Vehicle handling improvement by fuzzy explicit nonlinear forces parametrization Said Mammar, Andre Benine-Neto, Sebastien Glaser, Naima Ait Oufroukh

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Said Mammar, Andre Benine-Neto, Sebastien Glaser, Naima Ait Oufroukh. Vehicle handling im- provement by fuzzy explicit nonlinear tire forces parametrization. Chinese Control and Decision Con- ference (CCDC 2011), May 2011, Mianyang, China. pp.1560-1565, ￿10.1109/CCDC.2011.5968442￿. ￿hal-00654099￿

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Distributed under a Creative Commons Attribution| 4.0 International License Vehicle Handling Improvement by Fuzzy Explicit Nonlinear Tire Forces Parametrization

Sa¨ıd Mammar, Andre´ Benine-Neto, Sebastien´ Glaser, Na¨ıma Ait Oufroukh

Abstract— This paper presents the design and the simula- also investigated through active steering [5], [7]. Even if the tion test of a Takagi-Sugeno (TS) fuzzy output feedback for mechanical linkage between them is still a limiting factor, yaw motion control. The control synthesis is conducted on a solutions have been already implemented in series production nonlinear model in which tire-road interactions are modeled using Pacejka’s magic formula. Using sector approximation, a [17]. In such systems, the additional steering angle is limited TS fuzzy model is obtained. It is able to handle explicitly the and it is expected that real gain from active steering will nonlinear Pacejka lateral tire forces including the decreasing or come with steer-by-wire systems which will offer additional saturated region. The controller acts through the steering of the freedom-factors for the controller intervention [16]. front and the differential braking torque generation. The Using both steering angle rate and differential braking, computation of the controller takes into account the constraints that the trajectories of the controlled vehicle remain inside an this paper proposes a dynamic fuzzy output feedback. The invariant set. This is achieved using quadratic boundedness Takagi-Sugeno [8] fuzzy formalism allows modeling of the theory and Lyapunov stability. Some design parameters can be nonlinear behavior of the lateral forces described by the adjusted to handle the trade-off between safety constraints and pacejka formula [6]. The nonlinearity includes the decreasing comfort specifications. The solution to the associated problem is region. The dynamic output feedback formulation considered obtained using Linear and Bilinear Matrix Inequalities (LMI- BMI) methods. Simulation tests show the controlled is able in this paper presents three main advantages: the use of to well achieve standard maneuvers such as the ISO3888-2 only the yaw rate and the steering angle as controller input, transient maneuver and the sine with dwell maneuver. better flexibility to formulate the stabilization conditions and Index Terms— Vehicle handling, Fuzzy control, Output feed- the ability to handle input or state constraints and bounded back, LMI, BMI. disturbances. This controller uses the property of quadratic boundedness and invariant set [4]. This allows the constraints I. INTRODUCTION that the trajectories of the controlled vehicle remain inside Ground vehicles experience instabilities that are difficult an invariant set. In fact, during control intervention, it is for the driver to control. In fact, bifurcation analysis have important to ensure a good safety level by bounding state shown that the stability region, given for example in the variables. The chosen strategy fulfills this requirement and sideslip angle - yaw rate phase plane is limited [1]. In consists of building an invariant set for the system state. It addition its size is function of the driver input on the guarantees that each trajectory that starts in the invariant set steering speed, the road adhesion and the longitudinal speed. will not exceed it, hence the trajectories will be bounded Notice that instabilities are mainly due to lateral tire-road inside it [9]. Some design parameters can be adjusted to forces saturation. It is thus important de help the driver in handle the trade-off between safety constraints and comfort maintaining control of the vehicle is extreme dynamics and specifications. The solution to the associated problem ob- even prevent that the vehicle enters them. tained using Linear and Bilinear Matrix Inequalities (LMI- In this context, electronic stability control systems (ESC) BMI) methods. is the subject of intense research while solutions are already The paper is organized as follows: the next Section gives a available and become more and more popular on commercial description of the developed vehicle lateral dynamics Takagi- vehicles in Europe. They have largely contributed during the Sugeno model of the vehicle. The fuzzy output feedback syn- last decade to accident and death reduction [12]. Today’s thesis, including the requirements concerning the quadratic systems act on the vehicle lateral dynamics mainly through boundedness, the state constraints and control limitation independent braking. Recent studies have demon- are then presented in Section 3. In Section 4, simulation strated that differential braking may have a better effect on results for the ISO 3888-2 and the sine with dwell transient yaw dynamics than independent active wheel braking [13], maneuvers which excite the nonlinear tire dynamics are [14]. Optimal strategies for braking forces allocation have provided. The conclusions wrap up the paper. been explored in [15]. In parallel, vehicle handling has been II. VEHICLE LATERAL DYNAMICS T-S MODEL S. Mammar and N. Ait Oufroukh are with Universited’´ Evry´ Val As lateral control is concerned, a simple nonlinear model d’Essonne, France. IBISC: Informatique, Biologie Integrative´ et Systemes` Complexes - EA 4526, 40 rue du Pelvoux CE1455, 91020, Evry, Cedex, of a vehicle is obtained by neglecting the roll and pitch France, (e-mail: naima.aitoufroukh, [email protected]). motions. This model includes the lateral translational motion A. Benine-Neto and S. Glaser are with IFSTTAR - LIVIC Laboratoire and the yaw motion (Fig. 1). The two wheels of each sur les Interactions Vehicule-Infrastructure-Conducteur.´ 14, route de la Miniere,` Batˆ 824, 78000, Versailles, France, e-mail: (sebastien.glaser, are lumped into one located at its center. This leads to the [email protected]). vehicle bicycle model. The lateral forces between each tire

1 10

8 C α f1 f f f 6 C α f2 f 4

(kN) 2 f

0

−2 Lateral force f −4

−6

−8

−10 −20 −15 −10 −5 0 5 10 15 20 Front tire sideslip angle α (deg) f Fig. 1. Vehicle model.

Fig. 2. Tire lateral force given by the pacejka model and sector based approximation. and the road surface are added at each axle leading to two resulting forces f f α f = fy1 + fy2 and fr (αr)= fy3 + fy4 at the front and rear wheels of the bicycle model respectively. based on the mathematical representation of the tire dynamic These forces which( will) be detailed below are function of behavior using analytical functions having a particular struc- ture. Lateral forces of front and rear are function of the the front and rear tires sideslip angle, denoted α f and αr respectively. side angle αi at the tire-road contact location. The effect The lateral translation and rotational yaw motion equations of the camber angle is neglected. Here, the index i stands written in the vehicle fixed frame take the following form for f (front) or r (rear) :

˙ f f α f mv β + r 110 fi(αi)= = fr (αr) (1) 1 1 (3) Jr˙ l f lr 1 ⎡ ( ) ⎤ di sin ci tan− (bi(1 ei)αi + ei tan− (biαi)) [ ( ) ] [ − ] Tz ⋅ − ⋅ ⎣ ⎦ Notice that the adhesion coefficient and the normal force where β is the vehicle side , ψ˙ = r is the yaw ( ) acting on each tire are embedded inside the parameters b , c , rate and T is the yaw moment input applied by differential i i z d and e . See [7] for further details. The definition and the wheel braking. m is the vehicle mass while J is the vehicle i i value of the above parameters are described in the appendix moment of inertia. The vehicle center of gravity is located at at the end of the paper. a distance l f from the front axle and a distance lr from the rear axle. The vehicle parameters values are listed in Table The goal now is to achieve a Takagi-Sugeno fuzzy model I in the Appendix. which covers the entire operating domain (linear and non- Assuming that the angles remain small, the front and the linear) of the forces [2]. rear sideslip angles are given by: B. Four rules Takagi-Sugeno vehicle fuzzy model

l f α f = δ f β + r The nonlinear vehicle model is transformed into a four − v (2) lr rules Takagi-Sugeno (T-S) fuzzy model according to the αr = β +( r ) − v values of the front and rear cornering stiffnesses: A. Lateral tire forces model f f = c f1 α f if α f is m1 and αr is n1 then Several types of models of the forces of tire-pavement ∙ ∣ ∣ fr = cr αr { 1 interaction have been proposed in the literature [6]. They are   f f = c f2 α f if α f  is m2 and αr is n1 then usually derived from experimental data, as for the Pacejka ∙ ∣ ∣ fr = cr αr { 1 model, and have as parameters the adhesion, the speed v and   f f = c f1 α f if α f  is m1 and αr is n2 then the vertical load f . The shape of the lateral force is often ∙ ∣ ∣ fr = cr αr ni { 2 similar from one model to another. A first linear domain for   f f = c f2 α f if α f  is m2 and αr is n2 then small sideslip angle allows to define a slope factor called the ∙ ∣ ∣ fr = cr αr { 2 tire cornering stiffness coefficient. When the sideslip angle The membership  functions m and n (i = 1,2) are deter-   i i increases, the tire enters a nonlinear operating zone where mined by the approximation method of nonlinear function the lateral force saturates. The maximum value defines the by linear sectors. Coefficients c fi and cri (i = 1,2) represent limit of the vehicle maneuverability, resulting in a loss of the tire cornering stiffnesses associated to each sector. In controllability that can cause an understeering phenomenon fact they represent also the slope of the limits of the sectors or an unusual oversteering which may surprise the driver. which include the tire forces (Fig. 2). For example, given two

Here, the Pacejka magic formula [10], [11] is used to coefficients c f1 and c f2 , chosen according to the expected represent the efforts exerted on each tire. This model is road adhesion and driving conditions, one can determine

2 the membership functions m1 α f and m2 α f . The evo- C. Reference yaw rate tracking lution of the two functions m and m as functions of the 1 2 Ideally, the vehicle should respond to driver’s steering sideslip angle are shown in Figure( ) 3. They( are) obtained angle δ as a speed depended yaw rate reference steady state with numerical values: c = 1.2c and c = 0.6c .Itis d f1 f f2 f value with almost constant settling time. Let T be the desired important to outline that this sector representation is an exact 0 transfer function between δ and r. In order to ensure at approximation of the nonlinear system. d nominal speed, the same steady state value for the controlled and the conventional car, the reference model is chosen as a 1 first order transfer function with the same steady state gain Kd (v) 0.9 as the conventional car. It is of the form rd = τs+1 δd.The m 1 speed dependent steady state gain is Kd(v), derived from the 2 0.8 nominal linear bicycle model, and τ = 0.2 sec. 0.7 and m In order to ensure that the yaw rate reference value is 1 0.6 achieved in steady state, the integral z of the yaw rate tracking error is added as state a variable: 0.5

0.4 δ f + αr α f z˙ = r rd = − v rd (7) 0.3 − l f + lr − Membership functions m m 0.2 2 This variable is thus added to the previous third order 0.1 model while the desired yaw rate is considered as a distur-

0 bance. The fuzzy model is finally discretized at a sample −15 −10 −5 0 5 10 15 Front tire sideslip angle α (deg) time of T = 0.005sec. The final fuzzy model is of the form: f x(t + 1)=∑4 h α ,α A x(t)+Bu(t)+Ew(t) i=1 i f r i (8) Fig. 3. Membership functions m1 and m2 associated to the front tire contact y(t)=Cx(t)+Dw(t) forces. ( ) T T where x =[α f ,αr,δ f ,z] and y(t)=[r,z] . The disturbance The membership functions n and n for the rear tire forces T 1 2 w(t)=rd(t) εQ = w R/w Qw 1 is bounded. Ma- are obtained by the same procedure. Finally, one can write: ∈ ∈ ≤ trices Ai and B can be easily derived from equations (6) and (7). This discrete{ time fuzzy system} is characterized f f = (h1 + h3)c f1 +(h2 + h4)c f2 α f (4) by common B, E and C matrices. This property simplifies fr =[(h1 + h2)cr1 +(h3 + h4)cr2 ]αr { [ ] drastically the stability and performance conditions as only with h1 = m1 n1, h2 = m2 n1, h3 = m1 n2 and h4 = × × × simple summations are involved. m2 n2. In× order to have the front and the rear sideslip angle III. DYNAMIC OUTPUT FEEDBACK FUZZY CONTROLLER as state vector components, let us define the statex ¯ = T ˙ T [α f ,αr,δ f ] and the control input u =[δ f ,Tz] , the fuzzy In the following, a dynamic output feedback fuzzy con- system takes the form: troller is sought. It has the form:

4 4 i ¯ ¯ xc(t + 1)=∑ hi α f ,αr A xc(t)+Bcy(t) x¯˙ = ∑ hi α f ,αr Aix¯+ Bu (5) i=1 c (9) i=1 u(t)=Ccxc(t)+Dcy(t) ( ) ( ) where R4 i where xc is the controller state; Ac,Bc,Cc,Dc are a a a l f ∈ { } 11i 12i 13 1 Jv matrices to be designed. −l A¯i = a21i a22i a23 ,B¯ = 0 r (6) This controller uses the parallel distributed compensation ⎡ ⎤ ⎡ Jv ⎤ 000 10 (PDC) concept of the fuzzy system control. In this concept, where ⎣ ⎦ ⎣ ⎦ each control rule is distributively designed for the corre- v 1 1 lrl f sponding rule of a T-S fuzzy model. Linear control theory can a11i = + c′ , − l f +lr − v m J fi then be used to design controllers for each of the consequent ⎧ v 1 1 l f lr a12i = ( c)r′ , part of the fuzzy system while ensuring the same properties  l f +lr v m J i  − − 2  v 1 1 lr for the fuzzy system.  a21i = ( ) c′ ,  l f +lr v m J fi  − − −2 As pointed out in [4], Dc is an important parameter for  v 1 (1 lr )  a22i = + cr′ , stabilization, and the controller structure is able to handle ⎨ l f +lr − v m J i v constraints on the input and the state. By combining (8) and a13 = l +l , ( )  f r  a = v . (9), the augmented closed-loop fuzzy model is given by  23 l +lr  f  4 where c′ =c f 1 for i = 1,3 and c′ = c f 2 for i = 2,4. fi ⎩ fi x˜(t + 1)=∑ hi α f ,αr Φix˜(t)+Γw(t). (10) Similarly, cri′ = cr1 for i = 1,2 and cri′ = cr2 for i = 3,4. i=1 ( )

3 x A + BD C BC wherex ˜ = , Φ = i c c and the existence of the controller is ensured if there exist x i B CAi c c c matrices P1, P2, M1, M2 and a positive scalar α such that BD [D + E] [ ] Γ = c . the following condition holds BcD [ 4 ] Let Φz = ∑i=1 hi α f ,αr Φi, the closed loop system takes 4 the form:x ˜(t + 1)=Φzx˜(t)+Γw(t). ∑ hi α f ,αr ϒi 0 (16) ( ) i=1 ≥ A. Invariant set and output feedback PDC control ( ) where Assume that there exists a quadratic function V(x˜)=x˜T Px˜, where P is a symmetric, positive definite matrix that satisfies, (1 α)P1 T − ∗∗∗∗ for allx ˜, w satisfying (10), w Qw 1, V(x˜) 1, the (1 α)I (1 α)M1 ≤ ≥ ⎡ − − ∗∗∗⎤ condition [3]: ϒ = 00αQ i ∗∗ V(x˜+ 1) V(x˜) (11) ⎢ Ai + BDˆ cCAiM1 + BCˆc BDˆ cD + EM1 ⎥ ⎢ ∗ ⎥ ≤ ⎢ P A + Bˆ C Aˆi Bˆ D + P EIP⎥ ⎢ 1 i c c c 1 1 ⎥ Consider the reachable set Λ defined by: ⎣ (17)⎦ Notice that the matrices M , M , P and P verify: Λ ≜ x˜(T) x˜, w satisfying (10), 1 2 1 2 { ∣ T (12) x˜(0)=0, w Qw 1, T 0 T ≤ ≥ } M P2 = I M1P1 (18) 2 − The set εP defined by: 8 T In addition, it is possible to handle constraints on the control εP = x˜(t) R x˜(t) Px˜(t) 1 , (13) { ∈ ∣ ≤ } signal and the state: is an invariant set for the system (10) with w R, wT Qw 1. ∈ ≤ u¯ u(t) u¯, Ψ¯ Ψx(t + 1) Ψ¯ , t 0 (19) This means that every trajectory that starts inside εP remains − ≤ ≤ − ≤ ≤ ∀ ≥ inside it for t ∞. → T ¯ The existence of such a function V(x˜) means that the set whereu ¯ =[u¯1,u¯2] withu ¯1 > 0,u ¯2 > 0 and Ψ := [Ψ¯ ,...,Ψ¯ ]T with Ψ¯ > 0, j = 1,...,q, Ψ Rq 4 and εP is an outer approximation of the reachable set Λ. 1 q j × q is the number of imposed constraints. Notice∈ that the εP is also and outer approximation of the reachable set bounds are provided separately on each state variables or Λ∗ ≜ x˜(T) x˜, w satisfying equation (10), a combination of state variables. { ∣ T (14) x˜(0) εP, w Qw 1, T 0 The control input limitation is verified if for a pre-specified ∈ ≤ ≥ } scalar η (0,1], the additional inequality In this section the control law and the invariant set εP are ∈ synthesized. This is achieved using BMI (Bilinear Matrix ηP1 Inequalities) optimization method such that the system with- ∗∗∗ ηI ηM1 out the disturbance is asymptotically stable and at the same ⎡ ∗∗⎤ 0 (20) 00Q ≥ time, the reachable set for an initial state values inside the ∗ ⎢ √2Dˆ C √2Cˆ √2Dˆ DZ ⎥ invariant set is contained in this invariant set. ⎢ c c c ⎥ ⎣ ⎦ 2 2 2 2 B. Invariant set - quadratic boundedness holds with Z R is such that Z11 u¯ and Z22 u¯ . ∈ × ≤ 1 ≤ 1 According to the previous considerations, the closed loop A similar procedure can be applied for the constraints on the state variables. One can achieve from the convex property linear systemx ˜(t + 1)=Φzx˜(t)+Γw(t) is strictly quadrati- cally bounded with a common Lyapunov matrix P > 0for conditions (21): T all allowable w(t) εQ,fort > 0, ifx ˜(t) Px˜(t) > 1 implies 4 T ∈ T ∑i=1 hi α f ,αr ϒi 0, t 0, (Φzx˜(t)+Γw(t)) P(Φzx˜(t)+Γw(t)) < x˜ Px˜, for any w 2 ≥ ≥ (21) ∈ Ξkk Ψ¯ , k 1,...,q εQ. ≤ (k ) ∈ { } The corresponding condition is obtained using the ˜ where Ξ is a symmetric matrix and S procedure and invoking the Schur complement, using that − T T the satisfaction of w εQ andx ˜ Px˜ 1 implies w Qw ∈ ≥ ≤ ηP1 x˜T Px˜. ∗∗∗ T T ηI ηM1 P1 P2 1 M1 M2 ϒ = ∗∗ (22) Defining P = and P− = .As- i ⎡ 00Q ⎤ P2 P3 M2 M3 ∗ suming that P and[ M are] full rank matrices[ and setting:] ⎢ ϒ ϒ ϒ Ξ ⎥ 2 2 ˜ ⎢ i41 i42 i43 ⎥ ⎣ ⎦ Dˆ D c = c and ˜ ˜ ˜ Cˆc = DcCM1 +CcM2 ⎧ ϒi = √2Ψ Ai + BDˆ cC Bˆ = P BD + PT B (15) 41  c 1 c 2 c √ ˆ  ˆi T ϒi42 = 2Ψ AiM1 + BCc ⎨ Ac = P1AiM1 + P1BDcCM1 + P2 BcCM1 ( ) T i ˜ϒi = √2Ψ BDˆ cE + D +P1BCcM2 + P2 AcM2 43 ( )  ˜  ( ) ⎩ ˜

4 C. Controller synthesis 6 6 5 4

Under the proposed modeling approach, the desired yaw 4 2 rate could be seen as an input disturbance under which 3 0 the closed-loop system should remain stable with bounded 2 −2

Lateral distance (m) 1 Steering angle (deg) values for the state vector components. More generally, the −4 0 state variables should not exceed the bounds of a “safety −6 M M M −1 zone”, namely α α , α α and δ δ . Thus, −8 f r r f −2 f f 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 ≤ ∣ ∣ ≤ ≤ Longitudinal distance (m) Time (sec) the state vector x has to be confined to a hypercube L(ZM)     defined by the above bounds. Finally, the control input, the (a) (b) steering angle rate and the yaw moment, have to be bounded 30 50 M M δ˙ δ˙ and T T . 20 f f z z 0 ≤ ∣ ∣ ≤ 10  According to the equation (19), control limitation is given −50 M M 0 Steering angle rate (deg/s) 0 0.5 1 1.5 2 2.5 3 3.5   ˙ ¯ Time (sec) by u ¯ =[δ f ,Tz ], while state limitation is given by Ψ = −10 15

M M M T Yaw rate (deg/s) [α ,α ,δ ] and Ψ = I3 0 . 10 f r f −20 5

The PDC output feedback controller was synthesized with 0 −30 [ ] −5 the following numerical values: Yaw moment (KN.m) −40 −10 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Time (sec) Time (sec) ˙M α = 0.02, η = 0.02, δ f = 100deg/s, (c) (d) M M M M Tz = 10KN α f = αr = 13deg δ f = 6deg, Fig. 4. ISO3888-2 maneuver: Yaw rate, steering angle, steering angle rate These design parameters could be adjusted to handle the and trajectory for the uncontrolled and the controlled vehicles. trade-off between safety constraints and comfort specifica- tions. The achieved Q is 5, which ensures that the constraints are verified for a disturbance of a magnitude less than 0.447rad/s at the considered longitudinal speed of 20m/s. In fact, the maximum value is constrained by [18]: g r = 0.85 dmax v .

IV. SIMULATION TESTS In order to proof the assistance ability to maintain the dynamic vehicle stability in extreme conditions, several type Fig. 5. ISO3888-2 maneuver: Coefficients hi reflecting the contribution of of maneuvers have been defined to test the ESC systems. each sub-controller for the controlled vehicles. Among them, double lane-change manoeuvre defined in ISO 3888-2 standard and the sine with dwell transient maneuver. The controller is tested below for the two maneuvers. plot of Figure 4-d. Figure 4-c shows that the controlled car yaw rate is closer to the reference one than the yaw rate of the A. Testing for the ISO 3888-2 maneuver uncontrolled vehicle (dashed line). The contribution of each The ISO3888-2 double lane-change maneuver setup is sub-controller to according to the actual depicted in Figure 4-a. The maneuver is carried out with and are shown in Figure 5. Finally, Figures 6-a and 6-b provide without the controller at the same speed of 80km/h.During the developed sideslip angles at the front and rear tires. The the maneuver, the throttle is released. corresponding front and rear forces are shown in Figures 6- The driver initiates the maneuver by applying the steering c and 6-d. It is clear that the saturation zones are reached angle shown in dashed line in Figure 4-b. Figures 4-a by the uncontrolled vehicle during the maneuver while the highlights that the controlled vehicle is able to perform the controller avoids that these zones are reached. maneuver (solid line) while the uncontrolled vehicle fails (dashed line). Figure 4-d shows that the controller shares B. Sine with dwell maneuver the effort on the steering angle rate an the yaw moment, The sine with dwell is a transient maneuver considered by respectively. In this situation the driver applied steering angle NHTSA (National Highway Traffic Safety Administration) is too high (dashed line in Figure 4-b) while the the steering for electronic stability control evaluation. Such type of angle of the controlled vehicle is limited to the admissible maneuver is suited for the excitation of the vehicle oversteer safety value of few degrees, as shown by the solid line in response. The maneuver is conducted at the same speed of Figure 4-b for the angle value. The steering angle rate is 80km/h. It corresponds to a 0.7Hz frequency sine wave form depicted in the top plot of Figure 4-d and is limited. The the with dwell steering angle of 500ms. During the maneuver, the yaw moment handles the main effort as shown in the bottom throttle is released. Figure 7 shows both the control sharing

5 6 10 APPENDIX 4

5 2

0 TABLE I 0 −2 VEHICLE PARAMETERS. −4 −5 m Vehicle total mass 1600 kg. −6 c Front cornering stiffness 40000 N/rad. Rear wheel sideslip Angles (deg) Front wheel sideslip Angles (deg) −8 f −10

−10 cr Rear cornering stiffness 35000 N/rad. 2 −12 −15 J Vehicle yaw moment of inertia 2454 kg m . 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Time (sec) Time (sec) ⋅ l f Distance form CG to front axle 1.22m. (a) (b) lr Distance from CG to rear axle 1.44m. v Longitudinal velocity.

8 6

6 4 4

2 2 TABLE II

0 0 TIRE MODEL PARAMETERS. −2

−4 −2 Tire bi ci di ei Rear tire lateral force (KN) Front tire lateral force (KN) −6 Front (i = f ) 8.3278 1.1009 4536.0 -1.661 −4 −8 Rear (i = r) 11.6590 1.1009 3671.6 -1.542

−10 −6 −12 −10 −8 −6 −4 −2 0 2 4 6 −15 −10 −5 0 5 10 Front tire sideslip angle (deg) Rear tire sideslip angle (deg) (c) (d) REFERENCES Fig. 6. ISO3888-2 maneuver: Front and rear tire sideslip angle and [1] H.D. Tuan, E. Ono, S. Hosoe, S. Doi, Bifurcation in Vehicle Dynamics corresponding lateral forces for the uncontrolled (dashed) and the controlled and Robust Front Wheel Steering Control. IEEE Transactions on vehicles (solid), with vertical offset for the uncontrolled one for better Control Systems Technology, 6, pp. 412-421, 1998. display. [2] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applica- tion to modeling and control, IEEE Trans. Systems Man Cybern. 15, pp. 116-132, 1985.

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10 for uncertain linear systems using quadratic boundedness. Automatica, 20 0 42, 497-502, 2006.

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