1
Second Order Sliding Mode Observer for Estimation of Road Profile A. Rabhi1 , N.K. M’sirdi1 , L. Fridman2 and Y. Delanne3 Abstract – This paper deals with an approach to estimate the road profile, by use of second order sliding mode observer. The method is based on a robust observer designed with a nominal dynamic model of vehicle. The estimation accuracy of observer has been validated experimentally using a trailer equiped with position sensors and accelerometers. Keywords – Road Profile, Vehicle dynamics, Sliding Modes observer, Robust nonlinear observers.
This paper is organized as follows: section 2 deals with the vehicle description and modelling. The design of the second order sliding mode observer is presented in section 3. Some results about the states observation and road profile estimation by means of the proposed method are presented in section 4. Finally, some remarks and perspectives are given and commented in a concluding section.
I. INTRODUCTION Road profile unevenness through road/vehicle dynamic interaction and vehicle vibration a ects safety (Tyre contact forces), ride confort, energy consumption and wear of tire. Thus, an overview of the road profile (over a wide distance) appears to be necessary to qualify the serviceability of a road pavement. The road profile unevenness is consequently a basic information for road maintenance management systems. For the purpose of road serviceability, surveillance and road maintenance, several profilometers have been developped. For instance, [1] have proposed a method based on direct measurements of the road roughness. However, some drawbacks of this method and some limitations of its capabilities have been pointed out in [2]. The Road and Bridges Central Laboratory in French (LCPC) has developped a Longitudinal Profile Analyser (LPA) [3]. It is equiped with a laser sensor to measure the elevation of road profile. A profilometer is an instrument used to produce series of numbers related in a well-defined way to the true profile [1]. However, this instrument produces biased and corrupted measures. Other geometrical methods using many sensors (distance sensor, accelerometers...) were also developped [4]. However, these methods depend directly on the sensors reliability and cost. It is worthwhile to mention that these methods do not take into consideration the dynamic behaviour of the vehicle. In a previous work, M’Sirdi and all [5][7][8] have presented an observer to estimate the road profile by means of sliding mode observers designed from a dynamic modelization of the vehicle. But in the previous method the vehicle rolling velocity is constant and steering angle is assumed zero. For estimation of the road profile, slope and inclinason are also neglected. The main contribution is here to extend this observer. 1 LSIS, CNRS UMR 6168. Dom. Univ. St Jérôme, Av Escadrille Normandie-Niemen 13397 Marseille France 2 UNAM Dept of Control, Division of Electrical Engineering,Faculty of Engineering, Ciudad Universitaria, Universidad Nacional Autonoma de Mexico, 04510, Mexico, D.F., Mexico 3 LCPC Nantes: Division ESAR BP 44341 44 Bouguenais Cedex
[email protected]
II. VEHICLE DYNAMIC MODEL In litterature, many studies deal with vehicle modelling [9][10][11]. The objective may be either confort analysis or design or increase of safety and maniability of the car. The dynamic equations of the motion of the vehicle body are obtained by applying the fundamental principle of mechanics. Xc
zf2
zf1
2pf
rf
r1
z4
Yc
m4
rr r
2
2pr B2
K2
Kr2
Karr Br2 u2
m3 Kf1
zr1
Bf1 u3
B1
K1 z2
m2
z3
Karf
B
Bf2 u4
Kf2
zr2
B3
K3
B4
K4
z1
m1 Kr1
Br1 u1
Fig. 1. vehicle modele
The system under consideration is a vehicle represented as depicted in figure 1. This vehicle is composed by a car body, four suspensions and four wheels. }, and ! represent the displacements of the vehicle body, roll angle, and pitch angle respectively. }l l = 1==4 is the displacement of the wheel l. £ ¤W X = x1 x2 x3 x4 is the vector of the unknown inputs wich characterizes the road profile and will be estimated.
2
When considering the vertical displacement along the vertical axis }, the dynamic of the system can be written as: P t¨ + F t˙ + Nt = DX (1) = ==
where (t> t) represent the velocities and accelerations vector respectively. P 5 U7×7 is the inertia matrix, F 5 U7×7 is related to the damping e ects, N 5 U7×7 is the springs sti ness vector (see Figure 1). The car body is assumed rigid. The matrix P , F, N and D are defined in appendix. t 5 U7 is the coordinates vector defined by: t = [}1 > }2 > }3 > }4 > > !]
(2)
III. ESTIMATION OF THE ROAD PROFILE The dynamical model (1) can be written in the state form as follows: ; { =t A A ? 1 {˙ 1 = {2 = (3) {˙ 2 = { ¨1 = t¨ = P 1 (F{2 N{1 + DX ) A A = | = {1 =
where the state vector { = ({1 > {2 )W = (t> t)W > and | = t (| 5 U7 ) is the vector of measured outputs of the system. |=
£
}1
Thus, we obtain: ½
}2
}3
}4
}
!
¤W
{˙ 1 = {2 = {˙ 2 = i ({1 > {2 ) +
(4)
(5)
with i ({1 > {2 ) = P 1 (F{2 N{1 )
(6)
The unknown input component is = P 1 DX
(7)
In order to estimate the state vector { and to deduce the unknown inputs vector X , we propose the following second order sliding mode observer [15]: { ˆ˙ 1 = { ˆ2 + }1 ˙{ ˆ2 ) + }2 ˆ2 = i (w> {1 > {
=
{ b2 = i (w> {1 > { ˆ2 ) + vljq({1 { b1 )
{ ˜˙ 1 = { ˜2 |˜ {1 |1@2 vljq(˜ {1 ) ˙{ ˆ2 ) vljq(˜ {1 ) ˜2 = I (w> {1 > {2 > {
ˆ2 = 0, are taken to The initial moment { ˆ1 = {1 and { ensures observer convergence.
(12)
Let us recall that I (w> {1 > {2 > { ˆ2 ) = i (w> {1 > {2 ) i (w> {1 > { ˆ2 ) + (w> {1 > {2 ) In our case, the system states are bounded, then the existence of a constant bound i + is ensured such that ˆ2 )| ? i + |I (w> {1 > {2 > {
(13)
{2 | 2ymax . holds for any possible w, {1 , {2 and |ˆ ypd{ and {pd{ are defined such that ;w 5 R+ ;{2 > {1 |{2 | ymax and {1 {max The state boundedness is true, because the mechanical system (5) is BIBS stable, and the control input x is bounded. The maximal possible acceleration in the system is a priori known and it coincides with the bound i + . In order to define the bound i + let us consider the system physical properties. We have: - pL P pL - fL P fL - nL P nL where p, f and n are the minimal respective eigenvalues and p, f and n the maximal ones. 1 Then we obtain pd{(P 1 ) = p L and i + can be written as 1 i + = (fymax + n{max ) (14) p Let and satisfy the following inequalities, where s is some chosen constant, 0 ? s ? 1 A
(9)
(11)
where { bl represent the observed state vector and , and are the observer gains. It is important to note that in a first step, input e ects on the dynamic are rejected by the proposed observer like a perturbation. Taking { ˜1 = {1 { ˆ1 and { ˜2 = {2 { ˆ2 we obtain the equations for the estimation error dynamics
(8)
where { ˆ1 and { ˆ2 are the state estimations, and the correction variables }1 and }2 are calculated by the super-twisting algorithm }1 = |{1 { ˆ1 |1@2 vljq({1 { ˆ1 ) }2 = vljq({1 { ˆ1 )=
We assume {1 available for measurement and we propose the following sliding mode observer: = p { b1 = { b2 + |{1 { b1 |vljq({1 { b1 ) (10)
+ q Ai >+
(+i )(1+s) 2 > i + (1s)
(15)
Theorem 1: The observer (8),(9) for the system (5) ensures the finite time convergence to estimate the system, i.e. (ˆ {1 > { ˆ2 ) $ ({1 > {2 )= The proof is given by Davila and Fridman in [16]. The previous observers ensures that in finite time we have { e2 = 0 then { ˜˙ 2 = i ({1 > {2 ) i ({1 > { ˆ2 ) + }2 = 0
(16)
3
Let us take a low pass filtering of }2 which is defined in equation 8 and 9, then we obtain in the mean average: = }2
IV. EXPERIMENTAL RESULTS In this section, we present some experimental results to validate our approach. Several trials have been done with a vehicle (Peugeot 406 of LCPC) equipped with di erent sensors.
(17)
Note that } 2 is the filtred version of }2 . In order to estimate the elements xl l = 1===4 of the unknown input vector X and according 7 we can write
with = given by:
£
1 = D11 Xx + E11 X˙ x ¤W 0 0 0 , and the matrices D11 and E11
1
6
5
nu1 9 p1 9 9 9 0 9 A11 =9 9 0 9 9 7 0
0
0
0
nu2 p2
0
0
ni 1 p3
0
0 0
0
ni 2 p4
: : : : : :, : : : 8
6
5
Eu1 9 p1 9 9 9 0 9 B11 =9 9 0 9 9 7 0
0
0
0
Eu2 p2
0
0
0 0
Ei 1 p3 0
0 Ei 2 p4
: : : : : : : : : 8
for l = 1==4 we have =
1l = dll xl + ell xl
(18)
where dll and ell are respectively the elements of D11 and E11 . To solve this system we can take an approach simpler that the one in [7] which uses a standard observer. We can write: ½ = xl = j(xl > 1l ) (19) 1l = k(xl ) with:
1 (dll xl + 1l ) ell The observer proposed here is the: j(xl > 1l ) =
(20)
Fig. 2. Vehicle used for experiments
Some tests were carried out at the Road and Bridges Centaral Laboratory (LCPC) test track with an instrumented vehicle. Measures have been acquired with the vehicle rolling at several speeds. The signal measured by a Longitudinal Profile Analyser (LPA) constitutes in this experiment our reference profile. The figure 3 shows the longitudinal vehicle speed variations.
=
{ bl = i (b {l > |el ) + l (|l |bl )
(21)
Let us not the the observation error: bl x el = xl x
(22)
The observation error dynamics is then obtained from equation (19) and (21). =
x el = j(e xl > 1l ) + l (e 1l ) The convergence is proved by the following Lyapunov condidate function: 1 2 (23) Yl = x e 2 l The time derivative of Y is then: =
=
Yl =x el x el from 20, we obtain: ¸ = 1 Yl =x el (dll x el + e 1l ) l (e 1l ) ell
(24) Fig. 3. Longitudinal velocity of the vehicle
(25)
and then as 1l is measured or reconstructured by a ob= server we choose: l = e1ll , on Y l ? 0,
The Figure 4 shows clearly that the estimated displacements of the four wheels converge quickly to the measured ones. The curves are superposed. The figure 5, presents the estimation errors for displacements of the wheels. The estimation error stay less than
4
Fig. 4. Displacements of weels: estimated and measured Fig. 5. Estimation errore for displacements of wheels
0> 5pp anyway. In the figure 6, we present the roll angle and the pitch angles. A good reconstruction of state enables the estimation of the unknown inputs of the system. Figure 7 presents both the measured road profile (coming fromthe LPA instrument) and the estimated one. We can observe that the estimated values are quite close the measured ones. V. CONCLUSION In this paper, we present enhancement of previously proposed method to estimate the road profile elevation based on second-order sliding-mode. The gains of the proposed observer are chosen very easily ignoring the system parameters. This observer is compared, using experimental data. This observer is better than the previous one in convergence and do not assume that velocity is constant. This is due to robustness of the second order Sliding mode observer wich allows better rejection of perturbation and then a better reconstruction of the unknown inputs. The latter reconstruction has been also enhanced. The estimation scheme build up using a Second Order Sliding Mode observers has been tested on experimental data (acquired with a P406 vehicle) and shown to be very e!cient. The experimental results prove e ectiveness and robustness of the proposed method. In our further investigations the estimations produced on line will be used to define a predictive control to enhance the safety.
[2] [3] [4] [5] [6] [7] [8]
[9] [10] [11]
[12]
[13]
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E.B. Spangler and W.J. Kelly, GMR Road Profilometer Method for Measuring Road Profile.
[14]
Meau- Fuh Hong, The Development of an Extensive-Range Dynamic Road Profile and Roughness Measuring System", May 1992. Vincent Legea "Localisation et détection des défauts d’uni dans le signal APL ", Bulletin de liason du laboratoire Central des Ponts et Chaussées, n 192> juillet août 1994. T.D Gillespie and al., Methodology for Road Roughness Profiling and Rut Depth Measurement, Federal Highway Administration Report FHWA/RD-87042, 1987 50 p. Nacer K. M’Sirdi. Observateurs robustes et estimateurs pour l’estimation de la dynamique des véhicules et du contact pneu route. JAA. Bordeaux, 5-6 Nov 2003 H. Imine, M’Sirdi, Y.Delanne, Adaptive Observers and Estimation of the Road Profile, SAE Word Congress, vehicle dynamics and simulation Mars 2003, pp. 175-180, Detroit, Michigan. H. Imine. Observation d’états d’un véhicule pour l’estimation du profil dans les traces de roulement. PHD thesis Université de Versailles, 2003. A. Rabhi, H. Imine, N. M’ Sirdi and Y. Delanne. Observers With Unknown Inputs to Estimate Contact Forces and Road Profile AVCS’04 International Conference on Advances in Vehicle Control and Safety Genova -Italy, October 28-31 2004 U. Kiencken, L. Nielsen. Automotive Control Systems. Springer, Berlin, 2000. A. Rabhi, N. K. M’sirdi, N. Zbiri and Y. Delanne. Modélisation pour l’estimation de l’état et des forces d’Interaction VéhiculeRoute, CIFA2004, Tunisie. R. Ramirez-Mendoza. Sur la modélisation et la commande des véhicules automobiles. PHD thesis Institut National Polytechnique de Grenoble, Laboratoire d Automatique de Grenoble 1997. J.P. Barbot, M. Djemai, and T. Boukhobza. Sliding Mode Observers; in Sliding Mode Control in Engineering, ser.Control Engineering, no. 11, W. Perruquetti and J.P. Barbot, Marcel Dekker: New York, 2002, pp. 103-130. G. Bartolini, A. Pisano, E. Punta, and E. Usai. "A survey of applications of second-order sliding mode control to mechanical systems", International Journal of Control, vol. 76, 2003, pp. 875-892. A. Pisano, and E. Usai, "Output-feedback control of an under-
5
Appendix 5 9 9 9 9 M=9 9 9 7
5
Fig. 7. Comparison between observers approach and LPA profile
[15]
[16] [17] [18]
[19] [20] [21] [22]
water vehicle prototype by higher-order sliding modes ", Automatica, vol. 40, 2004, pp. 1525-1531. J. Davila and L. Fridman. “Observation and Identification of Mechanical Systems via Second Order Sliding Modes”, 8th. International Workshop on Variable Structure Systems,September 2004, Espana J. Davila and L. Fridman., A Levant. "Second Order Sliding Mode Observer for Mechanical Systems", IEEE Transaction on Automatic Control, 50(11), 2005, pp. 1785-1789. A. Levant, "Sliding order and sliding accuracy in sliding mode control," International Journal of Control, vol. 58, 1993, pp 12471263. J. Alvarez, Y. Orlov, and L. Acho, "An invariance principle for discontinuous dynamic systems with application to a coulomb friction oscillator", Journal of Dynamic Systems, Measurement, and Control, vol. 122, 2000, pp 687-690. Y.B. Shtessel, I.A. Shkolnikov, and M.D.J. Brown, "An asymptotic second-order smooth sliding mode control", Asian Journal of Control, vol. 5(4), 2003, pp 498-504. A. Levant, "Robust exact di erentiation via sliding mode technique", Automatica, vol. 34(3), 1998, pp 379-384. A.F. Filippov, Di erential Equations with Discontinuous Righthand Sides, Dordrecht, The Netherlands:Kluwer Academic Publishers; 1988. V. Utkin, J. Guldner, J. Shi, Sliding Mode Control in Electromechanical Systems, London, UK:Taylor & Francis; 1999.
0 p2 0 0 0 0 0
0 0 p3 0 0 0 0
0 0 0 p4 0 0 0
0 0 0 0 p 0 0
0 0 0 0 0 M{{ 0
0 0 0 0 0 0 M||
6 : : : : : : : 8
(E1 + Eu1 ) 0 0 0 (E2 + Eu2 ) 0 0 0 (E3 + Ei 1 ) 0 0 0 E1 E2 E3 E1 su E2 su E3 si E1 u2 E2 u2 E3 u1 6 0 E1 F16 F17 0 E2 F26 F27 : : 0 E3 F36 F37 : : (E4 + Ei 2 ) E4 F46 F47 : : E4 F55 F56 F57 : : E4 si F65 F66 F67 8 E4 u1 F75 F76 F77 5 n1 + nu1 0 0 9 0 n + n 0 2 u2 9 9 0 0 n3 + ni 1 9 0 0 0 K= 9 9 9 n1 n n 2 3 9 7 n1 su n2 su n3 si n1 u2 n2 u2 n3 u1 6 0 n1 n1 su n1 u2 0 n2 n2 su n2 u2 : : 0 n3 n3 si n3 u1 : : n4 + ni 2 n4 n4 si n4 u1 : : n4 N55 N56 N57 : : n4 si N65 N66 N67 8 n4 u1 N75 N76 N77
9 9 9 9 C= 9 9 9 9 7
Fig. 6. Estimation of the roll and the pich angles
p1 0 0 0 0 0 0
