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Center of Gravity Estimation and RolloverPrevention Using Multiple Models & Controllers

Selim Solmaz, Mehmet Akar and Robert Shorten

Abstract — In this paper, we present a methodology based onmultiple models and switching for real–time estimation of centerof gravity (CG) position and rollover prevention in automotivevehicles. Based on a linear vehicle model in which the unknownparameters appear nonlinearly, we propose a novel sequentialidentication algorithm to determine the vehicle parametersrapidly in real time. The CG height estimate is further coupledwith a switching controller to prevent un–tripped rollover inautomotive vehicles. The efcacy of the proposed switched multimodel/controller estimation and control scheme is demonstratedvia numerical simulations.

I. INTRODUCTION

It is well known that vehicles with a high center of gravitysuch as light trucks (vans, pickups, and SUVs) are moreprone to rollover accidents than other types of passengervehicles. According to recent statistical data [1], this classof vehicles were involved in nearly 70% of all the rolloveraccidents in the USA during 2004, with SUVs alone wereresponsible for about half of this total. The fact that the

composition of the current automotive eet in the U.S.consists of nearly 36% light trucks [2], along with the recentincrease in their popularity worldwide, makes rollover animportant safety problem.

There are two distinct types of vehicle rollover: trippedand un-tripped. A tripped rollover commonly occurs whena vehicle slides sideways and digs its tires into soft soilor strikes an object such as a curb or guardrail. Driverinduced un-tripped rollover however, can occur during typicaldriving situations and it poses a real threat for top-heavyvehicles. Examples are excessive speed during cornering,obstacle avoidance and severe lane change maneuvers, whererollover may occur as a direct result of the lateral wheelforces induced during these maneuvers [2]. Rollover has beenthe subject of intensive research in recent years, especiallyby the major automobile manufacturers, and the majoritythis work is geared towards the development of rolloverprediction schemes and robust occupant protection systems.While robust active rollover control systems achieve thegoal of preventing this type of accidents, such a controlapproach may be too conservative, and it can potentiallycompromise the performance of the vehicle under non-critical driving situations. It is however, possible to prevent

S. Solmaz and R. Shorten are with the Hamilton Institute, NUIM, Ireland(Email:selim.solmaz@nuim.ie, robert.shorten@nuim.ie). M. Akar is with the

Department of Electrical and Electronic Engineering, Bo˘ gazici University,Bebek, Istanbul, 34342, Turkey. Tel: +90 212 3596466. Fax: +90 2122872465 (E-mail: mehmet.akar@boun.edu.tr).

rollover accidents in an effective and efcient manner bycontinuously monitoring the car dynamics and applying theproper and sufcient control action to recover handling of the vehicle in emergency situations.

The height of CG along with the lateral acceleration are themost important parameters affecting the rollover propensityof an automotive vehicle; while the former is available as part

of standard sensor packs, the CG height can not be measureddirectly [3]. With this background in mind, we rst proposeour CG estimation method based on multiple models, andthen use the technique in designing a switching rollovercontroller. As part of the feedback implementation we utilizemultiple simplied linear models, which are parameterizedto cover uncertainty in the vehicle parameters. Switchingbetween these models yields a rapid estimation of unknownand time-varying vehicle parameters through the selectedmodels, which is then used to switch among a set of suitablecontrollers in order to improve the performance of activerollover mitigation systems.

Our motivation for considering a switching controller im-plementation is twofold. Firstly, switching controllers are thealternative option to the robust ones and they can potentiallyprovide higher performance. Robust controllers have xedgains that are chosen considering the worst-case that theplant undergoes; for the rollover problem, the worst operatingcondition translates to operating the vehicle with the highestpossible CG position. While choosing the controller gainsfor the worst-case guarantees the performance (i.e., safety)under the designed extreme operating condition, the feedback performance of the robustly controlled systems under lesssevere or even normal operating conditions are suboptimal.Our second motivation is related to the time constant of

rollover accidents, which is on the order of seconds. Whileconventional adaptive controllers are known to have slowconvergence rates and large transient control errors when theinitial parameter errors are large (a factor that renders thesecontrol approaches unsuited for use in rollover mitigationapplications), utilization of MMST type algorithms [4] mayovercome these problems and provide high performanceadaptive controllers. Therefore, when improving the con-troller performance and speed for the rollover problem isconsidered, MMST framework becomes an ideal choice as itcan provide rapid identication of the unknown parametersas part of the closed loop implementation. This way wecan rapidly switch to a controller that is more suitable forthe vehicle operating conditions, thus improving the overallsafety of the vehicle without sacricing its performance.

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TABLE I

M ODEL VARIABLES

Variable Description Value Unitm Vehicle mass 1300 kgg Gravitational acceleration 9.81 m/s 2

vx Initial longitudinal speed 30 m/sJ xx Roll moment of inertia at the CG 400 kgm 2

J zz Yaw moment of inertia at the CG 1200 kgm 2

L Axle separation 2.5 mT Track width 1.5 mlv long. CG position w.r.t. front axle 1.2 mlh long. CG position w.r.t. rear axle 1.3 mh CG height over ground 0.51 mc suspension damping coefcient 5000 Nms/radk suspension spring st iffness 36000 Nm/radC v Front tire stiffness coefcient 60000 N/radC h Rear tire stiffness coefcient 90000 N/radδ,β,φ Steering angle, Side–slip angle (at

CG), and roll angle, respectivelyvarying rad

α v , α h Side–slip angles at the front andrear tire, respectively

varying rad

˙ψ, ˙φ Yaw rate and roll rate, respectively varying rad/s

Fig. 1. Single track model with roll degree of freedom.

I I . S YSTEM D ESCRIPTION

In this section we present the mathematical model captur-ing the lateral and vertical dynamics of a car. We also denethe load transfer ratio as the rollover assessment criterion,and further state our assumptions regarding the actuators andvehicle parameters. For related notation, refer to Table I.

A. Single track model with roll degree of freedomThis is the simplest model with combined roll and lateral

dynamics, and is used to represent the real vehicle in oursimulations. By assuming that the left and right tires arelumped into a single one at the axle centerline as shown onthe left hand side of Fig. 1, the combined horizontal and rolldynamics of the vehicle can be compactly characterized byx = Ax + B δ δ + B u u with

A =

− σJ x eq

mJ xx vρJ x eq

mJ xx v 2− 1 − hc

J xx vh ( mgh − k )

J xx vρ

J zz

− κJ zz v

0 0− hσ

J xx

hρJ xx v

− cJ xx

mgh − kJ xx

0 0 1 0

B δ = C v J x eq

mJ xx vC v l v

J zzhC v

J xx0

T

,

B u = 0 − T 2J zz

0 0T

,

(1)

Fig. 2. Differential braking force as control input.

where x = [ β, ψ, φ, φ]T is the state; β is the side–slip angle;ψ is the yaw–rate; φ is the roll–angle and σ,ρ, and κ areauxiliary parameters that are dened as follows: σ C v +C h , ρ C h lh − C v lv , κ C v l2

v + C h l2h . Also J x eq =

J xx + mh 2 denotes the equivalent roll moment of inertia. Inthe model u represents the total effective differential braking

force acting on the wheels about the vertical axis. This forceis parallel to the road, and it is positive if the effective brakingis on the right wheels and negative if the effective braking ison the left wheels. Differential braking force as the controlinput is depicted in Fig. 2. In order to model the changein the vehicle longitudinal speed as a result of braking, weassume that the longitudinal wheel forces generated by theengine counteract the rolling resistance and the aerodynamicdrag at all times. Under this assumption, the vehicle speedis approximately governed by

v = −|u |m

. (2)

B. The Load Transfer Ratio, LT R d

The vehicle load transfer ratio ( LT R ) is dened by

LT R =Load on Right Tires-Load on Left Tires

Total Load. (3)

Clearly, LT R varies within [− 1, 1], and it is equal to zero fora symmetric car that is driving straight. The bounds LT R ∈

{− 1, 1} are reached in the case of a wheel lift-off on eitherside of the vehicle. This indication capability of the LT R isuseful in design of rollover prevention schemes. A dynamical

approximation for the load transfer ratio, denoted LT R d , isgiven as follows [2]

LT R d = −2(cφ + kφ)

mgT . (4)

C. Actuators and Vehicle parameters

1) Actuators: Rollover prevention techniques may relyon several actuation mechanisms including active steering,active suspension, active roll stabilizer bars and differentialbraking. Among these techniques, differential braking sys-tems can be found in almost every class of passenger vehiclesthrough ABS (Anti-lock Braking System) and thus it hasbeen used extensively for rollover prevention [5], [6]. It isthe most effective way to manipulate the tire forces, and

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it is the only one that can reduce the vehicle longitudinalspeed among the afore-mentioned actuator types. Althoughthe switching rollover controller to be described in this papercan easily be extended to other types of actuators, differentialbraking will be adopted in the sequel.

2) Parameters: We assume that vehicle mass m is known,which can be estimated as part of the braking system [2]. Fur-thermore C v , C h , lv , k , c and h are all assumed be unknownparameters of the vehicle and are estimated through themultiple model identication algorithm. We further assumethat these parameters vary within certain closed intervalsC v ∈ C v , C h ∈ C h , lv ∈ Lv , c ∈ C , k ∈ K and h ∈ H,and these intervals can be found via accurate numericalsimulations as well as eld tests.

I I I . V EHICLE PARAMETER IDENTIFICATION

The problem in hand is to determine the vehicle parametersthat have been described in the previous section. While linearregression techniques, including least squares identicationcan be tried, such methods require persistently exciting inputsignals [7], which might impose unrealistic and dangerousmaneuvers. Moreover, note that the linear model introducedin Section II is nonlinear in the unknown vehicle parametersfurther complicating the formulation of the estimation prob-lem using the traditional approaches. Thus, there is a needfor alternative techniques for vehicle parameter identication,which imposes no restriction on the driver input, has fastconvergence rates and requires minimum additional output

information (sensors).With the above motivation in mind, we now introduce ourmultiple model based identication algorithm to determinethe unknown vehicle parameters rapidly in real–time. To thisend, a rst step approach would be to setup the multipleidentication models using (1) with u = 0 . However inthis case, the resulting parameter space will be too complexto handle. Instead we adapt a modular estimation strategyof decoupling the vehicle dynamics into subsystems byassuming a weak relationship from the roll dynamics ontothe lateral.

A. Identication of lateral dynamics parametersThe identication of the longitudinal CG location lv and

the lateral tire stiffness parameters C v , C h makes use of theyaw–rate equation in (1), i.e.,

ψ =C h lh − C v lv

J zzβ −

C v l2v + C h l2

h

J zz vψ +

C v lv

J zzδ. (5)

Dene the ltered signals ωl ∈ ℜ3

ωl = λ1ωl + [ β, ψ, δ]T , (6)

where λ1 < 0. By following the standard arguments in

identication [7], (5) can be rewritten asψ = θ∗T ωl , (7)

where θ∗∈ ℜ3 represents the parameter vector, from whichthe vehicle parameters can be determined as follows:

l∗v =L(θ∗1 + θ∗3 ) − v(− λ1 − θ∗2 )

θ∗1, (8)

C ∗v =J zz θ∗3

l∗v, C ∗h =

J zz (θ∗1 + θ∗3 )L − l∗v

. (9)

The multiple model based identication algorithm to de-termine the longitudinal CG location lv and the tire stiffnessparameters C v , C h assumes that each unknown parameterbelongs to a closed interval such that C v ∈ C v , C h ∈

C h , and lv ∈ Lv . These intervals are divided into certainnumber of grid points that can be represented as C v ={C v 1 , C v 2 , C v 3 , . . . , C v p }, C h = {C h 1 , C h 2 , C h 3 , . . . , C h q },and Lv = {lv 1 , lv 2 , lv 3 , . . . , l v r } with dimensions p, q and r ,respectively. These grid points form the p × q × r xed iden-

tication models. Additionally, we employ one free runningadaptive model , and one re–initialized adaptive model [4].The identication error, ei , corresponding to the i th modelis dened as

ei = ψ − ˆψ = ( θ∗− θi )T ωl , (10)

where θi denotes the parameter of the i–th model. Thevehicle parameters are estimated as the parameters of thei⋆–th model

i⋆ = arg mini∈{1,2,...,n }

J i (t), (11)

where J i (t) is the cost corresponding to the i th identication

error and is given by [4]

J i (t) = α c || ei (t)|| + β c t

0e− λ c ( t − τ ) || ei (τ )|| dτ. (12)

In (12), α c and β c are non–negative design parameterscontrolling the relative weights given to transient and steadystate measures respectively, whereas λ c is the non–negativeforgetting factor.

B. Identication of roll dynamics parameters

Given the vehicle lateral dynamics parameters, l∗v , C ∗v , andC ∗h , we now proceed to determine the suspension parameters

k, c and the CG height h by utilizing the roll dynamicsequation in (1) which is given below

J xx φ + cφ + kφ = J xx hδ, (13)

where the auxiliary input signal δ ∈ ℜis given by

δ =1

J xx− (C v + C h )β +

J zz θ1

vψ + mgφ + C v δ (14)

We further dene the ltered signal ωv ∈ ℜ3 as

ωv = λ2ωv + [ φ,φ, δ]T , (15)

where λ2 < 0. Hence, the roll–rate equation (13) can beparameterized as

φ = Ξ∗T ωv , (16)

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where Ξ∗∈ ℜ3 represents the parameter vector. The vehicleparameters suspension parameters and the CG height arerelated to Ξ∗ as follows:

c∗= ( − λ2 − Ξ∗1 )J xx , k∗= − Ξ∗2J xx , h∗= Ξ ∗

3 . (17)

Analogous to the lateral vehicle parameter estimation, eachunknown parameter belongs to a closed interval such thath ∈ H, k ∈ K, and c ∈C . These intervals are divided intosufcient number of grid points and are represented as H ={h1 , h2 , h3 , . . . , h p}, K = {k1 , k2 , k3 , . . . , k q }, and C ={c1 , c2 , c3 , . . . , c r } with dimensions p, q and r respectively.Hence we employ these p × q× r xed models together withone free running adaptive and one re–initialized adaptivemodel in the multiple model extension. Once again, theidentication error, ei , corresponding to the i th model isdened as

ei = φ − ˆφ = (Ξ ∗− Ξi )T ωv , (18)

where Ξi denotes the parameter of the i–th model. Sub-sequently, one can compute the associated cost value (12)corresponding to each identication error (18). Finally, themodel that is obtained from (11) yields the roll–plane param-eters, including the CG height and suspension parameters.

C. Numerical Analysis

In this section, we combine the identication schemesdescribed above as a two step algorithm, whose rst stepestimates the lateral vehicle parameters C v , C h and lv ateach instant, and passes these values to the second step in

which we determine the suspension parameters c, k and thecenter of gravity height h .

Now we investigate whether the multiple model schemeusing the proposed two step algorithm has any advantagesover the same two step algorithm that employs conventionaltype adaptation. To this end, suppose we choose the vehicleparameter grid points for the xed candidate models forthe lateral dynamics as lv ∈ {1.01, 1.11, . . . , 1.61}, C v ∈{57600, 60100, 62600}, and C h ∈ {87600, 90100, 92600}.Similarly, we choose grid points for the xed candidateroll plane models as h ∈ {0.5, 0.52, . . . , 0.84}, k ∈

{35500, 36100, 36700}, and c ∈ {4760, 5010, 5260}. Wenote that the simulated reference vehicle parameters of h∗=0.51, k∗= 36000 , c∗= 5000 , l∗v = 1 .2, C ∗v = 60000 , C ∗h =90000 are not in the xed candidate model parameter set.As shown in Figs. 3–4, the free running adaptive model thatis initialized to the lower bounds of the intervals does worsethan the proposed adaptive multiple model identicationalgorithm. In particular, we observe that the free runningadaptive model can have a signicant transient estimationerror.

IV. S WITCHING ROLLOVER C ONTROLLER

We now combine the multiple model identication schemediscussed in the previous section with a paired set of controllers in order to improve the performance of activerollover mitigation systems. The controller design described

0 5 10 15 20−100

−50

0

50

100

δ [ d e g

]

time [sec]

0 5 10 15 201

1.1

1.2

1.3

1.4

1.5

l v [ m ]

time [sec]

0 5 10 15 20

5

5.5

6

6.5

7x 10

4

C v

[ N / r a d ]

time [sec]

0 5 10 15 206

7

8

9

10x 10

4

C h

[ N / r a d ]

time [sec]

RLSMMST

RLSMMST

RLSMMST

Fig. 3. CG longitudinal position and tire stiffness estimations.

0 5 10 15 200.3

0.4

0.5

time [sec]

h [ m ]

0 5 10 15 20

2.5

3

3.5

4x 10

4

k [ N m

/ r a d ]

time [sec]

0 5 10 15 203000

4000

5000

6000

time [sec]

c [ N m s / r a

d ]

RLSMMST

RLSMMST

RLSMMST

Fig. 4. CG height and suspension parameter estimations.

in the sequel is based on differential braking actuators only;however, the results can be extended to other actuator typessuch as the active steering and active/semi-active suspensionwith ease.

We emphasize that for the rollover prevention problem asingle robust control mechanism may be designed for theworst case scenario, i.e., for the highest possible CG height.While such a strategy makes sense, in that, safety comes rstin rollover prevention, it also takes away from performanceconsiderably as the controller will always be on. In orderto possibly reduce the degradation in system performancewhile still preventing rollover, we therefore propose the multimodel/controller implementation shown in Figure 5. Beforewe synthesize a paired set of controllers corresponding toeach CG height conguration, we discuss the design of asingle rollover controller.

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Fig. 5. Multiple model switched adaptive control structure.

A. Rollover controller based on a single model

Due its simplicity and its adequate performance in rolloverprevention [5], [6], we adopt a proportional feedback con-troller of the form

u = K 0ay , (19)

where ay = vx (β + ψ) is the measured lateral acceleration,and the feedback gain K 0 is chosen to maximize somesystem performance criterion. Suppose that the CG heighth0 is known. In this paper, we use the single track modelwith roll degree of freedom in (1) to choose the feedback

gain, K 0 , such that the peak value of LT R d is below somepre–specied level. In other words, we want to keep

|LT R d | ≤ 1, (20)

for the largest possible steering inputs, which is equivalentto keeping all four wheels in contact with the road andthus preventing rollover. This design is done such that (20)is satised for a given maximum speed vmax and a givenmaximum steering input δmax . This in turn will guaranteethat |LT R d | ≤ 1 for all |δ| < |δmax | and v < vmax

corresponding to the CG height h0 . In this respect, u = K 0ay

is a robust controller for all CG height h ≤ h0 as well.Comment: Note that a disadvantage of the controller

designed as above is that it is always active. In other words,it will always attempt to limit the LT R d , even in non-criticalsituations, thus potentially interfering with, and annoyingthe vehicle driver. It therefore makes sense to activate thecontroller in situations only when the potential for rolloveris signicant [2]. One can limit this by putting a thresholdoutput for the activation of the controllers. Since the systemoutput considered here is the lateral acceleration, we adoptthe following rule for activating the switched controllers

u = K 0ay , if |LT R d | ≥ LT R thr ,0, if |LT R d | ≤ LT R thr , (21)

where LT R thr is a positive threshold that depends on thevehicle type and parameters (for the simulations of this paper,LT R thr = 0 .6 has been used).

B. Switched Rollover Prevention

In the proposed multiple model/controller architectureshown in Figure 5, n identication models are paired upwith n locally robust controllers. For each combination of

h ∈ {h1 , . . . , h p}, k ∈ {k1 , . . . , k q }, and c∈ {c1 , . . . , cd }, apaired local controller C i ∈ {C 1 , C 2 , . . . , C n } is designedas discussed above; hence we have

C i : u i = K i ay , i ∈ {1, 2, . . . , n }, (22)

which yields higher performance for the current values of h ,k, and c. In the execution of the proposed rollover scheme,the best model is identied based on the 2nd order roll planemodel (13) and the corresponding controller C i is used inrollover prevention.

V. N UMERICAL A NALYSIS

In this section, we investigate the performance of theCG estimation algorithm, and its application in rolloverprevention. The simulation results to be presented assume theset of synthetic parameters given in Table I. In the numericalsimulations we assume that the lateral vehicle parameterslv , C v , C h are xed and known, but roll dynamics parametersk,c,h are unknown. The rationale for this is twofold; rstlyit reduces the controller implementation complexity, thushelping with exposing the benets of the control approachdiscussed in this paper. Secondly, the major parametersaffecting roll dynamics behavior are k, c and h , whichnecessitates continuous monitoring of these parameters inrollover situations, whereas the estimation of the lateraldynamics parameters can be achieved during normal drivingconditions, long before a rollover situation is likely to occurat freeway speeds. With these in mind, we use the same xedcandidate model set as in Section III-C. We emphasize thatthe simulated vehicle roll dynamics parameters of h∗= 0 .51,k∗= 36000 , c∗= 5000 are not in the xed candidate modelparameter set. Motivated by the ease of exposition, we furtherassume that controller switching is based on the estimatedCG height only.

For the design of local controllers, we assume a peak vehicle speed of vmax = 30[m/s ] (i.e. 108[km/h ]), whichrepresents typical freeway driving condition for a compact

passenger vehicle. The peak steering wheel input of δmax =100◦ (with steering ratio of 1/ 18) is used to design theswitched controllers such that, when the vehicle is operatingat δmax and vmax , the condition (20) satised for eachCG height conguration, which is sufcient for mitigatingrollover. We choose the controller gains K η as small aspossible to minimize the control effort. The resulting 8controller gains are calculated as follows:

K h> 0. 8 = − 1550 , K 0.75<h ≤ 0.8 = − 1350K 0.7<h ≤ 0.75 = − 1170 , K 0.65<h ≤ 0.7 = − 1000K 0.6<h ≤ 0.65 = − 850 , K 0.55<h ≤ 0.6 = − 700K 0.5<h ≤ 0.55 = − 580 , K h ≤ 0.5 = − 480

(23)

For the numerical simulations, we use a typical obstacleavoidance maneuver known as the Elk test with a peak

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0 2 4 6 8 10 12 14 16 18 20

−100

−50

0

50

100

D r i v e r

S t e e r

I n p u

t [ d e g

]

time [sec]

0 2 4 6 8 10 12 14 16 18 200.5

0.6

0.7

0.8

0.9

C G

h e

i g h t e s t i m a

t i o n

[ m ]

time [sec]

Fig. 6. Steering input and the resulting CG height estimation.

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

L T R

time [sec]

LTRd−uncontrolledLTRd−Robust ControlLTRd−Adaptive Control

Fig. 7. LT R d for the controlled and uncontrolled vehicles.

driver steering input of magnitude δmax = 100 ◦ and withan initial speed of v = 108[km/h ]. The steering prolecorresponding to this maneuver and the resulting CG heightestimation is shown in Fig. 6, where the worst case CGheight (i.e., hmax = 0 .85[m]) is assumed until the initiationof the steering maneuver. After the maneuver starts, theCG height has been estimated to be 0.51[m] as seen fromthe gure. Fig. 7 depicts the resulting LT R d values forthe controlled and the uncontrolled vehicles. Clearly, the

uncontrolled vehicle rolls over as |LT R d | > 1 during themaneuver. Moreover, both of the robust (i.e., xed gain) andthe switched adaptive controllers prevent rollover by keeping|LT R d | < 1. However, the adaptive controller does it in aless conservative way which is favorable. In Fig. 8 we com-pare the vehicle states of the controlled and the uncontrolledvehicles, where we observe that due to smaller attenuationobtained by the adaptive (switched) controller, the resultingstates trajectories are closer to the uncontrolled vehicle statesas compared to the robust one. Again, this is favorable as theadaptive controller causes smaller driver intervention, andmaintains a natural response of the vehicle. Finally, Fig. 9depicts the vehicle speed and the normalized braking forcevariations for the controlled and the uncontrolled vehicles.We observe that the adaptive controller results in much

0 5 10 15 20−5

0

5

S i d e s l i p a n g

l e ,

β [ d e g

]

time [sec]

no control with robust control with adaptive control

0 5 10 15 20−40

−20

0

20

40

Y a w r a

t e ,

d ψ / d t [ d e g

]

time [sec]

0 5 10 15 20

−60

−40

−20

0

20

40

R o

l l r a

t e ,

d φ / d t [ d e g

/ s ]

time [sec]0 5 10 15 20

−20

−10

0

10

20

R o

l l a n g

l e ,

φ [ d e g

]

time [sec]

Fig. 8. Vehicle states for the controlled and uncontrolled vehicles.

0 5 10 15 200

10

20

30

S p e e

d [ m / s ]

time [sec]

uncontrolledwith robust controlwith adaptive control

0 5 10 15 20−1.5

−1

−0.5

0

0.5

C o n

t r o l F o r c e

/ w e

i g h t

time [sec]

robust controlleradaptive controller

Fig. 9. Vehicle speed and the normalized control force.

less controller actuation and less drop in vehicle speed; thisclearly shows the performance benet of using the suggestedswitched controller as compared to the xed robust controlalternative.

ACKNOWLEDGEMENT

This work was jointly supported by the following grants:SFI 04/IN3/I478, TUBITAK 107E276, and EI PC/2007/0128.

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[3] M. Akar, S. Solmaz, and R. Shorten. Method for determining thecenter of gravity for an automotive vehicle, (WO 2007/098891 A1).

[4] K.S. Narendra and J. Balakrishnan. Adaptive control using multiplemodels. IEEE Trans. on Automatic Control , 42(2):171–187, 1997.

[5] B. Chen and H. Peng. Differential-braking-based rollover preventionfor sport utility vehicles with human-in-the-loop evaluations. VehicleSystem Dynamics , 36:359–389, 2001.

[6] Wielenga T.J., 1999, A Method for Reducing On-Road Rollovers: Anti

-Rollover Braking.SAE Paper

No. 1999-01-0123.[7] Narendra K.S., Annaswamy A. M., 1989, Stable Adaptive Systems ,(Prentice Hall International, Englewood Cliffs - NJ, U.S.A.).