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Int. J. Heavy Vehicle Systems, Vol. 14, No. 3, 2007

Rollover risk prediction of heavy vehicle in interaction with infrastructure H. Imine* and V. Dolcemascolo Laboratoide Central des Ponts et Chaussées, 58, bld Lefebvre, 75732 Paris Cedex 15, France Fax: +33 01 40 43 54 99 E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: This paper presents a concept of a warning system to prevent rollover of heavy vehicles. The system consists of several sensors which measure the heavy vehicle dynamics. Initially, these dynamics states are observed using observers, and then, knowing the infrastructure characteristics, the risk of rollover is predicted and assessed using the lateral acceleration of the vehicle, the height of the centre of gravity, and the Load Transfer Ratio (LTR). This coefficient is calculated using the estimated load on each wheel. Keywords: rollover; warning system; heavy vehicle; sliding mode observers; loads. Reference to this paper should be made as follows: Imine, H. and Dolcemascolo, V. (2007) ‘Rollover risk prediction of heavy vehicle in interaction with infrastructure’, Int. J. Heavy Vehicle Systems, Vol. 14, No. 3, pp.294–307. Biographical notes: Hocine Imine received his Diploma and his PhD in Robotics and Automation from the Versailles University, France, in 1996 and 2003 respectively. From 2003 to 2004, he was an Assistant Professor at the Versailles University. He was involved in National projects like ARCOS (Action de recherche pour une Conduite Sécurisée). In 2005, he joined the Central laboratory of roads and bridges (LCPC in French, Laboratoire Central des Ponts et Chaussées), France, as a researcher. He is involved in different projects related to vehicle modelling, interaction between trucks and infrastructure, transport safety. Vittorio Dolcemascolo works in the Division for Road Operation, Signalling and Lighting. He graduated from Ecole Nationale Supérieure des Télécommunications de Bretagne. He has been working for 15 years in the field of Weigh In Motion (WIM), especially in the design of multisensor WIM systems.

1

Introduction

In recent years, many authors have studied the problem of the rollover of heavy vehicles (Wang and Tomizuka, 1999; Chen and Tomizuka, 1995; Sanchez et al., 2004). However the interaction with the infrastructure is not taken into account. The originality of our method developed in this paper is to consider the infrastructure data such as road profile, Copyright © 2007 Inderscience Enterprises Ltd.

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radius of curvature, lateral inclination of the road, etc., to show their influence on the proposed warning system. In this study, we present an observer to estimate the dynamics of a heavy vehicle in interaction with the infrastructure and an estimator for vertical loads (Gillespie and Karamihas, 1993; Imine et al., 2005a). The designed observer is based on the sliding mode approach (Utkin and Drakunov, 1995; Slotine et al., 1987). If a high risk is detected, a warning is sent to the driver to prevent the rollover. Several simulations with various scenarios are presented, performed with the PROSPER simulator (www.sera-cd.com), to assess the accuracy of our method and then the efficiency and reliability of the whole system. Simulations are presented on the estimation of vertical forces, and discussed to evaluate the robustness of our approach and to assess the impact of the infrastructure. This paper is organised as follows: Section 2 deals with the heavy vehicle description and modelling. The design of the observer to estimate the rollover risk is presented in Section 3. Some results about the states observation, loads estimation and rollover risk evaluation are presented in Section 4. Finally, some remarks and perspectives are given in the concluding section.

2

Model description

Different studies have reported heavy vehicle modelling that did not take into account the infrastructure data (Bouteldja, 2005; Ibrahim, 2004; Allen et al., 1992). However, these data are very important to study risks such as rollover or jackknifing. In this work we develop a heavy vehicle model using the infrastructure data (road profile, radius of curvature, lateral road inclination) as inputs of the system (Cebon, 1993; Gillespie et al., 1992) (see Figure 1). Figure 1

Heavy vehicle model on the road with the lateral inclination

These infrastructure data are taken from measures done at Lyon (in France) within the framework of a European project. The road profile inputs are measured using LPA (Longitudinal Profile Analyser) system (Legeay et al., 1994; Imine et al., 2005b) developed at LCPC and presented in the Figure 2.

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Figure 2

Longitudinal Profile Analyser (APL in French)

In this paper, we consider the tractor-semitrailer model (with two axles for the trailer and one axel for the semitrailer) presented in the Figure 3 which has 12 Degrees of Freedom (DOF). This model is derived using Lagrangian equations. Figure 3

Heavy vehicle model

In this paper, we neglect the relative yaw angle and the longitudinal movement of the heavy vehicle. Then, we can define a dynamic model of the vehicle as: M (q )q + C (q, q)q + K (q ) = Fg

(1)

where M ∈ ℜ11×11 is the inertia matrix (mass matrix), C ∈ ℜ11×11 is related to the damping effects, K ∈ ℜ11 is the spring stiffness vector and Fg ∈ ℜ11 is a vector of generalised forces. q ∈ ℜ11 is the coordinates vector defined by: q = [q1 , q2 , q3 , q4 , q5 , q6 , x, y, z, φ , ψ ]T

(2)

q and q represent respectively the speed and acceleration vector, φ is the tractor roll angle, ψ is the yaw angle of the tractor, q1, q2 are respectively the left and right front suspension deflection of the tractor, q3, q4 are respectively the left and right rear suspension deflection of the tractor, q5, q6 are respectively the left and right suspension deflection of the semi-trailer, z is the vertical displacement of the tractor sprung mass (height of the gravity centre), y is the lateral displacement of the tractor.

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The suspension is modelled as the combination of a non-linear spring and damper elements. The tractor chassis (with the mass M) is suspended on its axles through two suspension systems. We can also shown that the tyre is modelled by spring and damper elements. The road profile inputs, which are considered as the heavy vehicle inputs. zri, i = 1 … 6 represent the vertical displacements of the wheels with respect to coordinate system fixed to the ground (road). These can be calculated using the following equations: Tw sin(φ ) − r cos(φ + ς ) 2 T zr 2 = z + q2 − w sin(φ ) − r cos(φ + ς ) 2 zr1 = z − q1 −

(3)

where Tw is the tractor track width and r is the wheel’s radius. The lateral road inclination is represented by ς. The normal forces Fni, i = 1 … 6 acting on the wheels are calculated using the following expression: Fni = Fci + ki (ui − zri ), i = 1… 6

(4)

where Fci is the static load and ui is the road profile input under the wheel i. In this study, we suppose that the force generated by damping effect is neglected comparing to the spring forces. The dynamic rolling of the vehicle is described using the following differential equations. I xxφ = ma y h + mg sin(φ + ς ) − CRφ − K Rφ

(5)

where Ixx is the inertia moment in the roll axis, CR represents a damping coefficient of the roll motion, KR is spring coefficient of the roll motion, φ is the roll rate and φ is the roll acceleration according to the road. h is the centre height of gravity in relation to the roll axis (see Figure 1). g represents the gravity acceleration and ay is the lateral acceleration of the heavy vehicle centre. This can be calculated by the following equation: a y = vψ + g sin(φ + ς )

(6)

where v is the vehicle speed and ψ is the yaw rate.

3

Rollover risk estimation

In this section, we develop the triangular sliding mode observers to evaluate the heavy vehicle states and the vertical loads. This observer allows to perform a finite time convergence of the positions and speeds (see more details in Barbot et al., 1996; Imine and Delanne, 2005). The Load Transfer Ratio (LTR) is depending on the load on each wheel and calculated as follows:

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Fnr − Fnl . Fnr + Fnl

(7)

Many works (Bouteldja, 2005) have shown that this coefficient can be approximated (we assume that the roll angle φ is small) by the following expression: R=

2(h + hR ) a y h + φ T g T

(8)

where hR is the roll axis height in relation to the ground (see Figure 1). When Fnr = 0 (Fnl = 0) the right (left) wheels lift off the road and the rollover coefficient takes on the value R = –1 (R = 1). For straight driving on a horizontal road, the tyres vertical forces are equal for the left and right wheel, then we obtain Fnr = Fnl which means that R = 0. The rollover occurs when the moment created by the lateral acceleration acting on the vehicle exceeds a threshold value ayc. This instantaneous value is depending on the vehicle parameters, centre of gravity height and infrastructure database. In this paper, the critical LTR value is about 0.9. Then ayc is calculated from the equation (8) as following: a yc =

(0.9Tg − 2 ghφ ) 2(h + hR )

.

(9)

To evaluate the LTR, we must, first, estimate the heavy vehicle states and the vertical wheels forces. Then we write the dynamic model (equation (1)) in the state form as follows:  x = f ( x) + Fg   y = h( x )

(10)

where the state vector x = ( x1 , x2 )T = (q, q)T , and y = q and y = q is the vector of measured outputs of the system. Thus, we obtain:  x1 = x2  −1  x2 = q = M ( Fg = C ( x1 , x2 ) x2 = K ( x1 )).   y = x1

(11)

Before developing the sliding mode observer, let us consider the following assumptions: •

the state is bounded ||x(t)|| < ∞, ∀ t ≥ 0

•

the system is the inputs bounded (∃ a constant µ ∈ ℜ such as: ui < µ)

•

the generalised forces Fg are bounded (∃ a constant η ∈ ℜ such as: Fgi < η)

•

in the vector C(x1, x2)x2, we find quadratics elements in x2. We can then bounded this vector as:

Rollover risk prediction of heavy vehicle in interaction with infrastructure 2

C ( x1 , x2 ) x2 ≤ c1 x2 .

299 (12)

This last assumption arises from the mechanical and physical properties of our system and because of the real signals are bounded (position, speed, acceleration).

3.1 Triangular observer design In order to estimate the state vector x and to deduce both the vertical loads vector Fni, we propose the following triangular sliding mode observer (Imine, 2003; Imine et al., 2005a):  xˆ1 = xˆ2 + H1sign( x1 )  −1  xˆ2 = M ( Fg − C ( xˆ1 , xˆ2 ) xˆ2 − K ( xˆ1 )) + H 2sign( x2 − xˆ2 )

(13)

where xˆi represents the observed state vector and: x2 = xˆ2 + H1sign( x1 )

(14)

H1 ∈ ℜ11×11 and H2 ∈ ℜ11×11 represent positive diagonal gain matrices. Let us now define another observer to estimate the vertical loads vector. It has the following form: Fˆni = Fci + f ( xˆ1 ).

(15)

3.2 Convergence analysis We can write the dynamics estimation errors as following:  x1 = x2 − H1sign( x1 )  −1 −1  x2 = M Fˆg − M (C ( x1 , x2 ) x2 − C ( xˆ1 , xˆ2 ) xˆ2 ) .  − M −1 ( K ( x1 ) − K ( xˆ1 )) − H 2 sign( x2 − xˆ2 ) 

(16)

In order to study the observer stability and to find the gain matrices H1 and H2, first, we have to prove the convergence of x1 to the sliding surface x1 = 0, in finite time t1. Then, we deduce some conditions about x2 to ensure its convergence towards 0. Let us consider the following Lyapunov function V1 = (1/ 2) x1T x1 . The time derivative of this function is given by: V1 = x1T ( x2 − H1sign( x1 )).

(17)

By considering gains matrix H1 = diag(hi1) with hi1 > | xi 2 |, i = 1 … 8, then V1 < 0. Therefore, from sliding mode theory (Drakunov, 1992), the surface defined by x1 = 0 is attractive, leading x1 to converge towards x1 in finite time t0. Moreover, we have x1 = 0∀t ≥ t0 . Consequently and according to equation (14), we have the convergence of x2 towards x2.

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Then, equation system (16) can be written as follows:  x1 = x2 = H1sign eq ( x1 ) → 0 .  −1 −1  x2 = M Fg − M (C ( x1 , x2 ) x2 − C ( x1 , x2 ) x2 ) − H 2 sign( x2 )

(18)

Now, let us consider a (second) Lyapunov function V2 and its time derivative V2: V2 = x2T Mx2 .

(19)

Then, from equation (18), V2 becomes: V2 = x2T Fg − x2T (C ( x1 , x2 ) x2 − C ( x1 , xˆ2 ) xˆ2 ) − x2T MH 2 sign( x2 ).

(20) (21)

Using the assumption (equation (4)), we have: C ( x1 , x2 ) x2 − C ( x1 , xˆ2 ) xˆ2 = −C ( x1 , x2 ) x2 − C ( x1 , xˆ2 ) x2 .

(22)

Finally the equation (20), becomes: V2 = x2T Fg − x2T (−(C ( x1 , x2 ) + C ( x1 , xˆ2 )) + MH 2 sign( x2 )).

(23)

Recalling that M and H1 are positive definite matrices, Fg bounded and by choosing h2i > ζ, we obtain V2 < 0. Therefore, the surface x2 = 0 is attractive, leading xˆ2 to converge towards x2 in finite time t1. According to equations (4) and (15), after convergence of the position x1, we can estimate from the equation (3), the vertical displacements zri of the wheels and consequently, we deduce the vertical forces Fni. Finally, we obtain the convergence of all system states (positions, centre height of gravity, speeds and accelerations) and we can calculate the LTR coefficient using the equation (7) or (8).

4

Rollover risk prediction

It is very important to detect the risk in sufficient time to prevent rollover of heavy vehicles. Then the knowledge of this Time to Rollover (TTR) is very interesting (Chen and Peng, 1999). In this paper, we choose this time at 3 seconds. As in the previous section, in order to calculate the LTR, we use the values of the infrastructure data base at the time (t + 3) seconds and we try to estimate at the moment t, the heavy vehicle states (displacements, the height of the gravity centre and speeds). The knowledge of these states permits us to calculate the LTR at the current moment to know if we have a rollover risk in the following 3 seconds. From equation (9), and if the risk is detected (LTR ≥ 0.9), we calculate the critical lateral acceleration of the heavy vehicle and we can deduce the maximal speed of the vehicle. This information can be also send to the driver to prevent the rollover.

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301

Main results

In this section, we give some results in order to test our approach. These simulations results are considered as a reference in our work to compare with estimation using mode observers. An infrastructure data base is used in simulation. We begin with heavy vehicle rolling on a straight line. Figure 4 shows that the observed suspension deflection (and speed) are good compared to the true one computed by the PROSPER simulator. Figure 4

Suspension deflection: observed and true one

In Figure 5, we notice that the centre of gravity height is well observed. We have also shown that the vertical force of the semi-trailer is quite close to the true one estimated by the PROSPER simulator. This allows us to determine the LTR. Since this coefficient does not reach the value 0.9, we can conclude that no rollover risk is detected. This result

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is confirmed by the estimated lateral acceleration which stays very low in comparison to the critical lateral acceleration. Figure 5

LTR comparing to the lateral acceleration and vertical load

Now, we suppose that the driver is rolling on the curvature making the steering angle represented in the Figure 6. Figure 6

Steering angle

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Figure 7 shows that the centre of gravity height is correctly evaluated. Figure 7

Height centre of gravity

In Figure 8, we compare the estimated vertical forces and the true ones given by the PROSPER estimator. We can show that the estimation is correct for the tractor’s forces. Figure 8

Estimated vertical forces

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However, after 10 seconds, the estimated semi-trailer vertical force becomes bad. This arises because the truck is in a rollover situation. We can confirm that, in the Figure 9 by comparing the corresponding calculated LTR, using equation (7) to 0.9. Figure 9

LTR comparing to the vertical load

This figure shows that the left vertical force of the semi-trailer is near to 0 at time 11 seconds. We notice that at this same time, the LTR is near to 0.9. That means that the rollover of the semi-trailer occurs before that of the tractor. In Figure 10, we show the difference between the left and right vertical forces of the semi-trailer. When the right vertical force near to 0, the left one converges toward its maximum value. This is due to an axle load transfer situation. In Figure 11, we show the good estimation of the centre of gravity height. We also notice that the estimated lateral acceleration converges toward the limiting one (critical one) after only 10 seconds. This means that, using the acceleration, the risk is detected before the LTR method (11 seconds).

Rollover risk prediction of heavy vehicle in interaction with infrastructure Figure 10 Left and right semi-trailer estimated vertical load

Figure 11 Critical lateral acceleration and estimated height centre gravity

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Figure 11 Critical lateral acceleration and estimated height centre gravity (continued)

6

Conclusion

In this paper, we present an original method about rollover risk detection and estimation. A model of a heavy vehicle taking into account the infrastructure data base (road profile input, curvature radius, road lateral inclination) is developed and tested using the PROSPER simulator. The sliding mode observers are used to estimate heavy vehicle dynamics states and the vertical loads estimator is developed and the convergence is studied. We show through different results that the states are well observed and the centre of gravity height is estimated. Then we estimate the vertical forces in order to calculate LTR coefficient. In the first example, when a heavy vehicle is rolling on a straight line, we have seen that no risk is detected and the LTR stays lower than 0.9. In the second example, the driver is rolling on the curvature. The results show that a risk is detected and the left vertical force of the semi-trailer is near to 0 while the vertical force of the tractor stays different from 0.We can then conclude that the semi-trailer rolls over before the tractor. Another important result is risk detection using the lateral acceleration. We have seen that the estimated lateral acceleration reaches its limit in 10 seconds so before the LTR reaches 0.9. In this paper, we choose arbitrarily the TTR. In our next works, we will test our approach using sliding mode observers to estimate this TTR.

References Allen, R.W., Szostak, H.T., Klyde, D.H., Rosenthal, T.J. and Owens, K.J. (1992) Vehicle Dynamic Stability and Rollover, Tech. Rep., Systems Technology, Inc., Hawthorne, CA, U.S.-D.O.T., NHTSA. USA. Barbot, J.P., Boukhobza, T. and Djemai, M. (1996) ‘Triangular input observer form and sliding mode observer’, IEEE Conf. on Decision and Control, pp.1489–1491. Bouteldja, M. (2005) Modélisation des Interactions Dynamiques Poids Lourd/Infrastructures Pour la Sécurité et les Alertes, PhD Thesis, l’Université de Versailles Saint Quentin en Yvelines.

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Cebon, D. (1993) Interaction Between Heavy Vehicles and Roads, Society of Automotive Engineers, SP-931, p.81. Chen, B.C. and Peng, H. (1999) ‘A real-time rollover threat index for sports utility vehicles’, American Control Conference, June, San Diego, USA, pp.1233–1237. Chen, C. and Tomizuka, M. (1995) ‘Dynamic modeling of articulated vehicles for automated highway systems’, American Control Conference, pp.653–657. Drakunov, S.V. (1992) ‘Sliding-mode observers based on equivalent control method’, Proc. 31st IEEE Conf., Decision and Control, Tucson, Arizona, pp.2368–2369. Gillespie, T.D. and Karamihas, S.M. (1993) ‘Characterising trucks for dynamic load prediction’, Int. J. Vehicle Design, Vol. 1, No. 1, pp.3–19. Gillespie, T.D., Karamihas, S.M., Cebon, D., Sayers, M., Nasim, M., Hansen, W. and Ehsan, N. (1992) Effects of Heavy Vehicle Characteristics on Pavement Response and Performance, Trans. Research Board, National Cooperative Highway Research Program, Report No. 1–25(1). Ibrahim, I.M. (2004) ‘A generally applicable 3D truck ride simulation with coupled rigid bodies and finite element models’, Heavy Vehicle Systems, Int. J. Vehicle Design, Vol. 11, No. 1, pp.67–85. 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,. Imine, H. and Delanne, Y. (2005) ‘Triangular observers for road profiles inputs estimation and vehicle dynamics analysis’, ICRA’05, International Conference on Robotics and Automation, April 18–22, Barcelona, Spain, pp.349–354. Imine, H., Delanne, Y. and M’Sirdi, N.K. (2005a) ‘Road profiles inputs to evaluate loads on the wheels’, International Journal of Vehicle System Dynamics, Supplement, November, Vol. 43, pp.359–369. Imine, H., M’Sirdi, N.K. and Delanne, Y. (2005b) ‘Sliding mode observers for systems with unknown inputs: application to estimate the road profile’, Proceedings of the I MECH E Part D Journal of Automobile Engineering, August, Vol. 219, No. 8, pp.989–997. Legeay, V., Daburon, P. and Gourraud, C. (1994) Laboratoire Central des Ponts et Chaussées, DGER/IRVAR, Comparaison de mesures de l’uni par L’Analyseur de Profil en Long et par Compensation Dynamique, Bulletin Interne, Décembre. Sanchez, E.N., Ricalde, L.J., Langari, R. and Shahmirzadi, D. (2004) ‘Rollover prediction and control in heavy vehicles via recurrent neural networks’, IEEE Conference on Decision and Control, December, Atlantis, Bahamas, pp.5210–5215. Slotine, J.J.E., Hedrick, J.K. and Misawa, E.A. (1987) ‘On sliding observer for nonlinear systems’, Journal of Mathematical System, Estimation and Control, Vol. 109, No. 3, pp.245–252. Utkin, V.I. and Drakunov, S. (1995) ‘Sliding mode observer’, IEEE Conference on Decision and Control, Orlando, Florida USA, pp.3376–3378. Wang, J-Y. and Tomizuka, M. (1999) ‘Robust H¥ lateral control of heavy-duty vehicles in automated highway system’, IEEE American Control Conference, June, San Diego, CA, USA.

Website PROSPER, Sera-Cd., Technical Report, www.sera-cd.com.