Nonlinear observer of sideslip angle using a particle filter estimation methodology

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Nonlinear observer of sideslip angle using a particle filter estimation methodology ⋆ Qi Cheng, Alessandro Correa Victorino and Ali Charara Laboratoire Heudiasyc, CNRS UMR 6599, Universit´e de Technologie de Compi`egne, Centre de recherche Royallieu, 60205 Compi`egne, France. (e-mail: [email protected] ). Abstract: The knowledge of the vehicle dynamic is important to improve the stability control of modern automotive engineering. Some key variables of the vehicle dynamic, such as the sideslip angle, are difficult to measure directly for technology or economic reasons. Lots of algorithms have been proposed to estimate these variables. To the best of our knowledge, most of these algorithms are based on linearization techniques, among which the extended Kalman filter is the most frequently used algorithm in the unmeasurable variable estimation for automotive control. In this paper, we propose two new nonlinear observers which uses the particle filter and the modified bootstrap filter to estimate the sideslip angle, respectively. These observers are based on the nonlinear double track model, in which the Dugoff model is used to describe the relation between the tire road forces and the sideslip angle. The good performances of these two observers are demonstrated by two classic experiments. Keywords: Vehicle dynamics, Nonlinear models, Automotive control, State estimation 1. INTRODUCTION Improving the vehicle security is one of the most important topics in the modern automotive engineering. It is reported that losing control is the main reason of the car accidents. To avoid this emergency situation, lots of electronic control systems, like Electronic Stability Program (ESP) and Dynamic Stability Control (DSC) have been equipped in vehicles. These electronic control systems need the vehicle motion information to detect and minimize skids. The sideslip angle, is one of the key variables to understand the vehicle dynamics, which is the deviation between the vehicle’s longitudinal axis and the direction of travel at the center of gravity. However, it is too expensive to measure directly the sideslip angle for common vehicles. A low cost solution which still provides accurate estimation results, is necessary for the automotive engineering. A large literature has been devoted to estimate the sideslip angle from analytical and experimental studies, see Baffet et al. (2008), Cheli et al. (2007), Doumiati et al. (2010), Jazar (2008), Kiencke and Nielsen (2005), Vietinghoff et al. (2005), Kim (2009), Piyabongkarn et al. (2009), You et al. (2009) and R. Rajamani and Grogg (2006). In Baffet et al. (2008), the estimation process is separated into two blocks. In the first block, it assumes that the sideslip angle is given, and the tire forces are modeled as a random walk model. In the second block, the sideslip angle is unknown and estimated by the extended Kalman filter (EKF). The similar idea is presented by Doumiati et al. (2010), in which these two blocks are integrated into one block. In Kim (2009), the vehicle model with four degrees of freedom is presented and the sideslip angle is estimated by the extended Kalman filter (EKF). In Vietinghoff et al. (2005), ⋆ This work is supported by the French ANR project PERCOIVE.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

a nonlinear double track model for the sideslip angle and the corresponding nonlinear observer with adaption of a quality function is proposed. Other details about the vehicle dynamics can be found in some excellent books, e.g. Kiencke and Nielsen (2005), R. Rajamani and Grogg (2006) and Jazar (2008). In You et al. (2009) and Piyabongkarn et al. (2009), the sideslip angle is calculated directly from the lateral acceleration, the cornering stiffness and the tire steer angle and the lateral tire forces are modeled linearly with the tire sideslip angles. But they use different sideslip angle estimation methods, the former uses the yaw-ratedynamics-based approach, while the latter employs a new estimation methodology to estimate the sideslip angle, which combines the yaw-rate-dynamics estimation and the kinematics based estimation. The kinematic formula is applied to obtain the sideslip angle in Cheli et al. (2007). In Baffet et al. (2008), Kim (2009) and Cheli et al. (2007), the sideslip angle is modeled by the state space models and estimated by the EKF. State space models are popular for the following reasons. The vehicle dynamic models are not exactly the same as the true physical processes. Generally, the practical physical processes are more complex than the supposed models. They contain many incertitudes. In the state space models, these incertitudes in the physical processes can be treated as the noises, which makes the results gotten by the state space models being more flexible than the one of the direct computation. In this paper, we estimate the sideslip angle from the nonlinear state spaces models. The EKF is widely used to estimate the sideslip angle from the state space models for its simplicity. The EKF first uses a first-order Taylor series expansion to approximate

6266

10.3182/20110828-6-IT-1002.01162

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

the nonlinear state transition equation as well as the measurement equation. After the approximation, the classical Kalman filter can be used to estimate the states. When the nonlinearity of the models is small, the deviation of the linearisation is not big, the estimation results of the EKF are acceptable. Therefore the EKF provides an accurate sideslip angle estimation if the sideslip angle is small. But the EKF estimation results become more and more biased, as the sideslip angle increases and the nonlinearity of the vehicle model is more obvious. Another successful nonlinear filter is the particle filter (PF), which can be regarded as an approximation of a recursive Bayesian filter. The principle of PF is to implement a recursive Bayesian filter by Monte Carlo simulations. The densities involved in the Bayesian filter are represented by a set of random samples with associated weights (Arulampalam et al. (2002)). The advantage of PF is its ability to process the nonlinear model with satisfactory accuracy. An excellent tutorials of PF is given by Arulampalam et al. (2002). How to choose the important distribution (ID) is one of the key problem in the design of a PF. In classic PF, the transition density is chosen as the ID. This choice can not use the latest information of the observation and when the variance of the state noise is large, the transition density contains little information to predict new state, a huge number of particles is needed in the PF. To better use the latest information of the observation in the state space models, a modified bootstrap filter (MBF), see Cheng and Bondon (2009) and Cheng and Bondon (2011), is chosen to estimate the hidden state.

gravity, while L2 is the one between the rear axle and the center of gravity. E is the vehicle track. Vg is the velocity at the center of gravity. β is the sideslip angle at the center of gravity. The αij , (i, j = 1, 2) is the sideslip angle at each wheel.

α11 Fy11 Fy11

Fx21

Vg α21

Fy22

α12

β Fy12

ψ

E

Fx11

δ12

Fx12

α22 L1 Fx22

δ22 L2

Fig. 1. Double track model A reduced nonlinear double track model is described by the following equations: 1 V˙g = [Fx11 cos(β − δ11 ) + Fx12 cos(β − δ12 ) m + Fx21 cos(β − δ21 ) + Fx22 cos(β − δ22 ) + Fy11 sin(β − δ11 ) + Fy12 sin(β − δ12 ) + Fy21 sin(β − δ21 ) + Fy22 sin(β − δ22 )] 1 ψ¨ = [L1 (Fy11 cos(δ11 ) + Fy12 cos(δ12 ) Iz + Fx11 sin(δ11 ) + Fx12 sin(δ12 )) − L2 (Fy21 cos(δ21 ) + Fy22 cos(δ22 )) E + (Fy11 sin(δ11 ) − Fy12 sin(δ12 ) 2 + Fx12 cos(δ12 ) − Fx11 cos(δ11 )

2. VEHICLE MODEL

An illustration of this nonlinear double model is given in Fig 1, where the aerodynamic drag force and the rolling resistance are ignored. Further more, we suppose the road is plane. Fy11 , Fy12 , Fy21 and Fy22 are the lateral forces at the front left, front right, rear left and rear right wheel, respectively. The Fxij , (i, j = 1, 2) is the corresponding longitudinal force at each wheel. ψ˙ is the yaw rate and δij , (i, j = 1, 2) are the steering angles at each wheel. L1 is the distance between the front axle and the center of

δ21

Fy21

In this paper we choose a nonlinear double track model to describe the vehicle dynamics and use the Dugoff model to calculate the lateral tire forces. The PF and the MBF are employed to estimate the sideslip angle in this nonlinear model. The remainder of this paper is organized as follows. The vehicle model and the lateral tire forces models are introduced in Sections 2. In Section 3, we present two new nonlinear observers which use the PF and the MBF to estimate the sideslip angle, respectively. The simulation results are illustrated in Section 4. Finally, some conclusions are given in Section 5.

The single track model has been widely used to describe the vehicle dynamics for its simplicity, see Baffet et al. (2006) and Baffet et al. (2008). However, the single track model could not provide the detailed information of each wheel, a sufficient accurate model is needed. Therefore, we choose the nonlinear double track model which is more accurate than the single track model. A detailed presentation of the nonlinear double track model can be found in Vietinghoff et al. (2005).

δ11

+ Fx22 cos(δ22 ) − Fx21 cos(δ21 ))], 1 [Fx11 sin(δ11 − β) + Fx12 sin(δ12 − β) β˙ = mVg + Fy11 cos(δ11 − β) + Fy12 cos(δ12 − β) + Fy21 cos(δ21 − β) + Fy22 cos(δ22 − β) ˙ + Fx21 sin(δ21 − β) + Fx22 sin(δ22 − β)] − ψ, where m is the mass of the vehicle and Iz is the yaw moment of the inertia. In this model, the velocity Vg and the yaw rate ψ˙ can be measured by sensors equipped in most of modern vehicles. These sensors are much cheaper than the sensors used for detecting the sideslip angle β. The exact model of the tire-road contact forces is complex, see Pacejka (2005). When the sideslip angle is small, the lateral force is linear with the wheel slip angle. Fyij =

6267

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Cij αij , i, j = 1, 2, where Cij is the cornering stiffness of each wheel. When the sideslip angle increases, the lateral force increases nonlinearly with it. After some threshold, the road-tire forces begin to be saturated. For the reason of simplicity, the linear model is popular used in the literatures, see Kiencke and Nielsen (2005) and Pacejka and Bakker (1993). The Pacejka Model (also called “the magic formula”) contains some specific parameters determined by the tire and the road conditions. When these specific parameters are unknown, other models like the Dugoff model (Dugoff et al. (1969)) are expected. In this paper, we choose the Dugoff model. The Dugoff model is more complex than the linear model, defined as Fyij = −Cij tan αij f (λ), (1) where Cij is the cornering stiffness which is assumed to be known, αij is the sideslip angle at each wheel and is calculated by L1 ψ˙ α11 = δ11 − β − , Vg L1 ψ˙ , α12 = δ12 − β − Vg L2 ψ˙ , α21 = −β + Vg L2 ψ˙ . α22 = −β + Vg and f (λ) is a nonlinear function of the sideslip angle and the vertical force,  (2 − λ)λ if λ < 1, f (λ) = 1 if λ ≥ 1, and

µFzij , 2Cij tan βi,j where µ is the friction coefficient between the tire and the road, Fzij is the vertical force on the wheel. λ=

3. THE DESIGN OF THE NONLINEAR OBSERVER 3.1 Nonlinear Observers We propose two nonlinear observers which use the PF and the MBF to estimate the sideslip angle β, respectively. The state variables, the observation variables and the input variables are the following: ˙ β], X = [Vg , ψ, ˙ ay ], Y = [Vg , ψ, (2) U = [δ11, δ12, δ21, δ22, Fx11, Fx12 , Fx21 , Fx22 , Fz11 , Fz12 , Fz21 , Fz22 ], where ay is the lateral acceleration measured directly by inertial sensors. It is calculated by 1 ay = (Fy11 cos(δ11 ) + Fy12 cos(δ12 ) + Fy21 cos(δ21 ) m + Fy22 cos(δ22 ) + Fx11 sin(δ11 ) + Fx12 sin(δ12 )) + Fx21 sin(δ21 ) + Fx22 sin(δ22 ). All the lateral tire forces are calculated by the Dugoff model (Equation (1)).

In the following, we use an Euler discretization for all continuous variables in (2) to obtain a discrete time system. Then, the state variables can be estimated by the PF and the MBF, respectively. 3.2 Particle filter Consider a dynamic nonlinear discrete time system described by the following state-space model xt = f (xt−1 ) + ut , (3) yt = h(xt ) + vt , (4) where xt is the hidden state, yt is the observation, ut and vt are the state and observation noises. Both noises are independent and identically distributed sequences and are mutually independent. When we write (3), we always assume implicitly that ut is independent of {xt−k , k ≥ 1}. This condition is natural when the process (xt ) is generated from the model in the increasing time order. Then, xt is a homogeneous Markov chain, i.e., the conditional probability density of xt given the past states x0:t−1 = (x0 , . . . , xt−1 ) depends only on xt−1 through the transition density p(xt |xt−1 ). The conditional probability density of yt given the states x0:t and the past observations y1:t−1 depends only on xt through the conditional likelihood p(yt |xt ). We further assume that the initial state x0 is distributed according to a density function p(x0 ). The objective of filtering is to estimate the posterior density of the state given the past observations p(xt |y1:t ). A recursive update of the posterior density as new observations arrive, is given by the recursive Bayesian filter defined by Z p(xt |y1:t−1 ) = p(xt |xt−1 )p(xt−1 |y1:t−1 )dxt−1 , p(yt |xt )p(xt |y1:t−1 ) , p(yt |y1:t−1 ) where the conditional density p(yt |y1:t−1 ) can be calculated by Z p(yt |y1:t−1 ) = p(yt |xt )p(xt |y1:t−1 )dxt . p(xt |y1:t ) =

The difficulty to implement the recursive Bayesian filter is that the integrals are intractable, except for a linear Gaussian system in which case the closed-form solution of the integral equations is the well known Kalman filter. The PF uses Monte Carlo methods to calculate the integrals. The posterior density p(x0:t |y1:t ) is represented by a set of N random samples xi0:t (particles) drawn from i p(x 1:t ) with associated normalized positive weights ωt P0:t |y i ( i ωt = 1). The posterior density is approximated by the discrete distribution, N X ωti δxi0:t , p(x0:t |y1:t ) ≃ i=1

where δ is the Dirac function. In general, it is difficult to sample directly from the full posterior density. To overcome this difficulty, the method of importance sampling (Robert and Casella (2004)) is used. We draw particles xi0:t , i = 1, . . . , N from an easy sampling ID q(x0:t |y1:t ) and define the non-normalized weights as

6268

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

ωti =

p(xi0:t |y1:t ) . q(xi0:t |y1:t )

(2011)) is developed in which a two-stage sampling technique is used :

For the purpose of sequential implementation of the filtering, we suppose that the ID could factorize as q(x0:t |y1:t ) = q(xt |x0:t−1 , y1:t )q(x0:t−1 |y1:t−1 ), where the density p(xt |y1:t ) is a marginal of the full posterior density p(x0:t |y1:t ). Then, the weight could be updated sequentially as p(yt |xit )p(xit |xit−1 ) i ωti ∝ ωt−1 . (5) q(xit |xit−1 , yt ) We can implement recursively a basic sequential importance sampling (SIS) PF in the following steps (Arulampalam et al., 2002) : (1) Sample the particles xit ∼ q(xt |xit−1 , yt ); (2) Update the weights according to (5). A serious problem of the PF is the degeneracy problem. After some iterations, only few particles have non negligible weights so that the estimation may become unreliable. The sampling importance resampling filter has been developed by Gordon et al. (1993) to overcome this drawback. The objective of resampling is to eliminate samples with low importance weights and multiply samples with high importance weights. Several methods of resampling have been developed, such as the multinomial resampling, the residual resampling and the systematic resampling. Here, we use the residual resampling (Capp´e et al. (2005)). 3.3 A modified bootstrap filter Another problem of PF is the ID choice. The optimal ID should satisfy q(xt |x0:t−1 , y1:t ) = p(xt |xt−1 , yt ) and fully exploit the information both in xt−1 and yt , see e.g. Liu and Chen (1998) and Doucet et al. (2000). In practice, this distribution is unknown for a general nonlinear model and therefore, it is impossible to sample from it. The second choice of ID is the transition density, since it is easy to sample. This choice leads to the standard bootstrap filter (BF). The BF just uses the transition density as the ID but omits the information of the observation. When the likelihood lies in the tails of the transition density or it is too narrow, most particles drawn from the transition density have small weights and the estimation is not reliable. In this case, the likelihood provides more information than the transition density does.

(1) For j = 1, . . . , M , draw xti,j ∼ p(xt |xit−1 ) and compute the conditional likelihood p(yt |xi,j t ). i,j ⋆ (2) Select the particle xt whose conditional likelihood ⋆ . is maximum and set xit = xi,j t In the first step, the particles move randomly according to the prior information like in the BF, and in the second step, the information yt is used to select the particle with high conditional likelihood. The corresponding MBF is described in the following algorithm: Algorithm: Modified bootstrap filter. Initialization, t = 0 for i = 0 to N do Draw particle xi0 ∼ p(x0 ) and set t = 1 end for for t = 1 to T do for i = 1 to N do for j = 1 to M do i Draw particle xi,j t ∼ p(xt |xt−1 ) Compute the conditional likelihood p(yt |xi,j t ) end for ⋆ ⋆ ) is maximum such that p(yt |xi,j Select xi,j t t ⋆ Set xit = xi,j t end for Resample particle from the xit according to the weights p(yt |xit ) end for

As we mentioned before, the state variables are estimated by the PF and the MBF. We choose the transition density as the ID in the design of the PF. The covariances of the measurement noise and of the state noise are determined empirically according to the experimental conditions. The main disadvantage of the PF is that there is no general rule for choosing the number of particle. To achieve accurate result, a large number of particles should be used in the PF and the MBF. At the same time, increasing the number of particles means adding computational complexity. A compromise between estimation precise and computational time should be made carefully . 4. SIMULATION RESULTS

To improve the performance of the PF, we have two choices. The first one is drawing more particles. The second is using the information of yt and let the particles move towards the region of high likelihood. In the practical application, the variance of the state noise is larger than the variance of the observation noise, the transition density (prior information) contains little information to predict the future state. In this case, as pointed out in Van der Merwe et al. (2000) : “It is therefore of paramount importance to move the particles towards the regions of high likelihood”.

In our simulations, all the data are generated by the CALLAS software, which is a professional vehicle dynamic simulator developed by OKTAL Company. The CALLAS can take into account lots of practical influences like vertical suspension, tires type, engine model, kinematics, tire adhesion and aerodynamics. The simulation environment given by the CALLAS is very close to the practical experimental environment. To test the performance of different observers, two classic experiments are given by the CALLAS software: The Chicane test and the Slalom test.

To better use the observation information, the MBF (Cheng and Bondon (2009) and Cheng and Bondon

The performances of sideslip estimator based on the PF and the MBF are compared with the one based on the

6269

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Filter

RMSE (degree)

EKF PF MBF

2.37 1.95 1.33

1 0.8

Table 1. Estimation results of the sideslip angle: Chicane test

0.6

Lateral accleration

0.4

EKF. To measure the performance of the state estimation xt for t = 1, . . . , T (T is the number of the total sampling number), we introduce the root mean-squared error v u T u1 X RM SE = t (xt − x ˆt )2 . T t=1

0.2 0 −0.2 −0.4

4.1 Chicane test

−0.6

First, a Chicane test is doing to compare the performance of different observers. The road condition is set to be dry (µ = 1.1). The initial velocity of vehicle is 80km/h, the sampling interval is 0.01 second. We use 3000 particles in the PF and in the MBF. We choose M = 3 in the MBF. The sideslip angle estimation results are shown in Fig 3. When the sideslip angle is small (β < 4 degrees), all the estimators could provide accurate estimation. In Fig 3, in time interval 0 to 2, the dot line (EKF estimation), the dash line (PF estimation), dot-dash line (the MBF estimation) and the solid line (true value) are almost overlap. When the sideslip angle increases, from time interval 2 to 6, the four lines diverge obviously, the dotdash line is closer to the solid line than the other two lines, the dot line deviates from the solid line. So the EKF can not track the true angle when sideslip angle β > 4. The PF and the MBF give better estimation than the EKF. The MBF could further improve the performances of the PF. The RMSE of the different estimators is listed in Table 1. The MBF has a smallest RMSE among the 3 filters, and its RMSE is less than 60% of EKF’s RMSE. The PF follows the MBF and its RMSE is 80% of EKF’s RMSE. The “g-g” diagram of longitudinal and lateral accelerations is shown in Fig 2. It is clear that the acceleration exceeds the stability limit of 0.4g. The vehicle is spining and losing of control.

−0.8 −1 −0.04

−0.02

0 0.02 0.04 Longitudinal accleration

0.08

Fig. 2. ”g-g” diagram: Chicane test

12 true PF MBF EKF

10 8

sideslip angle

6 4 2 0 −2 −4 −6 −8

0

1

2

4.2 Slalom test Then, we do a Slalom test. The road condition is set to be sliding (µ = 0.4 and is supposed to be known). The initial velocity of vehicle is also 80km/h, the sampling interval is 0.01 second. The sideslip angle estimation results are presented in Fig 5. We use 2000 particles in the PF and in the MBF. We set M = 3 in the MBF. When the sideslip angle is small, see the time interval 0 to 4 in Fig 5, all the estimators could provide accurate estimation. But as the sideslip angle increases, see time interval 4 to 10, the sideslip angle is over-estimated by the EKF. The PF and the MBF can still provide accurate estimate in this situation. The performance of the MBF is a little better than the PF. The RMSEs of the different estimators are listed in Table 2. The RMSEs of the MBF and the PF are only 10% of that of the EKF and the MBF has a smaller RMSE than the PF does. Since the estimation accuracy of PF is good enough, the MBF can only increase a little

0.06

3 time

4

5

6

Fig. 3. Sideslip angle estimation results: Chicane test Filter

RMSE (degree)

EKF PF MBF

4.121 0.427 0.394

Table 2. Estimation results of the sideslip angle: Slalom test this accuracy in this experiment compared with the first one. This means in the cases the PF works well, the MBF has performance at least as good as the PF does; while in the cases the PF has medium performance, the MBF can enhance the estimation precision. The ”g-g” diagram of the longitudinal and the lateral accelerations is shown in Fig 4.

6270

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

REFERENCES 0.6

0.4

Lateral accleration

0.2

0

−0.2

−0.4

−0.6

−0.8 −0.2

−0.15

−0.1

−0.05 0 0.05 Longitudinal accleration

0.1

0.15

Fig. 4. ”g-g” diagram: Slalom test

15 true PF MBF EKF

10

sideslip angle

5 0 −5 −10 −15 −20 −25

0

2

4

6 time

8

10

12

Fig. 5. Sideslip angle estimation results: Slalom test 5. CONCLUSIONS Two nonlinear observers have been proposed to estimate the sideslip angle . These observers are based the nonlinear double track model which is more accurate than the single track model. The Dugoff model has been used to describe the relation of the slip angle and the tire road forces. Two nonlinear filters, the PF and MBF are employed to conveniently process the nonlinearly structure and to obtain satisfactory results. This is well demonstrated when the sideslip angle is large. The estimation results given by the PF and the MBF are more accurate than the one given by the EKF. What’s more, the MBF can provide better results than the PF, specially when the PF can not guarantee precise results. ACKNOWLEDGEMENTS

Arulampalam, S., Maskell, S., Gordon, N., and Clapp, T. (2002). A tutorial on particle filter for on-line nonlinear/non-gaussian Bayesian tracking. IEEE Trans. Signal Process., 50(2), 174–188. Baffet, G., Charara, A., Lechner, D., and Thomas, D. (2008). Experimental evaluation of observers for tire-road forces, sideslip angle, and wheel cornering stiffness. Vehicle Sytem Dynamics, 46(6), 501–520. Baffet, G., Charara, A., and St´ ephant, J. (2006). Sideslip angle, lateral force and road friction estimation in simulation and experiments. In IEEE International Conference on Control Applications, 903–908. Munich, Germany. Capp´ e, O., Douc, R., and Moulines, E. (2005). Comparison of resampling schemes for particle filtering. In International Symposium on Image and Signal Processing and Analysis. Zagreb, Croatia. Cheli, F., Sabbioni, E., Pesce, M., and Melzi, S. (2007). A methodology for vehicle sideslip angle identification: comparison with experimental data. Vehicle Sytem Dynamics, 45(6), 549–563. Cheng, Q. and Bondon, P. (2009). A modified bootstrap filter. In IEEE International Workshop on Robotic and Sensors Enviromments. Lecco, Italy. Cheng, Q. and Bondon, P. (2011). A new sampling method in particle filter. Submitted. Doucet, A., Godsill, S., and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3), 197–208. Doumiati, M., Victorino, A., Charara, A., and Lechner, D. (2010). On-board real-time estimation of vehicle lateral tire-road forces and sideslip angle. Unpublished paper. Dugoff, H., Fancher, P.S., and Segal, L. (1969). Tire performance characteristics affecting vehicle response to steering and braking control inputs. Final report, Office of Vehicle System Research, National Bureau of Standards. Gordon, N.J., Salmond, D.J., and Smith, A.F.M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F, 140(2), 107–113. Jazar, R. (2008). Vehicle Dynamics: Theory and Applications. Springer, New York, USA. Kiencke, U. and Nielsen, L. (2005). Automotive Control Systems. Springer, New York, USA, second edition. Kim, J. (2009). Identification of lateral tyre force dynamics using an extended kalman filter from experimental road test data. Control Engineering Practice, 17(3), 357–367. Liu, J. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc., 93(443), 1032–1044. Pacejka, H.B. (2005). Tire and vehicle dynamics. SAE International, second edition. Pacejka, H.B. and Bakker, E. (1993). The magic formula type model. Vehicle Sytem Dynamics, 21(supplment), 1–18. Piyabongkarn, D., Rajamani, R., Grogg, J.A., and Lew, J.Y. (2009). Development and experimantal evalution of a slip angle estimator for vehicle stability control. IEEE Trans. Control Syst. Tech., 17(1), 78–88. R. Rajamani, D. Piyabongkarn, J.L. and Grogg, J. (2006). Algorithms for real-time estimation of individual wheel tire road friction coeficients. In American Control Conference. Minnesota, USA. Robert, C.P. and Casella, G. (2004). Monte Carlo statistical methods. Springer-Verlag, New York, second edition. Van der Merwe, R., de Freitas, N., Doucet, A., and Wan, E. (2000). The unscented particle filter. In Advances in Neural Information Processing Systems. Vietinghoff, A.V., Hiemer, M., and Kiencke, U. (2005). Nonlinear observer design for lateral vehicle dynamics. In IFAC World Congress. Prague, Czech Republic. You, S., Hahn, J., and Lee, H. (2009). New adaptive approaches to real-time estimation of vehicle sideslip angle. Control Engineering Practice, 17(12), 1367–1379.

The authors are grateful to two anonymous referees for their helpful comments. 6271