudinal slip and

the tire sideslip angle of the 119894th wheel The definition andcalculation method of these variables are shown in [16] Thevariable 120582

119894and the function 119891(120582

119894) are defined as

120582

119894=

120583119865

119911119894(1 + 120590

119894)

2

radic

(119862

120590120590

119894)

2

+ (119862

120572tan120572119894)

2

119891 (120582

119894) =

1 120582

119894ge 1

(2 minus 120582

119894) 120582

119894120582

119894lt 1

(5)

where 120583 is the tire-road friction coefficient 119865119911119894is the normal

force for each wheel and is calculated as

119865

1199111=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199112=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199113=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

119865

1199114=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

(6)

where ℎ denotes the height of vehicle COGFor four-wheel independent drive electric vehicle the

torque balance of the wheel 119894 is

119869

119877

119894= minus119877eff119865119909119894 + 119879119894 (7)

where 119869119877is the moment of inertia of the wheel 119877eff is the

effective radius of the wheel 120596119894and 119879

119894denote the angular

velocity and the wheel torque of the 119894th wheel

3 Observer Design

31 Estimation of Friction Force In the wheel longitudinaldynamic equation (7) the wheel angular velocity 120596

119894is

measurable and the wheel torque 119879119894can be obtained directly

from the motor control system of four-wheel independentdrive electric vehicle So (7) can be described in state spaceform as follows

= 119861119906 + 119875119889 119910 = 119909 (8)

where 119909 = 120596

119894is the system state and 119906 = 119879

119894is the control

input 119889 = 119865

119909119894is unknown and bounded input 119910 is the

measurement output 119861 = 1119869119877 and 119875 = minus119877eff119869119877

Obviously the unknown input 119889 is observable withrespect to the measurement output 119910 in the system (8) Sothe estimation problem of longitudinal friction force canbe described as to reconstruct the unknown input 119889 of thesystem (8) from the measurement output 119910

The slidingmode observer through sliding surface designand equivalent control concept has been proven to be an

4 Mathematical Problems in Engineering

effective approach for handling the systemswith disturbancesand modeling uncertainties Based on the unknown inputestimation technique of the slidingmode observer this paperproposed the following observer for longitudinal frictionforce estimation

_119909 = 119861119906 + 119897 (119910minus

_119909) + 119875120592

120592 = minus119875

minus1120588 sign (

_119909 minus119909)

(9)

where 120588 is the sliding mode gain 119897 denotes the feedback gainand the Luenberger type of feedback loop in the observer isused to ensure the stability of the observer

If we define the estimation error 119909 = 119909minus

_119909 and the

Lyapunov function 119881 = 119909

22 the stability results for error

dynamics of the observer (9) then can be obtained directly Ifthe gains 120588 and 119897 are chosen such that

120588 gt minus119897119909 + 119875119889 (10)

it will have

119881 lt 0 Hence the dynamics 119909 reaches the slidingmode in finite time and stays thereafter

Once the state of the observer (9) converges to the actualstate the longitudinal friction force can be reconstructedaccording to (9) as follows

_119865119909119894

=

_119889asymp (minus119875

minus1120588 sign (

_119909 minus119909))

eq

=

119869

119877

119877eff120588

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(11)

where 120578 is a small positive real number and affects theaccuracy of the unknown input estimation

Actually the longitudinal and lateral friction forces can becalculated directly using theDugoff tiremodel But the calcu-lation of the longitudinal and lateral friction forces needs toknow the tire-road friction coefficient which usually cannotbe measured based on the sensors equipped in modern carsThe tire-road friction coefficient not only depends on theroad conditions (such as asphalt ice and snow etc) but alsois affected by tire materials ambient temperature and otherfactors The relationship between them is very difficult to bedescribed by mathematical model So the estimation of thetire-road friction coefficient is not an easy task [17]

According to the Dugoff tire model (4) and (5) therelationship between the longitudinal and lateral frictionforces can be derived as follows

119865

119910119894=

119862

120572tan120572119894

119862

120590120590

119909119894

119865

119909119894 (12)

Based on this relationship the lateral friction force can becalculated using the estimated longitudinal friction forceand vehicle states directly which did not need the tire-roadfriction coefficient

32 Estimation of Roll Angle According to the vehicle rollmodel shown in Figure 2(b) and (1) the model used in thispaper for roll angle estimation is given by

120601V = 120593

= minus

119862roll120593

119869

119909

minus

119870roll120601V119869

119909

+

119898

119904ℎroll119886119910

119869

119909

(13)

where 120593 denotes the roll rate of the vehicleSince the lateral acceleration and roll rate usually are

measurable in modern cars the observer for roll angleestimation in this paper is proposed as follows

_120601V = 120593 + 119896120601 (120593 minus 120593)

_120601 = minus

119862roll120593

119869

119909

minus

119870roll_120601V

119869

119909

+

119898

119904ℎroll119886119910

119869

119909

+ 119896

120593(120593minus

_120601)

(14)

where 119896120601and 119896120593are the feedback gain and can be determined

by the pole placement method

33 Estimation of Vehicle Velocity According to the vehiclekinematic model (1) and dynamic model (2) the modelused in this paper for vehicle velocity estimation is givenby

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin120601V + 119892 sin120601119903

119903 =

119872

119911

119869

119911

(15)

Since the roll angle of the vehicle and the road bank angleusually are small the assumption sin(120601V+120601119903) = sin120601V+sin120601119903is made in the above equation The obtaining of the roadgrade angle and bank angle is not an easy task based onthe sensors equipped in modern cars The terms 119892 sin 120579

119903and

119892 sin120601119903are considered as unknown and bounded inputs of

the system in this paperIn modern cars the longitudinal acceleration lateral

acceleration and yaw rate usually are measurable Takinginto account the model mismatch of the nonlinear vehicledynamics and the presence of measurement errors suchas biases and noise from the existing sensors equipped invehicle the difference between the calculation value of lon-gitudinal and lateral accelerations based on nonlinear vehicledynamic model and the measurement value provided by thesensor as a feedback term is introduced to improve the esti-mation accuracy of the vehicle velocity And the estimation

Mathematical Problems in Engineering 5

method of vehicle velocity based on the slidingmode observ-er is proposed in this paper as follows

_V119909= 119903

_V119910+ 119886

119909minus 119897

119909(119886

119909minus

_119865119909

119898

) minus 120588

119909sign(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892 sin120601V minus 119897119910(119886119910 minus

_119865119910

119898

)

minus 120588

119910sign(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119897

119903(119903minus

_119903 ) + 120588

119903sign (119903minus

_119903 )

(16)where 119897

119894and 120588

119894 119894 = 119909 119910 119903 are the feedback gain and siding

mode gain of the observer_119865119909

_119865119910

and_119872119911

are estimatedvalues of 119865

119909 119865119910 and 119872

119911 which can be calculated based

on (3) using the estimated longitudinal friction forces andvehicle states The estimated roll angle of the vehicle fromthe observer (14) is considered as an input of the observer(16) The observer defined by (16) copies the structure ofthe Luenberger observer with disturbances replaced by theircorresponding sliding mode terms The main aim is to showthat state estimationwill be totally insensitive to disturbancesif and only if there exists a sliding mode term to track everyunknown input

The structure of the roll angle and vehicle velocityobservers proposed in this paper for four-wheel independentdrive electric vehicle is illustrated in Figure 3 Based on (11)the longitudinal friction force of the four wheels can beobtained using the measured wheel angular velocity andwheel torque and then the lateral friction forces are achievedaccording to (12) which does not need to know the tire-road friction coefficient In other words the calculation ofthe friction forces in the vehicle velocity observer can adaptwith road friction condition At the same time the road gradeangle and bank angle can be constructed together with theestimation of the vehicle velocity

To avoid excessive chattering the sign(sdot) in the observer(16) is replaced by signeq(sdot) which is defined as follows

signeq (119890 120578) =119890

|119890| + 120578

(17)

where 119890 is the estimation error and 120578 is a small positive scalarto adjust the slope of the function signeq(sdot)

In the following the selection of the feedback gains andsliding mode gains to guarantee the stability of the observer(16) should be discussed Define

119896

119909= 119897

119909+

120588

119909

1003816

1003816

1003816

1003816

1003816

1003816

119886

119909minus

_119865119909119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119910= 119897

119910+

120588

119910

1003816

1003816

1003816

1003816

1003816

1003816

119886

119910minus

_119865119910119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119903= 119897

119903+

120588

119903

1003816

1003816

1003816

1003816

1003816

1003816

119903minus

_119903

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(18)

Ti Longitudinalfriction force

observerFxi

120593

Rollangle

observer

ax

rr

120575

ayy

xwi

Vehiclevelocity

observer

120601

(i = 1 4)

Figure 3 The structure of the proposed roll angle and vehiclevelocity observers

Substituting (17) and (18) into (16) the vehicle velocityobserver equations can be rewritten as

_V119909= 119903

_V119910+ 119886

119909minus 119896

119909(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892120601 minus 119896

119910(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119896

119903(119903minus

_119903 )

(19)

Define the estimation errors V119909= V119909minus

_V119909 V119910= V119910minus

_V119910

and 119903 = 119903minus_119903 the error dynamics of the observer (19) is given

by

V119909= 119903V119910+ 119896

119909(119886

119909minus

_119865119909

119898

) + 119906

1

V119910= minus119903V

119909+ 119896

119910(119886

119910minus

_119865119910

119898

) + 119906

2

=

(119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

(20)

with 1199061= minus119892 sin 120579

119903 1199062= 119892 sin120601

119903 The values of 119906

1and 119906

2

are bounded because the road grade angle 120579119903and the road

bank angle 120601119903usually are small according to the real road

condition

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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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

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

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

2 Mathematical Problems in Engineering

estimate vehicle sideslip angle while a linear lateral frictionmodel is used and longitudinal friction forces are treatedas inputs of the observer However the linear relationshiprepresenting lateral friction force is not sufficiently accurateanymore for lateral acceleration above 4ms2 and large tiresideslip angle and longitudinal friction force is not alwaysavailable in modern cars [9] However the researches aboveassume that the road grade angle and bank angle are zeroor are known for accurate estimation of vehicle velocity Theroad angles are nonzero if the road is nonflat and then havesignificant effect in both vehicle dynamics and accelerationmeasurements [10] Amethod for identifying road bank angleand vehicle roll angle separately using measurements fromglobal positioning system and inertial navigation systemsensors is developed in [11] In [12] the lateral tire forcesensors are used to estimate the vehicle sideslip angle androll angle for four-wheel independent drive electric vehicleand high estimation accuracy is achieved But the sensorsmentioned above are expensive for modern cars In [13] theestimation of the road grade and bank angles is discussedbased on vehicle longitudinal and lateral dynamics and theassumption that the angles vary slowly enough compared tothe dynamics of the system the effect of the roll angle has notbeen considered in this paper

In this paper using the measurements from the existingsensors equipped in the four-wheel independent drive elec-tric vehicle including the wheel angular velocities longitudi-nal and lateral accelerations yaw rate roll rate wheel steeringangles and wheel torques a method for vehicle velocity androll angle estimation with road and friction adaptation is pro-posed based on the nonlinear vehicle dynamics and DugofftiremodelThe proposedmethod is evaluated experimentallyunder a variety of maneuvers and road conditions

2 Vehicle Model

As shown in Figures 1 and 2 a vehicle maneuvering androll model is used to design the vehicle velocity and rollangle observers This model has four degrees of freedomfor longitudinal motion lateral motion yaw motion androll motion of the vehicle During handling maneuvers onsmooth roads the vehicle roll motion is primarily inducedby lateral acceleration [14] Based on this assumption a body-fixed coordinate systemwith the origin at the vehicle center ofgravity (COG) is used to set up the model and the kinematicrelationships among the vehicle velocity yaw rate roll angleand acceleration are as follows

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin (120601V + 120601119903)

119869

119909

120601V + 119862roll

120601V + 119870roll120601V = 119898119904119886119910ℎroll

119903 =

119872

119911

119869

119911

(1)

where V119909and V

119910are the longitudinal and lateral velocities

of vehicle COG 119903 is the yaw rate 120579119903and 120601

119903denote the

road grade and bank angles The effect between the road

bR2

bF2

bF

bR

Fx1 Fx2

Fy1x lF

r

y COG

120575

lRFx3 Fx4

Fy3 Fy4

Fy2

Figure 1 Vehicle maneuvering model

grade and bank angle is not considered in this paper 119892 isthe gravitational constant 120601V is the roll angle as a result ofvehicle lateral motion by steering maneuvers 119886

119909and 119886

119910are

the longitudinal and lateral accelerationsmeasurements fromthe sensors attached to the vehicle body and aligned with thebody-fixed coordinate system and they will be different fromthe actual accelerations of the COG when there are nonzeroroad grade angle and bank angle 119869

119909is the vehicle moment

of inertia about longitudinal axis 119862roll is the roll stiffnessand 119870roll is the roll damping 119898

119904is the sprung mass of the

vehicle and ℎroll is the height of the roll center 119869119911 is the vehiclemoment of inertia about vertical axis119872

119911is the torque about

vertical axisDue to the presence of measurement errors such as

biases and noise from the sensors some feedback should beused to make the estimated results converge and then thevehicle dynamics is introduced into the vehicle model (1)According to Figure 1 the force balances in the direction ofthe longitudinal and lateral axis as well as the torque balanceabout the vertical axis are given by

119886

119909=

119865

119909

119898

119886

119910=

119865

119910

119898

119872

119911= 119897

119865((119865

1199091+ 119865

1199092) sin 120575 + (119865

1199101+ 119865

1199102) cos 120575)

minus 119897

119877(119865

1199103+ 119865

1199104)

+

119887

119865((119865

1199101minus 119865

1199102) sin 120575 minus (119865

1199091minus 119865

1199092) cos 120575)

2

minus

119887

119877(119865

1199093minus 119865

1199094)

2

(2)

Mathematical Problems in Engineering 3

X

Z

COG

120579r

(a) Force induced by road grade angle

Y

Z

COG

120601

120601r

(b) Vehicle roll model

Figure 2 Force induced by road grade angle and vehicle roll model

with

119865

119909= (119865

1199091+ 119865

1199092) cos 120575

minus (119865

1199101+ 119865

1199102) sin 120575 + 119865

1199093+ 119865

1199094minus 119862

119909V2119909

119865

119910= (119865

1199091+ 119865

1199092) sin 120575

+ (119865

1199101+ 119865

1199102) cos 120575 + 119865

1199103+ 119865

1199104

(3)

where the effect of wheel rolling resistance forces is ignored119898 denotes the total mass of the vehicle 120575 is the frontwheel steering angle and can be obtained from the measuredsteering wheel angle directly 119862

119909represents the aerodynamic

resistances coefficient 119897119865and 119897119877are the distances from the

vehicle COG to the front and rear axle and 119887119865and 119887119877are the

front and rear track width 119865119909119894and 119865

119910119894are the longitudinal

and lateral friction forces of the 119894th wheel and 119894 is 1 2 3 and4 and represents four wheels respectively

The nonlinearity of vehicle tires will become a criticalfactor during emergency maneuvers in which the linear tiremodel is not sufficiently accurate anymore To account for thenonlinearities of the tire friction force Dugoff tire model [15]is used in this paper and the longitudinal friction force 119865

119909119894

and the lateral friction force 119865119910119894

of the 119894th wheel in (3) aregiven as

119865

119909119894=

119862

120590120590

119894

1 + 120590

119894

119891 (120582

119894)

119865

119910119894=

119862

120572tan120572119894

1 + 120590

119894

119891 (120582

119894)

(4)

where 119862120590and 119862

120572are the longitudinal and cornering stiffness

of the tire 120590119894and 120572

119894denote the tire longitudinal slip and

the tire sideslip angle of the 119894th wheel The definition andcalculation method of these variables are shown in [16] Thevariable 120582

119894and the function 119891(120582

119894) are defined as

120582

119894=

120583119865

119911119894(1 + 120590

119894)

2

radic

(119862

120590120590

119894)

2

+ (119862

120572tan120572119894)

2

119891 (120582

119894) =

1 120582

119894ge 1

(2 minus 120582

119894) 120582

119894120582

119894lt 1

(5)

where 120583 is the tire-road friction coefficient 119865119911119894is the normal

force for each wheel and is calculated as

119865

1199111=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199112=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199113=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

119865

1199114=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

(6)

where ℎ denotes the height of vehicle COGFor four-wheel independent drive electric vehicle the

torque balance of the wheel 119894 is

119869

119877

119894= minus119877eff119865119909119894 + 119879119894 (7)

where 119869119877is the moment of inertia of the wheel 119877eff is the

effective radius of the wheel 120596119894and 119879

119894denote the angular

velocity and the wheel torque of the 119894th wheel

3 Observer Design

31 Estimation of Friction Force In the wheel longitudinaldynamic equation (7) the wheel angular velocity 120596

119894is

measurable and the wheel torque 119879119894can be obtained directly

from the motor control system of four-wheel independentdrive electric vehicle So (7) can be described in state spaceform as follows

= 119861119906 + 119875119889 119910 = 119909 (8)

where 119909 = 120596

119894is the system state and 119906 = 119879

119894is the control

input 119889 = 119865

119909119894is unknown and bounded input 119910 is the

measurement output 119861 = 1119869119877 and 119875 = minus119877eff119869119877

Obviously the unknown input 119889 is observable withrespect to the measurement output 119910 in the system (8) Sothe estimation problem of longitudinal friction force canbe described as to reconstruct the unknown input 119889 of thesystem (8) from the measurement output 119910

The slidingmode observer through sliding surface designand equivalent control concept has been proven to be an

4 Mathematical Problems in Engineering

effective approach for handling the systemswith disturbancesand modeling uncertainties Based on the unknown inputestimation technique of the slidingmode observer this paperproposed the following observer for longitudinal frictionforce estimation

_119909 = 119861119906 + 119897 (119910minus

_119909) + 119875120592

120592 = minus119875

minus1120588 sign (

_119909 minus119909)

(9)

where 120588 is the sliding mode gain 119897 denotes the feedback gainand the Luenberger type of feedback loop in the observer isused to ensure the stability of the observer

If we define the estimation error 119909 = 119909minus

_119909 and the

Lyapunov function 119881 = 119909

22 the stability results for error

dynamics of the observer (9) then can be obtained directly Ifthe gains 120588 and 119897 are chosen such that

120588 gt minus119897119909 + 119875119889 (10)

it will have

119881 lt 0 Hence the dynamics 119909 reaches the slidingmode in finite time and stays thereafter

Once the state of the observer (9) converges to the actualstate the longitudinal friction force can be reconstructedaccording to (9) as follows

_119865119909119894

=

_119889asymp (minus119875

minus1120588 sign (

_119909 minus119909))

eq

=

119869

119877

119877eff120588

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(11)

where 120578 is a small positive real number and affects theaccuracy of the unknown input estimation

Actually the longitudinal and lateral friction forces can becalculated directly using theDugoff tiremodel But the calcu-lation of the longitudinal and lateral friction forces needs toknow the tire-road friction coefficient which usually cannotbe measured based on the sensors equipped in modern carsThe tire-road friction coefficient not only depends on theroad conditions (such as asphalt ice and snow etc) but alsois affected by tire materials ambient temperature and otherfactors The relationship between them is very difficult to bedescribed by mathematical model So the estimation of thetire-road friction coefficient is not an easy task [17]

According to the Dugoff tire model (4) and (5) therelationship between the longitudinal and lateral frictionforces can be derived as follows

119865

119910119894=

119862

120572tan120572119894

119862

120590120590

119909119894

119865

119909119894 (12)

Based on this relationship the lateral friction force can becalculated using the estimated longitudinal friction forceand vehicle states directly which did not need the tire-roadfriction coefficient

32 Estimation of Roll Angle According to the vehicle rollmodel shown in Figure 2(b) and (1) the model used in thispaper for roll angle estimation is given by

120601V = 120593

= minus

119862roll120593

119869

119909

minus

119870roll120601V119869

119909

+

119898

119904ℎroll119886119910

119869

119909

(13)

where 120593 denotes the roll rate of the vehicleSince the lateral acceleration and roll rate usually are

measurable in modern cars the observer for roll angleestimation in this paper is proposed as follows

_120601V = 120593 + 119896120601 (120593 minus 120593)

_120601 = minus

119862roll120593

119869

119909

minus

119870roll_120601V

119869

119909

+

119898

119904ℎroll119886119910

119869

119909

+ 119896

120593(120593minus

_120601)

(14)

where 119896120601and 119896120593are the feedback gain and can be determined

by the pole placement method

33 Estimation of Vehicle Velocity According to the vehiclekinematic model (1) and dynamic model (2) the modelused in this paper for vehicle velocity estimation is givenby

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin120601V + 119892 sin120601119903

119903 =

119872

119911

119869

119911

(15)

Since the roll angle of the vehicle and the road bank angleusually are small the assumption sin(120601V+120601119903) = sin120601V+sin120601119903is made in the above equation The obtaining of the roadgrade angle and bank angle is not an easy task based onthe sensors equipped in modern cars The terms 119892 sin 120579

119903and

119892 sin120601119903are considered as unknown and bounded inputs of

the system in this paperIn modern cars the longitudinal acceleration lateral

acceleration and yaw rate usually are measurable Takinginto account the model mismatch of the nonlinear vehicledynamics and the presence of measurement errors suchas biases and noise from the existing sensors equipped invehicle the difference between the calculation value of lon-gitudinal and lateral accelerations based on nonlinear vehicledynamic model and the measurement value provided by thesensor as a feedback term is introduced to improve the esti-mation accuracy of the vehicle velocity And the estimation

Mathematical Problems in Engineering 5

method of vehicle velocity based on the slidingmode observ-er is proposed in this paper as follows

_V119909= 119903

_V119910+ 119886

119909minus 119897

119909(119886

119909minus

_119865119909

119898

) minus 120588

119909sign(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892 sin120601V minus 119897119910(119886119910 minus

_119865119910

119898

)

minus 120588

119910sign(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119897

119903(119903minus

_119903 ) + 120588

119903sign (119903minus

_119903 )

(16)where 119897

119894and 120588

119894 119894 = 119909 119910 119903 are the feedback gain and siding

mode gain of the observer_119865119909

_119865119910

and_119872119911

are estimatedvalues of 119865

119909 119865119910 and 119872

119911 which can be calculated based

on (3) using the estimated longitudinal friction forces andvehicle states The estimated roll angle of the vehicle fromthe observer (14) is considered as an input of the observer(16) The observer defined by (16) copies the structure ofthe Luenberger observer with disturbances replaced by theircorresponding sliding mode terms The main aim is to showthat state estimationwill be totally insensitive to disturbancesif and only if there exists a sliding mode term to track everyunknown input

The structure of the roll angle and vehicle velocityobservers proposed in this paper for four-wheel independentdrive electric vehicle is illustrated in Figure 3 Based on (11)the longitudinal friction force of the four wheels can beobtained using the measured wheel angular velocity andwheel torque and then the lateral friction forces are achievedaccording to (12) which does not need to know the tire-road friction coefficient In other words the calculation ofthe friction forces in the vehicle velocity observer can adaptwith road friction condition At the same time the road gradeangle and bank angle can be constructed together with theestimation of the vehicle velocity

To avoid excessive chattering the sign(sdot) in the observer(16) is replaced by signeq(sdot) which is defined as follows

signeq (119890 120578) =119890

|119890| + 120578

(17)

where 119890 is the estimation error and 120578 is a small positive scalarto adjust the slope of the function signeq(sdot)

In the following the selection of the feedback gains andsliding mode gains to guarantee the stability of the observer(16) should be discussed Define

119896

119909= 119897

119909+

120588

119909

1003816

1003816

1003816

1003816

1003816

1003816

119886

119909minus

_119865119909119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119910= 119897

119910+

120588

119910

1003816

1003816

1003816

1003816

1003816

1003816

119886

119910minus

_119865119910119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119903= 119897

119903+

120588

119903

1003816

1003816

1003816

1003816

1003816

1003816

119903minus

_119903

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(18)

Ti Longitudinalfriction force

observerFxi

120593

Rollangle

observer

ax

rr

120575

ayy

xwi

Vehiclevelocity

observer

120601

(i = 1 4)

Figure 3 The structure of the proposed roll angle and vehiclevelocity observers

Substituting (17) and (18) into (16) the vehicle velocityobserver equations can be rewritten as

_V119909= 119903

_V119910+ 119886

119909minus 119896

119909(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892120601 minus 119896

119910(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119896

119903(119903minus

_119903 )

(19)

Define the estimation errors V119909= V119909minus

_V119909 V119910= V119910minus

_V119910

and 119903 = 119903minus_119903 the error dynamics of the observer (19) is given

by

V119909= 119903V119910+ 119896

119909(119886

119909minus

_119865119909

119898

) + 119906

1

V119910= minus119903V

119909+ 119896

119910(119886

119910minus

_119865119910

119898

) + 119906

2

=

(119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

(20)

with 1199061= minus119892 sin 120579

119903 1199062= 119892 sin120601

119903 The values of 119906

1and 119906

2

are bounded because the road grade angle 120579119903and the road

bank angle 120601119903usually are small according to the real road

condition

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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 3

X

Z

COG

120579r

(a) Force induced by road grade angle

Y

Z

COG

120601

120601r

(b) Vehicle roll model

Figure 2 Force induced by road grade angle and vehicle roll model

with

119865

119909= (119865

1199091+ 119865

1199092) cos 120575

minus (119865

1199101+ 119865

1199102) sin 120575 + 119865

1199093+ 119865

1199094minus 119862

119909V2119909

119865

119910= (119865

1199091+ 119865

1199092) sin 120575

+ (119865

1199101+ 119865

1199102) cos 120575 + 119865

1199103+ 119865

1199104

(3)

where the effect of wheel rolling resistance forces is ignored119898 denotes the total mass of the vehicle 120575 is the frontwheel steering angle and can be obtained from the measuredsteering wheel angle directly 119862

119909represents the aerodynamic

resistances coefficient 119897119865and 119897119877are the distances from the

vehicle COG to the front and rear axle and 119887119865and 119887119877are the

front and rear track width 119865119909119894and 119865

119910119894are the longitudinal

and lateral friction forces of the 119894th wheel and 119894 is 1 2 3 and4 and represents four wheels respectively

The nonlinearity of vehicle tires will become a criticalfactor during emergency maneuvers in which the linear tiremodel is not sufficiently accurate anymore To account for thenonlinearities of the tire friction force Dugoff tire model [15]is used in this paper and the longitudinal friction force 119865

119909119894

and the lateral friction force 119865119910119894

of the 119894th wheel in (3) aregiven as

119865

119909119894=

119862

120590120590

119894

1 + 120590

119894

119891 (120582

119894)

119865

119910119894=

119862

120572tan120572119894

1 + 120590

119894

119891 (120582

119894)

(4)

where 119862120590and 119862

120572are the longitudinal and cornering stiffness

of the tire 120590119894and 120572

119894denote the tire longitudinal slip and

the tire sideslip angle of the 119894th wheel The definition andcalculation method of these variables are shown in [16] Thevariable 120582

119894and the function 119891(120582

119894) are defined as

120582

119894=

120583119865

119911119894(1 + 120590

119894)

2

radic

(119862

120590120590

119894)

2

+ (119862

120572tan120572119894)

2

119891 (120582

119894) =

1 120582

119894ge 1

(2 minus 120582

119894) 120582

119894120582

119894lt 1

(5)

where 120583 is the tire-road friction coefficient 119865119911119894is the normal

force for each wheel and is calculated as

119865

1199111=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199112=

119898119892119897

119877minus 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119877ℎ

(119897

119865+ 119897

119877) 119887

119865

119865

1199113=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

minus

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

119865

1199114=

119898119892119897

119865+ 119898119886

119909ℎ

2 (119897

119865+ 119897

119877)

+

119898119886

119910119897

119865ℎ

(119897

119865+ 119897

119877) 119887

119877

(6)

where ℎ denotes the height of vehicle COGFor four-wheel independent drive electric vehicle the

torque balance of the wheel 119894 is

119869

119877

119894= minus119877eff119865119909119894 + 119879119894 (7)

where 119869119877is the moment of inertia of the wheel 119877eff is the

effective radius of the wheel 120596119894and 119879

119894denote the angular

velocity and the wheel torque of the 119894th wheel

3 Observer Design

31 Estimation of Friction Force In the wheel longitudinaldynamic equation (7) the wheel angular velocity 120596

119894is

measurable and the wheel torque 119879119894can be obtained directly

from the motor control system of four-wheel independentdrive electric vehicle So (7) can be described in state spaceform as follows

= 119861119906 + 119875119889 119910 = 119909 (8)

where 119909 = 120596

119894is the system state and 119906 = 119879

119894is the control

input 119889 = 119865

119909119894is unknown and bounded input 119910 is the

measurement output 119861 = 1119869119877 and 119875 = minus119877eff119869119877

Obviously the unknown input 119889 is observable withrespect to the measurement output 119910 in the system (8) Sothe estimation problem of longitudinal friction force canbe described as to reconstruct the unknown input 119889 of thesystem (8) from the measurement output 119910

The slidingmode observer through sliding surface designand equivalent control concept has been proven to be an

4 Mathematical Problems in Engineering

effective approach for handling the systemswith disturbancesand modeling uncertainties Based on the unknown inputestimation technique of the slidingmode observer this paperproposed the following observer for longitudinal frictionforce estimation

_119909 = 119861119906 + 119897 (119910minus

_119909) + 119875120592

120592 = minus119875

minus1120588 sign (

_119909 minus119909)

(9)

where 120588 is the sliding mode gain 119897 denotes the feedback gainand the Luenberger type of feedback loop in the observer isused to ensure the stability of the observer

If we define the estimation error 119909 = 119909minus

_119909 and the

Lyapunov function 119881 = 119909

22 the stability results for error

dynamics of the observer (9) then can be obtained directly Ifthe gains 120588 and 119897 are chosen such that

120588 gt minus119897119909 + 119875119889 (10)

it will have

119881 lt 0 Hence the dynamics 119909 reaches the slidingmode in finite time and stays thereafter

Once the state of the observer (9) converges to the actualstate the longitudinal friction force can be reconstructedaccording to (9) as follows

_119865119909119894

=

_119889asymp (minus119875

minus1120588 sign (

_119909 minus119909))

eq

=

119869

119877

119877eff120588

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(11)

where 120578 is a small positive real number and affects theaccuracy of the unknown input estimation

Actually the longitudinal and lateral friction forces can becalculated directly using theDugoff tiremodel But the calcu-lation of the longitudinal and lateral friction forces needs toknow the tire-road friction coefficient which usually cannotbe measured based on the sensors equipped in modern carsThe tire-road friction coefficient not only depends on theroad conditions (such as asphalt ice and snow etc) but alsois affected by tire materials ambient temperature and otherfactors The relationship between them is very difficult to bedescribed by mathematical model So the estimation of thetire-road friction coefficient is not an easy task [17]

According to the Dugoff tire model (4) and (5) therelationship between the longitudinal and lateral frictionforces can be derived as follows

119865

119910119894=

119862

120572tan120572119894

119862

120590120590

119909119894

119865

119909119894 (12)

Based on this relationship the lateral friction force can becalculated using the estimated longitudinal friction forceand vehicle states directly which did not need the tire-roadfriction coefficient

32 Estimation of Roll Angle According to the vehicle rollmodel shown in Figure 2(b) and (1) the model used in thispaper for roll angle estimation is given by

120601V = 120593

= minus

119862roll120593

119869

119909

minus

119870roll120601V119869

119909

+

119898

119904ℎroll119886119910

119869

119909

(13)

where 120593 denotes the roll rate of the vehicleSince the lateral acceleration and roll rate usually are

measurable in modern cars the observer for roll angleestimation in this paper is proposed as follows

_120601V = 120593 + 119896120601 (120593 minus 120593)

_120601 = minus

119862roll120593

119869

119909

minus

119870roll_120601V

119869

119909

+

119898

119904ℎroll119886119910

119869

119909

+ 119896

120593(120593minus

_120601)

(14)

where 119896120601and 119896120593are the feedback gain and can be determined

by the pole placement method

33 Estimation of Vehicle Velocity According to the vehiclekinematic model (1) and dynamic model (2) the modelused in this paper for vehicle velocity estimation is givenby

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin120601V + 119892 sin120601119903

119903 =

119872

119911

119869

119911

(15)

Since the roll angle of the vehicle and the road bank angleusually are small the assumption sin(120601V+120601119903) = sin120601V+sin120601119903is made in the above equation The obtaining of the roadgrade angle and bank angle is not an easy task based onthe sensors equipped in modern cars The terms 119892 sin 120579

119903and

119892 sin120601119903are considered as unknown and bounded inputs of

the system in this paperIn modern cars the longitudinal acceleration lateral

acceleration and yaw rate usually are measurable Takinginto account the model mismatch of the nonlinear vehicledynamics and the presence of measurement errors suchas biases and noise from the existing sensors equipped invehicle the difference between the calculation value of lon-gitudinal and lateral accelerations based on nonlinear vehicledynamic model and the measurement value provided by thesensor as a feedback term is introduced to improve the esti-mation accuracy of the vehicle velocity And the estimation

Mathematical Problems in Engineering 5

method of vehicle velocity based on the slidingmode observ-er is proposed in this paper as follows

_V119909= 119903

_V119910+ 119886

119909minus 119897

119909(119886

119909minus

_119865119909

119898

) minus 120588

119909sign(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892 sin120601V minus 119897119910(119886119910 minus

_119865119910

119898

)

minus 120588

119910sign(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119897

119903(119903minus

_119903 ) + 120588

119903sign (119903minus

_119903 )

(16)where 119897

119894and 120588

119894 119894 = 119909 119910 119903 are the feedback gain and siding

mode gain of the observer_119865119909

_119865119910

and_119872119911

are estimatedvalues of 119865

119909 119865119910 and 119872

119911 which can be calculated based

on (3) using the estimated longitudinal friction forces andvehicle states The estimated roll angle of the vehicle fromthe observer (14) is considered as an input of the observer(16) The observer defined by (16) copies the structure ofthe Luenberger observer with disturbances replaced by theircorresponding sliding mode terms The main aim is to showthat state estimationwill be totally insensitive to disturbancesif and only if there exists a sliding mode term to track everyunknown input

The structure of the roll angle and vehicle velocityobservers proposed in this paper for four-wheel independentdrive electric vehicle is illustrated in Figure 3 Based on (11)the longitudinal friction force of the four wheels can beobtained using the measured wheel angular velocity andwheel torque and then the lateral friction forces are achievedaccording to (12) which does not need to know the tire-road friction coefficient In other words the calculation ofthe friction forces in the vehicle velocity observer can adaptwith road friction condition At the same time the road gradeangle and bank angle can be constructed together with theestimation of the vehicle velocity

To avoid excessive chattering the sign(sdot) in the observer(16) is replaced by signeq(sdot) which is defined as follows

signeq (119890 120578) =119890

|119890| + 120578

(17)

where 119890 is the estimation error and 120578 is a small positive scalarto adjust the slope of the function signeq(sdot)

In the following the selection of the feedback gains andsliding mode gains to guarantee the stability of the observer(16) should be discussed Define

119896

119909= 119897

119909+

120588

119909

1003816

1003816

1003816

1003816

1003816

1003816

119886

119909minus

_119865119909119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119910= 119897

119910+

120588

119910

1003816

1003816

1003816

1003816

1003816

1003816

119886

119910minus

_119865119910119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119903= 119897

119903+

120588

119903

1003816

1003816

1003816

1003816

1003816

1003816

119903minus

_119903

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(18)

Ti Longitudinalfriction force

observerFxi

120593

Rollangle

observer

ax

rr

120575

ayy

xwi

Vehiclevelocity

observer

120601

(i = 1 4)

Figure 3 The structure of the proposed roll angle and vehiclevelocity observers

Substituting (17) and (18) into (16) the vehicle velocityobserver equations can be rewritten as

_V119909= 119903

_V119910+ 119886

119909minus 119896

119909(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892120601 minus 119896

119910(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119896

119903(119903minus

_119903 )

(19)

Define the estimation errors V119909= V119909minus

_V119909 V119910= V119910minus

_V119910

and 119903 = 119903minus_119903 the error dynamics of the observer (19) is given

by

V119909= 119903V119910+ 119896

119909(119886

119909minus

_119865119909

119898

) + 119906

1

V119910= minus119903V

119909+ 119896

119910(119886

119910minus

_119865119910

119898

) + 119906

2

=

(119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

(20)

with 1199061= minus119892 sin 120579

119903 1199062= 119892 sin120601

119903 The values of 119906

1and 119906

2

are bounded because the road grade angle 120579119903and the road

bank angle 120601119903usually are small according to the real road

condition

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

effective approach for handling the systemswith disturbancesand modeling uncertainties Based on the unknown inputestimation technique of the slidingmode observer this paperproposed the following observer for longitudinal frictionforce estimation

_119909 = 119861119906 + 119897 (119910minus

_119909) + 119875120592

120592 = minus119875

minus1120588 sign (

_119909 minus119909)

(9)

where 120588 is the sliding mode gain 119897 denotes the feedback gainand the Luenberger type of feedback loop in the observer isused to ensure the stability of the observer

If we define the estimation error 119909 = 119909minus

_119909 and the

Lyapunov function 119881 = 119909

22 the stability results for error

dynamics of the observer (9) then can be obtained directly Ifthe gains 120588 and 119897 are chosen such that

120588 gt minus119897119909 + 119875119889 (10)

it will have

119881 lt 0 Hence the dynamics 119909 reaches the slidingmode in finite time and stays thereafter

Once the state of the observer (9) converges to the actualstate the longitudinal friction force can be reconstructedaccording to (9) as follows

_119865119909119894

=

_119889asymp (minus119875

minus1120588 sign (

_119909 minus119909))

eq

=

119869

119877

119877eff120588

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

_120596

119894minus 120596

119894

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(11)

where 120578 is a small positive real number and affects theaccuracy of the unknown input estimation

Actually the longitudinal and lateral friction forces can becalculated directly using theDugoff tiremodel But the calcu-lation of the longitudinal and lateral friction forces needs toknow the tire-road friction coefficient which usually cannotbe measured based on the sensors equipped in modern carsThe tire-road friction coefficient not only depends on theroad conditions (such as asphalt ice and snow etc) but alsois affected by tire materials ambient temperature and otherfactors The relationship between them is very difficult to bedescribed by mathematical model So the estimation of thetire-road friction coefficient is not an easy task [17]

According to the Dugoff tire model (4) and (5) therelationship between the longitudinal and lateral frictionforces can be derived as follows

119865

119910119894=

119862

120572tan120572119894

119862

120590120590

119909119894

119865

119909119894 (12)

Based on this relationship the lateral friction force can becalculated using the estimated longitudinal friction forceand vehicle states directly which did not need the tire-roadfriction coefficient

32 Estimation of Roll Angle According to the vehicle rollmodel shown in Figure 2(b) and (1) the model used in thispaper for roll angle estimation is given by

120601V = 120593

= minus

119862roll120593

119869

119909

minus

119870roll120601V119869

119909

+

119898

119904ℎroll119886119910

119869

119909

(13)

where 120593 denotes the roll rate of the vehicleSince the lateral acceleration and roll rate usually are

measurable in modern cars the observer for roll angleestimation in this paper is proposed as follows

_120601V = 120593 + 119896120601 (120593 minus 120593)

_120601 = minus

119862roll120593

119869

119909

minus

119870roll_120601V

119869

119909

+

119898

119904ℎroll119886119910

119869

119909

+ 119896

120593(120593minus

_120601)

(14)

where 119896120601and 119896120593are the feedback gain and can be determined

by the pole placement method

33 Estimation of Vehicle Velocity According to the vehiclekinematic model (1) and dynamic model (2) the modelused in this paper for vehicle velocity estimation is givenby

V119909= 119903V119910+ 119886

119909minus 119892 sin 120579

119903

V119910= 119886

119910minus 119903V119909+ 119892 sin120601V + 119892 sin120601119903

119903 =

119872

119911

119869

119911

(15)

Since the roll angle of the vehicle and the road bank angleusually are small the assumption sin(120601V+120601119903) = sin120601V+sin120601119903is made in the above equation The obtaining of the roadgrade angle and bank angle is not an easy task based onthe sensors equipped in modern cars The terms 119892 sin 120579

119903and

119892 sin120601119903are considered as unknown and bounded inputs of

the system in this paperIn modern cars the longitudinal acceleration lateral

acceleration and yaw rate usually are measurable Takinginto account the model mismatch of the nonlinear vehicledynamics and the presence of measurement errors suchas biases and noise from the existing sensors equipped invehicle the difference between the calculation value of lon-gitudinal and lateral accelerations based on nonlinear vehicledynamic model and the measurement value provided by thesensor as a feedback term is introduced to improve the esti-mation accuracy of the vehicle velocity And the estimation

Mathematical Problems in Engineering 5

method of vehicle velocity based on the slidingmode observ-er is proposed in this paper as follows

_V119909= 119903

_V119910+ 119886

119909minus 119897

119909(119886

119909minus

_119865119909

119898

) minus 120588

119909sign(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892 sin120601V minus 119897119910(119886119910 minus

_119865119910

119898

)

minus 120588

119910sign(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119897

119903(119903minus

_119903 ) + 120588

119903sign (119903minus

_119903 )

(16)where 119897

119894and 120588

119894 119894 = 119909 119910 119903 are the feedback gain and siding

mode gain of the observer_119865119909

_119865119910

and_119872119911

are estimatedvalues of 119865

119909 119865119910 and 119872

119911 which can be calculated based

on (3) using the estimated longitudinal friction forces andvehicle states The estimated roll angle of the vehicle fromthe observer (14) is considered as an input of the observer(16) The observer defined by (16) copies the structure ofthe Luenberger observer with disturbances replaced by theircorresponding sliding mode terms The main aim is to showthat state estimationwill be totally insensitive to disturbancesif and only if there exists a sliding mode term to track everyunknown input

The structure of the roll angle and vehicle velocityobservers proposed in this paper for four-wheel independentdrive electric vehicle is illustrated in Figure 3 Based on (11)the longitudinal friction force of the four wheels can beobtained using the measured wheel angular velocity andwheel torque and then the lateral friction forces are achievedaccording to (12) which does not need to know the tire-road friction coefficient In other words the calculation ofthe friction forces in the vehicle velocity observer can adaptwith road friction condition At the same time the road gradeangle and bank angle can be constructed together with theestimation of the vehicle velocity

To avoid excessive chattering the sign(sdot) in the observer(16) is replaced by signeq(sdot) which is defined as follows

signeq (119890 120578) =119890

|119890| + 120578

(17)

where 119890 is the estimation error and 120578 is a small positive scalarto adjust the slope of the function signeq(sdot)

In the following the selection of the feedback gains andsliding mode gains to guarantee the stability of the observer(16) should be discussed Define

119896

119909= 119897

119909+

120588

119909

1003816

1003816

1003816

1003816

1003816

1003816

119886

119909minus

_119865119909119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119910= 119897

119910+

120588

119910

1003816

1003816

1003816

1003816

1003816

1003816

119886

119910minus

_119865119910119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119903= 119897

119903+

120588

119903

1003816

1003816

1003816

1003816

1003816

1003816

119903minus

_119903

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(18)

Ti Longitudinalfriction force

observerFxi

120593

Rollangle

observer

ax

rr

120575

ayy

xwi

Vehiclevelocity

observer

120601

(i = 1 4)

Figure 3 The structure of the proposed roll angle and vehiclevelocity observers

Substituting (17) and (18) into (16) the vehicle velocityobserver equations can be rewritten as

_V119909= 119903

_V119910+ 119886

119909minus 119896

119909(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892120601 minus 119896

119910(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119896

119903(119903minus

_119903 )

(19)

Define the estimation errors V119909= V119909minus

_V119909 V119910= V119910minus

_V119910

and 119903 = 119903minus_119903 the error dynamics of the observer (19) is given

by

V119909= 119903V119910+ 119896

119909(119886

119909minus

_119865119909

119898

) + 119906

1

V119910= minus119903V

119909+ 119896

119910(119886

119910minus

_119865119910

119898

) + 119906

2

=

(119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

(20)

with 1199061= minus119892 sin 120579

119903 1199062= 119892 sin120601

119903 The values of 119906

1and 119906

2

are bounded because the road grade angle 120579119903and the road

bank angle 120601119903usually are small according to the real road

condition

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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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

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 5

method of vehicle velocity based on the slidingmode observ-er is proposed in this paper as follows

_V119909= 119903

_V119910+ 119886

119909minus 119897

119909(119886

119909minus

_119865119909

119898

) minus 120588

119909sign(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892 sin120601V minus 119897119910(119886119910 minus

_119865119910

119898

)

minus 120588

119910sign(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119897

119903(119903minus

_119903 ) + 120588

119903sign (119903minus

_119903 )

(16)where 119897

119894and 120588

119894 119894 = 119909 119910 119903 are the feedback gain and siding

mode gain of the observer_119865119909

_119865119910

and_119872119911

are estimatedvalues of 119865

119909 119865119910 and 119872

119911 which can be calculated based

on (3) using the estimated longitudinal friction forces andvehicle states The estimated roll angle of the vehicle fromthe observer (14) is considered as an input of the observer(16) The observer defined by (16) copies the structure ofthe Luenberger observer with disturbances replaced by theircorresponding sliding mode terms The main aim is to showthat state estimationwill be totally insensitive to disturbancesif and only if there exists a sliding mode term to track everyunknown input

The structure of the roll angle and vehicle velocityobservers proposed in this paper for four-wheel independentdrive electric vehicle is illustrated in Figure 3 Based on (11)the longitudinal friction force of the four wheels can beobtained using the measured wheel angular velocity andwheel torque and then the lateral friction forces are achievedaccording to (12) which does not need to know the tire-road friction coefficient In other words the calculation ofthe friction forces in the vehicle velocity observer can adaptwith road friction condition At the same time the road gradeangle and bank angle can be constructed together with theestimation of the vehicle velocity

To avoid excessive chattering the sign(sdot) in the observer(16) is replaced by signeq(sdot) which is defined as follows

signeq (119890 120578) =119890

|119890| + 120578

(17)

where 119890 is the estimation error and 120578 is a small positive scalarto adjust the slope of the function signeq(sdot)

In the following the selection of the feedback gains andsliding mode gains to guarantee the stability of the observer(16) should be discussed Define

119896

119909= 119897

119909+

120588

119909

1003816

1003816

1003816

1003816

1003816

1003816

119886

119909minus

_119865119909119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119910= 119897

119910+

120588

119910

1003816

1003816

1003816

1003816

1003816

1003816

119886

119910minus

_119865119910119898

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

119896

119903= 119897

119903+

120588

119903

1003816

1003816

1003816

1003816

1003816

1003816

119903minus

_119903

1003816

1003816

1003816

1003816

1003816

1003816

+ 120578

(18)

Ti Longitudinalfriction force

observerFxi

120593

Rollangle

observer

ax

rr

120575

ayy

xwi

Vehiclevelocity

observer

120601

(i = 1 4)

Figure 3 The structure of the proposed roll angle and vehiclevelocity observers

Substituting (17) and (18) into (16) the vehicle velocityobserver equations can be rewritten as

_V119909= 119903

_V119910+ 119886

119909minus 119896

119909(119886

119909minus

_119865119909

119898

)

_V119910= minus119903

_V119909+ 119886

119910+ 119892120601 minus 119896

119910(119886

119910minus

_119865119910

119898

)

_119903 =

_119872119911

119869

119911

+ 119896

119903(119903minus

_119903 )

(19)

Define the estimation errors V119909= V119909minus

_V119909 V119910= V119910minus

_V119910

and 119903 = 119903minus_119903 the error dynamics of the observer (19) is given

by

V119909= 119903V119910+ 119896

119909(119886

119909minus

_119865119909

119898

) + 119906

1

V119910= minus119903V

119909+ 119896

119910(119886

119910minus

_119865119910

119898

) + 119906

2

=

(119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

(20)

with 1199061= minus119892 sin 120579

119903 1199062= 119892 sin120601

119903 The values of 119906

1and 119906

2

are bounded because the road grade angle 120579119903and the road

bank angle 120601119903usually are small according to the real road

condition

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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

6 Mathematical Problems in Engineering

Vehicleand wheeldynamic

model

Vehiclevelocityand roll

angleobserver

External IO

io task

veDYNA Modeldata

Vehicle trace dataMD trafo

Animation data

Trace data

vdy mdl CPT MDL

CPT IO

Standardanimation data

TESIS DYNAware GmbH

0

120593

r

ax

ay

T

120575

wi

ax

ay

Mz

x

y

r

simulationmodel

veDYNA 392

Control deskcontrol

Triggeredvehicle animation

PulseVehicle animationdata

Figure 4 The structure of the simulation system

Define the Lyapunov function

119881 =

1

2

V2119909+

1

2

V2119910+

1

2

119903

2 (21)

its time derivative along the trajectories of (18) is given by

119881 = 119896

119909V119909(119886

119909minus

_119865119909

119898

) + 119906

1V119909+ 119896

119910V119910(119886

119910minus

_119865119910

119898

)

+ 119906

2V119909+

119903 (119872

119911minus

_119872119911

)

119869

119885

minus 119896

119903119903

2

(22)

According to the analysis method in the authorsrsquo previouspublished literature [16] the following results can be obtaineddirectly

119881 le minus|x|119879119860 |x| + x u (23)

where |x| = (|V119909| |V119910| |119903|)

119879 u = (119906

1 119906

2)

119879 and the matrix 119860is defined as

119860 =

[

[

[

[

[

[

[

119896

119909119888

1minus

119896

119909119888

2+ 119896

119910119888

4

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119909119888

2+ 119896

119910119888

4

2

119896

119910119888

5minus

119896

119910119888

6+ 119888

8

2

minus

119896

119909119888

3+ 119888

7

2

minus

119896

119910119888

6+ 119888

8

2

119896

119903minus 119888

9

]

]

]

]

]

]

]

(24)

and 119888119894 119894 = 1 2 9 is positive constants So if the observer

gains 119896119895 119895 = 119909 119910 119903 are chosen as

119896

1gt 0

2119888

1119888

5minus 119888

2119888

4minus 119888

10

119888

2

4

119896

1lt 119896

2lt

2119888

1119888

5minus 119888

2119888

4+ 119888

10

119888

2

4

119896

1

119896

3gt 119888

9+

119888

11

4119896

1119896

2119888

1119888

5minus (119896

1119888

2+ 119896

2119888

4)

2

(25)

with

119888

10=radic4119888

1119888

5(119888

1119888

5minus 119888

2119888

4)

119888

11= 119896

1119888

1(119896

2119888

6+ 119888

8)

2

+ 119896

2119888

5(119896

1119888

3+ 119888

7)

2

+ (119896

1119888

2+ 119896

2119888

4) (119896

1119888

3+ 119888

7) (119896

2119888

6+ 119888

8)

(26)

the matrix 119860 is positive definite and then

119881 le minus120582min (119860) x2+ x u

le minus120582min (119860) (1 minus 120579) x2

forall x ge u120582min (119860) 120579

(27)

where 0 lt 120579 lt 1 and 120582min(119860) denotes the minimumeigenvalue of the matrix 119860

From the above discussion it is clear that if the road is flatthe road grade angle and bank angle are zero and u = 0 theinequality (23) can be rewritten as

119881 le minus|x|119879119860|x| then theobserver error dynamics (20) is asymptotically stable For u =

0 the inequality (23) clearly implies that the observer errordynamics (20) is input-to-state stability (ISS) with respect tou

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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 7

0 5 10 15 20 25 30

0

1

2

t (s)

Stee

ring

whe

el an

gle (

rad)

minus1

minus20 5 10 15 20 25 30

0

1000

2000

3000

FL wheelFR wheel

RL wheelRR wheel

Whe

el to

rque

(Nmiddotm

)

minus2000

minus1000

t (s)

Figure 5 The steering wheel angle and wheel torque in the sudden steering maneuver

0 5 10 15 20 25 30

0

01

02

Road

gra

de an

gle (

rad)

minus02

minus01

t (s)0 5 10 15 20 25 30

0

01

Road

ban

k an

gle (

rad)

minus01

t (s)

Figure 6 The road grade angle and bank angle in the sudden steering maneuver

4 Simulation Results

As shown in Figure 4 by modifying some related part basedon the existing model in veDYNA a high-precision vehicledynamics simulation system for four-wheel independentdrive electric vehicle has been built in this paper TheveDYNA simulator developed by the group of companiesTESIS is the software which provides an integrated develop-ment environment for quickly conducting vehicle simulationand control algorithm design especially for chassisdrivelinemodeling and simulation The vehicle parameters used inthe veDYNA and observers proposed in this paper are thesame as in [11]

To represent the most likely cause of error in true dataacquisition zero-mean-value random measure noises withGaussian distribution are introduced into the vehicle acceler-ation yaw rate and roll rate measurements during the course

of simulation The performance of the observers is evaluatedunder a sudden steering maneuver on a high friction surface(120583 = 09) and a slalom maneuver on a low friction surface(120583 = 045) and the test vehicle drives on nonflat roads ineach maneuver respectively

In the sudden steeringmaneuver the longitudinal vehiclevelocity strongly varies As shown in Figure 5 the durationof vehicle acceleration and brake has been included andthe steering wheel angle changes from zero to 90 degreesduring 1 second when vehicle is running at high speed Theroad grade angle and bank angle are shown in Figure 6 InFigures 7 and 8 the estimated roll angle yaw rate and vehiclevelocities from the proposed observers are compared to thoseof the veDYNA simulator respectively

The second test is a slalom maneuver on a low frictionsurface The steering wheel angle and wheel torque mea-surements are shown in Figure 9 In this test the steering

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30

0

005

01

015Ro

ll an

gle (

rad)

MeasuredEstimated

minus015

minus01

minus005

t (s)0 5 10 15 20 25 30

0

05

1

Yaw

rate

(rad

s)

MeasuredEstimated

minus05

t (s)

Figure 7 The measured and estimated vehicle roll angle and yaw rate in the sudden steering maneuver

0 5 10 15 20 25 300

10

20

30

40

Long

itudi

nal v

eloci

ty (m

s)

MeasuredEstimated

t (s)0 5 10 15 20 25 30

0

1

2

3

4

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus2

minus1

t (s)

Figure 8 The measured and estimated vehicle velocities in the sudden steering maneuver

wheel angle turns quickly during the vehicle running and theamplitude of the steering wheel angle is 30 degrees The roadgrade angle and bank angle in this test are shown in Figure 10Figures 11 and 12 show the roll angle yaw rate and vehiclevelocities estimation results from the proposed observer andthe veDYNA simulator

As shown in Figures 8 and 12 the estimated vehiclevelocities from the proposed observer with respect to dif-ferent road angles and surface conditions are very close tothe values of the veDYNA simulator and the vehicle velocityobserver does not need to know the road angles and thetire-road friction coefficient This is to be expected since theestimated longitudinal friction forces and the relationshipbetween the longitudinal and lateral friction forces are usedin the vehicle velocity observer proposed in this paper At

the same time the sliding mode terms are used in the vehiclevelocity observer as a ldquotracking elementrdquo for the unknowninputs caused by road grade angle and bank angle which aredifficult to be measured From Figures 7 and 11 it can be seenthat the estimation of the roll angle has also achieved goodperformance It also can be seen that the estimation of yawrate contains relatively more noise This is mainly induced bythe noise of measurement which can be decreased by a low-pass filter such as one with a transfer function of 1(119904 + 119886)where 119886 is a constant

5 Conclusion

In this paper a roll angle observer and a vehicle velocityobserver have been proposed for four-wheel independent

Mathematical Problems in Engineering 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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 9

0 5 10 15 20

0

03

06

Stee

ring

whe

el an

gle (

rad)

minus06

minus03

t (s)

FL wheelFR wheel

RL wheelRR wheel

0 5 10 15 20

0

1000

2000

3000

Whe

el to

rque

(Nmiddotm

)

minus1000

t (s)

Figure 9 The steering wheel angle and wheel torque in the slalom maneuver

0 5 10 15 20

0

005

01

015

Road

gra

de an

gle (

rad)

minus01

minus005

t (s)0 5 10 15 20

0

005

01

Road

ban

k an

gle (

rad)

minus01

minus005

t (s)

Figure 10 The road grade angle and bank angle in the slalom maneuver

0 5 10 15 20

0

004

Roll

angl

e (ra

d)

MeasuredEstimated

minus008

minus004

t (s)0 5 10 15 20

MeasuredEstimated

03

02

01

0

minus01

minus02

Yaw

rate

(rad

s)

t (s)

Figure 11 The measured and estimated vehicle roll angle and yaw rate in the slalom maneuver

10 Mathematical Problems in Engineering

0 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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 5 10 15 200

5

10

15

20

25

30Lo

ngitu

dina

l velo

city

(ms

)

MeasuredEstimated

t (s)0 5 10 15 20

0

02

04

06

Late

ral v

eloci

ty (m

s)

MeasuredEstimated

minus02

t (s)

Figure 12 The measured and estimated vehicle velocities in the slalom maneuver

drive electric vehicle without needing to measure or estimatethe road angles and surface conditions using the availablemeasurements in modern cars including the wheel angularvelocities longitudinal and lateral accelerations yaw rateroll rate wheel steering angles and wheel torques Theproposed observers have been validated on a high-precisionvehicle dynamics simulation system based on veDYNA andthe simulation results show that good performance of theproposed observers has been achieved Using the estimatedvehicle velocities the vehicle body sideslip angle can becalculated directly which is also useful for the automotivechassis control systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61104060) and the China Postdoc-toral Science Foundation (no 2014M551244)

References

[1] H F Grip L Imsland T A Johansen T I Fossen J CKalkkuhl andA Suissa ldquoNonlinear vehicle side-slip estimationwith friction adaptationrdquoAutomatica vol 44 no 3 pp 611ndash6222008

[2] S-E Zhao Y Li and X Qu ldquoVehicle chassis integrated controlbased on multimodel and multilevel hierarchical controlrdquoMathematical Problems in Engineering vol 2014 Article ID248676 13 pages 2014

[3] S Melzi and E Sabbioni ldquoOn the vehicle sideslip angle esti-mation through neural networks numerical and experimental

resultsrdquoMechanical Systems and Signal Processing vol 25 no 6pp 2005ndash2019 2011

[4] K T Leung J F Whidborne D Purdy and P Barber ldquoRoadvehicle state estimation using low-cost GPSINSrdquo MechanicalSystems and Signal Processing vol 25 no 6 pp 1988ndash2004 2011

[5] R Wang G Yin and J Wang ldquoVehicle lateral velocity and tire-road friction coefficient estimationrdquo in Proceedings of the 5thASME Annual Dynamic Systems and Control Conference pp495ndash502 Fort Lauderdale Fla USA October 2012

[6] M A Wilkin D C Crolla M C Levesley et al ldquoDesign of arobust tyre force estimator using an extended Kalman filterrdquo inSociety of Automotive Engineers (SAE) World Congress DetroitMich USA 2005 Paper no 2005-01-0402

[7] L Imsland T A Johansen T I Fossen H F Grip J CKalkkuhl and A Suissa ldquoVehicle velocity estimation usingnonlinear observersrdquoAutomatica vol 42 no 12 pp 2091ndash21032006

[8] J Stephant A Charara and D Meizel ldquoEvaluation of a slidingmode observer for vehicle sideslip anglerdquo Control EngineeringPractice vol 15 no 7 pp 803ndash812 2007

[9] H Shraim B Ananou L Fridman H Noura and M Oulad-sine ldquoSliding mode observers for the estimation of vehicleparameters forces and states of the center of gravityrdquo inProceedings of the 45th IEEE Conference onDecision and Control(CDC rsquo06) pp 1635ndash1640 San Diego Calif USA December2006

[10] Y Sebsadji S Glaser S Mammar and J Dakhlallah ldquoRoadslope and vehicle dynamics estimationrdquo in Proceedings ofthe American Control Conference (ACC rsquo08) pp 4603ndash4608Washington DC USA June 2008

[11] J Ryu and J CGerdes ldquoEstimation of vehicle roll and road bankanglerdquo in Proceedings of the American Control Conference vol 3pp 2110ndash2115 IEEE Boston Mass USA July 2004

[12] K Nam S Oh H Fujimoto and Y Hori ldquoEstimation of sideslipand roll angles of electric vehicles using lateral tire force sensorsthrough RLS and kalman filter approachesrdquo IEEE Transactionson Industrial Electronics vol 60 no 3 pp 988ndash1000 2013

Mathematical Problems in Engineering 11

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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

[13] H F Grip L Imsland T A Johansen J C Kalkkuhl andA Suissa ldquoEstimation of road inclination and bank angle inautomotive vehiclesrdquo in Proceedings of the American ControlConference (ACC rsquo09) pp 426ndash432 St Louis Mo USA June2009

[14] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoPaper 2004-01-1757 Society of Automotive Engineers (SAE)World Congress Michigan Mich USA 2004

[15] H Dugoff P S Fancher and L Segel ldquoAn analysis of tiretraction properties and their influence on vehicle dynamicperformancerdquo SAE Transactions vol 79 pp 341ndash366 1970

[16] L H Zhao Z Y Liu and H Chen ldquoDesign of a nonlinearobserver for vehicle velocity estimation and experimentsrdquo IEEETransactions on Control Systems Technology vol 19 no 3 pp664ndash672 2011

[17] R Wang and J Wang ldquoTire-road friction coefficient and tirecornering stiffness estimation based on longitudinal tire forcedifference generationrdquo Control Engineering Practice vol 21 no1 pp 65ndash75 2013

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