Sliding Mode Observers for Parameters Estimation in Vehicle Belgacem JABALLAH1, 2, Nacer Kouider M’SIRDI1, Hassani MESSAOUD 2, Aziz NAAMANE1 1
LSIS, CNRS UMR 6168, Domaine Univ. St Jerome, Av. Escadrille Normandie - Niémen. 13397, Marseille Cedex 20, France. {belgacem.jaballah, nacer.msirdi, aziz.naamane}@Lsis.org http://www.lsis.org 2 ATSI, Ecole Nationale d'Ingénieurs de Monastir, Rue Ibn El Jazzar, 5019 Monastir, Tunisie.
[email protected]
Abstract. In this paper, we present a nominal dynamic model for the vehicle composed with 16 Dof and we propose a Robust Sliding Mode Observer to estimate the slope of road and roll angle in vehicle. These parameters affect the vehicle dynamic performance and behavior properties. However, slope and banking of road are difficult to measure directly.
Keywords. Vehicle dynamics model, Sliding mode observer, Road Slope Estimation, Vehicle roll angle.
1. Introduction Car accidents occur for several reasons which may involve either the driver or components of the vehicle or environment. Vehicle controllability along road admissible trajectories still remains an open problem. It is extremely important to detect (on time) a tendency towards instability or faults. This must be done without adding expensive sensors, so robust observers are needed to estimate input variables like road slope and vehicle roll angle. In applications such as vehicles dynamic control precise models are difficult to obtain and interactions with environment cannot be modeled. Only nominal models can be deduced and their parameters approximately estimated. Despite these conditions we try to produce robust estimations of the system states and the pertinent but no measurable inputs like pneumatic contact forces and road profile or characteristics.
STA′2009 – Topics …, pages 1 à X
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STA′2009 – Topics …, pages 1 à X
This challenging problems gave help to develop some kind of, what we can call, software sensors or observers. The weak knowledge of the system model and the bad measurement conditions require robustness and reduced time convergence. Recently, many analytical and experimental studies have been performed on estimation of the road slope [10] [11]. In this paper we propose robust sliding mode observers to tackle problems due to unknown inputs and uncertainties of modeling interactions with environment. In a first stage first order sliding mode observers are proposed to estimate road slope in three steps: estimate the velocities and accelerations then estimate contact forces and finally estimate. Secondly we use the same technical to estimate road bank angle for vehicle in two steps: estimation of the roll angle in vehicle then estimation of road bank angle.
2. Vehicle Modeling Consider a fixed reference frame R as in Figure (1) and represent the vehicle by the
q ∈ 16 are defined as: qT = [ x, y, z , θ , φ ,ψ , q31 , q32 , q33 , q34 , δ 3 , δ 4 , ϕ1 , ϕ 2 , ϕ3 , ϕ4 ] Where x , y and z represent respectively longitudinal, lateral and vertical displace-
scheme below [1][2][6][7]. The generalized coordinates
ment. Angles of roll, pitch and yaw are θ ,
φ
and
ψ respectively. The suspensions
elongations are noted q3i (i = 1..4) . δ i : Stands for the steering angles ( i = 3, 4 ) .
ϕi : are angles of wheels rotations ( i = 1..4 ) .
∈ 16 are respectively velocities q , q
and corresponding accelerations. The nominal model of the vehicle with uncertainties is developed in assuming and reduced to one point for each wheel [9]. The 16 Degrees of Freedom model is then equivalent to [3][7] :
τ = M (q )q + C (q, q )q + V (q, q ) + η0 (t , q, q )
(1)
τ =Γ e + Γ =Γ e + J F
(2)
T
Where:
M (q ) is the inertia matrix, it is Symmetric Positive Definite (SPD). M 1,1 M 2 ,1 M = M 3 ,1 0 0
M 1, 2
M 1,3
0
M 2,2
M 2 ,3
M 2 ,4
M 3,2
M 3,3
0
M 4 ,2
0
M 4 ,4
M 52
0
0
0 M 2 ,5 0 0 M 55
Paper title 3
C (q, q ) is Coriolis and Centrifugal forces. 0 0 C = 0 0 0
C 1,3
0
C2 , 2
C 2 ,3
C 2 ,4
C 3,2
C 3,3
0
C 4 ,2
0
0
C 5,2
0
0
0 C 2 ,5 0 0 C 5 ,5
J (q ) ∈ × is the Jacobian matrix depending on the contact points. 16
12
JT
C 1, 2
J1,1 J 2,1 = J 3,1 0 0
J1,2 J 2,2
J1,3 J 2,3
J 3,2
J 3,3
0
0
0
0
J1,4 J 2,4 0 J1,1 0
F is the input forces vector acting on the wheels. It has 12 components (longitudinal, lateral and normal
(F
xi
, Fyi , Fzi ) forces for each one of the 4
wheels). Forces applied to wheel i are expressed in a frame attached to this wheel.
F = Fx1 , Fy1 , Fz1 , Fx 2 , Fy 2 , Fz 2 , Fx 3 , Fy 3 , Fz 3 , Fx 4 , Fy 4 , Fz 4 Γ represent extra inputs for perturbations. V ( q, q ) = ξ ( K v q + K p q ) + G ( q ) are suspensions and gravitation force: Respectively damping and stiffness matrices K v , K p . G ( q ) is the gravity term. ξ is assumed equal to unity when the corresponding wheel is in contact with the ground and zero if not.
Fig. 1. Vehicle dynamics and reference frames
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STA′2009 – Topics …, pages 1 à X
3. Estimation of the Slope of road Among the parameters that largely influence a vehicle's performance, road slope angle is the most important. It represents also necessary information for the systems of help to driving. This angle depends on the longitudinal dynamics of the vehicle. It is therefore important to estimate on line to the road slopes. The Figure (2) represents a vehicle in position of road slope α .
Fig. 2. Vehicle in a road slope
Fig.3. The effective ray of wheel
3.1. Modeling Like the slope angle depends primarily on longitudinal dynamics of vehicle, we use here the simple longitudinal model to estimate this angle [8]. 1 Fx − Cax ρ xVx2 − M v g sin(α ) M vVx = (3) 2 Jω T f − Re Fx = Where M v ,
Vx , Cax , ρ x , ω , T f and Re are respectively the vehicle mass, the
linear velocity, the drag coefficient, air density, the angular velocity of the considered wheel, the accelerating torque and the effective ray of wheel. (See Figure (3)) 3.2. Observer In this section we develop a First Order Sliding Mode observer to estimate the road slope. The estimation will be produced in three steps as cascade observers and estimation in order to reconstruct information and system states step by step. The first step produces estimations of velocities and accelerations of the wheels. The second one estimates the longitudinal tire forces and the last step reconstruct the slope. The wheels are assumed available for measurements. 1st Step: Estimation of velocities and accelerations By choosing x1 = ( x, ϕ ) and x= x= (Vx , ω ) , the previous First Order Sliding 2 1 Mode is useful for retrieval of the velocities and accelerations. This approach is robust
Paper title 5 versus the model and the parameters uncertainties for state estimation and is able to reject perturbations and uncertainties effects.
xˆ1= xˆ2 − Λ1sign( xˆ1 − x1 ) ˆ xˆ2= x2 − Λ 2 sign( xˆ1 − x1 )
(4)
Λ1 and Λ 2 are observer gains to be adjusted for convergence and sign is the vector of sign(i ) function (i = 1, 2) .
Where
For convergence analysis, we have to express the state estimation error xˆ1 − x1 ) and ( x= xˆ2 − x2 ) dynamics equation. Then the equations (4) give ( x= 1 2 the observation error dynamics: x1= x2 − Λ1sign( x1 ) x2= x2 − Λ 2 sign( x1 ) The Lyapunov function V1 = 1 x1T x1 , help to show that the sliding surface x1 2 attractive surface if V1 < 0 :
(5)
= 0 is
= V1 x1T ( x2 − Λ1sign( x1 ) )
(6)
If we choose Λ1 =diag (λ1i ) such as x2i < λ1i (for i
= 1, 2 ), the convergence in
finite time so x1
t0 for the system state is obtained: xˆ1 goes to x1 in finite time t0 ,
= 0 ∀ t > t0 . For the state x2 , we use the same technic of analysis to proof the
convergence in finite time t1 . 2nd Step: Estimation of
Fx
In the second step we can estimate the longitudinal force Fx . In the simplest way, assuming the input torques known, we use the following equation to estimate Fx :
T f − J ωˆ Fˆx = Re
(7)
3rd Step: Estimation of the Slope α After those estimations, their uses in the same time with the system equations allow us to retrieve the slope angle α as follows: 1 ˆ ˆ ˆ2 Fx − 2 Cax ρ xVx − M vVx α = arcsin Mvg
(8)
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STA′2009 – Topics …, pages 1 à X
3.3. Simulation results In order to validate our approach proposed observer, we give some realistic simulation results. In simulation, the state and forces are generated by use of a car simulator with Matlab-Simulink (see Figure (4)).
Fig. 4. Modular concept of the simulator The validation of this simulator was made for the laboratory LCPC of Nantes by an instrumented car (Peugeot 406). The figures (5a), (5b), (5c) and (5d) represent respectively the real and estimated angular velocity ω of the wheel, the longitudinal velocity Vx , the longitudinal force
Fx and the slope angle α . We note that a good conver-
gence for all parameters.
Fig. 5. Real and observed parameters
Paper title 7
4. Estimation of the Road Bank Angle
4.1. Modeling In this application, we use a simple model already used and presented in literature. Where χ , θ and θ x are respectively the road bank angle, the roll angle made between the vehicle body and the road, and the roll angle made between the vehicle body and the fixed reference frame R (see Figure (6)) [4][5][9], we have:
θ x= χ − θ
(9)
Then the lateral acceleration can be measured with an accelerometer by the following equation:
a y= ay + ,s
g sin( χ − θ )
(10)
The rotation movement for vehicle following the axis x is presented by the equation (11). Where J xx , Cr ,
K r are respectively the inertia moment around the axis x , the
rumble coefficient of vehicle and the reminder stiffness.
J xxθ + Crθ + K rθ= M v a y h + M v g sin −1 ( χ − θ )
(11)
The two equations (9) and (11) give:
J xx χ + Cr χ + K r χ = J xxθx + Crθx + K rθ x + M v a y , s h = By supposing χ
χ= 0 , we obtain: 1 χ J xxθx + Crθx + K rθ x + M v a y , s h ) = ( Kr
Fig. 6. Vehicle Roll Model
(12)
(13)
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STA′2009 – Topics …, pages 1 à X
4.2. Observer In this section we develop a First Order Sliding Mode observer to estimate the road bank angle. The estimation will be produced in two steps as cascade observers and estimation in order to reconstruct information and system states step by step. The first step produces estimations of the roll angle
θx
made between the vehicle body and the
fixed reference frame R. The second one estimates the road bank angle.
θx
1st Step: Estimation of
For estimate the roll angle, we suppose x3
= θ x and x4 = θx . Then by using the First
Order Sliding Mode observer, we obtain the following system:
xˆ3= xˆ4 − Λ 3 sign( xˆ3 − x3 ) ˆ xˆ4= x4 − Λ 4 sign( xˆ3 − x3 )
(14)
Λ 3 and Λ 4 are observer gains to be adjusted for convergence and sign is the vector of sign(i ) function (i = 1, 2) . For convergence analysis, we use the same technic (Lyapunov function) to proof the convergence of xˆ3 and xˆ4 in finite time t3 .
Where
2nd Step: Estimation of
χ
By using the equation (13) and in finite time t
> t3 , we can estimate the road bank
angle by the following equation:
χˆ =
(
1 J xx xˆ2 + Cr xˆ2 + K r xˆ1 + M v a y , s h Kr
)
(15)
4.3. Simulation results To validate the previous results, we use car simulator presented in section 3.3. Then with Λ 3 = 1.5 and Λ 4 = 1.8 , we obtain a good convergence of the roll angle θ x and its derivative
θx
presented by the Figures (7b) and (7c). We remark finally that
the estimated value of the road bank angle given converges towards the simulated value in Figures (7d).
Paper title 9
Fig. 7. Real and observed parameters
5. Conclusion In this paper, we have presented a nominal dynamic model of the vehicle and we developed a robust sliding mode observer to estimate the attributes of the road. The first observer for estimation slope of road is presented in three steps; estimation of velocities and accelerations, estimation of the longitudinal force and finally estimation of road slope. Then a second robust sliding mode observer presented to estimate road bank angle in two steps: estimation of roll angle then estimation of road bank angle. The simulation results involve a complete vehicle model to emphasize performance of the method.
References 1. B. Jaballah, N.K. M’Sirdi, A. Naamane, H. Messaoud: Model Splitting and First Order Sliding Mode Observers for Estimation and Diagnosis in Vehicle, in:12th IFAC Symposium on Control in Transportation Systems, CTS’09, pp: 407-412, September 2009, California, USA. 2. B. Jaballah, N.K. M’Sirdi, A. Naamane, H. Messaoud: "Estimation of Longitudinal and Lateral Velocity of Vehicle", in: 17th Mediterranean Conference on Control and Automation, MED'09, IEEE, pp:582-587, June 24-26, 2009, Thessaloniki, Greece. 3. N.K. M'sirdi, B. Jaballah, A. Naamane, H. Messaoud: "Robust Observers and Unknown Input Observers for estimation, diagnosis and control of vehicle dynamics", IEEE/RSJ International Conference on Intelligent RObots and Systems, Invited paper in the Workshop on Modeling, Estimation, Path Planning and Control of All Terrain Mobile Robots, IROS,
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http://wwwlasmea.univ-bpclermont.fr/MEPPC08/, pp: 49-57, September22th 2008, Nice, France. 4. A. Rabhi, K. N. M’sirdi, A. Naamane, B. Jaballah: Vehicle Velocity Estimation Using Sliding Mode Observers, in: 9th international conference on Sciences and Techniques of Automatic control and computer engineering, STA'08, Decembre2008, Sousse, Tunisia. 5. N.K. M'sirdi, A. Rabhi, L. Fridman, J. Davila, Y. Delanne: Second Order Sliding Mode Observer for Estimation of Velocities, Wheel Sleep, Radius and Stiffness, in: ACC, Proceedings of the American Control Conference Minneapolis, pp:3316-3321, June 14-16, 2006 , Minnesota, USA 6. N.K. M'Sirdi, A. Rabhi, Aziz Naamane: Vehicle models and estimation of contact forces and tire road friction". Invited paper KCINCO: 351-358, 2007. 7. N.K. M'Sirdi, L. H. Rajaoarisoa, J.F. Balmat, J. Duplaix: Modeling for Control and diagnosis for a class of Non Linear complex switched systems. Advances in Vehicle Control and Safety AVCS 07, February 8-10, 2007, Buenos Aires, Argentine. 8. Lingman Peter, Schmidtbauer Bent: Road Slope and mass estimation using Kalman Filtering". Vehicle System dynamics, ISSN: 0042-3114. 9. N.K. M'sirdi, A. Rabhi, N. Zbiri, and Y. Delanne: VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces. TMVDA’04. 3rd Int. Tyre Colloquium Tyre Models For Vehicle Dynamics Analysis August 30-31, 2004 University of Technology Vienna, Austria. 10. A. Rabhi and N.K. M’sirdi: "Vehicle State and Parameters Estimation Using Sliding Mode Observer" in: The 3rd International Conference on Advances in Vehicle Control and Safety AVCS’07, February 8th to 10th, 2007, Buenos Aires, Argentina. 11. A.Y. Ungoren, H. Peng, and H. E. Tseng: "Experimental verification of lateral speed estimation methods". In: Proceedings of AVEC 2002 6th Int. Symposium on Advanced Vehicle Control, pp: 361-366, 2002, Hiroshima, Japan.
