research article a study of coordinated vehicle traction

Research ArticleA Study of Coordinated Vehicle Traction Control System Basedon Optimal Slip Ratio Algorithm

Gang Liu12 and LiQiang Jin1

1State Key Laboratory of Automotive Simulation and Control Jilin University Changchun Jilin 130025 China2Department of Automatic Control Henan Institute of Technology Xinxiang Henan 453000 China

Correspondence should be addressed to LiQiang Jin jinlqjlueducn

Received 5 March 2016 Revised 31 May 2016 Accepted 21 June 2016

Academic Editor Luis J Yebra

Copyright copy 2016 G Liu and L Jin 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

Under complicated situations such as the low slippery road surface and split-120583 road surface traction control system is the key issueto improve the performance of vehicle acceleration and stability In this paper a novel control strategy with engine controller andactive pressure controller is presented First and foremost an ideal vehicle model is proposed for simulation then a method forthe calculation of optimal slip ratio is also brought Finally the scheme of control method with engine controller and active brakecontroller is presented From the results of simulation and road tests it can be concluded that the acceleration performance andstability of a vehicle equipped with traction control system (TCS) can be improved

1 Introduction

Vehicle traction control system plays an important role inelectronic stability control (ESC) system In order to improvethe vehicle traction performance on low 120583 surface road andmaximize the longitudinal friction coefficient TCS is used tocontrol the slip ratio within the range of optimal slip ratio byregulating engine output torque or braking pressure

In recent years many theoretical studies based on wheelslip rate are conducted all around the world In literatures[1 2] the logic threshold controller is realized by regulatingthe braking pressure It is easy to implement this methodbut the development cycles for different vehicles are longReference [3] introduces the throttle actuator systemwith thetime delay scheme and the vehicle test result shows the goodperformance for the TCS In [4] the proportion integrationdifferentiation (PID) controller is implemented to adjustslip ratio Due to high nonlinearity of the vehicle dynamicsystem these conventional linear controllers cannot meetthe requirements of TCS under complicated road conditionsIn [5] a fuzzy controller is proposed to control the drivingwheel slip for robust traction control The simulation indi-cates that the controllers can further maximize the tractionforce Reference [6] introduces a coordinated cascade control

strategy which includes two sliding mode controllers Boththe engine torque and the active brake pressure are tunedby sliding mode controllers Reference [7] proposes a slidingmode observer for road adhesion coefficient estimation andthen the fuzzy controller is used to regulate the engine outputtorque by adjusting the fuel supply The control strategyis implemented on Volvo 76 Turb Reference [8] presentsa model predictive control strategy for TCS Reference [9]introduces a second-order sliding mode of TCS controllerand the TCS is coupled with a nonlinear observer to estimatethe road condition

From the researchesmentioned above we learn that thereare two requirements for TCS controller algorithm One isrobustnessWheel slip should bemaintained at around targetslip ratio under any complicated road surface conditionsTheother one is that the TCS controller strategy should meetdifferent demands in different road surface conditionsWhenthe vehicle is accelerating on the low 120583 road surface thestability should be considered as a key point When it isdriving on 120583 split (one driven wheel on the ice) the tractionissue should be considered first All of these referencesmentioned above are on single controller to regulate theslip ratio However the single controller cannot maintainthe vehicle stability under complicated road conditions It

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3413624 10 pageshttpdxdoiorg10115520163413624

2 Mathematical Problems in Engineering

Fy

Fy

Fy

Fy

L

rl

120572rl

120573

120573

fl

flFx

Fx

120572fl

120574X

Y

rr

120572rr

120572fr

frfr

b a

w

120575

120575

Figure 1 7-DOF vehicle model

is necessary to combine the active brake pressure controllerwith the engine torque controller This paper presents atraction control system based on the engine output torqueadjustment and brake pressure controlThe optimal slip ratiois estimated via Varied Forgetting Factor Recursive LeastSquares (VFFRLS) And the optimal slip ratio is consideredas the desired value The engine output torque is adjusted byfuzzy PID controller The active brake pressure controller isbased on sliding mode control theory The control strategy isto coordinate these two controllers according to road surfaceconditions

Section 2 introduces the vehicle dynamicmodel Section 3introduces the estimation algorithm of desired slip ratioSection 4 explains the coordination strategy between theoutput torque and the brake pressure Section 5 presentsthe results of simulation and the winter road experimentsFinally conclusions are introduced in Section 6

2 Vehicle Model

21 Vehicle Model As shown in Figure 1 a nonlinear 7-DOF(degree of freedom) vehicle model is used to predict vehicleresponse under different road conditions during variousdriving maneuvers [10 11] Some factors such as the air dragresistance and the resistance of slope are not consideredThevehicle model includes longitudinal motion lateral motionand yaw motion The vehicle dynamic equations can bedescribed as follows

119898(V119909minus V119910119903) = (119865

119909fl+ 119865119909fr) cos 120575 minus (119865

119910fl+ 119865119910fr) sin 120575

+ 119865119909rl+ 119865119909rr

119898 (V119910+ V119909119903) = (119865

119909fl+ 119865119909fr) sin 120575 minus (119865

119910fl+ 119865119910fr) cos 120575

+ 119865119910rl+ 119865119910rr

119868V 119903 = (119865119910fl + 119865119910fr) 119886 cos 120575

+ (119865119910flminus 119865119910fr)119882

2sin 120575

minus (119865119910fl+ 119865119910fr) 119887

+ (119865119909fl+ 119865119909fr) 119886 sin 120575

minus (119865119909flminus 119865119909fr)119882

2cos 120575

minus (119865119909flminus 119865119909fr)119882

2

(1)where V

119909is the longitudinal velocity V

119910is the lateral velocity

119903 is the yaw angle 120575 is the steer angle 119898 represents thevehicle mass 119868V is the yaw inertia moment 119865

119909denotes the

longitudinal forces of wheels 119865119910denotes the lateral forces of

the wheels 119898 is the vehicle mass 119886 is the distance betweenfront axle and center gravity 119887 is the distance between rearaxle and center gravity119882 is width of vehicle

Wheel rotational equations can be expressed as

119868119908

119889119908

119889119905= 119879119902minus 119865119889119903119908minus 119879119887 (2)

where 119868119908denotes wheels inertia moment 119879

119902is the longitudi-

nal driving torque 119879119887is the breaking torque 119903

119908is the wheel

radius 119865119889is driving force between road and tire 119908 is wheel

speedThe equations for each wheel can be described as

119865119911fl= 119865119911fr=119898119892

2[119887

119871minus(V119909minus V119910119903) ℎ119892

119892119871]

119865119911rl= 119865119911rr=119898119892

2[119886

119871+(V119909minus V119910119903) ℎ119892

119892119871]

(3)

Mathematical Problems in Engineering 3

Table 1 Values of the vehicle parameters

Parameter ValueVehicle mass 1545mkgGravity 981ms2

119886 114m119887 163mWheel radius 032mWheel width 156mHeight of center gravity 052mInertia moment 2300 kglowastm2

Wheel inertia moment 1 kglowastm2

where 119865119911denotes the vertical force of each wheel ℎ

119892denotes

the distance between the gravity center and the ground 119871 isthe distance between the front axle and the rear axle

Other parameters of the vehicle model used for simula-tion are listed in Table 1

22 Tire Model Pacejkarsquos tire model is used to simulate thecontact forces between the road and the tire [12]The equationof the tire longitudinal forces 119865

119909and 119865

119910can be defined as

follows

119910 = 119863 sin 119862 arctan [119861120582 (1 minus 119864) + 119864 arctan119861120582] (4)

where 119861 is the tire stiffness factor 119862 represents the shapefactor 119863 represents the peak factor 119864 represents the curva-ture factor and 120582 is the slip ratio which can be expressed asfollows

120582 =

V119909minus 119908119903119908

V119909

(V119909gt 119908119903119908)

119908119903119908minus V119909

V119909

(V119909lt 119908119903119908)

(5)

23 Engine Model In this paper the dynamics model of theengine can be depicted as

119879119890= 119879119904minus 119891(

119889120572

119889119909) minus 119869119890

119889120596119890

119889119905 (6)

where 119879119890is engine output torque 119879

119904denotes the static output

torque of engine which is according to engine map chart119889120572119889119909 denotes the function of the differential of throttleangle 120572 120596

119890denotes the engine rotation speed 119869

119890denotes

crankrsquos inertia moment The engine output torque dependedon the throttle angle and the engine speed of rotation

3 Parameter Estimation

The precondition of slip ratio control is to estimate the opti-mal slip ratio exactly The method of estimating the optimalslip ratio is based on seeking extreme value of 120583-120582 curve(120583-120582 curve denotes the relation between the road frictioncoefficient and the slip ratio)The relationship between 120583 and

Table 2 The values of Kiencke mode on different road conditions

Road condition 1199011

1199012

Dry asphalt 105104 345987Wet asphalt 183410 584155Dry concrete 112732 390633Dry cobblestone 145401 62497Wet cobblestone 582343 510124Snow 1183411 2778144Ice 5360750 10108

120582 can be expressed by Kiencke model [13] The equation forKiencke model is as follows

120583 (120582) =30120582

1 + 1199011120582 + 11990121205822 (7)

where 1199011and 119901

2denote the parameters for different road

conditions Table 2 shows the values of 1199011and 119901

2in different

road surface conditions Optimal slip ratio is available byseeking the extreme value of Kiencke model (8) Optimal slipratio 120582

119901is shown as follows

120582119901= 1199012

minus12

120583119901(120582119901) =

30

1199011+ 21199012

12

(8)

It is obvious that optimal slip ratio is available by determi-nation of 119901

1and 119901

2on different road surfaces Considering

that the coefficients of friction 120583(120582) and slip 120582 are availableinstantly 119901

1and 119901

2mentioned in (8) can thus be formulated

by the Variable Forgetting Factors Recursive Least Square(VFFRLS) algorithm [14 15]

119910 (119896) = Φ119879

(119896)Θ (119896) (9)

where Θ(119896) Φ119879(119896) and 119910(119896) are given as

Θ (119896) = [1199011(119896) 119901

2(119896)]119879

119910 (119896) = 30120582 (119896) minus 120583 (119896)

Φ119879

(119896) = [120583 (119896) 120582 (119896) 120583 (119896) 1205822

(119896)]

(10)

The recursive process of VFFRLS algorithm is shown asfollows

Θ (119896) = Θ (119896 minus 1) + 119866 (119896) [119910 (119896) minus Φ119879

(119896)Θ (119896 minus 1)]

119866 (119896) =119901 (119896 minus 1)Φ (119896)

Λ + Φ119879 (119896) 119901 (119896 minus 1)Φ (119896)

119901 (119896) =1

Λ[119868 minus 119866 (119896)Φ

119879

(119896)] 119901 (119896 minus 1)

(11)

where 119868 is the identity 119901(119896) denotes the covariance matrix119866(119896) denotes the gain matrix Λ denotes the variable forget-ting factor which is in the range (09 1) Seen from (11) thevariable forgetting factor Λ is directly related to the dynamic

4 Mathematical Problems in EngineeringRo

ad fr

ictio

n co

effici

ent

1

08

06

04

02

0

Time (s)0 10 20 30 40

Estimation resultTrue value

Figure 2 Simulation results of road friction coefficient

tracking ability of Variable Forgetting Factors Recursive LeastSquare (VFFRLS) algorithm [16] A low value of Λ meansthe low weight of the past data Owing to the different roadconditions (ie varying-120583 road) the value of Λ is to decreaseduring road transient conditions and then restore back to itsoriginal value under steady state The value of Λ varies asfollows

Λ =

Λ0

119880 lt |120576|

Λ1+ (Λ0minus Λ1) (1 minus exp (minus120591119896)) 119880 gt |120576|

(12)

whereΛ0denotes the value ofΛ under steady state 120591 denotes

a sensitive coefficient which regulates the adjustment speedof forgetting factor 120576 denotes the differential equation oflongitudinal acceleration

Figure 2 is the simulation result of road friction coefficientestimationThe proposedmethod can estimate the coefficientquickly and the error between the true value and theestimation value is less than 01

4 Controller Design

41 Control Scheme The overall structure for TCS can bedepicted as in Figure 3 When the driving vehicle accelerateson a complicated road surface if the driving force of drivingwheels exceeds the friction force the wheel begins to skidThe TCS begins to estimate the optimal slip ratio The valueof optimal slip ratio is considered as the desired value ofTCS The engine torque controller and the brake torquecontroller are activated to adjust the output torque and thebrake pressure

Generally speaking vehicles may get into accelerationtrouble when driving on two typical road conditions One isthe low 120583 slippery road and the other one is the split 120583 roadsurface This paper proposes a control scheme to solve thetraction problem by maintaining the slip ratio of wheels

Under the low 120583 slippery road condition engine torquecontroller is used to regulate the slip ratio of driving wheelsUnder the split-120583 road condition the engine torque controllerand the active brake controller are coordinately used to avoid

Traction control system

Desiredslip ratio

estimation

Brake torquecontroller

Engine torqueoutput controller

Brakerequest

Engine torquereduction

request

Hydraulicbrake

modulatorEngine

managementsystem

Pedal way

Acceleratorpedal

Vehicle model

Sensorinformation

Figure 3 The structure of TCS

the phenomenon of spin in two steps In the first step theactive brake controller is used to regulate the slip ratio ofdrivingwheels on the low120583 road surfaceThedesired pressurecan be calculated by the fuzzy controller In the second stepthe engine output torque is chosen to increase the outputtorque and the wheel angular accelerationThe engine outputtorque is calculated by the sliding mode controller

42 Engine Output Torque Controller As known to all thethrottle is used to adjust the output torque by regulating theamount of airThe adaptive sliding mode controller is chosenas the control algorithm of engine output torque The errorbetween the desired slip ratio and the current slip ratio canbe defined as follows

119890119894= 120582119894minus 120582119901 (13)

where 120582119894denote the current slip ratio 120582

119901is the desired slip

ratio described in Section 3The sliding surface is as follows

119904119894= 119890119894+ 119888119890119894 (14)

where 119888 is a constant and meets the Hurwitz theoremAnd the differential version of (14) is as follows

119904119894= 119890119894+ 119888 119890119894 (15)

Then choose the control law

119904119894= 119890119894+ 119888 119890119894= minus1198960119904119894minus 1205760sat(

119904119894

120601) (16)

where

sat(119904119894

120601) =

119904119894

120601

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816le 1

sgn(119904119894

120601)

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816gt 1

(17)

120601 is the thickness about the boundary layer of the saturationfunction and the value of 120601 is greater than zero 119904

119894= 0 can be


achieved in finite time both 1198960gt 0 and 120576

0gt 0 are the sliding

coefficients which can be tuned according to the effectivenessof control

The differential version of (2) can be defined as

119868119908 =

119902minus 119889119903119908 (18)

Then based on (5) (18) and (16) the following formula canbe deduced

= minus (119888 + 1198960) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)

(19)

Based on (18) and (19) the control torque can be deduced as

119879119902= int 119868119908(minus (119888 + 119896

0) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)) +

119889119903119908

(20)

43 Active Brake Controller As for the design of active brakecontroller the fuzzy control algorithm is adopted to keep theslip ratio within the range of optimal slip ratio There are twoinput signals and one output signal for fuzzy logic controllerOne input signal is 119890 which denotes the error between thecurrent slip ratio and the optimal slip ratio The other one isec which represents the first derivative of 119890 The output signalof fuzzy logic controller regulates the opening of the solenoidvalve by the PWM of ECU Then the target brake pressure isavailable119890 is divided into six fuzzy subsets [NB NM Z PS PM

and PB] ec is divided into seven fuzzy subsets [NB NM NSZ PS PM and PB] the output signal is divided into fourfuzzy subsets [NB Z PS and PB]

As shown in Table 3 the rule base of active brakecontroller is based on the experience knowledge and expertknowledge Then the gravity center method is used fordefuzzification If the output signal is zero the brake pressureis in pressure-hold condition if the output signal is NBthe brake pressure is in pressure-increasing condition ifthe output signal is PB the brake pressure is in pressure-decreasing condition

5 Experiment and Results

51 Simulation Result For validation some typical simula-tions were carried out by using MatlabSimulink softwareThe basic simulation conditions are listed as follows

(i) Target path straight no steering angle input(ii) Accelerator pedal opening 100

Table 3 Table of fuzzy control rules

119890119890119888

NB NM NS Z PS PM PBNB PB PB PB PS PS PS PSNM PB PB PB PS PS Z ZZ PS PS PS Z NS NS NSPS PS Z Z Z NS NS NSPM NS NB NB NB NB NB NBPB NB NB NB NB NB NB NB

Vehi

cle sp

eed

100

80

60

40

20

0

Acceleration pedal opening0 02 04 06 08 1

Figure 4 Control input

Case 1 (low 120583 slippery road surface) Figure 4 shows therelation between the opening degree of accelerator pedal andthe vehicle speed The opening degree of accelerator pedal isproportional to the vehicle speed However the vehicle speedceases to increase when the opening degree reaches 90

Figure 5 shows the simulation result of the vehicledriving under a low 120583 slippery road surface condition Thefriction coefficient of road surface is 02 From Figures 5(a)and 5(b) from the comparison between the noncontrolledone and the controlled one we can find that the speedof uncontrolled vehicle is 50 kmh at 11 s but that of thecontrolled one is 50 kmh at 7 s We can also find that there isabout 45 improvement in the performance of longitudinalacceleration From Figure 5(c) the one under control keepsthe slip ratio within the range of optimal slip ratio (about016) However the driving wheels of the uncontrolled oneslipped excessively

Case 2 (120583-jump road surface) Figure 6 shows the simulationresult of the vehicle driving under a 120583-jump road surfacecondition The phenomenon of road transition from high 120583(120583 = 08) to low 120583 (120583 = 02) happens at 4 s From Figure 6(c)the slip ratio of uncontrolled one is soaring after 4 s It isobvious that the controller could identify the change of roadconditions and the slip ratio is controlled to the new desiredvalue

52 Road Test Result As shown in Figure 7 the TCScontroller and a suite of standard sensors are added to thevehicle The controller is a 32-bit microcontroller unit which


Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8

Driving wheel speedVehicle speed

(a) Velocity versus time in Case 1 with no control

Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60


(b) Velocity versus time in Case 1 under control

Slip

ratio

Under TCS controlWithout control

0

02

04

06

08

1

2 4 6 80 10

Time (s)

(c) Comparison of slip ratio between the one under control and thenoncontrolled one

Figure 5 Simulation result of Case 1

is produced by Freescale Semiconductor The C code isobtained from the TCS control strategy of Simulink modelthen C code is downloaded to microcontroller unit Andthe microcontroller is mounted to the hydraulic unit Thetesting system consists of Vector CANalyzer and a SpeedboxThe CANalyzer captures sensory information and engineinformation fromCANbusThe Speedbox is used tomeasurethe speed of the test vehicle

Case 1 (ice road test) Figure 8 shows that the road test wascarried out on an ice road The friction coefficient of the iceroad was approximately 02 Figure 9 shows the test result ofnoncontrolled vehicle The velocity of the vehicle was only17 kmh at the end of the road test Furthermore the slip ratioincreased excessively

Figure 10 shows the test result of the controlled one on theice road surface FromFigure 10(a) the angular velocity of thedriving wheel was reduced effectively by the engine outputtorque controller The improvement of vehicle accelerationwas 43 The gear changed at 12 s The optimal slip ratiowas approximately 01 Figure 10(b) shows that the slip ratio

was controlled within the range of 01 From Figure 10(c)when the vehicle began to accelerate at 2 s the currentengine torque decreased quickly to the desired oneThen theengine output torque followed the tendency of vehicle speedchange To conclude the engine torque controller played animportant role in restraining the slip ratio of the drivingwheels on the icy road surface

Case 2 (split-120583 road test) Figure 11 shows the vehicle test onthe split-120583 road surface The right side was ice road (120583 =02) and the left side was dry concrete pavement (120583 =

08) Figure 12 shows the test results of parameters FromFigure 12(a) the angular velocity of the wheel on the slipperyside was close to vehicle speed

When the vehicle began to accelerate at 3 s the slip ratioof front right wheel increased excessively Then the valuewas kept close to the desired slip ratio (approximately 01)as shown in Figure 12(b) From Figure 12(c) when the frontright wheel began to skid at 32 s the engine torque controllerreduced the torque quicklyThen the slip ratio was controlledby the brake pressure properly To sum up in the spilt-120583 road


Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio

Without controlUnder TCS control

0

02

04

06

08

2 4 6 80 10

Time (s)



Figure 7 The MCU controller

test the proper switch between the engine controller and theactive pressure controller can improve the performance ofacceleration and stability

Figure 8 Winter test

6 Conclusions

To sum up this paper presents a novel traction controlsystem which consists of an engine torque controller and a


Velo

city

(km

h)

Gear informationDriving wheel speedVehicle speed

0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)

Figure 9 The winter test result with no TCS controller

Velo

city

(km

h)

Time (s)0 5 10 15

Driving wheel speedVehicle speedTCS active flag

0

10

20

30

40

(a) Wheel speed versus vehicle speed

Slip

ratio

Time (s)0 5 10 15

Slip ratioTCS active flag

0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15

TCS active flagActual torqueDesired torque

0

50

100

150

200

(c) Torque

Figure 10 The winter test results obtained on the ice road


Figure 11 The vehicle test under the split-120583 road surface

Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)

FL driving wheel speedFR driving wheel speedTCS active flag

Gear informationVehicle speed


Slip

ratio

FR wheel slip ratioFL wheel slip ratioTCS active flag

0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150

TCS active flagDesired torque

Active torqueCylinder pressure

(c) Torque

Figure 12 The winter test results obtained on split-120583 road surface

pressure controller for the evaluation of vehicle accelerationand stability The characteristics of the TCS can be describedas follows

(1) A method for real-time calculation of optimal slipratio is realized by Variable Forgetting Factors Recur-sive Least Square (VFFRLS) algorithm And then

the optimal slip ratio is considered as the desired valueof slip control

(2) The cascade control method with fuzzy control algo-rithm and sliding mode control algorithm can beeffectively adapted to the complicated road surfaceconditions


(3) The algorithm takes 2ms to run a time and runs onceevery 20ms so that the TCS controller can discoverand correct the vehicle wheel slipping phenomenonin time

(4) Simulation which is based on Matlab software andtypical road tests were carried outThe results indicatethat the control scheme is fit for complicated workingcircumstances

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by China Postdoctoral ScienceFoundation (2013M540248)

References

[1] A T Van Zanten R Erhardt G Pfaff and F Kost ldquoControlaspects of Bosch-VDCrdquo in Proceedings of the 3rd InternationalSymposium on Advanced Vehicle Control pp 573ndash607 AachenGermany June 1996

[2] H Chen X Gong Y-F Hu Q-F Liu B-Z Gao andH-Y GuoldquoAutomotive control the state of the art and perspectiverdquo ActaAutomatica Sinica vol 39 no 4 pp 322ndash346 2013

[3] J-B Song and K-S Byun ldquoThrottle actuator control system forvehicle traction controlrdquoMechatronics vol 9 no 5 pp 477ndash4951999

[4] H-Z Li L Li L He et al ldquoPID plus fuzzy logic method fortorque control in traction control systemrdquo International Journalof Automotive Technology vol 13 no 3 pp 441ndash450 2012

[5] G Yin S Wang and X Jin ldquoOptimal slip ratio based fuzzycontrol of acceleration slip regulation for four-wheel inde-pendent driving electric vehiclesrdquo Mathematical Problems inEngineering vol 2013 Article ID 410864 7 pages 2013

[6] M-X Kang L Li H-Z Li J Song and Z Han ldquoCoordinatedvehicle traction control based on engine torque and brakepressure under complicated road conditionsrdquo Vehicle SystemDynamics vol 50 no 9 pp 1473ndash1494 2012

[7] J H Park and C Y Kim ldquoWheel slip control in traction controlsystem for vehicle stabilityrdquoVehicle SystemDynamics vol 31 no4 pp 263ndash278 1999

[8] F Borrelli A Bemporad M Fodor and D Hrovat ldquoAnMPChybrid system approach to traction controlrdquo IEEE Trans-actions on Control Systems Technology vol 14 no 3 pp 541ndash5522006

[9] H Lee and M Tomizuka ldquoAdaptive vehicle traction forcecontrol for intelligent vehicle highway systems (IVHSs)rdquo IEEETransactions on Industrial Electronics vol 50 no 1 pp 37ndash472003

[10] Y-J Huang T-C Kuo and S-H Chang ldquoAdaptive sliding-mode control for nonlinear systemswith uncertain parametersrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 38 no 2 pp 534ndash539 2008

[11] L Li J Song H Wang and C Wu ldquoFast estimation andcompensation of the tyre force in real time control for vehicledynamic stability control systemrdquo International Journal of Vehi-cle Design vol 48 no 3-4 pp 208ndash229 2008

[12] M Amodeo A Ferrara R Terzaghi and C Vecchio ldquoWheelslip control via second-order sliding-mode generationrdquo IEEETransactions on Intelligent Transportation Systems vol 11 no 1pp 122ndash131 2010

[13] U Kiencke ldquoReal-time estimation of adhesion characteristicbetween tyres and roadrdquo in Proceeding of the IFAC WorldCongress pp 136ndash159 Sydney Australia 1993

[14] 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

[15] M Doumiati A C Victorino A Charara and D LechnerldquoOnboard real-time estimation of vehicle lateral tire-road forcesand sideslip anglerdquo IEEEASME Transactions on Mechatronicsvol 16 no 4 pp 601ndash614 2011

[16] C Paleologu J Benesty and S Ciochina ldquoA robust variableforgetting factor recursive least-squares algorithm for systemidentificationrdquo IEEE Signal Processing Letters vol 15 no 3 pp597ndash600 2008

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Fy

Fy

Fy

Fy

L

rl

120572rl

120573

120573

fl

flFx

Fx

120572fl

120574X

Y

rr

120572rr

120572fr

frfr

b a

w

120575

120575

Figure 1 7-DOF vehicle model

is necessary to combine the active brake pressure controllerwith the engine torque controller This paper presents atraction control system based on the engine output torqueadjustment and brake pressure controlThe optimal slip ratiois estimated via Varied Forgetting Factor Recursive LeastSquares (VFFRLS) And the optimal slip ratio is consideredas the desired value The engine output torque is adjusted byfuzzy PID controller The active brake pressure controller isbased on sliding mode control theory The control strategy isto coordinate these two controllers according to road surfaceconditions

Section 2 introduces the vehicle dynamicmodel Section 3introduces the estimation algorithm of desired slip ratioSection 4 explains the coordination strategy between theoutput torque and the brake pressure Section 5 presentsthe results of simulation and the winter road experimentsFinally conclusions are introduced in Section 6

2 Vehicle Model

21 Vehicle Model As shown in Figure 1 a nonlinear 7-DOF(degree of freedom) vehicle model is used to predict vehicleresponse under different road conditions during variousdriving maneuvers [10 11] Some factors such as the air dragresistance and the resistance of slope are not consideredThevehicle model includes longitudinal motion lateral motionand yaw motion The vehicle dynamic equations can bedescribed as follows

119898(V119909minus V119910119903) = (119865

119909fl+ 119865119909fr) cos 120575 minus (119865

119910fl+ 119865119910fr) sin 120575

+ 119865119909rl+ 119865119909rr

119898 (V119910+ V119909119903) = (119865

119909fl+ 119865119909fr) sin 120575 minus (119865

119910fl+ 119865119910fr) cos 120575

+ 119865119910rl+ 119865119910rr

119868V 119903 = (119865119910fl + 119865119910fr) 119886 cos 120575

+ (119865119910flminus 119865119910fr)119882

2sin 120575

minus (119865119910fl+ 119865119910fr) 119887

+ (119865119909fl+ 119865119909fr) 119886 sin 120575

minus (119865119909flminus 119865119909fr)119882

2cos 120575

minus (119865119909flminus 119865119909fr)119882

2

(1)where V

119909is the longitudinal velocity V

119910is the lateral velocity

119903 is the yaw angle 120575 is the steer angle 119898 represents thevehicle mass 119868V is the yaw inertia moment 119865

119909denotes the

longitudinal forces of wheels 119865119910denotes the lateral forces of

the wheels 119898 is the vehicle mass 119886 is the distance betweenfront axle and center gravity 119887 is the distance between rearaxle and center gravity119882 is width of vehicle

Wheel rotational equations can be expressed as

119868119908

119889119908

119889119905= 119879119902minus 119865119889119903119908minus 119879119887 (2)

where 119868119908denotes wheels inertia moment 119879

119902is the longitudi-

nal driving torque 119879119887is the breaking torque 119903

119908is the wheel

radius 119865119889is driving force between road and tire 119908 is wheel

speedThe equations for each wheel can be described as

119865119911fl= 119865119911fr=119898119892

2[119887

119871minus(V119909minus V119910119903) ℎ119892

119892119871]

119865119911rl= 119865119911rr=119898119892

2[119886

119871+(V119909minus V119910119903) ℎ119892

119892119871]

(3)







119892denotes




119909and 119865


follows



120582 =

V119909minus 119908119903119908

V119909

(V119909gt 119908119903119908)

119908119903119908minus V119909

V119909

(V119909lt 119908119903119908)

(5)


119879119890= 119879119904minus 119891(

119889120572

119889119909) minus 119869119890

119889120596119890

119889119905 (6)





119890denotes






1199012



120583 (120582) =30120582

1 + 1199011120582 + 11990121205822 (7)

where 1199011and 119901



2in different



120582119901= 1199012

minus12

120583119901(120582119901) =

30

1199011+ 21199012

12

(8)


1and 119901



1and 119901



119910 (119896) = Φ119879

(119896)Θ (119896) (9)


Θ (119896) = [1199011(119896) 119901

2(119896)]119879

119910 (119896) = 30120582 (119896) minus 120583 (119896)

Φ119879

(119896) = [120583 (119896) 120582 (119896) 120583 (119896) 1205822

(119896)]

(10)


Θ (119896) = Θ (119896 minus 1) + 119866 (119896) [119910 (119896) minus Φ119879

(119896)Θ (119896 minus 1)]

119866 (119896) =119901 (119896 minus 1)Φ (119896)

Λ + Φ119879 (119896) 119901 (119896 minus 1)Φ (119896)

119901 (119896) =1

Λ[119868 minus 119866 (119896)Φ

119879

(119896)] 119901 (119896 minus 1)

(11)



ad fr

ictio

n co

effici

ent

1

08

06

04

02

0

Time (s)0 10 20 30 40




Λ =

Λ0

119880 lt |120576|


(12)




4 Controller Design





Desiredslip ratio

estimation



Brakerequest


request

Hydraulicbrake

modulatorEngine

managementsystem

Pedal way

Acceleratorpedal

Vehicle model

Sensorinformation




119890119894= 120582119894minus 120582119901 (13)




119904119894= 119890119894+ 119888119890119894 (14)


119904119894= 119890119894+ 119888 119890119894 (15)


119904119894= 119890119894+ 119888 119890119894= minus1198960119904119894minus 1205760sat(

119904119894

120601) (16)

where

sat(119904119894

120601) =

119904119894

120601

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816le 1

sgn(119904119894

120601)

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816gt 1

(17)


119894= 0 can be






119868119908 =

119902minus 119889119903119908 (18)


= minus (119888 + 1198960) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)

(19)


119879119902= int 119868119908(minus (119888 + 119896

0) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)) +

119889119903119908

(20)








119890119890119888


Vehi

cle sp

eed

100

80

60

40

20

0








Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8



Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60



Slip

ratio


0

02

04

06

08

1

2 4 6 80 10

Time (s)











Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119892denotes




119909and 119865


follows



120582 =

V119909minus 119908119903119908

V119909

(V119909gt 119908119903119908)

119908119903119908minus V119909

V119909

(V119909lt 119908119903119908)

(5)


119879119890= 119879119904minus 119891(

119889120572

119889119909) minus 119869119890

119889120596119890

119889119905 (6)





119890denotes






1199012



120583 (120582) =30120582

1 + 1199011120582 + 11990121205822 (7)

where 1199011and 119901



2in different



120582119901= 1199012

minus12

120583119901(120582119901) =

30

1199011+ 21199012

12

(8)


1and 119901



1and 119901



119910 (119896) = Φ119879

(119896)Θ (119896) (9)


Θ (119896) = [1199011(119896) 119901

2(119896)]119879

119910 (119896) = 30120582 (119896) minus 120583 (119896)

Φ119879

(119896) = [120583 (119896) 120582 (119896) 120583 (119896) 1205822

(119896)]

(10)


Θ (119896) = Θ (119896 minus 1) + 119866 (119896) [119910 (119896) minus Φ119879

(119896)Θ (119896 minus 1)]

119866 (119896) =119901 (119896 minus 1)Φ (119896)

Λ + Φ119879 (119896) 119901 (119896 minus 1)Φ (119896)

119901 (119896) =1

Λ[119868 minus 119866 (119896)Φ

119879

(119896)] 119901 (119896 minus 1)

(11)



ad fr

ictio

n co

effici

ent

1

08

06

04

02

0

Time (s)0 10 20 30 40




Λ =

Λ0

119880 lt |120576|


(12)




4 Controller Design





Desiredslip ratio

estimation



Brakerequest


request

Hydraulicbrake

modulatorEngine

managementsystem

Pedal way

Acceleratorpedal

Vehicle model

Sensorinformation




119890119894= 120582119894minus 120582119901 (13)




119904119894= 119890119894+ 119888119890119894 (14)


119904119894= 119890119894+ 119888 119890119894 (15)


119904119894= 119890119894+ 119888 119890119894= minus1198960119904119894minus 1205760sat(

119904119894

120601) (16)

where

sat(119904119894

120601) =

119904119894

120601

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816le 1

sgn(119904119894

120601)

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816gt 1

(17)


119894= 0 can be






119868119908 =

119902minus 119889119903119908 (18)


= minus (119888 + 1198960) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)

(19)


119879119902= int 119868119908(minus (119888 + 119896

0) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)) +

119889119903119908

(20)








119890119890119888


Vehi

cle sp

eed

100

80

60

40

20

0








Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8



Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60



Slip

ratio


0

02

04

06

08

1

2 4 6 80 10

Time (s)











Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ad fr

ictio

n co

effici

ent

1

08

06

04

02

0

Time (s)0 10 20 30 40




Λ =

Λ0

119880 lt |120576|


(12)




4 Controller Design





Desiredslip ratio

estimation



Brakerequest


request

Hydraulicbrake

modulatorEngine

managementsystem

Pedal way

Acceleratorpedal

Vehicle model

Sensorinformation




119890119894= 120582119894minus 120582119901 (13)




119904119894= 119890119894+ 119888119890119894 (14)


119904119894= 119890119894+ 119888 119890119894 (15)


119904119894= 119890119894+ 119888 119890119894= minus1198960119904119894minus 1205760sat(

119904119894

120601) (16)

where

sat(119904119894

120601) =

119904119894

120601

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816le 1

sgn(119904119894

120601)

10038161003816100381610038161003816100381610038161003816

119904119894

120601

10038161003816100381610038161003816100381610038161003816gt 1

(17)


119894= 0 can be






119868119908 =

119902minus 119889119903119908 (18)


= minus (119888 + 1198960) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)

(19)


119879119902= int 119868119908(minus (119888 + 119896

0) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)) +

119889119903119908

(20)








119890119890119888


Vehi

cle sp

eed

100

80

60

40

20

0








Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8



Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60



Slip

ratio


0

02

04

06

08

1

2 4 6 80 10

Time (s)











Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119868119908 =

119902minus 119889119903119908 (18)


= minus (119888 + 1198960) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)

(19)


119879119902= int 119868119908(minus (119888 + 119896

0) +

119908

V119909

(119888V119909+ 2119903

119908

minus 1205760sat(

119904119894

120601)119908119903119908+ 1198960119888V119909+ 1198960V119909minus 1198960119888119908119903119908

+ 1198960119888119908119903119908120582119901)) +

119889119903119908

(20)








119890119890119888


Vehi

cle sp

eed

100

80

60

40

20

0








Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8



Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60



Slip

ratio


0

02

04

06

08

1

2 4 6 80 10

Time (s)











Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Velo

city

(km

h)

0

100

200

300

400

500

0

16

32

48

64

80

0 2

Time (s)4 6 8



Velo

city

(km

h)

Time (s)0 2 4 6 8 10

0

20

40

60



Slip

ratio


0

02

04

06

08

1

2 4 6 80 10

Time (s)











Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Velo

city

(km

h)


0

50

100

50 1510

Time (s)



0

10

20

30

40

50

60

Velo

city

(km

h)

2 4 6 80 10

Time (s)


Slip

ratio


0

02

04

06

08

2 4 6 80 10

Time (s)






6 Conclusions



Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Velo

city

(km

h)


0

20

40

60

80

14 16 18 20 22 24 2612

Time (s)


Velo

city

(km

h)

Time (s)0 5 10 15


0

10

20

30

40


Slip

ratio

Time (s)0 5 10 15


0

02

04

06

08

(b) Slip ratio

Torq

ue (N

m)

Time (s)0 5 10 15


0

50

100

150

200

(c) Torque




Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Velo

city

(km

h)

0

20

40

60

80

50 1510

Time (s)




Slip

ratio


0

02

04

06

08

2 4 6 80 1210

Time (s)

(b) Slip ratio

Torq

ue (N

m)

50 1510

Time (s)

0

50

100

150



(c) Torque









Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Competing Interests


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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