THE vehicle is a highly complex system comprising a large

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1 278 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 Virtual Sensor: Application to Vehicle Sideslip Angle and Transversal Forces Joanny Stéphant, Ali Charara, Member, IEEE, and Dominique Meizel Abstract This paper compares four observers (virtual sensors) of vehicle sideslip angle and lateral forces. The first is linear and uses a linear vehicle model. The remaining observers use an extended nonlinear model. The three nonlinear observers are: extended Luenberger observer, extended Kalman filter and sliding-mode observer. Modeling, model simplification, and observers are described, and an observability analysis is performed for the entire vehicle trajectory. The paper also deals with three different sets of sensors to see the impact of observers results. Comparison is first done by simulation on a valid vehicle simulator, and then observers are used on experimental data. Our study shows that observers are more accurate than simple models as regards unmeasurable variables such as sideslip angle and transversal forces. It also shows that speed of center of gravity is not an indispensable variable here. Index Terms Linear system, nonlinear system, observers, state estimation, vehicle dynamics. NOMENCLATURE Input noise vector ( ). Model noise vector ( ). Output/measurement noise vector ( ). Front, rear wheel cornering stiffness (N rad ). Longitudinal front, rear force in the vehicle frame (N). Transversal front, rear force in the vehicle frame (N). Longitudinal force in the front wheel frame (N). Transversal force in the front wheel frame (N). CG to front, rear axle distance (m). Speed of center of gravity ( m s ). State vector. Measurement vector. Yaw rate only ( ). Speed of center of gravity only ( ). Speed of center of gravity and yaw rate ( ). Steering angle (rad). Vehicle sideslip angle (rad). Front, rear wheel sideslip angle (rad). Yaw rate (rad s ). Manuscript received January 31, 2003; revised October 16, Abstract published on the Internet January 13, This work was performed in collaboration with the research group DIagnostic et Véhicules Avancés DIVA supported by the Picardie region, and with the help of SERA-CD ( within the framework of the Action de Recherche pour une COnduite sécurisée ARCOS2004 project, supported by the PREDIT program. J. Stéphant and A. Charara are with the Laboratoire Heudiasyc (UMR CNRS 6599), Centre de Recherche de Royallieu, Université de Technologie de Compiègne, Compiegne Cedex, France, ( joanny.stephant@hds.utc.fr; ali.charara@hds.utc.fr). D. Meizel was with the Laboratoire Heudiasyc (UMR CNRS 6599), Centre de Recherche de Royallieu, Université de Technologie de Compiègne, Compiegne Cedex, France. He is now with the ENSIL Parc d Ester Technopole, Limoges Cedex, France ( meizel@ensil.unilim.fr). Digital Object Identifier /TIE Fig. 1. Schematic representation of the problem. I. INTRODUCTION THE vehicle is a highly complex system comprising a large number of mechanical, electronic, and electromechanical elements. To describe all the movements of the vehicle, numerous measurements and a very precise mathematical model are required. This paper presents methods which use simple models and a certain number of measurements in order to estimate particular unmeasurable variables: sideslip angle and lateral forces. In vehicle development, knowledge of wheel ground contact forces is important. The information is useful for security actuators, for validating vehicle simulators, and for advanced vehicle control systems. Braking systems and control systems must be able to stabilize the car during cornering. When subject to transversal forces, such as when cornering, or in the presence of a camber angle, tire torsional flexibility produces an aligning torque which modifies the original wheel direction. The difference is characterized by an angle known as sideslip angle. This is a significant signal in determining the stability of the vehicle [1], and it is the main transversal force variable. Measuring sideslip angle would represent a disproportionate cost in the case of an ordinary car, and it must therefore be observed or estimated. The aim of an observer or virtual sensor is to estimate a particular unmeasurable variable from available measurements and a system model. This is an algorithm which describes the movement of the unmeasurable variable by means of statistical conclusions from the measured inputs and outputs of the system. A schematic representation of the observer method is presented in Fig. 1. This algorithm is applicable only if the system is observable. The literature describes several observers for sideslip angle. For example, Kiencke in [2] and [3] presents linear and nonlinear observers using a bicycle model. Venhovens [4], uses a Kalman filter for a linear vehicle model /04$ IEEE

2 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 279 Fig. 2. Heudiasyc Laboratory experimental vehicle: STRADA. The present study compares four observers for sideslip angle on a conventional test with three different speeds. We are particularly concerned with the stability of observers and models when the vehicle approaches the linear dynamic limits. Results for three different sets of sensors: yaw rate; vehicle speed; yaw rate and vehicle speed together are presented. Results concerning observability are also included. Finally, this study presents some experimental results obtained with the Heudiasyc experimental vehicle. All simulations were performed using Callas software developed by SERA-CD (Vehicle Engineering Research and Development Company), and all data processed with MATLAB software. II. VEHICLE AND SIMULATOR A. Simulation Software: Callas Simulator Callas software is a realistic simulator validated by vehicle manufacturers including PSA and research institutions including the Institut national de recherche sur les transports et leur sécurité (INRETS). The Callas model takes into account numerous factors including vertical dynamics (suspension, tires), engine model, kinematics, elasto-kinematics, tire adhesion, and aerodynamics. B. Experimental Vehicle: STRADA STRADA (see Fig. 2) is the Heudiasyc Laboratory s test vehicle, a Citroën Xantia estate car equipped with a number of sensors: GPS vehicle localization; accelerometer longitudinal, lateral, and vertical acceleration; odometry rotation speeds of the four wheels (ABS sensors); gyrometer yaw and pitch rates; steering angle. Tests use GPS with longitudinal and lateral acceleration to trace the path and to determine whether the vehicle reaches linear approximation limits. The speed of the center of gravity is calculated as the mean of the longitudinal speeds of the two rear wheels (odometry). Fig. 3. Nonlinear bicycle model. III. SIMPLIFIED VEHICLE MODELS Lateral vehicle dynamics has been studied since the 1950s. In 1956, Segel [5] presented a vehicle model with three degrees of freedom in order to describe lateral movements including roll and yaw. If roll movement is neglected, a simple model known as the bicycle model is obtained. This model is currently used for studies of lateral vehicle dynamics (yaw and sideslip). A nonlinear representation of the bicycle model is shown in Fig. 3. Some simplifications are available for the different models featured in this study. Cornering stiffness is taken to be constant. But cornering stiffness increases with tire pressure. When the car turns, the mass transfer onto the external wheels increases tire pressure. Fig. 4 presents variations in cornering stiffness for simulation (see Fig. 7 and Section V-B for simulation conditions). Variations are less than 10%. Tire/road forces are highly nonlinear. Various wheel ground contact force models are to be found in the literature, including a comparison between three different models by Stéphant in [6]. In this paper, transversal forces are taken to be linear. This assumption is reasonable when lateral acceleration of the vehicle is less than 0.4 g [7]. Consequently, transversal forces can be written as Rear and front tire sideslip angles are calculated as This study uses three bicycle models with constant cornering stiffness. Models are described in linear or nonlinear state-space formulation. (1) (2)

3 280 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 Fig. 4. Rear and front cornering stiffness at 20, 60, and 90 km 1 h. A. Nonlinear Model (NLM) The nonlinear bicycle model is described as where the state vector is. (3) and the input vector B. Linear Model (LM) Fig. 5. Linear and nonlinear observer methods. Given the assumption of cornering at constant speed, with small steering angle and sideslip angle, the linear model is with,, (4)

4 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 281 Fig. 6. Sideslip angle error, 60 km 1 h, Z. C. Extended NLM (ENLM) In the extended nonlinear model, longitudinal forces and their first derivatives become state variables with a random walk dynamic. This can be used for estimating longitudinal forces, as in [8]. The state vector becomes and the input vector D. Remarks All models were implemented in a discrete form with MATLAB software. The sampling rate is 20 ms. The nonlinear and extended nonlinear systems are undefined when m s. In practice, there is a problem of divergence when m s. When speed is less than 1 m s, sideslip angle effects on the vehicle dynamics are negligible in comparison to the yaw rate. IV. OBSERVERS VIRTUAL SENSORS Four different observers are featured in this paper. The method applied is presented in Fig. 5. A description of the different observers is presented in this section. A. Linear Observer (LO) The linear observer featured here is a Luenberger observer [3]. It is applied to system (4) (5) (6)

5 282 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 TABLE I MAXIMUM VALUES FOR SIDESLIP ANGLE, YAW RATE, AND SPEED OF CENTER OF GRAVITY. SIMULATION MODEL RESULTS. (a) MAXIMUM VALUES FOR SIMULATION. (b) SIMULATION MODEL RESULTS MAX/MEAN. B. Nonlinear Observer (NLO) This nonlinear observer could be called an extended Luenberger observer. It is applied to ENLM (5) and are nonlinear functions in state and input. After linearization around the estimated variable, the error dynamic could be written as With a pole-placement technique, it is possible to impose error dynamics. The system matrix of the closed-loop system has constant poles, since the observer is stable The gain matrix of the observer is calculated by (7) (8) (9) (10) the first step of the EKF is to linearize the evolution equation around the estimated state and the input (12) The second step is the prediction of the next state, from the previous state and the measured input (13) The covariance matrix of state estimation uncertainty is (14) The third step is to calculate the Kalman gain matrix from the linearization of the measurement matrix (15) (16) The prediction phase is (17) The covariance matrix of state estimation uncertainty becomes (18) with the pseudoinverse The EKF is used with the ENLM (5). C. Extended Kalman Filter (EKF) The Kalman filter is used and described in many references. For example, Mohinder and Andrews [9] present a broad overview of Kalman filtering. Given a system with measured input D. Sliding-Mode Observer (SMO) From [10], this kind of observer is useful when working with reduced observation error dynamics, for a finite time convergence for all observable states, and for robustness under parameter variations (with respect to conditions) (19) (11)

6 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 283 Fig. 7. Simulation conditions at 20, 60, and 90 km 1 h path and acceleration. To cover chattering effects [11], the function as follows: The SMO is used with the ENLM (5). V. SIMULATION RESULTS OBSERVERS AND MODELS is (20) A. Remarks on the Reading of Results Values in the different tables and figures are calculated from the maximum error between the observer and the sensor and the error mean along the path. For example, Fig. 6 gives the error max and mean for the sideslip angle. In Fig. 9(b) the normalized maximum error of the SMO for sensor set for sideslip angle is 30%. The normalized mean error is therefore 9%. The normalization was done by the maximum value of sideslip angle along the path [Table I(a)]. B. Simulation Conditions Simulations were performed using three sets of sensors for observers: yaw rate only; speed of center of gravity only; speed of center of gravity and yaw rate together. The tests took place in a double lane change at three different speeds: 20, 60, and 90 km h. Fig. 7 presents the simulation path and acceleration for the different speeds. Table I(a) gives the maximum values for the speed of the center of gravity, yaw rate, and sideslip angle at the different speeds. Fig. 8(a) shows the sideslip angle calculated using the linear and nonlinear ( ) models at 60 km h. C. Model Results In Fig. 7, it can be seen that longitudinal acceleration is close to zero, meaning that longitudinal forces are virtually nonexistent. The nonlinear model with zero force input seems to be a good simulation. Table I(b) shows that a good approximation of speed is obtained from the nonlinear model. The error is less than 1% for the three speeds (mean and max). The Callas simulator driver aims to maintain a constant speed throughout the path. There are only small speed variations. The greater the speed, the greater the yaw rate estimation error. At 60 km h, the mean error is 5%. At 90 km h, it is 10%. This indicates that the models are valid with respect to lateral movements. As regards sideslip angle, neither model is accurate. It would appear that observers are necessary to correct estimations. D. Observer Results Fig. 9(a) shows the results of sideslip angle observation at 20 km h. All nonlinear observers are highly accurate with respect to sideslip angle (less than 1% in max and mean).

7 284 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 Fig. 8. Models and observers for sideslip angle simulation, and lateral forces, 60 km 1 h, sensor set Z. Figs. 8(a) and 9(b) show the results of sideslip angle observation at 60 km h. A comparison of Table I(b) and Fig. 9(b) shows that observers give a better approximation of sideslip angle than

8 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 285 Fig. 9. Observer results for double lane change at 20, 60, and 90 km 1 h max/mean. models. If the measurement is only the speed of the center of gravity ( ), observers improve the accuracy of the sideslip angle computed by models only. But yaw rate measurement, with its substantially better mean accuracy (10%), would appear indispensable. Fig. 9(c) shows the results of sideslip angle observation at 90 km h. The same remarks can be made as for 60 km h. Two explanations can be given for the errors. Table I(b) shows that the accuracy of the model decreases as speed increases. The second explanation is that at 90 km h demands on tires are high. The maximum error occurs at maximum transversal acceleration, when we reach the limit of linear approximation. Fig. 8(b) presents the lateral forces calculated during the double lane change at 60 km h by using (1) and (2). Some differences can be noted during lane change. It could be explained by linear force model and neglected load transfer or by neglected geometrical parameters like camber angle. Maximum error is around 13% for path at 60 km h.

9 286 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 Fig. 9. (Continued.) Observer results for double lane change at 20, 60, and 90 km 1 h max/mean. VI. ROAD TESTS A. Test Conditions Fig. 10 presents the STRADA trajectory and acceleration during the tests. Vehicle lateral acceleration is under 0.4 g. Linear assumptions for transversal forces are valid. Table II(a) gives the maximum values for vehicle speed, taken as the mean of longitudinal speeds of the two rear wheels, and maximum yaw rate. TABLE II MAXIMUM VALUES FOR YAW RATE AND SPEED OF CENTER OF GRAVITY. TEST MODEL RESULTS. (a) MAXIMUM VALUES FOR TEST. (b) TEST MODEL RESULTS MAX/MEAN B. Results Table II(b) gives maximum and mean error, normalized by maximum value, for linear and extended nonlinear systems along the test path. Fig. 10 shows that longitudinal acceleration is not negligible, and that longitudinal forces are present. The NLM simulation has zero force input. This explains the error obtained in the nonlinear model. The approximation of yaw rate obtained from the models has a mean error lower than 17%. Because STRADA does not have a sideslip angle sensor, we do not have a validation measurement for sideslip angle. Fig. 11 gives the maximum and mean error for yaw rate and velocity estimations. This figure shows that yaw rate estimation error is reduced when the center of gravity is added to the measurement vector ( ) Fig. 12(a) presents the sideslip angle observed during the tests, and the confidence interval at for the EKF. This figure shows that the linear observer is the least accurate. All observers are in the bandwidth of EKF. The real sideslip angle is in this confidence interval. The three nonlinear observers are close to each other. This is the same for estimations of lateral forces presented in Fig. 12(b). VII. OBSERVABILITY The state of a state-space model is observable if it is possible to compute the state from the model, supposing that both input and output sequences are known [12]. This property is often presented as a rank condition on the observability matrix.

10 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 287 Fig. 10. Test conditions path and acceleration. Fig. 11. Observers results max/mean.

11 288 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 2, APRIL 2004 Fig. 12. Models and observers for sideslip angle and lateral forces, test, sensor set Z. A. Linear System The observability condition for the linear system (4) is given by (21) This condition is equivalent to saying that steering is not neutral (understeering gradient different to 0). It corresponds to a state of equilibrium where the force of the lateral acceleration at the center of gravity causes an identical increase in slip angle at the front and rear wheels [13].

12 STÉPHANT et al.: VIRTUAL SENSOR: APPLICATION TO VEHICLE SIDESLIP ANGLE AND TRANSVERSAL FORCES 289 Numerical application of (21) says that the car has an understeering behavior and the linear model is observable. B. Nonlinear System An observability analysis was undertaken in [14]. It shows that system NLM (3) and system ENLM (5) are observable under the sensor set along the experimental and simulation paths. VIII. CONCLUSION This paper has dealt with four different sideslip angle observers with three sets of sensor. It consists of two parts. The first part includes simulation results. We can see from the results that the measurement of the speed of center of gravity is not a determinant variable in the estimation of sideslip angle. However, this measurement does enable a slight improvement in estimation. The EKF applied with the sensor set gives less accurate estimations than the NLO and the SMO. There are some convergence problems with the NLO with nonoptimal initial states. Nonlinear sideslip angle observers (NLO, EKF, and SMO) give approximately the same results. All observers are satisfactory when lateral acceleration is low. In normal driving conditions, lateral acceleration is often low. Observers can, therefore, provide a good estimation. At high acceleration, assumptions made for models (linearity) are false. Observers need to have better models. Along the different paths, all observers are stable. They all represent transients qualitatively. Future studies will take into account the four wheels and vertical dynamics, as well as providing a better model for longitudinal forces. REFERENCES [1] S. Mammar and D. Koenig, Vehicle handling improvement by active steering, Veh. Syst. Dyn., vol. 38, no. 3, pp , July [2] U. Kiencke and A. Daiß, Observation of lateral vehicle dynamics, Control Eng. Practice, vol. 5, no. 8, pp , [3] U. Kiencke and L. Nielsen, Automotive Control System. Berlin, Germany: Springer-Verlag, [4] P. J. T. Venhovens and K. Naab, Vehicle dynamics estimation using Kalman filters, Veh. Syst. Dyn., vol. 32, pp , [5] M. Segel, Theorical prediction and experimental substantiation of the response of the automobile to steering control, in Proc. Automobile Div. Inst. Mech. Eng., vol. 7, 1956, pp [6] J. Stéphant, A. Charara, and D. Meizel, Force model comparison on the wheel-ground contact for vehicle dynamics, in Proc. IEEE Intelligent Vehicle Symp., Versailles, France, June 2002, pp [7] D. Lechner, Analyse du comportement dynamique des véhicules routiers légers: Développement d une méthodologie appliquée à la sécurité primaire, Ph.D. dissertation, École Centrale de Lyon, Lyon, France, Oct [8] L. R. Ray, Nonlinear tire force estimation and road friction identification: Simulation and experiments, Automatica, vol. 33, no. 10, pp , [9] S. G. Mohinder and P. A. Angus, Kalman Filtering Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, [10] W. Perruquetti and J.-P. Barbot, Sliding Mode Control in Engineering. New York: Marcel Dekker, [11] H. Imine, Observation d états d un véhicule pour l estimation du profil dans les traces du roulement,, Univ. Versailles-Saint-Quentin-en-Yvelines, Versailles, France, Dec [12] D. Meizel, Observabilité et observation d état en robotique mobile, in Proc. Journées Nationales de la Recherche en Robotique, Presqu île de Giens, France, Oct. 2001, pp [13] J. Stéphant, A. Charara, and D. Meizel, Vehicle sideslip angle observers, in Proc. European Control Conf. (ECC 2003), Cambridge, U.K., Sept. 2003, pp [14] J. Y. Wong, Theory of Ground Vehicles, 2nd ed. New York: Wiley- Interscience, Joanny Stéphant was born in Lorient, France, in He received the Masters degree in 2001 from the Université de Technologie de Compiègne, Compiègne, France, where he is currently working toward the Ph.D. degree in the Laboratoire Heudiasyc (UMR CNRS 6599). His research areas are observers, observability, diagnosis, and vehicle dynamics. Ali Charara (M 95) was born in Bent-Jbeil, Lebanon, in He received the B.S. degree in electrical engineering from the Lebanese University, Beirut, Lebanon, in 1987, the M.S. degree in automatic control from ENSIEG-INPG, Grenoble, France, in 1988, and the Ph.D. degree in automatic control from the University of Savoie, Savoie, France, in Since 1992, he has been with the Department of Information Processing Engineering, Université de Technologie de Compiègne, Compiègne, France, where he became an Assistant Professor in 1992 and a Professor in He is also conducting research in the Laboratoire Heudiasyc (UMR CNRS 6599). His current research interests include control of nonlinear systems, vehicular control systems, fault detection, and diagnosis of electromechanical systems. Dominique Meizel was born in He received the Engineer degree, the Ph.D. degree in 1979, and the Doctorat d Etat degree in 1984 from the Ecole Centrale de Lille, Lille, France. From 1988 to 2003, he was a Professor at the Université de Technologie de Compiègne, Compiègne, France, where he was deeply involved in mobile robotics and intelligent vehicles. He has also been involved in the analysis of the relevance of various advanced driver assistance systems from both human factors and technological viewpoints. His main field of interest is the application of state observer techniques to data fusion problems. Since September 2003, he has been the Head of the Mechatronics Department, ENSIL Parc d Ester Technopole, Limoges, France.

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