Steering Control for Rollover Avoidance of Heavy ... - Semantic Scholar
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 8, OCTOBER 2012
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Steering Control for Rollover Avoidance of Heavy Vehicles Hocine Imine, Leonid M. Fridman, and Tarek Madani
Abstract—The aim of this paper is to develop an active steering assistance system to avoid the rollover of heavy vehicles (HV). The proposed approach is applied on a single body model of HV presented in this paper. An estimator based on the high-order sliding mode observer is developed to estimate the vehicle dynamics, such as lateral acceleration limit and center height of gravity. Lateral position and lateral speed are controlled using a twisting algorithm to ensure the stability of the vehicle and avoid accidents. At the same time, the identification of unsprung masses and suspension stiffness parameters of the model have been computed to increase the robustness of the method. Some simulation and experimental results are given to show the quality of the proposed concept. Index Terms—Estimation, identification, rollover, sliding mode control, sliding mode observers, steering control, twisting algorithm, vehicle modeling.
I. I NTRODUCTION
T
HE ROLL stability of the heavy vehicles (HV) is affected by the center height of gravity, the track width, and the kinematic properties of suspensions. A more destabilizing moment arises during the cornering maneuver when the center of gravity of the vehicle shifts laterally. The roll stability of the vehicle can be guaranteed if the sum of the destabilizing moment is compensated during a lateral maneuver. Recently, several solutions have been proposed to reduce the number of HV rollover accidents. Several systems have been developed to assist the driver to avoid the rollover. Some of them are installed in the infrastructure before a dangerous curvature. 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. They use informative measures coming from sensors, such as vehicle
Manuscript received August 26, 2011; revised November 14, 2011 and March 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 by the French Ministry of Industry and in part by the Lyon Urban Trucks & Bus competitiveness cluster. The work of L. Fridman was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACyT) under Grant 132125 and in part by Programa de Apoyo a Proyectos de Investigación e Innovación Tecnología (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, Simulators and Simulations, Laboratory for Road Operation, Perception, Simulations and Simulators (LEPSIS), Paris 75732, France (e-mail: [email protected]). L. M. Fridman is with the Departamento de Ingeniería de Control y Robótica, División de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad Nacional 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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2012.2206837
speed and lateral acceleration, to send a warning signal to the driver when he goes beyond some risk thresholds [3], [4]. In the case of active systems, the concept is to minimize the lateral acceleration by braking action, steering action, suspension action, antiroll action, or a combination of all [5]–[9]. However, most of the existing methods require full information on the state that may limit their practical utility. In fact, even if all the state measurements are possible, they are typically corrupted by noises. Moreover, the increased number of sensors makes the overall system more complex in implementation and expensive in realization. Thus, design of an observer becomes an attractive approach to estimate the unmeasured states of the HV. A second-order sliding mode observer, providing theoretically exact state observation, parameter identification, and impact force estimation, is presented in this paper. In most of the recent research, the parameters of the vehicle are supposed to be known and constant. Some of them were given by vehicle constructors and manufacturers, and other unknown parameters are taken from literature. In the present work, suspension stiffness and unsprung masses have been identified. This allows improving the quality of the proposed system. The impact forces affect vehicle dynamic performance and behavior properties. These forces are very important for evaluating the rollover risk of HV by computing the load transfer ratio (LTR), studying and evaluating the damage of the vehicle on the road or bridges, and controling the weights of the vehicles [10]–[12]. These forces can also be used in control systems to assist the driver. However, the exiting sensors to measure these forces are very expensive and difficult to install [13]. Compared to the previous works, the proposed method is based on a sliding mode observer to estimate the vertical forces and at the same time identify the dynamic parameters of the vehicle [14]–[18]. Design of such observers requires a dynamic model of HV that is build up in a first step of this paper. This model is coupled with an appropriate wheel road contact model. It has been validated using a simulator [19] and by real measurements carried out with an instrumented tractor [20]. In this paper, an observer–controller law using the sliding mode technique is developed. The proposed method has been adopted to: 1) Achieve good tracking of desired trajectories by ensuring the convergence of the lateral acceleration of the vehicle toward the estimated acceleration limit. This allows the limitation of the load transfer between the right and left sides of the vehicle to its limited value, which is set to 0.9.
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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 vehicle description and modeling. The design of the observer and the estimation of the vertical forces and parameter identification are presented in Section III. Section IV is devoted to steering control method design. Some experimental results are presented in Section V. Finally, some remarks and perspectives are given in a concluding section. II. V EHICLE M ODELING The vehicle studied in this paper and developed by reference [5] is a nonarticulated HV with two axles. Therefore, some assumptions were considered: the roll angle is assumed to be small, and the suspension and tire dynamics are assumed to be linear. The corresponding model has five degrees of freedom and used to represent the roll and lateral dynamics, as shown in Fig. 1. The vehicle consists of two bodies. Body 1, which is composed of two axles, is represented by the unsprung mass mu and center of gravity CG1. Body 2 is the sprung mass m and center of gravity CG2. The center CG1 is assumed to be in the road plane above CG2. In this case, the lateral forces acting on the system can be linearized and computed as follows: Fyr = μcr αr (1) Fyf = μcf αf where cf and cr are, respectively, the front and rear cornering stiffness after assuming the equality of the cornering stiffness for the front and rear wheels. The road adhesion is represented by the variable μ. In this paper, the dry road is assumed (μ = 1). The parameters αf and αr are, respectively, the front and rear tire slip angles computed by using the following formula: ⎧ ⎨ αr = − β − lr ψ˙ v (2) ⎩ αf = δ − β + lf ψ˙ v
Fig. 2.
Front view of HV model.
where β is the side slip angle, which is assumed to be equal for the two front wheels and the rear ones, lf and lr , and are, respectively, the distance from CG1 to the front axle and rear axle, v is the vehicle speed, ψ˙ is the yaw rate, and δ is the front wheel 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 vector composed of, respectively, left and right front suspension deflections, lateral displacement, roll angle, and yaw angle; M ∈ 5×5 is the inertia matrix; C ∈ 5×5 is the matrix related to the damping effects; K ∈ 5 is the spring stiffness vector; Fg ∈ 3 is a vector of generalized forces; and u is the system input. The suspension is modeled as a combination of spring and damper elements, as shown in Fig. 2. The vertical acceleration of the chassis is obtained as follows:
T m z¨ = k1 q1 + k2 q2 + (k1 − k2 ) sin(φ) 2 T ˙ + B1 q˙1 + B2 q˙2 − (B1 − B2 ) cos(φ)φ m (4) 2 where φ is the tractor roll angle, T is the tractor track width, q1 and q2 are, respectively, the left and right front suspension deflections of the tractor, and z is the vertical displacement of the tractor sprung mass (center of gravity height). The tractor chassis with mass m is suspended on its axles through two suspension systems. The suspensions stiffness and damping coefficients are represented, 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 represented by the variables u1 and u2 , which are considered as HV inputs. The variables zr1 and zr2 correspond, respectively, to the vertical displacement of the left and right wheels of the
IMINE et al.: STEERING CONTROL FOR ROLLOVER AVOIDANCE OF HEAVY VEHICLES
tractor front axle, which can be computed by means of using the following equations: zr1 = z − q1 − T2 sin(φ) . (5) zr2 = z − q2 + T2 sin(φ) The lateral and yaw accelerations are computed as follows: (c +c ) (−cf lf +cr lr )−mv 2 ˙ c y¨l = − fmv r y˙ l + ψ + mf δ mv 2 (6) (cf lf2 +cr lr2 ) ˙ lf cf (−cf lf +cr lr ) β ψ + Iz δ + ψ¨ = − Iz v Iz where Iz is the inertia along the Z axis. The vertical accelerations of the wheels are given by ⎧ z¨r1 = k1 q1 − k1 T2 sin(φ) /m1 + B q ˙ 1 1 ⎪ ⎪ ⎪ ⎪ T ⎨ − B1 cos(φ)φ˙ + Fzl /m1 2 ⎪ /m2 z ¨ == k2 q2 + k2 T2 sin(φ) + B q ˙ r2 2 2 ⎪ ⎪ ⎪ ⎩ + B T cos(φ)φ˙ − F 22 zr /m2 .
(7)
The accelerations of suspensions are given by the following equation system: ⎧ k2 q2 1 1 T 1 1 ⎪ q ¨ k = − q + +(k −k ) sin(φ)+ − ⎪ 1 1 1 1 2 m m1 2 ⎪ ⎪ m m m1 ⎪ ⎪ B2 q˙2 T ⎪ ˙ ⎪ ×B1 q˙1 + m − (B1 − B2 ) 2 cos(φ)φ /m − Fmzl1 ⎪ ⎪ ⎪ ⎨ 1 1 q¨2 = m − m12 k2 q2 + k1mq1 +(k1 −k2 ) T2 sin(φ)+ m − m12 ⎪ ⎪ B1 q˙1 T ˙ ⎪ ×B q ˙ + − (B − B ) cos(φ) φ /m − Fmzr2 ⎪ 2 2 1 2 2 m ⎪ ⎪ ⎪¨ ⎪ T ⎪ + B1 q˙1 ⎪ ⎪ φ = (k1 q1 − k2 q2 − (k1 +Tk2 ) 2 sin(φ) ⎩ T ˙ −B2 q˙2 − (B1 + B2 ) 2 cos(φ)φ) 2 /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. S LIDING M ODE O BSERVER D ESIGN This section is devoted to observer design using the sliding mode technique. To develop it, let us rewrite (3) in state form as follows: x˙ = x 1
2
x˙ 2 = f (x1 + x2 ) + Fg (x1 + u) y = x1
(10)
˙ T ∈ 10 is the state vector, y = where x = (x1 , x2 )T = (q, q) 5 q ∈ is the measured outputs vector of the system, f is a vector of nonlinear analytical function, and Fg is an unknown input vector computed as follows: ⎤ ⎡ −Fzl /m1 Fg = ⎣ −Fzr /m2 ⎦ . 0
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Before developing the sliding mode observer, one supposes the 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 been adapted to the presented model to estimate in finite time states and to identify parameters. It has the following form: ˆ 2 + z1 x ˆ˙ 1 = x (11) x1 + x ˆ 2 ) + z2 x ˆ˙ 2 = f (ˆ ˆ2 are, respectively, the estimations of vectors x1 where x ˆ1 and x and x2 . The correction variables z1 and z2 represent the output injections of the form x1 )Sign(˜ x1 ) z1 = λδ(˜ (12) x1 ) z2 = αSign(˜ ˆ1 ∈ 5 is a vector of the state estimation where x 1 = x1 − x x1 ), error. The gain matrices (λ and α) ∈ 5×5 , the matrix Δ( and the vector “Sign” are defined as follows: ⎧ λ = diag{λ1 + λ2 + · · · λ5 } ⎪ ⎪ ⎪ ⎨ α = diag{α1 + α2 + · · · α5 } 1 1 1 2 + |˜ 2 + · · · |˜ 2 δ(˜ x ) = diag |˜ x | x | x | 1 1 1 1 ⎪ 1 2 5 ⎪ ⎪ ⎩ Sign(˜ x1 ) = [sign(˜ x11 ) + sign(˜ x12 ) + · · · sign(˜ x15 )]T . (13) The dynamic estimation errors are calculated as follows: ⎧ ⎨x ˜2 − λδ(˜ x1 )Sign(˜ x1 ) ˜˙ 1 = x (14) x1 + x ˆ2 )+ x ˜˙ 2 = f (x1 + x2 ) − f (ˆ ⎩ x1 ) Fg (x1 + u) − αSign(˜ where x 2 = x2 − x ˆ2 ∈ 5 is the estimation error of the vector x2 . Considering the accelerations of the system as bounded, the elements of the diagonal matrix α can be selected, satisfying the following inequality: ˙ (15) ˆ2i , i = 1 . . . 5. αii > 2 x On the other hand, from [24], the elements of the diagonal matrix λ can be selected as ˙ ˆ2i (1+pi ) α +2 x ii 2 λii > , i = 1, . . . , 5 ˙ 1−pi ˆ2 αii −2 x i
(16) where pi ∈]0, 1[ are some constants to be chosen (see proof in [24]). To study the observer stability, first, the convergence of x 1 in finite time t0 is proved. Then, some conditions about x 2 to ensure its convergence to 0 are deduced. Therefore, for t ≥ t0 ,
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the surface x 2 = 0 is attractive, leading x ˆ2 to converge toward x2 , satisfying the inequalities (15) and (16). The convergence proof of the second-order observer can be found in [24]. B. Vertical Forces Estimation and Parameters Identification
To eliminate the high frequency component, a filter of the following form is used: τ z¯˙ 2 (t) = z¯2 (t) + z2 (t)
(21)
where τ ∈ R and h τ 1, h being a sampling step. The variable z2 is then rewritten as follows:
To estimate the vertical forces and to identify the parameters of the system, let us rewrite system (10) by reducing its order as follows: ⎧ x˙ 11 = x21 ⎪ ⎨ x˙ 21 = a1 ϕ(x1 + x2 ) − Fzl /m1 (17) ⎪ ⎩ x˙ 12 = x22 x˙ 22 = a2 ϕ(x1 + x2 ) − Fzr /m2
where z 2 (t) is the filtered version of z2 (t) and ξ(t) ∈ R is the difference caused by the filtration. Nevertheless, as shown in [25] and [26]
with x11 = q1 , x21 = q˙1 , x12 = q2 , and x22 = q˙2 . The unknown vectors of parameters represented by a1 and a2 and the vector ϕ are defined as follows: 1 −m k a1 = k1 mm 2 1 2 −m a2 = k1 k2 mm 2
Thus, it is possible to assume that the equivalent output injection is equal to the output of the filter. 1) Vertical Forces Estimation: To estimate the vertical force Fgi (x1 , u), the vector of parameter a is supposed to be known. In this case, a = 0. Therefore, using (20), the vertical force is obtained as follows:
ϕ = [ qm1
q2 T m ]
z2 (t) = z¯2 (t) + ξ(t)
lim z¯2 (τ + h) = z2 (t).
τ ∞0 h/τ ∞0
x1 ). Fgi = αsign(˜
.
To simplify the system, the vector of unknown parameters a is computed as follows: ⎧ ⎨ = [ a11 a21 ] = k1 m1 −m k2 m1 a= ⎩ = [ a12 a22 ] = k1 k2 m2 −m . m2 To estimate the vertical forces and to identify the unknown inputs, the second-order sliding mode observer defined in (11) is rewritten as x ˆ˙ 1 = x ˆ2 + λδ(˜ x1 )sign(˜ x1 ) (18) ¯ϕ(ˆ x1 + x ˆ2 ) + αsign(˜ x1 ) x ˆ˙ 2 = a 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: ˜2 − λδ(˜ x1 )sign(˜ x1 ) x ˜˙ 1 = x (19) ˜ϕ(x1 , x ˜2 ) + Fgi (x1 , u) − αsign(˜ x1 ) x ˜˙ 2 = a where a = a − a is the estimation error of the vector parameters a, Fgi = −Fzl /m1 , or Fgi = −Fzr /m2 . After convergence of the observer (18), the variable x 2 convergences toward 0 in finite time t ≥ t0 . In this case, and from (19), one obtains x1 ) = a ˜ϕ(x1 , x ˜2 ) + Fgi (x1 , u). z2 = αsign(˜
(20)
Theoretically, the equivalent output injection is the result of an infinite switching frequency of the discontinuous term αsign( x1 ). Nevertheless, the realization of the observer produces a high switching frequency that makes the application of a filter necessary.
(22)
(23)
(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 method as follows: 1) The measuring of the road profiles u1 and u2 is not necessary. 2) The estimation of the vertical displacements of the wheels and its derivative is also not necessary to obtain. 2) Parameter Identification: To identify the parameters of the system, the impact force vector is supposed to be zero, i.e., Fg (x1 , u) = 0. That means that the road profile is supposed to be very small. There are no irregularities on the road that can vertically affect the vehicle. In this case, the vertical displacements of the wheels are close to zero. Using (20), it follows that ˜ϕ(x1 + x2 ) = αsign(˜ x1 ). z2 = a
(25)
Considering the unknown parameter vector a as a constant vector, and to identify it, a linear regression algorithm, namely the least square method, is applied [17], [18]. The time integration is given by 1 t
t
1 z2 (σ)ϕ(σ) dσ = a ˜ t
t
T
ϕ(σ)ϕ(σ)T dσ.
(26)
The vector a is then estimated by ⎡ t ⎤⎡ t ⎤−1 ˆ˜ = ⎣ z2 (σ)ϕ(σ)T dσ ⎦ ⎣ ϕ(σ)ϕ(σ)T dσ ⎦ a
(27)
0
0
0
0
ˆ is the estimation of where a a. t Let us define Γ = [ 0 ϕ(σ)ϕ(σ)T dσ]−1 .
IMINE et al.: STEERING CONTROL FOR ROLLOVER AVOIDANCE OF HEAVY VEHICLES
Its derivative gives Γ˙ = −Γϕ(σ)ϕ(σ)T Γ. ˆ using (27) gives The derivative of the vector a ⎤ ⎡ t ˙ˆ ⎣ z2 (σ)ϕ(σ)T dσ ⎦ Γ˙ + z2 ϕT Γ. a ˜=
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(28)
0
Replacing Γ˙ by its value given before and using (28), it follows that ˆ˙ = − a ˆ a ˜ ˜ϕϕT + z2 ϕT Γ ˆ = (−a ˜ϕ + z2 )ϕT Γ.
(29)
ˆ toward This ensures the asymptotic convergence of a a and, therefore, this allows the identification of the real value of the vector a. To obtain the unsprung masses and after identification of k1 = a12 and k2 = a21 , the expression of the variable a defined earlier is used here. The identified values are then as follows: a11 = k1 (m1 − m)/m1 and a22 = k2 (m2 − m)/m2 .
high values of lateral accelerations. In the case of a small roll angle, one can assume that ay2 h sin φ < (hR + h cos φ) . g
Finally, the unsprung masses are deduced as follows: m1 = mk1 /(k1 − a11 ) and m2 = mk2 /(k2 − a22 ).
From (30), the acceleration limit can be obtained and approximated by
IV. S TEERING C ONTROL Rollover risk evaluation is based on LTR, which corresponds to the difference in normal tire forces acting on each side of the vehicle. It can be computed as follows: Fzr − FzL Fzr + Fzl 2m2 ay2 + h sin φ = (hR + h cos φ) mT g
Fig. 3. Controller diagram.
LT R =
(30)
where ay2 = ay − hφ¨ is the lateral acceleration of the sprung mass, ay = v(ψ + β) is the lateral acceleration of the unsprung mass, T is the track width, and Fzl and Fzr are the normal forces acting on, respectively, the left and right sides of the vehicle. When LT R is equal to 0, the HV has a stable roll dynamic. The risk becomes higher when this indicator goes toward ±1. Both extreme values characterize wheel lift-off. The same model developed before has been used in this section. However, to perform the controller, only the lateral part of the model is important. Therefore, the suspension deflection variables are not used here. In this section, an active steering control is developed to assist the vehicle in case of rollover risk. In addition to the steering angle commanded by the driver and noted δd , an auxiliary steering angle δa is set by an actuator. Therefore, the control input u = δ = δa + δd . In this paper, the limit value of LTR is set to 0.9. This chosen value is used arbitrarily, less than the limit 1, to give sufficient time to the controller/driver to react, before one of the wheels lifts off the road. In this case, the controller has time to avoid the rollover before to obtain
ay2 lim ≈
0.9T gm ; 2m2 H
H = h + HR .
(31)
The aim of the developed steering control is to ensure the convergence of the lateral acceleration ay2 of the vehicle to its acceleration limit ay2lim . This will allow the limitation of load transfer between the right and left sides of the vehicle to its limited 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 = y˜˙ l + η y˜l
(32)
where y˙ l = y˙ l − y˙ ld and yl = yl − yld are, respectively, the lateral speed error and the lateral offset, η is a positive constant, and y˙ ld and yld are, respectively, the desired velocity and the desired lateral displacement obtained by the first and double integration of the acceleration limit ay2lim , computed earlier using (31). Deriving the variable S and using (6), it follows that cf S˙ = y¨˜l + η y˜˙ l = u + (η + a55 )y˜˙ l . m
(33)
The equivalent control ueq is obtained by resolving the equation S˙ = 0 as follows: ueq = −
m (η + a55 )y˜˙ l − δd . cf
(34)
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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 systems to avoid chattering. In this case, the trajectories are characterized by twisting around the origin, as shown in Fig. 4. The continuous control law u is composed of two terms. The first 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 (35) u˙ 1 = −G2 sign(S) 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. E XPERIMENTAL R ESULTS The tractor in Fig. 5 has been instrumented on behalf of the Véhicule Lourd Interactif du Future project [27]. The vehicle was equipped with different sensors to measure the dynamic states of the vehicle, such as gyrometers, accelerometers, linear variable differential transformer (LVDT), lasers, etc., as shown in Fig. 6. The installation and positions of these sensors in the tractor are illustrated in 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 to validate and calibrate the whole system: 1) Four accelerometers installed on the chassis to measure the vertical accelerations of wheels. 2) Four sensors for LVDT to measure the suspension deflections (q1 and q2 have been used in the observer). 3) Three axial gyroscopes installed on the chassis to measure the angular rates (roll, pitch, and yaw rates). The roll angle is deduced from integration of the measured roll rate or by computation using the following formula: q1 − q2 φ = arcsin . T In this paper, this last method was preferred and used to avoid errors that can result from roll rate integration. 4) Two lasers to measure the vertical displacements of the chassis. Many tests and scenarios have been performed with the instrumented vehicle driving at various speeds. In this paper, the results obtained from zigzag and ramp tests are presented to show the robustness of the estimation and identification, whereas the presented results for steering control to avoid rollover are obtained from simulation. The nominal dynamic parameters and the static vertical forces are measured before the tests. The static values of front left and front right vertical forces are, respectively, 24200N and 25250N. The static values of the rear left and rear right vertical forces are, respectively, 9450N and 12050N.
IMINE et al.: STEERING CONTROL FOR ROLLOVER AVOIDANCE OF HEAVY VEHICLES
Fig. 8.
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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 of suspension stiffness k1 and k2 are, respectively, 194 680 and 188 540 N/m. A. Zigzag Test A zigzag test is presented in this section. This test is very important to show the reaction of the controller when the driver changes direction in a short time, often, and brutally changes the direction of the vehicle. The simulation results are shown in Fig. 8. One remarks that the rollover risk appears at 14 s and lasts 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 the controller is OFF (dashed line). Otherwise, and when the controller 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 the LTR at its limit of 0.9, the computed lateral acceleration limit is 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 decreases until −12 m/s2 (dashed line). During the risk time interval [14, 15] s, the controller is activated and the LTR is stabilized to −0.9 (solid line). Without control, the rollover risk decreases and the LTR tends toward −1.8 (dashed line). The same situation occurs at the time interval [16, 17] s. In this case, the LTR is also stabilized to the limited value 0.9 when the controller is active. Otherwise, the LTR tends toward 1.5. The use of a sliding mode observer allowed a good and quick estimation of the roll angle, as presented in Fig. 8. Without control (dashed line), the absolute value of this variable increases from 0.06 to 0.11 rad during the first time interval [14, 15] s and from 0.06 to 0.09 rad in the time interval [16, 17] s. In Fig. 9, the identification results
are shown. The suspension stiffness k1 and k2 are identified with success. In effect, compared to their nominal values of, respectively, 194 680 and 188 540 N/m, these parameters have been identified with errors, which can be neglected as shown in Fig. 10. The percentage of error is less than 0.025%. However, one notes some variations at the time interval [13, 20] s. This is due to the fact that, at this time, the driver abruptly changes the direction of the vehicle. In the right side of Fig. 10, the unsprung masses m1 and m2 have been identified with a percentage of error less than 0.1% compared to their nominal values of, respectively, 100 and 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, during this time, the estimated steering angle coming from the control block is the same as that coming from the model without control. The LTR in this case is less than 1, as shown in the right side of Fig. 11. After 1.5 s, the risk appears and the LTR reaches the limit value 1, which corresponds to the situation where one of the wheels of the same axle lifts off. To avoid this risk situation, the controller is then activated to avoid the rollover of the vehicle. With the active controller, the steering angle is reduced and its value becomes less than the original one coming from the model. In this case, the lateral acceleration limit is estimated and shown in Fig. 11 (solid line). Without control (dashed line), the lateral acceleration increases until 4 m/s2 . Otherwise, when the control is activated (solid line), this acceleration does
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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 reduced and it becomes less than 1. On the other hand, the sliding mode observer allows us to estimate in finite time and quickly determine the different variables of the system. In Fig. 11, one notices also the well estimation of the side slip angle when it is compared to the variable coming from the model. The identification results are shown in Fig. 12. In this test, the suspension stiffness k1 and k2 have been identified better than in the first one. Indeed, one notes that the signals are smooth, and only small variations of these identified values occur around their nominal values of, respectively, 194 680 and 188 540 N/m.
In the right side of Fig. 12, the identified unsprung masses m1 and m2 are shown. The variations of these parameters occur around their nominal values of, respectively, 100 and 95 kg, with an error quite close to 0, as shown in Fig. 13. VI. C ONCLUSION In this paper, a steering control system has been developed to avoid the rollover of HV. It is based on control of lateral acceleration of the vehicle. Previously, an estimator based on the sliding mode was implemented, which made it possible to estimate the nonmeasured dynamics of the vehicle. Lateral
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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 this paper. Compared to existing method, the proposed approach is based on robust controller estimator. The identification of unsprung masses and suspension stiffness has been computed to increase the robustness of the controller. A real zigzag scenario is tested and presented. The results showed that this system is effective and made it possible to control the vehicle and to avoid its rollover. The lateral acceleration limit is stabilized. Therefore, the steering angle of the vehicle is modified to force this latter to stay on a safety trajectory. The LTR is maintained around a maxi-
mum value of 0.9. The experimental tests done on an instrumented truck showed the quality of this approach since the convergence of the observer is quick and it is done in finite time. The identification of the parameters was a success with small variations around the nominal values at the time interval [13 20] s. These variations are due to the abrupt direction change of the driver. A second ramp test is presented in this paper. Compared to the first one, the identification process has been carried out with errors quite close to zero and the controller is activated quickly to avoid the accident.
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Fig. 13. Identification errors for a driver steering input ramp.
The originality of this approach is the use of the equivalent control, which provides a linear regression algorithm to identify the unknown parameters of the system. An example of identification of the unsprung mass and the stiffness is given in this paper. The result is of quality. In future work, it will be interesting to test this approach in real time with an instrumented tractor semitrailer and using the dynamo wheel to measure the impact forces, which will be the reference for a better validation of the estimation. In this case, the controller will be applied to stabilize the semitrailer, which is, in this type of HV, the first unit that can lose control. In this paper, the unsprung masses and the suspension stiffness have been identified. In future work, one will focus on the identification of other important dynamic parameters, namely, damping coefficients, roll, and yaw inertia moments. ACKNOWLEDGMENT This work was developed by the French Institut Français des Sciences et Technologies des Transports laboratory (Laboratoire Central des Ponts et Chausséees) in collaboration with French industrial partners, Renault Trucks, Michelin, and Sodit in the framework of French project Véhicule Lourd Interactif du Future (VIF). R EFERENCES [1] B. Johansson and M. Gafvert, “Untripped SUV rollover detection and prevention,” in Proc. 43rd IEEE Conf. Decision Control, Atlantis, Bahamas, Dec. 14–17, 2004, pp. 103–109. [2] H. Imine, S. Srairi, D. Gil, and J. Receveur, “Heavy vehicle, vehicle management center and infrastructure interactivity to manage the access to infrastructure,” in Proc. ICHV, Paris, France, May 19–22, 2008. [3] H. Imine, A. Benallegue, T. Madani, and S. Srairi, “Rollover risk prediction of an instrumented heavy vehicle using high order sliding mode observer,” in Proc. ICRA, Kobe, Japan, May 12–17, 2009, pp. 64–69. [4] H. Imine and V. Dolcemascolo, “Rollover risk prediction of heavy vehicle in interaction with infrastructure,” Int. J. Heavy Veh. Syst., vol. 14, no. 3, pp. 294–307, 2007.
[5] J. Ackermann and D. Odenthal, “Damping of vehicle roll dynamics by speed-scheduled active steering,” in Proc. Eur. Control Conf., Karlsruhe, Germany, Aug. 31–Sep. 3, 1999. [6] D. Odenthal, T. Bunte, and J. Ackermann, “Nonlinear steering and braking control for vehicle rollover avoidance,” in Proc. Eur. Control Conf., Karlsruhe, Germany, Aug. 31–Sep. 3, 1999. [7] A. J. P. Miège, “Active roll control of an experimental articulated vehicle,” Ph.D. dissertation, Univ. Cambridge, Dept. Eng., Cambridge, U.K., 2003. [8] C. Kim and P. I. Ro, “A sliding mode controller for vehicle active suspension systems with non-linearities,” Proc. Inst. Mech. Eng. D—J. Autom. Eng., vol. 212, no. 2, pp. 79–92, Feb. 1998. [9] S. Mammar, “Tractor-semitrailer latera1 control robust reduced order two-degree-of-freedom,” in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp. 3158–3162. [10] H. Imine and V. Dolcemascolo, “Sliding mode observers to heavy vehicle vertical forces estimation,” Int. J. Heavy Veh. Syst. (IJHVS), vol. 15, no. 1, pp. 53–64, 2008. [11] O. Khemoudj, H. Imine, M. Djemai, and L. Fridman, “Variable gain sliding mode observer for heavy duty vehicle contact forces,” in Proc. IEEE VSS, Mexico City, Mexico, June 26–28, 2010, pp. 522–527. [12] O. Khemoudj, H. Imine, and M. Djemai, “Robust observation of tractortrailer vertical forces using model inverse and numerical differentiation,” Int. J. Mater. Manuf., vol. 3, no. 1, pp. 278–289, Aug. 2010. [13] Kistler, Suisse. [Online]. Available: www.kistler.com [14] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Control, vol. 58, no. 6, pp. 1247–1263, 1993. [15] V. I. Utkin and S. Drakunov, “Sliding mode observer,” in Proc. IEEE Conf. Decision Control, Orlando, FL, 1995, pp. 3376–3379. [16] H. Imine, Y. Delanne, and N. K. MSirdi, “Road profiles inputs to evaluate loads on the wheels,” Int. J. Veh. Syst. Dyn., vol. 43, pp. 359–369, Nov. 2005. [17] T. Soderstrom and P. Stoica, System Identification. Cambridge, U.K.: Prentice-Hall, 1989. [18] F. Floret-Pontet and F. Lamnabhi-Lagarrigue, “Parameter identification methodology using sliding mode observers,” Int. J. Control, vol. 74, no. 18, pp. 1743–1753, 2001. [19] PROSPER, Sera-Cd. Tech. Rep., Meudon, France. [Online]. Available: www.sera-cd.com [20] H. Imine, Plan d’expérience, instrumentation, traitement des signaux et vérification du système au laboratoire, June 2007. Rapport Projet VIF2. [21] L. Fridman, A. Levant, and J. Davila, “High-order sliding-mode observation and identification for linear systems with unknown inputs,” in Proc. 45th Conf. Decision Control, San Diego, CA, 2006, pp. 5567–5572.
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[22] I. Boiko and L. Fridman, “Analysis of chattering in continuous slidingmode controllers,” IEEE Trans. Autom. Control, vol. 50, no. 9, pp. 1442– 1446, Sep. 2005. [23] H. Imine, L. Fridman, H. Shraim, and M. Djemai, “Sliding mode based analysis and identification of vehicle dynamics,” in Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 2011. [24] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, Mar. 1998. [25] L. Fridman, “The problem of chattering: An averaging approach,” in Variable Structure, Sliding Mode and Nonlinear Control, number 247 in Lecture Notes in Control and Information Science, K. D. Young and U. Ozguner, Eds. London, U.K.: Springer-Verlag, 1999, pp. 363–386. [26] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992. [27] H. Imine, S. Srairi, and T. Madani, “Experimental validation of rollover risk prediction of heavy vehicles,” in Proc. TRA, Ljubljana, Slovenia, Apr. 21–25, 2008.
Hocine Imine received the Master’s degree and the Ph.D. degree in robotics and automation from Versailles University, Versailles, France, in 2000 and 2003, respectively. From 2003 to 2004, he was a Researcher with the Robotic Laboratory of Versailles (LRV in French) and an Assistant Professor with Versailles University. In 2005, he joined Institut Français des Sciences et Technologies des Transports, where he is currently a Researcher. He received the Accreditation to Supervise Research (Habilitation à Diriger des Recherches—HDR) in March 2012. He is involved in different projects like véhicule interactif du futur (VIF) and heavyroute (intelligent route guidance for heavy traffic). He is also responsible for the research project PLInfra on heavy vehicle safety and the assessment of their impacts on pavement and bridges. He has published two books, over 60 technical papers, and several industrial technical reports. His research interests include intelligent transportation systems, heavy vehicle modeling and stability, diagnosis, nonlinear observation, and nonlinear control. Dr. Imine was a member of the organization committee of the International Conference on Heavy Vehicles (HVParis’2008, merging HVTT10 and ICWIM5 in May 2008). He was a Guest Editor of the International Journal of Vehicle Design, Special Issue on “Variable Structure Systems in Automotive Applications.” He is a member of the IFAC Technical Committee on Transportation systems (TC 7.4).
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Leonid M. Fridman received the M.S. degree in mathematics from Kuibyshev (Samara) State University, Samara, Russia, in 1976, the Ph.D. degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the Dr.Sci. degree in control science from the Moscow State University of Mathematics and Electronics, Moscow, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil 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 joined the Department of Control Engineering and Robotics, Division of Electrical Engineering of Engineering Faculty, National Autonomous University of Mexico (UNAM), México, México. He was an Invited Professor in 15 universities and research centers in France, Germany, Italy, Israel, and Spain. He is the author 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 is variable structure systems. Dr. Fridman is Associate Editor of the International Journal of System Science and the Journal of the Franklin Institute, Nonlinear Analysis: Hybrid Systems. He was on the Conference Editorial Board of the IEEE Control Systems Society, a member of the Technical Committee (TC) in the Variable Structure Systems and Sliding Mode Control of IEEE Control Systems Society and TC in the Discrete Events and Hybrid Systems of IFAC. He is a winner of the Scopus prize for the best cited Mexican Scientists in Mathematics and Engineering 2010 award.
Tarek Madani received the Engineering and Magister degrees in automatic control for electrical engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, in 1997 and 2000, respectively, and the Ph.D. degree in automatic control and robotics from Versailles University, Versailles, France, in 2005. He is currently with the Laboratoire d’Ingénierie des Systèmes de Versailles. His main research interests include nonlinear control.
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Steering Control for Rollover Avoidance of Heavy Vehicles Hocine Imine, Leonid M. Fridman, and Tarek Madani
Abstract—The aim of this paper is to develop an active steering assistance system to avoid the rollover of heavy vehicles (HV). The proposed approach is applied on a single body model of HV presented in this paper. An estimator based on the high-order sliding mode observer is developed to estimate the vehicle dynamics, such as lateral acceleration limit and center height of gravity. Lateral position and lateral speed are controlled using a twisting algorithm to ensure the stability of the vehicle and avoid accidents. At the same time, the identification of unsprung masses and suspension stiffness parameters of the model have been computed to increase the robustness of the method. Some simulation and experimental results are given to show the quality of the proposed concept. Index Terms—Estimation, identification, rollover, sliding mode control, sliding mode observers, steering control, twisting algorithm, vehicle modeling.
I. I NTRODUCTION
T
HE ROLL stability of the heavy vehicles (HV) is affected by the center height of gravity, the track width, and the kinematic properties of suspensions. A more destabilizing moment arises during the cornering maneuver when the center of gravity of the vehicle shifts laterally. The roll stability of the vehicle can be guaranteed if the sum of the destabilizing moment is compensated during a lateral maneuver. Recently, several solutions have been proposed to reduce the number of HV rollover accidents. Several systems have been developed to assist the driver to avoid the rollover. Some of them are installed in the infrastructure before a dangerous curvature. 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. They use informative measures coming from sensors, such as vehicle
Manuscript received August 26, 2011; revised November 14, 2011 and March 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 by the French Ministry of Industry and in part by the Lyon Urban Trucks & Bus competitiveness cluster. The work of L. Fridman was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACyT) under Grant 132125 and in part by Programa de Apoyo a Proyectos de Investigación e Innovación Tecnología (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, Simulators and Simulations, Laboratory for Road Operation, Perception, Simulations and Simulators (LEPSIS), Paris 75732, France (e-mail: [email protected]). L. M. Fridman is with the Departamento de Ingeniería de Control y Robótica, División de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad Nacional 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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2012.2206837
speed and lateral acceleration, to send a warning signal to the driver when he goes beyond some risk thresholds [3], [4]. In the case of active systems, the concept is to minimize the lateral acceleration by braking action, steering action, suspension action, antiroll action, or a combination of all [5]–[9]. However, most of the existing methods require full information on the state that may limit their practical utility. In fact, even if all the state measurements are possible, they are typically corrupted by noises. Moreover, the increased number of sensors makes the overall system more complex in implementation and expensive in realization. Thus, design of an observer becomes an attractive approach to estimate the unmeasured states of the HV. A second-order sliding mode observer, providing theoretically exact state observation, parameter identification, and impact force estimation, is presented in this paper. In most of the recent research, the parameters of the vehicle are supposed to be known and constant. Some of them were given by vehicle constructors and manufacturers, and other unknown parameters are taken from literature. In the present work, suspension stiffness and unsprung masses have been identified. This allows improving the quality of the proposed system. The impact forces affect vehicle dynamic performance and behavior properties. These forces are very important for evaluating the rollover risk of HV by computing the load transfer ratio (LTR), studying and evaluating the damage of the vehicle on the road or bridges, and controling the weights of the vehicles [10]–[12]. These forces can also be used in control systems to assist the driver. However, the exiting sensors to measure these forces are very expensive and difficult to install [13]. Compared to the previous works, the proposed method is based on a sliding mode observer to estimate the vertical forces and at the same time identify the dynamic parameters of the vehicle [14]–[18]. Design of such observers requires a dynamic model of HV that is build up in a first step of this paper. This model is coupled with an appropriate wheel road contact model. It has been validated using a simulator [19] and by real measurements carried out with an instrumented tractor [20]. In this paper, an observer–controller law using the sliding mode technique is developed. The proposed method has been adopted to: 1) Achieve good tracking of desired trajectories by ensuring the convergence of the lateral acceleration of the vehicle toward the estimated acceleration limit. This allows the limitation of the load transfer between the right and left sides of the vehicle to its limited value, which is set to 0.9.
0018-9545/$31.00 © 2012 IEEE
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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 vehicle description and modeling. The design of the observer and the estimation of the vertical forces and parameter identification are presented in Section III. Section IV is devoted to steering control method design. Some experimental results are presented in Section V. Finally, some remarks and perspectives are given in a concluding section. II. V EHICLE M ODELING The vehicle studied in this paper and developed by reference [5] is a nonarticulated HV with two axles. Therefore, some assumptions were considered: the roll angle is assumed to be small, and the suspension and tire dynamics are assumed to be linear. The corresponding model has five degrees of freedom and used to represent the roll and lateral dynamics, as shown in Fig. 1. The vehicle consists of two bodies. Body 1, which is composed of two axles, is represented by the unsprung mass mu and center of gravity CG1. Body 2 is the sprung mass m and center of gravity CG2. The center CG1 is assumed to be in the road plane above CG2. In this case, the lateral forces acting on the system can be linearized and computed as follows: Fyr = μcr αr (1) Fyf = μcf αf where cf and cr are, respectively, the front and rear cornering stiffness after assuming the equality of the cornering stiffness for the front and rear wheels. The road adhesion is represented by the variable μ. In this paper, the dry road is assumed (μ = 1). The parameters αf and αr are, respectively, the front and rear tire slip angles computed by using the following formula: ⎧ ⎨ αr = − β − lr ψ˙ v (2) ⎩ αf = δ − β + lf ψ˙ v
Fig. 2.
Front view of HV model.
where β is the side slip angle, which is assumed to be equal for the two front wheels and the rear ones, lf and lr , and are, respectively, the distance from CG1 to the front axle and rear axle, v is the vehicle speed, ψ˙ is the yaw rate, and δ is the front wheel 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 vector composed of, respectively, left and right front suspension deflections, lateral displacement, roll angle, and yaw angle; M ∈ 5×5 is the inertia matrix; C ∈ 5×5 is the matrix related to the damping effects; K ∈ 5 is the spring stiffness vector; Fg ∈ 3 is a vector of generalized forces; and u is the system input. The suspension is modeled as a combination of spring and damper elements, as shown in Fig. 2. The vertical acceleration of the chassis is obtained as follows:
T m z¨ = k1 q1 + k2 q2 + (k1 − k2 ) sin(φ) 2 T ˙ + B1 q˙1 + B2 q˙2 − (B1 − B2 ) cos(φ)φ m (4) 2 where φ is the tractor roll angle, T is the tractor track width, q1 and q2 are, respectively, the left and right front suspension deflections of the tractor, and z is the vertical displacement of the tractor sprung mass (center of gravity height). The tractor chassis with mass m is suspended on its axles through two suspension systems. The suspensions stiffness and damping coefficients are represented, 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 represented by the variables u1 and u2 , which are considered as HV inputs. The variables zr1 and zr2 correspond, respectively, to the vertical displacement of the left and right wheels of the
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tractor front axle, which can be computed by means of using the following equations: zr1 = z − q1 − T2 sin(φ) . (5) zr2 = z − q2 + T2 sin(φ) The lateral and yaw accelerations are computed as follows: (c +c ) (−cf lf +cr lr )−mv 2 ˙ c y¨l = − fmv r y˙ l + ψ + mf δ mv 2 (6) (cf lf2 +cr lr2 ) ˙ lf cf (−cf lf +cr lr ) β ψ + Iz δ + ψ¨ = − Iz v Iz where Iz is the inertia along the Z axis. The vertical accelerations of the wheels are given by ⎧ z¨r1 = k1 q1 − k1 T2 sin(φ) /m1 + B q ˙ 1 1 ⎪ ⎪ ⎪ ⎪ T ⎨ − B1 cos(φ)φ˙ + Fzl /m1 2 ⎪ /m2 z ¨ == k2 q2 + k2 T2 sin(φ) + B q ˙ r2 2 2 ⎪ ⎪ ⎪ ⎩ + B T cos(φ)φ˙ − F 22 zr /m2 .
(7)
The accelerations of suspensions are given by the following equation system: ⎧ k2 q2 1 1 T 1 1 ⎪ q ¨ k = − q + +(k −k ) sin(φ)+ − ⎪ 1 1 1 1 2 m m1 2 ⎪ ⎪ m m m1 ⎪ ⎪ B2 q˙2 T ⎪ ˙ ⎪ ×B1 q˙1 + m − (B1 − B2 ) 2 cos(φ)φ /m − Fmzl1 ⎪ ⎪ ⎪ ⎨ 1 1 q¨2 = m − m12 k2 q2 + k1mq1 +(k1 −k2 ) T2 sin(φ)+ m − m12 ⎪ ⎪ B1 q˙1 T ˙ ⎪ ×B q ˙ + − (B − B ) cos(φ) φ /m − Fmzr2 ⎪ 2 2 1 2 2 m ⎪ ⎪ ⎪¨ ⎪ T ⎪ + B1 q˙1 ⎪ ⎪ φ = (k1 q1 − k2 q2 − (k1 +Tk2 ) 2 sin(φ) ⎩ T ˙ −B2 q˙2 − (B1 + B2 ) 2 cos(φ)φ) 2 /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. S LIDING M ODE O BSERVER D ESIGN This section is devoted to observer design using the sliding mode technique. To develop it, let us rewrite (3) in state form as follows: x˙ = x 1
2
x˙ 2 = f (x1 + x2 ) + Fg (x1 + u) y = x1
(10)
˙ T ∈ 10 is the state vector, y = where x = (x1 , x2 )T = (q, q) 5 q ∈ is the measured outputs vector of the system, f is a vector of nonlinear analytical function, and Fg is an unknown input vector computed as follows: ⎤ ⎡ −Fzl /m1 Fg = ⎣ −Fzr /m2 ⎦ . 0
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Before developing the sliding mode observer, one supposes the 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 been adapted to the presented model to estimate in finite time states and to identify parameters. It has the following form: ˆ 2 + z1 x ˆ˙ 1 = x (11) x1 + x ˆ 2 ) + z2 x ˆ˙ 2 = f (ˆ ˆ2 are, respectively, the estimations of vectors x1 where x ˆ1 and x and x2 . The correction variables z1 and z2 represent the output injections of the form x1 )Sign(˜ x1 ) z1 = λδ(˜ (12) x1 ) z2 = αSign(˜ ˆ1 ∈ 5 is a vector of the state estimation where x 1 = x1 − x x1 ), error. The gain matrices (λ and α) ∈ 5×5 , the matrix Δ( and the vector “Sign” are defined as follows: ⎧ λ = diag{λ1 + λ2 + · · · λ5 } ⎪ ⎪ ⎪ ⎨ α = diag{α1 + α2 + · · · α5 } 1 1 1 2 + |˜ 2 + · · · |˜ 2 δ(˜ x ) = diag |˜ x | x | x | 1 1 1 1 ⎪ 1 2 5 ⎪ ⎪ ⎩ Sign(˜ x1 ) = [sign(˜ x11 ) + sign(˜ x12 ) + · · · sign(˜ x15 )]T . (13) The dynamic estimation errors are calculated as follows: ⎧ ⎨x ˜2 − λδ(˜ x1 )Sign(˜ x1 ) ˜˙ 1 = x (14) x1 + x ˆ2 )+ x ˜˙ 2 = f (x1 + x2 ) − f (ˆ ⎩ x1 ) Fg (x1 + u) − αSign(˜ where x 2 = x2 − x ˆ2 ∈ 5 is the estimation error of the vector x2 . Considering the accelerations of the system as bounded, the elements of the diagonal matrix α can be selected, satisfying the following inequality: ˙ (15) ˆ2i , i = 1 . . . 5. αii > 2 x On the other hand, from [24], the elements of the diagonal matrix λ can be selected as ˙ ˆ2i (1+pi ) α +2 x ii 2 λii > , i = 1, . . . , 5 ˙ 1−pi ˆ2 αii −2 x i
(16) where pi ∈]0, 1[ are some constants to be chosen (see proof in [24]). To study the observer stability, first, the convergence of x 1 in finite time t0 is proved. Then, some conditions about x 2 to ensure its convergence to 0 are deduced. Therefore, for t ≥ t0 ,
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the surface x 2 = 0 is attractive, leading x ˆ2 to converge toward x2 , satisfying the inequalities (15) and (16). The convergence proof of the second-order observer can be found in [24]. B. Vertical Forces Estimation and Parameters Identification
To eliminate the high frequency component, a filter of the following form is used: τ z¯˙ 2 (t) = z¯2 (t) + z2 (t)
(21)
where τ ∈ R and h τ 1, h being a sampling step. The variable z2 is then rewritten as follows:
To estimate the vertical forces and to identify the parameters of the system, let us rewrite system (10) by reducing its order as follows: ⎧ x˙ 11 = x21 ⎪ ⎨ x˙ 21 = a1 ϕ(x1 + x2 ) − Fzl /m1 (17) ⎪ ⎩ x˙ 12 = x22 x˙ 22 = a2 ϕ(x1 + x2 ) − Fzr /m2
where z 2 (t) is the filtered version of z2 (t) and ξ(t) ∈ R is the difference caused by the filtration. Nevertheless, as shown in [25] and [26]
with x11 = q1 , x21 = q˙1 , x12 = q2 , and x22 = q˙2 . The unknown vectors of parameters represented by a1 and a2 and the vector ϕ are defined as follows: 1 −m k a1 = k1 mm 2 1 2 −m a2 = k1 k2 mm 2
Thus, it is possible to assume that the equivalent output injection is equal to the output of the filter. 1) Vertical Forces Estimation: To estimate the vertical force Fgi (x1 , u), the vector of parameter a is supposed to be known. In this case, a = 0. Therefore, using (20), the vertical force is obtained as follows:
ϕ = [ qm1
q2 T m ]
z2 (t) = z¯2 (t) + ξ(t)
lim z¯2 (τ + h) = z2 (t).
τ ∞0 h/τ ∞0
x1 ). Fgi = αsign(˜
.
To simplify the system, the vector of unknown parameters a is computed as follows: ⎧ ⎨ = [ a11 a21 ] = k1 m1 −m k2 m1 a= ⎩ = [ a12 a22 ] = k1 k2 m2 −m . m2 To estimate the vertical forces and to identify the unknown inputs, the second-order sliding mode observer defined in (11) is rewritten as x ˆ˙ 1 = x ˆ2 + λδ(˜ x1 )sign(˜ x1 ) (18) ¯ϕ(ˆ x1 + x ˆ2 ) + αsign(˜ x1 ) x ˆ˙ 2 = a 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: ˜2 − λδ(˜ x1 )sign(˜ x1 ) x ˜˙ 1 = x (19) ˜ϕ(x1 , x ˜2 ) + Fgi (x1 , u) − αsign(˜ x1 ) x ˜˙ 2 = a where a = a − a is the estimation error of the vector parameters a, Fgi = −Fzl /m1 , or Fgi = −Fzr /m2 . After convergence of the observer (18), the variable x 2 convergences toward 0 in finite time t ≥ t0 . In this case, and from (19), one obtains x1 ) = a ˜ϕ(x1 , x ˜2 ) + Fgi (x1 , u). z2 = αsign(˜
(20)
Theoretically, the equivalent output injection is the result of an infinite switching frequency of the discontinuous term αsign( x1 ). Nevertheless, the realization of the observer produces a high switching frequency that makes the application of a filter necessary.
(22)
(23)
(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 method as follows: 1) The measuring of the road profiles u1 and u2 is not necessary. 2) The estimation of the vertical displacements of the wheels and its derivative is also not necessary to obtain. 2) Parameter Identification: To identify the parameters of the system, the impact force vector is supposed to be zero, i.e., Fg (x1 , u) = 0. That means that the road profile is supposed to be very small. There are no irregularities on the road that can vertically affect the vehicle. In this case, the vertical displacements of the wheels are close to zero. Using (20), it follows that ˜ϕ(x1 + x2 ) = αsign(˜ x1 ). z2 = a
(25)
Considering the unknown parameter vector a as a constant vector, and to identify it, a linear regression algorithm, namely the least square method, is applied [17], [18]. The time integration is given by 1 t
t
1 z2 (σ)ϕ(σ) dσ = a ˜ t
t
T
ϕ(σ)ϕ(σ)T dσ.
(26)
The vector a is then estimated by ⎡ t ⎤⎡ t ⎤−1 ˆ˜ = ⎣ z2 (σ)ϕ(σ)T dσ ⎦ ⎣ ϕ(σ)ϕ(σ)T dσ ⎦ a
(27)
0
0
0
0
ˆ is the estimation of where a a. t Let us define Γ = [ 0 ϕ(σ)ϕ(σ)T dσ]−1 .
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Its derivative gives Γ˙ = −Γϕ(σ)ϕ(σ)T Γ. ˆ using (27) gives The derivative of the vector a ⎤ ⎡ t ˙ˆ ⎣ z2 (σ)ϕ(σ)T dσ ⎦ Γ˙ + z2 ϕT Γ. a ˜=
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(28)
0
Replacing Γ˙ by its value given before and using (28), it follows that ˆ˙ = − a ˆ a ˜ ˜ϕϕT + z2 ϕT Γ ˆ = (−a ˜ϕ + z2 )ϕT Γ.
(29)
ˆ toward This ensures the asymptotic convergence of a a and, therefore, this allows the identification of the real value of the vector a. To obtain the unsprung masses and after identification of k1 = a12 and k2 = a21 , the expression of the variable a defined earlier is used here. The identified values are then as follows: a11 = k1 (m1 − m)/m1 and a22 = k2 (m2 − m)/m2 .
high values of lateral accelerations. In the case of a small roll angle, one can assume that ay2 h sin φ < (hR + h cos φ) . g
Finally, the unsprung masses are deduced as follows: m1 = mk1 /(k1 − a11 ) and m2 = mk2 /(k2 − a22 ).
From (30), the acceleration limit can be obtained and approximated by
IV. S TEERING C ONTROL Rollover risk evaluation is based on LTR, which corresponds to the difference in normal tire forces acting on each side of the vehicle. It can be computed as follows: Fzr − FzL Fzr + Fzl 2m2 ay2 + h sin φ = (hR + h cos φ) mT g
Fig. 3. Controller diagram.
LT R =
(30)
where ay2 = ay − hφ¨ is the lateral acceleration of the sprung mass, ay = v(ψ + β) is the lateral acceleration of the unsprung mass, T is the track width, and Fzl and Fzr are the normal forces acting on, respectively, the left and right sides of the vehicle. When LT R is equal to 0, the HV has a stable roll dynamic. The risk becomes higher when this indicator goes toward ±1. Both extreme values characterize wheel lift-off. The same model developed before has been used in this section. However, to perform the controller, only the lateral part of the model is important. Therefore, the suspension deflection variables are not used here. In this section, an active steering control is developed to assist the vehicle in case of rollover risk. In addition to the steering angle commanded by the driver and noted δd , an auxiliary steering angle δa is set by an actuator. Therefore, the control input u = δ = δa + δd . In this paper, the limit value of LTR is set to 0.9. This chosen value is used arbitrarily, less than the limit 1, to give sufficient time to the controller/driver to react, before one of the wheels lifts off the road. In this case, the controller has time to avoid the rollover before to obtain
ay2 lim ≈
0.9T gm ; 2m2 H
H = h + HR .
(31)
The aim of the developed steering control is to ensure the convergence of the lateral acceleration ay2 of the vehicle to its acceleration limit ay2lim . This will allow the limitation of load transfer between the right and left sides of the vehicle to its limited 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 = y˜˙ l + η y˜l
(32)
where y˙ l = y˙ l − y˙ ld and yl = yl − yld are, respectively, the lateral speed error and the lateral offset, η is a positive constant, and y˙ ld and yld are, respectively, the desired velocity and the desired lateral displacement obtained by the first and double integration of the acceleration limit ay2lim , computed earlier using (31). Deriving the variable S and using (6), it follows that cf S˙ = y¨˜l + η y˜˙ l = u + (η + a55 )y˜˙ l . m
(33)
The equivalent control ueq is obtained by resolving the equation S˙ = 0 as follows: ueq = −
m (η + a55 )y˜˙ l − δd . cf
(34)
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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 systems to avoid chattering. In this case, the trajectories are characterized by twisting around the origin, as shown in Fig. 4. The continuous control law u is composed of two terms. The first 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 (35) u˙ 1 = −G2 sign(S) 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. E XPERIMENTAL R ESULTS The tractor in Fig. 5 has been instrumented on behalf of the Véhicule Lourd Interactif du Future project [27]. The vehicle was equipped with different sensors to measure the dynamic states of the vehicle, such as gyrometers, accelerometers, linear variable differential transformer (LVDT), lasers, etc., as shown in Fig. 6. The installation and positions of these sensors in the tractor are illustrated in 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 to validate and calibrate the whole system: 1) Four accelerometers installed on the chassis to measure the vertical accelerations of wheels. 2) Four sensors for LVDT to measure the suspension deflections (q1 and q2 have been used in the observer). 3) Three axial gyroscopes installed on the chassis to measure the angular rates (roll, pitch, and yaw rates). The roll angle is deduced from integration of the measured roll rate or by computation using the following formula: q1 − q2 φ = arcsin . T In this paper, this last method was preferred and used to avoid errors that can result from roll rate integration. 4) Two lasers to measure the vertical displacements of the chassis. Many tests and scenarios have been performed with the instrumented vehicle driving at various speeds. In this paper, the results obtained from zigzag and ramp tests are presented to show the robustness of the estimation and identification, whereas the presented results for steering control to avoid rollover are obtained from simulation. The nominal dynamic parameters and the static vertical forces are measured before the tests. The static values of front left and front right vertical forces are, respectively, 24200N and 25250N. The static values of the rear left and rear right vertical forces are, respectively, 9450N and 12050N.
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Fig. 8.
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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 of suspension stiffness k1 and k2 are, respectively, 194 680 and 188 540 N/m. A. Zigzag Test A zigzag test is presented in this section. This test is very important to show the reaction of the controller when the driver changes direction in a short time, often, and brutally changes the direction of the vehicle. The simulation results are shown in Fig. 8. One remarks that the rollover risk appears at 14 s and lasts 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 the controller is OFF (dashed line). Otherwise, and when the controller 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 the LTR at its limit of 0.9, the computed lateral acceleration limit is 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 decreases until −12 m/s2 (dashed line). During the risk time interval [14, 15] s, the controller is activated and the LTR is stabilized to −0.9 (solid line). Without control, the rollover risk decreases and the LTR tends toward −1.8 (dashed line). The same situation occurs at the time interval [16, 17] s. In this case, the LTR is also stabilized to the limited value 0.9 when the controller is active. Otherwise, the LTR tends toward 1.5. The use of a sliding mode observer allowed a good and quick estimation of the roll angle, as presented in Fig. 8. Without control (dashed line), the absolute value of this variable increases from 0.06 to 0.11 rad during the first time interval [14, 15] s and from 0.06 to 0.09 rad in the time interval [16, 17] s. In Fig. 9, the identification results
are shown. The suspension stiffness k1 and k2 are identified with success. In effect, compared to their nominal values of, respectively, 194 680 and 188 540 N/m, these parameters have been identified with errors, which can be neglected as shown in Fig. 10. The percentage of error is less than 0.025%. However, one notes some variations at the time interval [13, 20] s. This is due to the fact that, at this time, the driver abruptly changes the direction of the vehicle. In the right side of Fig. 10, the unsprung masses m1 and m2 have been identified with a percentage of error less than 0.1% compared to their nominal values of, respectively, 100 and 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, during this time, the estimated steering angle coming from the control block is the same as that coming from the model without control. The LTR in this case is less than 1, as shown in the right side of Fig. 11. After 1.5 s, the risk appears and the LTR reaches the limit value 1, which corresponds to the situation where one of the wheels of the same axle lifts off. To avoid this risk situation, the controller is then activated to avoid the rollover of the vehicle. With the active controller, the steering angle is reduced and its value becomes less than the original one coming from the model. In this case, the lateral acceleration limit is estimated and shown in Fig. 11 (solid line). Without control (dashed line), the lateral acceleration increases until 4 m/s2 . Otherwise, when the control is activated (solid line), this acceleration does
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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 reduced and it becomes less than 1. On the other hand, the sliding mode observer allows us to estimate in finite time and quickly determine the different variables of the system. In Fig. 11, one notices also the well estimation of the side slip angle when it is compared to the variable coming from the model. The identification results are shown in Fig. 12. In this test, the suspension stiffness k1 and k2 have been identified better than in the first one. Indeed, one notes that the signals are smooth, and only small variations of these identified values occur around their nominal values of, respectively, 194 680 and 188 540 N/m.
In the right side of Fig. 12, the identified unsprung masses m1 and m2 are shown. The variations of these parameters occur around their nominal values of, respectively, 100 and 95 kg, with an error quite close to 0, as shown in Fig. 13. VI. C ONCLUSION In this paper, a steering control system has been developed to avoid the rollover of HV. It is based on control of lateral acceleration of the vehicle. Previously, an estimator based on the sliding mode was implemented, which made it possible to estimate the nonmeasured dynamics of the vehicle. Lateral
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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 this paper. Compared to existing method, the proposed approach is based on robust controller estimator. The identification of unsprung masses and suspension stiffness has been computed to increase the robustness of the controller. A real zigzag scenario is tested and presented. The results showed that this system is effective and made it possible to control the vehicle and to avoid its rollover. The lateral acceleration limit is stabilized. Therefore, the steering angle of the vehicle is modified to force this latter to stay on a safety trajectory. The LTR is maintained around a maxi-
mum value of 0.9. The experimental tests done on an instrumented truck showed the quality of this approach since the convergence of the observer is quick and it is done in finite time. The identification of the parameters was a success with small variations around the nominal values at the time interval [13 20] s. These variations are due to the abrupt direction change of the driver. A second ramp test is presented in this paper. Compared to the first one, the identification process has been carried out with errors quite close to zero and the controller is activated quickly to avoid the accident.
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Fig. 13. Identification errors for a driver steering input ramp.
The originality of this approach is the use of the equivalent control, which provides a linear regression algorithm to identify the unknown parameters of the system. An example of identification of the unsprung mass and the stiffness is given in this paper. The result is of quality. In future work, it will be interesting to test this approach in real time with an instrumented tractor semitrailer and using the dynamo wheel to measure the impact forces, which will be the reference for a better validation of the estimation. In this case, the controller will be applied to stabilize the semitrailer, which is, in this type of HV, the first unit that can lose control. In this paper, the unsprung masses and the suspension stiffness have been identified. In future work, one will focus on the identification of other important dynamic parameters, namely, damping coefficients, roll, and yaw inertia moments. ACKNOWLEDGMENT This work was developed by the French Institut Français des Sciences et Technologies des Transports laboratory (Laboratoire Central des Ponts et Chausséees) in collaboration with French industrial partners, Renault Trucks, Michelin, and Sodit in the framework of French project Véhicule Lourd Interactif du Future (VIF). R EFERENCES [1] B. Johansson and M. Gafvert, “Untripped SUV rollover detection and prevention,” in Proc. 43rd IEEE Conf. Decision Control, Atlantis, Bahamas, Dec. 14–17, 2004, pp. 103–109. [2] H. Imine, S. Srairi, D. Gil, and J. Receveur, “Heavy vehicle, vehicle management center and infrastructure interactivity to manage the access to infrastructure,” in Proc. ICHV, Paris, France, May 19–22, 2008. [3] H. Imine, A. Benallegue, T. Madani, and S. Srairi, “Rollover risk prediction of an instrumented heavy vehicle using high order sliding mode observer,” in Proc. ICRA, Kobe, Japan, May 12–17, 2009, pp. 64–69. [4] H. Imine and V. Dolcemascolo, “Rollover risk prediction of heavy vehicle in interaction with infrastructure,” Int. J. Heavy Veh. Syst., vol. 14, no. 3, pp. 294–307, 2007.
[5] J. Ackermann and D. Odenthal, “Damping of vehicle roll dynamics by speed-scheduled active steering,” in Proc. Eur. Control Conf., Karlsruhe, Germany, Aug. 31–Sep. 3, 1999. [6] D. Odenthal, T. Bunte, and J. Ackermann, “Nonlinear steering and braking control for vehicle rollover avoidance,” in Proc. Eur. Control Conf., Karlsruhe, Germany, Aug. 31–Sep. 3, 1999. [7] A. J. P. Miège, “Active roll control of an experimental articulated vehicle,” Ph.D. dissertation, Univ. Cambridge, Dept. Eng., Cambridge, U.K., 2003. [8] C. Kim and P. I. Ro, “A sliding mode controller for vehicle active suspension systems with non-linearities,” Proc. Inst. Mech. Eng. D—J. Autom. Eng., vol. 212, no. 2, pp. 79–92, Feb. 1998. [9] S. Mammar, “Tractor-semitrailer latera1 control robust reduced order two-degree-of-freedom,” in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp. 3158–3162. [10] H. Imine and V. Dolcemascolo, “Sliding mode observers to heavy vehicle vertical forces estimation,” Int. J. Heavy Veh. Syst. (IJHVS), vol. 15, no. 1, pp. 53–64, 2008. [11] O. Khemoudj, H. Imine, M. Djemai, and L. Fridman, “Variable gain sliding mode observer for heavy duty vehicle contact forces,” in Proc. IEEE VSS, Mexico City, Mexico, June 26–28, 2010, pp. 522–527. [12] O. Khemoudj, H. Imine, and M. Djemai, “Robust observation of tractortrailer vertical forces using model inverse and numerical differentiation,” Int. J. Mater. Manuf., vol. 3, no. 1, pp. 278–289, Aug. 2010. [13] Kistler, Suisse. [Online]. Available: www.kistler.com [14] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Control, vol. 58, no. 6, pp. 1247–1263, 1993. [15] V. I. Utkin and S. Drakunov, “Sliding mode observer,” in Proc. IEEE Conf. Decision Control, Orlando, FL, 1995, pp. 3376–3379. [16] H. Imine, Y. Delanne, and N. K. MSirdi, “Road profiles inputs to evaluate loads on the wheels,” Int. J. Veh. Syst. Dyn., vol. 43, pp. 359–369, Nov. 2005. [17] T. Soderstrom and P. Stoica, System Identification. Cambridge, U.K.: Prentice-Hall, 1989. [18] F. Floret-Pontet and F. Lamnabhi-Lagarrigue, “Parameter identification methodology using sliding mode observers,” Int. J. Control, vol. 74, no. 18, pp. 1743–1753, 2001. [19] PROSPER, Sera-Cd. Tech. Rep., Meudon, France. [Online]. Available: www.sera-cd.com [20] H. Imine, Plan d’expérience, instrumentation, traitement des signaux et vérification du système au laboratoire, June 2007. Rapport Projet VIF2. [21] L. Fridman, A. Levant, and J. Davila, “High-order sliding-mode observation and identification for linear systems with unknown inputs,” in Proc. 45th Conf. Decision Control, San Diego, CA, 2006, pp. 5567–5572.
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[22] I. Boiko and L. Fridman, “Analysis of chattering in continuous slidingmode controllers,” IEEE Trans. Autom. Control, vol. 50, no. 9, pp. 1442– 1446, Sep. 2005. [23] H. Imine, L. Fridman, H. Shraim, and M. Djemai, “Sliding mode based analysis and identification of vehicle dynamics,” in Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 2011. [24] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, Mar. 1998. [25] L. Fridman, “The problem of chattering: An averaging approach,” in Variable Structure, Sliding Mode and Nonlinear Control, number 247 in Lecture Notes in Control and Information Science, K. D. Young and U. Ozguner, Eds. London, U.K.: Springer-Verlag, 1999, pp. 363–386. [26] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992. [27] H. Imine, S. Srairi, and T. Madani, “Experimental validation of rollover risk prediction of heavy vehicles,” in Proc. TRA, Ljubljana, Slovenia, Apr. 21–25, 2008.
Hocine Imine received the Master’s degree and the Ph.D. degree in robotics and automation from Versailles University, Versailles, France, in 2000 and 2003, respectively. From 2003 to 2004, he was a Researcher with the Robotic Laboratory of Versailles (LRV in French) and an Assistant Professor with Versailles University. In 2005, he joined Institut Français des Sciences et Technologies des Transports, where he is currently a Researcher. He received the Accreditation to Supervise Research (Habilitation à Diriger des Recherches—HDR) in March 2012. He is involved in different projects like véhicule interactif du futur (VIF) and heavyroute (intelligent route guidance for heavy traffic). He is also responsible for the research project PLInfra on heavy vehicle safety and the assessment of their impacts on pavement and bridges. He has published two books, over 60 technical papers, and several industrial technical reports. His research interests include intelligent transportation systems, heavy vehicle modeling and stability, diagnosis, nonlinear observation, and nonlinear control. Dr. Imine was a member of the organization committee of the International Conference on Heavy Vehicles (HVParis’2008, merging HVTT10 and ICWIM5 in May 2008). He was a Guest Editor of the International Journal of Vehicle Design, Special Issue on “Variable Structure Systems in Automotive Applications.” He is a member of the IFAC Technical Committee on Transportation systems (TC 7.4).
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Leonid M. Fridman received the M.S. degree in mathematics from Kuibyshev (Samara) State University, Samara, Russia, in 1976, the Ph.D. degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the Dr.Sci. degree in control science from the Moscow State University of Mathematics and Electronics, Moscow, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil 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 joined the Department of Control Engineering and Robotics, Division of Electrical Engineering of Engineering Faculty, National Autonomous University of Mexico (UNAM), México, México. He was an Invited Professor in 15 universities and research centers in France, Germany, Italy, Israel, and Spain. He is the author 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 is variable structure systems. Dr. Fridman is Associate Editor of the International Journal of System Science and the Journal of the Franklin Institute, Nonlinear Analysis: Hybrid Systems. He was on the Conference Editorial Board of the IEEE Control Systems Society, a member of the Technical Committee (TC) in the Variable Structure Systems and Sliding Mode Control of IEEE Control Systems Society and TC in the Discrete Events and Hybrid Systems of IFAC. He is a winner of the Scopus prize for the best cited Mexican Scientists in Mathematics and Engineering 2010 award.
Tarek Madani received the Engineering and Magister degrees in automatic control for electrical engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, in 1997 and 2000, respectively, and the Ph.D. degree in automatic control and robotics from Versailles University, Versailles, France, in 2005. He is currently with the Laboratoire d’Ingénierie des Systèmes de Versailles. His main research interests include nonlinear control.
