L Nehaoua
email: lamri.nehaoua@iup.univ-evry.fr

H Arioui

N Seguy

S Mammar

Dynamic Modeling of a Two Wheeled Vehicle : Jourdain Formalism

Keywords: Motorcycle modeling, motorcycle control, Jourdain's dynamics principle

This paper presents a motorcycle direct dynamic formulation by the Jourdain's principle approach. This vehicle is considered as an assembly of six rigid bodies and the resulting equation of motion allows to simulate 11 Degrees of Freedom (DoF). The vehicle geometry is described and a step by step procedure is introduced to evaluate the kinematics and the generalized efforts of the considered vehicle. In addition, to simulate the equation of motion, a Lyapunov based stabilization is developed to assess the vehicle behavior in response to a propulsion/braking torque applied on the vehicle's wheels and a rider torque exerted on the motorcycle's handlebar. Simulation results reveal some dynamics features like load transfer and counter-steering phenomena.

Introduction

During last decades, motorcycles have become a popular transportation tool. Indeed, the number of two-wheeled vehicles is in constant increase, specifically in urban areas. However, the risk associated with driving such vehicles is important and a road accident, even trivial, can have fatal consequences on the two-wheeled users.

Motorcyclists are the most vulnerable road users. They represent less than 2% of the traffic, but 18% of those killed in 2010 in France [1]. There are many causes to these fatalities: alcohol, non-compliance with the road rules, poor visibility, lack of assistive devices and especially the overvaluation of the self mastering capabilities of such a vehicle. Indeed, with the same power, the motorcycle represents more aggressive dynamics compared to those of a car vehicle. Riding a two-wheeled vehicle is neither simple nor intuitive like driving a car vehicle. Involved accelerations, critical equilibrium, trajectory anticipation are, among others, many points that make riding a 1 motorcycle a complicated exercise especially for inexperienced riders who get carried away by the riding sensations and pleasure.

Fortunately, recent preventive as well as suppressive laws have significantly contributed to decrease number of fatalities. It must be said that motorcyclists' road safety has attracted the attention of all academic, industrial and political institutions. In France, several research projects were undertaken: SUMOTORI 1 (2003-2006), SIMA-COM 2 (2006-2009) [START_REF] Arioui | Mechatronics, Design, and Modeling of a Motorcycle Riding Simulator[END_REF], SAFERIDER 3 (2008-2010), DAMOTO 4 (2008-2011), and others with a common point: understand the rider behavior.

To deal with this aim, it is necessary to predict the dynamics of the motorcycle vehicle and model it. In the study of mechanical systems, modeling usually leads to a set of differential/algebraic equations (DAE) derived from laws of mechanics. In this field, two approaches are proposed: the analytical approach and the numerical one. In the later one, many tools were developed such MSC ADAMS [START_REF]ADvAnced Multi-body Simulation[END_REF], simMechanics [4] and Modelica [START_REF] Tanelli | Object-Oriented Multibody Motorcycle Modelling for Control Systems Prototyping[END_REF]. The alternative approach is the development of symbolic equations of motion which is a time consuming method, but it offers more flexibilities in the system description. In that case, an algebra package like MBsymba [START_REF] Lot | A Symbolic Approach for Automatic Generation of the Equations of Motion of Multibody Systems[END_REF] is highly recommended.

Many works have addressed the modeling of the two-wheeled vehicles. Indeed, the first study on the stability and balance of a bicycle vehicle was performed by Whipple [START_REF] Whipple | Stability of the motion of a bicycle[END_REF]. This bicycle is represented by two bodies linked via the steering mechanism. Wheels are modeled as circular thin disks where the gyroscopic effect is considered. Other research works have been undertaken [START_REF] Bower | Steering and stability of single track vehicles[END_REF][START_REF] Pearsall | The stability of the bicycle[END_REF][START_REF] Kondo | Theoretical study on the ruuning stability of twowheelers[END_REF]. In all these studies, the tireroad interaction has been neglected. In 1971, Sharp integrated tires' efforts [START_REF] Sharp | The stability and control of motorcycles[END_REF]. The objective of this study is to examine the effect of parameters uncertainties on the vehicle stability. The vehicle is modeled as a set of two rigid bodies connected by the steering mechanism, simulating 4 DoF: the lateral displacement and the yaw, the roll and the steering rotations.

So far, in all these models, the longitudinal velocity was assumed to be constant and the vehicle suspension motions were ignored. It was interesting to examine how these factors influenced the frequency characteristics of the different stability modes. These aspects have been investigated in a further work of Sharp in 1974 [START_REF] Sharp | The influence of frame flexibility on the lateral stability of motorcycles[END_REF]. In 1994, Sharp published a paper, in which, the motorcycle is represented by a set of four rigid bodies to simulate 8 DoF. The originality of this model lies in the integration of flexibilities and the rider tilting motion [START_REF] Sharp | Vibrational modes of motorcycles and their design parameter sensitivities, Vehicle NVH and Refinement[END_REF].

In the same perspective, a model incorporating the suspension travels and a more complex representation of the tire-road interaction is presented in [START_REF] Sharp | A Motorcycle Model for Stability and Control Analysis[END_REF]. The advances made by this model have allowed exploring the dynamics of such a vehicle for a large motion around equilibrium. In that case, the interaction between the in-plane and the out-of-plane motion is well described. More recently, the FastBike program presents a nonlinear model of a motorcycle described as a set of 6 bodies and simulating 11 DoF. Each body is identified by its natural coordinates (a vector of six coordinates: three Cartesian coordinates of its center of mass and three rotations) and the assembly 1 Safety of motorcyclist against risky environment 2 Simulator for learning to drive motorized two-wheeled vehicles 3 Advanced telematics for enhancing the SAFETY and comfort of motorcycle RIDERS 4 Detection of accident situations on motorcycle 2 is done by introducing geometric constraints [START_REF] Cossalter | A motorcycle multibody model for real time simulation based on the natural coordinates approach[END_REF]. Finally, a modeling technique based on the recursive Newton-Euler approach is adapted to derive a motorcycle dynamics model intended for a two-wheeled simulator application [START_REF] Hima | Motorcycle Dynamic Model Synthesis for Two Wheeled Driving Simulator[END_REF][START_REF] Hima | Suitable Two Wheeled Vehicle Dynamics Synthesis for Interactive Motorcycle Simulator[END_REF].

Among the vehicle models, the simulation of single-track one turns out to be particularly difficult. While a car vehicle remains stable, the stabilization of two-wheeled vehicles is required and a rider model must be included. The literature of two-wheeled vehicle stabilization is recent [START_REF] Lot | A nonlinear rider model for motorcycles[END_REF][START_REF] Lot | Advanced motorcycle virtual rider[END_REF][START_REF] Sharp | Advances in the Modelling of Motorcycle Dynamics[END_REF]. Generally, a two-layer controller is adopted to track the reference road trajectory and an inner controller stabilizes the longitudinal and lateral dynamics. For this, several strategies are used: the optimal maneuver method [START_REF] Biral | Optimization of the Layout of a Racing Motorcycle using the Optimal Maneuver Method[END_REF][START_REF] Saccon | A virtual rider for motorcycles: An approach based on optimal control and maneuver regulation[END_REF], robust H ∞ and LPV 1 control techniques [START_REF] Corno | On LPV Roll Angle Controller Design for Two-Wheeled Vehicles[END_REF].

In this paper, a step by step modeling approach based on the Jourdain's principle is presented. This method was firstly used in [START_REF] Rill | Simulation von Kraft-fahrzeugen[END_REF] to derive a car dynamics model and the modeling prerequisites are inspired from this work. It offers an efficient tool for deriving motion equations in a systematic way and with less theoretical skills comparing to the Lagrangian and the Newton/Euler approaches. The great efforts concern the computation of Jacobian matrices and the virtual power of the different external applied forces. Our aim is to derive a mathematical description of a motorcycle vehicle with middle complexity, which captivates the essential dynamics issues. This model is intended for control, state observation and security devices development applications. In section 2, a brief outline to the mathematical modeling of the motorcycle kinematics and dynamics is introduced. Section 3 focuses on the computation of the generalized vector efforts. Section 4 is intended to demonstrate the basics concepts of the tire/road contact. Finally, a virtual rider controller is introduced in order to regulate the longitudinal speed towards a reference profile and to stabilize the lateral dynamics by using the roll motion.

Motorcycle kinematics and dynamics

Motorcycle description

In the present work, the motorcycle is composed of six bodies (see Figure 1). A rear body G r including the saddle, the engine and the fuel tank. The upper front body G f includes the handlebar and the upper part of the suspension assembly. The lower front body G l represents the lower part of the suspension assembly and the brake system. The swing arm body G s contains the swing arm mass and the rear brake system. Finally, R f and R r represent respectively, the front and the rear wheel bodies.

Most of these bodies are connected via revolute link except the upper and lower parts of the front body which are interconnected with a prismatic link (telescopic suspension). In order to completely define the configuration of the motorcycle, a set of 11 DoF has been considered. A full characterization of motorcycle's kinematics includes :

• the longitudinal, lateral and vertical position (x, y, z) of the rear body,

• the roll, pitch and yaw orientation (ϕ, θ, ψ) of the rear body, 1 LPV: Linear Parameter Varying

• the steering angle δ of the front body and the front wheel,

• the front suspension travel λ f and the swing arm rotation µ,

• and finally, the front and rear wheels spinning, ζ f and ζ r .

R f R r G r G l G f G s S l

Kinematic background

Consider a right-handed, orthogonal axis reference frame ℜ v (v, i v , j v , k v ) attached to the vehicle at point v in which the motion of the overall mechanical system will be expressed. Point v is defined as the intersection of the longitudinal plane of symmetry, the ground plane, and the vertical transverse plane containing point P , attach point of the swing arm with the rear body. The motion of the motorcycle is referred to the inertial reference frame ℜ o (O, i o , j o , k o ), by the velocity vector components v x , v y of point v, and the yaw rotation ψ around the z-axis k o of the inertial reference frame ℜ o (see Figure 2). Starting from the reference frame ℜ v , the orientations of the other reference frames are defined as follows (see Figure 3). A roll rotation ϕ about i v gives ℜ ϕ (v, i ϕ , j ϕ , k ϕ ). Next, a pitch rotation θ followed by a castor rotation ǫ about j ϕ gives ℜ ǫ (P, i ǫ , j ǫ , k ǫ ). Then, a steer rotation δ about k ǫ gives ℜ δ (B, i δ , j δ , k δ ). In the other side, a swing rotation µ about j ϕ gives ℜ µ (P, i µ , j µ , k µ ). In the movable reference frame ℜ v , the position of a body i is given by:

c r • v • c f • O • i o j o ℜ o ℜ v i v j v ψ
r oi = r ov + r vi (1)
From equation (1), the velocity and the acceleration of a body i are derived as following:

v oi = v ov + ω ov × r vi + ṙvi (2) a oi = a ov + ǫ ov × r vi + ω ov × (ω ov × r vi + 2 ṙvi ) + rvi
and, the angular velocity ω oi and the angular acceleration ǫ oi vectors of body i are derived in the same way:

ω oi = ω ov + ω vi (3) ǫ oi = ǫ ov + ω ov × ω vi + ωvi
Since the kinematic configuration of each body is well defined in its local reference frame, it remains to define the kinematic configuration of the movable reference frame ℜ v , it means, to find the expressions of a ov and ǫ ov in equations (2,3) w.r.t vov and ωov . Suppose that the linear and angular velocity vectors of ℜ v w.r.t the inertial one ℜ o are given by v o ov = R vo v v ov and ω o ov = R vo ω v ov . It goes that the linear and angular acceleration vectors of ℜ v , expressed in ℜ v , can be derived as following: From equations (2-4), the linear and angular acceleration vectors of each body i are expressed in the movable reference frame ℜ v by the following equations:

a ov = ω ov × v ov + vov (4) ǫ ov = ωov v • i ϕ k ϕ ℜ ϕ v • j v k v ℜ v j ϕ i ϕ ℜ ϕ ϕ P • k ǫ i ǫ ℜ ǫ k µ i µ ℜ µ µ θ + ǫ P • i ǫ j ǫ ℜ ǫ B • i δ j δ ℜ δ δ
a oi = vov + ωov × r vi + rvi + ω ov × (v oi + ṙvi ) (5)
ǫ oi = ωov + ω ov × ω vi + ωvi
At present, all kinematic quantities are defined, then we introduce the principle of the virtual power called also Jourdain's principle.

Jourdain's principle

Jourdain's principle states that the virtual power done by motion compatible constraint efforts F i,c /M i,c is null [START_REF] Rill | Simulation von Kraft-fahrzeugen[END_REF], then:

n i=1 {(∆v oi ) • F i,c + (∆ω oi ) • M i,c } = 0 ( 6
)
where n is the number of bodies and ∆v describes a virtual velocity. By introducing the Newton/Euler dynamics principle, the constraint efforts F i,c /M i,c can be expressed from the system motion and the external applied efforts F i,a /M i,a as:

F i,c = m i a oi -F i,a (7)
M i,c = I i ǫ oi + ω oi × I i ω oi -M i,a
Moreover, from equations (2, 3, 5), the velocity and acceleration vectors can be written as following:

v oi = ∂v oi ∂ϑ ϑ ω oi = ∂ω oi ∂ϑ ϑ a oi = ∂v oi ∂ϑ θ + a R ǫ oi = ∂ω oi ∂ϑ θ + ǫ R (8)
where:

ϑ = v x v y ψ ż φ θ δ λf μ ζf ζr T (9)
denotes the vector of generalized velocities and the partial derivatives are called Jacobian matrices of the velocity vector w.r.t the generalized velocities vector ϑ. Vectors a R and ǫ R are referred as the residual acceleration terms. Consequently, from equations [START_REF] Whipple | Stability of the motion of a bicycle[END_REF][START_REF] Bower | Steering and stability of single track vehicles[END_REF] and after some algebraic manipulations, the motion dynamics can be expressed as:

M θ = Q ( 10
)
where M is the mass matrix obtained from the direct computation of Jacobian matrices:

M = n i=1 m i ∂v oi ∂ϑ T ∂v oi ∂ϑ + ∂ω oi ∂ϑ T I i ∂ω oi ∂ϑ (11)
and the generalized efforts vector Q can be split as following:

Q = n i=1 ∂v oi ∂ϑ T F i,a + ∂ω oi ∂ϑ T M i,a Q a - (12)
n i=1 m i ∂v oi ∂ϑ a R + ∂ω oi ∂ϑ T (I i ǫ R + ω oi × I i ω oi ) Q R (13)
The next subsection presents a step-by-step procedure to derive the Jacobian matrices. These matrices allow the calculation of the mass matrix M and the residual generalized effort vector Q.

Motorcycle Kinematics

Rear body

The rear body G r has 3 DoF w.r.t ℜ v which are the roll, pitch rotations and vertical displacement. Then, its center of mass position vector is:

r vGr = r vP + r P Gr ( 14
)
where the coordinates of vectors r vP and r P Gr in the reference frame ℜ v are computed by:

r vP = R ϕ   0 0 z + h p   r P Gr = R ϕ,θ   x Gr 0 z Gr   (15)
By differentiating, the relative linear and angular velocity vectors are:

ṙvGr = ṙvP + ṙP Gr = (ω ϕ × r vP + żk ϕ ) + ω ϕ,θ × r P Gr (16) ω vGr = ω ϕ,θ = ω ϕ + ω θ (17) (18)
where the coordinates of the unit vector k ϕ in ℜ v are obtained by k ϕ = R ϕ [0, 0, 1] T and the components of the angular velocity vectors ω ϕ and ω θ are respectively given by ω ϕ = [ φ, 0, 0] T , ω θ = R ϕ [0, θ, 0] T . By using equations [START_REF] Tanelli | Object-Oriented Multibody Motorcycle Modelling for Control Systems Prototyping[END_REF][START_REF] Bower | Steering and stability of single track vehicles[END_REF], the Jacobian matrices and the residual acceleration vectors can be deduced by simple algebraic manipulations.

Front upper body

The front upper body G f is belonging to the fixed-body reference frame ℜ δ and has 1 DoF w.r.t the rear body which represents the steer rotation. Then, its center of mass position vector is governed by:

r vG f = r vP + r P B + r BG f ( 19
)
where the coordinates of the vectors r P B and r BG f are given in the reference frame ℜ v as following:

r P B = R ϕ,θ,ǫ   l P B 0 0   r BG f = R ϕ,θ,ǫ,δ   x Gf 0 z Gf   (20)
By differentiating, the relative linear and angular velocity vectors are:

ṙvG f = ṙvP + ṙP B + ṙBG f = ṙvP + ω ϕ,θ × r P B + ω ϕ,θ,δ × r BG f (21) ω vG f = ω ϕ,θ,δ = ω vGr + ω δ (22) (23)
where

ω δ = R ϕ,θ,ǫ [0, 0, δ] T .

Lower front body

The front lower body has 1 DoF w.r.t G f representing the suspension travel λ f . Its center of mass position vector is:

r vG l = r vB + r BG l ( 24
)
where the coordinates of the vector r BG l are:

r BG l = R ϕ,θ,ǫ,δ   x Gl 0 λ f + z Gl   ( 25
)
By differentiating, the relative linear and angular velocity vectors are:

ṙvG l = ṙvB + ṙBG l = ṙvB + ω ϕ,θ,δ × r BG l + λf k δ (26) ω vG l = ω vGr + ω δ ( 27
)
where the coordinates of the unit vector

k δ in ℜ v are k δ = R ϕ,θ,ǫ,δ [0, 0, 1] T .

Swing arm body

The swing arm body has 3 DoF w.r.t ℜ v which represents the roll, swing rotations and vertical displacement. Then, its center of mass position vector is: r vGs = r vP + r P Gs [START_REF] Evangelou | Influence of Road Camber on Motorcycle Stability[END_REF] in which the coordinates of r P Gs in the reference frame ℜ v are calculated using the following expression:

r P Gs = R ϕ,µ   x Gs 0 z Gs   (29)
By differentiating, the relative linear and angular velocity vectors are:

ṙvGs = ṙvP + ṙP Gs = ṙvP + ω ϕ,µ × r P Gs (30) ω vGs = ω ϕ,µ = ω ϕ + ω µ ( 31
)
where

ω µ = R ϕ,µ [0, μ, 0] T .

Rear-wheel bodies

The rear-wheel body has three independent DoF w.r.t ℜ v including roll rotation, vertical displacement and the tire spin rotation around wheel spin axis j ζ,r . It follows that:

ω vRr = ω ϕ + ω ζ,r (32)
where the components of the angular velocity are ω ζ,r = R ϕ [0, ζr , 0]. On the other hand, the position of the rear wheel center of mass is:

r vRr = r vP + r P Rr (33)
where the coordinates of the vector r P Rr are computed by:

r P Rr = R ϕ,µ   x Rr 0 z Rr   (34)
By differentiating, the relative linear velocity vector is:

ṙvRr = ṙvP + ṙP Rr = ṙvP + ω ϕ,µ × r P Rr (35)
(36)

Front-wheel body

The front-wheel body has one independent DoF w.r.t G l which represents the tire spin rotation around wheel spin axis j ζ,f , consequently:

ω vR f = ω vG l + ω ζ f (37) (38)
where

ω ζ f = R ϕ,θ,ǫ,δ [0, ζf , 0] T .
In the same way as the rear wheel, the position of the front wheel center of mass is:

r vR f = r vB + r BR f (39)
in which the coordinates of the vector r BR f are:

r BR f = R ϕ,θ,ǫ,δ   x Rf 0 λ f + z Rf   (40)
By differentiating, the relative linear velocity vector is:

ṙvR f = ṙvB + ṙBR f = ṙvB + ω ϕ,θ,δ × r BR f + λf k δ (41)
At this level, the whole motorcycle kinematics is defined. Matrix mass M and the residual generalized efforts Q R are fully defined. In the next section, we look for the development of the generalized efforts vector associated to external applied efforts.

Vector of Generalized Efforts

The motorcycle system is subject to various forces and torques elements. One enumerates gravity forces, suspensions forces due to springs and shock-absorbers, tire forces and torques at the tire/road contact point, rider steering torque and steer damper torque applied on motorcycle's handlebar, rear and front brake torques and finally the driving torque applied on the rear wheel. In this work, the rider is considered to be a rigid body attached to the rear assembly G r where its lean movement is ignored.

To express the generalized efforts vector Q a associated to external applied efforts, one can make use of equation [START_REF] Sharp | The influence of frame flexibility on the lateral stability of motorcycles[END_REF]. This method may be tedious, especially when the applied efforts are numerous. So, it is most convenient to find the virtual power done by each effort and hence its associated contribution.

Gravity force

The contribution of this force in Q a is directly computed from equation ( 12):

Q a,g = i ∂v oi ∂ϑ T   0 0 -m i g   ( 42
) for i ∈ (G r , G f , G l , G s , R f , R r ).

Rider torque

The rider's torque τ r applied on the motorcycle's handlebar is considered to be an important input for the vehicle riding. This torque is applied around the steering axis, which constitutes the interconnection link between the rear body G r and the front upper body G f . The contribution of this torque is defined by:

Q a,h = ∂ δ ∂ϑ T   0 0 τ r -C δ δ   =   0 6 τ r -C δ δ 0 4   (43)
where -C δ δ is the steer damping torque.

In [START_REF] Zhu | Motion analysis of a motorcycle taking into account the rider's effects[END_REF], a more detailed rider-motorcycle interaction is described by taking into account the rider's upper torso leaning motion and the connection of his/her arms with the handlebars.

Propulsion and braking torque

For the front-wheel, the braking force acts between the wheel knuckle and the front lower body which contains the braking disk. For the rear wheel, the braking torque is applied between the rear-wheel body and the swing arm assembly where the engine propulsion torque is applied around the rear-wheel spin axis, hence:

Q a,pb = ∂ ζf ∂ϑ T   0 τ B,f 0   + ∂ ζr ∂ϑ T   0 τ D + τ B,r 0   + ∂ μ ∂ϑ T   0 -τ B,r 0   (44) =     0 8 -τ B,r τ B,f τ D + τ B,r    

Front and rear suspension

The front suspension force F s,f is applied between the front upper and the front lower bodies. The contribution of this force in Q a can be expressed by using the virtual power principle, therefore:

Q a,sf = ∂ λf ∂ϑ T F s,f =   0 7 F s,f 0 3   (45)
In the same way, the contribution of the rear suspension can be computed as following:

Q a,sr = ∂ λr ∂ϑ T F s,r (46)
where λ r is the rear suspension travel given by λ 2 r = r S l Su • r S l Su . By differentiating, the suspension travel rate λr is deduced and written by the following equation: where e S l Su is the rear suspension unit vector. Consequently, the contribution of the rear suspension in the total generalized effort is given by:

λr = e S l Su • {ω ϕ,θ × r P Su -ω ϕ,µ × r P S l } (47) R f Rr k δ -F s,f eS l Su -Fs,r
Q a,sr =       0 5 e S l Su • (j θ × r P Su ) 0 2 -e S l Su • (j µ × r P S l ) 0 2       F s,r (48)
where the coordinates of the unit vectors j θ and j µ in ℜ v are respectively equal to R ϕ [0, 1, 0] T .

Equations (42-48) show the contribution in Q a of each external force/moment applied on the motorcycle vehicle. In the next section, the tire-road interaction is exposed and its contribution is demonstrated.

Tires' kinematics and dynamics

Tire-road interaction is the most important phenomena that characterize ground vehicles. Due to this importance, several works are undertaken leading to a mathematical description of friction forces which arises from tire and carcass deformation [START_REF] Cossalter | A motorcycle multibody model for real time simulation based on the natural coordinates approach[END_REF][START_REF] Zhu | Dynamical Analysis of Motorcycle by Multibody Dynamics Approach[END_REF].

In the present paper, the motorcycle's wheels are considered as rigid, thin disks where the road/tire contact is dot shaped at point c. At this point, a system of longitudinal, lateral and vertical forces is implemented by using an empirical tire model type named Pacejka model [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF] with respect to kinematic slip quantities and the normal load. Moreover, carcass deformations are not taken into account but reproduced by introducing the tire relaxation equations [START_REF] Sharp | Advances in the Modelling of Motorcycle Dynamics[END_REF].

In following, we expose the tire kinematics to define slip variables required while computing the tire/road contact forces and hence, deduce their contribution in the generalized effort vector Q a .

Tires kinematics

To describe the tire motion, a new reference frame ℜ T (c, i T , j T , k T ) is introduced at the contact point c of each wheel's tire. k T is the normal vector to the road surface. Vector i T is obtained by the cross product j ζ × k T and j T completes the reference axis. As shown in Figure 5, the camber angle is an important variable in the motorcycle safety studies [START_REF] Evangelou | Influence of Road Camber on Motorcycle Stability[END_REF]. This variable is expressed by the following scalar product:

ρ • j ζ γ i T j T F x F y F z v oc α c •
sin γ = j ζ • k T (49)
Besides, the position of the rear and front road/tire contact point c w.r.t the movable frame reference ℜ v is given by:

r vcr = r vRr + r Rr cr (50) r vc f = r vR f + r R f c f (51)
in which the coordinates of r Rr cr and r R f c f in ℜ v are respectively given by:

r Rr cr = R ϕ   0 0 -ρ r   r R f c f = R ϕ,θ,ǫ,δ   -ρ f sin ǫ 0 ρ f cos ǫ   (52)
By differentiating, the relative linear velocity vector of each contact point can be deduced as:

ṙvcr = ṙvRr + ω ϕ × r Rrcr ṙvc f = ṙvR f + ω vG f × r R f c f
By using equation ( 2), the contact point velocity vector is computed and the slip variables are defined by the following equations:

α r = -atan j v • v ocr i v • v ocr (53)
α f = -atan j v • v oc f i v • v oc f + δ cos ǫ κ i = - V c,i -r D,i ζi max (V c,i , r D,i ζi )
where, V c,i = v oci is the contact point speed, r D,i is the wheel's dynamic radius and i = r, f . Finally, knowing that the contact point c i must always belong to the road surface then, the vertical deformations δ z,i of the wheels' tire can be derived by:

δ z,i = k T • r vci (54)
These deformations help to determine the normal load necessary to maintain contact with the road plane [START_REF] Jalon | Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time challenge[END_REF].

Tires force/moment system

Once the tire's kinematics is established, the equivalent tire forces/moments wrench at the center of each wheel is given by the following expressions:

F T = F x i T + F y j T + F z k T M T = M x i T + M y j T + M z k T + F T × r cR
where F x = F x (κ) is the longitudinal force, F y = F y (α, γ) is the lateral force and F z = F z (δ z ) is the vertical force. M x is the tire torque about i T , M y is the rolling resistance torque and M z = M z (α, γ) is the alignment torque. These forces and torques are obtained from equation (53) for the longitudinal slip κ, the lateral slip α and from equation (54) for the vertical tire deformation δ z . Afterwards, the contribution of the tire/road contact forces/moments in the generalized efforts vector is given by:

Q a,T = i=f,r ∂v oRi ∂ϑ T F T,i + ∂ω oRi ∂ϑ T M T,i (55)

Motorcycle model control

To simulate the motorcycle model, a convenient control is to be synthesized to track a reference longitudinal speed profile and to stabilize the lateral dynamics around trim trajectories. This is a hard task since the roll DoF is unstable and, for some aggressive driving maneuvers conditions, system nonlinearities cannot be neglected. In addition, it is impossible to exploit the whole nonlinear model to develop a suitable controller, even by using a nonlinear control method. For these reasons, a linearized version of the dynamics model representing small motions in the neighborhood of the straight motion is considered.

In this paper, the speed and the roll controllers are based on Lyapunov method [START_REF] Dinga | An adaptive integrated algorithm for active front steering and direct yaw moment control based on direct Lyapunov method[END_REF]. This technique avoids the gain-scheduling, necessary for the linear method control to consider longitudinal speed variations. For each mode, longitudinal and lateral, the controller acts on the speed error to produce a torque to be applied on the rear wheel and on the roll angle error to generate a rider steering torque to be applied on the motorcycle's handlebar.

Longitudinal control

Consider the following longitudinal simplified model:

m i vx -v y ψ = F x,f + F x,r -F y,f sin δ (56) i y,f ζf = τ b,f -ρ f (F x,f -F y,f sin δ) i y,r ζr = τ D + τ b,r -ρ r F x,r
With the assumption of zero longitudinal slip, it goes that v x = ρ r ζr = ρ f ζf and by replacing in equation ( 56) the simplified longitudinal mode becomes:

m eq vx = τ b,f ρ f + τ D + τ b,r ρ r + m i v y ψ (57)
where m eq = m i + i y,f /ρ 2 f + i y,r /ρ 2 r is the equivalent mass. Let us introduce the Lyapunov function V = m eq e 2 /2, where e = v x -v x,ref is the speed tracking error, its derivative is given by:

V = m eq e ė = e τ b,f ρ f + τ D + τ b,r ρ r + m i v y ψ -m eq vx,ref (58)
In order to the Lyapunov function derivative to be negative, the propulsion torque τ D must satisfy:

τ D = ρ r -k x e x - τ b,f ρ f - m i v y ψ + m eq vx,ref -τ b,r (59)
where k x is a positive constant. With this input control, the Lyapunov function derivative is always negative ( V = -k x e 2 ) and hence the system is also, always stable.

Remark With the control of equation ( 59), the error dynamics becomes ė = -k x e, then, k x is just a proportional coefficient which is tuned until the desired tracking error level is achieved. However, a high value introduces an oscillatory behavior while stability remains preserved. The following simulations are carried-out using the developed nonlinear model and the controller of equation (59). Starting from 5ms -1 , the desired speed profile consists of two main phases, an acceleration phase to reach 23m/s -1 followed by a deceleration phase to keep a constant speed (see Figure 6-a). In Figure 6-b, the tire/road contact point speed is shown to give an overview of the longitudinal slip where Figure 6-c presents the simulation results of the in-plane mode variables (vertical displacement, front suspension travel, pitch and swing angles). Figure 6-d, describes the evolution of the nominal load applied on each tire w.r.t to speed variations (and so longitudinal acceleration). Lastly, requested propulsion torque at the rear-wheel and the resulting acceleration are sketched in Figure 7.

The second simulation aims to highlight the motorcycle behavior when subjected to a braking torques applied separately on the front and the rear wheel. Here in, it is not question to apply a particular braking model such as the ABS system. It will be shown that when applying a braking torque, significant load transfers take place from the rear wheel to the front one. If suspensions are assumed to be rigid, an approximation of the vertical load on each wheel when braking [START_REF] Cossalter | Optimum Suspension Design for Motorcycle Braking[END_REF] is given by:

F z,f = mg (x Gr -x Rr ) + F B z Gr p (60)
F z,r = mg (p -x Gr -x Rr ) -F B z Gr p
where, p is the motorcycle wheelbase and F B is the total braking force at the tire/road contact. Based on these two equations, a limit braking force F B,lim which causes the rear wheel lift can be computed.

Figure 8 represents the rear and the front brake profile applied separately on each wheel. In the first scenario, a rear braking torque with a maximum value of -600 N.m is applied between 4-7 seconds. The motorcycle dynamics behavior is demonstrated in Figure 9. From an initial forward speed of 20m/s, the motorcycle brakes hardly up to 2m/s with a deceleration of 0.4g. In that case, the vertical position z of point P is lowered which introduces a clockwise and hence a negative pitch rotation θ. The same vertical movement produces an anti-clockwise and hence a positive swing rotation µ. It is clear that the rear wheel does not lift due to the applied vertical force (600N which is equivalent to a payload of 60kg. See Figure 9-d). In the second scenario, the same torque profile as before is applied but with minus a half amplitude (-280Nm). The motorcycle dynamics behavior is shown in Figure 10. Initially the motorcycle was traveling at 20m/s and brakes up to 12 m/s with a deceleration of 0.4g. A first conclusion concerns the amplitude of the applied torque which is not sufficient to stop the motorcycle. In addition, such front brake can easily create the limit braking force F B,lim and hence causes the loss of the rear wheel contact with the ground. Indeed, the vertical force at the rear wheel is about 100N (a payload of only 10kg. See Figure 10-d). With the same justification, the vertical position z of point P goes up which introduces an anti-clockwise and hence a positive pitch rotation θ and a clockwise and hence a negative swing rotation µ. As is recognized in literature, this simulation presents the braking efficiency where an optimal braking must be split between the two wheels with more brake ratio at the front wheel (Figure 11).

Cornering maneuver control

As the longitudinal mode, a linearized version of the lateral dynamics model representing small motions in the neighborhood of the straight motion is derived which leads to the following linear state-space representation:

ẋ = A L x + Bτ r (61)
where x = [v y , ψ, φ, δ, ϕ, δ] T is the state vector and τ r is the rider torque input. In Figure 12, the stability of the uncontrolled motorcycle model of equation ( 61) is presented. This eigenvalues plot shows the existence of stable modes (in • symbol) and three instability modes, the first concerns the steering motion which appears from 8m/s (in ⋆ symbol), the second concerns the roll motion at very low speed until 7m/s (in ▽ symbol) and finally, a coupled steering-roll instable motion generated at very low speed (in + symbol). In [START_REF] Sharp | The stability and control of motorcycles[END_REF], these instability modes are named capsize, weave and wobble. 62), it can be seen that φ acts as a virtual input to the roll error dynamics. By using backstepping technique [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF], the required rider torque to track a reference roll angle is given by:

B 3 τ r = -k z z -A L,(3,:) x + β (63) β = -k ϕ e + φref (64)
where z = φ -β is the error between the real roll rate and the virtual input β. The following simulation are carried-out using the nonlinear model and the controllers of equations (59,63) where Figure 13 shows the road curvature. From this curvature, the reference roll profile is calculated. In Figure 14-a, the performance of backstepping controller in reference tracking is described. Obviously, the linear modelbased control of equation ( 63) is able to stabilize the nonlinear dynamics. However, for an aggressive cornering maneuver where the system nonlinearities are important, the stability of the motorcycle is not guaranteed. Otherwise, from figures 14-(a,b,c), the main features of the proposed dynamic model are illustrated mainly the counter-steer phenomena. Indeed, for a given applied torque τ r , the vehicle leans in the same direction as torque sign whereas the vehicle is steered in the opposite direction. Furthermore, steering angle and curvature have approximately the same plot. Finally, the requested rider torque at the motorcycle's handlebar is illustrated in Figure 14-d.

Conclusion

Based on the virtual power principle, called also Jourdain's principle, a middle complexity motorcycle dynamics model is proposed. The multi-body model has 11 DoF and includes the main features of motorcycle subsystems. This approach is chosen for its simplicity of synthesis and its fine analytical derivation with an acceptable calculation load, in particular, when the equations of motion are developed by using generalized coordinates approach. This model exhibits many of interesting behaviors of a real motorcycle such non-minimum phase steering response and load transfer resulting from acceleration and braking.

Besides, a virtual rider is synthesized for longitudinal and cornering modes respectively by using Lyapunov theory. This technique allows the calculation of speed and roll controllers which are valid for a wide range of forward speeds and hence, avoids controller scheduling.

From Figure 14.b, it is clear that using only roll stabilization cannot lead to a good trajectory tracking. For this, additional cost functions must be included to minimize the relative yaw angle and lateral deviation errors of the motorcycle from the road center line. These issues will be the scope of future works. Moreover, as perspective, experimental validation of the proposed motorcycle model and the simulation of more limit riding maneuvers are expected.

R i =   1 0 0 0 cos(i) -sin(i) 0 sin(i) cos(i)   , for i = ϕ R i =   cos(i) 0 sin(i) 0 1 0 -sin(i) 0 cos(i)   , for i = θ, ǫ, µ R i =   cos(i) -sin(i) 0 sin(i) cos(i) 0 0 0 1   , for i = δ R i,j,••• ,k = R i R j • • • R k C δ
damping ratio of the handlebar around its axis k s,i , c s,i stiffness and damping of the motorcycle suspensions F i,c /M i,c constraint forces/moments applied on body i F i,a /M i,a external forces/moments applied on body i F s,i spring and shock absorber force

F T /

Numerical values

All numerical values are taken from [START_REF] Sharp | Advances in the Modelling of Motorcycle Dynamics[END_REF]. However, the coordinate system convention are different (SAE with z axis points downward in [START_REF] Sharp | Advances in the Modelling of Motorcycle Dynamics[END_REF] and ISO with z axis points upward in this paper). Consequently, the sign of some Pacejka tire parameters must be inverted (p Ky1 , p Ky6 , p Ky7 , q Bz10 , q Dz3 , q Dz8 , q Dz9 , q Ez2 , q Ez5 , q Hz3 , q Hz4 ).

Linear and angular velocity Jacobian matrices

Let be the set of coordinates e x = [1, 0, 0] T , e y = [0, 1, 0] T and e z = [0, 0, 1] T . The expressions of the linear and angular Jacobian matrices for each body is given by:

• For the rear body:

Figure 1 :

 1 Figure 1: The motorcycle geometrical description

Figure 2 :

 2 Figure 2: Configuration of the vehicle system axis ℜ v w.r.t the inertial reference frame ℜ o

Figure 3 :

 3 Figure 3: Axis systems and angular displacements configuration

Figure 4 :

 4 Figure 4: Suspensions configuration, (a) front, (b) rear

Figure 5 :

 5 Figure 5: Tire reference frame

Figure 7 :

 7 Figure 6: Speed profile tracking

Figure 8 :

 8 Figure 8: Rear and front brake torque profiles

Figure 9 :Figure 10 :

 910 Figure 9: Simulation results according to only hard rear brake torque

Figure 11 :

 11 Figure 11: Simulation results according to 70% front brake torque and 30% rear brake torque of previous torques

Figure 12 :

 12 Figure 12: Eigenvalues real part of the uncontrolled linearized motorcycle model w.r.t the forward speed

Figure 13 :

 13 Figure 13: Road curvature

Figure 14 :

 14 Figure 14: Simulation results for roll profile tracking

•• 3 ∂ω oG l ∂ϑ = 0 3 ,2 e z 0 3 e x 0 3 e δ 0 3, 4 •••

 334 ∂v oGr ∂ϑ = e x e y e z × r vGr e ′ z e x × r vGr e θ × r P Gr 0 3,5 ∂ω oGr ∂ϑ = 0 3,2 e z 0 3 e x e θ 0 3,5 e ′ z = R ϕ e z e θ = R ϕ e y For the front upper body: ∂v oG f ∂ϑ = e x e y e z × r vG f e ′ z e x × r vG f e θ × r P G f e δ × r BG f 0 3,4 ∂ω oG f ∂ϑ = 0 3,2 e z 0 3 e x e θ e δ 0 3,4 e δ = R ϕ,θ,ǫ e z For the front lower body: ∂v oG l ∂ϑ = e x e y e z × r vG l e ′ z e x × r vG l e θ × r P G l e δ × r BG l e δ 0 3,For the swing arm body: ∂v oGs ∂ϑ = e x e y e z × r vGs e ′ z e x × r vGs 0 3,3 e µ × r P Gs 0 3,2 ∂ω oGs ∂ϑ = 0 3,2 e z 0 3 e x 0 3,3 e µ 0 3,2 e µ = e θ For the rear wheel: ∂v oRr ∂ϑ = e x e y e z × r vRr e ′ z e x × r vRr 0 3,3 e µ × r P Rr 0 3,2 ∂ω oRr ∂ϑ = 0 3,2 e z 0 3 e x 0 3,5 e ζr e ζr = e θ For the front wheel: ∂v oR f ∂ϑ = e x e y e z × r vR f e ′ z e x × r vR f e θ × r P R f e δ × r BR f e δ 0 3,3 ∂ω oR f ∂ϑ = 0 3,2 e z 0 3,1 e x 0 3,1 e δ 0 3,2 e ζ f 0 3 e ζ,f = R ϕ,θ,ǫ,δ e y

  M T tire/road contact efforts Q a,g contribution of the gravity force in Q a Q a,hcontribution of the rider torque on the handlebar inQ a Q a,sf contribution of the front suspension force in Q a Q a,sr contribution of the rear suspension force in Q a Q a,pb contribution of the propulsion/braking torques in Q a Q a,Tcontribution of the tire/road interaction efforts in Q a

	r	scalar variable
	0 n	zero vector of dimension n
	0 m,n	zero matrix of dimension m by n
	r	vector variable
	R	matrix variable
	x T	Transposition operator
	ẋ, ẍ

first and second differentiation of a variable x w.r.t time variable

Table 1 :

 1 Motorcycle geometric specification (lengths are in [m] and angles in [rad]) x

	Gr	z Gr	x Gf	z Gf	x Gl	z Gl
	0.1289	0.1116	0.0452 0.1237 0.0679 -0.263
	x Gs	z Gs	x Rr	z Rr	x Rf	z Rf
	-0.353 -0.0495 -0.549 -0.0638 0.0474 -0.3655
	x Sl	z Sl	x Su	z Su	h p	l P B
	-0.1047 -0.1826 -0.062	0.128	0.3608 0.6831
	ρ f	ρ r	ǫ
	0.324	0.297	-0.4189

Table 2 :

 2 Motorcycle mass specification [Kg] m Gr m Gf m Gl m Gs m Rr m Rf 165.13 9.99 7.25 8 14.7 11.9

Table 3 :

 3 Motorcycle inertia specification [Kg.m 2 ]

		11.085	I Gr 0	3.691	I Gf	I Gl
		0	22.013	0
		3.691	0	14.982

Table 4 :

 4 Motorcycle handlebar and suspensions specificationC δ [N.m.rad -1 .s] F s,f,0 [N] F s,r,0 k s,f [N.m -1 ] k s,r c s,f [N.m -1 .s]

	c s,r


H Slimi

D Ichalal

H Arioui

S Mammar

Motorcycle Maximal Safe Speed in Cornering Situation

In this paper, we present a new method for the computation of the maximal authorized motorcycle speed in curves. The three main actors which are the vehicle, the driver and the infrastructure are taken into account. The vehicle dynamics are represented by a dynamic four degrees of freedom model which includes the vehicle's longitudinal slip and side slip angle. The driver behavior model considers his ability in deceleration maneuvers according to the mobilized friction. The infrastructure characteristics introduce a precise handling of the road geometry and of the maximal available friction.

I. INTRODUCTION

The number of death in road accidents has experienced a huge reduction of about 50% during the last decade. However analysis of accident statistics shows that the number of death when a motorcycle is involved has not decreased and even increased. In fact, the motorcycle remains a particularly dangerous mode of transport: the number of death is still very high, and if one takes account of the number of traveled kilometers, the risk of death for a motorcycle rider is 21 times higher than that of other transportation modes. In additions, motorcycles and scooters become more and more popular for urban an suburban displacement in European countries. Associated traveled kilometers and number of vehicles sold is continuously increasing.

Thus France as well as Europe have launched several research programs in order to study motorcycle dynamics and detect precursor indicators of imminent accident. Notice that good interactions between the rider-the motorcycle and the infrastructure is an important factor. This paper is focused on the development of a computational method of the maximal authorized speed for a motorcycle driving the upcoming curve.

The interaction of the three actors which are the vehicle, the rider and the infrastructure are taken into account. The motorcycle dynamics are represented by a model which includes longitudinal load transfer. The rider behavior considers his ability in deceleration while the infrastructure characteristics introduce a precise handling of the road geometry (curvature, slope and road bank angle) and the maximal available friction. These parameters even if they are not actually available for measurement could be provided through ADASE maps for example.

The remaining of the paper is organized as follow: the problem statement is provided in section [START_REF] Amans | RIDER : Recherche sur les accidents impliquant un deux roues motoris[END_REF]. Existing speed models in curves are reviewed in section (3) while the proposed speed profile model is detailed in section (4). hichem.arioui@ibisc.univ-evry.fr

Simulation results are provided is section [START_REF] Limebeer | Bicycles, motorcycles and models[END_REF]. The conclusion is given in section (6) wrap up the paper.

II. OVERVIEW ON LIMIT SAFE MOTORCYCLE PROFILE IN

CURVE

The models on safe profile we discuss in this section are mostly specific to cars (four-wheeled ones). Nevertheless, few models exist for motorcycles. These last were initially concerned the admissible roll angle and speed limits to perform a turn safely [START_REF] Evangelou | Infuence of Road Camber on Motorcycle Stability[END_REF].

It should be noted that the speed limits that we find on our roads and highways are based on studies conducted for cars than any other category must fulfill (motorcycles) or even reduce it (truck). Also, we can find this type of requirement in terms of infrastructure, sometimes poorly suited for motorcycles, [START_REF] Amans | RIDER : Recherche sur les accidents impliquant un deux roues motoris[END_REF].

Among the earliest models, limiting the car's safe speed, is the one proposed by [START_REF] Karnopp | Optimum Roll Angle behavior for Tilting Ground Vehicle[END_REF] (we call it : simple model). This model is based on a quasi-static study of a car with a mass M moving at v x longitudinal speed taking a turn of ρ curvature. The limit safe speed, neglecting any parameters other than the curvature, is governed by:

v x max = gµ lat ρ (1)
Where µ lat is the maximum available lateral friction and g is the gravity acceleration.

Unfortunately, this speed model does not contain any other element such as: instant road adhesion, road geometry (slope and elevation), etc. In addition, the curvature radius is assumed constant, what is wrong, at least at the beginning and the exit of turn. However, the centrifugal force depends on the curvature dynamics, which we try to proof afterwards.

In the US, National Highway Traffic Safety Administration (NHTSA) calculates the car's limit speed when cornering based on the following formula [START_REF] Pomerleau | Run-off-road collision avoidance using ivhs countermeasures[END_REF]:

v x max = g ρ • φ d + µ lat 1 -φ d µ lat (2)
Unlike the previous formula takes into account another aspect of the road geometry, ie the bank angle. However, the linearized formula seems to ignore the vehicle angle roll. For motorcycle, the roll angle can reach 30 • in some cases, and therefore absorbs a large part of the lateral acceleration or centrifugal force. Thereafter, we propose a model adapted for bike, taking into account the roll angle and the road slope also.

As mentioned previously, very few models for setting the speed limit or eligible roll angle, when cornering, have been proposed. Generally, a roll angle providing an outlet to safely turn depends strongly on the road geometry (curvature, slope and bank angles), the road conditions (adhesion, weather), the driver's experience and dynamic states of the vehicle, [START_REF] Evangelou | Infuence of Road Camber on Motorcycle Stability[END_REF], [START_REF] Karnopp | Optimum Roll Angle behavior for Tilting Ground Vehicle[END_REF]. Based on the same assumptions (under constant longitudinal speed, v x ) than for cars, [START_REF] Evangelou | Infuence of Road Camber on Motorcycle Stability[END_REF] proposed a model for limited the admissible roll angle φ des :

|φ des -φ d | < arctan(µ limit ) (3)

III. MODELS ENHANCEMENT A. Simple Model

In this section, we present improvements to previous models. For the simple one (1), we illustrate the impact of curvature radius and its "dynamics" on the safe speed limitation. Indeed, the impact of this road parameter (first and second derivative) is particularly important for two-wheelers, because, for example, of the ability of drivers to change the trajectory abruptly.

Consider a material point P, hence, the influence of moments is ignored, with mass M rotating around the z axis with angular velocity ω, Figure [START_REF] Slimi | Advanced Motorcycle-Infrastructure-Driver Roll Angle Profile for Loss Control Prevention[END_REF]. The trajectory is defined, among others, by its curvature ρ(t) = 1 R(t) and its first derivative ρ(t) (assumed continuous and differentiable). The point P coordinates are given by:

OP = R(t) y = 1 ρ(t) y ( 5
)
We can express the point motion, speed v x = v P/O and acceleration a y = ÿ = a P/O , by the successive derivatives, as follows:

   v P/O = d R(t) dt y + R(t) ψ x = -ρ(t) ρ(t) 2 y + 1 ρ(t) ψ x a P/O = d 2 R(t) dt 2 -R(t) ψ2 y + 2 dR(t) dt ψ + R(t) ψ x (6)
This equation can be rewritten as a function of v x speed by substituting ψ by 1 R(t) v x . The equation ( 6) has two important terms, namely:

• The first term obtained by projection on y axis represents the centrifugal acceleration (force):

d 2 R(t) dt 2 -v 2 R(t)
. It is noted that in the case of a circular road type (constant curvature ρ = 0) or a clothoid, ρ = constant, acceleration (or force) depends only on the centrifugal speed and the instantaneous curvature value. Moreover, the curvature dynamics can affect the safe speed limit, especially at the beginning and the end of turn, where the radius and its first derivative are variable.

• The second term obtained by projection on x axis represents the tangential acceleration: 2 dR(t) dt ψ + R(t) ψ . The highest safe speed in a curve, can be rewritten as follows:

ÿ = - d 2 R(t) dt 2 + V 2 x R(t) = gµ lat ⇒ v x max = R(t) + g µ lat ρ(t) (7)
with:

d 2 R(t) dt 2 = 2ρ ρ2 -ρ 2 ρ ρ 4 (8)
Also, the impact of curvature dynamics may be more important in the determination of the under or over-steering cases.

B. NHTSA Model

As stated before, equation (2) takes into account the slope angle, road curvature and adherence and seems to ignore the roll angle of the vehicle itself.

If we consider a single track, the equilibrium will be governed by the following equations:

   F x = R sin(φ p )cos(α + φ + φ d ) R cos(φ p ) sin(α + φ + φ d ) = M ρ v 2 x R cos(φ p ) cos(α + φ + φ d ) = M g (9)
where : tan(α) = µ lat . Then, we can express, after light calculations, the maximum safe speed by:

v x max = g ρ tan(φ d ) + tan(φ )+µ lat 1-tan(φ )µ lat 1 -tan(φ d ) tan(φ )+µ lat 1-tan(φ )µ lat ( 10
)
We can notice that the slope angle has no effect on the speed limit calculation v x max . Also, the bank and the roll angles have the same effect on v x max synthesis. This relationship gives highest safe speed limit for an admissible roll angle which could be calculated by various methods [START_REF] Slimi | Advanced Motorcycle-Infrastructure-Driver Roll Angle Profile for Loss Control Prevention[END_REF], [START_REF] Pomerleau | Run-off-road collision avoidance using ivhs countermeasures[END_REF].

IV. NEW APPROACH FOR LIMIT SPEED CALCULATION

The proposed approach takes into account the driver behavior and the road geometry. Also, the main idea of the limit speed calculation is based on motorcycle dynamics:

• A two bodies motorcycle dynamic model which takes into account the forces of tire/road contact and the load transfer due to the longitudinal accelerations. • Lateral and longitudinal mobilized frictions on each wheel are computed. The highest safe speed is defined considering the maximum friction. As we consider a four degrees (roll, steer, yaw and longitudinal dynamics) of freedom model, its speed is computed by resolving the maximum of the two tires mobilized frictions. • The driver behavior (acceleration/deceleration profile).

A. Motorcycle Dynamic Model

This model has two joined rigid bodies and is inspired from those presented in [START_REF] Koenen | The dynamic behaviour of motorcycles when running straight ahead and when cornering[END_REF], [START_REF] Limebeer | Bicycles, motorcycles and models[END_REF], [START_REF] Sharp | A motorcycle model for stability and control analysis[END_REF], [START_REF] Cossalter | A motorcycle multi-body model for real time simulations based on the natural coordinates approach[END_REF], [START_REF] Slimi | Preventive Safety: Warning System for Control Loss of Two-Wheeled Vehicles[END_REF]. The longitudinal dynamics is given by: [START_REF] Pomerleau | Run-off-road collision avoidance using ivhs countermeasures[END_REF] Where, ∑ F xi represents the longitudinal forces acting on the tires according to the longitudinal slip. F aerolong is the aerodynamic force in the longitudinal direction.

M( ẍ -ẏ ψ) -(M f j + M r h) φ ψ2 -(M f e cos ε) δ 2 -F aerolong -M g sin φ d sin φ p = ∑ F xi
∑ F xi = F x f + F xr = ∑ K pi λ i (12)
Where i = 1, 2. K pi is the longitudinal stiffness coefficient of tires, λ i is the tires longitudinal slip.

The lateral dynamics is described by [START_REF] Slimi | Advanced Motorcycle-Infrastructure-Driver Roll Angle Profile for Loss Control Prevention[END_REF]:

M( ÿ + ẋ ψ) + (M f j + M r h) φ + M f k ψ + M f e δ -F aerolat + M g cos φ p sin φ d = ∑ F yi (M f j + M r h) ÿ + a 1 φ + a 2 ψ + a 3 δ + a 4 ẋ ψ + a 5 ẋ δ = ∑ M x M f k( ÿ + ẋ ψ) + a 2 φ + b 1 ψ + b 2 δ -b 3 ẋ φ -a 5 ẋ δ = ∑ M z M f e ÿ + a 3 φ + c 1 ψ + c 2 δ -a 5 ẋ φ + c 3 ẋ ψ = ∑ M s (13)
Where a i , b j and c k are defined in the annex section (VIII). We have:

∑ M x = (M f j + M r h)g sin φ -(ηF z f -M f eg) sin δ ∑ M z = l f F f -l r F r -τ cos ε ∑ M s = -(η F z f -M f e g) sin φ -(η F z f -M f e g) sin ε sin δ + τ (14)
Where, ∑ F yi are the lateral forces exerted on each tire. We can express the lateral forces on each tire as, [START_REF] Slimi | Advanced Motorcycle-Infrastructure-Driver Roll Angle Profile for Loss Control Prevention[END_REF]:

F y f = -C f 1 α f +C f 2 γ f F yr = -C r1 α r +C r2 γ r ( 15
)
Where, C f i and C r j are the lateral stiffness and the camber coefficients of the tire.

On the other hand, the slip angles of the front and rear tires are given by:

     α f = v y +L f ψ-η δ v x -δ cos ε α r = v y -L r ψ v x (16)
L f (resp. L r ) represents the distance between the center of mass and the front axis (resp. rear axis).

The camber angles of front and rear tires are given by [START_REF] Slimi | Advanced Motorcycle-Infrastructure-Driver Roll Angle Profile for Loss Control Prevention[END_REF]:

γ f = φ + δ sin ε γ r = φ (17)
Finally, we can express the lateral forces on the two wheels as follows:

     Y f = C f δ cos ε - v y +L f ψ-η δ v x +C f 2 (φ + δ sin ε) Y r = C r1 L r ψ-v y v x
+C r2 φ (18) Under dynamical conditions, load can transfer on the front wheel when braking or on the rear wheel during accelerations/decelerations phases. In this cases, normal forces on the two tires can be obtained as follows:

       F z f = M f + M r L f + L r (L r g cos φ p cos φ d -h ẍ) F zr = M f + M r L f + L r (L f g cos φ p cos φ d + h ẍ) (19)
Where, h is the height of the center of mass, L f (resp. L r ) corresponding to the dimensions of vehicle configuration (horizontal dimensions of the overall center of gravity relative to the center of the front wheel, rear respectively). F z f (resp. F z f ) represent normal forces applied on the front tire (resp. rear tire).

B. Tire Road Friction

The tire dynamic behavior is very complex and is linear only under certain restricted conditions of drive. We can observe various phenomena like skidding and blocking.

The Coulomb friction model is used to obtain the mobilized friction µ mob . Thus the transversal force F t and the normal force F n = F z f + F zr are such that:

µ mob = F t F n ≤ µ max (20)
Where µ max is the maximum available friction. Since:

F 2 t = F 2 lat + F 2 long (21)
We have:

       µ 2 mob = F 2 lat +F 2 long F 2 n = µ 2 lat + µ 2 long µ lat = F lat F n µ long = F long F n (22)
To calculate µ lat and µ long , we use the simplified model of the motorcycle in absolute frame based on previous model, including lateral and longitudinal equations, as follows:

   M ÿ = (Y f +Y r ) -Mg sin(φ + φ d ) F x = M ẍ -Mg sin φ p I z ψ = (L f Y f -L r Y r ) cos(φ + φ d ) (23)
Then, front and rear lateral forces expressed by:

   Y f = L r M ÿ cos(φ +φ d )+L r Mg sin(φ +φ d ) cos(φ +φ d )+I z ψ (L f +L r ) cos(φ +φ d ) Y r = L f M ÿ cos(φ +φ d )+L f Mg sin(φ +φ d ) cos(φ +φ d )-I z ψ (L f +L r ) cos(φ +φ d ) (24)
After light calculations and using the vertical forces (19), lateral and longitudinal frictions on each wheel are given by:

         µ lat f = Y f /F z f = L r M ÿ cos(φ +φ d )+L r Mg sin(φ +φ d ) cos(φ +φ r )+I z ψ (L r Mg cos(φ p )-hF x ) cos(φ +φ d ) µ latr = Y r /F zr = L f M ÿ cos(φ +φ d )+L f Mg sin(φ +φ d ) cos(φ +φ r )-I z ψ (L f Mg cos(φ p )+hF x ) cos(φ +φ d ) (25)
The previous system can be simplified by:

             µ lat f = µ lat cos(φ + φ d ) + I z ψ L r Mg (cos φ p -hF x L r Mg ) cos(φ + φ d ) µ latr = µ lat cos(φ + φ d ) -I z ψ L f Mg (cos φ p + hF x L f Mg ) cos(φ + φ d ) (26)
We assume here that

I z ψ L r Mg cos(φ +φ d )
is very small against µ lat . Then, we can rewrite the previous system as :

           µ lat f = µ lat (1-hax Lr g ) tan(φ p ) µ latr = µ lat (1+ hax L f g ) tan(φ p ) µ lat = a y g + sin(φ + φ d ) µ long = a x g -sin(φ p ) (27)
The highest safe speed is defined considering the maximum mobilized friction. As we consider a two-wheeled vehicle model, the aim is to find the maximum between the two mobilized frictions, so:

µ max = max µ 2 lat f + µ 2 long f , µ 2 latr + µ 2 longr (28)

C. Driver Behavior

The driver is unceasingly obliged, according to the information taken from the environment, to define an appropriate speed and position to the encountered situation [START_REF] Glaser | Modelisation et analyse d'un vehicule en trajectoire limites Application au developpement de systemes d'aide a la conduite[END_REF]. Here, the capacities of the driver in acceleration are taken into account. The driver cannot mobilize the same level of acceleration into longitudinal and lateral directions. For this reason, we have to distinguish between the maximum friction in longitudinal and lateral modes:

1 = max( µ lat f µ latmax 2 + µ long f µ longmax 2 , µ latr µ latmax 2 + µ longr µ longmax 2 ) (29)
For comfort and safety reasons, and under good weather conditions, a driver generally does not mobilize all the available lateral and/or longitudinal friction. Therefore we represent the driver behavior with through two dimensionless coefficients λ lat , λ long and a maximum friction µ max [START_REF] Glaser | Modelisation et analyse d'un vehicule en trajectoire limites Application au developpement de systemes d'aide a la conduite[END_REF].

µ latmax = λ lat • µ max and µ longmax = λ long • µ max (30)
We consider that the driver behavior varies proportional to 1 µ max .

λ long = ∆λ long µ max -λ slip-long λ lat = ∆λ lat µ max -λ slip-lat (31)
With:

∆λ long = λ dry-long -λ slip-long ∆λ lat = λ dry-lat -λ slip-lat (32)
Where, λ dry (resp. λ slip ) represents the maximum friction that the driver can mobilize on a dry road (resp. slipping one).

D. Generation of Speed Profile

Considering the equilibrium state, we can make the following approximations:

     φ = tan -1 ( ρv 2 x g ) a x = ẍ = dv x dt = v x dv x ds ψ = ρv x (33)
Where, s is the road curvilinear coordinate. Finally µ lat f and µ latr may be given in a more compact form, in braking and accelerating cases, as: In a simpler form:

   1 = ( ÿ g +sin(φ +φ d ) η 1 ) 2 + η 2 2 1 = ( ÿ g +sin(φ +φ d ) η 3 ) 2 + η 2 2 (34)
With:

     η 1 = (λ lat µ max )(1 + h L f ( v x g dv x ds -tan φ p )) η 2 = ( 1 λ lat µ max )( v x g dv x ds -tan φ p ) η 3 = (λ lat µ max )(1 -h L r ( v x g dv x ds -tan φ p ))
By developing the previous equations (34), the maximum speed v xi at each wheels is given by solving the following equation system of equations as shown by the following block diagram (Figure 3). Then, the highest safe speed v limit , in curve, is obtained by solving:

v limit = min(v x1 , v x2 ).
This model speed is function of intrinsic motorcycle parameters, roadway parameters and driver behavior. Also, it takes into account the vehicle side slip, and longitudinal dynamics by considering the longitudinal slip on each wheel.

V. ROAD PARAMETERS ESTIMATION AND DYNAMIC STATES RECONSTRUCTION

To deploy such safety systems or function on motorcycles, one has to know reliably what the current state of the vehicle and road parameters are. Various sensors can be used (e.g. lateral acceleration, yaw rate, wheel speeds), which are already used for four-wheeled vehicle applications. However, the major difference concerns the roll angle which can exceed 40 • in some limit cases or lateral speed (due to initial condition issue).

Roll angle is the main characteristic of the Motorcycle lateral dynamics. A good control of motorcycle motions requires an accurate assessment of this quantity and for safety applications also the risk of sliding or friction loss need to be considered. Direct measurement of roll angle and tire slip is not available or very expensive as for the lateral speed (corevit sensor). Some previous work [START_REF] Teerhuis | Motorcycle State Estimation for Lateral Dynamics[END_REF] and [START_REF] Stefano | Attitude Estimation of a Motorcycle Via Unscented Kalman Filter[END_REF] have addressed these challenges based on estimation and/or observation techniques of these dynamic parameters.

Regarding to road parameters estimation, two approaches can be implemented. The fist one uses observation techniques based on unknown input estimation. Indeed, as shown previously, the function calculating the safe authorized maximal speed depends on some motorcycle parameters. Due to economic and/or technique reasons these parameters are not accessible to measure. Observers can then provide an estimation of them. For that purpose, in future works, we consider a nonlinear model of the motorcycle in order to provide an accurate and robust estimation of the parameters which allows reducing false alarms and to give a good estimation of the safe authorized maximal speed in curves. The second approach is based on available data given by cartographies.

This issues and their real time implementation will be addressed in future works.

VI. SIMULATION RESULTS

In this section, we test the new safe speed model we have proposed and the impact of inherent parameters. Also, we compare it with existing models (1 and 2). simulation tests were conducted under the same conditions: the motorcycle moves on a road with two curves in the same direction with positive curvature radius of different sizes (blue line on Figure 4). The variations of slope and bank are chosen to highlight the impact of each on the limit of the safe speed (Figure 4). We also limited the longitudinal velocity to about 40m/s in straight line. For all simulation conditions, adhesions coefficients µ max are fixed around 0.7. Figure [START_REF] Limebeer | Bicycles, motorcycles and models[END_REF] illustrates the various speed limits obtained by the three models (equations: 1, 2 and 34). The difference between the speed synthesized by our approach and those given by the other two approaches is significant. Indeed, this speed difference (between the first and third model) can reach, in some cases, 16 m/s. The first observation can be done on the impact of the bank angle on the NHTSA and the proposed models in this paper. The second remark is made on the discrepancy in these two models, where the motorcycle dynamics (load transfer, geometry and inertia of the bike, etc.) and the driver behavior parameters are very important. Maximum Speed Profile For Different λ x , λ y V New (λ x =0.5 , λ y =0.5) V New (λ x =0.1 , λ y =0.1) Fig. 7. Maximum Speed Profile For Different λ longx , λ lat Figure [START_REF] Cossalter | A motorcycle multi-body model for real time simulations based on the natural coordinates approach[END_REF] shows, the influence of the driver behaviour (λ long , and λ lat ) on limit safe speed. When λ long and λ lat (λ long = 0.1, and λ lat = 0.1) decrease the speed drops signif- icantly.

VII. CONCLUSIONS AND FUTURE WORKS

In this paper, we studied and adapted some existing models for the synthesis of the profile of the highest safe speed when cornering. Moreover, we proposed a new model taking into account the motorcycle dynamics, the road's geometry and driver behavior.

We have shown that existing models (single or NHTSA one) give a safe speed profiles not adapted or suitable for two-wheeled vehicles. These models take into account little important information, for example adherence and geometry of the road (curvature and/or the bank angle). This means that whatever the vehicle be light or heavy (not taking into account the mass and inertia), would not change the safe speed limit. This finding is of course false, since the centrifugal force is proportional to the vehicle mass. On the other hand, we found that a safe speed profile giving by the simple model can depend on curvature dynamics (second derivative) and thus impacting, according to the road shape, the safe speed profile. This issue will be studied soon.

In addition, we proposed a new "dynamic" model to calculate the motorcycle speed limit when cornering. This model does not take into account the quasi-static conditions as the two previous models. The motorcycle dynamics (load transfer, roll and camber angles), road infrastructure (adhesion, curvature, bank angle) and driver behavior are supported in the proposed model. The effect of these parameters is significant, since we have a safe speed difference, under the same conditions of simulation, about 16 m/s with the simple model and about 9 m/s with NHTSA model.

In future work, we plan to synthesize a limit roll profile in cornering situations. It should be noted that the roll dynamics is one of the most important degrees of freedom. In addition, dynamic states and road profile estimation will be addressed thus the impact of road traffic.

Fig. 1 .

 1 Fig. 1. Kinematic parameters of a point P in circular motion

Fig. 2 .

 2 Fig. 2. Schematic diagram of the Motorcycle

Fig. 3 .

 3 Fig. 3. Block diagram for limit speed calculation

  Fig. 4. Road Parameters

Fig. 5 .

 5 Fig. 5. Maximum Speed Profile synthesized by Simple (1), NHTSA (2) and the proposed (34) approaches

Fig. 6 .

 6 Fig. 6. Maximum Speed Profile For Different Road Maximum Friction


Dalil Ichalal
email: dalil.ichalal@ibisc.univ-evry.fr

Hichem Arioui
email: hichem.arioui@ibisc.univ-evry.fr

Saïd Mammar
email: said.mammar@ibisc.univ-evry.fr

Observer Design for Motorcycle's Lean and Steering Dynamics Estimation: a Takagi-Sugeno Approach

In this paper, a nonlinear motorcycle model is considered in order to estimate both the lean and steering dynamics. The model is transformed into a Takagi-Sugeno (T-S) form using the well-known sector nonlinearity approach. The first contribution of this work is the exactness of the obtained T-S model compared to the considered nonlinear model, where the weighting functions of the T-S model depend on unmeasured state variables. A novel approach to construct a nonlinear unknown input fast observer is proposed. The objective is the simultaneous reconstruction of the state variables and the rider's torque. The observer's convergence is studied using Lyapunov theory guaranteeing boundedness of the state and unknown input estimation errors which is expressed by the Input to State Practical Stability (ISpS). Stability conditions are then expressed in terms of Linear Matrix Inequalities (LMI). Finally, simulation results are provided to confirm the suitability of the proposed nonlinear observer.

I. INTRODUCTION

Currently, powered two-wheeled (PTW) vehicles is a mean of transport increasingly sought after, especially for the opportunities it offers to sidestep traffic congestion. This increase in vehicle fleet to PTW is accompanied by the blast of the number of accidents. For a long time, industrial societies record on their roadways thousands of deaths and other fatalities per year. Road safety institutions have launched several preventive actions (radar, tickets, etc.) and research programs headed for safety systems, mainly for cars. Some of these steps have brought their results, since the number of fatalities recorded a significant decrease over the past few years. However, if the number of fatalities experiences an overall decrease, the PTW remains a particularly dangerous transportation, [START_REF]characteristics and causes of road accidents[END_REF], [START_REF] Amans | Rider project . research on accidents involving powered two-wheelers final report n rider[END_REF].

Development of safety systems for cars has reached a certain maturity. Unfortunately, for motorcycles it is not the case. The most prominent example is that of the ABS, that exists for 20 years, but is not still standard equipment for PTW and its use remains marginal, [START_REF] Evangelos | Saferider project. final report n rider[END_REF]. The direct transposition of security systems, from cars to motorcycles, is not obvious because of the complex motorcycle dynamics (highly nonlinear, [START_REF] Cossalter | A motorcycle multibody model for real time simulation based on the natural coordinates approach[END_REF], [START_REF] Hima | Suitable two wheeled vehicle dynamics synthesis for interactive motorcycle simulator[END_REF]), for example. In addition, ahead of the design of a safety system, we must ensure the availability of the relevant dynamic states in order to quantify the risk (loss of control, skidding, etc.). PTW dynamics can be estimated through suitable sensors and/or observers. The first way is generally avoided for several reasons: price, noise measurement, feasibility, etc. On the other hand, there are only few works dealing with the problem of motorcycle state observation.

In the literature, most studies have mainly concerned the estimation of the lean dynamics unlike the steering one. Different techniques have been proposed to estimate the roll angle: for example, frequency separation filtering [START_REF] Boniolo | Estimate of the lean angle of motorcycles[END_REF] or extended Kalman filters, [START_REF] Teerhuis | Motorcycle state estimation for lateral dynamics[END_REF]. These techniques, performed under restrictive assumptions (dynamic steering is neglected, tire-road forces are linear, etc.), are not robust against the variations of the forward velocity.

The topic of estimation of the steering angle (not the rider's torque reconstruction) is not well covered in literature as the lean angle estimation problem. However, a few results have been obtained in [START_REF] Filippi | Singlesensor control strategies for semi-active steering damper control in two-wheeled vehicles[END_REF], where an LPV observer has been used to design single-sensor control strategies for a semi-active steering damper. The approach is a simple gain scheduling for an LTI motorcycle model under three constant forward velocities (50, 100 and 140 m/s). Unfortunately, no guarantees for stability or convergence of the LPV observer are given outwards these constant velocities.

To the best knowledge of the authors, the simultaneous estimation of the lean and the steering dynamics (rider's torque) have never been addressed.

II. PROBLEM STATEMENT

Our long term objective points the identification of all pertinent inputs and dynamic states improving the risk quantification of the loss-of-control during cornering. Indeed, inadequate cornering is responsible for most motorcycle fatalities, especially for single motorcycle crashes. For safe cornering, riders should respect: 1) a suitable speed before starting the corner, 2) the road conditions (under weak friction) and 3) weather conditions do not allow optimal visibility when riding. Early warning systems are based generally on related work carried out for standard cars [START_REF] Slimi | Advanced motorcycle-infrastructure-driver roll angle profile for loss control prevention[END_REF]. The goal is the synthesis of a function estimating the maximum safe speed at which a vehicle can be kept stable on the road while moving at a constant longitudinal velocity on a circular section. This velocity depends, among other factors, on the lateral friction µ lat whose computation involves all the dynamic states of the PTW and a good modeling of the tireroad contact [START_REF] Slimi | Advanced motorcycle-infrastructure-driver roll angle profile for loss control prevention[END_REF]. This makes the success of such warning systems strongly dependent on the availability of dynamic states of the motorcycle. Correlating this fact with the highly nonlinear dynamics, standard observation approaches are questionable and less efficient. To answer this real challenge, we address in this paper a nonlinear observer synthesis based on Takagi-Sugeno approach for PTW vehicles.

Takagi-Sugeno fuzzy structure is one of the most interesting approach to model nonlinear behaviors as proven in recent years [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. Indeed, it offers a way to represent nonlinear complex behaviors by a more tractable mathematical formulation inspired by linear models. It consists on decomposing the operating state space on several regions and each region is modeled by a linear model. Thanks to nonlinear weighting functions which satisfy the convex sum property, the overall nonlinear behavior of a system is characterized in a compact set of the state space. In recent years, some works are dedicated to the study of nonlinear systems via T-S models, especially in observers design [START_REF] Ichalal | Fault tolerant control for Takagi-Sugeno systems with unmeasurable premise variables by trajectory tracking[END_REF], [START_REF] Bergsten | Observers for Takagi-Sugeno fuzzy systems[END_REF], [START_REF] Akhenak | Design of sliding mode unknown input observer for uncertain Takagi-Sugeno model[END_REF]. The proposed observer in this paper is inspired from the linear adaptive one in [START_REF] Wang | Actuator fault diagnosis: an adaptive observerbased technique[END_REF] and adapted to T-S models with unmeasurable premise variables which constitutes an open and interesting field of research. Indeed, the most developed work in the literature are limited only to the case of T-S models with measurable premise variables. T-S models with unmeasurable premise variables have an interesting properties, namely, the ability to transform a general nonlinear model into a T-S model by the use of nonlinear sector nonlinearity transformation with no loss of information (see [START_REF] Ichalal | State estimation of Takagi-Sugeno systems with unmeasurable premise variables[END_REF] and references therein), and it is pointed out in [START_REF] Yoneyama | H ∞ filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach[END_REF] that T-S models with unmeasurable premise variables can describe a wider class of nonlinear systems.

The paper is organized as follows: Section III presents the lateral dynamics model of a motorcycle and a Takagi-Sugeno formulation of the considered model. In Section IV, a unknown input and state observer for estimating the motorcycle lateral dynamics and the steering torque is synthesized. The convergence of the observer is studied with Lyapunov theory and an optimization problem under LMI constraints is provided to design the observer in such a way to guarantee the ISpS Property which illustrates the boundedness of the state and unknown input errors. Finally, Section V provides some simulations results and discussions on the proposed observer. The lateral dynamics of a motorcycle are represented by a model with four equations [START_REF] Sharp | The stability and control of motorcycle[END_REF], [START_REF] Slimi | Advanced motorcycle-infrastructure-driver roll angle profile for loss control prevention[END_REF] describing the lateral motion, due essentially to the effect of lateral forces from the front and rear wheels (F y f et F yr ) and the yaw and roll motions under rider's steering actions. The study of such a model aims to reconstruct the dynamic states of a motorcycle in cornering situation.

III. MOTORCYCLE MODEL DESCRIPTION A. Nonlinear model of the motorcycle

G r G f v x v y F y f F yr δ , τ φ ψ τ d -τ b ε ε
These movements expressed by the following equations (all the parameters are defined in Annex 2):

• Lateral motion

M vy + M f k ψ + M f j + M r h φ + M f e δ = F y f + F yr (1)
• Roll motion

M f j + M r h vy + a 1 φ + a 2 ψ + a 3 δ + a 4 v x ψ + a 4 v x δ = ∑ M x
(2)

• Yaw motion

M f k vy + v x ψ + a 2 φ + b 1 ψ + b 2 δ -b 3 v x φ -a 5 v x δ = ∑ M z (3)
• Steering motion (4) where:

M f e vy + a 3 φ + c 1 ψ + c 2 δ -a 5 v x φ + c 3 v x ψ = ∑ M s
∑ M x = M f j + M r h g sin(φ ) + M f eg -ηF z f sin(δ ) (5)
∑ M z = l f F y f -l r F yr -τ cos(ε) (6) ∑ M s = M f eg -ηF z f sin(φ ) + M f eg -ηF z f sin(ε) sin(δ ) + τ (7)
The lateral forces F y f and F yr acting, respectively, on the front and rear wheels depend on the sideslip angles α f and α r and camber angles γ f and γ r , are expressed by:

F y f = -C f 1 α f +C f 2 γ f F yr = -C r1 α r +C r2 γ r ( 8
)
where [START_REF] Ichalal | Fault tolerant control for Takagi-Sugeno systems with unmeasurable premise variables by trajectory tracking[END_REF] and

     α f = v y +l f ψ-η δ v x -δ cos(ε) α r = v y -l r ψ v x
γ f = φ + δ sin(ε) γ r = φ (10)
In this work, a normal riding is considered (without taking into account the limit situations) which justifies the linear form of the lateral forces F y f and F yr with respect to both sideslip and camber angles. By replacing the mathematic expressions of the forces in the dynamics model and by choosing the state vector as x(t) = [v y ψ φ φ δ δ ] T , the system is rewritten as follows:

E ẋ(t) = A(x(t), v x )x(t) + Bτ(t) ( 11
)
where E is a constant nonsingular matrix, B is a constant matrix and A(x) is a nonlinear matrix given by:

A(x) =         a 11 a 12 (v x ) 0 a 14 a 21 a 22 (v x ) a 23 (v x ) a 24 0 a 32 (v x , φ ) 0 a 34 (v x , φ ) 0 0 1 0 a 51 (v x ) a 52 (v x ) a 53 (v x ) a 54 (v x , φ ) 0 0 0 0 a 15 (v x ) a 16 a 25 (v x ) a 26 (v x ) a 35 (v x ) a 36 (v x ) 0 0 a 54 (v x ) a 56 (v x ) 1 0         E =         M e
        B = 0 -cos(ε) 0 0 1 0 T
The components a i j and e i j of, respectively, the matrix A(x) and E are given in Annex 1.

B. Exact T-S model of a motorcycle nonlinear model

In order to express the model in T-S fuzzy structure, let us consider the following nonlinearities:

z 1 = v x , z 2 = 1 v x , z 3 = sin(φ ) φ , z 4 = φ ( 12
)
It is important to notice that the motorcycle, contrarily to a standard vehicle with four wheels, is stable only for a range longitudinal velocities v x . Consequently, taking v x in the interval where the motorcycle is stable, the bounds of the premise variables are given by:

z min 1 ≤ z 1 ≤ z max 1 z min 2 ≤ z 2 ≤ z max 2 z min 3 ≤ z 3 ≤ z max 3 z min 4 ≤ z 4 ≤ z max 4
By following the well-known sector nonlinearity approach (see [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] for more details), the obtained model is:

E ẋ(t) = 16 ∑ i=1 µ i (v x , x)A i x(t) + Bτ(t) (13)
The number 16 of sub-models comes from the fact that 4 nonlinearities are chosen (see [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] for more details). Since the matrix E is nonsingular, its inverse E -1 exists and allows to write the model [START_REF]characteristics and causes of road accidents[END_REF] in the form:

ẋ(t) = 16 ∑ i=1 µ i (v x , x)A i x(t) + Bτ(t) ( 14
)
where

A i = E -1 A i and B = E -1 B.

C. Measurement equation

Sensors generally available allow to measure different state variables, as the roll and yaw rates φ , ψ and both the steering angle and the steering angle rate δ and δ , which leads to the output equation:

y(t) = Cx(t) (15)
where:

C =     0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1     IV. OBSERVER DESIGN
The following lemma is used in the proof of the observer's convergence study.

Lemma 1: Consider two matrices X and Y with appropriate dimensions, G a positive definite matrix. The following property holds

X T Y +Y T X ≤ X T GX +Y T G -1 Y G > 0 (16)
The objective of this section is to provide a new approach in order to design observers for T-S fuzzy systems with unmeasurable premise variables which is the case for the considered motorcycle model. The observer aims to estimate, simultaneously, the state variables and the rider's torque, especially, the lateral velocity v y , the roll angle φ and the steering torque τ which constitute the most important variables in synthesizing risk function for riding assistance.

The following nonrestrictive assumptions can be made Assumption 1: In the remaining it is supposed that

• the state x(t) is bounded (stable or stabilized motorcycle) • rank(CB) = n u (n u is the dimension of the unknown input vector) • the torque derivative τ(t) is bounded by τ 1max In the case of the considered motorcycle model, dim(τ) = n u = 1. The third assumption is not restrictive since the variation of the rider's steering torque is bounded.

A. State estimation

Let us consider the T-S model given by the equations ( 14) and [START_REF] Slimi | Advanced motorcycle-infrastructure-driver roll angle profile for loss control prevention[END_REF]. The state observer is then given by the following equations:

           ẋ(t) = 16 ∑ i=1 µ i (v x , x) (A i x(t) + L i (y(t) -ŷ(t)) + B τ(t) ŷ(t) = C x(t) τ(t) = ΓF( ėy (t) + σ e y (t)) e y (t) = y(t) -ŷ(t) (17)
Let us consider the state estimation error e(t) = x(t)x(t) and torque estimation errors e τ (t) = τ(t)τ(t). The matrices L i , F and the scalars Γ and σ , of the observer, are to be determined in such a way to ensure state and unknown input estimation errors with a minimal bound to have an accurate estimations.

Using equations ( 14), ( 15) and ( 17), the state estimation error obeys the following differential equation:

ė(t) = 16 ∑ i=1 µ i (v x , x)Φ i e(t) + Be τ (t) + ∆(t) (18)
where:

Φ i = A i -L i C ( 19
)
and:

∆(t) = 16 ∑ i=1 (µ i (v x , x) -µ i (v x , x)) A i x(t) ( 20
)
Notice that if the state estimation errors converge to zero, the term ∆(t) converges also towards zero. In addition, since the weighting functions are bounded and the state vector x(t) is also bounded (see assumption 1), the term ∆(t) is then bounded. The objective is to design the matrices L i , F and the scalars Γ and σ guaranteeing an accurate estimation of the state and the unknown input by minimizing the bound of the state and unknown estimation errors.

B. Observer's convergence study

From Assumption 1 and the fact that the functions µ i are bounded, the term ∆(t) is bounded. Indeed, the system is stable which provides bounded states for bounded input τ(t). Under bounded perturbation term ∆(t), the observer ( 17) is synthesized by solving the optimization problem under LMI constraints given in the Theorem 1.

Definition 1: ( [START_REF] Lazar | On input-to-state stability of minmax nonlinear model predictive control[END_REF], [START_REF] Jiang | Small gain theorem for ISS systems and applications[END_REF]) The system [START_REF] Teerhuis | Motorcycle state estimation for lateral dynamics[END_REF] verify the Input To State Practical Stability (ISpS) if there exists a K L function β : R 6+1 × R -→ R, a K function α : R -→ R and a constant d such that for each input ∆(t) satisfying ∆(t) ∞ < ∞ and each initial conditions e(0), the trajectory of ( 18) associated to e(0) and ∆(t) satisfies

e(t) 2 ≤ β ( e(0) ,t) + α ( ∆(t) ∞ ) + d (21
) Theorem 1: Under the Assumption 1, given a positive scalars σ and α and a ∈ [0, 1], if there exists a symmetric and positive definite matrix P, gain matrices M i and positive scalars G, γ, η and S solution to the following optimization problem, i 1, ..., 16 min

P,M i ,η,γ aη + (1 -a)γ s.t. B T P = FC (22)   Ω i + αP -1 σ Φ T i PB -PB P -1 σ B T PΦ i -B T P Ψ + α S σ -1 σ B T P P -1 σ PB -γ   < 0 (23) αηI Q Q αηI > 0 (24) Q ≥ I ( 25
)
where:

Ω i = A T i P + PA i -M i C -C T M T i (26) Ψ = - 2 σ B T PB + 1 σ G (27) Q = diag(P, S σ ) (28)
and Γ = S -1 , then the state and torque estimation errors are bounded. The gains of the observer are computed from L i = P -1 M i and F is obtained directly by solving the above optimization problem. The attenuation level of the transfer from ∆(t) to state and unknown input estimation errors is bounded and given by the quantity

λ max (Q)γ
λ min (Q)α . In addition, If ∆(t) = 0 the state and unknown input estimation errors (norms of errors) converge to a set around the origin with a size defined by the quantity λ max (Q)δ λ min (Q)α where:

δ = 1 σ τ 2 1 max λ max Γ -1 G -1 Γ -1 (29)
The parameter Γ is then chosen sufficiently large in order to have a minimal value of δ which guaranteeing a more accurate estimations. Otherwise, if ∆(t) = 0, the state estimate errors guaranty the Input To State Practical Stability (ISpS) given by the following property

e(t) 2 ≤ λ max (Q) λ min (Q) e -α 2 t + γ α ∆(t) ∞ + δ α (30)
Proof: In order to prove the convergence of the state and torque estimation errors, let us consider the following Lyapunov function:

V (t) = e T (t)Pe(t) + 1 σ e T τ (t)Γ -1 e τ (t) (31)
where P = P T > 0 and Γ > 0. In the framework of the motorcycle model, the dimension of τ is 1 then the parameter Γ is a scalar. (For more general case, we have Γ ∈ R n u where n u is the dimension of the unknown input vector).

According to the equation ( 18), the time derivative of V (t) is given by:

V (t) = 16 ∑ i=1 µ i (v x , x(t))e T (t) Φ T i P + PΦ i e(t) + 2 σ e T τ (t)Γ -1 ėτ (t) + 2e T (t)P∆(t) + 2e T τ (t)B T Pe(t) (32)
Knowing that ėτ (t) = τ -τ(t) and given the expression of τ(t) [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], the time derivative of the Lyapunov function (32) becomes: where Ω i = Φ T i P + PΦ i . Using the output equations, we have e y (t) = Ce(t) and ėy (t) = C ė, then, equation (33) can be rewritten under the following form:

V (t) = 16 ∑ i=1 µ i (v x , x(t))e T (t
V (t) = 16 ∑ i=1 µ i (v x , x(t))(e T (t)Ω i e(t) - 2 σ e T τ (t)FCΦ i e(t)) - 2 σ e T τ (t)FC∆(t) -2e T τ (t)FCe(t) + 2 σ e T τ (t)Γ -1 τ(t) - 2 σ e T τ (t)FCBe τ (t) + 2e T (t)P∆(t) (34)
lemma 1 and the fact that the first derivative of τ is bounded by τ 1max (assumption 1), we obtain 35) and using assumption 1. It is possible to obtain matrices F and P such that B T P = FC holds. The time derivative of the Lyapunov function ( 34) is then bounded as follows

2 σ e T τ (t)Γ -1 τ(t) ≤ 1 σ e T τ (t)Ge τ (t) + 1 σ τT (t)Γ -1 G -1 Γ -1 τ(t) ≤ 1 σ e T τ (t)Ge τ (t) + 1 σ τ 2 1 max λ max Γ -1 G -1 Γ -1 (
V (t) ≤ 16 ∑ i=1 µ i (v x , x(t))e T (t)Ω i e(t) - 2 σ e T τ (t)B T PΦ i e(t) - 2 σ e T τ (t)B T P∆(t) -2e T τ (t)B T Pe(t) - 2 σ e T τ (t)B T PBe τ (t) + 1 σ e T τ (t)Ge τ (t) + 1 σ τ 2 1 max λ max Γ -1 G -1 Γ -1 + 2e T (t)P∆(t) (36)
Let us define x(t) = [e T (t) e τ (t) ∆(t)] T . The inequality (36) is equivalent to:

V (t) ≤ xT (t) 16 ∑ i=1 µ i (v x , x(t))Ξ i x(t) -αe T a (t)Qe a (t) + γ∆ T (t)∆(t) + δ ( 37
)
where α and γ are positive scalars, Q = diag(P, Γ -1 σ ), e a (t) = [e T (t) e τ (t)] T and

Ξ i =   Ω i + αP -1 σ Φ T i PB -PB P -1 σ B T PΦ i -B T P Ψ + α Γ -1 σ -1 σ B T P P -1 σ PB -γ 2   (38) and δ = 1 σ τ 2 1 max λ max Γ -1 G -1 Γ -1 (39) Ψ = - 2 σ B T PB + 1 σ G Ω i = Φ T i P + PΦ i Now, if xT (t) 16 ∑ i=1 µ i (v x , x (
t))Ξ i x(t) < 0 then the inequality (37) can be bounded as follows

(t) ≤ -αe T a (t)Qe a (t) + γ∆ T (t)∆(t) + δ (40)
which is equivalent to

V (t) ≤ -αV (t) + γ∆ T (t)∆(t) + δ (41)
It follows

V (t) ≤ V (0)e -αt + γ t 0 e -α(t-s) ∆(s) 2 2 ds + δ t 0 e -α(t-s) ds (42) ≤ V (0)e -αt + γ α ∆(t) 2 ∞ + δ α (43) Knowing that λ min (Q) e a (t) 2 2 ≤ V (t) ≤ λ max (Q) e a (t) 2 2
, where:

Q = P 0 0 Γ -1 σ one obtains the inequality e a (t) 2 2 ≤ λ max (Q) λ min (Q) e -αt + γ α ∆(t) 2 ∞ + δ α (44)
By using the square root on (44), on obtains

e a (t) 2 ≤ λ max (Q) λ min (Q) e -α 2 t + γ α ∆(t) ∞ + δ α
(45) According to Lyapunov formulation of Input To State Practical Stability (ISpS), the state and unknown input errors converge to a region which will be minimized in order to achieve a more accurate estimation of the states of the vehicle and the torque applied on the handlebar. This ball is smaller as the constant δ and the attenuation level of the transfer from ∆(t) to the state estimation errors are smaller. To enhance the performances of the observer, a minimal values of these quantities are obtained by the following reasoning: Let us consider the quantity

λ max (Q) λ min (Q)α ≤ √ η
where η is a positive scalar. It is then sufficient to minimize the term η and assuming λ min (Q) ≥ 1 (Q > I), on obtains:

λ max (Q) α ≤ √ η (46)
which is transformed easily into:

(αη) 2 I -Q T Q > 0 (47)
Using Shur's complement lemma:

αηI Q Q αηI > 0 (48)
The second quantity

λ max (Q)δ
λ min (Q)α is minimized by choosing the parameters α and G sufficiently large for minimizing the term δ and the the term

λ max (Q) λ min (Q) is treated above.
Finally, the bound of this term will be δ /α multiplied by η. Always in the purpose of minimizing the two quantities, in theorem 1, the chosen objective function is a linear combination between η and γ. By choosing the change of variables S = Γ -1 and M i = PL i , the linear matrix inequalities in theorem 1 are obtained.

V. SIMULATION RESULTS

The nonlinear system, including longitudinal and lateral dynamics of two-wheeled vehicle is used. It requires three inputs: the rider's steering torque applied on the handlebars τ (see Figure 2) and the angular velocities of both front and rear wheels ω f and ω r . The observer estimating the lateral dynamics and steering torque using only the measured states ψ, φ given by the inertial unit and δ and δ obtained from a suitable encoder. The gains of the observer were calculated by solving the optimization problem under LMI constraints proposed in theorem 1. The obtained attenuation level, from the term ∆(t) to the state and unknown input estimation errors is γ = 2.6119. The initial conditions of the system are x(0) = [0 0 0 0 0 0] T and those of the observer are x(0) = [1 1 0.1 1 0 0.1] T . The adaptation law providing an estimation of the steering torque is designed in such a way to have a fast convergence of τ(t) to τ(t). The chosen parameters are α = 1, σ = 5 and a = 0.99. If ∆(t) = 0, the state and unknown input estimation errors e(t) and e a (t) converge to a ball around the origin having the size λ max (Q)δ λ min (Q)α ≈ 0.1, for τ 1max = 40. In the steady state ∆(t) ∞ is less than 0.07, the attenuation of ∆(t) is then given by In order to test the observer in the presence of measurement noise, let us consider the same observer's parameters and assume that the measurement signals are affected by a centered and random noise with magnitude 5% of the maximal values of the measured variables. The obtained results are depicted in the figure 5, the states are then estimated accuratly. The estimation of the steering torque is also acceptable, but the effect of the measurement noise is visible. This is due to the presence of a derivation in the adaptation law leading to estimate the steering torque and also due to the large value of Γ. It is then possible to reduce the effect of the measurement noise by reducing the value of Γ. A compromize can then be obtaind by an adequate Γ selected sufficiently large to ensure an accurate estimation of the unknown input and sufficiently less to reduce the effect of the measurement noise.

λ max (Q) λ min (Q)α ≤ √ ηγ ∆(t) ∞ ≈ 2.
Of course the ISpS property is weaker than ISS or asymptotic stability, but the proposed observer with the estimation's Nevertheless, the simulation results are satisfactory. Furthermore, some parameters in the proposed design approach are fixed a priori, in future work, this will be treated in order to take them into account in the optimization problem leading to optimal values which give better results and less theoretic bounds values.

VI. CONCLUSION

In this paper, an unknown input and state observer for estimating the lateral dynamics and the steering torque in motorcycles is proposed. A nonlinear model is then considered with some nonlinearities and time-varying longitudinal velocity. The proposed approach is based on the transformation of the nonlinear lateral dynamics model using sector nonlinearity transformation in Takagi-Sugeno structure, the specificity of this part is that the weighting functions of the T-S model depend on the state variables which are not totally measured. The study of this type of model is more difficult compared to T-S models with measured premise variables. Based on the obtained model, an adaptive observer is then proposed. The observer's convergence is studied using Lyapunov theory and LMI conditions are given to ease the design of the observer with dedicated softwares. It is pointed out that this observer guaranty the ISpS property. Simulation results are provided which illustrate the effectiveness of the proposed observer in estimating both the states and the steering torque. In future work, the T-S model will be refined by taking into account some other nonlinear behaviors, namely, the nonlinear form of the lateral forces (Pacejka's or Dugoff's model) and taking into account the longitudinal motion. The observer will be then redesigned for more accuracy. In addition, the lateral and longitudinal forces estimation will be considered in order to estimate the road adhesion. This is in the perspective to synthesis risk functions which inform the rider on dangerous situations. Some validation results in real situations will be published in future papers.

VII. ANNEX 1

The components of the matrix A(x(t)) are defined as follows:

a 11 = -C f 1 -C r1 (49) a 12 (v x ) = 1 v x -C f 1 L f +C r1 L r -Mv x (50) a 14 = C f 2 +C r2 (51) a 15 (v x ) = 1 v x ηC f 1 ( 52
)
a 16 = C f 1 cos(ε) +C f 2 sin(ε) (53) a 21 = -C f 1 L f +C r1 L r (54) a 22 (v x ) = 1 v x -C f 1 L 2 f -C r1 L 2 r -M f kv x (55) a 23 (v x ) = i f y R f + i ry R r v x ( 56
)
a 24 = C f 2 L f -C r2 L r ( 57
)
a 25 (v x ) = 1 v x ηL f C f 1 + i f y R f sin(ε)v x ( 58
)
a 26 = L f C f 1 cos(ε) + L f C f 2 sin(ε) (59) a 32 (v x , φ ) = - 1 v x M f j + M r h + i f y R f + i ry R r (60)
+ (M f j 2 + M r h 2 + I f y + I ry -I rz -I f z cos(ε) 2 -I f y sin(ε) 2 ) φ (61)

a 34 (φ ) = M f j + M r h g sin(φ ) φ (62)
(63)

a 35 (v x ) = i f y R f cos(ε)v x ( 64
)
a 36 = M f eg -ηF z f (65)

a 51 (v x ) = 1 v x ηC f 1 (66) a 52 (v x ) = 1 v x ηL f C f 1 -v x M f e i f y R f sin(ε) (67) a 53 (v x ) = i f y R f cos(ε)v x ( 68
)
a 53 (φ ) = (M f eg -ηF z f ) sin(φ ) φ -ηC f 2 (69)

a 55 (v x ) = η 2 C f 1 v x ( 70

Fig. 1 .

 1 Fig. 1. Geometrical representation of the Sharp's motorcycle model

12 e 13 0 e 15 0 e 12 e

 12 22 e 23 0 e 25 0 e 13 e 23 e 33 0 e 35

Fig. 2 .Fig. 3 .

 23 Fig. 2. Rider's torque (blue) and estimated torque (red)

Fig. 4 .Fig. 5 .

 45 Fig. 4. State estimation errors on the time range [0 5](s)

Fig. 6 .

 6 Fig. 6. Steering torque (blue) and estimated steering torque (red) in noised measurement situation

) a 56 =

 56 -ηC f 1 cos(ε) -ηC f 2 sin(ε) + M f eg -ηF z f sin(ε) e 12 = M f k (71) e 13 = M f j + M r h (72) e 15 = M f e (73)e 22 = M f k 2 + I rz + I f x sin(ε) 2 + I f z cos(ε) 2 (74) e 23 = M f jk +C rxz + (I f z + I f x ) cos(ε) sin(ε) (75) e 25 = M f ek + I f z cos(ε) (76)e 33 = M f j 2 + M r h 2 + I rx + I f x cos(ε) 2 + I f z sin(ε) 2 (77) e 35 = M f e j + I f z sin(ε)(78)e 55 = M f e 2 + I f z (79) VIII. ANNEX 2

  Front and rear tire relaxation lengths respectively M total mass of the motorcycleM = M f + M r C f 1 C f 2 Frontand rear tire cornering stiffnesses respectively C r1 C r2Front and rear tire camber stiffnesses respectively η

	M f	Mass of front frame
	M r	Mass of rear frame
	j	Distance between the center of gravity
		of the front frame and ground
	k	Distance between the centers of gravity
		of each frame
	L f	Distance between the center of gravity
		and the front wheel
	L r	Distance between the center of gravity
		and the rear wheel
	e	Distance between the fork and the center
		of gravity
	h ε	Height of the center of gravity Steering head angle
	i f y	Polar moment of inertia of front wheel
	i ry	Polar moment of inertia of rear wheel
	I rz	Camber inertia of rear wheel.
	R f	radius of the front wheel
	R r	radius of the rear wheel
	g σ f σ r	Acceleration due to gravity.
		Pneumatic trail
	v x σ r , σ f	Forward speed Front and rear tyre relaxation lengths respectively
	Z f ρ	Front wheel load road's curvature