Vehicle Parameter and States Estimation Via Sliding Mode Observers

In the vehicle dynamics, the estimations of vehicle side slip angle and wheel interaction forces with the ground are very important, because of their influence on the stability of the veh

Sliding Mode Observers

Hassan Shraim1, Mustapha Ouladsine1, and Leonid Fridman2

1 Laboratory of sciences of informations and of systems, SIS UMR 6168 University of Paul C´ezanne, Aix-Marseille III, Av escadrille de Normandie Niemen 13397 Marseille Cedex 20

{hassan.shraim, mustapha.Ouladsine}@lsis.org

2 Department of Control, Engineering Faculty, National Autonomous University of Mexico (UNAM)

lfridman@servidor.unam.mx

Preliminaries and motivations In the recent years, important research has been

undertaken to investigate safe driving conditions in both normal and critical situations But due to the fact that safe driving requires the driver to react ex-tremely quickly in dangerous situations, which is generally very difficult unless for experts, then this reaction may influence the stability of the system Con-sequently, the improvement of the vehicle dynamics by active chassis control is necessary for such catastrophic situations Increasingly, commercial vehicles are being fitted with micro processor based systems to enhance the safety and to improve driving comfort, increase traffic circulation, and reduce environmental pollution associated with vehicles

Considerable attention has been given to the development of the control sys-tems over the past few years, authors have investigated and developed different methods and different strategies for enhancing the stability and the handling of the vehicle such as, the design of the active automatic steering [30] and the wheel ABS control [16], [25], or the concept of a four-wheel steering system (4WS) which has been introduced to enhance vehicle handling Some researchers have shown disadvantages on (4WS) vehicles [15]

In terms of vehicle safety, and in order to develop a control law for the vehicle chassis, accurate and precise tools such as sensors should be implemented on the vehicle, to give a correct image of its comportment Difficulties in measuring all vehicle states and forces, due to high costs of some sensors, or the non existence

of others, make the design and the construction of observers necessary In the vehicle dynamics, the estimations of vehicle side slip angle and wheel interaction forces with the ground are very important, because of their influence on the stability of the vehicle Many researchers have studied and estimated vehicle side slip angle, using a bicycle model as in [23], or by using an observer with adaptation of a quality function as in [29] which requires a certain linearised form

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 345–362, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

of the model Moreover, an extended Kalman filter is used for the estimation of wheel forces [19]

Methodology In this chapter sliding mode observers are used for estimation of the

vehicle parameters These observers are developed and used (see, for example, the corresponding chapter in the textbooks by [5] and [26] and the recent tutorials

by [4] and [17] They are widely used due to their attractive features:

• insensitivity (more than robustness) with respect to unknown inputs;

• possibilities to use the values of the equivalent output injection for the

un-known inputs identification ([26]);

• finite time convergence to the reduced order manifold.

In [12], [14] robust exact differentiators were designed ensuring finite time convergence to the real values of derivatives, as an application of super-twisting algorithm ([13]) New generation of observers based on the high order sliding mode differentiators are recently developed (see [3], [1], [2], [7] and [8]) Those observers:

• provide a convergence to the exact values of states variables;

• allow the exact identification the unknown inputs without filtration;

• guarantee the best possible accuracy of the state estimation w.r.t to the

sampling steps and deterministic noises

The results of the chapter are validated by the simulator ve-dyna (see [24]) developed by the group of companies TESIS, which is an independent expert team for the simulation of virtual vehicles in real time Their software comprises higher precision models for simulation of vehicle dynamics ve-dyna, engine dy-namics en-dyna and brake hydraulics (RT Brake Hydraulics) The simulator ve-dyna that we use in our study is a software especially designed for the fast simulation of vehicle dynamics both in real-time applications (Hardware-in-the-Loop, Software-in-the-Loop) and concept studies on a standard desktop

PC, its computational performance enables its usage for the optimization e.g for the parameter identification ve-dyna vehicle model is fully parametric and has a modular architecture with the following program modules: Vehicle multi-body system (chassis), axles (axle kinematics, compliance), tire model TM-Easy, transmission and drive line, engine, aerodynamics and braking system All ve-hicle degrees of freedom (DoF) are nonlinearly modeled The number of overall DoF depends on the number of additional bodies and the suspension type The global vehicle model and each sub model are validated by real experiments with different operation conditions

Contribution The main contributions of this work reside in the estimations of

wheels contact forces with the ground, side slip angle and the velocity of the vehicle By these estimations, we avoid the use of expensive sensors and we can preview some critical situations that may occur while driving, such as

exces-sive rotation around Z axis and also excesexces-sive side slipping, inappropriate lateral

acceleration, The proposed observers are characterized by the fast convergence

to real values, robustness and they don’t require extensive computation load

As we have mentioned before, the proposed observers are validated by the simulator ve-dyna Several validations were made to cover most of the driving cases, such as a double lane trajectory, straight line motion with strong variation

in acceleration and deceleration, strong change in the steering angle

Structure of the chapter This chapter is organized as follows: In section 2,

prob-lem statement is proposed and the model used for the vehicle is shown In sec-tion 3, two steps are presented In the first one, we present the estimasec-tion of the longitudinal forces and wheels angular velocities In this step, comparison

of two high order sliding mode observers is made and the third order one is chosen for the estimation In the second step, a sliding mode observer based

on the super-twisting algorithm for the estimation of the side slip angle and the velocity of the vehicle is proposed Finally a conclusion for the work is presented

The presented problem is how to assist the driver in critical situations, i.e., when the vehicle goes out from its safety regions and enters in the dangerous situations

So accurate tools to represent vehicle states and parameters are required These accurate representations need many precise and expensive sensors The use of many sensors requires an important diagnosis system to avoid false data To overcome these problems, robust virtual sensors are proposed which estimate vehicle parameters, forces and states The model used for estimation and control

is a non-linear one obtained by applying the fundamental principles of dynamics

at the center of gravity on figure 1 (see [29]):

˙v COG= 1

M



cos(β)

F L − sin(β)F S



(1)

˙

M v COG



cos(β)

F S − sin(β)F L



with

F L = F xwind +cos(δ f )(F x1+F x2)+cos(δ r )(F x3+F x4)−sin(δ f )(F y1+

F y2)− sin(δ r )(F y3+ F y4),

and



F S = sin(δ f )(F x1+ F x2) + sin(δ r )(F x3+ F x4) + cos(δ f )(F y1+ F y2)+

cos(δ r )(F y3+ F y4).

I Z ψ = t¨ f {cos(δ f )(F x2− F x1) + sin(δ f )(F y1− F y2)} + L1{sin(δ f)

(F x2+ F x1) + cos(δ f )(F y1+ F y2)} + L2{sin(δ r )(F x3+ F x4)− cos(δ r )(F y3+ F y4)} + t r {cos(δ r )(F x4− F x3) + sin(δ r )(F y3− F y4)}

(3)

Table 1 Nomenclature

Symbol Physical signification

Ω i angular velocity of the wheel i

M total mass of the vehicle

r i radius of the wheel i

COG centre of gravity of the vehicle

r 1i dynamical radius of the wheel i

F z i vertical force at wheel i

F x i longitudinal force applied at the wheel i

F y i lateral force applied at the wheel i

C f i braking torque applied at wheel i

C mi motor torque applied at wheel i

torque i C mi − C f i

I Z moment of inertia around the Z axis

ψ yaw angle

˙

ψ yaw velocity

δ f front steering angle

δ r rear steering angle

δ i deflection in the tire i

V x longitudinal velocity of the center of gravity

V y lateral velocity of the center of gravity

I ri moment of inertia of the wheel i

v COG total velocity of the center of gravity

L1 distance between COG and the front axis

L2 distance between COG and the rear axis

L L1+ L2

h COG height of COG

t f front half gauge

t r rear half gauge

l t f + t r

A L front vehicle Area

ρ air density

C aer coefficient of aerodynamic drag

α i slip angle at the wheel i

β side slip angle at the COG

μ i friction coefficient at the wheel i

X t length of the contact patch for the wheel i

X ad,i length of the adhesion patch for the i-th wheel

X sl,i length of the sliding patch for the i-th wheel

V x i Longitudinal velocity of the wheel i

p i inflation pressure of the tire i

K 1i constant depending on the deformation of the tire

K constant depending on the deformation of the tire

L

2

L

1

x

F

2

F

3

3

F

1

F

2

F

2

z

F

1

z

F

3

z

F

4

z

F

OG

C

r

t

× 2

f

t

× 2

ψ

o

x

o

y

1 α

4

F

Fig 1 2D vehicle representation

The model representing the dynamics of each wheel i is found by applying

New-ton’s law to the wheel and vehicle dynamics figure 2:

Fig 2 Wheel and its contact with the ground

I ri Ω˙i =−r 1i F x i + torque i i = 1 : 4 (4)

In this paper, the task is to design a virtual sensors (observer) for the vehicle

to estimate the states, parameters and forces which need expensive sensors for their measurement But due to the fact that it is not easy to apply an observer for the global model, equations 4 are taken at first, a high order sliding mode

observer is proposed for each equation 4 to estimate the longitudinal force of each wheel After having the longitudinal forces, we substitute their values in equations 1, 2 and 3 From these equations, it is seen that if we substitute the estimated longitudinal forces, we will still have as complex terms the lateral forces In order to model the lateral force, we use a brush model for the contact with the ground [21] The brush model divides the surface of contact into two parts: a sliding part and an adhesion part Then the lateral force generated at the surface of contact will be the sum of forces generated at each part of the surface These lateral forces are represented as in [21]:

F y i=  tan(α i )μ i F z i X sl,i

tan(α i)2+



Ω i

V x i



K 1i+K 2i

p i



× F z i

1000



− 12



× X ti

+A i

3 × X3

ad,i

(5) with

A i= 1

B 2K i tan(α i)− F y sl,i × ϕ

X ad,i X sl,i

(6) and

C = 1

β1(1 + exp ( −β1X ad,i ) (sin(β1X ad,i))− cos(β1X ad,i)) (8)

D = X (1 + exp ( −β1X ad,i ) cos(β1X ad,i)) (9) with

β1= 4

K i

E × I ri

(10) and

F = −exp (−β1X sl,i ) cos(β1X sl,i) (12)

G = −exp (−β1X ad,i ) cos(β1X ad,i) (13)

H = exp ( −β1X ti ) cos(β1X ti) (14)

F y sl,i=  tan(α i )μ i F z i X sl,i

tan(α i)2+



Ω i

V x i



K1+K2

p i



× F z i

1000



− 12



× X ti

(15)

From this representation of the force, we need to define some of the variables, all the equations for these variables are detailed in [22] and [28], and they are given by:

• The coefficient of adherence μ i:

μ i= F x i

F z i

(16)

• The side slip angle of each wheel:

α1= tan −1 v COG sin(β) + L1ψ˙

v COG cos(β) − t f ψ˙

α2= tan −1 v COG sin(β) + L1ψ˙

v COG cos(β) + t f ψ˙

α3= tan −1 v COG sin(β) − L2ψ˙

v COG cos(β) − t r ψ˙

α4= tan −1 v COG sin(β) − L2ψ˙

v COG cos(β) + t r ψ˙

• The velocity of each wheel:

V x1=



v COG cos(β) − t f ψ˙

cos(δ f) +



v COG sin(β) + L1ψ˙

sin(δ f) (21)

V x2=



v COG cos(β) + t f ψ˙

cos(δ f) +



v COG sin(β) + L1ψ˙

sin(δ f) (22)

V x3=



v COG cos(β) − t r ψ˙

cos(δ r) +



v COG sin(β) − L2ψ˙

sin(δ r) (23)

V x4=



v COG cos(β) + t r ψ˙

cos(δ r) +



v COG sin(β) − L2ψ˙

sin(δ r) (24) For the vertical forces, cheap sensors can be found for their measure (or they can be estimated as shown in [28]), the determination of the contact patch and its repartition into a sliding part and adhesion part can be found as shown

in [22]

The system described by the equations 1, 2 and 3 is observable if we consider that we measure only the yaw rate (and by supposing the longitudinal forces as inputs) A sliding mode observer based on hierarchical super twisting algorithm

is used to estimate vehicle side slip angle and velocity By these estimations, the longitudinal and lateral velocities of the center of gravity, the lateral forces of the wheels are then directly deduced

By these estimations, the driver (or the controller) knows if the vehicle is in the safe region or not These regions depend on the velocity, the coefficient of friction and the steering angle [28] and [9]

In this study it is supposed that we have as inputs (estimated or measured):

• angular positions of the wheels (used if angular positions sensors are

imple-mented on the vehicle)

• front wheel angle (measured)

• yaw rate (measured)

• torque applied at each wheel (estimated see [11] and [18])

And it is required to estimate:

• angular velocity of the wheels (to be observed only if it is not measured)

• contact forces

• vehicle velocity

• vehicle side slip angle

If the wheels angular velocities are measured instead of wheels angular posi-tions, then in the first step we have to estimate only the longitudinal forces For this goal, a sliding mode observer based on the super twisting algorithm may be used

3.1 Estimation of Wheels Angular Velocities and Longitudinal Forces

In this part, sliding mode observers are proposed to observe the angular velocity

Ω i and to identify the longitudinal force of each wheel F x i

Case when the angular positions of the wheels are measured

Dynamical equations of wheels (4) are written in the following form:

˙x1= x2

where x1and x2are respectively θ i (which is measured) and Ω i(to be observed)

(appears implicitly in F x i ), and u is the torque i In fact this torque may be measured as shown in [18] and it can also be estimated by estimating the motor and the braking torque, the motor torque may be estimated as in [11], while the braking torque is estimated by measuring the hydraulic pressure applied at each wheel (existing on the most of the vehicles)

Broken super-twisting observer

In the first part of this section, the so-called broken super-twisting algorithm observer proposed in [3] in the form:

˙ˆx1= ˆx2+ z1

˙ˆx2= f1(x1, ˆ x2, u) + z2

(26)

where ˆx1and ˆx2are the state estimations of the angular positions and the

angu-lar velocities of the four wheels respectively, f1is a nonlinear function containing

only the known terms (which is only the torque in our case), z and z are the

correction factors based on the super twisting algorithm having the following forms:

z1= λ |x1− ˆx1| 1/2 sign(x1− ˆx1)

In the above equations the function |.|1

and sign(.) should be thought of as

componentwise extensions of their traditional scalar coordinates We will define the solutions of (26)-(27) and all equations below in the Filippov sense [6] Taking ˜x1= x1− ˆx1and ˜x2= x2− ˆx2we obtain the equations for the error

˙˜x1= ˜x2− λ|˜x1| 1/2 sign(˜ x1)

˙˜x2= F (t, x1, x2, ˆ x2)− λ0sign(˜ x1) (28)

where F (t, x1, x2, ˆ x2) = f (x1, x2, u) − f1(x1, ˆ x2, u) is the unknown function to

be identified

Suppose that the system states can be assumed bounded, then the exists a

such a constant f+ that the inequality:

|F (t, x1, x2, ˆ x2)| < f+ (29)

holds for any possible t, x1, x2 and |ˆx2| ≤ 2sup |x2| As described in [3], it is

sufficient to choose λ0= 1.1f+and λ = 1.5

f+ A sliding motion occurs in the error system (27) which makes ˜x1 ≡ 0 and ˜x2 ≡ 0 in finite time Furthermore

we get:

α sign(x1− ˆx1) = F (t, x1, x2, ˆ x2) (30) where the left hand of the above equation represents the average value of the discontinous term which must be taken in order to maintain a sliding motion

The value of sign(x1− ˆx1) can be obtained by appropriate low-pass filtering of

the discontinous injection signal sign(x1− ˆx1) and so is available in real time

So in the broken super twisting observer suggested here allowing to estimate

of the longitudinal forces by filtering the discontinuous injection signal (which may cause a certain delay)

Third order sliding mode Observer

To avoid filtration let us consider s third order sliding mode observer in the form:

˙ˆθ i = ˆΩ i + λ0|θ i − ˆθ i| 2/3 sign(θ i − ˆθ)

v o

˙ˆ

Ω i=torque i

I ri + z1

(31)

where ˆθ iand ˆΩ iare the state estimations of the angular positions and the angular

velocities of the four wheels respectively, z1is the correction factor based on the super twisting algorithm having the following forms:

z1= λ1| ˆ Ω i − v o | 1/2 sign( ˆ Ω i − v o ) + Z1

v1

˙

Z1= λ2sign(Z1− v1).

(32)

with λ2 = 3√3

λ0, λ1 = 1.5 √

λ0 et λ0 = 2f+ The second-order sliding-mode occurs in (32) in finite time Levant (2003) In particular, this means that the longitudinal forces can be estimated in a finite time without the need for low pass filter of a discontinuos switched signal

Case when the angular velocities of the wheels are measured

In this case, we need to estimate only the longitudinal forces The proposed observer in this case is:

˙ˆx 2i =torque i

I ri + λ1|x 2i − ˆx 2i | 1/2 sign(x 2i − ˆx 2i ) + Z 1i

˙

The second-order sliding-mode occurs in system (33) in finite time [14]

-400

-200

0

200

400

600

800

t ime(s ec )

Fig 3 Motor and braking torque (N.m) applied at the two rear wheels

-700

-600

-500

-400

-300

-200

-100

0

t ime(s ec )

Fig 4 Motor and braking torque (N.m) applied at the two front wheels

Third order sliding mode Observer

To avoid filtration let us consider s third order sliding mode observer in the form:

˙ˆθ i... z1

(31)

where ˆθ iand ˆΩ iare the state estimations of the angular positions and the angular

velocities of the four wheels...

f+ A sliding motion occurs in the error system (27) which makes ˜x1 ≡ and ˜x2 ≡ in finite time Furthermore