1

Vehicle Model and Estimation Models using Bond Graph S.A. Federici§, N.K. M’Sirdi*, A. Naamane*, S. Junco§ *LSIS, CNRS UMR 6168. Dom. Universitaire St Jérôme, Avenue Escandrille Normandie–Niemen 13397 § Facultad de Ciencias Exactas, Ingenieria y Agrimensura. Universidad Nacional de Rosario, Av. Pellegrini 250, S2000EKE Rosario, Argentina – [email protected]

Abstract—Modeling nonlinear vehicle dynamics is made using bond graphs and implemented in the 20-Sim simulation environment. Then some partial models are considered and justified for the design of robust estimators to identify the input variables and tire-road friction coefficient. Sliding Mode Observers for systems with unknown Inputs are developed. Index Terms—Bond Graphs, Multibody system, Vehicle dynamics, Sliding Modes observer, Robust nonlinear observer, tire forces estimation.

T

I.

INTRODUCTION

HE vehicles are a complex Dynamics System with unknown inputs (like contact forces, road profile, external perturbations…). Their behaviour is affected by several factors that may depend or not on its structure. The external influence depends mainly on the contact between the pneumatic tyre and the road and the aerodynamic forces introduced by the wind flowing around it. Tire forces affect the vehicle dynamic, performance and behavior properties. The dynamic model of the whole vehicle is composed by different sub models belonging to the different parts in which the vehicle may be decomposed. Among them we can mention the chassis, the suspension system, the engine, the transmission, the pneumatic tyres, the several control elements including the driver and so on. The vehicle is a mechanical system composed by a lot of elements. The engine and the transmission give the necessary traction to the vehicle. These efforts are sending to the pneumatic tyres to produce the movement of the vehicle. The accelerator pedal, the brakes and the steer wheel are the components by which the driver can control the vehicle to guide it along the specified path. These inputs are modified by the steering and suspension systems and their effects are very complex to predict without a mathematical model. The chassis is designed to transmit the forces necessary for the vehicle operation being its flexibility a crucial key to this target.

Santiago A. Federici wants to be grateful to the ME. German Filippini, Dr. Norberto Nigro y Dr. Monica Romero, for his supports to this investigation.

The acceleration and the braking mechanism allow the production of traction and braking efforts to control de vehicle speed. Several reason which involve vehicles components, environment or the driver, produce car accidents. This has produced an increase on safety demand in vehicles and motive research of active safety and development of new safety system. All of these systems are installed on vehicles for the monitoring and controlling different variables on the dynamics vehicles to get better stability (like ABS, ESB, ESP). Robust observer with unknown inputs is shown efficient for estimation of road profile and for estimation of the contact forces. Different dynamic controls on the vehicle like tracking, braking and cornering, reduce the friction coefficient. The traction and braking control reduces the wheel slip, and this can be done by the utilization of sliding mode approach for observation and control. For vehicles and road safety analysis, it is necessary to take into account the contact force characteristics. However, the friction coefficient and different force (like traction force) cannot be directly measured. They are complex to precisely represent by some deterministic model equations. Usually some experimentally fitted and approximated model are used to deduce their values. The friction coefficient and forces are utilized on vehicles control, and they are difficult to estimate on line. Those are important for the detection and diagnosis application for the monitoring and surveillance. In this work we develop a method to observe tire forces. The proposed estimation procedures have to be robust, and can then be used to improve the security detecting some critical driving situation. This estimation can be used in several vehicles control system such as Anti-look Brake System (ABS), traction control system (TCS), diagnosis systems, etc. An observer is proposed to estimate the forces and friction coefficient. The estimations are produced using only the angular wheel position and longitudinal velocity as measurement and they are the input to the specially designed robust observer based on the Second Order Sliding Modes (SOSM). The method of estimation is verified through simulation with as contact model a “Pacejka Model” (Magic

2 Formula) [5], [19], [20], [21]. II. VEHICLE DYNAMICS MODEL A. Vehicle Chassis The vehicle chassis is modeled as a rigid body with local coordinate reference frame (x, y, z) attached to the center of mass and aligned with the inertial principal axes. It has mass m, and the following principal inertia moments: roll (Jr) respect to the body x-axis, pitch (Jp) respect to the body y-axis and Yaw (Jy) respect to the body z-axis.

Figure 2: Forces and Angle on Tire

For all forces moments acting on the tire, the Pacejka model, inspired by a lot of experiments carried out using different types of pneumatics tires, is used.

Figure 1: SAE Vehicle Axis System

The vehicle body receives as excitations external forces and moments following the three axes: Longitudinal, Lateral and Vertical. This comes from interaction between wheel and road, from perturbation like wind, gravity and vehicle drive line. B. Suspension Systems Suspension system connects the vehicle chassis with the wheel. It is possible by the ensemble of mechanical elements like the spring, shock absorbers and linkage. This system is used to support the suspended mass, absorb and damper road shock and help to maintain tire contact as well as proper wheel-to-chassis relationship. C. Pneumatic Tires Aerodynamic, gravitational and all other forces and moments affecting the motion of a ground vehicle are applied through the running gear-ground contact. On ground vehicles it is important understand the characteristic of interaction between the running gear and the ground for the study of performance characteristics, ride quality and handling behaviour. The figure 2 shows forces, moments, angles and speeds that play a major role on the modelling of pneumatics tire. Each one is associated with a corresponding local axis located at the centre of the contact patch of the wheel (C). Traction (Fx), lateral (Fy) and vertical (Fz) forces along the x, y and z local axis respectively. Overturning (Mx), rolling resistance (My) and aligning (Mz) torques along the same axis.

D. Engine and Transmission Most vehicles are propelled by internal (spark or compression ignites) combustion engines which may be modelled through a given static curve relating the engine speed and the load with its torque and its power. The transmission is composed by the mechanical members connecting the engine crankshaft with the traction wheels, the gearbox, and the differential. E. Aerodynamic Forces The aerodynamics force is simplified in this works through the usage of empirical aerodynamics coefficients. Only the drag forces (x axis) is included. The lift (z axis) and side force (y axis) are considered zero but could be utilized on future works. III. MATHEMATICAL AND BOND GRAPH MODELING A. Vehicle Chassis Euler’s equations [9] are used to determine the spatial motion of the rigid body, which appear in (1) and (2). The first one represents the conservation of linear momentum: _

_

∂p ∂p ∑ F= ∂t = ∂t _

_

_

+ ω× p

( 1)

Rel

Fi = _

_

∂pi ∂pi = ∂t ∂t

+ε ijk ω j pk Rel

_

where F, ω , p represents the external forces, the angular velocity and the linear momentum vector. The

∂

∂t

, ∂ ∂t rel , ε ijk

represent the derivative with respect to the inertial frame, the derivative with respect to the body attached frame and the Levi-Civita tensor used for expressing the cross product in tensor notation. The second Euler’s equation represents the conservation of

3 angular momentum: _

_

∂h ∂h ∑ M= ∂t = ∂t _

_

Traslacion A B

_

+ ω× h

I

(2)

Traslacion B

I3

JuntaSuspDelDer

A

JuntaSuspTraDer

Rel

∂h ∂h Mi = i = i ∂t ∂t

1

Gyro

+ε ijk ω j hk

Se

OneJunction4

Transf ormacion

Rel

Local

OneJunction5 _

_

_

_

M i =ε ijk F j rk _

_

To transform the dynamics equation from the body attached frame of reference (roll, pitch and yaw) to a spatially fixed frame of reference (X, Y, Z: Inertial frame) [9] it is necessary choose some parameterization like Euler angles (used on this work). 0 co s φ sin φ

⎡ co s θ Θ= ⎢ 0 ⎢ ⎢⎣ − sin θ

⎡ cos ψ Ψ = ⎢⎢ sin ψ ⎢⎣ 0

0 1 0

0 ⎤ − sin φ ⎥⎥ co s φ ⎦⎥

(5)

sin θ ⎤ 0 ⎥ ⎥ co s θ ⎥⎦

(6)

− sin ψ cos ψ 0

0⎤ 0 ⎥⎥ 1 ⎥⎦

JuntaSuspDelIzq

I

B

I4

JuntaSuspTrasIzq

A

B. Suspension Systems The suspension systems is adopted for the four wheel, each one composed by a spring with stiffness ‘ks’ and a damper with damping coefficient ‘bs’. The unsprung mass is modeled with inertia ‘I’ bond graph component. We add the weight of this type of mass like source of effort ‘Se’ bond graph component. I

e

IRueda

R R

RollingResist

TF

1 f

FlowSensor1

e

Torque

OneJunction1 ZeroJunction8

1

1 PowerSplitter7

0

PowerSplitter21

1 Transf ormacion

OneJunction6

Local

1 C

⎡ Fx ⎤ ⎢ F ⎥ (9) ⎢ y⎥ ⎢⎣ Fz ⎥⎦

where the power variables are transformed from (x,y,z) axes to the rotated ones (X,Y,Z). The vehicle chassis is composed by a rigid body model, a four coordinate translation to the pivot of the suspension at each wheel, a coordinate system transformation (LocalGlobal) for the vehicle weight.

Se7 Se3

Sf

Sf1 Tyre

TransGlobLocPeso

I R

Spring

Se 0 C

GLOBAL

1

mNs

Se

OneJunction17 Se9

Shock

1

Se

PowerSplitter6 1

Se8

Rot

Se Se2

Figure 4: BG Suspension System

⎡ vx ⎤ ⎡ω X ⎤ ⎡ω x ⎤ ⎡ v X ⎤ ⎢ ω ⎥ =ΨΘΦ ⎢ω ⎥ ; ⎢ v ⎥ =ΨΘΦ ⎢v ⎥ (8) ⎢ y⎥ ⎢ Y⎥ ⎢ y⎥ ⎢ Y ⎥ ⎣⎢ vz ⎦⎥ ⎣⎢ ω Z ⎦⎥ ⎣⎢ω z ⎦⎥ ⎣⎢ vZ ⎦⎥

Se

1

1

⎡τ x ⎤ ⎡ FX ⎤ ⎢τ ⎥ ; ⎢ F ⎥ = ΨΘΦ T ) ⎢ y⎥ ⎢ Y ⎥ ( ⎢⎣τ z ⎥⎦ ⎢⎣ FZ ⎥⎦

1

1

0

OneJunction12

PowerSplitter1

(7)

f

EffortSensor1 Pacejka RadRueda FlowSensor2 RuedaDD

Trasl

then

⎡τ X ⎤ ⎢τ ⎥ = ΨΘΦ T ) ⎢ Y⎥ ( ⎢⎣τ Z ⎥⎦

Se4

Figure 3: BG Vehicle Chassis

v = ω× r (4) vi =ε ijk ω j rk

⎡1 Φ = ⎢⎢ 0 ⎣⎢ 0

Se

Traslacion

(3)

For the flow variables the equation are the following: _

Traslacion A B

_

M = F× r

TransGlobLocPeso

1

Gyro

where M, h represent the external torque and the angular momentum vector. For the translation between a point ‘A’ and ‘B’ (see figure 2) [9] the equation relating the linear and rotational effort are the following:

Weigth

GLOBAL

C. Pneumatic Tires The pneumatics tires are modeled with the Pacejka model (Magic Equation) [5], [19], [20], [21], inspired in a lot of experiments using different types of pneumatics tires, which is described by the expression (10) and (12): μ x = sign ( s ) i ⎡ A i 1 − e − B s + C i s 2 − D i s ⎤ ⎣ ⎦ 1 (10) n K B = ( d)

(

)

A = 1.12; C = 0.625; D =1 K = 46; d = 5; n = 0.6 where ‘s’ is the longitudinal slip (-1 λ1 > λ2

5 the third order sliding mode motion will be established in a finite time. ii

i

The obtained estimates are z1 = s1 = s w and z2 = s2 = s w can be used in the estimation of the state variables and also in control. B. Cascaded Observers and Estimator This work utilize the previous approach to build the observer and obtain an estimation scheme in 20-Sim [12], [15], [16]. We produce the estimation in steps using like input the wheel angular position and the longitudinal body speed. The inputs are considered available for measurements. The robust differentiation observer is used for estimation of the velocities and acceleration of the four wheels. 1st Step: produces estimation of angular velocity of the wheel. The convergence of these estimates is guaranteed in finite time t0.

θ

i 0

2 3

= v0 = ω − λ0 θ − θ

i

i

ω = v1 = ω − λ 1 ω − v 0 ⎛

ii

i

1 2

(

s ig n θ − θ s ig n

(ω

− v0

) (17) )

⎞

ω = − λ 2 s i g n ⎜⎜ ω − v 1 ⎟⎟ 2

nd

⎝ ⎠ Step: Estimation of the forces Fx (longitudinal) and Fz

(vertical). To estimate the Fx we used the following equation, i ⎛ ⎞ R e f (18) = ⎜⎜ T − J i ω ⎟⎟ ⎝ ⎠ The torque could be also estimated by means of use additional equation from engine behavior or measured. To estimate the Fz we use the following equation,

F

F

zf

=

x

2 i (l

m f

i ⎛ i⎜ g ilr − h iv + lr ) ⎝

x

i

zr

=

2 i (l

Rrolling = Rrolling1 + Rrolling 2 + Rrolling 3 + Rrolling 4 are the total traction forces and rolling resistance (each by tire-four wheels). C. Implementation in 20-Sim The observer is build in 20-Sim with blocks element. 20-Sim admits the interaction between bond graph and signal components [17]. We used the bond graph vehicle model to get the powers variables utilizing sensors (flow and effort sensors), and are introduced in the observer and estimator modules like signal mode. The inputs in the observer module are the tire angle position and the longitudinal body velocity measured. The observer module produces the variables estimates where are utilized on the estimator module to get the longitudinal and vertical force, and finally get the friction coefficient estimated. Torque is other variables measured with a sensor on the bond graph model where it is an input for the estimator module. Figure 7 shows the general observer module where it has like input variable measured and the outputs are the first and the second derivative estimated. In our work the inputs are the angular position and the longitudinal body velocity, and the outputs are the angular acceleration estimated and longitudinal acceleration estimated. Before to get the angle acceleration estimated, we obtain the angle velocity estimated. K VelAngRueda

Signo

VariableMeasured

2

K

Potencia

Lambda0

x PlusMinus1

FirstDerivEstimated MultiplyDivide1 PlusMinus2

Ø Integrate1

K

i

m

+ lr

i

⎞ ⎟ (24) ⎟ ⎠

where F x = F x1 + F x 2 + F x 3 + F x 4

⎞ (19) ⎟ ⎠

⎛ ⎞ i ⎜ g i l f + h i v x ⎟ (20) )⎝ f ⎠ where (21) is for the front axis, and (22) is for the rear axis. F

⎛ F x − 12 C x ρ air A frontal v xprom − R rolling ⎜ Mvg ⎝

α = arc sin ⎜

AcelAngRueda

Signo2 2

K

Potencia1

Lambda1

x PlusMinus5

SecondDerivEstimated MultiplyDivide6 PlusMinus6

Ø Integrate3

v x and ω x are produced by the Robust Estimator (RE).

3rd Step: We estimated the friction coefficient or pneumatic adherence. F x (21) μx = F z 4th Step: We estimated the Rolling Resistance and Aerodynamics Resistance Force. R

r o llin g

⎛ 3 .6 i v = 0 .0 1i⎜ 1 + ⎜ 160 ⎝

x

⎞ ⎟⎟ i F ⎠

zi

(22)

i= 1 ..4

1 i ρ a ir i C x i A f i v 2 5th Step: We estimated the Slope Angle. F

A er x

=

2 x

(23)

Figure 7: General Observer module with blocks diagram

Figure 8 shows the Fx estimate module, where it has like input the tire angular acceleration estimated (estimated in the observer module) and the torque measured (or estimated). On this module other parameters are needed to estimate the longitudinal force ( Fx ) by the equations (18).

6 LongVelocityEstimated

DataFxEstimated

RollingResistance MultiplyDivide4 PlusMinus2 MultiplyDivide5 MultiplyDivide6

FxEstimada

1

AccelAngularEstimated

K InertiaTire

Constant7

FxEstimated

One

Constant8

PlusMinus3MultiplyDivide2 Logical5

FzEstimated

RadioTire

Figure 11: Rolling Resistance Estimate Module

Figure 12 shows the Aerodynamics Resistance estimated

Torque

Figure 8: Fx Estimate Module

Figure 9 shows the Fz estimate module, where it has like input the longitudinal acceleration estimated (estimated in the observer module). On this module other parameters are needed to estimate the vertical force ( Fz ) by the equations (19) and (20).

module, where it has like input the Longitudinal Velocity ( v x ) by the equation (23). Other coefficients like Cx, Area, ρair are inputs for the module. LongVelocityProm

x

2

ResAerodynamics MultiplyDivide9

Function2

DataFzEstimated Cax glr

AccelLongEstimated

FzEstimada

FzEstimated

K halCG

PlusMinus4 MultiplyDivide3

masa

MultiplyDivide4

RoAir

Medio

Area

Figure 12: Aerodynamics Resistance Force Estimate Module

Figure 13 shows the complete estimate module for the slope angle, where we use like inputs the estimated variables from each wheel. Then we estimate each rolling resistance, and the aerodynamics resistance force. Finally we can obtain the Slope angle estimated by the equation (24). Other coefficients are necessary like inputs for the estimate module. FzTotalEstimated

chassis

Promedio

Figure 9: Fz Estimate Module

Figure 10 shows the μ estimate module, where it has like

DataAlfaEstimated

AcelTotalEstimated m

input the Longitudinal and Vertical Forces estimated ( F x , Fz ) by the equation (21). FxEstimated

MultiplyDivide10

sin PlusMinus9 MultiplyDivide8 ArcSin alfaEstimated

Promedios

ResistanceAerodynamics VelXTotalEstimated mg

RollingResistanceTfr

EstimatedTfr SignalSplitter1

MultiplyDivide5

RollingResistanceTfl

EstimatedTfl

mu

FzEstimated

SignalSplitter2

EstimatedTrr SignalSplitter3

Figure 10: µ Estimate Module

RollingResistanceTfl

PlusMinus2

RollingResistanceTrr RollingResistanceTrr

RollingResistanceTrl

EstimatedTrl SignalSplitter4

RollingResistanceTrl

Figure 11 shows the Rolling resistance estimate module, where it has like input the Longitudinal Velocity measured

Figure 13: Complete Slope Angle Estimate Module

and Vertical Forces estimated ( v x , Fz ) by the equation (22). This module is used in each wheel to estimate all different rolling resistance. These modes of use make an optimal estimated slope angle, so we can make individual estimation for each wheel.

Figure 14 shows the complete bond graph model for the suspension system, where it is the mechanical suspension system modeled with bond graph, sensor placed on the bond graph model and the observer and estimate modules with their signal inputs.

7 muEstimated

FzEstimated

DATA

FxEstimated Variables

Observer2

Observer

Variables

EstimatedTfr

I

Ø

e

IRueda Integrate3

R R3

1 delt

f

FlowSensor5

1

1

GLOBAL

Torque

e

1

Se Se10

1 PowerSplitter8

R

Transf ormacion

Submodel3

0

1

f

FlowSensor4 EffortSensor2 OneJunction24

Se

OneJunction33

PowerSplitter9

Rot

TF2

1

f

Local

TF

OneJunction26 ZeroJunction9

OneJunction23

FlowSensor3

Trasl

R Pacejka

0

e

PacejkaLat

Transf ormacion

1

Local

OneJunction25

c13

C

GLOBAL

Se1

0

Sf

Sf2 c14

Submodel9

1 C

PowerSplitter101

I R

r12

1

i12

Se

OneJunction28 Se12

1

PowerSplitter5 1

Se

Se11

Se

Se13

Figure 14: Suspension System with Observer and Estimate modules

Figure 15 shows the complete bond graph model for the vehicle, where we have the BG vehicle model and the Slope Angle Estimate Module. DATA

First Gearbox ratio 3.731 Second Gearbox ratio 2.049 Third Gearbox ratio 1.321 Four Gearbox ratio 0.967 Five Gearbox ratio 0.795 Reverse Gearbox ratio 3.171 Maximum Speed 146km/h Acceleration 0-100km/h 17s Tire, type and dimensions 145 70 R13 S Pneumatics Tire Radius 0.28261 Unsprung Masses (at each wheel) 40kg Tire Vertical Stiffness 150000N/m Tire Inertia 1.9kg/m2 Damper Coefficient 475Ns/m Suspension Stiffness 14900N/m The simulation begins with zero velocity (vehicle stopped). On the second 8, the accelerator changes his position from 0 to 1 (the butterfly valve obtains maximum position) and begins the acceleration. It produces a torque from the engine to the traction tires.

AlfaEstimated 80

AlfaEstimated FrontRightSusp

75

Traslacion B

I

JuntaSuspDelDer

I3

A

Traslacion B

A

JuntaSuspTraDer

1

Gyro

Motor

R ResAero

OneJunction4

0

1 Local

ZeroJunction1

OneJunction5

1

Se

OneJunction2 Se4

Traslacion B

I

B

JuntaSuspDelIzq

I4

JuntaSuspTrasIzq

A

75

76,8 78

78,5 78,1

77,2

70

Engine Torque 75,8 72 68

65

65

60

60

60

55

GLOBAL

TransGlobLocPeso

1

Gyro

Se

OneJunction1 Weight

Transf ormacion

Torque [Nm]

RearRightSusp

50 400

900

1400

1900

2400

2900

3400

RPM

3900

4400

4900

5400

5900

6400

Traslacion A

Figure 16: Engine Torque RearLef tSusp

FrontLef tSusp

Figure 15: Vehicle BG Model and Slope Estimate Module

V. SIMULATION In this part we show the results obtained with simulation on 20-Sim. The simulation shows us the behavior of vehicle dynamic and validates our approach and the proposed observers. The state and forces are generated by the Bond Graph Vehicle dynamics proposed. The data from a car Renault Clio RL 1.1 are, Aerodynamics Coefficient 0.33 Frontal Area 1.86m2 Air Density 1.225kg/m3 Distance between axes 2.472m Vehicle Weight (Sprung mass) 1100kg Sprung Mass – Roll Inertia 637.26kgm2 Sprung Mass – Pitch Inertia 2443.26kgm2 Sprung Mass – Yaw Inertia 2345.53kgm2 Centre of Mass 0.6m Maximum Engine Torque 78.5Nm at 2500RPM Maximum Engine Power 48CV at 5250RPM Planetary drive train ratio 3.171 (Differential)

When time gets 100 second, the accelerator passes from 1 to 0 and begins the deceleration from the vehicle. Figure 17 shows the accelerator profile between 0 and 1 position. model 1.5

Accelerator Profile

1

0.5

0

0

20

40

60 time {s}

80

100

120

Figure 17: Accelerator Profile

We propose a variable slope angle, where we have: Time Slope Angle Proposed 0s-10s 0° 10s-20s 5° 20s-50s 10° >50s 3°

8 Table 1: Slope Angle Proposed

CG Vertical Position

0

Figure 18 shows the engine behavior, where we have the RPM, Torque and speed gearbox.

Z Position Chassis

-20 -40

Engine

-60

5000 4000 3000 2000 1000 0 100

-80 -100 -120

RPM

-140

Engine Torque

-160

50 0

0

500

1000 1500 X Position Chassis

-50

2000

Figure 20: Z Chassis Position

5

Figure 21 shows the tire angular velocity estimated and measured. The figure shows the good convergence to tire angular velocity.

4 3 2

GearBox speed

1 0

20

40

60 time {s}

80

100

Tire Angle Speed 100

120

Figure 18: Engine Behavior

80

Figure 19 shows the principal behavior of the vehicle, where we have Speed on axis ‘X’, Speed on axis ‘Z’, Pitch Angular Speed, Longitudinal and Vertical Position of the Centre Gravity.

60

40

model

Tire Ang Speed Estimated Tire Ang Speed Measured (Real)

20 10

20

X CG Velocity

0 1 Y CG Velocity

0

0

-1 0.5

0

20

40

60 time {s}

Z CG Velocity

80

100

120

-1

Figure 21: Tire Angular velocity Estimated and Measured

-2.5 1 Roll Angular Velocity

0 -1 1

Figure 22 shows the angular acceleration estimated and linear acceleration estimated.

Pitch Angular Velocity

0

Yaw Angular Velocity

0

AccelLongEstimated

2 1

0

1000

-5

0 -1 -2

0

0

1

20

40

Pychasis

0

60 time {s}

80

100

120

0

20

40

60 time {s}

80

100

120

Figure 22: Angular Acceleration and Linear Acceleration Estimated

-1 -40 -100 -160 1

Pzchasis

Gxchasis

0 -1 1

Gychasis

0 -1 1

Gzchasis 0 -1

3

Angle Tire Acel Estimated Acel Real

5

-1 2000 Pxchasis

model

model 10

-1 1

0

20

40

60 time {s}

80

100

120

Figure 19: Principal behavior Centre of Gravity

Figure 20 shows the Chassis Position on the Z global axis.

The 2dn step on the observer model gives us the estimated longitudinal force Fx and the vertical estimated force Fz Figure 23 shows the Fx estimated and real (vehicle simulation)

9 Rolling Resistance Module

Fx Traction Force 2500 40

2000

35

1500

Fx Estimated Fx Real (Not Measured)

30

1000

25

500

20

0

15

Rolling Resistance Estimated Rolling Resistance Real

10

-500

5

-1000 0

20

40

60 time {s}

80

100

120

0

0

20

40

60 time {s}

80

100

120

Figure 23: Longitudinal Force Estimated and Simulated

Figure 24 shows the Fz estimated and real (vehicle simulation) model

4000

Figure 26: Rolling Resistance Estimated and Simulated

Figure 27 shows the Aerodynamics resistance force estimated and real. Aerodynamics Resistance Module 300

Fz Estimated FzN Real (Not Measured)

3500

250

3000 200

2500

150

100

2000

Aerodynamics Resistance Estimated Aerodynamcis Resistance Real

50

0

20

40

60 time {s}

80

100

120 0

0

20

40

Figure 24: Vertical Force Estimated and Simulated

The 3rd step on the observer model gives us the friction coefficient estimated. Figure 25 shows the friction coefficient tire estimated and the friction coefficient obtained with the Pacejka model. model

60 time {s}

80

100

120

Figure 27: Aerodynamics Resistance Force Estimated and Real

The 5th step on the observer model gives us the estimate slope angle. Figure 28 shows the Slope Angle Estimated and Real proposed by table (1) to the vehicle model. Slope Angle

0.9 10

0.8

Slope Angle Estimated (alfa) Slope Angle Real

mu Friction Coefficient Estimated mu Friction Coefficient Real

0.7

8

0.6

6

0.5

4

0.4 2

0.3 0

0.2 0.1 0

-2

0

20

40

60 time {s}

80

100

120

Figure 25: Friction Coefficient Estimated and with the Pacejka Model

The 4th step on the observer model gives us the estimates rolling resistance and aerodynamics resistance force. Figure 26 shows the Rolling resistance estimated and real.

0

20

40

60 time {s}

80

100

120

Figure 28: Slope Angle Estimated and Real

VI. CONCLUSION This work presents a vehicle model composed by different modules. We have the chassis like rigid body, the suspension system, the pneumatics tire (Pacejka), transmission and

10 engine. This model was constructed and simulated with the bond graph model in 20-Sim. Then we used the proposed efficient robust estimator, based on the second order sliding mode differentiator, to build an estimation scheme to identify the longitudinal and vertical forces, the friction coefficient, the tire rolling resistance and the slope angle. The estimation converging finite time produced allows us to obtain virtual measurements of model inputs, in five steps by cascade observers and estimators. Using the vehicle model we could compare the estimation variables with real variables (not measurable). These robust estimations on line are necessary for use on vehicle control dynamics. The observer and estimators were constructed and simulated in 20-Sim with the vehicle model. This form of construction of models allows obtaining more finished and complex models, making possible to improve the definition of variables. It is possible to obtain estimations for different components specifically and individually, achieving a major precision in the estimation of variables. All this is possible with the relative simple construction of models. The plots obtained on the simulation in 20-Sim verify the correct works of the proposed robust observer for the estimation of variables. We developed different modules where we have a general observer module, a longitudinal and vertical force module, friction coefficient estimation module, the tire rolling resistance and aerodynamics estimation module and finally the slope angle estimation module. REFERENCES [1]

F. Aparicio Izquierdo, C. Vera Alvarez, V. Dias Lopez, “Teoria de los vehiculos automoviles”. U.P. de Madrid. [2] G. Rill, “Vehicle Dynamics”. Fachhochschule Regensburg University of Applied Sciences, 2005. [3] W.F. Milliken, D.L. Milliken, “Race Car Vehicle Dynamics”. Society of Automotive Engineers, Inc., 1995. [4] T.D. Gillespie, “Fundamentals of Vehicle Dynamics” - SAE. [5] H.B. Pacejka, “Tyre and Vehicle Dynamics”. Delft University of Technology – Second Edition. [6] J. Katz, “Race car Aerodynamics, Designing for Speed”. Bentley Publishers. [7] Cátedra DSF, “Introducción a la Modelización con Bond Graphs”. Departamento de Electrónica – FCEIA – UNR. [8] G. Filippini, “Dinamica Vehicular mediante Bond Graph”, Proyecto final de carrera de grado. Escuela de Ingenieria Mecanica, Universidad Nacional de Rosario, Argentina 2004. [9] D.C.Karnopp, D.L.Margolis, R.C.Rosenberg, “Modeling and Simulation of Mechatronic System”. John Wiley & Sons, Inc. - Third Edition. [10] A. J. Blundell, “Bond Graphs for modeling Engineering Systems”. Department of Systems and Control Lanchester Polytechnic Coventry [11] I Hocine, “Observation d’états d’un véhicule pour l’estimation du profil dans les traces de roulement”. 2003. [12] N.K. M’Sirdi, A. Naamane, A. Rabhi, “A nominal model for vehicle dynamics and estimation of input forces and tire friction”. Marrakech CSC 2007 [13] L. Fridman.“Observadores en modos deslizantes”, available : http://verona.fi-p.unam.mx/~lfridman/ [14] A. Levant, ”Sliding order and sliding accuracy in sliding mode control,” International Journal of Control, vol. 58, 1993, pp 1247-1263. [15] A. Rabhi, H. Imine, N.K. M’Sirdi, et Yves Delanne. Observers with Unknown Inputs to Estimate Contact Forces and Road Profile. pp 188193. AVCS’04. International Conference on Advances in Vehicle Control and Safety Genova -Italy, October 28-31 2004 [16] N.K. M’sirdi, A. Rabhi, N. Zbiri and Y. Delanne. VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces. TMVDA04. 3rd

Int. Tyre Colloquium Tyre Models For Vehicle Dynamics Analysis August 30-31, 2004 University of Technology Vienna, Austria. [17] “Getting Started with 20-Sim 3.6”, Controllab Products B.V., Enschede, Netherlands. Available: www.20sim.com, 2005 [18] Nacer K. M'Sirdi. Observateurs robustes et estimateurs pour l'estimation de la dynamique des véhicules et du contact pneu - route. JAA. Bordeaux, 5-6 Nov 2003 [19] H.B.Pacejka, I.Besseling. Magic Formula Tyre Model with Transient Properties. 2nd International Tyre Colloquium on Tyre Models for Vehicle Dynamic Analysis, Berlin, Germany (1997). Swets and Zeitlinger. [20] H.B. Pacejka.: Simplified behavior of steady state turning behavior of motor vehicles, part 1: Handling diagrams and simple systems. Vehicle System Dynamics 2, 1973, pp. 162 - 172. [21] H.B. Pacejka.: Simplified behavior of steady state turning behavior of motor vehicles, part 2: Stability of the steady state turn. Vehicle System Dynamics 2, 1973, pp. 173 - 183.