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Oct 28, 2014

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011 4217Design of a Preview Controller forVehicle Rollover PreventionSeongjin YimAbstractThis paper presents a method for designing a previewcontrollerforvehiclerolloverprevention. It is assumedthat adriverssteeringinputispreviewablewithaGlobal PositioningSystem(GPS) and an inertial measurement unit (IMU), or with anautomatic steering system for collision avoidance. Based on a lin-ear vehicle model, a linear optimal preview controller is designed.To avoid the full-state measurement of a linear quadratic regulator(LQR), linear quadratic static output feedback (LQ SOF) controlis adopted. To compare with several types of controllers such asLQR or LQ SOF with respect to rollover prevention capabilities,Bode plot analysis based on a linear vehicle model is performed.To show the effectiveness of the proposed controller, simulationsare performed on a vehicle simulation package CarSim.Index TermsLinear quadratic static output feedback(LQ SOF), preview control, vehicle rollover prevention.NOMENCLATUREaxLongitudinal acceleration (in meters per squaresecond).ayLateral acceleration (in meters per square second).Cf, CrCorneringstiffnessof front/rear tire(inNewtonsper radian).CRoll damping coefcient (in Newton-meter-secondper radian).Fyf, FyrLateral tire force of front/rear wheel (in Newtons).FRz, FLzSum of right/left vertical tire forces (in Newtons).Fx,brakeBraking force to decelerate a vehicle (in Newtons).f1, f2Active suspension forces (in Newtons).g Gravitational acceleration constant (9.81 m/s2).heHeight of a roll center from ground (in meters).hsHeight of CG from a roll center (in meters).Ix, Izroll/yaw moment of inertia about roll axis (kg m2).KBPressureforce constant (in Newtons perMegapasca).KSteady-state gain of the reference yaw rate.KRoll damping coefcient (in Newton-metersper radian).lf, lrDistance from CG to front/rear axle (in meters).MBControl yaw moment (in Newton-meters).Manuscript received December 27, 2010; revised May 10, 2011 andAugust 1, 2011; accepted September 18, 2011. Date of publicationSeptember26, 2011; dateofcurrent versionDecember9, 2011. Thisworkwas supported by Advanced Institutes of Convergence Technology under Grant2011-P3-08. The review of this paper was coordinated by Prof. J. Wang.The author is with Advanced Institutes of Convergence Technology, SeoulNational University, Suwon 443-270, Korea (e-mail: [email protected]).Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identier 10.1109/TVT.2011.2169687m Vehicle total mass (in kilograms).msSprung mass (in kilograms).rwRadius of a wheel (in meters).R Lateral load transfer (LTR).R LTR threshold.tfTrack width of a front axle (in meters).TsSampling rate (in seconds).vx, vyLongitudinal/lateral velocity of a vehicle (in metersper second).f, rTire slip angle of front/rear wheel (in radians). Yaw rate (in radians per second).dReference yaw rate (in radians per second). Roll angle (in radians). Roll rate (in radians per second).fFront steering angle (in radians).maxMaximumsteering angle of shhook maneuver(in degrees).REmergency steering angle (in radians). Tireroad friction coefcient.I. INTRODUCTIONIN THE LAST decade, a widespread supply of sports utilityvehicles (SUVs) with a high CGhas led to increased rolloveraccidents.Mostrolloveraccidentsarefatalsincetherolloverrateinfatal crashes is quitehigh. Althoughtheportionoffatalitiesinall crasheshasslightlydecreasedfrom36.3%to33.7%inthelast eight years, vehiclerollover still accountsforalargeportionofalldeathscausedbypassengervehiclecrashes[1]. Hence, vehiclerollover shouldbepreventedforpassenger safety.Untripped rollover occurs due to the large lateral accelerationby excessive steering at high speed. Therefore, it is necessarytoreducelateral accelerationtoprevent rollover. Followingthisidea, several control schemeswereproposed. Themostcommon scheme is to reduce the reference yaw rate or the lon-gitudinal velocity through differential braking or active steeringto make a vehicle exhibit understeer characteristics [2][8]. Asa measure of rollover threat, lateral acceleration, LTR, or time-to-rollover has been used [7], [8]. However, this reduction in thereferenceyawratemaycauseanotherkindofaccident, suchasacrashoratrippedrollover.Theotherapproachistouseactivesuspensionoractiveantirollbartoattenuatetheeffectof lateral acceleration on the roll motion of the vehicle, underthe assumption that the lateral acceleration is an uncontrollabledisturbance [9], [10]. In this paper, a rollover prevention con-troller is designed with a linear optimal control methodology.Differential braking and active suspension are used as actuators.0018-9545/$26.00 2011 IEEE4218 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011Fig. 1. Three-degree-of-freedomvehiclemodel. (a)2-DOFbicyclemodel.(b) 1-DOF roll model.Toavoidfull-statemeasurementoflinearquadraticregulator(LQR), alinear quadraticstaticoutput feedback(LQSOF)control methodology is adopted.Linearoptimalpreviewcontrolwasadoptedforridecom-fort enhancement in designing an active suspension controller[11]. Indevelopingthe driver model, a previewcontrol isinherentsinceadriverpreviewsthefuturepathtogenerateasteeringinput. Basedonthepreviewedpath, aproportionalintegraldifferentialorLQpreviewcontrolmethodologywasadopted in designing a driver model [12], [13]. By virtue of therecent development of sensor technologies such as Global Po-sitioning System (GPS) and inertial measurement unit (IMU),a drivers steering input can be previewable by a model predic-tivecontrolifitiscombinedwithmapinformation[14]. Onthe other hand, a drivers steering input is also previewable ifan automatic steering system for collision avoidance is adopted[15]. Under thesesituations, it is possibletodesignapre-view controller, which shows better performance in preventingvehicle rollover.This paper is organized as follows: In Section II, the designprocedureof apreviewcontroller for rollover preventionispresented. In Section III, Bode plot analysis is performedonalinearvehiclemodel, andsimulationsareperformedonthe commercial multibodydynamics software Carsim[16].Section IV concludes this paper.II. DESIGNOFA PREVIEW CONTROLLERFORROLLOVER PREVENTIONA. Vehicle ModelThe vehicle model used in this paper is a 3-degree-of-freedom (DOF) model, as shown in Fig. 1. This model consistsof a 2-DOF bicycle and a 1-DOF roll model to describe the yawand the lateral motion, and the roll motion, respectively.The equations of motion for this vehicle model are given asfollows [17]:Lateral motionmaymshs = Fyf + Fyr. (1)Yaw motionIz + Izx Ixy 2= lfFyf lrFyr + MB. (2)Roll motionIx + Ixz Iyz2mshsay= C K + msghs +t2 f1t2 f2. (3)In these equations, MB, f1, and f2 are the control yaw mo-ment by differential braking and the active suspension forces,respectively. In(2)and(3), thesquareterms2and2, andthe cross moment of inertia,Ixz,Ixy, andIyzwere neglectedbecause they have slight effect on the model accuracy. The roadbank angle is not considered in this model. In (1) and (3), lateralacceleration ay is dened as follows:ay = vy + vx. (4)In (1) and (2), it is assumed that the lateral tire forcesFyfandFyrare proportional to the tire slip angle for small, asshown inFyf= Cff, Fyr = Crr(5).Tire slipangle is denedas the difference betweenthedirection of wheel velocity and the steering angle. The tire slipangles of the front and rear wheels can be obtained through theapproximation tan1() as follows:f=vy + lfvxf, r =vylrvx. (6)Constants CfandCrarevalidwithinthelinear regionswhereissmall. IfgoesoverthesaturatedregionofFy,the constant cornering stiffness assumption is no longer valid.Moreover, Fy varies according to the variation of .The reference yaw rate d generated by the drivers steeringinput fis modeled with a rst-order system as follows:d =_Ks + 1_ f=Cf Cr (lf + lr) vxCf Cr (lf + lr)2+ m v2x (lr Crlf Cf)_fs + 1_(7)whereisthetimeconstant,andKisthesteady-stateyawrate gain determined by the speed of vehicle [18].The state-space representation of (7) is given as d = 1 d +Kf. (8)The error of yaw rateeis dened as the difference betweenthe actual yaw rate and the reference d, as shown ine = d. (9)Statex, control input u, anddisturbancewaredenedasfollows:x=[vy d]Tu=[MBf1f2]Tw=f. (10)YIM: DESIGN OF PREVIEW CONTROLLER FOR VEHICLE ROLLOVER PREVENTION 4219From these denitions and equations of motion, thecontinuous-timestate-spaceequationofthevehiclemodel isobtained as x = Ax +B1w+B2u. (11)Thedetaileddescriptiononthederivationof thestate-spacemodel can be found in [19].Thediscrete-timeequivalent of(11)canbeobtainedwiththezero-order holdtechniqueandthesamplingtime Tsasfollows [20]:x(k + 1) = Adx(k) +B1,dw(k) +B2,du(k) (12)where Ad= eATs, B1,d= (_Ts0Ad()d)B1, and B2,d=(_Ts0Ad()d)B2.B. Discrete-Time LQ SOF Controller Design for RolloverPreventionThe discrete-time LQ cost function for rollover prevention isgiven as follows:J =

k=1_q1e2(k) + q2a2y(k) + q32(k) + q4 2(k)+ q5M2B(k) + q6f21(k) + q6f22(k). (13)In (13), qi is the weight of each objective. By tuning the valueof qi, it is possible to emphasize each objective. The weights qiof the LQ cost function are set by the relation qi = 1/2ifromBrysons rule, where irepresents themaximumallowablevalue of each term [21].The LQ cost function (13) can be converted into the follow-ing discrete-time equivalent form:J =

k=1[C2x(k) +D2u(k)]T[C2x(k) +D2u(k)]=

k=1_xT(k)Qx(k) +uT(k)NTx(k)+ xT(k)Nu(k) +uT(k)Ru(k)(14)whereQ=CT2C2N= CT2D2R= DT2D2C2=q1 a11q1( a12+vx)q1 a13q1 a14q1 a150q20 0 q20 0q30 00 0 0q400 0 0 0 00 0 0 0 00 0 0 0 0D2=q1b2,11q1b2,12q1b2,130 0 00 0 00 0 0q50 00q600 0q6.In(14), a1iand b2,1iaretheelementsoftherst rowofmatrices Ad and B2,d from (12).Assuming that the static output feedback controller u(k) =Ky(k) is used, the discrete-time optimal LQ SOF problem istondKthat minimizestheLQcost functionJ[22]. Theava