Sliding Mode Observers for the Estimation of Vehicle Parameters, Forces and States of the Center of Gravity

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1 Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006 Sliding Mode Observers for the Estimation of Vehicle Parameters, Forces and States of the Center of Gravity Hassan Shraim, Bouchra Ananou, Leonid Fridman, Hassan Noura and Mustapha Ouladsine Abstract In this paper, sliding mode SM observers are proposed to replace expensive sensors used for the measurement of tires forces, vehicle side slip angle and vehicle velocity. These estimations are done for two important purposes: The first is for the estimation of the forces and parameters needed for vehicle control, while the second is for the diagnosis preview based on the safety region for each parameter and state. For that purpose, the model of the vehicle is divided in two parts, in the first, the dynamical equations of the wheels are used to estimate their longitudinal forces and angular velocities. The estimations of the longitudinal forces are realized using a second order SM observer based on a super-twisting algorithm. In the second part, the estimated longitudinal forces are injected in the reduced state space equations representing the vehicle, which are the equations of side slip angle, yaw rate and the velocity of the vehicle. Estimations in this part are based on the principles of the classical SM observer. In this part the observability of the model is studied. The model takes as input the yaw rate and estimates the side slip angle and the velocity. Validation with the simulator VE-DYNA, at each step, pointed out the good performance and the robustness of the proposed observers. I. INTRODUCTION Recently, a great deal of research has been performed on the study of the traction control [1], [2]. But for the complicated analytical models representing the vehicle [4], it is seen that the study of the observation and the control for the global vehicle is not evident due to the complicated form of the contact forces. These complications make the identification and the estimation of such forces of substantial interest, especially that, sensors for these forces are expensive. In [7] an extended kalman filter is used for the estimation of wheel forces, in which a simplified case is treated. Moreover, and due to the importance of the side slip angle in the determination of the stability of the vehicle, many researchers have studied and estimated the side slip angle [5] in which a bicycle model is used for the vehicle, and in [6], an observer with adaptation of a quality function is used for the estimation of the side slip angle. SM observers designs have been proposed by various authors [8], [9], they have received much attention recently and have been shown to be extremely effective when applied to non linear systems [5]. These types of observers are widely This work was supported by ST Microelectronics, Rousset-France and the Conseil Général des Bouches-du-Rhône. H. Shraim, M. Ouladsine and H. Noura are with Laboratory LSIS, Faculty of Saint Jerôme, University of paul cézanne, Avenue Escadrille, Normandie-Niemen,13397 MARSEILLE CEDEX 20, FRANCE hassan.shraim@lsis.org Leonid Fridman is with Department of Control, Division of Electrical Engineering, Engineering Faculty, National Autonomous University of MexicoUNAM, Ciudad Universitaria, 04510,D.F, Mexico, lfridman@verona.fi-p.unam.mx used due to the finite time convergence, robustness with respect to uncertainties and the possibility of uncertainty estimation. New generation of differentiators and observers based on the second order SM algorithms are recently developed and used as observers with asymptotic convergence of error developed in [10]. In [11], a robust exact differentiator was designed ensuring finite time convergence, as application of super twisting algorithm [12]. These differentiators are, for example, successfully used in [13], [14]. Recently, another type of observer that also uses the SM have been reported. In the early work of [15], the observer was constructed for a second order non linear dynamic system involving only single measurement. Further development in this field of sliding observers was made by [16] and [17] which have applied such observers in robot manipulators. In this paper two classes of the SM observers are used: In the first part, the estimation of angular velocities of the four wheels of the vehicle and the identification of the longitudinal forces, which are supposed as unknown elements, are made using a second order SM observer based on the super-twisting algorithm with the finite time convergence. Only partial knowledge of the system model is required. In the second part, the identified longitudinal forces are used as inputs. The vehicle side slip angle and velocity are estimated by a classical SM observer. All the other forces and parameters are then deduced by using validated equations. Simulation results are compared by the results of the simulator VE-DYNA [21], which is developed by the group of companies TESIS. This simulator specially designed for fast simulation of vehicle dynamics and it is validated by real measures. The main contributions of this work reside in the estimations of the contact forces with the ground, vehicle side slip angle and velocities. These estimations preview some critical situations that may occur while rolling such as excessive rotation around Z axis and also excessive side slipping, un appropriate lateral acceleration,... For all these estimations, SM observers are used. This observer is characterized by its rapid convergence to real values, its robustness for bounded modeling errors can be guaranteed and extensive computation load is not required. The paper is organized as follows: In section 2, problem statement is proposed. In this section, the model of the vehicle is presented to show the complexity that it has and which influences the observability. In section 3 steps of /06/$ IEEE. 1635

2 TABLE I NOMENCLATURE symbol physical signification w i angular velocity of the wheel M total mass of the vehicle COG center of gravity of the vehicle r 1i effective radius of the wheel i Fx i longitudinal force applied at the wheel i Fy i lateral force applied at the wheel i C fi braking torque applied at wheel i C mi motor torque applied at wheel i torque i C mi C fi I Z moment of inertia around the Z axis ψ yaw angle ψ yaw velocity δ f front steering angle δ r rear steering angle Vx longitudinal velocity of the center of gravity Vy lateral velocity of the center of gravity I ri moment of inertia of the wheel i total velocity of the center of gravity L 1 distance between COG and the front axis L 2 distance between COG and the rear axis L L 1 + L 2 C ij tire side slip constants i:front F, rearr,j:rightr, leftl α ij slip angle of the wheel i h COG height of COG h 1 distance between the COG and the Pitch axis t f front half gauge t r rear half gauge l t f + t r F xwind air resistance in the longitudinal direction F ywind air resistance in the lateral direction A L front vehicle Area ρ air density C aer coefficient of aerodynamic drag n L1 caster effect front n L2 caster effect rear c presi parameter to correct for tire pressure distribution β side slip angle at the COG work are proposed by a graph showing at each step the proposed solution. In section 4, observer design is made, in this section, two classes of SM observer are proposed and at each step validation with the simulator VE-DYNA and comments are made. Finally in section 5 a conclusion of the work will be shown. Fig. 1. 2D schema representation 1 β = cosβ F S sinβ F L Mv ψ 2 COG With FL = F xwind + cosδ f Fx 1 + Fx 2 +cosδ r Fx 3 + Fx 4 sinδ f Fy 1 + Fy 2 sinδ r Fy 3 + Fy 4 And FS = sinδ f Fx 1 + Fx 2 +sinδ r Fx 3 + Fx 4 + cosδ f Fy 1 + Fy 2 +cosδ r Fy 3 + Fy 4 ψ = t f I Z {cosδ f Fx 2 Fx 1 +sinδ f Fy 1 Fy 2 } + L 1 {sinδ f Fx 2 + Fx 1 +cosδ f Fy 1 + Fy 2 } + L 2 {sinδ r Fx 3 + Fx 4 cosδ r Fy 3 + Fy 4 } + t r {cosδ r Fx 4 Fx 3 +sinδ r Fy 3 Fy 4 } 3 The model representing the dynamics of each wheel i is found by applying Newton s law to the wheel and vehicle dynamics Fig. 2: II. PROBLEM STATEMENT Increasing demands of safety require accurate tools to represent states and parameters of the vehicle. These accurate representations need many precise and expensive sensors, which mean also, an important diagnosis system should be implemented to avoid false data. For these reasons, robust virtual sensors are proposed, these virtual sensors are based on a non linear model which can be found by applying the fundamental principles of dynamics at the center of gravity [3]onFig.1: = 1 M cosβ F L sinβ F S 1 Fig. 2. Wheel and its contact with the ground I ri ω i = r 1i Fx i + torque i i =1:4 4 In this paper, the task is to design a virtual sensors observer for the vehicle to estimate the states, parameters and forces which need expensive sensors for their measurement. But due to the fact that it is not easy to apply an observer for the global model due to the observability study, equations 4 are taken at first, a second order sliding mode is proposed to observe the angular velocity and to identify the longitudinal force of each wheel. After having the longitudinal forces, we inject their values in equations 1, 2 and 3. From these 1636

3 equations, it is seen that if we inject the longitudinal forces, The system described by the equations 1, 2 and 3 is we will still have as complex terms the lateral forces. For observable if we consider that we measure only the yaw that reason, the wheel side forces are approximated to be rate and by supposing the longitudinal forces as inputs. A proportional to the tire side slip angles α ij [18]: classical sliding mode observer is used to the estimate the side slip angle and the velocity of the center of gravity. By Fy 1 = C FL α FL = C FL δ f β L 1 ψ these estimations, the longitudinal and lateral velocities of 5 the center of gravity, the lateral forces of the wheels are then directly deduced. Fy 2 = C FR α FR = C FR δ f β L 1 ψ The first order SM observer is not used in the first part 6 because for the uncertainties and parameter identification two successive filtrations are necessary leading to a bigger Fy 3 = C RL α RL = C RL β + L 2 ψ corruption of results. 7 So, realizing the standard first order SM observer one filtration is needed to reconstruct the velocity and two successive filtrations are necessary to identify unknown inputs, Fy 4 = C RR α RR = C RR β + L 2 ψ 8 while for the second order SM observer the filtration is needed once when we would like to identify the unknown But for lateral accelerations above 4m/sec 2 and large tire inputs. side slip angles, the linear relation ship representing lateral forces are not sufficiently accurate any more. In this case, III. OBSERVER DESIGN the tire side slip constant becomes time variant and the A. ESTIMATION OF WHEELS ANGULAR VELOCITIES parameters C FL, C FR,C RL and C RR will be written as: AND LONGITUDINAL FORCES C FL t, C FR t,c RL t and C RR t. As described in [18], the tire side slip constant is calculated at every time step: In this part, a second order SM observer based on the super twisting algorithm is proposed to observe the angular C ij t k = k 0k 1 FZijt k k 2 F Zij t k arctank 3 α ij t k velocity and to identify the longitudinal force of each wheel. α ij t k For that reason, dynamical equations of wheels 4 are 9 written in the following form: k 0 contains the actual friction coefficient; k 1, k 2 and k 3 ẋ depends on the tires by this equation [22], the tire side 1 = x 2 13 ẋ slip constants are adapted and injected in the lateral forces 2 = ft, x 1,x 2,u equations. Where x 1 and x 2 are respectively the angular position which Then, the model of the vehicle represented by 1, 2 and is measured and the angular velocity to be observed of 3 is rewritten [18] as: each wheel they may appear implicitly in Fx i, and u is the input torque. = 1 M {Fx 1 + Fx 2 cosδ f β C FL t+c FR t As described in [19], the proposed super-twisting observer δ f β sinδ f β+ has the form: F x3 + F x4 C a era L vcog 2 ρ 2 cosβ+ } L2 ψ ˆx 1 =ˆx 2 + z 1 C RL t+c RR t β + sinβ 14 ˆx 2 = f 1 t, x 1, ˆx 2,u+z 2 10 Where ˆx 1 and ˆx 2 are the state estimations of the angular 1 β = M {F x1 + F x2 sinδ f β+c FL t+c FR t positions and the angular velocities of the four wheels respectively, f 1 is a nonlinear function containing only the δ f β cosδ f β F x3 + F x4 C a era L vcog 2 ρ 2 sinβ+ known terms which is only the torque in our case, z 1 and } L2 ψ C RL t+c RR t β + cosβ ψ z 2 are the correction factors based on the super twisting algorithm having the following forms: 11 z 1 = λ x 1 ˆx 1 1/2 signx 1 ˆx 1 15 z ψ = 1 2 = αsignx 1 ˆx 1. I Z {L 1 n fl cosδ f F x1 + F x2 sinδ f + δ f β cosδ f C FL t+c FR t Suppose that x 1 = x 1 ˆx 1 and x 2 = x 2 ˆx 2, we obtain the equations for the error: L 1 n lf cosδ f + t f F x2 F x1 cosδ f t f C FR t C FL t δ f β sinδ f x 1 = x 2 λ x 1 1/2 sign x 1 16 L2 ψ x L 2 + n lr C RL t+c RR t β + 2 = F t, x 1,x 2, ˆx 2 αsign x 1 +t r F x4 F x3 } Where F t, x 1,x 2, ˆx 2 is the difference between the unknown and the known 12 function. 1637

4 Suppose that the system states can be assumed bounded, then the existence is ensured of a constant f +, such that the inequality: F t, x 1,x 2, ˆx 2 <f + 17 holds for any possible t, x 1, x 2 and ˆx 2 2sup x 2. Which means it is sufficient to give for f + a value greater than the sum of the maximum values of the longitudinal force and the torque that may be applied. Let α and λ satisfy the inequalities: α>f +, λ> 2 α+f + 1+p 18 α f + 1 p, Where p is some chosen constant, 0 <p<1. To proof that the observer 14, 15 for the system 13 ensures the finite time convergence of estimated states to the real states, i.e. ˆx 1, ˆx 2 x 1,x 2. Same steps are considered as in [19]. 1 SIMULATIONS AND RESULTS: In this section, we illustrate the performance of the proposed observer, in order to show firstly how to estimate angular velocity of each wheel without differentiating the angular position, and secondly, to identify its longitudinal force. Simulations are made and results are compared by the results of the simulator VE-DYNA, inthissectiontheshown simulation corresponds to the case where the longitudinal force and the angular velocity strongly vary acceleration, constant velocity, deceleration, constant velocity, acceleration, constant velocity, deceleration, constant velocity with a zero steering angle. The same observer is applied on the four wheels, front left wheel. A run of 80 seconds is made. The simulator uses a car which has two rear wheel drives. The input torques given by the simulator are shown in Fig. 3 and Fig. 4. In Fig. 5, we see the angular position given by the simulator VE-DYNA and that which is the output of the observer. In this part we see the rapid convergence of the observer in spite of the initial value of the angular position is put 50 radians while the simulator begins from zero, its time of convergence is very small. In Fig. 6, the estimated angular velocity of the front left wheel and that given by the simulator VE-DYNA is shown. The same thing is also made, initial value of 100 rad/sec is put, which is very far from that of the simulator which begins by zero, it also converges to the real state in a short period. In Fig. 7, the unknown functions are filtered by a smooth filter, which is a low pass filter, we see that the filtered function approximately coincides with the longitudinal force given by the simulator. In Fig. 7 we see some peak values, these peak values are that of the simulator VE-DYNA. Fig. 3. Motor and braking torque N.m applied at the two rear wheels Fig. 4. Motor and braking torque N.m applied at the two front wheels Fig. 5. Angular position radians by the simulator VE-DYNA, and that estimated by the observer Fig. 6. Angular velocity radians/sec by the simulator VE-DYNA, and that estimated by the observer Fig. 7. The unknown input after filtration N and the longitudinal force from the simulator VE-DYNA 1638

5 B. ESTIMATION OF THE SIDE SLIP ANGLE, VELOCITY OF THE VEHICLE AND RECONSTRUCTION OF THE YAW RATE In this part, a classical first order SM observer is used to estimate vehicle velocity and vehicle side slip angle, with the only one measure which is the yaw rate. As described in 10, 11 and 12, the model of the vehicle is a non linear model and it can be can be written as follows: slip angle and the yaw rate with that of the simulator, they converge in a short period of time: Where x = ẋ = fx, u y = cx [ ] β ψ Fig. 8. Front steering angle radians The input: u =[F x1 F x2 F x3 F x4 δ f ] 21 And the measurement vector [ y = ψ] 22 Before the design of the SM observer for the model of 19, the observability of the model must be investigated and tested. The observability definition is local and uses the Lie derivative [24]. It is a function of state trajectory and inputs applied to model. For the system described by 19 the observability function is: cx observabilityx, u = L f cx, u L 2 f cx, u Where L f cx, u = dcjx dx fx, u The system is observable if its Jacobian matrix J observability has a full rank which is three in our case. J observability = d dxobservabilityx, u Fig. 9. Fig. 10. DYNA Motor and braking torque N.m applied at the two rear wheels Estimated vehicle velocity m/sec and that of the simulator VE- By applying these notions to the system described by 19, we see that its rank is three and then it is observable. So, the proposed SM observer is: { ˆx = fˆx, u+ signy ŷ 23 ŷ = C ˆx Where is the gain of the SM observer. Fig. 11. Estimated side slip angle radians using SM and that of the simulator VE-DYNA 1 SIMULATIONS AND RESULTS: In this part, the estimated vehicle velocity and side slip angle using SM observer are compared to that of the simulator VE-DYNA, it is seen that the errors are practically small. Simulations are made for many different driving conditions, but here one significant simulation is shown where the side slip angle varies strongly, which is the case of changing in the steering angle and torques Fig. 8 and Fig. 9 respectively. A run of 22 seconds is made, The longitudinal forces are estimated from the first part, and then injected as inputs to this observer. In Fig. 10, Fig. 11 and Fig. 12 we see the observed velocity, side Fig. 12. VE-DYNA Reconstructed yaw rate radians/sec and that of the simulator 1639

6 C. LONGITUDINAL VELOCITY, LATERAL VELOCITY, WHEEL SIDE SLIP ANGLE AND THE LATERAL FORCES In this section, and by estimating vehicle slip angle and velocity, the velocities of the center of gravity in X, Y can be directly found by [18]: The lateral velocity Fig. 13: Fig. 13. Vy= sinβ 24 Estimated Vym/sec and that of the simulator VE-DYNA And the longitudinal velocity which coincides with that of the simulator Fig. 14: Fig. 14. Vx= cosβ 25 Estimated Vxm/sec and that of the simulator VE-DYNA The lateral forces can be found by applying 5, 6, 7 and 8. IV. CONCLUSIONS The SM observers are used to estimate vehicle velocities, parameters and forces of contact with the ground. These observers show a short time of convergence and robustness while applying them to vehicle model. Their short time of convergence and their robustness make their usage very encouraged in the automotive applications, especially that the automotive is subjected to outside disturbance and some uncertain parameters. [4] H. Shraim, M. Ouladsine, M. ElAdel, H. Noura. Modeling and simulation of vehicles dynamics in presence of faults, 16th IFAC World Congress, 2005, pp [5] J. Stephant, Contribution à l étude et à la validation expérimentale d observateurs appliqués à la dynamique du véhicule, thesis presented to have the doctor degree from the UTC, university of technology Compiègne, [6] A. Von Vietinghoff, M. Hiemer, U. Kiencke, Non Linear Observer Design for Lateral Vehicle Dynamics, 16th IFAC World Congress, [7] B. Samadi, R. Kazemi, K. Y. Nikravesh and M. Kabganian, Real- Time Estimation of Vehicle State and Tire Friction Forces. Proceedings of the American Control Conference ACC, 2001, pp [8] S. Drakunov, sliding Mode observers based on equivalent control method, IEEE conference on Decision and ControlCDC,Tucson, Arizona, 1992, pp [9] S. Drakunov and U. Ozguner, decentralized sliding observers for interconnected non linear systems, IEEE International workshop on variable structure and and Lyapunov Control of Uncertain Dynamical Systems, Sheffield, England. [10] Y.B. Shtessel, I.A. Shkolnikov, MDJ. Brown, An invariance principle for discontinuous dynamic systems with application to a coulomb friction oscillator, An asymptotic second order smooth sliding mode control, 2003, pp [11] A. Levant, Robust exact differentiation via sliding mode technique, Automatica, 343, 1998, pp [12] A. Levant, sliding order and sliding accuracy in sliding mode control, international Journal of control, vol.58, 1993, pp [13] H. Sira-Ramirez, dynamic second order sliding mode control of the hovercraft vessel, IEEE trans.on control system technology, vol.10, 2002, pp [14] G. Bartolini,A. Pisano, E. Punta, and E. Usai, A survey of applications of second order sliding control to mechanical systems, International Journal of control, vol.76, 2003, pp [15] J. J. Slotine, J. K. Hedrik, and E. A. Misawa, On sliding observers for nonlinear systems, ASME J. Dynam. Syst. Measvol. 109,, 1987, pp [16] E.A Misawa, Non linear state estimation using sliding observers-a state of the art survey, ASME J.Dyn.System Measurment Control vol. 109,,1988, pp [17] J. J. Slotine and W. Li, Applied Nonlinear Control, Prentice- Hall,Inc.,Englewood Cliffs,NJ,USA, [18] K. UWE and L. NIELSEN, Automotive control system, Springer, [19] J. Davila, L. Fridman, A. Levant, Second-Order Sliding Mode Observer for Mechanical Systems, IEEE Transactions on Automatic Control, vol: 50, no. 2, 2005, pp [20] Zeitz, the extended luenberger observer for nonlinear systems, Systems and Control Letters, [21] [22] M. Hiemer et al. Cornering stiffness adaptation for improved side slip angle observation, Proceedings of the First IFAC Symposium on advances in Automotive Control AAC04, [23] Thomas D. Gillespie, Fundamentals of Vehicle Dynamics, Published by Society of Automotive Engineers, Inc, [24] H. Nijmeijer, A.J. Van der Schaft, Nonlinear Dynamical Control Systems, Springer-verlag, REFERENCES [1] I. Petersen, Wheel Slip Control in ABS Brakes using Gain Scheduled Optimal Control with Constraints, thesis submitted for the degree of doctor engineer Department of Engineering Cybernetics, Norwegian University of Science and Technology Trondheim, Norway, [2] C. Unsa and P. Kachroo, Sliding Mode Measurement Feedback Control for Antilock Braking Systems, IEEE Transactions on Control System Technology,vol. 7,1999, NO.2. [3] H. Shraim, M. Ouladsine, H. Noura, B.Annanou. Modeling and Validation of Ground Contact Forces and their Influence on the Movement of the center of gravity, Integrated Modeling and Analysis in Applied Control and Automation IMAACA, 2005, pp