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Road profile inputs for evaluation of the loads on the wheels H. Imine

a b

b

, Y. Delanne & N.K. M'sirdi

a

a

Laboratoire de Robotique de Versailles, Université de Versailles, 10 avenue de l’Europe, 78140, Vélizy, France b

Laboratoire Central des Ponts et Chaussées, Centre de Nantes, route de Bouaye, BP 4129-44341, 44, Bouguenais Cedex, France Published online: 12 Oct 2011.

To cite this article: H. Imine , Y. Delanne & N.K. M'sirdi (2005) Road profile inputs for evaluation of the loads on the wheels, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 43:sup1, 359-369, DOI: 10.1080/00423110500108945 To link to this article: http://dx.doi.org/10.1080/00423110500108945

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Vehicle System Dynamics Vol. 43, Supplement, 2005, 359–369


Road profile inputs for evaluation of the loads on the wheels H. IMINE*†‡, Y. DELANNE‡ and N.K. M’SIRDI† †Laboratoire de Robotique de Versailles, Université de Versailles, 10 avenue de l’Europe, 78140 Vélizy, France ‡Laboratoire Central des Ponts et Chaussées, Centre de Nantes, route de Bouaye, BP 4129-44341, 44 Bouguenais Cedex, France

This paper presents a method for estimating the loads on the wheels using road profiles. Regarding road profiles, a new method based on sliding mode observers has been developed and is compared with longitudinal profile analyser measurements. Experimental results are shown and discussed to evaluate the robustness of our approach.

Keywords: Road profile; Longitudinal profile analyser; Vehicle modelling; Loads on the wheels; Sliding-mode observers

1.

Introduction

The dynamics of a vehicle are directly dependent on the tyre–road contact forces and torques which are themselves dependent on loads on the wheels and tyre–road friction characteristics. To evaluate the frictional forces and torques in the tyre contact patch well, it is necessary to evaluate the wheel loads accurately. This can be achieved only if relevant road profiles are input to the vehicle dynamic model. For the purpose of road serviceability, survey and road maintenance, several profilometers have been developed in a recent European program called FILTER. Some of these have proved to give reliable measurements compared with the profiles obtained with the reference device [1]. In this paper, we present two methods to evaluate the road profile, namely the longitudinal profile analyser, which is an instrument developed by the Laboratoire Central des Ponts et Chaussées (LCPC) in the 1960s [2], and a recently developed robotic approach based on sliding-mode observers [3, 4]. The objective of this research in that regard was to develop an easily implemented method based on the dynamic response of a vehicle instrumented with cheap sensors so as to give an accurate estimation of the profile along the actual wheel tracks. The method based on *Corresponding author. Email: [email protected]

Vehicle System Dynamics ISSN 0042-3114 print/ISSN 1744-5159 online © 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00423110500108945

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sliding-mode observers considers the road profile as unknown inputs of the vehicle dynamic system to be estimated. Consequently, the loads on the wheels are evaluated. In this work, some experimental results related to the estimation of the road profile by these methods are reported and discussed to evaluate the robustness of our approach. A comparison of their relevance to obtaining a good estimate of loads on the wheels is carried out firstly from the dynamic model developed in the framework of the observers approach and secondly in a validation procedure comparing the measured and computed dynamic responses of an instrumented vehicle.

2. The longitudinal profile analyser In this section we present an instrument to measure the road profile, namely the longitudinal profile analyser (LPA). This system includes one or two single-wheel trailers towed at constant speed by a car and employs a data acquisition system. A ballasted chassis supports an oscillating beam holding a feeler wheel that is kept in permanent contact with the pavement by a suspension and damping system. The chassis is linked to the towing vehicle by a universaljointed hitch. Vertical movements of the wheel result in angular travel of the beam, measured with respect to the horizontal arm of an inertial pendulum, independently of movements of the towing vehicle (figure 1). This measurement is made by an angular displacement transducer associated with the pendulum; the induced electrical signal is amplified and recorded. Rolling surface undulations in a range of ±100 mm are recorded with wavelengths in ranges from 0.5–20 m to 1–50 m, depending on the speed of the vehicle [5]. This device has proved to give very precise measurements of profile elevation. Rough measurements have to be processed to obtain a reliable estimation of the road profile in the measured waveband (phase distortion correction).

3.

Estimation of the road profile

To implement the sliding-mode method, a vehicle model must be assumed [6, 7].

Figure 1.

LPA.

Road profile inputs for evaluation of loads on wheels

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3.1 Vehicle modelling The vehicle model is shown in figure 2. In this section, we are interested in the excitations of pavement and the vehicle–road interaction [15–18]. The model is established while making the following simplifying hypotheses.


(i) The vehicle is rolling with constant speed. (ii) The wheels are rolling without slip and without contact loss.

3.1.1 Vertical model. The vertical motion of the vehicle model can be described by the following equation: M q¨ + B˙q + Kq = [ζ T 0 0 0 0]T , (1) q ∈ R8 is the coordinate’s vector defined by q = [z1 , z2 , z3 , z4 , z, θ, φ, ψ]T ,

(2)

where zi , i = 1, . . . , 4, is the displacement of the wheel i. The variables z, θ , φ and ψ represent the displacement of the vehicle body, the roll angle, the pitch angle and the yaw angle respectively. q˙ and q¨ represent the velocities and accelerations vectors respectively. M ∈ 8×8 is the inertia matrix given by M1 0 , M= 0 M2 with M1 = diag(m1 , m2 , m3 , m4 ) and M2 = diag(m, Jx , Jy , Jz ), where mi , i = 1, . . . , 4, represent the mass of wheel i, coupled to the chassis with mass m. Jx , Jy and Jz are the inertia moments of the vehicle along the x, y and z axes, respectively.

Figure 2.

Full-car model.

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B ∈ 8×8 is related to the damping effects: B B = 11 B21

B12 , B22

where B11 = diag(B1 + Br1 , B2 + Br2 , B3 + Bf1 , B4 + Bf2 ) is a diagonal positive matrix and     −B2 −B3 −B4 −B1 C16 C17 0 −B1 −B2 C26 C27 0 B1 pr −B2 pr B3 pf −B4 pf  ,   B12 =  −B3 C36 C37 0 , B21 =  B1 r1 B2 r2 −B3 r1 −B4 r1  −B4 C46 C47 0 C81 C82 C83 C84   C55 C56 C57 0 C65 C66 C67 0  . B22 =  C75 C76 C77 0  C85 C86 C87 C88 The elements of these matrices are defined in appendix A. K ∈ 8×8 is the springs stiffness vector given by K11 K12 , K= K21 K22 where K11 = diag(k1 + kr1 , k2 + kr2 , k3 + kf1 , k4 + kf2 ) is a diagonal positive matrix and     k 1 r2 0 −k2 −k1 k1 pr −k1 −k3 −k4 −k2 −k2 pr k2 r2 0 k1 pr −k2 pr k3 pf −k4 pf   ,  K12 =  −k3 k3 pf −k3 r1 0 , K21 =  k1 r2 k2 r2 −k3 r1 −k4 r1  −k4 −k4 pf −k4 r1 0 K81 K82 K83 K84   0 K55 K56 K57 K65 K66 K67 0  . K22 =  K75 K76 K77 0  K85 K86 K87 K88 The elements of these matrices are defined in appendix A. ˙ let us define the variable ζ as In order to estimate the unknown vectors U and U, ˙ ζ = CU + DU,

(3)

where ζ ∈ 4 and U = [u1 , u2 , u3 , u4 ]T is the vector of unknown inputs which characterize the road profile. The system outputs are the displacements of the wheels and the chassis, corresponding to signals measured by the vehicle sensors. The matrices M, B, K, C and D are defined in appendix A. 3.1.2 Longitudinal model. The longitudinal force Fx is given by the following equation [8]: (4) Fxi = µi Fzi , where µi is the adhesion coefficient and Fzi represents the vertical tyre force. The variation in Fzi can be calculated using the following formula: Fzi = mi g + Kri (zri − ui ) + Bri (˙zri − u˙ i ), i = 1, . . . , 4, where i refers to the position of the wheel.

(5)


3.2

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Observer design

For our study, we put the model (1) in state equation form while taking as state vector x1 = q T T T ˙ φ, ˙ ψ] ˙ T . then we obtain and x2 = q˙ = (x21 , x22 ) where x21 = [˙z1 , z˙ 2 , z˙ 3 , z˙ 4 ]T and x22 = [˙z, θ, x1 = q, x˙ 1 = x2 ,


x˙ 21 = −M−1 1 (B11 x21 + B12 x22 + K11 x1 + ζ ),

(6)

x˙ 22 = −M−1 2 (B21 x21 + B22 x22 + K22 x1 ), y = [xT1 , xT22 ]T , The matrices K11 and K22 are defined in 4×8 . Before developing the sliding-mode observer [9, 10], let us consider the following hypotheses. (i) The state is bounded (x(t) < ∞∀t ≥ 0). ˙ < µ). (ii) The system is the inputs bounded (∃ a constant µ ∈ 4 such as U (iii) The vehicle rolls at a constant speed on a defect road of the order of millimetres, without bumps. The structure of the proposed observer is triangular [11] having the following form: x˙ˆ 1 = xˆ 2 + H1 sgn1 (˜x1 ), ˆ 21 + B12 x22 + K11 xˆ 1 + ζˆ ) + H21 sgn2 (¯x21 − xˆ 21 ), x˙ˆ 21 = −M−1 1 (B11 x

(7)

where xˆ i represents the observed state vector and ζˆ is the estimated value of ζ . The mean average of the variable x¯ 2 is given by x¯ 2 = xˆ 2 − H1 sgn1 (˜x1 ),

(8)

H1 ∈ R8×8 , H21 ∈ R4×4 and H22 ∈ R4×4 represent positive diagonal gain matrices. sgneq1 is the equivalent of the signum function in the slide surface (˜x1 = x1 − xˆ 1 = 0) [12]. The dynamics estimation errors are given by x˙˜ 1 = x˜ 2 − H1 sgn1 (˜x1 ), ˜ 21 + K11 x˜ 1 + ζˆ ) + H21 sgn2 (¯x21 − xˆ 21 ). x˙˜ 21 = −M−1 1 (B11 x

(9)

3.3 Convergence analysis In order to study the observer stability and to find the gain matrices Hi , i = 1, . . . , 2, we proceed, step by step, starting to prove the convergence of x˜ 1 to the sliding surface x˜ 1 = 0 in finite time t1 . Then we deduce some conditions about x˜ 2 to ensure its convergence towards zero. We consider the following Lyapunov function: V1 =

1 T x˜ • x˜ 1 2 1

(10)

It can be easily shown that, if hi1 > |˜xi2 |, i = 1, . . . , 8, then V˙1 < 0. Then the variable x˜ 1 converges towards zero in finite time t0 . We obtain in the sliding surface

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x˙˜ 1 = x˜ 2 − H1 sgneq1 (˜x1 ) = 0 ⇒ x˜ 2 = H1 sgneq1 (˜x1 ). Thus, according to equation (8), we have (11) x˜ 2 = x2 . Then, we obtain x¯ 21 = x21 (x22 is measured). The system (9) becomes x˙˜ 1 = 0,


˜ 21 + ζ˜ ) − H21 sgn2 (˜x21 ). x˙˜ 21 = −M−1 1 (B11 x

(12)

The second step is to study the convergence of xˆ 2 . To study the convergence of xˆ 21 , first consider the following Lyapunov function: V21 =

1 T x˜ · M1 x˜ 21 . 2 21

(13)

Deriving this function, we obtain V˙21 = −˜xT21 • B11 x˜ 21 + x˜ T21 · ζ˜ − x˜ T2 · M1 H2 sgn(˜x2 ).

(14)

Since ζ˜ is bounded and during this step the first condition stays true (x˙˜ 1 = 0) and B11 is a diagonal definite matrix, so we have, when choosing the terms of the matrix H2 to be very high, V˙21 < 0. Therefore, the variable x˜ 21 converges towards zero in finite time t1 > t0 and then x˙˜ 2 = 0 [13]. The system (9) becomes ∀t > t1 x˙˜ 1 = 0 ˜ x˙˜ 21 = M−1 x21 ) = 0 1 ζ − H21 sgn2 (˜

(15)

The estimation of the unknown vector ζ˜ is obtained according to equation (15). We have then ζ˜ = ζ − ζˆ = M−1 x21 ). 1 H21 sgn(˜

(16)

Finally, we obtain the variable ζ : x21 ). ζ = ζˆ + M−1 1 H21 sgn(˜

(17)

In order to estimate the elements ui , i = 1, . . . , 4, of the unknown vector U and according to equation (3), we resolve the following equation: ζ = CU + D

dU . dt

(18)

When we consider the initial conditions U(t = 0) = 0, we obtain from equation (15) the unknown input vector U so that ui =

ζi (1 − e(ci /di )t , ci

i = 1, . . . , 4,

(19)

where ci = kri , i = 1, . . . , 4, and di = Bri , i = 1, . . . , 4, are the elements of the matrices C and D given in appendix A. We have discussed in this paper, two methods to estimate the road profile, namely the LPA measure and the method using sliding-mode observers. In the next section, we compare the results obtained using these methods.



Figure 3.

Displacements of wheels: estimated and measured.

Figure 4.

Estimation of displacement of body and yaw angle.

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Figure 5.

Comparison between the observers approach and the LPA profile.

3.4 Estimation results Some tests were carried out at LCPC test track with an instrumented car towing two LPA trailers at a constant speed of 72 km h−1 . The signal measured by an LPA constitutes in this experiment our reference profile. Figure 3 shows clearly that the estimated displacements of the four wheels converge quickly to the measured displacements. In the upper two plots of figure 4, we present respectively the vertical displacement z and the yaw angle ψ of the chassis. The lower two plots of this figure represent the velocities. We

Figure 6.

Friction coefficient and load on the wheel.


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can see that the estimated vertical velocity (˙z) is very close to the measured signal. However, the estimation of ψ˙ is not very good. A good reconstruction of states enables the unknown inputs of the system to be estimated. Figure 5 presents both the measured road profile (from the LPA) and the estimated profile. We can then observe that the estimated values are quite close to the true values.


4.

Evaluation of the loads on the wheels

From equations (19) and when the unknown inputs ui , i = 1, . . . , 4, are estimated, we can deduce the loads on the wheels using equations (5). Figure 6 represents the friction coefficient and the load respectively on the wheel.

5.

Conclusions

In this paper, we developed a method to estimate the road profile elevation based on slidingmode observers. Compared with the LPA signal, our estimation is correct. It has been shown that by estimating the road profile, we can deduce the load on the wheels. Regarding our objective in this work, we consider that, if the output vector (vertical acceleration displacement of the wheels and vertical and rotational movement of the vehicle body) is accurate, the sliding-mode observers method constitutes a reliable and easily implemented method to estimate the road profile. Consequently, we have a good estimation of the load on the wheels.

References [1] http://www.vti.se [2] Legeay, V., 1994, Localisation et détection des défauts d’uni dans le signal APL. Bulletin de Liaison du Laboratoire Central des Ponts et Chaussées, 192. [3] Imine, H., 2003, Observation d’états d’un véhicule pour l’estimation du profil dans les traces de roulement. PhD thesis, l’Université de Versailles Saint Quentin en Yvelines. [4] Imine, H., Rabhi, H., M’Sirdi, N.K. and Delanne, Y., Observers with unknown inputs to estimate contact forces and road profile. Paper presented at the International Conference on Advances in Vehicle Control and Safety, 28–31 October 2004, Genoa, Italy. [5] Legeay, V., Daburon, P. and Gourraud, C., 1996, Comparaison de mesures de l’uni par l’analyseur de profil en long et par compensation dynamique. Bulletin de Liaison du Laboratoire Central des Ponts et Chaussées. [6] Ellis, J.R., 1960, Vehicle Handling, Vehicle Dynamics (Norwich: Page Bros). [7] Gwangun, G. and Nikravesh, E.N., 1990, An analytical model of pneumatic tyres of vehicle dynamic simulation. Part 1: pure slips. International Journal of Vehicles, 11, 589–618. [8] Bachmann, T., 1995, The importance of the integration of road, tyre and vehicle technologies. Paper presented at the 20th World Road Congress, Workshop on the Synergy of Road, Tyre and Vehicle Technologies, Montreal, Canada, 5 September 1995. [9] Drakunov, S.V., 1992, Sliding-mode observers based on equivalent control method. In Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, Arizona, USA 1992 (New York: IEEE), pp. 2368–2369. [10] Slotine, J.J.E., Hedrick, J.K. and Misawa, E.A., 1987, On sliding observer of nonlinear systems. Journal of Mathematical System, Estimation and Control, 109, 245–259. [11] Barbot, J.P., Boukhobza, T. and Djemai, M., 1996, Triangular input observer form and sliding mode observer. In Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996 (New York: IEEE), pp. 1489–1491. [12] Utkin, V.I., 1977, Variable structure systems with sliding mode. IEEE Transactions on Automatic Control, 26, 212–222. [13] Utkin, V.I. and Drakunov, S., Sliding mode observer. In Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, Florida, USA, 1994 (New York: IEEE), pp. 3376–3378.

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Appendix




kr1 0  0 kr2 C= 0 0 0 0  k1 + kr1  0   0   0 K=  −k1   k1 pr   k1 r2 K81

0 0 kf1 0

 0 0 , 0 kf2

0 k2 + kr2 0 0 −k2 −k2 pr k 2 r2 K82



0 Br2 0 0

Br1  0 D=  0 0

0 0 k4 + kf1 0 −k3 k 3 pf −k3 r1 K83

0 0 0 k4 + kf2 −k4 −k4 pf −k4 r1 K84

0 0 Bf1 0

 0 0  , 0  Bf2

−k1 −k2 −k3 −k4 K55 K65 K75 K85

k1 p r −k2 pr k3 p f −k4 pf K56 k66 K76 K86

k 1 r2 k 2 r2 −k3 r1 −k4 r1 K57 K67 K77 K87

 0 0   0   0  . 0   0   0  K88

C17 C27 C37 C47 C57 C67 C77 C87

 0 0   0   0  , 0   0   0  C88

The elements of this matrix are given by K55 = k1 + k2 + k3 + k4 , K56 = [(k1 − k2 )pr + (k3 − k4 )pf ], K57 = [(k1 + k2 )r2 + (k3 + k4 )r1 ], K65 = [(k1 − k2 )pr + (k3 − k4 )pf ], K66 = (k1 + k2 + k3 + k4 )pf pr + (karr + karf ), K67 = (k1 − k2 )r2 pr + (k3 − k4 )r1 pf , K75 = [(k1 − k2 )r2 + (k3 + k4 )r1 ], K76 = [−(k1 − k2 )r2 pr + (k3 − k4 )r1 pf ], K77 = (k1 + k2 )r22 + (k3 + k4 )r12 , 

B1 + Br1  0   0   0 B=  −B1   B1 pr   B1 r2 C81

0 B2 + Br2 0 0 −B2 −B2 pr B2 r 2 C82

0 0 B3 + Bf1 0 −B3 B3 p f −B3 r1 C83

where C16 = B1 pr cos(θ ), C17 = B1 r2 cos(φ), C26 = −B2 pr cos(θ ), C27 = B2 r2 cos(φ), C36 = B3 pf cos(θ ),

0 0 0 B4 + Bf2 −B4 −B4 pf −B4 r1 C84

−B1 −B2 −B3 −B4 C55 c65 C75 C85

C16 C26 C36 C46 C56 C66 C76 C86


C37 = B3 r1 cos(φ), C46 = B4 pf cos(θ ), C47 = −B4 r1 cos(φ), C55 = B1 + B2 + B3 + B4 , C56 = [(B1 − B2 )pr + (B3 − B4 )pf ] cos(θ ),


C57 = −[(B1 + B2 )r2 + (B3 + B4 )rf ] cos(φ), C65 = [(B1 − B2 )pr + (B3 − B4 )pf ], C66 = −(B1 + B2 + B3 + B4 )pf pr cos(θ ), C67 = −[−(B1 − B2 )r2 pr − (B3 − B4 )r1 pf ] cos(φ), C75 = [(B1 + B2 )r1 − (B3 + B4 )r2 ], C76 = −[−(B1 − B2 )r2 pr + (B3 − B4 )r1 pf ] cos(θ ), C77 = [(B1 + B2 )r22 + (B3 + B4 )r12 ] cos(φ), C88 =

2(Cyf r12 + Cyr r22 ) . v

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