steering control for rollover avoidance of heavy...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 8, OCTOBER 2012 3499

Steering Control for Rollover Avoidanceof Heavy Vehicles

Hocine Imine, Leonid M. Fridman, and Tarek Madani

Abstract—The aim of this paper is to develop an active steeringassistance system to avoid the rollover of heavy vehicles (HV). Theproposed approach is applied on a single body model of HV pre-sented in this paper. An estimator based on the high-order slidingmode observer is developed to estimate the vehicle dynamics, suchas lateral acceleration limit and center height of gravity. Lateralposition and lateral speed are controlled using a twisting algorithmto ensure the stability of the vehicle and avoid accidents. At thesame time, the identification of unsprung masses and suspensionstiffness parameters of the model have been computed to increasethe robustness of the method. Some simulation and experimentalresults are given to show the quality of the proposed concept.

Index Terms—Estimation, identification, rollover, sliding modecontrol, sliding mode observers, steering control, twisting algo-rithm, vehicle modeling.

I. INTRODUCTION

THE ROLL stability of the heavy vehicles (HV) is affectedby the center height of gravity, the track width, and

the kinematic properties of suspensions. A more destabilizingmoment arises during the cornering maneuver when the centerof gravity of the vehicle shifts laterally. The roll stability ofthe vehicle can be guaranteed if the sum of the destabilizingmoment is compensated during a lateral maneuver.

Recently, several solutions have been proposed to reduce thenumber of HV rollover accidents. Several systems have beendeveloped to assist the driver to avoid the rollover. Some ofthem are installed in the infrastructure before a dangerous cur-vature. In the case of speeding leading to rollover in cornering,a warning is sent to the driver to decrease the speed [1], [2].

Other systems have been installed onboard the HV. Theyuse informative measures coming from sensors, such as vehicle

Manuscript received August 26, 2011; revised November 14, 2011 andMarch 15, 2012; accepted June 9, 2012. Date of publication June 29, 2012;date of current version October 12, 2012. This work was supported in part bythe French Ministry of Industry and in part by the Lyon Urban Trucks & Buscompetitiveness cluster. The work of L. Fridman was supported in part by theConsejo Nacional de Ciencia y Tecnología (CONACyT) under Grant 132125and in part by Programa de Apoyo a Proyectos de Investigación e InnovaciónTecnología (PAPIIT) Universidad Nacional Autónoma de México (UNAM)117211. The review of this paper was coordinated by Dr. K. Deng.

H. Imine is with the Laboratory for Road Operation, Perception, Simulatorsand Simulations, Laboratory for Road Operation, Perception, Simulations andSimulators (LEPSIS), Paris 75732, France (e-mail: [email protected]).

L. M. Fridman is with the Departamento de Ingeniería de Control yRobótica, División de Ingeniería Eléctrica, Facultad de Ingeniería, UniversidadNacional Autónoma de México (UNAM), México 04510, México (e-mail:[email protected]).

T. Madani is with the Laboratoire d’Ingénierie des Systèmes de Versailles,Vélizy 78140, France (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.2012.2206837

speed and lateral acceleration, to send a warning signal to thedriver when he goes beyond some risk thresholds [3], [4].

In the case of active systems, the concept is to minimize thelateral acceleration by braking action, steering action, suspen-sion action, antiroll action, or a combination of all [5]–[9].

However, most of the existing methods require full infor-mation on the state that may limit their practical utility. Infact, even if all the state measurements are possible, they aretypically corrupted by noises. Moreover, the increased num-ber of sensors makes the overall system more complex inimplementation and expensive in realization. Thus, design ofan observer becomes an attractive approach to estimate theunmeasured states of the HV. A second-order sliding modeobserver, providing theoretically exact state observation, pa-rameter identification, and impact force estimation, is presentedin this paper.

In most of the recent research, the parameters of the vehicleare supposed to be known and constant. Some of them weregiven by vehicle constructors and manufacturers, and otherunknown parameters are taken from literature. In the presentwork, suspension stiffness and unsprung masses have beenidentified. This allows improving the quality of the proposedsystem.

The impact forces affect vehicle dynamic performance andbehavior properties. These forces are very important for evalu-ating the rollover risk of HV by computing the load transfer ra-tio (LTR), studying and evaluating the damage of the vehicle onthe road or bridges, and controling the weights of the vehicles[10]–[12]. These forces can also be used in control systems toassist the driver. However, the exiting sensors to measure theseforces are very expensive and difficult to install [13].

Compared to the previous works, the proposed method isbased on a sliding mode observer to estimate the vertical forcesand at the same time identify the dynamic parameters of thevehicle [14]–[18].

Design of such observers requires a dynamic model of HVthat is build up in a first step of this paper. This model iscoupled with an appropriate wheel road contact model. It hasbeen validated using a simulator [19] and by real measurementscarried out with an instrumented tractor [20].

In this paper, an observer–controller law using the slidingmode technique is developed.

The proposed method has been adopted to:

1) Achieve good tracking of desired trajectories by ensuringthe convergence of the lateral acceleration of the vehicletoward the estimated acceleration limit. This allows thelimitation of the load transfer between the right and leftsides of the vehicle to its limited value, which is set to 0.9.

0018-9545/$31.00 © 2012 IEEE

3500 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 8, OCTOBER 2012

Fig. 1. HV model.

2) Estimate the nonmeasurable states of the HV.3) Identify some unknown parameters of HVs.

This paper is organized as follows. Section II deals with ve-hicle description and modeling. The design of the observer andthe estimation of the vertical forces and parameter identificationare presented in Section III. Section IV is devoted to steeringcontrol method design. Some experimental results are presentedin Section V. Finally, some remarks and perspectives are givenin a concluding section.

II. VEHICLE MODELING

The vehicle studied in this paper and developed by reference[5] is a nonarticulated HV with two axles. Therefore, someassumptions were considered: the roll angle is assumed to besmall, and the suspension and tire dynamics are assumed to belinear. The corresponding model has five degrees of freedomand used to represent the roll and lateral dynamics, as shown inFig. 1.

The vehicle consists of two bodies. Body 1, which is com-posed of two axles, is represented by the unsprung mass mu

and center of gravity CG1. Body 2 is the sprung mass m andcenter of gravity CG2. The center CG1 is assumed to be in theroad plane above CG2.

In this case, the lateral forces acting on the system can belinearized and computed as follows:{

Fyr = μcrαr

Fyf = μcfαf(1)

where cf and cr are, respectively, the front and rear corneringstiffness after assuming the equality of the cornering stiffnessfor the front and rear wheels. The road adhesion is representedby the variable μ. In this paper, the dry road is assumed (μ = 1).The parameters αf and αr are, respectively, the front and reartire slip angles computed by using the following formula:⎧⎨⎩αr = −

(β − lr

ψv

)αf = δ −

(β + lf

ψv

) (2)

Fig. 2. Front view of HV model.

where β is the side slip angle, which is assumed to be equalfor the two front wheels and the rear ones, lf and lr, and are,respectively, the distance from CG1 to the front axle and rearaxle, v is the vehicle speed, ψ is the yaw rate, and δ is the frontwheel steering angle.

The model is derived using Lagrangian’s equations

M(q)q + C(q + q)q +Kq = Fg(q + u) (3)

where q = [q1, q2, yl, φ, ψ]T represents the coordinate’s vec-

tor composed of, respectively, left and right front suspensiondeflections, lateral displacement, roll angle, and yaw angle;M ∈ �5×5 is the inertia matrix; C ∈ �5×5 is the matrix relatedto the damping effects; K ∈ �5 is the spring stiffness vector;Fg ∈ �3 is a vector of generalized forces; and u is the systeminput.

The suspension is modeled as a combination of spring anddamper elements, as shown in Fig. 2.

The vertical acceleration of the chassis is obtained as follows:

z =

(k1q1 + k2q2 + (k1 − k2)

T

2sin(φ)

)/m

+

(B1q1 +B2q2 − (B1 −B2)

T

2cos(φ)φ

)/m (4)

where φ is the tractor roll angle, T is the tractor track width,q1 and q2 are, respectively, the left and right front suspensiondeflections of the tractor, and z is the vertical displacement ofthe tractor sprung mass (center of gravity height). The tractorchassis with mass m is suspended on its axles through twosuspension systems.

The suspensions stiffness and damping coefficients are repre-sented, respectively, by ki, i = 1, 2, and Bi, i = 1, 2, in Fig. 2.

The tire is modelized by springs represented by ki and i =3, 4, in Fig. 2. The left and right wheels masses are, respectively,m1 and m2. At the tire contact, the road profile is representedby the variables u1 and u2, which are considered as HVinputs. The variables zr1 and zr2 correspond, respectively, tothe vertical displacement of the left and right wheels of the

IMINE et al.: STEERING CONTROL FOR ROLLOVER AVOIDANCE OF HEAVY VEHICLES 3501

tractor front axle, which can be computed by means of usingthe following equations:{

zr1 = z − q1 − T2 sin(φ)

zr2 = z − q2 +T2 sin(φ)

. (5)

The lateral and yaw accelerations are computed as follows:{yl = − (cf+cr)

mv yl +(−cf lf+crlr)−mv2

mv2 ψ +cfm δ

ψ = − (cf l2f+crl2r)

Izvψ +

lf cfIz

δ +(−cf lf+crlr)

Izβ

(6)

where Iz is the inertia along the Z axis.The vertical accelerations of the wheels are given by⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

zr1 =(k1q1 − k1

T2 sin(φ) +B1q1

)/m1

−(B1

T2 cos(φ)φ+ Fzl

)/m1

zr2 ==(k2q2 + k2

T2 sin(φ) +B2q2

)/m2

+(B2

T2 cos(φ)φ− Fzr

)/m2.

(7)

The accelerations of suspensions are given by the followingequation system:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

q1=(

1m− 1

m1

)k1q1+

k2q2m +(k1−k2)

T2 sin(φ)+

(1m− 1

m1

)×B1q1 +

B2q2m −

((B1 −B2)

T2 cos(φ)φ

)/m− Fzl

m1

q2=(

1m− 1

m2

)k2q2+

k1q1m +(k1−k2)

T2 sin(φ)+

(1m− 1

m2

)×B2q2 +

B1q1m −

((B1 −B2)

T2 cos(φ)φ

)/m− Fzr

m2

φ = (k1q1 − k2q2 − (k1 + k2)T2 sin(φ) +B1q1

−B2q2 − (B1 +B2)T2 cos(φ)φ)T2 /Ix

(8)

where Ix is the inertia moment in the roll axis and Fzl and Fzr

are, respectively, the vertical forces of the left and right wheels,which are considered unknown perturbations to be estimated.

These forces can be computed by using the following:{Fzl = k3(zr1 − u1)Fzr = k4(zr2 − u2).

(9)

III. SLIDING MODE OBSERVER DESIGN

This section is devoted to observer design using the slidingmode technique. To develop it, let us rewrite (3) in state formas follows: { x1 = x2

x2 = f(x1 + x2) + Fg(x1 + u)y = x1

(10)

where x = (x1, x2)T = (q, q)T ∈ �10 is the state vector, y =

q ∈ �5 is the measured outputs vector of the system, f is avector of nonlinear analytical function, and Fg is an unknowninput vector computed as follows:

Fg =

⎡⎣ −Fzl/m1

−Fzr/m2

0

⎤⎦ .

Before developing the sliding mode observer, one supposesthe following:

1) The state is bounded.2) The inputs of the system are bounded.

A. States Observation

The second-order observer developed in [21]–[23] has beenadapted to the presented model to estimate in finite time statesand to identify parameters.

It has the following form:{˙x1 = x2 + z1˙x2 = f(x1 + x2) + z2

(11)

where x1 and x2 are, respectively, the estimations of vectors x1

and x2.The correction variables z1 and z2 represent the output

injections of the form{z1 = λδ(x1)Sign(x1)z2 = αSign(x1)

(12)

where x1 = x1 − x1 ∈ �5 is a vector of the state estimationerror. The gain matrices (λ and α) ∈ �5×5, the matrix Δ(x1),and the vector “Sign” are defined as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩

λ = diag{λ1 + λ2 + · · ·λ5}α = diag{α1 + α2 + · · ·α5}δ(x1) = diag

{|x11 |

12 + |x12 |

12 + · · · |x15 |

12

}Sign(x1) = [sign(x11) + sign(x12) + · · · sign(x15)]

T .(13)

The dynamic estimation errors are calculated as follows:⎧⎨⎩˙x1 = x2 − λδ(x1)Sign(x1)˙x2 = f(x1 + x2)− f(x1 + x2)+Fg(x1 + u)− αSign(x1)

(14)

where x2 = x2 − x2 ∈ �5 is the estimation error of thevector x2.

Considering the accelerations of the system as bounded, theelements of the diagonal matrix α can be selected, satisfyingthe following inequality:

αii > 2∣∣∣ ˙x2i

∣∣∣ , i = 1 . . . 5. (15)

On the other hand, from [24], the elements of the diagonalmatrix λ can be selected as

λii>

√√√√ 2

αii−2∣∣∣ ˙x2i

∣∣∣(αii+2

∣∣∣ ˙x2i

∣∣∣) (1+pi)

1−pi, i=1, . . . , 5

(16)

where pi ∈]0, 1[ are some constants to be chosen (see proofin [24]).

To study the observer stability, first, the convergence of x1

in finite time t0 is proved. Then, some conditions about x2 toensure its convergence to 0 are deduced. Therefore, for t ≥ t0,


the surface x2 = 0 is attractive, leading x2 to converge towardx2, satisfying the inequalities (15) and (16).

The convergence proof of the second-order observer can befound in [24].

B. Vertical Forces Estimation and Parameters Identification

To estimate the vertical forces and to identify the parametersof the system, let us rewrite system (10) by reducing its orderas follows: ⎧⎪⎨⎪⎩

x11 = x21

x21 = a1ϕ(x1 + x2)− Fzl/m1

x12 = x22

x22 = a2ϕ(x1 + x2)− Fzr/m2

(17)

with x11 = q1, x21 = q1, x12 = q2, and x22 = q2.The unknown vectors of parameters represented by a1 and

a2 and the vector ϕ are defined as follows:

a1 =[k1

(m1−mm1

)k2

]a2 =

[k1 k2

(m2−mm2

) ]ϕ = [ q1

mq2m ]T .

To simplify the system, the vector of unknown parameters ais computed as follows:

a =

⎧⎨⎩= [ a11 a21 ] =[k1

(m1−mm1

)k2

]= [ a12 a22 ] =

[k1 k2

(m2−mm2

) ].

To estimate the vertical forces and to identify the unknowninputs, the second-order sliding mode observer defined in (11)is rewritten as{

˙x1 = x2 + λδ(x1)sign(x1)˙x2 = aϕ(x1 + x2) + αsign(x1)

(18)

where (x1, x2) = (x11, x21) or (x1, x2) = (x12, x22).The variable a represents a vector of the nominal values of

the vector parameters a.In this case, the dynamic estimation errors are calculated as

follows:{˙x1 = x2 − λδ(x1)sign(x1)˙x2 = aϕ(x1, x2) + Fgi(x1, u)− αsign(x1)

(19)

where a = a− a is the estimation error of the vector parame-ters a, Fgi = −Fzl/m1, or Fgi = −Fzr/m2.

After convergence of the observer (18), the variable x2

convergences toward 0 in finite time t ≥ t0.In this case, and from (19), one obtains

z2 = αsign(x1) = aϕ(x1, x2) + Fgi(x1, u). (20)

Theoretically, the equivalent output injection is the resultof an infinite switching frequency of the discontinuous termαsign(x1). Nevertheless, the realization of the observer pro-duces a high switching frequency that makes the applicationof a filter necessary.

To eliminate the high frequency component, a filter of thefollowing form is used:

τ ˙z2(t) = z2(t) + z2(t) (21)

where τ ∈ R and h � τ � 1, h being a sampling step.The variable z2 is then rewritten as follows:

z2(t) = z2(t) + ξ(t) (22)

where z2(t) is the filtered version of z2(t) and ξ(t) ∈ R is thedifference caused by the filtration.

Nevertheless, as shown in [25] and [26]

limτ∞0

h/τ∞0

z2(τ + h) = z2(t). (23)

Thus, it is possible to assume that the equivalent outputinjection is equal to the output of the filter.

1) Vertical Forces Estimation: To estimate the vertical forceFgi(x1, u), the vector of parameter a is supposed to be known.In this case, a = 0. Therefore, using (20), the vertical force isobtained as follows:

Fgi = αsign(x1). (24)

One remembers that this vector is composed of the forces Fzl

or Fzr, which can be computed using the system equation (9).One can then mention the advantages of the proposed methodas follows:

1) The measuring of the road profiles u1 and u2 is notnecessary.

2) The estimation of the vertical displacements of the wheelsand its derivative is also not necessary to obtain.

2) Parameter Identification: To identify the parameters ofthe system, the impact force vector is supposed to be zero, i.e.,Fg(x1, u) = 0. That means that the road profile is supposed tobe very small. There are no irregularities on the road that canvertically affect the vehicle. In this case, the vertical displace-ments of the wheels are close to zero.

Using (20), it follows that

z2 = aϕ(x1 + x2) = αsign(x1). (25)

Considering the unknown parameter vector a as a constantvector, and to identify it, a linear regression algorithm, namelythe least square method, is applied [17], [18].

The time integration is given by

1t

t∫0

z2(σ)ϕ(σ)T dσ = a

1t

t∫0

ϕ(σ)ϕ(σ)T dσ. (26)

The vector a is then estimated by

ˆa =

⎡⎣ t∫0

z2(σ)ϕ(σ)T dσ

⎤⎦⎡⎣ t∫0

ϕ(σ)ϕ(σ)T dσ

⎤⎦−1

(27)

where ˆa is the estimation of a.Let us define Γ = [

∫ t

0 ϕ(σ)ϕ(σ)T dσ]−1.


Its derivative gives Γ = −Γϕ(σ)ϕ(σ)TΓ.The derivative of the vector â using (27) gives

˙a =

⎡⎣ t∫0

z2(σ)ϕ(σ)T dσ

⎤⎦ Γ + z2ϕTΓ. (28)

Replacing Γ by its value given before and using (28), itfollows that

˙a = − âϕϕT + z2ϕ

TΓ

= (−âϕ+ z2)ϕTΓ. (29)

This ensures the asymptotic convergence of â toward a and,therefore, this allows the identification of the real value of thevector a. To obtain the unsprung masses and after identificationof k1 = a12 and k2 = a21, the expression of the variable adefined earlier is used here.

The identified values are then as follows:

a11 = k1(m1 −m)/m1 and a22 = k2(m2 −m)/m2.

Finally, the unsprung masses are deduced as follows:

m1 = mk1/(k1 − a11) and m2 = mk2/(k2 − a22).

IV. STEERING CONTROL

Rollover risk evaluation is based on LTR, which correspondsto the difference in normal tire forces acting on each side of thevehicle. It can be computed as follows:

LTR =Fzr − FzL

Fzr + Fzl

=2m2

mT

((hR + h cosφ)

ay2g

+ h sinφ

)(30)

where ay2 = ay − hφ is the lateral acceleration of the sprungmass, ay = v(ψ + β) is the lateral acceleration of the unsprungmass, T is the track width, and Fzl and Fzr are the normalforces acting on, respectively, the left and right sides of thevehicle. When LTR is equal to 0, the HV has a stable rolldynamic. The risk becomes higher when this indicator goestoward ±1. Both extreme values characterize wheel lift-off. Thesame model developed before has been used in this section.However, to perform the controller, only the lateral part ofthe model is important. Therefore, the suspension deflectionvariables are not used here.

In this section, an active steering control is developed toassist the vehicle in case of rollover risk. In addition to thesteering angle commanded by the driver and noted δd, anauxiliary steering angle δa is set by an actuator. Therefore, thecontrol input u = δ = δa + δd. In this paper, the limit value ofLTR is set to 0.9. This chosen value is used arbitrarily, less thanthe limit 1, to give sufficient time to the controller/driver toreact, before one of the wheels lifts off the road. In this case,the controller has time to avoid the rollover before to obtain

Fig. 3. Controller diagram.

high values of lateral accelerations. In the case of a small rollangle, one can assume that

h sinφ <

((hR + h cosφ)

ay2g

).

From (30), the acceleration limit can be obtained and approx-imated by

ay2 lim ≈ 0.9Tgm2m2H

; H = h+HR. (31)

The aim of the developed steering control is to ensure theconvergence of the lateral acceleration ay2 of the vehicle to itsacceleration limit ay2lim. This will allow the limitation of loadtransfer between the right and left sides of the vehicle to itslimited value 0.9.

The steering control diagram is shown in Fig. 3.To be able to achieve the aim, let us consider the following

sliding mode surface:

S = ˙yl + ηyl (32)

where ˙yl = yl − yld and yl = yl − yld are, respectively, thelateral speed error and the lateral offset, η is a positive constant,and yld and yld are, respectively, the desired velocity and thedesired lateral displacement obtained by the first and doubleintegration of the acceleration limit ay2lim, computed earlierusing (31).

Deriving the variable S and using (6), it follows that

S = ÿl + η ˙yl =cfm

u+ (η + a55) ˙yl. (33)

The equivalent control ueq is obtained by resolving theequation S = 0 as follows:

ueq = −m

cf(η + a55) ˙yl − δd. (34)


Fig. 4. Supertwisting algorithm trajectory.

Fig. 5. Instrumented vehicle.

The proposed control is based on a supertwisting algorithm.That algorithm has been developed to control dynamic systemsto avoid chattering.

In this case, the trajectories are characterized by twistingaround the origin, as shown in Fig. 4.

The continuous control law u is composed of two terms. Thefirst is defined by means of its discontinuous time derivative,whereas the other is a continuous function of a sliding variable.

Therefore, the proposed control law is defined as follows:{δa = ueq −G1 |S|1/2 sign(S) + u1

u1 = −G2sign(S)(35)

where G1 and G2 are the positive control gains.The convergence proof and analysis of the used supertwisting

algorithm can be found in [14] and [15].

V. EXPERIMENTAL RESULTS

The tractor in Fig. 5 has been instrumented on behalfof the Véhicule Lourd Interactif du Future project [27].The vehicle was equipped with different sensors to mea-sure the dynamic states of the vehicle, such as gyrome-ters, accelerometers, linear variable differential transformer(LVDT), lasers, etc., as shown in Fig. 6. The installationand positions of these sensors in the tractor are illustratedin Fig. 7.

Fig. 6. Sensors in the vehicle.

Fig. 7. Sensor positions in the tractor.

As shown in Fig. 6, different sensors have been installed tovalidate and calibrate the whole system:

1) Four accelerometers installed on the chassis to measurethe vertical accelerations of wheels.

2) Four sensors for LVDT to measure the suspension deflec-tions (q1 and q2 have been used in the observer).

3) Three axial gyroscopes installed on the chassis to mea-sure the angular rates (roll, pitch, and yaw rates). The rollangle is deduced from integration of the measured rollrate or by computation using the following formula:

φ = arcsin

(q1 − q2

T

).

In this paper, this last method was preferred and used toavoid errors that can result from roll rate integration.

4) Two lasers to measure the vertical displacements of thechassis.

Many tests and scenarios have been performed withthe instrumented vehicle driving at various speeds. In thispaper, the results obtained from zigzag and ramp tests arepresented to show the robustness of the estimation andidentification, whereas the presented results for steeringcontrol to avoid rollover are obtained from simulation.

The nominal dynamic parameters and the static verticalforces are measured before the tests. The static values of frontleft and front right vertical forces are, respectively, 24200N and25250N. The static values of the rear left and rear right verticalforces are, respectively, 9450N and 12050N.


Fig. 8. Simulation results for a driver steering input zigzag.

The nominal values of the unsprung masses m1 and m2

are, respectively, 100 and 95 kg, and the nominal values ofsuspension stiffness k1 and k2 are, respectively, 194 680 and188 540 N/m.

A. Zigzag Test

A zigzag test is presented in this section. This test is veryimportant to show the reaction of the controller when the driverchanges direction in a short time, often, and brutally changesthe direction of the vehicle. The simulation results are shownin Fig. 8. One remarks that the rollover risk appears at 14 s andlasts approximately 1 s.

The value of the steering angle at this time is about−0.06 rad. This value decreases until −0.25 rad, when thecontroller is OFF (dashed line). Otherwise, and when the con-troller is ON (solid line), this value is decreased to −0.15 rad.At the critical time of 14 s, and to stabilize the value of theLTR at its limit of 0.9, the computed lateral acceleration limitis around −8 m/s2. When the controller is activated (solid line),the lateral acceleration is still equal to −8 m/s2. Otherwise,if the controller is still OFF, the lateral acceleration decreasesuntil −12 m/s2 (dashed line). During the risk time interval[14, 15] s, the controller is activated and the LTR is stabilizedto −0.9 (solid line). Without control, the rollover risk decreasesand the LTR tends toward −1.8 (dashed line). The same situa-tion occurs at the time interval [16, 17] s. In this case, the LTRis also stabilized to the limited value 0.9 when the controller isactive.

Otherwise, the LTR tends toward 1.5. The use of a slidingmode observer allowed a good and quick estimation of the rollangle, as presented in Fig. 8. Without control (dashed line), theabsolute value of this variable increases from 0.06 to 0.11 radduring the first time interval [14, 15] s and from 0.06 to 0.09 radin the time interval [16, 17] s. In Fig. 9, the identification results

are shown. The suspension stiffness k1 and k2 are identifiedwith success. In effect, compared to their nominal values of,respectively, 194 680 and 188 540 N/m, these parameters havebeen identified with errors, which can be neglected as shown inFig. 10. The percentage of error is less than 0.025%. However,one notes some variations at the time interval [13, 20] s. This isdue to the fact that, at this time, the driver abruptly changes thedirection of the vehicle.

In the right side of Fig. 10, the unsprung masses m1 andm2 have been identified with a percentage of error less than0.1% compared to their nominal values of, respectively, 100and 95 kg.

B. Ramp Test

The results of the ramp test are presented in this section.The steering angle increased until a maximum value of 0.5 rad,during 5 s before stabilization, as shown in Fig. 11.

During the time interval [0, 1.5] s, there is no rollover risk,as one can see in the left side of Fig. 11. Although, duringthis time, the estimated steering angle coming from the controlblock is the same as that coming from the model withoutcontrol. The LTR in this case is less than 1, as shown in theright side of Fig. 11.

After 1.5 s, the risk appears and the LTR reaches the limitvalue 1, which corresponds to the situation where one of thewheels of the same axle lifts off. To avoid this risk situation,the controller is then activated to avoid the rollover of thevehicle.

With the active controller, the steering angle is reduced andits value becomes less than the original one coming from themodel. In this case, the lateral acceleration limit is estimatedand shown in Fig. 11 (solid line). Without control (dashedline), the lateral acceleration increases until 4 m/s2. Otherwise,when the control is activated (solid line), this acceleration does


Fig. 9. Identification results for a driver steering input zigzag.

Fig. 10. Identification errors for a driver steering input zigzag.

not exceed 3 m/s2. Therefore, the value of LTR is reducedand it becomes less than 1. On the other hand, the slidingmode observer allows us to estimate in finite time and quicklydetermine the different variables of the system.

In Fig. 11, one notices also the well estimation of the sideslip angle when it is compared to the variable coming fromthe model. The identification results are shown in Fig. 12. Inthis test, the suspension stiffness k1 and k2 have been identifiedbetter than in the first one. Indeed, one notes that the signalsare smooth, and only small variations of these identified valuesoccur around their nominal values of, respectively, 194 680 and188 540 N/m.

In the right side of Fig. 12, the identified unsprungmasses m1 and m2 are shown. The variations of these parame-ters occur around their nominal values of, respectively, 100 and95 kg, with an error quite close to 0, as shown in Fig. 13.

VI. CONCLUSION

In this paper, a steering control system has been developedto avoid the rollover of HV. It is based on control of lateralacceleration of the vehicle. Previously, an estimator based onthe sliding mode was implemented, which made it possibleto estimate the nonmeasured dynamics of the vehicle. Lateral


Fig. 11. Simulation results for a driver steering input ramp.

Fig. 12. Identification results for a driver steering input ramp.

acceleration and roll angle estimation are presented in thispaper. Compared to existing method, the proposed approach isbased on robust controller estimator.

The identification of unsprung masses and suspension stiff-ness has been computed to increase the robustness of thecontroller.

A real zigzag scenario is tested and presented. The resultsshowed that this system is effective and made it possibleto control the vehicle and to avoid its rollover. The lateralacceleration limit is stabilized. Therefore, the steering angleof the vehicle is modified to force this latter to stay ona safety trajectory. The LTR is maintained around a maxi-

mum value of 0.9. The experimental tests done on an instru-mented truck showed the quality of this approach since theconvergence of the observer is quick and it is done in fi-nite time.

The identification of the parameters was a success withsmall variations around the nominal values at the time interval[13 20] s. These variations are due to the abrupt directionchange of the driver.

A second ramp test is presented in this paper. Compared tothe first one, the identification process has been carried out witherrors quite close to zero and the controller is activated quicklyto avoid the accident.


Fig. 13. Identification errors for a driver steering input ramp.

The originality of this approach is the use of the equivalentcontrol, which provides a linear regression algorithm to identifythe unknown parameters of the system. An example of identi-fication of the unsprung mass and the stiffness is given in thispaper. The result is of quality.

In future work, it will be interesting to test this approach inreal time with an instrumented tractor semitrailer and using thedynamo wheel to measure the impact forces, which will be thereference for a better validation of the estimation. In this case,the controller will be applied to stabilize the semitrailer, whichis, in this type of HV, the first unit that can lose control.

In this paper, the unsprung masses and the suspension stiff-ness have been identified. In future work, one will focus on theidentification of other important dynamic parameters, namely,damping coefficients, roll, and yaw inertia moments.

ACKNOWLEDGMENT

This work was developed by the French Institut Françaisdes Sciences et Technologies des Transports laboratory (Lab-oratoire Central des Ponts et Chausséees) in collaboration withFrench industrial partners, Renault Trucks, Michelin, and Soditin the framework of French project Véhicule Lourd Interactifdu Future (VIF).

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Hocine Imine received the Master’s degree andthe Ph.D. degree in robotics and automation fromVersailles University, Versailles, France, in 2000 and2003, respectively.

From 2003 to 2004, he was a Researcher with theRobotic Laboratory of Versailles (LRV in French)and an Assistant Professor with Versailles Univer-sity. In 2005, he joined Institut Français des Scienceset Technologies des Transports, where he is cur-rently a Researcher. He received the Accreditationto Supervise Research (Habilitation à Diriger des

Recherches—HDR) in March 2012. He is involved in different projects likevéhicule interactif du futur (VIF) and heavyroute (intelligent route guidance forheavy traffic). He is also responsible for the research project PLInfra on heavyvehicle safety and the assessment of their impacts on pavement and bridges.He has published two books, over 60 technical papers, and several industrialtechnical reports. His research interests include intelligent transportation sys-tems, heavy vehicle modeling and stability, diagnosis, nonlinear observation,and nonlinear control.

Dr. Imine was a member of the organization committee of the Interna-tional Conference on Heavy Vehicles (HVParis’2008, merging HVTT10 andICWIM5 in May 2008). He was a Guest Editor of the International Journalof Vehicle Design, Special Issue on “Variable Structure Systems in AutomotiveApplications.” He is a member of the IFAC Technical Committee on Trans-portation systems (TC 7.4).

Leonid M. Fridman received the M.S. degree inmathematics from Kuibyshev (Samara) State Uni-versity, Samara, Russia, in 1976, the Ph.D. degreein applied mathematics from the Institute of ControlScience, Moscow, Russia, in 1988, and the Dr.Sci.degree in control science from the Moscow StateUniversity of Mathematics and Electronics, Moscow,in 1998.

From 1976 to 1999, he was with the Depart-ment of Mathematics, Samara State Architecture andCivil Engineering University, Samara. From 2000 to

2002, he was with the Department of Postgraduate Study and Investigations,Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joinedthe Department of Control Engineering and Robotics, Division of Electri-cal Engineering of Engineering Faculty, National Autonomous University ofMexico (UNAM), México, México. He was an Invited Professor in 15 univer-sities and research centers in France, Germany, Italy, Israel, and Spain. He is theauthor and editor of five books and ten special issues on sliding mode control.He is an author of more than 300 technical papers. His main research interest isvariable structure systems.

Dr. Fridman is Associate Editor of the International Journal of SystemScience and the Journal of the Franklin Institute, Nonlinear Analysis: HybridSystems. He was on the Conference Editorial Board of the IEEE ControlSystems Society, a member of the Technical Committee (TC) in the VariableStructure Systems and Sliding Mode Control of IEEE Control Systems Societyand TC in the Discrete Events and Hybrid Systems of IFAC. He is a winnerof the Scopus prize for the best cited Mexican Scientists in Mathematics andEngineering 2010 award.

Tarek Madani received the Engineering andMagister degrees in automatic control for electricalengineering from the Ecole Nationale Polytechnique,Algiers, Algeria, in 1997 and 2000, respectively, andthe Ph.D. degree in automatic control and roboticsfrom Versailles University, Versailles, France,in 2005.

He is currently with the Laboratoire d’Ingénieriedes Systèmes de Versailles. His main research inter-ests include nonlinear control.

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