Unscented Kalman Filter for Real-Time Vehicle Lateral Tire Forces And

Unscented Kalman ﬁlter for real-time vehicle lateral tire forces and sideslip angle estimation

Moustapha Doumiati, Alessandro Victorino, Ali Charara and Daniel Lechner

Abstract— Knowledge of vehicle dynamic parameters is im- trajectories, and to better vehicle control. Moreover, it makes portant for vehicle control systems that aim to enhance vehicle possible the development of a diagnostic tool for evaluating handling and passenger safety. Unfortunately, some fundamen- the potential risks of accidents related to poor adherence or tal parameters like tire-road forces and sideslip angle are difficult to measure in a car, for both technical and economic dangerous maneuvers. reasons. Therefore, this study presents a dynamic modeling and Vehicle dynamic estimation has been widely discussed in observation method to estimate these variables. The ability to the literature. Several studies have been conducted regarding accurately estimate lateral tire forces and sideslip angle is a crit- the estimation of tire-road forces, sideslip angle and the ical determinant in the performances of many vehicle control friction coefficient. For example, in [9], the authors esti- system and making vehicle’s danger indices. To address system nonlinearities and unmodeled dynamics, an observer derived mate longitudinal and lateral vehicle velocities and evaluate from unscented Kalman filtering technique is proposed. The sideslip angle and lateral forces. In [13] and [5], sideslip estimation process method is based on the dynamic response of angle estimation is discussed in detail. More recently, the a vehicle instrumented with easily-available standard sensors. sum of front and rear lateral forces have been estimated in Performances are tested using an experimental car in real [4] and [6]. However, lateral forces at each tire/road level driving situations. Experimental results show the potentiel of the estimation method. were not estimated. Other studies present valid tire/road force observers, but these involve wheel-torque measurements re- I. INTRODUCTION quiring expensive sensors [15]. The goal of this study is to develop an estimation method that Today, automotive electronic technologies are developing uses a simple vehicle model and a certain number of valid for safe and comfortable travelling of drivers and passengers. measurements in order to estimate in real-time the lateral There are a lot of vehicle control system such as Rollover force at each tire-road contact point and the sideslip angle. Prevention System, Adaptive Cruise Control System, Elec- The observation system is highly nonlinear and presents tronic Stability Control (ESC) and so on. These control unmodeled dynamics. For this reason, an observer based on systems installation rate is increasing all around the world. UKF (the Unscented Kalman Filter) is proposed. Vehicle control algorithms have made great strides towards In order to show the effectiveness of the estimation method, improving the handling and safety of vehicles. For example, some validation tests were carried out on an instrumented experts estimate that ESC prevents 27% of loss-of-control vehicle in realistic driving situations. accidents by intervening when emergency situations are The remainder of the paper is organized as follows. Section 2 detected [1]. While nowadays vehicle control algorithms are describes briefly the estimation process algorithm. Section 3 undoubtedly a life-saving technology, they are limited by the presents and discusses the vehicle model and the tire-road available vehicle state information. interaction phenomenon. Section 4 describes the observer Vehicle control systems currently available on production and presents the observability analysis. In section 5 the cars rely on available inexpensive measurements such as results are discussed and compared to real experimental data, longitudinal velocity, accelerations and yaw rate. Sideslip and then in the final section we make some concluding rate can be evaluated using yaw rate, lateral acceleration remarks regarding our study and future perspectives. and vehicle velocity [2]. However, calculating sideslip angle from sideslip rate integration is prone to uncertainty and II. ESTIMATION PROCESS DESCRIPTION errors from sensor bias. Besides, these control systems use unsophisticated, inaccurate tire models to evaluate lateral The estimation process is shown in its entirety by the block tire dynamics. In fact, measuring tire forces and sideslip diagram in figure 1, where ax and aym are respectively the angles is very difficult for technical, physical and economic longitudinal and lateral accelerations, ψ˙ is the yaw rate, θ˙ is reasons. Therefore, these important data must be observed the roll rate, ∆ij (i represents the front(1) or the rear(2) and or estimated. If control systems were in possession of the j represents the left(1) or the right (2)) is the suspension complete set of lateral tire characteristics, namely lateral deflection, wij is the wheel velocity, Fzij and Fyij are forces, sideslip angle and the tire-road friction coefficient, respectively the normal and lateral tire-road forces, β is the they could greatly enhance vehicle handling and increase sideslip angle at the center of gravity (cog). The estimation passenger safety. process consists of two blocks, and its role is to estimate As the motion of a vehicle is governed by the forces sideslip angle, normal and lateral forces at each tire/road generated between the tires and the road, knowledge of contact point, and consequently evaluate the used lateral the tire forces is crucial when predicting vehicle motion. friction coefficient. The following measurements are needed: Accurate data about tire forces and sideslip angle leads to • yaw and roll rates measured by gyrometers, a better evaluation of road friction and the vehicles possible • longitudinal and lateral accelerations measured by accelerometers, M. Doumiati, A. Victorino and A. Charara are with Heudi- asyc Laboratory, UMR CNRS 6599, Universite´ de Technologie de • suspension deflections using suspension deflections sen- Compiegne,` 60205 Compiegne,` France [email protected], sors, [email protected] and [email protected] • steering angle measured by an optical sensor, D. Lechner is with Inrets-MA Laboratory, Departement of Accident Mechanism Analysis, Chemin de la Croix Blanche, 13300 Salon de • rotational velocity for each wheel given by magnetic Provence, France [email protected] sensors.

978-1-4244-3504-3/09/$25.00 ©2009 IEEE 901 Suspension ∆ij vehicle. Assuming front-wheel drive, rear longitudinal forces deflection are neglected relative to the front longitudinal forces; we sensors Block 1 assume that Fx1 = Fx11 + Fx12. Roll plane model + . The lateral dynamics of the vehicle can be obtained by θ coupling longitudinal/lateral summing the forces and moments about the vehicle’s center Inertial ax, aym dynamics of gravity. Consequently, the simplified FWVM is formulated sensors as the following dynamic relationship: . ψ F β δ F β δ Fzij ax, ay 1 x1 cos( − ) + y11 sin( − )+ V˙g = , m Fy12 sin(β − δ) + (Fy21 + Fy22) sin β Block 2 L F δ F δ F δ Magnetic wij 1[ y11 cos + y12 cos + x1 sin ]− sensor Four wheel vehicle ¨ 1 L [F + F ]+ ψ = I 2 y21 y22 , model + Dugoff tire z E " [Fy11 sin δ − Fy12 sin δ] # model 2 δij Optical 1 −Fx1 sin(β − δ) + Fy11 cos(β − δ)+ sensor β˙ = − ψ,˙ mVg Fy12 cos(β − δ) + (Fy21 + Fy22) cos β Fyij β 1 ay = [Fy11 cos δ + Fy12 cos δ + (Fy21 + Fy22) + Fx1sinδ], Fig. 1. Process estimation diagram m 1 ax = m [−Fy11 sin δ − Fy12 sin δ + Fx1 cos δ], (1) where m is the vehicle mass, and Iz the yaw moment of The first block aims to provide the vehicle’s weight, lateral inertia. load transfer, normal tire forces and the corrected lateral The tire slip angle (αij ), as shown in figure 2, is the acceleration ay (by canceling the gravitational acceleration difference between the tire’s longitudinal axis and the tire’s component that distorts the accelerometer signal aym). We velocity vector. The tire velocity vector can be obtained have looked in detail at the first block in a previous study from the vehicle’s velocity (at the cog) and the yaw rate. [11]. This article focuses only on the second block, whose Assuming that rear steering angles are approximately null, main role is to estimate lateral tire forces and sideslip angle. the direction or heading of the rear tires is the same as that The second block makes use of the estimations provided by of the vehicle. The heading of the front tires includes the the first block. In fact, it is of major importance to include steering angle (δ). The front steering angles are assumed to the impact of accurate normal forces in the calculation of be equal (δ11 = δ12 = δ). The vehicle velocity Vg, steering lateral forces. ˙ One specificity of this estimation process is the use of blocks angle δ, yaw rate ψ and the vehicle body slip angle β are in series. By using cascaded observers, the observability then used to calculate the tire slip angles αij , where: problems entailed by an inappropriate use of the complete Vg β+L1ψ˙ α11 = δ − arctan , modeling equations are avoided, enabling the estimation Vg −Eψ/˙ 2 process to be carried out in a simple and practical way. h i Vg β+L1ψ˙ α12 = δ − arctan , Vg +Eψ/˙ 2 III. VEHICLE-ROAD MODEL h i (2) ˙ α = − arctan Vg β−L2ψ , This section presents the vehicle model and tire-road 21 V −Eψ/˙ 2 interaction dynamics, especially the lateral tire forces. Since g h i the quality of the observer largely depends on the accuracy of Vg β−L2ψ˙ α22 = − arctan . the vehicle and tire models, the underlying models must be Vg +Eψ/˙ 2 precise. Taking real-time calculation requirement, the models h i should also be simple. B. lateral tire-force model A. Four-wheel vehicle model The model of tire-road contact forces is complex because a wide variety of parameters including environmental factors The Four-Wheel Vehicle model (FWVM) is chosen for and pneumatic properties (load, tire pressure, etc.) impact this study because it is simple and corresponds sufficiently the tire-road contact interface. Many different tire models to our objectives. The FWVM is widely used to describe are to be found in the literature, based on the physical transversal vehicle dynamic behavior [4], [9], [5]. nature of the tire and/or on empirical formulations deriving Figure 2 shows a simple diagram of the FWVM model from experimental data, such as the Pacejka, Dugoff and in the longitudinal-lateral plane. In order to simplify the Burckhardt models [8], [7], [5]. Dugoff’s model was selected lateral and longitudinal dynamics, drive force and rolling for this study because of the small number of parameters that resistance were neglected. Additionally, the front and rear are sufficient to evaluate the tire-road forces. The nonlinear track widths (E) are assumed to be equal. L1 and L2 lateral tire forces are given by: represent the distance from the vehicle’s center of gravity to the front and rear axles respectively. The sideslip at the Fyij = −Cαitanαij .f(λ) (3) vehicle center of gravity (β) is the difference between the where Cαi is the lateral stiffness, αij is the slip angle and velocity heading (Vg) and the true heading of the vehicle f(λ) is given by: (ψ). The yaw rate (ψ˙) is the angular velocity of the vehicle (2 − λ)λ, if λ < 1 about the center of gravity. The longitudinal and lateral f(λ) = (4) forces (Fx,y,i,j) are shown for front and rear tires of the 1, if λ ≥ 1

902 α11 δ11 length is the distance covered by the tire while the tire force is kicking in. Using the relaxation model presented in [3], Fy11 E lateral forces can be written as: Fx11 Fy11 Vg F˙yij = (−Fyij + Fyij ), (6) α12 σi Vg δ12 Fy21 β α21 Fy12 where Fyij is calculated from a Dugoff tire-force model, Vg Fx21 ψ Fx12 is the vehicle velocity and σi is the relaxation length. Fy11 IV. OBSERVERS DESIGN

Fy22 α22 This section presents a description of the observer devoted L1 to lateral tire forces and sideslip angle. The state-space formulation, the observability analysis and the estimation Fx22 method will be presented. L2 A. Stochastic state-space representation Fig. 2. Four-wheel vehicle model. The nonlinear stochastic state-space representation of the system described in previous section is given as:

2000 3000 Measurement X˙ (t) = f(X(t),U(t)) + w(t) Dugoff 2000 (7) 0 Y (t) = h(X(t),U(t)) + v(t)

(N) (N) 1000 11 12 0 Fy −2000 Fy The input vector U comprises the steering angle and the −1000 normal forces considered estimated by the ﬁrst block (see −4000 −2000 −0.04−0.02 0 0.02 0.04 −0.04−0.02 0 0.02 0.04 ﬁgure 1): α (rad) α (rad) 11 12 U = [δ, F , F ,, F , F ]. (8) 1000 z11 z12 z21 z22 2000 0 1000 The measure vector Y(t) comprises yaw rate, vehicle velocity (N) (N) 21 −1000 22 (approximated by the mean of the rear wheel velocities Fy Fy 0 −2000 calculated from wheel-encoder data), longitudinal and lateral −1000 −3000 accelerations: −0.02 0 0.02 0.04 −0.02 0 0.02 0.04 α (rad) α (rad) ˙ 21 22 Y = [ψ, Vg, ax, ay]. (9)

Fig. 3. Validation of the Dugoff model. The state vector comprises yaw rate, vehicle velocity, sideslip angle at the cog, lateral forces and the sum of the front longitudinal tire forces: µF λ = zij (5) X = [ψ,˙ V , β, F , F , F , F , F ]. (10) 2Cαi |tanαij | g y11 y12 y21 y22 x1 In the above formulation, µ is the coefficient of friction of the The process and measurements noise vectors, respectively road and Fzij is the normal load on the tire. This simplified w(t) and v(t), are assumed to be white, zero mean and tire model assumes no longitudinal forces, a uniform pressure uncorrelated. distribution, a rigid tire carcass, and a constant coefficient of The nonlinear functions (f) and (h) representing the state friction of sliding rubber. and the observation equations are calculated according to In order to validate the chosen model, many tests were equations (1) and (6). carried out using a laboratory car (see section V-A). Figure 3 compares measured forces and the Dugoff model. These B. Observability results prove the model validity. As shown in figure 3, the relationship between the lateral Observability is a measure of how well the internal states force and the slip angle is initially linear (low slip angles). of a system can be inferred from knowledge of its inputs and When operating in the linear region, a vehicle responds external outputs. This property is often presented as a rank predictably to the driver’s inputs. As the vehicle approaches condition on the observability matrix. Using the nonlinear the handling limits, for example during an evasive emergency state-space formulation of the system presented in section maneuver, or when a vehicle undergoes high accelerations, IV-A, the observability definition is local and uses the Lie high slip angles occur and the vehicle’s dynamic becomes derivative [12]. An observability analysis of this system highly nonlinear and its response becomes less predictable was undertaken in [14]. It was shown that the system is and potentially very dangerous. observable except when: • steering angles are null, C. Relaxation model • the vehicle is at rest (Vg = 0). When vehicle sideslip angle changes, a lateral tire force is For these situations, we assume that lateral forces and created with a time lag. This transient behavior of tires can sideslip angle are null, which approximately corresponds to be formulated using a relaxation length σ. The relaxation the real cases.

903 where Measurements m λ w0 = n+λ Inputs c λ 2 w0 = n+λ + (n − α + β)  m c 1 (14) + w = w = i = 1,..., 2n X=f(X,U) h(X,U) -  i i 2(n+λ) sensors  2 evolution + λ = n(α − 1) + K X  correction Kalman Gain Measurement update observer 2n c ¯ PY¯kY¯k = i=0 wi (γi,k|k−1 − Yk|k−1)× ¯ T Fig. 4. Process estimation diagram (γi,k|k−1 − Yk|k−1) + R  P2n c P ¯ ¯ = w (χ − X¯ )×  XkYk i=0 i i,k|k−1 k|k−1  ¯ T  (γi,k|k−1 − Yk|k−1) (15)  P −1  Kk = P ¯ ¯ P  XkYk Y¯kY¯k C. Estimation method T Pk = Pk|k−1 − KkPY¯kY¯k Kk The aim of an observer or a virtual sensor is to estimate  X¯ X¯ K Y Y¯  k = k|k−1 + k( k − k|k−1) a particular unmeasurable variable from available measure-   ¯ ¯ ments and a system model in a closed loop observation where the variables are defined as follows: Xk and Yk|k−1 scheme, as illustrated in figure 4. The observer was imple- are the estimations respectively of the state and of the real mented in a first-order Euler approximation discrete form. At measurement at each instant k. wi is a set of scalar weights, each iteration, the state vector is first calculated according and n is the state dimension; the parameter α determines to the evolution equation and then corrected online with the spread of the sigma points around X¯ and is usually set the measurement errors and the filter gain K in a recur- to 1e − 4 < α < 1. The constant β is used to incorporate sive prediction-correction mechanism. The gain is calculated part of the prior knowledge of the distribution of X, and using the Kalman filter technique (EKF, UKF, . . . ). The EKF for Gaussian distributions, β = 2 is optimal. Q and R are (Extended Kalman Filter) is probably the most commonly- respectively the disturbance and sensor-noise covariance: R used estimator for nonlinear systems. However, in this study takes into account the uncertainty in the measured data and the UKF (Unscented Kalman Filter) [16] is chosen for the Q is tuned depending on the model quality. following reasons: V. EXPERIMENTAL RESULTS • the high nonlinearities of the vehicle-road model, In this section we present the experimental car used to test • the calculation complexity of the Jacobian matrices which causes implementation difficulties. the observers potential, and we discuss and analyze the test conditions and the observers’ results. 1) UKF algorithm: In this subsection, the principle of the UKF is summarized. Consider the general discrete nonlinear A. Experimental car system: The experimental vehicle shown in figure 4 is the INRETS-MA (Institut National de la Recherche sur les X = f(X ,U ) + w Transports et leur Securit´ e´ - Departement´ Mecanismes´ k+1 k k k (11) Yk = h(Xk) + vk d’Accidents) Laboratory’s test vehicle [2]. It is a Peugeot 307 equipped with a number of sensors including accelerometers, n r gyrometers, steering angle sensors, linear relative suspension where Xk ∈ R is the state vector, Uk ∈ R is the known input vector, Y ∈ Rm is the output vector at time k. w sensors, three Correvits and four dynamometric wheels. One k k Correvit is located in chassis rear overhanging position and and vk are, respectively, the disturbance and sensor noise vector, which are assumed to be Gaussian white noise with it measures longitudinal and lateral vehicle speeds and two zero mean and uncorrelated. Correvits are installed on the front right and rear right tires The UKF can be formulated as follows: and they measure front and rear tires velocities and sideslip Initialization angles. The dynamometric wheels fitted on all four tires, which are able to measure tire forces and wheel torques in X¯ = E[X ] and around all three dimensions. 0 0 (12) We note that the correvit and the wheel-force transducer (see P = E[(X − X¯ )(X − X¯ )T ] 0 0 0 0 0 figure 5) are very expensive sensors. The sampling frequency of the different sensors is 100 Hz. where X¯0 and P0 are respectively the initial state and the initial covariance. B. Test conditions Sigma points calculation and time update Test data from nominal as well as adverse driving conditions were used to assess the performance of the observer χk−1 = [X¯k−1, X¯k−1 ± (n + λ)Pk−1] presented in section IV, in realistic driving situations. We ∗ report a lane-change manoeuvre where the dynamic contribu- χk|k−1 = f(χk−1,Uk−1)  2n m ∗ p tions play an important role. Figure 6 presents the Peugeot’s  x¯k|k−1 = i=0 wi χi,k|k−1  trajectory (on a dry road), its speed, steering angle and  P = 2n wc(χ∗ − X¯ )× ”g-g” acceleration diagram during the course of the test.  k|k−1 Pi=0 i i,k|k−1 k|k−1 (13)  (γ∗ − X¯ )T + Q The acceleration diagram, that determines the maneuvering  Pi,k|k−1 k|k−1  ¯ ¯ area utilized by the driver/vehicle, shows that large lateral χk|k−1 = [Xk−1, Xk−1 ± (n + λ)Pk|k−1] accelerations were obtained (absolute value up to 0.6g). This  γk|k−1 = h(χk|k−1) means that the experimental vehicle was put in a critical  ¯ 2n m p  Yk|k−1 = i=0 wi γi,k|k−1 driving situation.    P

904 Front left lateral force Fy (N) 11 2000 measurement 1000 Estimation

−1000

−2000 0 10 20 30 40 50

Front right lateral force Fy (N) 12 Wheel transducer Correvit 4000

Fig. 5. Wheel-force transducer and sideslip sensor installed at the tire 2000 level. 0

−2000 200 0 10 20 30 40 50

) 110 Time (s) 1 100 100 Fig. 7. Estimation of front lateral tire forces. 0 90 Y position (m) Speed (km.h 80 −100 Rear left lateral force Fy (N) 0 500 1000 0 20 40 21 X position (m) Time (s) measurement 1000 Estimation 0.04 0.1 0 0.02 0.05 −1000 0 0

−0.02 −0.05 0 10 20 30 40 50

Steering angle (rad) −0.1 −0.04 Rear right lateral force Fy (N) 0 20 40 Longitudinal acceleration (g) −0.4 −0.2 0 0.2 0.4 0.6 22

Time(s) Lateral acceleration (g) 3000 2000

Fig. 6. Experimental test: vehicle trajectory, speed, steering angle and 1000 acceleration diagrams for the lane-change test 0

−1000

0 10 20 30 40 50 The estimation process algorithm was written in C++ and Time (s) has been integrated into the laboratory car as a DLL (Dy- namic Link Library) that functions according to the software acquisition system. Fig. 8. Estimation of rear lateral tire forces. C. Validation of observers The observer results are presented in two forms: as tables of normalized errors, and as figures comparing the This result is clearly a consequence of the load transfer measurements and the estimations. The normalized error for produced during cornering from the left to the right-hand an estimation z is defined as: side of the vehicle. In fact, lateral force increases as normal force increases. Figure 9 shows how sideslip angle changes kzobs − zmeasuredk during the test. Reported results are relatively good. ǫz = 100 × (16) Table I presents maximum absolute values, normalized mean max z (k measuredk) errors and normalized std (standard deviations) for lateral where zobs is the variable calculated by the observer, tire forces and sideslip angles. Despite the simplicity of the zmeasured is the measured variable and max(kzmeasuredk) is model, we can deduce that for this test, the performance of the absolute maximum value of the measured variable during the observer is satisfactory, with normalized error globally the test maneuver. less than 8%. Figures 7 and 8 show lateral forces on the front and Given the vertical and lateral tire forces at each tire-road rear wheels. According to these plots, the observers are contact level, the estimation process is able to evaluate the relatively good with respect to measurements. Some small used lateral friction coefficient µ. This is defined as the ratio differences during the trajectory are to be noted. These of friction force to normal force, and is given by [10]: might be explained by neglected geometrical parameters, especially the camber angles, which also produce a lateral Fyij µij = (17) forces component [10]. It is also shown that the lateral forces Fzij on the right-hand tires exceed those on the left-hand tires.

905 Sideslip angle at the cog (rad) Max kk Mean % Std % Fy11 2180 (N) 8.23 4.80 0.01 measurement Fy12 4070 (N) 3.70 3.74 Estimation Fy21 1441 (N) 7.51 3.52 0.005 Fy22 2889 (N) 1.91 1.77 β 0.027 (rad) 8.32 6.41 0 TABLE I MAXIMUM ABSOLUTE VALUES, NORMALIZED MEAN ERRORS, AND −0.005 NORMALIZED STD.

−0.01

−0.015 risk of leaving the road. The developed observer is derived −0.02 from a simpliﬁed four-wheel vehicle model and is based on unscented Kalman ﬁltering technique. Tire-road interaction −0.025 is represented by the Dugoff model.

10 20 30 40 50 60 A comparison with real experimental data demonstrates the Time (s) potential of the estimation process. It is shown that it may be possible to replace expensive correvit and dynamometric Fig. 9. Estimation of the sideslip angle at the cog. hub sensors by real-time software observers. This is one of the important results of our work. Another important result concerns the estimation of individual lateral forces acting on each tire. This can be seen as an advance with respect to the µ11 µ12 1 current vehicle-dynamics literature. 0.5 Future studies will improve the vehicle/road model in order 0.5 to widen validity domains for the observer, and will take into 0 account road irregularities and road bank angle. Moreover, it will be of major importance to study the effect of coupling 0 longitudinal/lateral dynamics on lateral tire behavior. −0.5

0 20 40 0 20 40 REFERENCES Time (s) Time (s) [1] Y. Hsu and J. Chistian Gerdes, Experimental studies of using steering µ µ 21 22 torque under various road conditions for sideslip and friction esti- 1 0.5 mation, Proceedings of the 2007 IFAC Symposium on Advances in Automotive Control, Monterey, California. [2] D. Lechner, Embedded laboratory for vehicle dynamic measurements, 0.5 0 9th International symposium on advanced vehicle control, Kobe, Japan, Octobre 2008. [3] P. Bolzern, F. Cheli, G. Falciola and F. Resta, Estimation of the 0 nonlinear suspension tyre cornering forces from experimental road −0.5 test data, Vehicle System Dynamics, volume 31, pages 23-24, 1999.

0 20 40 0 20 40 [4] G. Baffet A. Charara, D. Lechner and D. Thomas, Experimental Time (s) Time (s) evaluation of observers for tire-road forces, sideslip angle and wheel cornering stiffness, Vehicle System Dynamics, volume 45, pages 191- 216, june 2008. Fig. 10. Used lateral friction coefficients developed by the tires [5] U. Kiencke and L. Nielsen, Automotive control systems, Springer, 2000. [6] G. Baffet, A. Charara and J. Stephant, An observer of tire-road forces and friction for active-security vehicle systems, IEEE/ASME Transactions on Mechatronics, volume 12, issue 6 pages 651-661, 2007. The lateral friction coefficients in figure 10 show that the [7] J. Dugoff, P. Fanches and L. Segel, An analysis of tire properties and estimated µij are close to the measured values. A closer their influence on vehicle dynamic performance, SAE paper (700377), investigation reveals that the used lateral friction coefficients 1970. [8] Hans B. Pacejka, Tyre and vehicle dynamics, Elsevier, 2002. µ12 and µ22 corresponding to the overloaded tires during [9] J. Dakhlallah, S. Glaser, S. Mammar and Y. SebsadjiTire-road forces cornering are lower than µ11 and µ21. This phenomenon is estimation using extended Kalman filter and sideslip angle evaluation, due to the tire load sensitivity effect: the lateral friction coef- American Control Conference, Seatle, Washington, U.S.A, June 2008. [10] W.F. Milliken and D.L. Milliken, Race car vehicle dynamics, Society ficient is normally higher for the lighter loads, or conversely, of Automotive Engineers, Inc, U.S.A, 1995. falls off as the load increases [10]. [11] M. Doumiati, A. Victorino, A. Charara, D. Lechner and G. Baffet, An This test also demonstrates that µ and µ are high, estimation process for vehicle wheel-ground contact normal forces, 11 21 IFAC WC’08, Seoul Korea, july 2008. especially for lateral accelerations up to 0.6, and that they [12] H. Nijmeijer and A.J. Van der Schaft, Nonlinear dynamical control attain the limit for the dry road friction coefficient. In fact, systems, Springer Verlag, 1991. dry road surfaces show a high friction coefficient in the range [13] J. Stephant,´ A. Charara and D. Meizel, Virtual sensor, application to vehicle sideslip angle and transversal forces, IEEE Transactions on 0.9-1.2 (implying that driving on these surfaces is safe), Industrial Electronics. vol.51, no. 2, April 2004. which means that for this test the limits of handling were [14] G. Baffet, Developpement´ et validation exprimentale d’observateurs reached. des forces de contact pneumatique/chaussee´ d’une automobile, Ph.D thesis, University of Technology of Compiegne,` France, 2007. [15] N. K. M’Sirdi, A. Rabhi, N. Zbiri and Y. Delanne, Vehicle-road interaction modelling for estimation of contact forces, Vehicle System VI. CONCLUSION Dynamics, volume 43, pages 403-411, 2005. [16] S. J. Julier and J. K. Uhlman, A new extension of the Kalman filter This paper presents a new method for estimating lateral to nonlinear systems, International Symposium Aerospace/Defense tire forces and sideslip angle, that is to say two of the Sensing, Simulation and Controls, Orlando, USA, 1997. most important parameters affecting vehicle stability and the

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