Research Article Smooth Sliding Mode Control for Vehicle Rollover Prevention Using ... · 2019. 7. 31. · Research Article Smooth Sliding Mode Control for Vehicle Rollover Prevention

Research ArticleSmooth Sliding Mode Control for Vehicle Rollover PreventionUsing Active Antiroll Suspension

Duanfeng Chu1 Xiao-Yun Lu2 Chaozhong Wu1 Zhaozheng Hu1 and Ming Zhong1

1 Intelligent Transport Systems Research Center Wuhan University of Technology Beijing Key Laboratory forCooperative Vehicle Infrastructure Systems and Safety Control and Engineering Research Center of Transportation SafetyMinistry of Education 1040 Heping Avenue Wuchang District Wuhan 430063 China2California PATH Institute of Transportation Studies University of California Berkeley 1357 S 46th StreetRichmond Field Station Building 452 Richmond CA 94804-4648 USA

Correspondence should be addressed to Duanfeng Chu chudfwhuteducn

Received 27 October 2014 Revised 9 April 2015 Accepted 11 May 2015

Academic Editor Tamas Kalmar-Nagy

Copyright copy 2015 Duanfeng Chu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The rollover accidents induced by severe maneuvers are very dangerous and mostly happen to vehicles with elevated center ofgravity such as heavy-duty trucks and pickup trucks Unfortunately it is hard for drivers of those vehicles to predict and prevent thetrend of the maneuver-induced (untripped) rollover ahead of time In this study a lateral load transfer ratio which reflects the loaddistribution of left and right tires is used to indicate the rollover criticality An antiroll controller is designed with smooth slidingmode control technique for vehicles in which an active antiroll suspension is installed A simplified second order roll dynamicmodel with additive sector bounded uncertainties is used for control design followed by robust stability analysis Combined withthe vehicle dynamics simulation package TruckSim MATLABSimulink is used for simulating experiment The results show thatthe applied controller can improve the roll stability under some typical steeringmaneuvers such as Fishhook and J-turnThis directantiroll control method could be more effective for untripped rollover prevention when driver deceleration or steering is too lateIt could also be extended to handle tripped rollovers

1 Introduction

Vehicle rollover accidents typically contribute to fatal damageeven though they account for a small percentage amongall accidents According to the statistics of US NationalHighway Traffic Safety Administration (NHTSA) more than280000 rollover accidents and 10000 injuries or deaths arereported annually It has been marked as the second mostdangerous accident following head-on collisions in theUnited States [1]

There are two fundamental problems in rollover pre-vention to detectpredict rollover and to controlprevent itGenerallymost active control systems for vehicle roll stabilityin previous studies are based on the assumption of a gooddetection or prediction for vehicle rollovers In other wordsone has to accurately predict the impendence of rolloverin time before sensible control actions The Static Stability

Factor (SSF) presented by the Society of Automotive Engi-neers (SAE) is a simple and practical method for vehicle rollstability analysis This factor which is the ratio between thehalf-track and height of the center of gravity (CG) indicatesthe maximum allowable lateral acceleration for vehicle cor-nering It could be used to indicate the rollover propensityHowever this method has some drawbacks due to neglectingimpacts of suspension and tire defections as well as vehicledynamic characteristics Therefore some dynamic rolloverpredicting methods are presented to remedy the limitation ofthe above static predictingmethod such as Time-To-Rollovermetric and Rollover Prevention Energy Reserve index

In previous studies some active antiroll control systemsusing braking or steering control are used to help in dealingwith vehicle rollover threat and improving the vehicle sta-bility and safety Admittedly those approaches are easy toimplement since steering and brake are the most common

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 478071 8 pageshttpdxdoiorg1011552015478071

2 Mathematical Problems in Engineering

control actuators installed on vehicles However they do notcontrol vehicle roll dynamics directly Instead the rollmotionis controlled through coupling with lateral and longitudinalmotions Therefore control bandwidth and performance ofthose approaches are limited It may be effective for rollovercontrol of passenger cars but it is not effective for roll stabi-lization of high CG vehicles such as trucks fully loaded withstrawfoamclothing bales or other low density products

Therefore the main contributions of this study includethe following (1) an effective rollover index which reflectsthe load distribution of left and right tires is used to indicatethe rollover criticality (2) a simpler model-based controlapproach is used for stabilization of roll motion (3) insteadof using steering (lateral) and braking (longitudinal) directantiroll control for high CG vehicles can be taken to improvecontrol bandwidth and to reduce time delay (4) the direct rollcontrol method using the smooth sliding mode control tech-nique with robust stability analysis is applied to a simplifiedroll dynamics model with uncertainties

This paper is structured as follows Section 2 discusses thecritical threshold indicating the vehicle rollover impendencein Section 3 a smooth sliding mode control design based ona roll dynamics model is presented the performance of thiscontrol system (controlled vehicle) is analyzed in comparisonwith simulated experiments of the uncontrolled system (con-ventional vehicle) in Section 4 and this paper is concludedin Section 5

2 Literature Review

This part presents a brief literature review on two core issuesof rollover problems for vehicles with elevated CG that isrollover prediction and antiroll control

Aiming at the problem of vehicle rollover predictionChen and Peng [2 3] propose a Time-To-Rollover (TTR)metric Antiroll control is activated to avoid the rollover ifthe TTR is less than a preset value The TTR metric whichhelps to indicate the level of rollover threat will automaticallyldquocount downrdquo toward rollover according to vehicle speed andsteering patterns

An algorithm that detects impending rollover evaluatedthe index including lateral acceleration roll angle androll rate information which can be implemented to detectbothmaneuver-induced and road-induced rollovers [4] Oneenergy method using Rollover Prevention Energy Reserve(RPER) index is developed for vehicle rollover prevention [5]The RPER index is the difference between the energy con-sumption of vehicle rollover and the rotational kinetic energyAnother energy method for vehicle rollover prevention ispresented The rollover index is defined from the vehiclelateral kinetic energy and a new concept of virtual gravity

Trent and Greene [6] present a method for predictingrollover propensity by calculating tire deflections based ongenetic algorithm Tire deflections calculated through tiremodel are compared with their limited deflections estimatedthrough genetic algorithm If tire deflections exceed thethreshold rollover is assumed likely to happen

Presently most methods for the rollover prediction arebased on the above static or dynamic roll stability analysis

Some of them are practically implemented on several typesof chassis control systems for rollover prevention At presentthe practical vehicle control systems for preventing rolloveraccident are using active steering control andor differentialbraking control Odenthal et al [7] implement steering andbraking control to avoid vehicle rolloverThree feedback con-trol loops including continuous operation steering controlemergency steering control and emergency braking controlare used Moreover a load transfer ratio (rollover coefficient)indicating rollover impendence is defined which basicallydepends on the lateral acceleration and roll angle at CGSchofield [1] presents a control strategy based on limitation ofthe roll angle following a yaw rate reference A new computa-tionally efficient control allocation strategy based on convexoptimization is used to map the controller commands toindividual braking forces with actuator constraints taken intoaccount

These active control systems using active steering controlor differential braking control can help in dealingwith vehiclerollover threat and improving the stability and safety of thevehicle However these systems do not control vehicle rolldynamics directly In other words they do not aim at vehicleroll stability primarily Hence the vehicle handling perfor-mance and riding quality brought about by active steeringbraking and suspension have to be compromised to someextent due to the system extension for roll stabilityThereforefew studies have considered the direct active roll control forimproving vehicle roll stability and reducing rollover hazardsthrough active roll or suspension systems

Darling et al [8] present a low cost active antiroll systemwith an ldquoidealizedrdquo roll control mechanism aiming at con-trolling roll dynamics during several steering maneuvers andthus it improves vehicle ride comfort and safety Shuttlewoodet al [9] evaluate the performance of a slow-active hydrop-neumatic suspension which is incorporated into a vehiclehandlingmodel for roll control during a corneringmaneuverIt uses feedback measurements of the body motions for theroll control without compromising the ride performance Luand Hedrick [10] present a novel design of active suspensionsystem using hydraulic actuator (a cylinder and a piston)for each suspension with a centralized control activated by apump and servo valves However the dynamical behavior ofthe closed loop system is not yet analyzed in detail

Yu et al [11] developed an active suspension system basedon the linear quadratic regulator (LQR) method Yim et al[12] designed an antirollover controller with both active sus-pension and ESP and they also used linear quadratic methodfor the control design Yang and Liu [13] presented an activesuspension control system with the control technique ofsimple feedback control

In this paper a practical active suspension actuator (ieactive antiroll suspension) is used in order to tackle theproblems related to rollover prediction and antiroll controlof vehicles with elevated CG It can provide an emergingeffective solution for reducing rollover accidents and improv-ing vehicle handling performance Specifically this paperpresents a rollover index and a sliding mode controller regu-lated this kind of suspension for rollover prevention

Mathematical Problems in Engineering 3

Ks

MA

ay

Ixx msg

Roll center

Cd ΔFLΔFR

T

FzL FzR

hr

h

Figure 1 The induced roll dynamic model

3 Methodology

For rollover prevention a rollover index based on lateralload transfer ratio to monitor the rollover critical conditiononline is presented When the rollover index exceeds a presetthreshold value the sliding mode controller will be triggeredand generate appropriate antiroll moment which acted on thecenter of gravity Therefore this generated moment can dragthe unstable vehicle to the equilibrium state

31 Impending Rollover Prediction The method using a loadtransfer ratio with measured vehicular lateral accelerationand roll rate can identify the situation of wheel contractionreductionloss This situation is often referred to as a ldquocriticalsituationrdquo which should be avoided for rollover prevention[7] Suppose the vertical motion has been neglected the sumof vertical forces is just the static vehicle weight Hence thelateral load transfer ratio (the so-called rollover index) can becalculated as (Figure 1)

119877 =119865119911119877minus 119865119911119871

119865119911119877+ 119865119911119871

(1)

where the subscripts ldquo119871rdquo and ldquo119877rdquo represent the left and righttires Moreover tire vertical forces are denoted by 119865

119911119871and

119865119911119877

which can be obtained using Newtonrsquos Law in the verticaldirection

In (1) tire vertical forces can be calculated as follows

119865119911119877+119865119911119871= 119898119892

119865119911119877minus119865119911119871= (119898119904119892ℎ sin 120593+119898

119904119886119910(ℎ cos 120593+ ℎ

119903)) sdot

2119879

(2)

Combining (1) and (2) as well as assuming a small rollangle the rollover index is approximated by the followingequation (4)

119877 =2119898119904

119898119879((ℎ119903+ ℎ)

119886119910

119892+ ℎ120593) (3)

During the straight driving the lateral acceleration androll angle are both zero which leads to zero rollover index In

Vehicle

Rollover detection

Active roll controller

trigger

120575

ay

120593

R

MA

Figure 2 Sliding mode controller design for rollover prevention

the case of a severe steeringmaneuver the lateral accelerationand roll angle become large which may cause the absolutevalue of 119877 to be saturated to one that is |119877| le 1

In the real world the lateral acceleration (unit rsquog) is oftenmuch greater than the roll angle (unit rad) that is 119886

119910119892 ≫ 120593

Moreover the sprung mass is often much greater than theunsprung mass for heavy-duty vehicles That is to say thesprung mass occupies the absolute proportion of the totalmass that is119898

119904asymp 119898 So the calculation of the above rollover

index can be transformed as follows

119877 =2 (ℎ119903+ ℎ) 119886

119910

119879119892 (4)

32 Control Design for Rollover Prevention The objective ofthe roll control system is to generate a roll moment fromactive antiroll suspension for maximizing the roll stability ofthe vehicle Consequently the following vehicle roll dynamicmodel is obtained for control design

=1119868119909119909

[minus119862119889 minus (119870

119904minus119898119892ℎ) 120593+119898119886

119910ℎ +119872

119860] +Δ (5)

where 119872119860represents the control input and Δ represents the

allowable uncertainties (such as road disturbances nonlin-earity and unmodeled dynamics) which is sector boundedas follows

Δ = Δ (120593 ) le 120590110038161003816100381610038161205931003816100381610038161003816 + 1205902

10038161003816100381610038161003816100381610038161003816 (6)

where 1205901gt 0 and 120590

2gt 0 are known positive real numbers

Control design with the vehicle dynamics model will beused to control a loaded two-axle vehicle model in TruckSimfor simulation According to (5) the system input is rollmoment from controlled suspension 119872

119860and the control

output is the roll angle Thus this active roll control systemcan be demonstrated in Figure 2

The system input and output can be expressed respec-tively as follows

119906 = 119872119860

119910 = 120593(7)


The uncertainty in inequality (6) can be represented as

Δ (119910 119910) le 120590110038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816 (8)

The main results regarding the stability are stated in thefollowing

Theorem 1 For the vehicle dynamics in (5) with sectorbounded uncertainty in inequality (6) if the controller and thecorresponding gains are selected as

119906 = minus ((120582 + 120578) 119868119909119909minus119862119889) minus (120582120578119868

119909119909minus (119870119904minus119898119892ℎ)) 120593

minus119898119886119910ℎ minus 119868119909119909(120576 sdot sgn ( + 120582120593) +Δ)

120578 gt 1205902 120576 gt (1 + 1205901 + 1205902120582) 119910max

(9)

the overall system is semiglobally asymptotically (exponen-tially) stable

Proof Define the sliding surface as

119904 = 119910 + 120582119910 = + 120582120593 (10)

where 120582 is a positive constant indicating the performance ofsystem converging to the equilibrium point

Thederivative of sliding surface along the trajectory of themodel in (5) is

119904 = + 120582

=(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582

+Δ

(11)

Choose the reaching condition which guarantees theasymptotic stability as follows

119904 = minus 120578 sdot 119904 minus 120576 sdot sgn (119904) (12)

where the positive constants 120578 and 120576 are the control gains to bedetermined which represent the speed reaching the slidingsurface and sgn(sdot) is a sign function Now the control inputcan be derived as

(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582 +Δ

= minus 120578 ( + 120582120593) minus 120576 sdot sgn ( + 120582120593)

(13)

that is


119909119909minus (119870119904minus119898119892ℎ)) 120593


(14)

With this control law the original dynamical systemthat is (5) and (10) is equivalent to the following system instability

119910 = minus 120582119910+ 119904

119904 = minus 120578119904 minus 120576 sdot sgn (119904) + Δ(15)

Without loss of generality it is assumed that the initialcondition for 119910(0) belongs to a compact domain 119910(0) isin 119884 sub

R (the real set) Then there exists 119910max gt 0 such that 119910(0) le119910max for all 119910 isin 119884

The following Lyapunov candidate function is selected

119881 =12(119910

2+ 119904

2) (16)

The time derivative of this Lyapunov function along thestate trajectory of (15) is as follows

= 119910 sdot 119910 + 119904 sdot 119904

= 119910 sdot (minus120582119910+ 119904) + 119904 sdot (minus120578 sdot 119904 minus 120576 sdot sgn (119904) +Δ)

= minus 120582 sdot 1199102+119910 sdot 119904 minus 120578 sdot 119904

2minus 120576 sdot |119904| + Δ sdot 119904

le minus 120582 sdot 1199102+119910 sdot 119904 minus 120578 sdot 119904

2minus 120576 sdot |119904| + 119904 (1205901

10038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816)

le minus 120582 sdot 1199102+10038161003816100381610038161199101003816100381610038161003816 sdot |119904| minus 120578 sdot 119904

2minus 120576 sdot |119904| + 1205901

10038161003816100381610038161199101003816100381610038161003816 |119904|

+ 1205902 (12058210038161003816100381610038161199101003816100381610038161003816 + |119904|) |119904|

le minus 120582 sdot 1199102minus (120578 minus 1205902) sdot 119904

2

minus (120576 minus (1+1205901 +1205902120582) 119910max) |119904|

(17)

which is strictly negative if

120578 gt 1205902

120576 gt (1+1205901 +1205902120582) 119910max(18)

We therefore have proved the above theorem regardingthe robustness stability

From the sliding mode control theory the sliding surface119904 defined by inequality (6) is attractive leading 119904 to convergetowards 0 exponentially

A saturation function is used to replace the sign functionin (15) Consider

sat( 119904119887) =

119904

119887|119904| le 119887

sgn( 119904119887) |119904| gt 119887

(19)

The verification of this control strategy will be presentedin the following simulating experiment and analysis

4 Experiment Results and Analysis

Several steering maneuvers have been developed by NHTSAto evaluate the roll stability of vehicles Two typical maneu-vers that is Fishhook in Figure 3(a) and J-turn in Figure 4(a)are used for verification of the sliding mode controllerdesigned in this paper It has been applied to the maneuversto test the dynamic responses of uncontrolled and controlledsystems at the speed of 80 kmh The dynamic responsesof the uncontrolled vehicle (solid lines) and the controlledvehicle (dotted lines) are shown in Figures 3(b)ndash3(f) and 4(b)ndash4(f) respectively Specifically Figures 3(b) and 4(b) show


0 05 1 15 2 25 3 35 4

0

100

200

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

minus300

minus200

minus100

(a) Fishhook maneuver

0 1 2 3 4 5 6

0

1

Time (s)

Rollo

ver i

ndex

minus1

minus08

minus06

minus04

minus02

08

06

04

02

(b) Rollover index

0

02

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

Time (s)

Roll

rate

(rad

s)

minus25

minus2

minus15

minus1

minus05

(d) Roll rate

ControlledUncontrolled

0 1 2 3 4 5 6

0

10

20

30

Time (s)

Yaw

rate

(rad

s)

minus40

minus30

minus20

minus10

(e) Yaw rate

05

15

0 1 2 3 4 5 6

0

1

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus15

minus1

minus05

times105


(f) Antiroll moment

Figure 3 Comparisons of the uncontrolled and the controlled vehicle dynamics responding to Fishhook experiment with sliding modecontrol parameters 120582 = 16 120578 = 15 120576 = 12 120590

1= 01 120590

2= 01 and 119910max = 02 and sliding surface bound 119887 = 001


0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment

Figure 4 Comparisons of the uncontrolled and the controlled vehicle dynamics responding to J-turn experiment with sliding mode controlparameters 120582 = 16 120578 = 15 120576 = 12 120590

1= 01 120590



the dynamic change of rollover index under steering maneu-vers Figures 3(c)-3(d) and 4(c)-4(d) show the dynamicresponses of vehicle roll state Figures 3(e) and 4(e) show thedynamic responses of vehicle yaw state and Figures 3(f) and4(f) show control outputs of the designed sliding mode con-trollerMoreover in the simulating experiment the thresholdof the rollover index (119877) has been set to 08 If there is nocontrol performed when 119877 has exceeded the threshold thevehicle will be about to roll over

Moreover 119910max is set to be 02 rad which is the limit ofroll angle of the test vehicle model For the simplificationof experiments 120590

1is equal to 120590

2 and they are both set to

be 01 mathematically Therefore inequality (18) is derived asfollows

120578 gt 01

120576 gt 022+ 002times120582(20)

During Fishhook and J-turn experiments if the aboveequation is satisfied the designed sliding mode controller forthe antirollover prevention system will be stable accordingto Lyapunov stability theory However according to thesliding mode control technique there exists the chatteringproblem which is caused by frequently switching controlalong the sliding surface The first method for smoothing outthe chattering effect to some extent is using the saturationfunction shown in (19) instead of the sign functionThis is thepopularly used boundary layer sliding mode control and theboundary parameter 119887 in this case is set to 001 for narrowingthe switching control boundary which helps in reducing thechattering effectTheother one is tuning slidingmode param-eters which are the positive constants 120582 120578 120576 and so forth

Actually these three constants determine the extent ofchattering and the appropriate combinations of these param-eters are chosen by trial and error Through repeated selec-tion it is found that parametric values around120582 = 16 120578 = 15and 120576 = 12 can reduce the chattering effect to an extentWhen 120582 lt 8 or 120582 gt 40 the chattering will become severeAccording to (18) 120578 gt 120590

1= 01 but an analogous chattering

situation happens when 120578 gt 5 It also seems that 120576 does notexert toomuch influence on the chattering effect but accord-ing to (18) 120576 has to be larger than 054 Therefore the appro-priate chosen combinations of these parameters are shownin Figures 3 and 4

For the Fishhook experiment shown in Figure 3 it canbe observed that rollover index goes to minus1 after 28 secondswhich indicates that tires on one side lift off the groundand vehicle rollover happens in 36 seconds Then after ashort time the roll and yaw dynamics diverge rapidly and theprogram stops

During the simulating experiment of the controlled vehi-cle the rollover index has been dragged towards the set valueWhen the vehicle roll stability is about to be lost the antirollmoment is applied As shown in the dotted lines of Figures3(c) and 3(e) the roll and yaw dynamics aremaintained stablein equilibrium states

Likewise the result of J-turn experiment is shown inFigure 4 After 2 seconds when the severe steering maneuver

is exerted the magnitude of the rollover index goes up to 1rapidly which indicates that tires on one side lift offWithoutthe antiroll control vehicle roll and yaw dynamic stabilitiesbe lost within 44 seconds and then the program stops Fromthe dotted lines of Figures 4(b)ndash4(e) antiroll moment inFigure 4(f) drags the rollover index to equilibrium state andboth roll and yaw dynamics are maintained stable as well

5 Conclusions and Future Work

This paper presents a new method for rollover preventionthat is the direct antiroll control It is different from themethods with vehicle braking or steering which essentiallycontrols the roll motion through coupling with lateral andlongitudinal motions Through direct roll motion control itcan potentially increase control bandwidth and reduce timedelay so that it could handle untripped rollovers or eventripped rollovers for vehicles with elevated CG

An antiroll slidingmode controller has been designed andverified through the joint simulation of TruckSim and MAT-LABSimulink The underneath vehicle dynamics model inTruckSim is a full vehicle dynamics model The simulationresults are the control responses of the full truck model inTruckSim with respect to the feedback SMC derived fromthe simplified model Vehicle robust roll stability is shownthrough simulations with Fishhook and J-turn steeringmaneuvers For practical implementations the lateral loadtransfer ratio which indicates the rollover impendence is usedfor control activation The roll moment could be adjustedwith the active antiroll suspension in real time Comparedwith other indirect rollover prevention systems using brakingor steering there are some advantages for applying this directroll control system

(1) Control strategies become more and more compli-cated using differential braking or active steering toimprove both the handling stability and the roll stabil-ity besides controlling roll motion through couplingwith steering (lateral motion) and braking (longi-tudinal and later motions) is indirect where con-trol performance and bandwidth are limited there-fore it is simpler and more effective to avoid themaneuver-induced rollover with direct roll control

(2) Direct roll control strategy could potentially beused for tripped rollover prevention caused by roadirregularities if tripped rollover could be accuratelydetectedpredicted

(3) Active antiroll suspension is a part of vehicle chassissystem therefore the active roll control could also beextended to improve the riding comfort

Practical implementations of this approach could im-prove both safety and riding comfort when control energyconsumptions of the antiroll moment generation are limitedwithin an allowable range which will be the focus of thefuture work


Nomenclature

ℎ Distance from CG to roll center 126mℎ119903 Distance from roll center to the ground 047m

119898 Vehicle total mass 12562 kg119898119904 Vehicle sprung mass 11257 kg

119892 Acceleration due to gravity 981 kgsdotms2119868119909119909 Moment of inertia about 119909-axis 5114 kgsdotm2

119879 Track width 203m119862119889 Roll damping of passive suspension 31000Nsdotmsdotsrad

119870119904 Roll stiffness of passive suspension 347000Nsdotmrad

119877 Rollover index119886119910 Lateral acceleration

120593 Roll angle

Conflict of Interests

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

Acknowledgments

The authors would like to thank PATH UC Berkeley for pro-viding an active research environment The main innovationand research contents are supported by the National NaturalScience Foundation of China (51105286) Key Projects in theNational Science amp Technology Pillar Program during theTwelfth Five-Year Plan Period (2014BAG01B03) the Funda-mental Research Funds for the Central Universities (WUT2014-IV-137) the Foundation of Key Laboratory of Roadand Traffic Engineering of the Ministry of Education TongjiUniversity (K201301) and the Foundation of Beijing Key Lab-oratory for Cooperative Vehicle Infrastructure Systems andSafety Control (KFJJ-201401)

References

[1] B Schofield Vehicle dynamics control for rollover prevention[PhD thesis] ISRN LUTFD2TFRT-3241-SE Department ofAutomatic Control Lund Institute of Technology LundUniver-sity Lund Sweden 2006

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

[3] B-C Chen and H Peng ldquoRollover warning for articulatedheavy vehicles based on a time-to-rollover metricrdquo Journal ofDynamic Systems Measurement and Control vol 127 no 3 pp406ndash414 2004

[4] AHac T Brown and JMartens ldquoDetection of vehicle rolloverrdquoSAE Technical Paper 2004-01-1757 2004

[5] S B Choi ldquoPractical vehicle rollover avoidance control usingenergy methodrdquo Vehicle System Dynamics vol 46 no 4 pp323ndash337 2008

[6] V Trent and M Greene ldquoA genetic algorithm predictor forvehicular rolloverrdquo in Proceedings of the IEEE 28th Annual Con-ference of the Industrial Electronics Society vol 3 pp 1752ndash1756November 2002

[7] D Odenthal T Bunte and J Ackermann ldquoNonlinear steeringand braking control for vehicle rollover avoidancerdquo in Proceed-ings of the European Control Conference Karlsruhe GermanySeptember 1999

[8] J Darling R E Dorey and T J Ross-Martin ldquoLow cost activeanti-roll suspension for passenger carsrdquo Journal of DynamicSystems Measurement and Control vol 114 no 4 pp 599ndash6051992

[9] D W Shuttlewood D A Crolla R S Sharp and I LCrawford ldquoActive roll control for passenger carsrdquoVehicle SystemDynamics vol 22 no 5-6 pp 383ndash396 1993

[10] X Y Lu and J K Hedrick ldquoImpact of combined longitudinallateral and vertical control on autonomous road vehicle designrdquoInternational Journal of Vehicle Autonomous Systems vol 2 no1-2 pp 40ndash70 2004

[11] H Yu L Guvenc and U Ozguner ldquoHeavy duty vehicle rolloverdetection and active roll controlrdquo Vehicle System Dynamics vol46 no 6 pp 451ndash470 2008

[12] S Yim Y Park andK Yi ldquoDesign of active suspension and elec-tronic stability program for rollover preventionrdquo InternationalJournal of Automotive Technology vol 11 no 2 pp 147ndash153 2010

[13] H Yang and L Y Liu ldquoA robust active suspension controllerwith rollover preventionrdquo SAE Technical Paper 2003-01-09592003

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Journal of


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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



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


control actuators installed on vehicles However they do notcontrol vehicle roll dynamics directly Instead the rollmotionis controlled through coupling with lateral and longitudinalmotions Therefore control bandwidth and performance ofthose approaches are limited It may be effective for rollovercontrol of passenger cars but it is not effective for roll stabi-lization of high CG vehicles such as trucks fully loaded withstrawfoamclothing bales or other low density products

Therefore the main contributions of this study includethe following (1) an effective rollover index which reflectsthe load distribution of left and right tires is used to indicatethe rollover criticality (2) a simpler model-based controlapproach is used for stabilization of roll motion (3) insteadof using steering (lateral) and braking (longitudinal) directantiroll control for high CG vehicles can be taken to improvecontrol bandwidth and to reduce time delay (4) the direct rollcontrol method using the smooth sliding mode control tech-nique with robust stability analysis is applied to a simplifiedroll dynamics model with uncertainties

This paper is structured as follows Section 2 discusses thecritical threshold indicating the vehicle rollover impendencein Section 3 a smooth sliding mode control design based ona roll dynamics model is presented the performance of thiscontrol system (controlled vehicle) is analyzed in comparisonwith simulated experiments of the uncontrolled system (con-ventional vehicle) in Section 4 and this paper is concludedin Section 5

2 Literature Review

This part presents a brief literature review on two core issuesof rollover problems for vehicles with elevated CG that isrollover prediction and antiroll control

Aiming at the problem of vehicle rollover predictionChen and Peng [2 3] propose a Time-To-Rollover (TTR)metric Antiroll control is activated to avoid the rollover ifthe TTR is less than a preset value The TTR metric whichhelps to indicate the level of rollover threat will automaticallyldquocount downrdquo toward rollover according to vehicle speed andsteering patterns

An algorithm that detects impending rollover evaluatedthe index including lateral acceleration roll angle androll rate information which can be implemented to detectbothmaneuver-induced and road-induced rollovers [4] Oneenergy method using Rollover Prevention Energy Reserve(RPER) index is developed for vehicle rollover prevention [5]The RPER index is the difference between the energy con-sumption of vehicle rollover and the rotational kinetic energyAnother energy method for vehicle rollover prevention ispresented The rollover index is defined from the vehiclelateral kinetic energy and a new concept of virtual gravity

Trent and Greene [6] present a method for predictingrollover propensity by calculating tire deflections based ongenetic algorithm Tire deflections calculated through tiremodel are compared with their limited deflections estimatedthrough genetic algorithm If tire deflections exceed thethreshold rollover is assumed likely to happen

Presently most methods for the rollover prediction arebased on the above static or dynamic roll stability analysis

Some of them are practically implemented on several typesof chassis control systems for rollover prevention At presentthe practical vehicle control systems for preventing rolloveraccident are using active steering control andor differentialbraking control Odenthal et al [7] implement steering andbraking control to avoid vehicle rolloverThree feedback con-trol loops including continuous operation steering controlemergency steering control and emergency braking controlare used Moreover a load transfer ratio (rollover coefficient)indicating rollover impendence is defined which basicallydepends on the lateral acceleration and roll angle at CGSchofield [1] presents a control strategy based on limitation ofthe roll angle following a yaw rate reference A new computa-tionally efficient control allocation strategy based on convexoptimization is used to map the controller commands toindividual braking forces with actuator constraints taken intoaccount

These active control systems using active steering controlor differential braking control can help in dealingwith vehiclerollover threat and improving the stability and safety of thevehicle However these systems do not control vehicle rolldynamics directly In other words they do not aim at vehicleroll stability primarily Hence the vehicle handling perfor-mance and riding quality brought about by active steeringbraking and suspension have to be compromised to someextent due to the system extension for roll stabilityThereforefew studies have considered the direct active roll control forimproving vehicle roll stability and reducing rollover hazardsthrough active roll or suspension systems

Darling et al [8] present a low cost active antiroll systemwith an ldquoidealizedrdquo roll control mechanism aiming at con-trolling roll dynamics during several steering maneuvers andthus it improves vehicle ride comfort and safety Shuttlewoodet al [9] evaluate the performance of a slow-active hydrop-neumatic suspension which is incorporated into a vehiclehandlingmodel for roll control during a corneringmaneuverIt uses feedback measurements of the body motions for theroll control without compromising the ride performance Luand Hedrick [10] present a novel design of active suspensionsystem using hydraulic actuator (a cylinder and a piston)for each suspension with a centralized control activated by apump and servo valves However the dynamical behavior ofthe closed loop system is not yet analyzed in detail

Yu et al [11] developed an active suspension system basedon the linear quadratic regulator (LQR) method Yim et al[12] designed an antirollover controller with both active sus-pension and ESP and they also used linear quadratic methodfor the control design Yang and Liu [13] presented an activesuspension control system with the control technique ofsimple feedback control

In this paper a practical active suspension actuator (ieactive antiroll suspension) is used in order to tackle theproblems related to rollover prediction and antiroll controlof vehicles with elevated CG It can provide an emergingeffective solution for reducing rollover accidents and improv-ing vehicle handling performance Specifically this paperpresents a rollover index and a sliding mode controller regu-lated this kind of suspension for rollover prevention


Ks

MA

ay

Ixx msg

Roll center

Cd ΔFLΔFR

T

FzL FzR

hr

h


3 Methodology



119877 =119865119911119877minus 119865119911119871

119865119911119877+ 119865119911119871

(1)


119911119871and

119865119911119877



119865119911119877+119865119911119871= 119898119892

119865119911119877minus119865119911119871= (119898119904119892ℎ sin 120593+119898

119904119886119910(ℎ cos 120593+ ℎ

119903)) sdot

2119879

(2)


119877 =2119898119904

119898119879((ℎ119903+ ℎ)

119886119910

119892+ ℎ120593) (3)


Vehicle

Rollover detection


trigger

120575

ay

120593

R

MA




119910119892 ≫ 120593




119877 =2 (ℎ119903+ ℎ) 119886

119910

119879119892 (4)


=1119868119909119909

[minus119862119889 minus (119870

119904minus119898119892ℎ) 120593+119898119886

119910ℎ +119872

119860] +Δ (5)



Δ = Δ (120593 ) le 120590110038161003816100381610038161205931003816100381610038161003816 + 1205902

10038161003816100381610038161003816100381610038161003816 (6)

where 1205901gt 0 and 120590






119906 = 119872119860

119910 = 120593(7)



Δ (119910 119910) le 120590110038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816 (8)




119909119909minus (119870119904minus119898119892ℎ)) 120593


120578 gt 1205902 120576 gt (1 + 1205901 + 1205902120582) 119910max

(9)



119904 = 119910 + 120582119910 = + 120582120593 (10)



119904 = + 120582

=(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582

+Δ

(11)




(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582 +Δ


(13)

that is


119909119909minus (119870119904minus119898119892ℎ)) 120593


(14)


119910 = minus 120582119910+ 119904





119881 =12(119910

2+ 119904

2) (16)


= 119910 sdot 119910 + 119904 sdot 119904



2minus 120576 sdot |119904| + Δ sdot 119904


2minus 120576 sdot |119904| + 119904 (1205901

10038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816)


2minus 120576 sdot |119904| + 1205901

10038161003816100381610038161199101003816100381610038161003816 |119904|

+ 1205902 (12058210038161003816100381610038161199101003816100381610038161003816 + |119904|) |119904|


2

minus (120576 minus (1+1205901 +1205902120582) 119910max) |119904|

(17)


120578 gt 1205902

120576 gt (1+1205901 +1205902120582) 119910max(18)




sat( 119904119887) =

119904

119887|119904| le 119887

sgn( 119904119887) |119904| gt 119887

(19)





0 05 1 15 2 25 3 35 4

0

100

200

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

minus300

minus200

minus100


0 1 2 3 4 5 6

0

1

Time (s)

Rollo

ver i

ndex

minus1

minus08

minus06

minus04

minus02

08

06

04

02

(b) Rollover index

0

02

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

Time (s)

Roll

rate

(rad

s)

minus25

minus2

minus15

minus1

minus05

(d) Roll rate


0 1 2 3 4 5 6

0

10

20

30

Time (s)

Yaw

rate

(rad

s)

minus40

minus30

minus20

minus10

(e) Yaw rate

05

15

0 1 2 3 4 5 6

0

1

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus15

minus1

minus05

times105


(f) Antiroll moment


1= 01 120590



0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment


1= 01 120590





1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ks

MA

ay

Ixx msg

Roll center

Cd ΔFLΔFR

T

FzL FzR

hr

h


3 Methodology



119877 =119865119911119877minus 119865119911119871

119865119911119877+ 119865119911119871

(1)


119911119871and

119865119911119877



119865119911119877+119865119911119871= 119898119892

119865119911119877minus119865119911119871= (119898119904119892ℎ sin 120593+119898

119904119886119910(ℎ cos 120593+ ℎ

119903)) sdot

2119879

(2)


119877 =2119898119904

119898119879((ℎ119903+ ℎ)

119886119910

119892+ ℎ120593) (3)


Vehicle

Rollover detection


trigger

120575

ay

120593

R

MA




119910119892 ≫ 120593




119877 =2 (ℎ119903+ ℎ) 119886

119910

119879119892 (4)


=1119868119909119909

[minus119862119889 minus (119870

119904minus119898119892ℎ) 120593+119898119886

119910ℎ +119872

119860] +Δ (5)



Δ = Δ (120593 ) le 120590110038161003816100381610038161205931003816100381610038161003816 + 1205902

10038161003816100381610038161003816100381610038161003816 (6)

where 1205901gt 0 and 120590






119906 = 119872119860

119910 = 120593(7)



Δ (119910 119910) le 120590110038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816 (8)




119909119909minus (119870119904minus119898119892ℎ)) 120593


120578 gt 1205902 120576 gt (1 + 1205901 + 1205902120582) 119910max

(9)



119904 = 119910 + 120582119910 = + 120582120593 (10)



119904 = + 120582

=(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582

+Δ

(11)




(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582 +Δ


(13)

that is


119909119909minus (119870119904minus119898119892ℎ)) 120593


(14)


119910 = minus 120582119910+ 119904





119881 =12(119910

2+ 119904

2) (16)


= 119910 sdot 119910 + 119904 sdot 119904



2minus 120576 sdot |119904| + Δ sdot 119904


2minus 120576 sdot |119904| + 119904 (1205901

10038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816)


2minus 120576 sdot |119904| + 1205901

10038161003816100381610038161199101003816100381610038161003816 |119904|

+ 1205902 (12058210038161003816100381610038161199101003816100381610038161003816 + |119904|) |119904|


2

minus (120576 minus (1+1205901 +1205902120582) 119910max) |119904|

(17)


120578 gt 1205902

120576 gt (1+1205901 +1205902120582) 119910max(18)




sat( 119904119887) =

119904

119887|119904| le 119887

sgn( 119904119887) |119904| gt 119887

(19)





0 05 1 15 2 25 3 35 4

0

100

200

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

minus300

minus200

minus100


0 1 2 3 4 5 6

0

1

Time (s)

Rollo

ver i

ndex

minus1

minus08

minus06

minus04

minus02

08

06

04

02

(b) Rollover index

0

02

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

Time (s)

Roll

rate

(rad

s)

minus25

minus2

minus15

minus1

minus05

(d) Roll rate


0 1 2 3 4 5 6

0

10

20

30

Time (s)

Yaw

rate

(rad

s)

minus40

minus30

minus20

minus10

(e) Yaw rate

05

15

0 1 2 3 4 5 6

0

1

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus15

minus1

minus05

times105


(f) Antiroll moment


1= 01 120590



0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment


1= 01 120590





1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ (119910 119910) le 120590110038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816 (8)




119909119909minus (119870119904minus119898119892ℎ)) 120593


120578 gt 1205902 120576 gt (1 + 1205901 + 1205902120582) 119910max

(9)



119904 = 119910 + 120582119910 = + 120582120593 (10)



119904 = + 120582

=(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582

+Δ

(11)




(119898119886119910ℎ minus ((119870

119904minus 119898119892ℎ) 120593 + 119862

119889))

119868119909119909

+119906

119868119909119909

+120582 +Δ


(13)

that is


119909119909minus (119870119904minus119898119892ℎ)) 120593


(14)


119910 = minus 120582119910+ 119904





119881 =12(119910

2+ 119904

2) (16)


= 119910 sdot 119910 + 119904 sdot 119904



2minus 120576 sdot |119904| + Δ sdot 119904


2minus 120576 sdot |119904| + 119904 (1205901

10038161003816100381610038161199101003816100381610038161003816 + 1205902

1003816100381610038161003816 1199101003816100381610038161003816)


2minus 120576 sdot |119904| + 1205901

10038161003816100381610038161199101003816100381610038161003816 |119904|

+ 1205902 (12058210038161003816100381610038161199101003816100381610038161003816 + |119904|) |119904|


2

minus (120576 minus (1+1205901 +1205902120582) 119910max) |119904|

(17)


120578 gt 1205902

120576 gt (1+1205901 +1205902120582) 119910max(18)




sat( 119904119887) =

119904

119887|119904| le 119887

sgn( 119904119887) |119904| gt 119887

(19)





0 05 1 15 2 25 3 35 4

0

100

200

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

minus300

minus200

minus100


0 1 2 3 4 5 6

0

1

Time (s)

Rollo

ver i

ndex

minus1

minus08

minus06

minus04

minus02

08

06

04

02

(b) Rollover index

0

02

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

Time (s)

Roll

rate

(rad

s)

minus25

minus2

minus15

minus1

minus05

(d) Roll rate


0 1 2 3 4 5 6

0

10

20

30

Time (s)

Yaw

rate

(rad

s)

minus40

minus30

minus20

minus10

(e) Yaw rate

05

15

0 1 2 3 4 5 6

0

1

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus15

minus1

minus05

times105


(f) Antiroll moment


1= 01 120590



0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment


1= 01 120590





1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 4

0

100

200

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

minus300

minus200

minus100


0 1 2 3 4 5 6

0

1

Time (s)

Rollo

ver i

ndex

minus1

minus08

minus06

minus04

minus02

08

06

04

02

(b) Rollover index

0

02

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus14

minus12

minus1

minus08

minus06

minus04

minus02

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

Time (s)

Roll

rate

(rad

s)

minus25

minus2

minus15

minus1

minus05

(d) Roll rate


0 1 2 3 4 5 6

0

10

20

30

Time (s)

Yaw

rate

(rad

s)

minus40

minus30

minus20

minus10

(e) Yaw rate

05

15

0 1 2 3 4 5 6

0

1

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus15

minus1

minus05

times105


(f) Antiroll moment


1= 01 120590



0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment


1= 01 120590





1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 60

50

100

150

200

250

300

Time (s)

Stee

ring

whe

el an

gle (

deg)

(a) J-turn maneuver

0 1 2 3 4 5 6

1

12

Time (s)

Rollo

ver i

ndex

minus02

0

08

06

04

02

(b) Rollover index

0 1 2 3 4 5 6Time (s)

Roll

angl

e (ra

d)

minus02

0

02

04

06

08

1

12

14

(c) Roll angle

0 1 2 3 4 5 6

0

05

1

15

2

25

Time (s)

Roll

rate

(rad

s)

minus05

minus1

(d) Roll rate

0 1 2 3 4 5 6

0

5

10

15

20

25

30

Time (s)

Yaw

rate

(rad

s)

minus5


(e) Yaw rate

0 1 2 3 4 5 6

0

Time (s)

Ant

iroll

mom

ent (

Nm

)

minus14

minus12

minus10

minus8

minus6

minus4

minus2

times104


(f) Antiroll moment


1= 01 120590





1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1is equal to 120590



120578 gt 01

120576 gt 022+ 002times120582(20)

















Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Nomenclature







120593 Roll angle



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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