IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011 4217

Design of a Preview Controller forVehicle Rollover Prevention

Seongjin Yim

Abstract—This paper presents a method for designing a previewcontroller for vehicle rollover prevention. It is assumed that adriver’s steering input is previewable with a Global 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 Terms—Linear quadratic static output feedback(LQ SOF), preview control, vehicle rollover prevention.

NOMENCLATURE

ax Longitudinal acceleration (in meters per squaresecond).

ay Lateral acceleration (in meters per square second).Cf , Cr Cornering stiffness of front/rear tire (in Newtons

per radian).Cφ Roll damping coefficient (in Newton-meter-second

per radian).Fyf , Fyr Lateral tire force of front/rear wheel (in Newtons).FR

z , FLz Sum of right/left vertical tire forces (in Newtons).

Fx,brake Braking force to decelerate a vehicle (in Newtons).f1, f2 Active suspension forces (in Newtons).g Gravitational acceleration constant (9.81 m/s2).he Height of a roll center from ground (in meters).hs Height of CG from a roll center (in meters).Ix, Iz roll/yaw moment of inertia about roll axis (kg · m2).KB Pressure–force constant (in Newtons per

Megapasca).Kγ Steady-state gain of the reference yaw rate.Kφ Roll damping coefficient (in Newton-meters

per radian).lf , lr Distance from CG to front/rear axle (in meters).MB Control yaw moment (in Newton-meters).

Manuscript received December 27, 2010; revised May 10, 2011 andAugust 1, 2011; accepted September 18, 2011. Date of publicationSeptember 26, 2011; date of current version December 9, 2011. This workwas 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 figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2169687

m Vehicle total mass (in kilograms).ms Sprung mass (in kilograms).rw Radius of a wheel (in meters).R Lateral load transfer (LTR).R LTR threshold.tf Track width of a front axle (in meters).Ts Sampling rate (in seconds).vx, vy Longitudinal/lateral velocity of a vehicle (in meters

per second).αf , αr Tire slip angle of front/rear wheel (in radians).γ Yaw rate (in radians per second).γd Reference yaw rate (in radians per second).φ Roll angle (in radians).φ Roll rate (in radians per second).δf Front steering angle (in radians).δmax Maximum steering angle of fishhook maneuver

(in degrees).δR Emergency steering angle (in radians).μ Tire–road friction coefficient.

I. INTRODUCTION

IN THE LAST decade, a widespread supply of sports utilityvehicles (SUVs) with a high CG has led to increased rollover

accidents. Most rollover accidents are fatal since the rolloverrate in fatal crashes is quite high. Although the portion offatalities in all crashes has slightly decreased from 36.3% to33.7% in the last eight years, vehicle rollover still accountsfor a large portion of all deaths caused by passenger vehiclecrashes [1]. Hence, vehicle rollover should be prevented forpassenger safety.

Untripped rollover occurs due to the large lateral accelerationby excessive steering at high speed. Therefore, it is necessaryto reduce lateral acceleration to prevent rollover. Followingthis idea, several control schemes were proposed. The mostcommon 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 thereference yaw rate may cause another kind of accident, suchas a crash or a tripped rollover. The other approach is to useactive suspension or active antiroll bar to attenuate the effectof 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 IEEE

4218 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011

Fig. 1. Three-degree-of-freedom vehicle model. (a) 2-DOF bicycle model.(b) 1-DOF roll model.

To avoid full-state measurement of linear quadratic regulator(LQR), a linear quadratic static output feedback (LQ SOF)control methodology is adopted.

Linear optimal preview control was adopted for ride com-fort enhancement in designing an active suspension controller[11]. In developing the driver model, a preview control isinherent since a driver previews the future path to generate asteering input. Based on the previewed path, a proportional–integral–differential or LQ preview control methodology wasadopted 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 driver’s steering input can be previewable by a model predic-tive control if it is combined with map information [14]. Onthe other hand, a driver’s steering input is also previewable ifan automatic steering system for collision avoidance is adopted[15]. Under these situations, it is possible to design a pre-view controller, which shows better performance in preventingvehicle rollover.

This paper is organized as follows: In Section II, the designprocedure of a preview controller for rollover prevention ispresented. In Section III, Bode plot analysis is performedon a linear vehicle model, and simulations are performed onthe commercial multibody dynamics software Carsim [16].Section IV concludes this paper.

II. DESIGN OF A PREVIEW CONTROLLER FOR

ROLLOVER PREVENTION

A. Vehicle Model

The 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 motion

may − mshsφ = Fyf + Fyr. (1)

Yaw motion

Izγ + Izxφ − Ixyφ2 = lfFyf − lrFyr + MB . (2)

Roll motion

Ixφ + Ixzγ − Iyzγ2 − mshsay

= −Cφφ − Kφφ + msghsφ +t

2· f1 −

t

2· 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), the square terms φ2 and γ2, andthe cross moment of inertia, Ixz , Ixy, and Iyz were 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 defined as follows:

ay = vy + γvx. (4)

In (1) and (2), it is assumed that the lateral tire forces Fyf

and Fyr are proportional to the tire slip angle for small α, asshown in

Fyf = −Cfαf , Fyr = −Crαr (5).

Tire slip angle α is defined as the difference between thedirection of wheel velocity and the steering angle. The tire slipangles of the front and rear wheels can be obtained through theapproximation tan−1(θ) � θ as follows:

αf =vy + lfγ

vx− δf , αr =

vy − lrγ

vx. (6)

Constants Cf and Cr are valid within the linear regionswhere α is small. If α goes over the saturated region of Fy ,the constant cornering stiffness assumption is no longer valid.Moreover, Fy varies according to the variation of μ.

The reference yaw rate γd generated by the driver’s steeringinput δf is modeled with a first-order system as follows:

γd =(

Kγ

τs + 1

)· δf

=Cf · Cr · (lf + lr) · vx

Cf · Cr · (lf + lr)2 + m · v2x · (lr · Cr − lf · Cf )

×(

δf

τs + 1

)(7)

where τ is the time constant, and Kγ is the steady-state yawrate gain determined by the speed of vehicle [18].

The state-space representation of (7) is given as

γd = −1τ

γd +Kγ

τδf . (8)

The error of yaw rate eγ is defined as the difference betweenthe actual yaw rate γ and the reference γd, as shown in

eγ = γ − γd. (9)

State x, control input u, and disturbance w are defined asfollows:

x Δ= [vy γ φ φ γd]T

u Δ= [MB f1 f2]T

w Δ= δf . (10)

YIM: DESIGN OF PREVIEW CONTROLLER FOR VEHICLE ROLLOVER PREVENTION 4219

From these definitions and equations of motion, thecontinuous-time state-space equation of the vehicle model isobtained as

x = Ax + B1w + B2u. (11)

The detailed description on the derivation of the state-spacemodel can be found in [19].

The discrete-time equivalent of (11) can be obtained withthe zero-order hold technique and the sampling time Ts asfollows [20]:

x(k + 1) = Adx(k) + B1,dw(k) + B2,du(k) (12)

where AdΔ= eATs , B1,d

Δ= (∫ Ts

0 Ad(τ)dτ)B1, and B2,dΔ=

(∫ Ts

0 Ad(τ)dτ)B2.

B. Discrete-Time LQ SOF Controller Design for RolloverPrevention

The discrete-time LQ cost function for rollover prevention isgiven as follows:

J =∞∑

k=1

[q1e

2γ(k) + q2a

2y(k) + q3φ

2(k) + q4φ2(k)

+ q5M2B(k) + q6f

21 (k) + q6f

22 (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 qi

of the LQ cost function are set by the relation qi = 1/η2i from

Bryson’s rule, where ηi represents the maximum allowablevalue 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)NT x(k)

+ xT (k)Nu(k) + uT (k)Ru(k)]

(14)

where

Q Δ=CT2 C2 N Δ= CT

2 D2 R Δ= DT2 D2

C2Δ=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

√q1a11

√q1(a12+vx)

√q1a13

√q1a14

√q1a15

0√

q2 0 0 −√q2

0 0√

q3 0 00 0 0

√q4 0

0 0 0 0 00 0 0 0 00 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

D2Δ=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

√q1b2,11

√q1b2,12

√q1b2,13

0 0 00 0 00 0 0√q5 0 00

√q6 0

0 0√

q6

⎤⎥⎥⎥⎥⎥⎥⎥⎦

.

In (14), a1i and b2,1i are the elements of the first row ofmatrices 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 isto find K that minimizes the LQ cost function J [22]. Theavailable outputs for SOF are the lateral acceleration, the rollrate, and the yaw rate error, which is given as (15), shownbelow. These signals are easily measured by sensors

y(k) Δ= Cx(k) =

⎡⎣ ay

φeγ

⎤⎦ . (15)

In this paper, a simple iterative algorithm proposed byRosinova et al. was used to find the optimal K [23].

C. Discrete-Time LQ SOF Preview Controller Design forRollover Prevention

Let the preview time Tp = p · Ts be a multiple of the sam-pling time Ts, and let w(k) be the vector containing all thepreview inputs at instant k, as given in (16), shown below.The discrete-time dynamics equation of vector w(k) can berepresented as (17), shown below:

w(k) = [w(k) w(k + 1) · · · w(k + p) ]T (16)w(k + 1) =Φw(k) + Γw(k + p + 1). (17)

In (17), Φ and Γ are defined as follows:

Φ Δ=

⎡⎢⎢⎢⎢⎢⎣

0 1 0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 0. . . 1

0 0 0 · · · 0

⎤⎥⎥⎥⎥⎥⎦

, Γ Δ=

⎡⎢⎢⎢⎢⎣

00...01

⎤⎥⎥⎥⎥⎦ . (18)

By augmenting (12) with (17), the augmented discrete-timestate-space equation is obtained as follows [18]:[

x(k + 1)w(k + 1)

]=

[Ad Ψ0 Φ

] [x(k)w(k)

]

+[0Γ

]w(k + p + 1) +

[B2,d

0

]u(k) (19)

where Ψ Δ= [B1,d 0 · · · 0].By defining new state and matrices, (19) is converted into the

following equation:

x(k + 1) = Ax(k) + B1w(k + p + 1) + B2u(k) (20)

where

x(k) Δ=[

x(k)w(k)

], A Δ=

[Ad Ψ0 Φ

]

B1Δ=

[0Γ

], B2

Δ=[B2,d.

0

].

Notice that matrix Φ is stable, and therefore, the augmentedsystem (19) preserves the stabilizability and detectability prop-erties of the original plant.

4220 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011

Corresponding to the augmented system, the discrete-timeLQ cost function (14) should be modified as follows:

J =∞∑

k=1

[xT (k)Qx(k) + uT (k)NT x(k)

+ xT (k)Nu(k) + uT (k)Ru(k)]

(21)

where

Q Δ=[Q 00 0

], N Δ=

[N0

], R Δ= R.

The output equation (15) should also be modified as

y(k) Δ= Cx(k) =[C 00 I

]x(k). (22)

The static output feedback control for the augmented systemis defined as

u(k) = Ky(k). (23)

The optimal LQ SOF preview control gain K can be easilycomputed by the algorithm proposed by Rosinova et al. [23].

D. Preview on Steering Input

Recent developments in sensor technologies make it possibleto obtain precise information on the states of a vehicle. Forexample, it is possible to obtain vehicle position and headingwith an error of 10 cm using a differential GPS (DGPS). Ifvehicle position and heading information obtained from DGPSare combined with map information, a future navigation path ofa vehicle can be obtained. Based on this information and the as-sumption that a driver wants to follow a future navigation path,a driver’s steering input can be estimated by model predictivecontrol [14]. Another method to obtain a future navigation pathis to use an automatic steering system for collision avoidance[15]. In [15], a future path is obtained with a sigmoid functionto represent a path for collision avoidance. Based on this path,the driver’s steering input can be obtained with a driver model[11]. In this paper, a driver’s steering input acts as a disturbance,as shown in (10). To obtain an LQ SOF preview controller,the original system is augmented with a previewed disturbancesignal [20].

If the future navigation path is straight, the previewed steer-ing input will be zero. Under this situation, the method usingDGPS and map information cannot preview abrupt steeringchange such as a double-lane change maneuver because thefuture navigation path is straight and it is assumed that adriver wants to follow the future navigation path. If there isabrupt steering change on a straight road, only the feedbackcontrol—LQR or LQ SOF—is activated because the previewedsteering inputs are zero.

III. SIMULATION

Linear controllers are designed based on the linear vehiclemodel given in (12). The values of parameters in the linear

TABLE IPARAMETERS AND VALUES OF THE SMALLSUV MODEL IN CARSIM

TABLE IIWEIGHTS IN THE LQ COST FUNCTION

vehicle model are referred from SmallSUV model in CarSim,as given in Table I.

To avoid rollover in cornering situations, a vehicle shouldfollow the reference yaw rate γd with a small lateral accelera-tion ay . To follow the reference yaw rate, the yaw rate error eγ

should be reduced. To prevent rollover, the lateral accelerationay , roll angle φ, and roll rate φ should be reduced. For thispurpose, weights q2, q3, and q4 in (13) should be set to highervalues. If the weights on the roll angle and the roll rate arehighly emphasized for rollover prevention, the yaw rate errorincreases due to the LTR, and this can cause a loss of themaneuverability or the lateral stability [10]. To cope with thisproblem, an electronic stability control (ESC) was adopted inmy previous work [10]. On the other hand, if the weight onthe yaw rate error is highly emphasized for the maneuverabilityor the lateral stability, the roll angle cannot decrease [10].Therefore, the LQ objective function given in this paper shouldsimultaneously consider these effects. For this reason, the rollangle, the roll rate, and the yaw rate error are highly emphasizedin the LQ objective function. The values of ηi for the weightsin the LQ cost function are given in Table II.

In the discrete-time control, the sampling time of a sensorand a control input is set to 5 ms. In designing a previewcontroller, the preview length of the steering input is set to100 samples (=0.5 s).

To verify the effectiveness of the proposed controller, LQR,LQ preview, LQ SOF, and the proposed LQ SOF previewcontroller are compared in terms of the lateral acceleration, theyaw rate error, and the roll angle. The steering input used inthis paper was the fishhook maneuver with the maximum angleof 270◦. Hereafter, the legends LQR, LQ Preview, LQ SOF, andLQ SOF Preview in the figures represent the controller designedby LQR, LQ preview, LQ SOF, and the proposed LQ SOFpreview controller, respectively.

A. Linear System Analysis for the Designed Controllers

Fig. 2 shows the Bode plots that are drawn based on thedesigned controllers. In these plots, the input is the steeringangle, and the outputs are the roll angle, the lateral acceleration,and the yaw rate error. Since the frequency of the fishhookmaneuver with the maximum steering angle of 270◦ is near0.5 Hz, the frequency responses near 0.5 Hz should be checked[24]. As shown in Fig. 2(b), each controller has slight effecton the lateral acceleration although it decreases the lateral

YIM: DESIGN OF PREVIEW CONTROLLER FOR VEHICLE ROLLOVER PREVENTION 4221

Fig. 2. Bode plots from the steering input to each output. (a) Bode plot fromδf to φ. (b) Bode plot from δf to ay . (c) Bode plot from δf to eγ .

acceleration over an uncontrolled vehicle. The preview con-trollers LQ Preview, and LQ SOF Preview show the best perfor-mance in controlling the roll angle, compared with LQR and LQSOF, as shown in Fig. 2(a). On the contrary, these controllersshow poor performance in reducing the yaw rate error, as shown

Fig. 3. Sensitivity function of LQR and LQ SOF.

in Fig. 2(c). These results indicate that the reduction in theroll angle increases the yaw rate error. This is caused by thefact that the yaw rate and the lateral velocity are reduced bythe controllers in order to decrease the lateral acceleration,according to (4). In other words, the controllers make thevehicle exhibit an understeer characteristic or a large yaw rateerror [19]. The preview control deepens this characteristic. Thelarge yaw rate error indicates that the controlled vehicle did notsufficiently follow the reference yaw rate. From these results,it can be concluded that the preview controllers LQ Previewand LQ SOF Preview concentrate its effort to reduce the lateralacceleration and the roll angle at the expense of the increasedyaw rate error.

Fig. 3 shows the sensitivity function of LQR and LQ SOF.As shown in this figure, the active suspension input of LQ SOFis smaller than that of LQR near 0.5 Hz. On the contrary, theyaw moment input of LQ SOF is larger than that of LQR. Thisfact indicates that the difference in the active suspension inputgenerates the difference in the reduction of roll angle, as shownin Fig. 3. The sensitivity functions of LQ Preview and LQ SOFPreview are identical to those of LQR and LQ SOF becausethe feedforward part cannot have an effect on the robustnessof the feedback part. On the other hand, noises in the previewsignal can have a significant effect on the performance of thecontrolled system. To cope with this problem, an observer-based preview control can be used [25]. However, that is beyondthe scope of this paper.

B. Rollover Prevention Control for CarSim Vehicle Model

To show the effectiveness of the proposed preview controllerin view of rollover prevention, simulations were performed ona nonlinear multibody dynamic simulation software CarSim.To simulate the designed controllers on CarSim, it is necessaryto distribute yaw moment MB to the four wheels. According tothe sign of MB , the braking pressure is applied to the front leftor front right wheel. For instance, if the sign of MB is positive,the brake pressure is applied to the front left wheel. The givenyaw moment MB is transformed into the braking force Fx as

Fx =2MB

tf. (24)

4222 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011

Fig. 4. Fishhook maneuver and its previewed steering input.

The relationship between the braking force Fx and brake pres-sure PB on a wheel is assumed as follows:

PB =rw

KB· Fx. (25)

With the designed controllers, simulations were performedon the nonlinear vehicle model SmallSUV in CarSim. In thesimulation, the steering input is assumed to be known a priorias the fixed fishhook maneuver with the maximum angle δmax

of 270◦, as shown in Fig. 4 [24]. In Fig. 4, the dotted linerepresents the previewed steering input. This steering input isassumed to be exactly computed by a model predictive control[14]. The sampling rate of the previewed steering input is 50 mswith the preview interval of 0.5 s. Hence, the preview length is10. The initial speed of the vehicle is set to 80 km/h, and thetire–road friction coefficient is set to 1.0. There are no speedcontrols to maintain a constant speed. The actuators of thebrake and the active suspension are modeled as a first-ordersystem with a time constant of 0.12 and 0.08, respectively.The sampling rate of the control command, i.e., yaw momentand active suspension force, and the sensor signals are set to5 ms. The previewed signals on the driver’s steering input areinterpolated to be fitted to the sampling rate of the controlcommand, which is 5 ms. To prevent the locking of the brake,the ABS provided in CarSim is used. The operating range of theslip ratio in the ABS is between 0.05 and 0.15.

To check the rollover characteristic of the target vehicle,simulations were performed under the fishhook maneuver withthe maximum steering angle of 270◦. Figs. 5 and 6 show thesimulation results for the uncontrolled vehicle for each initialspeed. As shown in Fig. 5(a), the lateral accelerations werenearly the same, regardless of the initial speed. As a result,the roll angles were nearly the same. This fact indicates thatlateral acceleration is hard to control. For this reason, previ-ous works have concentrated on controlling roll motion underthe assumption that lateral acceleration is an uncontrollabledisturbance [9], [10]. The uncontrolled vehicle with an initialspeed of 80 km/h was drifted, as shown in Figs. 5(c) and 6.This is a rear-sway phenomenon, which is caused by the factthat the inner wheels are lifted in cornering due to large lateralacceleration, and consequently, the lateral tire force of the rearouter wheel becomes small, compared with that of the frontouter one [10]. According to a previous work, this tendency gets

Fig. 5. Simulation result of the uncontrolled vehicle for each initial speed.(a) Lateral acceleration. (b) Roll angle. (c) Yaw rate.

Fig. 6. Trajectories of the uncontrolled vehicle for each initial speed.

more severe if an active suspension or an active antiroll bar isused for the control of the roll motion [10]. From these facts,it is obvious that active suspension plays a role in reducingthe roll motion under excessive lateral acceleration and that adifferential braking plays a role in preventing wheel lift.

Figs. 7–9 show the responses of the controlled vehicle foreach controller. As shown in Fig. 7(a)–(c), the responses of theroll angle, the yaw rate error, and the lateral acceleration wereexpected from the Bode plot analysis in the previous section. Asshown in Fig. 7, there were slight differences in the responsesof the lateral acceleration, and the roll angle decreases as theyaw rate error increases, just like the implication of the Bodeplots in Fig. 2.

YIM: DESIGN OF PREVIEW CONTROLLER FOR VEHICLE ROLLOVER PREVENTION 4223

Fig. 7. Responses of the SmallSUV model for each controller. (a) Rollangle (in degrees). (b) Yaw rate error (in degrees per second). (c) Lateralacceleration (m/s2).

As shown in Fig. 8(a), the preview controllers LQ Previewand LQ SOF Preview generated larger active suspension inputs,compared with LQR and LQ SOF. Due to the larger active sus-pension inputs of the preview controllers, the roll angles weresignificantly reduced, compared with LQR and LQ SOF. Fromthese results, it can be concluded that the main advantage of thepreview control is the enhancement of roll control capability.As shown in Fig. 8(b), the peak yaw moment inputs of thepreview controllers are larger than those of the LQR and LQSOF. This is caused by the increased yaw rate error of LQPreview and LQ SOF Preview, as pointed out in Fig. 7(b).Fig. 8(c) shows the applied brake pressures of the designedcontrollers. In these figures, the legends FL, FR, RL, and RRrepresent the front left, front right, rear left, and rear rightwheels, respectively. As pointed out in Fig. 8(c), the brakeinputs of the preview controllers are larger than those of LQRand LQ SOF, due to the large yaw moment inputs. Fig. 9 showsthe trajectories of the vehicles with each controller. As shown inthis figure, the controlled vehicles were not drifted because theinner wheels were not lifted under the large lateral acceleration.The vehicles with preview controllers LQ Preview and LQ SOFPreview have a smaller cornering radius and a travel distancethan those of LQR and LQ SOF as the preview controllers madethe controlled vehicle exhibit understeer characteristic, whichgenerated a larger braking input.

Fig. 8. Control inputs in CarSim simulation for each controller in Fig. 3.(a) Active suspension input. (b) Yaw moment input for each controller.(c) Applied brake pressures of each controller.

C. Comparison of Rollover Prevention Controllers

For comparison, a rollover prevention controller was de-signed by the LTR-based method proposed in [7]. This methodused the LTR as a measure of rollover threat and emergencysteering and braking control for rollover prevention. The LTRR is simply defined as

RΔ=

FRz − FL

z

FRz + FL

z

. (26)

In (26), Fz,R and Fz,L are the sum of the right and left verticaltire forces, respectively. The vertical tire forces can be easilyestimated using the longitudinal and lateral acceleration signals[26]. If |R| is unity, the left or right wheels are lifted off, andthis is regarded as a rollover.

The method proposed in [7] uses emergency steering andbraking control for rollover prevention. Emergency steering

4224 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011

Fig. 9. Vehicle trajectories of each controller based on the CarSim vehiclemodel. (a) Trajectories of each controller. (b) Reference and real trajectories ofeach controller.

control is to reduce the current steering angle by δR, which iscalculated by

δR ={

kR · sgn(R) · (R − R), |R| > R0, |R| ≤ R.

(27)

Emergency braking control is to decelerate the vehicle as soonas the LTR becomes critical. The required braking force iscalculated by

Fx,brake ={

0, |R| ≤ R−m · ax,max, |R| > R.

(28)

This control was originated from the fact that the lateral accel-eration decreases as the longitudinal velocity does, as given in(4). In the method of [7], there are no roll controls for rolloverprevention and yaw motion control for lateral stability. In thispaper, an active suspension control is adopted for roll control.The roll moment Mφ,R that is required to stabilize the rollmotion is calculated with the lateral acceleration, as given in

Mφ,R = −m · hs · ay. (29)

The calculated roll moment is distributed to active suspensionsin left and right wheels.

Fig. 10. Responses of the SmallSUV model for each controller. (a) Roll angle(in degrees). (b) Yaw rate (in degrees per second). (c) Yaw rate error (in degreesper second). (d) Lateral acceleration (m/s2).

As presented, the LTR-based method given in [7] is quitesimple because it uses only longitudinal and lateral accel-eration. Simulation was performed to compare the previewcontrollers with the LTR-based method. Simulation condi-tions were identical to those of the previous section. In (27)and (28), the values of R and kR were set to 0.6 and 1.0,respectively.

Figs. 10–12 show the responses of the controlled vehicle foreach controller. In these figures, the legend LTR indicates thecontroller by the LTR-based method. As shown in Fig. 10, theresults of LTR are equivalent to those of preview controllers LQPreview and LQ SOF Preview except that the yaw rate error ofLTR is larger than those of the preview controllers. This wascaused by the fact that there are no yaw motion controls in theLTR-based method. As shown in Fig. 11, the active suspensionand braking input of LTR is larger than those of the preview

YIM: DESIGN OF PREVIEW CONTROLLER FOR VEHICLE ROLLOVER PREVENTION 4225

Fig. 11. Control inputs in CarSim simulation for each controller. (a) Activesuspension input. (b) Applied brake pressures of each controller.

Fig. 12. Vehicle trajectories of each controller based on the CarSim vehiclemodel.

controllers. This indicates that the LTR-based method requiresmore control efforts to give an equivalent performance over thepreview controllers. As a result of the excessive braking of LTR,the trajectory of LTR shows oversteer characteristic, as shownin Fig. 12.

IV. CONCLUSION

In this paper, a preview controller design method for rolloverprevention has been proposed. Differential braking and ac-tive suspension have been adopted as actuators. Under theassumption that the steering input is previewable, the rolloverprevention controller has been designed with LQ SOF preview

control. From the Bode plot analysis, it has been determinedthat the preview controller reduces the roll angle and the lateralacceleration and increases the yaw rate error by making thecontrolled vehicle exhibit understeer characteristics. From thesimulations in CarSim, it has been shown that the previewcontrollers LQ Preview and LQ SOF Preview show superiorperformance in preventing rollover by reducing the roll angleand the lateral acceleration with the active suspension and thedifferential braking at the expense of the increased yaw rateerror.

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Seongjin Yim received the B.S. degree in me-chanical engineering from Yonsei University, Seoul,Korea, in 1995 and the M.S. and Ph.D. degrees inmechanical engineering from the Korea AdvancedInstitute of Science and Technology, Daejeon, Korea,in 1997 and 2007, respectively.

From 2008 to 2010, he was a Postdoctoral Re-searcher with the Brain Korea 21 (BK21) Schoolfor Creative Engineering Design of Next-GenerationMechanical and Aerospace Systems, Seoul NationalUniversity. He is currently a Research Professor with

Advanced Institutes of Convergence Technology, Seoul National University.His research interests are robust control, vehicle rollover prevention, and unifiedchassis control systems for hybrid and electric vehicles.