Research ArticleRollover Mitigation Controller Development forThree-Wheeled Vehicle Using Active Front Steering

Raja Amer Azim Fahad Mumtaz Malik and Waheed ul Haq Syed

NUST College of Electrical amp Mechanical Engineering National University of Sciences amp Technology Peshawar RoadRawalpindi 46000 Pakistan

Correspondence should be addressed to Raja Amer Azim raja amercemenustedupk

Received 29 August 2014 Revised 12 January 2015 Accepted 23 January 2015

Academic Editor Sabri Arik

Copyright copy 2015 Raja Amer Azim et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Three-wheeled vehicles are agile less complex but relatively more prone to rollover The current study focuses on the rollovermitigation control design using active front steering for such vehicles A lateral load transfer ratio (LLTR) adapted for a three-wheeled platform is presented Sliding mode control design strategy has been devised which results in pseudo-direct control forroll dynamics of the vehicleThe lag in vehicle roll angle response has beenmanaged using adaptive sliding surfaceThis concept canbe extended for other vehicle configurationsThe proposed control scheme is investigated for efficacy using a full vehicle simulationmodel of CarSim software and National Highway Traffic Safety Administrationrsquos proposed Fishhook maneuver The controller isable to limit the rollover propensity even with vehicle parameter uncertainties

1 Introduction

Typical aircrafts while taxing on ground many wheeledrobots and some off- and on-road vehicles use a three-wheeled configuration Due to agility and weight benefitsit is also used as a transport medium for short distancepublic transport In wheeled vehicles rollover accidents area very small fraction of the total accidents but account formore than one third of the total fatalities and severe injuries[1] Static stability factor is one of the benchmarks used byNational Highway Traffic Safety Administration (NHTSA)for comparison of relative rollover likelihood of vehicles Itis the ratio between the track width and height of centerof gravity (cg) [2] For a three-wheeled vehicle this ratio isreduced as compared to a four-wheeled vehicle of similar sizeby a factor equal to the dimensionless distance of cg from rearaxle [3] Hence with the same track width and cg height athree-wheeled configuration is more prone to rollover thana similar sized four-wheeled platform Roll dynamics of athree-wheeled vehicle is sensitive to the vehicle track widthheight and longitudinal position of cg as observed in [3ndash5]Mukherjee et al [6] evaluated the rollover propensity of athree-wheeled vehicle using J-Turn and Road Edge Recovery

(RER) maneuversThe wheel liftoff condition was used as therollover thresholdThey found that themaneuver entry speedat nominal loading conditions which corresponds to wheelliftoff was 798ms for J-Turn and 90ms for RER On theother hand the maneuver entry speed for J-Turn and RERfor a four-wheeled (SUV) vehicle in reduced rollover config-uration was found to be 1729ms and 1609ms respectivelyas reported by Forkenbrock et al [7] Hence the rolloversusceptibility of a three-wheeled vehicle is almost two timesmore than that of a least stable four-wheeled SUV Based uponlateral acceleration limits for rollover four-wheeled vehiclesappear more prone to directional instabilities than rollover ascompared to three-wheeled vehicles

Rollover of three-wheeled vehicles has resulted in seriousinjuries and even fatalities of passengers On the averageof the total traffic related fatalities reported in India in2009sim2011 [8ndash10] three-wheeled vehicles were involved in5 of the total accidents The share in serious injuries is8 Even with these limitations production of three-wheeledvehicle has doubled in India In Mumbai three-wheeledvehicles are 11 of the total vehicles whereas 20 of dailycommuters use three-wheeled vehicles [11] This highlightsan increased risk as compared to other forms of transport

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 918429 9 pageshttpdxdoiorg1011552015918429

2 Mathematical Problems in Engineering

Another important factor revealed by a report on accidentdata [8] was that 775 of reported accidents were due todriverrsquos mistake Conventional three-wheeled vehicles have a1 1 steering and road-wheel angle ratio This results in veryhigh rate steering input in emergency or other impulsivereactions by the driver The steering response of three-wheeled vehicles is also significantly fast as compared to four-wheeled vehicles [3 12] thus highlighting its vulnerability

Rollover events are classified as tripped and untripped[13] Abrupt bumps and soft soil patches induce trippedrollovers These rollovers may occur even while traveling ona straight path Maneuver induced rollovers are termed asuntripped rollovers which are mainly the result of improperdriver input that is steering and speed while corneringNHTSA in report [14] highlights that ESC (Electronic Sta-bility Control) systems reduce fatalities by 34 in multiplevehicle crashes for passenger cars and SUVs Single vehiclecrashes resulted in up to 74 less fatalities in SUVs using ESCsystems This signifies the importance and efficacy of suchsystems and therefore a considerable research has focused onthis area [15ndash18]

Untripped rollover prevention systems can be classifiedinto three main categories The first type utilizes activesuspension systems which directly controls the roll motionresulting in increased rollover threshold [19ndash21] The secondcategory is based upon the roll and yaw motion couplingBraking to reduce speed differential braking and steeringare employed which indirectly reduces lateral accelerationto manage vehicle roll motion [22ndash24] The third categoryis termed as integrated chassis control which is usually thecombination of direct and indirect control interventions [25ndash27] Active suspension systems are typically expensive andbraking control relies on existing antilock braking system(ABS) All these systems have been developed for four ormore wheeled vehicles with wheels on both sides of thelongitudinal mid-plane

In a typical one-front and two-rear wheel (delta) configu-ration a three-wheeled vehicle has no roll stiffness at the frontaxle The rear two-wheeled axle unloads in braking whichreduces the lateral load transfer margin for rollover Becauseof this difference in vehicle setup the above mentionedcontrol systems have not yet been incorporated for com-mercial three-wheeled vehicles For three-wheeled platformsactive leaning mechanisms for rollover mitigation have beenproposed and developed as one of the solutions to addressthis problem [28ndash30] Two main schemes for tilt controlexist One is the Direct Tilt Control (DTC) as proposed byPiyabongkarn et al [28] In this scheme dedicated actuatorswere used for tilting the vehicle based on lateral accelerationand input steering angle The whole vehicle is controlled tolean towards the turn center In order to effectively mitigaterollover threat it requires significant actuator effort to initiateleaning To overcome this problem a split chassis designusing steering tilt control (STC) was proposed by Barker etal [29] which incorporates leaning of front single wheel forsteering and passenger compartment while the rear enginepod with actuators does not lean A combination of both STCand DTC modes was adopted by Kidane et al [30] with anadditional tilt brake system

The proposed techniques for three-wheeled vehicles dis-cussed above though effective have significant limitationsfrom implementation perspective for a single unit chassiscommercial three-wheeled vehicleThe current study focuseson a rollover prevention system for existing single chassisthree-wheeled platforms Considering the limitations of abrake based system a steering based rollover preventionsystem is proposed A superposition (additive) steering canbe integrating into existing system making it a viable add-on A rollover mitigationprevention controller is designedusing sliding mode control based on a lower order fullvehicle roll dynamics model The model has the advantagethat the steering input directly affects the roll motion Suchlow fidelity models represent pre-liftoff roll dynamics withreasonable accuracy for four-wheeled vehicles as discussed in[31 32] To conserve power a control trigger rule was usedCarSim and Simulink environments are set for cosimulationto evaluate the dynamics of the vehicle and performance ofthe controller under NHTSA defined Fishhook maneuversEffect of vehicle mass properties and cg height variation onthe controller performance was investigated

The paper is organized as the following vehicle model ispresented in Section 2 lateral load transfer ratio adapted for athree-wheeled vehicle is presented in Section 3 the proposedcontroller is detailed in Section 4 followed by simulations inSection 5 and conclusions are presented in Section 6

2 Vehicle Model for Controller Design

In this study a full vehicle model adapted from [4] is usedSmall angle approximations were used neglecting unsprungmass and roll axis effects In a typical rollover controllerdesign roll-yaw motion coupling is exploited to controlroll dynamics However the employed model enables directcontrol of roll motion using steering input The modelingscheme used also facilitates finding equivalent control

A rigid chassis model with six degrees of freedom is usedUnsprung mass is lumped with the sprung mass

Using a vehicle fixed coordinate system at the center ofgravity as shown in Figure 1 equations ofmotions are derivedas given below

= 119910

120595 +

119875 (119905)

119898

minus

120583

119903

119898

(119865119911

119903119897+ 119865119911

119903119903+ 119865119911

119891)

minus

(119861119909

119903119897+ 119861119909

119903119903+ 119861119909

119891)

119898

119910 =

120595 +

(119865119910

119891+ 119865119910

119903119897+ 119865119910

119903119903)

119898

= minus119892 +

(119865119911

119891+ 119865119911

119903119897+ 119865119911

119903119903)

119898

minus

120601 119910

119898

120601 =

1

119868

119909119909

[119887 (119865119911

119903119897minus 119865119911

119903119903) minus (ℎ

0+ 119911) (119865119910

119891+ 119865119910

119903119897+ 119865119910

119903119903)

minus

120579

120595 (119868

119911119911minus 119868

119910119910)]

Mathematical Problems in Engineering 3

Rear left wheel

Rear right wheel

Front wheel

120595

x

y

z

120601

120579

cg

Figure 1 Vehicle layout used formodel developmentThe equationsof motion are derived with respect to the coordinate system placedat the center of gravity

120595 =

1

119868

119911119911

[119887120583

119903(119865119911

119903119903minus 119865119911

119903119897) + 119887 (119861119909

119903119903minus 119861119909

119903119897)

+ 119865119910

119891119897

119891minus 119897

119903(119865119910

119903119897+ 119865119910

119903119903) minus

120601

120579 (119868

119910119910minus 119868

119911119911)]

120579 =

1

119868

119910119910

[119897

119891119865119911

119891minus 119897

119903(119865119911

119903119897+ 119865119911

119903119903)

+ (ℎ

0+ 119911) (

119875 (119905)

minus 120583

119903sum119865119911 minussum119861119909)]

(1)

Linear tire model is used for calculating lateral forces asfollows

119865119910

119891= 119862

120572119891120572

119891

119865119910

119903119897= 119862

120572119903120572

119903119897

119865119910

119903119903= 119862

120572119903120572

119903119903

(2)

where 119862120572119903and 119862

120572119891are rear and front cornering stiffness and

the slip angles 120572119891 120572

119903119897 120572

119903119903are expressed in terms of vehicle

states as follows

120572

119891= 120575

119891minus

( 119910 + 119897

119897

120595)

120572

119903119897= minus

119910 minus 119897

119903

120595

minus 119887

120595

120572

119903119903= minus

119910 minus 119897

119903

120595

+ 119887

120595

(3)

where 120575

119891is the front steering angle The vertical forces on

each wheel are calculated as follows

119865119911

119891= 119865119911

119891119894+ 119870

119891(minus119911 + 119897

119891120579) + 119888

119891(minus + 119897

119891

120579) (4)

119865119911

119903119897= 119865119911

119903119897119894+ 119870

119903(minus119911 minus 119897

119903120579 minus 119887120601) + 119888

119903(minus minus 119897

119903

120579 minus 119887

120601)

(5)

119865119911

119903119903= 119865119911

119903119903119894+ 119870

119903(minus119911 minus 119897

119903120579 + 119887120601) + 119888

119903(minus minus 119897

119903

120579 + 119887

120601)

(6)

where subscript 119894 denotes initial static forces119870 is the effectivesuspension stiffness and 119888 is the effective suspension dampingas tire characteristics are lumped with the suspension com-pliance characteristics It may be noted here that the minus119897

119903120579

and minus119897

119903

120579 terms unload the rear wheels in breaking and hencereduce the total lateral load transfer required to unload therear wheels which is further defined in Section 3 of this paper

The roll dynamics equation in terms of state variables canbe written as follows

120601 =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601)

minus (ℎ + 119911) [

( 119910 + 119897

119891

120595) 119862

120572119891

]

minus

120579

120601 (119868

119911119911minus 119868

119910119910) +

(ℎ + 119911)

119868

119909119909

120575

119891

(7)

Nominal values for parameters 119897

119903 119897

119891 119887 119898 119897 are directly

used from previous studies on three-wheeled vehicle [45 33 34] while the parameters 119862

120572119891 119862

120572119903 ℎ are adapted by

minimizing a cost function 119869 as given in (8) which is thesummation of normalized errors of roll angle and roll ratebetween values obtained from (7) and CarSim This ensuresthat the model used for controller design represents vehiclebehavior with reasonable accuracy

119869 = sum(

1003816

1003816

1003816

1003816

120601CarSim minus 120601Model1003816

1003816

1003816

1003816

max 1003816100381610038161003816

120601CarSim1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

120601CarSim minus

120601Model1003816

1003816

1003816

1003816

1003816

max 1003816100381610038161003816

1003816

120601CarSim1003816

1003816

1003816

1003816

1003816

) (8)

The response in terms of roll angle and roll rate to a stepinput of 3 degrees at 40 kmhr speed is shown in Figure 2

The response clearly indicates a delay of almost 055seconds for the roll angle to evolve Although close to anominal step input of this type is uncommon in typical roadvehicles due to the rate at which an average person can turnthe wheel coupled with the reduction in steering angle inputfrom hand-wheel to the road wheel The steering input rateused in NHTSA J-turn test is 720 degsec which effectivelytranslates into 36 degrees per second for a typical 20 1reduction In the absence of any reduction in the steeringsystem of a common three-wheeled vehicle using motorcyclesteering assemblies a steering input close to step input is infact a typical input for an evasive maneuver

4 Mathematical Problems in Engineering

13

12

11

10

9

8

7

6

5

4

3

2

1

0

minus1

00 05 10 15 20

Time (s)

Steer angle (deg)Roll angle (deg)

Roll rate (degs)

Figure 2 Roll angle and roll rate of vehicle resulting from a stepsteering input of 3 degrees

3 Rollover Detection

During cornering the inner wheel(s) in a turn are unloadedand the load is transferred to the outer wheel(s) NHSTAdefines liftoff when all wheels of one side of a vehicleleave contact with the ground This loss of contact meanslesser influence of driver input and loss of roll damp-ing provided by the suspension It is for this reason thatwheel liftoff condition is taken as the threshold or lim-iting value for all control scheme developments Severalfactors such as track width height of center of gravityand suspension characteristics affect the rate at which thisload transfer occurs hence specific rollover propensity fora particular vehicle setup exists as elaborated in [35ndash37]The lateral load transfer ratio (LLTR) is given by (9) as in[25]

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903+ 119865119911

119897

1003816

1003816

1003816

1003816

(9)

where 119865119911

119903and 119865119911

119897are the total vertical forces on the right

and left side wheels The value varies from minus1 to 1 A valueof zero is for no lateral load transfer A maximum value ofplusmn1 signifying total load on wheels on one side of the vehiclehence represents wheel(s) liftoff

In a four-wheeled vehicle a single wheel maintainingcontact on either side would keep significant roll motiondamping in effect In a three-wheeled vehicle only the axlehaving two wheels provides the roll motion damping hence asingle wheel liftoff is critical enough In a delta configurationbraking unloads the rear wheels as shown in (5) and (6)reducing the threshold for lateral load transfer Hence forthree-wheeled vehicles the following lateral load transferratio is suggested and is used in this research depending onlyon wheel loads at the rear axle

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903119903+ 119865119911

119903119897

1003816

1003816

1003816

1003816

(10)

Driver inputSuperposition steering assy

Vehicle

Controller State feedback

Compensation steer

Required steeringinput

Figure 3 Layout suggested for superposition steering system forcontroller realization

where119865119911119903119903and119865119911

119903119897denote rear right and rear left vertical tire

forces

4 Rollover Mitigation Controller

A rollover mitigation controller based on steering alone isproposed Slidingmode control is one of the many promisingcontrol techniques being used for vehicle control Rollovermitigation controller in [17] uses reference lateral positionand its derivative as the tracking error which are found byintegrating lateral acceleration corresponding to thresholdLLTR Chen et al [18] have proposed a sliding manifoldusing yaw rate following error sideslip angle and lateralacceleration The controller exploits the yaw-roll couplingand differential braking is used as control input

In this study roll angle 120601th corresponding to the targetLLTR at steady state is set as limiting value A superpositionsteering system as suggested by Joachim and Boerner [38]can be utilized for realization of this system and the layoutis proposed as shown in Figure 3 This reduces the inputrequired for the controller and only compensation steeringinput is required from the controller

A direct mechanical linkage between steering wheel androad wheel along with feedback to the driver to an extent ismaintained conforming to existing road vehicle standardsfor steering systems To limit the LLTR in finite time thesliding variable 120590 is defined as follows

120590 = 119890 + 120572119890 (11)

where 120572 gt 0 and 119890 = 120601th minus 120601 The state feedback slidingmode control design is based upon roll dynamics equation(7) which in standard form including disturbances is writtenas follows

120601 = 119891 (120601

120601) + 119892 (119911 120601

120601) 119906 + 119889 (119911 120601

120601) (12)

where

119891 (120601

120601) =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601) minus (ℎ + 119911)

sdot [

( 119910 + 119897

119891

120595) 119862

120572119891

] minus

120579

120601 (119868

119911119911minus 119868

119910119910)

119892 =

(ℎ + 119911)

119868

119909119909

(13)

Mathematical Problems in Engineering 5

This disturbance term 119889(119911 120601

120601) represents modelinguncertaintiesThe control law has two components as follows

119906 = 119906eq + 119906

119904 (14)

where 119906eq is the equivalent control and 119906119904 is the conventionalswitching control The equivalent control is based upon thenominal model and is given by

119906eq =(minus119891 (120601

120601) minus 120572

120601)

119892 (119911 120601

120601)

(15)

The switching component 119906119904is given by

119906

119904= 120588 sign (120590) (16)

where

120588 = 119872 + 120581 (17)

and 119872 is the bound on the disturbance term 119889(119911 120601

120601) thatsatisfies

1003816

1003816

1003816

1003816

1003816

119889 (119911 120601

120601)

1003816

1003816

1003816

1003816

1003816

le 119872 (18)

for all 119911 120601

120601 in the domain of interestThe finite time convergence to the sliding surface is guar-

anteed by the positive constant 120581 and the rate of convergencedepends on its magnitude as discussed by Shtessel et al [39]

An important consideration in the controller implemen-tation is that the controller attempts to regulate the roll angleat the limiting value as soon it is activated A threshold LLTRactivates the controller As roll angle value lags behind theLLTR by approximately 01 seconds with increasing LLTRreaching a level of critical value the value of roll angleis still less than the critical value corresponding to steadystate LLTR If the controller is activated based on LLTRit will try to immediately push the roll angle to set valueincreasing the LLTR overshoot significantly The referencethreshold in this condition is set as the current roll anglewhich is less than the threshold steady state roll anglecorresponding to desired LLTRThe evolution of roll angle isthus restricted significantly Furthermore the driver steeringinput is compared with the value of required steering inputfor maintaining the vehicle roll angle at threshold value Toprovide for the switching control toworkwith less oscillationsdue to one sided fencing of the roll angle the controlleroutput value is used with correction even if it is up to 5less than steering input of the driver This requires activationof controller prior to reaching the critical LLTR value Theswitching relay part of the controller can have a larger gainfor robustness requiring a larger operating range of controlleroutput with respect to the driver input It would result indelayed switch-off of the controller input even if the driverhas turned the steering wheel back To cater for this thedynamic regularization of the relay part of the controllersigmoid approximation of sign(sdot) is used as shown in

sign (119904) = tanh(119904120576

) (19)

0 2 4 6 8 10 12

Time (s)

Stee

r ang

le (d

eg)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

Figure 4 NHTSA Fishhook steering profile used for evaluatingrollover propensity

5 Simulations

In order to evaluate the efficacy of the proposed controllera high fidelity model of a three-wheeled vehicle is set up inCarSim softwareThismodel includes nonlinear effects of rollcenter movement and tire force nonlinearities including tirerelaxation length causing a further lag in response of the vehi-cle as discussed in Section 2 of this paperThe suspensions arealso modeled using nonlinear dampers and stiffness whichare typical of light vehicle suspension systemsThe controlleris coded as a level-1 S-function in MatlabSimulink NHTSAJ-Turn test is used for evaluating the efficacy of the controllerA cosimulation is set up inMatlabSimulink using simulationmodel of CarSim

While assessing the available data on three-wheeledvehicles it was observed that most of the untripped rolloveraccidents were caused by sudden large steering to avoid anobstacle on the road The second major cause was over-speeding while cornering [9 10] Considering the steeringinput severity pattern and available standards NHTSA Fish-hook maneuver was selected for evaluation of the proposedcontroller A slowly increasing steering input at constantspeed was given A steering input corresponding to a lateralacceleration of 03 g is measured This steering angle wasscaled by a factor of 8 to account for maneuver severity andused as the amplitude of ramp steering on each directionThe angle was changed with a rate of 720 degrees per secondduring the first two ramps and a dwell period of three secondsafter the steering reversal is maintained The correspondingroad wheel steering angle profile for a typical Asian three-wheeled vehicle is as shown in Figure 4

A maneuver entrance speed of 35 kmhr is used Athreshold roll angle is set at a steady state LLTR of 08arbitrarily considering a margin of safetyThe LLTR with andwithout controller is shown in Figure 5 A value of 1 and minus1for LTTR indicates wheel liftoff while the controller is able tomaintain these values near 08 The transients are managed

6 Mathematical Problems in Engineering

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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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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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Another important factor revealed by a report on accidentdata [8] was that 775 of reported accidents were due todriverrsquos mistake Conventional three-wheeled vehicles have a1 1 steering and road-wheel angle ratio This results in veryhigh rate steering input in emergency or other impulsivereactions by the driver The steering response of three-wheeled vehicles is also significantly fast as compared to four-wheeled vehicles [3 12] thus highlighting its vulnerability

Rollover events are classified as tripped and untripped[13] Abrupt bumps and soft soil patches induce trippedrollovers These rollovers may occur even while traveling ona straight path Maneuver induced rollovers are termed asuntripped rollovers which are mainly the result of improperdriver input that is steering and speed while corneringNHTSA in report [14] highlights that ESC (Electronic Sta-bility Control) systems reduce fatalities by 34 in multiplevehicle crashes for passenger cars and SUVs Single vehiclecrashes resulted in up to 74 less fatalities in SUVs using ESCsystems This signifies the importance and efficacy of suchsystems and therefore a considerable research has focused onthis area [15ndash18]

Untripped rollover prevention systems can be classifiedinto three main categories The first type utilizes activesuspension systems which directly controls the roll motionresulting in increased rollover threshold [19ndash21] The secondcategory is based upon the roll and yaw motion couplingBraking to reduce speed differential braking and steeringare employed which indirectly reduces lateral accelerationto manage vehicle roll motion [22ndash24] The third categoryis termed as integrated chassis control which is usually thecombination of direct and indirect control interventions [25ndash27] Active suspension systems are typically expensive andbraking control relies on existing antilock braking system(ABS) All these systems have been developed for four ormore wheeled vehicles with wheels on both sides of thelongitudinal mid-plane

In a typical one-front and two-rear wheel (delta) configu-ration a three-wheeled vehicle has no roll stiffness at the frontaxle The rear two-wheeled axle unloads in braking whichreduces the lateral load transfer margin for rollover Becauseof this difference in vehicle setup the above mentionedcontrol systems have not yet been incorporated for com-mercial three-wheeled vehicles For three-wheeled platformsactive leaning mechanisms for rollover mitigation have beenproposed and developed as one of the solutions to addressthis problem [28ndash30] Two main schemes for tilt controlexist One is the Direct Tilt Control (DTC) as proposed byPiyabongkarn et al [28] In this scheme dedicated actuatorswere used for tilting the vehicle based on lateral accelerationand input steering angle The whole vehicle is controlled tolean towards the turn center In order to effectively mitigaterollover threat it requires significant actuator effort to initiateleaning To overcome this problem a split chassis designusing steering tilt control (STC) was proposed by Barker etal [29] which incorporates leaning of front single wheel forsteering and passenger compartment while the rear enginepod with actuators does not lean A combination of both STCand DTC modes was adopted by Kidane et al [30] with anadditional tilt brake system

The proposed techniques for three-wheeled vehicles dis-cussed above though effective have significant limitationsfrom implementation perspective for a single unit chassiscommercial three-wheeled vehicleThe current study focuseson a rollover prevention system for existing single chassisthree-wheeled platforms Considering the limitations of abrake based system a steering based rollover preventionsystem is proposed A superposition (additive) steering canbe integrating into existing system making it a viable add-on A rollover mitigationprevention controller is designedusing sliding mode control based on a lower order fullvehicle roll dynamics model The model has the advantagethat the steering input directly affects the roll motion Suchlow fidelity models represent pre-liftoff roll dynamics withreasonable accuracy for four-wheeled vehicles as discussed in[31 32] To conserve power a control trigger rule was usedCarSim and Simulink environments are set for cosimulationto evaluate the dynamics of the vehicle and performance ofthe controller under NHTSA defined Fishhook maneuversEffect of vehicle mass properties and cg height variation onthe controller performance was investigated

The paper is organized as the following vehicle model ispresented in Section 2 lateral load transfer ratio adapted for athree-wheeled vehicle is presented in Section 3 the proposedcontroller is detailed in Section 4 followed by simulations inSection 5 and conclusions are presented in Section 6

2 Vehicle Model for Controller Design

In this study a full vehicle model adapted from [4] is usedSmall angle approximations were used neglecting unsprungmass and roll axis effects In a typical rollover controllerdesign roll-yaw motion coupling is exploited to controlroll dynamics However the employed model enables directcontrol of roll motion using steering input The modelingscheme used also facilitates finding equivalent control

A rigid chassis model with six degrees of freedom is usedUnsprung mass is lumped with the sprung mass

Using a vehicle fixed coordinate system at the center ofgravity as shown in Figure 1 equations ofmotions are derivedas given below

= 119910

120595 +

119875 (119905)

119898

minus

120583

119903

119898

(119865119911

119903119897+ 119865119911

119903119903+ 119865119911

119891)

minus

(119861119909

119903119897+ 119861119909

119903119903+ 119861119909

119891)

119898

119910 =

120595 +

(119865119910

119891+ 119865119910

119903119897+ 119865119910

119903119903)

119898

= minus119892 +

(119865119911

119891+ 119865119911

119903119897+ 119865119911

119903119903)

119898

minus

120601 119910

119898

120601 =

1

119868

119909119909

[119887 (119865119911

119903119897minus 119865119911

119903119903) minus (ℎ

0+ 119911) (119865119910

119891+ 119865119910

119903119897+ 119865119910

119903119903)

minus

120579

120595 (119868

119911119911minus 119868

119910119910)]

Mathematical Problems in Engineering 3

Rear left wheel

Rear right wheel

Front wheel

120595

x

y

z

120601

120579

cg

Figure 1 Vehicle layout used formodel developmentThe equationsof motion are derived with respect to the coordinate system placedat the center of gravity

120595 =

1

119868

119911119911

[119887120583

119903(119865119911

119903119903minus 119865119911

119903119897) + 119887 (119861119909

119903119903minus 119861119909

119903119897)

+ 119865119910

119891119897

119891minus 119897

119903(119865119910

119903119897+ 119865119910

119903119903) minus

120601

120579 (119868

119910119910minus 119868

119911119911)]

120579 =

1

119868

119910119910

[119897

119891119865119911

119891minus 119897

119903(119865119911

119903119897+ 119865119911

119903119903)

+ (ℎ

0+ 119911) (

119875 (119905)

minus 120583

119903sum119865119911 minussum119861119909)]

(1)

Linear tire model is used for calculating lateral forces asfollows

119865119910

119891= 119862

120572119891120572

119891

119865119910

119903119897= 119862

120572119903120572

119903119897

119865119910

119903119903= 119862

120572119903120572

119903119903

(2)

where 119862120572119903and 119862

120572119891are rear and front cornering stiffness and

the slip angles 120572119891 120572

119903119897 120572

119903119903are expressed in terms of vehicle

states as follows

120572

119891= 120575

119891minus

( 119910 + 119897

119897

120595)

120572

119903119897= minus

119910 minus 119897

119903

120595

minus 119887

120595

120572

119903119903= minus

119910 minus 119897

119903

120595

+ 119887

120595

(3)

where 120575

119891is the front steering angle The vertical forces on

each wheel are calculated as follows

119865119911

119891= 119865119911

119891119894+ 119870

119891(minus119911 + 119897

119891120579) + 119888

119891(minus + 119897

119891

120579) (4)

119865119911

119903119897= 119865119911

119903119897119894+ 119870

119903(minus119911 minus 119897

119903120579 minus 119887120601) + 119888

119903(minus minus 119897

119903

120579 minus 119887

120601)

(5)

119865119911

119903119903= 119865119911

119903119903119894+ 119870

119903(minus119911 minus 119897

119903120579 + 119887120601) + 119888

119903(minus minus 119897

119903

120579 + 119887

120601)

(6)

where subscript 119894 denotes initial static forces119870 is the effectivesuspension stiffness and 119888 is the effective suspension dampingas tire characteristics are lumped with the suspension com-pliance characteristics It may be noted here that the minus119897

119903120579

and minus119897

119903

120579 terms unload the rear wheels in breaking and hencereduce the total lateral load transfer required to unload therear wheels which is further defined in Section 3 of this paper

The roll dynamics equation in terms of state variables canbe written as follows

120601 =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601)

minus (ℎ + 119911) [

( 119910 + 119897

119891

120595) 119862

120572119891

]

minus

120579

120601 (119868

119911119911minus 119868

119910119910) +

(ℎ + 119911)

119868

119909119909

120575

119891

(7)

Nominal values for parameters 119897

119903 119897

119891 119887 119898 119897 are directly

used from previous studies on three-wheeled vehicle [45 33 34] while the parameters 119862

120572119891 119862

120572119903 ℎ are adapted by

minimizing a cost function 119869 as given in (8) which is thesummation of normalized errors of roll angle and roll ratebetween values obtained from (7) and CarSim This ensuresthat the model used for controller design represents vehiclebehavior with reasonable accuracy

119869 = sum(

1003816

1003816

1003816

1003816

120601CarSim minus 120601Model1003816

1003816

1003816

1003816

max 1003816100381610038161003816

120601CarSim1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

120601CarSim minus

120601Model1003816

1003816

1003816

1003816

1003816

max 1003816100381610038161003816

1003816

120601CarSim1003816

1003816

1003816

1003816

1003816

) (8)

The response in terms of roll angle and roll rate to a stepinput of 3 degrees at 40 kmhr speed is shown in Figure 2

The response clearly indicates a delay of almost 055seconds for the roll angle to evolve Although close to anominal step input of this type is uncommon in typical roadvehicles due to the rate at which an average person can turnthe wheel coupled with the reduction in steering angle inputfrom hand-wheel to the road wheel The steering input rateused in NHTSA J-turn test is 720 degsec which effectivelytranslates into 36 degrees per second for a typical 20 1reduction In the absence of any reduction in the steeringsystem of a common three-wheeled vehicle using motorcyclesteering assemblies a steering input close to step input is infact a typical input for an evasive maneuver

4 Mathematical Problems in Engineering

13

12

11

10

9

8

7

6

5

4

3

2

1

0

minus1

00 05 10 15 20

Time (s)

Steer angle (deg)Roll angle (deg)

Roll rate (degs)

Figure 2 Roll angle and roll rate of vehicle resulting from a stepsteering input of 3 degrees

3 Rollover Detection

During cornering the inner wheel(s) in a turn are unloadedand the load is transferred to the outer wheel(s) NHSTAdefines liftoff when all wheels of one side of a vehicleleave contact with the ground This loss of contact meanslesser influence of driver input and loss of roll damp-ing provided by the suspension It is for this reason thatwheel liftoff condition is taken as the threshold or lim-iting value for all control scheme developments Severalfactors such as track width height of center of gravityand suspension characteristics affect the rate at which thisload transfer occurs hence specific rollover propensity fora particular vehicle setup exists as elaborated in [35ndash37]The lateral load transfer ratio (LLTR) is given by (9) as in[25]

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903+ 119865119911

119897

1003816

1003816

1003816

1003816

(9)

where 119865119911

119903and 119865119911

119897are the total vertical forces on the right

and left side wheels The value varies from minus1 to 1 A valueof zero is for no lateral load transfer A maximum value ofplusmn1 signifying total load on wheels on one side of the vehiclehence represents wheel(s) liftoff

In a four-wheeled vehicle a single wheel maintainingcontact on either side would keep significant roll motiondamping in effect In a three-wheeled vehicle only the axlehaving two wheels provides the roll motion damping hence asingle wheel liftoff is critical enough In a delta configurationbraking unloads the rear wheels as shown in (5) and (6)reducing the threshold for lateral load transfer Hence forthree-wheeled vehicles the following lateral load transferratio is suggested and is used in this research depending onlyon wheel loads at the rear axle

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903119903+ 119865119911

119903119897

1003816

1003816

1003816

1003816

(10)

Driver inputSuperposition steering assy

Vehicle

Controller State feedback

Compensation steer

Required steeringinput

Figure 3 Layout suggested for superposition steering system forcontroller realization

where119865119911119903119903and119865119911

119903119897denote rear right and rear left vertical tire

forces

4 Rollover Mitigation Controller

A rollover mitigation controller based on steering alone isproposed Slidingmode control is one of the many promisingcontrol techniques being used for vehicle control Rollovermitigation controller in [17] uses reference lateral positionand its derivative as the tracking error which are found byintegrating lateral acceleration corresponding to thresholdLLTR Chen et al [18] have proposed a sliding manifoldusing yaw rate following error sideslip angle and lateralacceleration The controller exploits the yaw-roll couplingand differential braking is used as control input

In this study roll angle 120601th corresponding to the targetLLTR at steady state is set as limiting value A superpositionsteering system as suggested by Joachim and Boerner [38]can be utilized for realization of this system and the layoutis proposed as shown in Figure 3 This reduces the inputrequired for the controller and only compensation steeringinput is required from the controller

A direct mechanical linkage between steering wheel androad wheel along with feedback to the driver to an extent ismaintained conforming to existing road vehicle standardsfor steering systems To limit the LLTR in finite time thesliding variable 120590 is defined as follows

120590 = 119890 + 120572119890 (11)

where 120572 gt 0 and 119890 = 120601th minus 120601 The state feedback slidingmode control design is based upon roll dynamics equation(7) which in standard form including disturbances is writtenas follows

120601 = 119891 (120601

120601) + 119892 (119911 120601

120601) 119906 + 119889 (119911 120601

120601) (12)

where

119891 (120601

120601) =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601) minus (ℎ + 119911)

sdot [

( 119910 + 119897

119891

120595) 119862

120572119891

] minus

120579

120601 (119868

119911119911minus 119868

119910119910)

119892 =

(ℎ + 119911)

119868

119909119909

(13)

Mathematical Problems in Engineering 5

This disturbance term 119889(119911 120601

120601) represents modelinguncertaintiesThe control law has two components as follows

119906 = 119906eq + 119906

119904 (14)

where 119906eq is the equivalent control and 119906119904 is the conventionalswitching control The equivalent control is based upon thenominal model and is given by

119906eq =(minus119891 (120601

120601) minus 120572

120601)

119892 (119911 120601

120601)

(15)

The switching component 119906119904is given by

119906

119904= 120588 sign (120590) (16)

where

120588 = 119872 + 120581 (17)

and 119872 is the bound on the disturbance term 119889(119911 120601

120601) thatsatisfies

1003816

1003816

1003816

1003816

1003816

119889 (119911 120601

120601)

1003816

1003816

1003816

1003816

1003816

le 119872 (18)

for all 119911 120601

120601 in the domain of interestThe finite time convergence to the sliding surface is guar-

anteed by the positive constant 120581 and the rate of convergencedepends on its magnitude as discussed by Shtessel et al [39]

An important consideration in the controller implemen-tation is that the controller attempts to regulate the roll angleat the limiting value as soon it is activated A threshold LLTRactivates the controller As roll angle value lags behind theLLTR by approximately 01 seconds with increasing LLTRreaching a level of critical value the value of roll angleis still less than the critical value corresponding to steadystate LLTR If the controller is activated based on LLTRit will try to immediately push the roll angle to set valueincreasing the LLTR overshoot significantly The referencethreshold in this condition is set as the current roll anglewhich is less than the threshold steady state roll anglecorresponding to desired LLTRThe evolution of roll angle isthus restricted significantly Furthermore the driver steeringinput is compared with the value of required steering inputfor maintaining the vehicle roll angle at threshold value Toprovide for the switching control toworkwith less oscillationsdue to one sided fencing of the roll angle the controlleroutput value is used with correction even if it is up to 5less than steering input of the driver This requires activationof controller prior to reaching the critical LLTR value Theswitching relay part of the controller can have a larger gainfor robustness requiring a larger operating range of controlleroutput with respect to the driver input It would result indelayed switch-off of the controller input even if the driverhas turned the steering wheel back To cater for this thedynamic regularization of the relay part of the controllersigmoid approximation of sign(sdot) is used as shown in

sign (119904) = tanh(119904120576

) (19)

0 2 4 6 8 10 12

Time (s)

Stee

r ang

le (d

eg)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

Figure 4 NHTSA Fishhook steering profile used for evaluatingrollover propensity

5 Simulations

In order to evaluate the efficacy of the proposed controllera high fidelity model of a three-wheeled vehicle is set up inCarSim softwareThismodel includes nonlinear effects of rollcenter movement and tire force nonlinearities including tirerelaxation length causing a further lag in response of the vehi-cle as discussed in Section 2 of this paperThe suspensions arealso modeled using nonlinear dampers and stiffness whichare typical of light vehicle suspension systemsThe controlleris coded as a level-1 S-function in MatlabSimulink NHTSAJ-Turn test is used for evaluating the efficacy of the controllerA cosimulation is set up inMatlabSimulink using simulationmodel of CarSim

While assessing the available data on three-wheeledvehicles it was observed that most of the untripped rolloveraccidents were caused by sudden large steering to avoid anobstacle on the road The second major cause was over-speeding while cornering [9 10] Considering the steeringinput severity pattern and available standards NHTSA Fish-hook maneuver was selected for evaluation of the proposedcontroller A slowly increasing steering input at constantspeed was given A steering input corresponding to a lateralacceleration of 03 g is measured This steering angle wasscaled by a factor of 8 to account for maneuver severity andused as the amplitude of ramp steering on each directionThe angle was changed with a rate of 720 degrees per secondduring the first two ramps and a dwell period of three secondsafter the steering reversal is maintained The correspondingroad wheel steering angle profile for a typical Asian three-wheeled vehicle is as shown in Figure 4

A maneuver entrance speed of 35 kmhr is used Athreshold roll angle is set at a steady state LLTR of 08arbitrarily considering a margin of safetyThe LLTR with andwithout controller is shown in Figure 5 A value of 1 and minus1for LTTR indicates wheel liftoff while the controller is able tomaintain these values near 08 The transients are managed

6 Mathematical Problems in Engineering

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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

Rear left wheel

Rear right wheel

Front wheel

120595

x

y

z

120601

120579

cg

Figure 1 Vehicle layout used formodel developmentThe equationsof motion are derived with respect to the coordinate system placedat the center of gravity

120595 =

1

119868

119911119911

[119887120583

119903(119865119911

119903119903minus 119865119911

119903119897) + 119887 (119861119909

119903119903minus 119861119909

119903119897)

+ 119865119910

119891119897

119891minus 119897

119903(119865119910

119903119897+ 119865119910

119903119903) minus

120601

120579 (119868

119910119910minus 119868

119911119911)]

120579 =

1

119868

119910119910

[119897

119891119865119911

119891minus 119897

119903(119865119911

119903119897+ 119865119911

119903119903)

+ (ℎ

0+ 119911) (

119875 (119905)

minus 120583

119903sum119865119911 minussum119861119909)]

(1)

Linear tire model is used for calculating lateral forces asfollows

119865119910

119891= 119862

120572119891120572

119891

119865119910

119903119897= 119862

120572119903120572

119903119897

119865119910

119903119903= 119862

120572119903120572

119903119903

(2)

where 119862120572119903and 119862

120572119891are rear and front cornering stiffness and

the slip angles 120572119891 120572

119903119897 120572

119903119903are expressed in terms of vehicle

states as follows

120572

119891= 120575

119891minus

( 119910 + 119897

119897

120595)

120572

119903119897= minus

119910 minus 119897

119903

120595

minus 119887

120595

120572

119903119903= minus

119910 minus 119897

119903

120595

+ 119887

120595

(3)

where 120575

119891is the front steering angle The vertical forces on

each wheel are calculated as follows

119865119911

119891= 119865119911

119891119894+ 119870

119891(minus119911 + 119897

119891120579) + 119888

119891(minus + 119897

119891

120579) (4)

119865119911

119903119897= 119865119911

119903119897119894+ 119870

119903(minus119911 minus 119897

119903120579 minus 119887120601) + 119888

119903(minus minus 119897

119903

120579 minus 119887

120601)

(5)

119865119911

119903119903= 119865119911

119903119903119894+ 119870

119903(minus119911 minus 119897

119903120579 + 119887120601) + 119888

119903(minus minus 119897

119903

120579 + 119887

120601)

(6)

where subscript 119894 denotes initial static forces119870 is the effectivesuspension stiffness and 119888 is the effective suspension dampingas tire characteristics are lumped with the suspension com-pliance characteristics It may be noted here that the minus119897

119903120579

and minus119897

119903

120579 terms unload the rear wheels in breaking and hencereduce the total lateral load transfer required to unload therear wheels which is further defined in Section 3 of this paper

The roll dynamics equation in terms of state variables canbe written as follows

120601 =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601)

minus (ℎ + 119911) [

( 119910 + 119897

119891

120595) 119862

120572119891

]

minus

120579

120601 (119868

119911119911minus 119868

119910119910) +

(ℎ + 119911)

119868

119909119909

120575

119891

(7)

Nominal values for parameters 119897

119903 119897

119891 119887 119898 119897 are directly

used from previous studies on three-wheeled vehicle [45 33 34] while the parameters 119862

120572119891 119862

120572119903 ℎ are adapted by

minimizing a cost function 119869 as given in (8) which is thesummation of normalized errors of roll angle and roll ratebetween values obtained from (7) and CarSim This ensuresthat the model used for controller design represents vehiclebehavior with reasonable accuracy

119869 = sum(

1003816

1003816

1003816

1003816

120601CarSim minus 120601Model1003816

1003816

1003816

1003816

max 1003816100381610038161003816

120601CarSim1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

1003816

120601CarSim minus

120601Model1003816

1003816

1003816

1003816

1003816

max 1003816100381610038161003816

1003816

120601CarSim1003816

1003816

1003816

1003816

1003816

) (8)

The response in terms of roll angle and roll rate to a stepinput of 3 degrees at 40 kmhr speed is shown in Figure 2

The response clearly indicates a delay of almost 055seconds for the roll angle to evolve Although close to anominal step input of this type is uncommon in typical roadvehicles due to the rate at which an average person can turnthe wheel coupled with the reduction in steering angle inputfrom hand-wheel to the road wheel The steering input rateused in NHTSA J-turn test is 720 degsec which effectivelytranslates into 36 degrees per second for a typical 20 1reduction In the absence of any reduction in the steeringsystem of a common three-wheeled vehicle using motorcyclesteering assemblies a steering input close to step input is infact a typical input for an evasive maneuver

4 Mathematical Problems in Engineering

13

12

11

10

9

8

7

6

5

4

3

2

1

0

minus1

00 05 10 15 20

Time (s)

Steer angle (deg)Roll angle (deg)

Roll rate (degs)

Figure 2 Roll angle and roll rate of vehicle resulting from a stepsteering input of 3 degrees

3 Rollover Detection

During cornering the inner wheel(s) in a turn are unloadedand the load is transferred to the outer wheel(s) NHSTAdefines liftoff when all wheels of one side of a vehicleleave contact with the ground This loss of contact meanslesser influence of driver input and loss of roll damp-ing provided by the suspension It is for this reason thatwheel liftoff condition is taken as the threshold or lim-iting value for all control scheme developments Severalfactors such as track width height of center of gravityand suspension characteristics affect the rate at which thisload transfer occurs hence specific rollover propensity fora particular vehicle setup exists as elaborated in [35ndash37]The lateral load transfer ratio (LLTR) is given by (9) as in[25]

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903+ 119865119911

119897

1003816

1003816

1003816

1003816

(9)

where 119865119911

119903and 119865119911

119897are the total vertical forces on the right

and left side wheels The value varies from minus1 to 1 A valueof zero is for no lateral load transfer A maximum value ofplusmn1 signifying total load on wheels on one side of the vehiclehence represents wheel(s) liftoff

In a four-wheeled vehicle a single wheel maintainingcontact on either side would keep significant roll motiondamping in effect In a three-wheeled vehicle only the axlehaving two wheels provides the roll motion damping hence asingle wheel liftoff is critical enough In a delta configurationbraking unloads the rear wheels as shown in (5) and (6)reducing the threshold for lateral load transfer Hence forthree-wheeled vehicles the following lateral load transferratio is suggested and is used in this research depending onlyon wheel loads at the rear axle

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903119903+ 119865119911

119903119897

1003816

1003816

1003816

1003816

(10)

Driver inputSuperposition steering assy

Vehicle

Controller State feedback

Compensation steer

Required steeringinput

Figure 3 Layout suggested for superposition steering system forcontroller realization

where119865119911119903119903and119865119911

119903119897denote rear right and rear left vertical tire

forces

4 Rollover Mitigation Controller

A rollover mitigation controller based on steering alone isproposed Slidingmode control is one of the many promisingcontrol techniques being used for vehicle control Rollovermitigation controller in [17] uses reference lateral positionand its derivative as the tracking error which are found byintegrating lateral acceleration corresponding to thresholdLLTR Chen et al [18] have proposed a sliding manifoldusing yaw rate following error sideslip angle and lateralacceleration The controller exploits the yaw-roll couplingand differential braking is used as control input

In this study roll angle 120601th corresponding to the targetLLTR at steady state is set as limiting value A superpositionsteering system as suggested by Joachim and Boerner [38]can be utilized for realization of this system and the layoutis proposed as shown in Figure 3 This reduces the inputrequired for the controller and only compensation steeringinput is required from the controller

A direct mechanical linkage between steering wheel androad wheel along with feedback to the driver to an extent ismaintained conforming to existing road vehicle standardsfor steering systems To limit the LLTR in finite time thesliding variable 120590 is defined as follows

120590 = 119890 + 120572119890 (11)

where 120572 gt 0 and 119890 = 120601th minus 120601 The state feedback slidingmode control design is based upon roll dynamics equation(7) which in standard form including disturbances is writtenas follows

120601 = 119891 (120601

120601) + 119892 (119911 120601

120601) 119906 + 119889 (119911 120601

120601) (12)

where

119891 (120601

120601) =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601) minus (ℎ + 119911)

sdot [

( 119910 + 119897

119891

120595) 119862

120572119891

] minus

120579

120601 (119868

119911119911minus 119868

119910119910)

119892 =

(ℎ + 119911)

119868

119909119909

(13)

Mathematical Problems in Engineering 5

This disturbance term 119889(119911 120601

120601) represents modelinguncertaintiesThe control law has two components as follows

119906 = 119906eq + 119906

119904 (14)

where 119906eq is the equivalent control and 119906119904 is the conventionalswitching control The equivalent control is based upon thenominal model and is given by

119906eq =(minus119891 (120601

120601) minus 120572

120601)

119892 (119911 120601

120601)

(15)

The switching component 119906119904is given by

119906

119904= 120588 sign (120590) (16)

where

120588 = 119872 + 120581 (17)

and 119872 is the bound on the disturbance term 119889(119911 120601

120601) thatsatisfies

1003816

1003816

1003816

1003816

1003816

119889 (119911 120601

120601)

1003816

1003816

1003816

1003816

1003816

le 119872 (18)

for all 119911 120601

120601 in the domain of interestThe finite time convergence to the sliding surface is guar-

anteed by the positive constant 120581 and the rate of convergencedepends on its magnitude as discussed by Shtessel et al [39]

An important consideration in the controller implemen-tation is that the controller attempts to regulate the roll angleat the limiting value as soon it is activated A threshold LLTRactivates the controller As roll angle value lags behind theLLTR by approximately 01 seconds with increasing LLTRreaching a level of critical value the value of roll angleis still less than the critical value corresponding to steadystate LLTR If the controller is activated based on LLTRit will try to immediately push the roll angle to set valueincreasing the LLTR overshoot significantly The referencethreshold in this condition is set as the current roll anglewhich is less than the threshold steady state roll anglecorresponding to desired LLTRThe evolution of roll angle isthus restricted significantly Furthermore the driver steeringinput is compared with the value of required steering inputfor maintaining the vehicle roll angle at threshold value Toprovide for the switching control toworkwith less oscillationsdue to one sided fencing of the roll angle the controlleroutput value is used with correction even if it is up to 5less than steering input of the driver This requires activationof controller prior to reaching the critical LLTR value Theswitching relay part of the controller can have a larger gainfor robustness requiring a larger operating range of controlleroutput with respect to the driver input It would result indelayed switch-off of the controller input even if the driverhas turned the steering wheel back To cater for this thedynamic regularization of the relay part of the controllersigmoid approximation of sign(sdot) is used as shown in

sign (119904) = tanh(119904120576

) (19)

0 2 4 6 8 10 12

Time (s)

Stee

r ang

le (d

eg)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

Figure 4 NHTSA Fishhook steering profile used for evaluatingrollover propensity

5 Simulations

In order to evaluate the efficacy of the proposed controllera high fidelity model of a three-wheeled vehicle is set up inCarSim softwareThismodel includes nonlinear effects of rollcenter movement and tire force nonlinearities including tirerelaxation length causing a further lag in response of the vehi-cle as discussed in Section 2 of this paperThe suspensions arealso modeled using nonlinear dampers and stiffness whichare typical of light vehicle suspension systemsThe controlleris coded as a level-1 S-function in MatlabSimulink NHTSAJ-Turn test is used for evaluating the efficacy of the controllerA cosimulation is set up inMatlabSimulink using simulationmodel of CarSim

While assessing the available data on three-wheeledvehicles it was observed that most of the untripped rolloveraccidents were caused by sudden large steering to avoid anobstacle on the road The second major cause was over-speeding while cornering [9 10] Considering the steeringinput severity pattern and available standards NHTSA Fish-hook maneuver was selected for evaluation of the proposedcontroller A slowly increasing steering input at constantspeed was given A steering input corresponding to a lateralacceleration of 03 g is measured This steering angle wasscaled by a factor of 8 to account for maneuver severity andused as the amplitude of ramp steering on each directionThe angle was changed with a rate of 720 degrees per secondduring the first two ramps and a dwell period of three secondsafter the steering reversal is maintained The correspondingroad wheel steering angle profile for a typical Asian three-wheeled vehicle is as shown in Figure 4

A maneuver entrance speed of 35 kmhr is used Athreshold roll angle is set at a steady state LLTR of 08arbitrarily considering a margin of safetyThe LLTR with andwithout controller is shown in Figure 5 A value of 1 and minus1for LTTR indicates wheel liftoff while the controller is able tomaintain these values near 08 The transients are managed

6 Mathematical Problems in Engineering

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

13

12

11

10

9

8

7

6

5

4

3

2

1

0

minus1

00 05 10 15 20

Time (s)

Steer angle (deg)Roll angle (deg)

Roll rate (degs)

Figure 2 Roll angle and roll rate of vehicle resulting from a stepsteering input of 3 degrees

3 Rollover Detection

During cornering the inner wheel(s) in a turn are unloadedand the load is transferred to the outer wheel(s) NHSTAdefines liftoff when all wheels of one side of a vehicleleave contact with the ground This loss of contact meanslesser influence of driver input and loss of roll damp-ing provided by the suspension It is for this reason thatwheel liftoff condition is taken as the threshold or lim-iting value for all control scheme developments Severalfactors such as track width height of center of gravityand suspension characteristics affect the rate at which thisload transfer occurs hence specific rollover propensity fora particular vehicle setup exists as elaborated in [35ndash37]The lateral load transfer ratio (LLTR) is given by (9) as in[25]

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903+ 119865119911

119897

1003816

1003816

1003816

1003816

(9)

where 119865119911

119903and 119865119911

119897are the total vertical forces on the right

and left side wheels The value varies from minus1 to 1 A valueof zero is for no lateral load transfer A maximum value ofplusmn1 signifying total load on wheels on one side of the vehiclehence represents wheel(s) liftoff

In a four-wheeled vehicle a single wheel maintainingcontact on either side would keep significant roll motiondamping in effect In a three-wheeled vehicle only the axlehaving two wheels provides the roll motion damping hence asingle wheel liftoff is critical enough In a delta configurationbraking unloads the rear wheels as shown in (5) and (6)reducing the threshold for lateral load transfer Hence forthree-wheeled vehicles the following lateral load transferratio is suggested and is used in this research depending onlyon wheel loads at the rear axle

LLTR =

119865119911

119903minus 119865119911

119897

1003816

1003816

1003816

1003816

119865119911

119903119903+ 119865119911

119903119897

1003816

1003816

1003816

1003816

(10)

Driver inputSuperposition steering assy

Vehicle

Controller State feedback

Compensation steer

Required steeringinput

Figure 3 Layout suggested for superposition steering system forcontroller realization

where119865119911119903119903and119865119911

119903119897denote rear right and rear left vertical tire

forces

4 Rollover Mitigation Controller

A rollover mitigation controller based on steering alone isproposed Slidingmode control is one of the many promisingcontrol techniques being used for vehicle control Rollovermitigation controller in [17] uses reference lateral positionand its derivative as the tracking error which are found byintegrating lateral acceleration corresponding to thresholdLLTR Chen et al [18] have proposed a sliding manifoldusing yaw rate following error sideslip angle and lateralacceleration The controller exploits the yaw-roll couplingand differential braking is used as control input

In this study roll angle 120601th corresponding to the targetLLTR at steady state is set as limiting value A superpositionsteering system as suggested by Joachim and Boerner [38]can be utilized for realization of this system and the layoutis proposed as shown in Figure 3 This reduces the inputrequired for the controller and only compensation steeringinput is required from the controller

A direct mechanical linkage between steering wheel androad wheel along with feedback to the driver to an extent ismaintained conforming to existing road vehicle standardsfor steering systems To limit the LLTR in finite time thesliding variable 120590 is defined as follows

120590 = 119890 + 120572119890 (11)

where 120572 gt 0 and 119890 = 120601th minus 120601 The state feedback slidingmode control design is based upon roll dynamics equation(7) which in standard form including disturbances is writtenas follows

120601 = 119891 (120601

120601) + 119892 (119911 120601

120601) 119906 + 119889 (119911 120601

120601) (12)

where

119891 (120601

120601) =

1

119868

119909119909

(ℎ + 119911) [minus

2 ( 119910 minus 119897

119903

120595) 119862

120572119903

(

2minus 119887

2

120595

2)

]

minus 2119887

2

(119870

119903120601 + 119888

119903

120601) minus (ℎ + 119911)

sdot [

( 119910 + 119897

119891

120595) 119862

120572119891

] minus

120579

120601 (119868

119911119911minus 119868

119910119910)

119892 =

(ℎ + 119911)

119868

119909119909

(13)

Mathematical Problems in Engineering 5

This disturbance term 119889(119911 120601

120601) represents modelinguncertaintiesThe control law has two components as follows

119906 = 119906eq + 119906

119904 (14)

where 119906eq is the equivalent control and 119906119904 is the conventionalswitching control The equivalent control is based upon thenominal model and is given by

119906eq =(minus119891 (120601

120601) minus 120572

120601)

119892 (119911 120601

120601)

(15)

The switching component 119906119904is given by

119906

119904= 120588 sign (120590) (16)

where

120588 = 119872 + 120581 (17)

and 119872 is the bound on the disturbance term 119889(119911 120601

120601) thatsatisfies

1003816

1003816

1003816

1003816

1003816

119889 (119911 120601

120601)

1003816

1003816

1003816

1003816

1003816

le 119872 (18)

for all 119911 120601

120601 in the domain of interestThe finite time convergence to the sliding surface is guar-

anteed by the positive constant 120581 and the rate of convergencedepends on its magnitude as discussed by Shtessel et al [39]

An important consideration in the controller implemen-tation is that the controller attempts to regulate the roll angleat the limiting value as soon it is activated A threshold LLTRactivates the controller As roll angle value lags behind theLLTR by approximately 01 seconds with increasing LLTRreaching a level of critical value the value of roll angleis still less than the critical value corresponding to steadystate LLTR If the controller is activated based on LLTRit will try to immediately push the roll angle to set valueincreasing the LLTR overshoot significantly The referencethreshold in this condition is set as the current roll anglewhich is less than the threshold steady state roll anglecorresponding to desired LLTRThe evolution of roll angle isthus restricted significantly Furthermore the driver steeringinput is compared with the value of required steering inputfor maintaining the vehicle roll angle at threshold value Toprovide for the switching control toworkwith less oscillationsdue to one sided fencing of the roll angle the controlleroutput value is used with correction even if it is up to 5less than steering input of the driver This requires activationof controller prior to reaching the critical LLTR value Theswitching relay part of the controller can have a larger gainfor robustness requiring a larger operating range of controlleroutput with respect to the driver input It would result indelayed switch-off of the controller input even if the driverhas turned the steering wheel back To cater for this thedynamic regularization of the relay part of the controllersigmoid approximation of sign(sdot) is used as shown in

sign (119904) = tanh(119904120576

) (19)

0 2 4 6 8 10 12

Time (s)

Stee

r ang

le (d

eg)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

Figure 4 NHTSA Fishhook steering profile used for evaluatingrollover propensity

5 Simulations

In order to evaluate the efficacy of the proposed controllera high fidelity model of a three-wheeled vehicle is set up inCarSim softwareThismodel includes nonlinear effects of rollcenter movement and tire force nonlinearities including tirerelaxation length causing a further lag in response of the vehi-cle as discussed in Section 2 of this paperThe suspensions arealso modeled using nonlinear dampers and stiffness whichare typical of light vehicle suspension systemsThe controlleris coded as a level-1 S-function in MatlabSimulink NHTSAJ-Turn test is used for evaluating the efficacy of the controllerA cosimulation is set up inMatlabSimulink using simulationmodel of CarSim

While assessing the available data on three-wheeledvehicles it was observed that most of the untripped rolloveraccidents were caused by sudden large steering to avoid anobstacle on the road The second major cause was over-speeding while cornering [9 10] Considering the steeringinput severity pattern and available standards NHTSA Fish-hook maneuver was selected for evaluation of the proposedcontroller A slowly increasing steering input at constantspeed was given A steering input corresponding to a lateralacceleration of 03 g is measured This steering angle wasscaled by a factor of 8 to account for maneuver severity andused as the amplitude of ramp steering on each directionThe angle was changed with a rate of 720 degrees per secondduring the first two ramps and a dwell period of three secondsafter the steering reversal is maintained The correspondingroad wheel steering angle profile for a typical Asian three-wheeled vehicle is as shown in Figure 4

A maneuver entrance speed of 35 kmhr is used Athreshold roll angle is set at a steady state LLTR of 08arbitrarily considering a margin of safetyThe LLTR with andwithout controller is shown in Figure 5 A value of 1 and minus1for LTTR indicates wheel liftoff while the controller is able tomaintain these values near 08 The transients are managed

6 Mathematical Problems in Engineering

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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 5

This disturbance term 119889(119911 120601

120601) represents modelinguncertaintiesThe control law has two components as follows

119906 = 119906eq + 119906

119904 (14)

where 119906eq is the equivalent control and 119906119904 is the conventionalswitching control The equivalent control is based upon thenominal model and is given by

119906eq =(minus119891 (120601

120601) minus 120572

120601)

119892 (119911 120601

120601)

(15)

The switching component 119906119904is given by

119906

119904= 120588 sign (120590) (16)

where

120588 = 119872 + 120581 (17)

and 119872 is the bound on the disturbance term 119889(119911 120601

120601) thatsatisfies

1003816

1003816

1003816

1003816

1003816

119889 (119911 120601

120601)

1003816

1003816

1003816

1003816

1003816

le 119872 (18)

for all 119911 120601

120601 in the domain of interestThe finite time convergence to the sliding surface is guar-

anteed by the positive constant 120581 and the rate of convergencedepends on its magnitude as discussed by Shtessel et al [39]

An important consideration in the controller implemen-tation is that the controller attempts to regulate the roll angleat the limiting value as soon it is activated A threshold LLTRactivates the controller As roll angle value lags behind theLLTR by approximately 01 seconds with increasing LLTRreaching a level of critical value the value of roll angleis still less than the critical value corresponding to steadystate LLTR If the controller is activated based on LLTRit will try to immediately push the roll angle to set valueincreasing the LLTR overshoot significantly The referencethreshold in this condition is set as the current roll anglewhich is less than the threshold steady state roll anglecorresponding to desired LLTRThe evolution of roll angle isthus restricted significantly Furthermore the driver steeringinput is compared with the value of required steering inputfor maintaining the vehicle roll angle at threshold value Toprovide for the switching control toworkwith less oscillationsdue to one sided fencing of the roll angle the controlleroutput value is used with correction even if it is up to 5less than steering input of the driver This requires activationof controller prior to reaching the critical LLTR value Theswitching relay part of the controller can have a larger gainfor robustness requiring a larger operating range of controlleroutput with respect to the driver input It would result indelayed switch-off of the controller input even if the driverhas turned the steering wheel back To cater for this thedynamic regularization of the relay part of the controllersigmoid approximation of sign(sdot) is used as shown in

sign (119904) = tanh(119904120576

) (19)

0 2 4 6 8 10 12

Time (s)

Stee

r ang

le (d

eg)

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

Figure 4 NHTSA Fishhook steering profile used for evaluatingrollover propensity

5 Simulations

In order to evaluate the efficacy of the proposed controllera high fidelity model of a three-wheeled vehicle is set up inCarSim softwareThismodel includes nonlinear effects of rollcenter movement and tire force nonlinearities including tirerelaxation length causing a further lag in response of the vehi-cle as discussed in Section 2 of this paperThe suspensions arealso modeled using nonlinear dampers and stiffness whichare typical of light vehicle suspension systemsThe controlleris coded as a level-1 S-function in MatlabSimulink NHTSAJ-Turn test is used for evaluating the efficacy of the controllerA cosimulation is set up inMatlabSimulink using simulationmodel of CarSim

While assessing the available data on three-wheeledvehicles it was observed that most of the untripped rolloveraccidents were caused by sudden large steering to avoid anobstacle on the road The second major cause was over-speeding while cornering [9 10] Considering the steeringinput severity pattern and available standards NHTSA Fish-hook maneuver was selected for evaluation of the proposedcontroller A slowly increasing steering input at constantspeed was given A steering input corresponding to a lateralacceleration of 03 g is measured This steering angle wasscaled by a factor of 8 to account for maneuver severity andused as the amplitude of ramp steering on each directionThe angle was changed with a rate of 720 degrees per secondduring the first two ramps and a dwell period of three secondsafter the steering reversal is maintained The correspondingroad wheel steering angle profile for a typical Asian three-wheeled vehicle is as shown in Figure 4

A maneuver entrance speed of 35 kmhr is used Athreshold roll angle is set at a steady state LLTR of 08arbitrarily considering a margin of safetyThe LLTR with andwithout controller is shown in Figure 5 A value of 1 and minus1for LTTR indicates wheel liftoff while the controller is able tomaintain these values near 08 The transients are managed

6 Mathematical Problems in Engineering

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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

minus08

minus06

minus04

minus02

00

02

04

06

08

10

minus10

0 2 4 6 8 10

Time (s)

LLTR

Without controllerWith controller

Figure 5 Evolution of LLTR during NHTSA Fishhook maneuverwith and without controller The maneuver entrance speed is set tocorrespond to wheel lift off during thismaneuver without controller

5

4

3

2

1

minus1

0

minus2

minus3

minus4

minus5

0 2 4 6 8 10

Time (s)

Ang

le (d

eg)

Roll angleRef roll rate

Figure 6 Proposed adapting reference roll angle and the actual rollangle during NHTSA Fishhook maneuverThe adaption is based onthe value of LLTR

in finite time as apparent in the dwell period from 17 to 42seconds

The overshoot of roll angle when reduced results in con-troller action which is restricted by the switching thresholdon the lower side with a value of 075 The adapted roll anglethreshold is shown in Figure 6

The adaption algorithm was used to compensate for thedelay between the steering angle and the evolving roll angleThe value of roll angle reference for the controller is as follows

120601ref = 120601set point + (120601setpoint minus 120601actual) (20)

minus4

minus2

0

2

4

6

minus60 2 4 6 8 10

Time (s)

Roll

angl

e

Without controllerWith controller

Figure 7 The evolution of roll angle of vehicle during NHTSAFishhook maneuver with and without control

0 2 4 6 8 10

Time (s)

Without controllerWith controller

minus50minus45minus40minus35minus30minus25minus20minus15minus10

minus505

10

Roll

angl

e (de

g)

Figure 8 Roll angle evolution during Fishhook maneuver with anelevated speed of 38 kmhrThe controller was successful in limitingroll angle and mitigating rollover

The value is updated only if the LLTR is greater thanthe threshold value which is 08 in this study and the actualroll angle is less than the set point steady state roll anglecorresponding to the threshold LLTR The evolution of rollangle with and without controller is shown in Figure 7 Athreshold value for roll angle has been set at 48 degrees

As the roll angle is the focus of the controller it iseffectively regulated at the set point value despite approximatemodel and model parameters Road uncertainties slightincrease in vehicle speed and lack of driving skills may resultin rollover as indicated in Figure 8 indicating a roll angleprofile with the same steering profile for amaneuver entrancespeed of 38 kmhr

Mathematical Problems in Engineering 7

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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

4

2

0

minus2

minus4

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline cg height

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline cg height

(b)Figure 9 Evolution of roll angle and LLTR with 10 variation in cg height Controller was able to limit both with reasonable efficacy

0

minus1

minus5

minus4

minus3

minus2

1

5

4

3

2

0 2 4 6 8 10

Time (s)

Roll

angl

e (de

g)

minus10

+10

Baseline mass

(a)

10

08

06

04

02

00

minus02

minus04

minus06

minus08

minus10

LLTR

0 2 4 6 8 10

Time (s)

minus10

+10

Baseline mass

(b)Figure 10 Evolution of roll angle and LLTR with 10 variation in vehicle mass Controller was able to limit both with reasonable efficacyWith less mass increase in roll angle results in more lateral load transfer

As evident from Figure 8 the vehicle did not recoverwithout controller after steering reversal with this slightincrease in vehicle speed However the controller was stillable to limit the roll angle and LLTR despite this change inlongitudinal speed signifying robustness Next the controllerwas evaluated with plusmn10 variations in height of cg from the

ground and mass The results are presented in Figures 9 and10 respectively

These results show that the controller adapted better tocg height changes as compared to mass increase but still wasable tomaintain LLTR below one that is maintaining contactwith ground With additional mass the static deflection of

8 Mathematical Problems in Engineering

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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

suspension increases reducing the suspension travel and athigher roll angles the bump stops start interacting resultingin sudden change in stiffness This results in oscillationsabout the steady state roll angle The mass increase resultsin increased lateral force on the sprung mass for the samelateral acceleration This results in an increase in roll angleWhen the controller tries to regulate the roll angle lateralload transfer is reduced hence a better rollover mitigationWith reduced mass the roll angle corresponding to the samemaneuver reducesWhen the controller tries to regulate it to ahigher level relatively more lateral load transfer results Thisis evident from Figure 10 To overcome this it is suggestedthat the controller should be tuned for lowest operationalvehicle total mass This would ensure robustness with loadvariations

6 Conclusion

In this study a controller design methodology was presentedfor rollover mitigation of three-wheeled vehicles using activefront steering A low order full vehicle model for roll dynam-ics was used with steering as a control input A conventionalsliding mode control with equivalent control was used Toaccount for the delay in vehicle response to control inputsan adaptive reference for roll angle has been proposed Thecontroller was evaluated for efficacy using a high fidelitysimulation model of CarSim The controller was able tomitigate rollover even in the presence of varying forwardspeed cg and mass variations Vehicle loading increasesthe mass properties and cg height of a vehicle Based onthe performance of controller with reduced vehicle massit is recommended that the baseline mass properties usedfor controller design should correspond to the minimumanticipated operational loading instead of the nominal fullyladen vehicle stateThis would lead to maximum efficacy androbustness of the proposed controller

Nomenclature

119909 119910 119911 Displacements in body attached frame m120601 Vehicle roll angle rad120579 Vehicle pitch angle rad120595 Vehicle yaw angle rad119897

119891 Distance from cg to front axle m

119897

119903 Distance from cg to rear axle m

119868

119909119909 Moment of inertia about roll axis kg-m2

119868

119910119910 Moment of inertia about pitch axis kg-m2

119868

119911119911 Moment of inertia about yaw axis kg-m2

119887 Track width mℎ

0 Static height of center of gravity m

119875(119905) Power applied at rear wheels W119861119909 Breaking force N120575 Steering angle rad119865119911 Vertical force at tire contact N119865119910 Tire lateral force N120572 Tire slip angle rad

Subscripts

119891 119903119903 119903119897 Wheel locations front rear right and rear left

Conflict of Interests

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

References

[1] NHTSA Initiatives to Address the Mitigation of Vehicle RolloverNational Highway Traffic Safety Administration 2003

[2] NHTSA ldquoTrends in the Static Stability Factor of PassengerCarsLight Trucks and Vansrdquo NHTSA Technical Report DOT HS809 868 NHTSA 2005

[3] P G Van Valkenburgh R H Klein and J Kanianthra ldquoThree-wheel passenger vehicle stability and handlingrdquo SAE TechnicalPaper 820140 1982

[4] A Raman J S Rao and S R Kale ldquoOverturning stability ofthree wheeled motorized vehiclesrdquo Vehicle System Dynamicsvol 24 no 2 pp 123ndash144 1995

[5] T R Gawade S Mukherjee and D Mohan ldquoRollover propen-sity of three-wheel scooter taxisrdquo SAE Technical Paper Series2004-01-1622 2004

[6] S Mukherjee D Mohan and T R Gawade ldquoThree-wheeledscooter taxi a safety analysisrdquo Sadhana vol 32 no 4 pp 459ndash478 2007

[7] G J Forkenbrock W R Garrott M Heitz and B C O HarraExperimental Examination of Test Maneuvers That May InduceOn-Road Untripped Light Vehicle Rollover 2003

[8] Government of India Road Accidents in India 2010 Govern-ment of India New Delhi India 2010

[9] Government of India Road Accidents in India 2011 Govern-ment of India New Delhi India 2011

[10] Government of India Road Accidents in India 2009 NewDelhiIndia 2009

[11] A Mani M Pai and R Aggarwal ldquoSustainable urban transportpolicy in India focus on Autorickshaw sectorrdquo Journal of theTransportation Research Board vol 2317 pp 104ndash110 2012

[12] J C Huston B J Graves and D B Johnson ldquoThree wheeledvehicle dynamicsrdquo Tech Rep SAE Technical Series Paper no820139 1982

[13] S Yim ldquoDesign of a preview controller for vehicle rolloverpreventionrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4217ndash4226 2011

[14] J N Dang ldquoPreliminary results analyzing the effectiveness ofelectronic stability control (ESC) systemsrdquo 2004

[15] M A R Licea and I Cervantes ldquoRobust switched predictivebraking control for rollover prevention in wheeled vehiclesrdquoMathematical Problems in Engineering vol 2014 Article ID356250 12 pages 2014

[16] Y Suetake M Oya P Shu and J Zhuo ldquoAdaptive rolloverprevention controller for driver-vehicle systemsrdquo Artificial Lifeand Robotics vol 19 no 1 pp 9ndash15 2014

[17] H Imine L M Fridman and T Madani ldquoSteering controlfor rollover avoidance of heavy vehiclesrdquo IEEE Transactions onVehicular Technology vol 61 no 8 pp 3499ndash3509 2012

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

[19] D J M Sampson and D Cebon ldquoActive roll control of singleunit heavy road vehiclesrdquo Vehicle System Dynamics vol 40 no4 pp 229ndash270 2003

Mathematical Problems in Engineering 9

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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

[20] B L J Gysen J J H Paulides J L G Janssen and E ALomonova ldquoActive electromagnetic suspension system forimproved vehicle dynamicsrdquo IEEE Transactions on VehicularTechnology vol 59 no 3 pp 1156ndash1163 2010

[21] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System DynamicsInternational Journal of Vehicle Mechanics and Mobility vol 46no 6 pp 451ndash470 2008

[22] J Tjonnas and T A Johansen ldquoStabilization of automotive vehi-cles using active steering and adaptive brake control allocationrdquoIEEE Transactions on Control Systems Technology vol 18 no 3pp 545ndash558 2010

[23] S C Peters J E Bobrow and K Iagnemma ldquoStabilizing avehicle near rollover an analogy to cart-pole stabilizationrdquo inProceedings of the IEEE International Conference onRobotics andAutomation (ICRA rsquo10) pp 5194ndash5200 May 2010

[24] M M Islam and C Ha ldquoRoad vehicle rollover avoidance usingactive steering controllerrdquo in Proceedings of the 14th Interna-tional Conference on Computer and Information Technology(ICCIT rsquo11) pp 298ndash302 Dhaka Bangladesh December 2011

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

[26] J Yoon W Cho J Kang B Koo and K Yi ldquoDesign andevaluation of a unified chassis control system for rolloverprevention and vehicle stability improvement on a virtual testtrackrdquo Control Engineering Practice vol 18 no 6 pp 585ndash5972010

[27] B Zhang and L Tong ldquoModeling and simulation of integratedsteering and braking control for vehicle active safety systemrdquoInternational Journal of Image Graphics and Signal Processingvol 3 no 2 p 1 2011

[28] D Piyabongkarn T Keviczky and R Rajamani ldquoActive directtilt control for stability enhancement of a narrow commutervehiclerdquo International Journal of Automotive Technology vol 5no 2 pp 77ndash78 2004

[29] M Barker B Drew J Darling K A Edge and G W OwenldquoSteady-state steering of a tilting three-wheeled vehiclerdquoVehicleSystem Dynamics vol 48 no 7 pp 815ndash830 2010

[30] S Kidane R Rajamani L Alexander P J Starr and MDonath ldquoDevelopment and experimental evaluation of a tiltstability control system for narrow commuter vehiclesrdquo IEEETransactions on Control Systems Technology vol 18 no 6 pp1266ndash1279 2010

[31] T Shim and C Ghike ldquoUnderstanding the limitations ofdifferent vehicle models for roll dynamics studiesrdquo VehicleSystem Dynamics vol 45 no 3 pp 191ndash216 2007

[32] N Zhang G-M Dong and H-P Du ldquoInvestigation intountripped rollover of light vehicles in the modified fishhookand the sine maneuvers Part I vehicle modelling roll and yawinstabilityrdquoVehicle System Dynamics vol 46 no 4 pp 271ndash2932008

[33] T R Gawade S Mukherjee and D Mohan ldquoSix-degree-of-freedom three-wheeled-vehicle model validationrdquo Proceedingsof the Institution of Mechanical Engineers Part D Journal ofAutomobile Engineering vol 219 no 4 pp 487ndash498 2005

[34] T R Gawade S Mukherjee and D Mohan ldquoWheel lift-off andride comfort of three-wheeled vehicle over bumprdquo Journal ofthe Institution of Engineers Part MC Mechanical EngineeringDivision vol 85 pp 78ndash87 2004

[35] A Hac ldquoRollover stability index including effects of suspensiondesignrdquo in Proceedings of the SAE 2002 World Congress ampExhibition No 724 Detroit Mich USA 2002

[36] J Jung T Shim and J Gertsch ldquoA vehicle roll-stability indicatorincorporating roll-center movementsrdquo IEEE Transactions onVehicular Technology vol 58 no 8 pp 4078ndash4087 2009

[37] R J Whitehead W E Travis D D M D M Bevly and GT Flowers ldquoA study of the effect of various vehicle propertieson rollover propensityrdquo in Proceedings of the SAE AutomotiveDynamics Stability amp Controls Conference and Exhibition pp3381ndash3386 2004

[38] F Joachim and J Boerner ldquoSuperposition-drive for steeringsystemrdquo US Patent No 20100331133A12010

[39] Y Shtessel C Edwards L Fridman andA Levant SlidingModeControl and Observation Springer New York NY USA 2014

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