Unknown input observation via sliding modes : application to vehicle

contact forces

O. Khemoudj, H. Imine and M. Djemaı

Abstract— In this paper, a method to estimate vehicle contactforces for trucks by the use of sliding mode observation toolsis presented. The purpose is to use non linear observers andconsider contact forces as unknown inputs for the model. Weuse both a bicycle model to derive the longitudinal and lateralforces and an axle model validated with a software simulatorto obtain the vertical forces.

I. INTRODUCTION

The aim of the estimation of the contact forces for vehi-

cles is to provide a sufficient knowledge of the interaction

between the vehicle and the pavement in order to avoid road

and vehicle damage (tyre, suspension, fragile payload) and

to enhance security by preventing rollover of trucks. The

problem of estimating vehicle contact forces is considered

in many previous works. We can cite [7] who applied an

Extended Kalman Filter (EKF) to estimate the longitudinal

and lateral forces in a vehicle bicycle model, the forces

are considered as additionnal states and their dynamics are

modelled by the use of a shaping filter driven by white noise.

In the same way [9] have developped an estimation algorithm

for an off-highway mining truck, an EKF is used to estimate

the longitudinal, lateral and vertical forces for a four-wheel

vehicle model. The main limitation of this approach is that

some assumptions have to be made on the behaviour of the

forces when tuning the shaping filter, this requires to know

exactly the nature of the forces.

Mainly in litterature, the developped methods consist of

estimating the forces through the estimation of dynamic

variables of the vehicle. [2], [6] developped sliding mode

observers for heavy vehicles. They estimate the height of

the wheel hub, then by measuring the road profile, the tyre

deflection is calculated by assuming the linear stiffness of the

tyre, they get an estimation of the vertical force, then they

calculate the longitudinal and lateral forces using the Pacejka

experimental tyre model. This method involves knowing

accurately the parametres of the vehicle and the pneumatic

and also to measure the road profile.

Another way to estimate the forces is to use strain gauges

either in a hub which is a precise but expensive solution or

in the axle bar, this last method is limited to straight constant

speed manoeuvres and doesn’t suite to cornering manœvres

O. KHEMOUDJ and H. IMINE are with Laboratoire Centraldes Ponts et Chaussees, LEPSIS, UMR LCPC/INRETS, 58 BdLefebvre, 75732, Paris 15, France. omar.khemoudj@lcpc.fr,hocine.imine@lcpc.fr

M. DJEMAI is with Univ Lille Nord de France, F-59000 Lille, France ; UVHC, LAMIH, F-59313 Valenciennes,France, CNRS, UMR 8530, F-59313 Valenciennes, France.mohamed.djemai@univ-valenciennes.fr

with large lateral forces [4].

The method we propose here is to use a sliding mode

observer to evaluate the forces. We then consider these

forces as unknown inputs of the system on this basis we use

sliding mode observers and the main embedded sensors [1]

in the vehicle and finally we estimate the vertical forces.

The advantage of this approach is to avoid the use of many

parameters of the vehicle (usually unkonwn), to make use

of the available sensors in the vehicle and to present a

good alternative to the use of strain gauges especially while

cornering. This paper is divided into three main sections. In

the first section, we describe the used models and give also

give their limitation and the taken assumptions, in the second

section we present the estimation methods and the simulation

results, we finally conclude with some perspectives in the last

section.

II. MODEL DESCRIPTION

To be able to develop observers and to validate our ap-

proach, we use an observable non linear model as described

in [9]. It is known as the bicycle model. We are interested in

2-axle rigid body trucks such as tractors and straight trucks

(example mining truck). A representation of the model is

given in the figure 1. As we can see in this latter figure,

the front and rear wheels are lumped into one front and

one rear wheel where total front and real longitudinal and

lateral forces are respectively applied to each wheel. Note

that this model is valid for lateral acceleration to 0.3g and

is used frequently in litterature to study vehicle trajectory

stability. As the validation of this model in the simulator

gives satisfactory results we can then use it for a state

estimation purpose.

Fig. 1. Bicycle model.

To complete this model, we have to introduce the vertical

forces. To achieve that, the idea is to isolate the axle and to

represent the forces acting over it. In figure 2 we represent a

truck-axle and the main acting forces. This model makes the

junction between the slow dynamics of the chassis (rolling

and pitching) which are captured by the suspension forces

18th IEEE International Conference on Control ApplicationsPart of 2009 IEEE Multi-conference on Systems and ControlSaint Petersburg, Russia, July 8-10, 2009

978-1-4244-4602-5/09/$25.00 ©2009 IEEE 1720

and the rapid dynamics due to irregularities in the pavement

surface which are captured by the accelerations.

Fig. 2. Forces applied on the truck-axle

Remark 1. the suspension forces can be measured by the

use of a pressure transducer because of the proportionality

between the force and pressure in air-springs [4].

The equations of movement of the free body are given by:

Fz,l + Fz,r = Fs,l + Fs,r + Ma.g + Ma.a (1)

and

Fz,l.ℓ + Fz,r.ℓs = Fz,r.ℓ + Fs,l.ℓs + Fy.r + Ia.aϕ (2)

with gravity acceleration g = 9.81m/s2. Fz,l and Fz,r

the vertical wheel forces at the left and at the right of the

axle. Fs,l and Fs,r the suspension forces at the left and the

right of the axle. Fy the resultant lateral force i.e. the sum

of the left and right lateral tyre-force applied to the axle.

Me the mass of the axle. Ia the moment of inertia of the

axle around its roll axle. a the axle-hop acceleration. aϕ the

axle-roll acceleration. ℓ the distance between the point of

application of the tyre force and the center of gravity of the

axle. ℓs the distance between the point of application of the

suspension force and the center of gravity of the axle and

finally r the distance between the ground and the center of

gravity of the axle. In order to simplify the problem, we have

taken some assumptions :

Assumption 1. The roll center and the center of gravity

of the axle are the same.

Assumption 2. The axle is perfectly symetric .

Assumption 3. The distance r is constant and is equal to

wheel radius.

Assumption 4. The effects of anti-roll bars is neglegted.

We have validated this model by using a vehicle sofware-

simulator called PROSPER1([5]). For a curve manoeuvre, we

note that using this model, the vertical forces are correctly

tracked compared to the simulator forces as shown in the

figure 3. The vertical forces are composed of two distin-

guished dynamics, the slow dynamics due to the load while

conrnering and the rapid dynamics which are oscillations

caused by the pavement.

In the next section, we develop observers to estimate the

unknown contact forces, we proceed in two steps, the first

1PROSPER, Oktal, www.callasprosper.com

0 5 10 151.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2x 10

4

time (s)

N

from prosperfrom model

0 5 10 151.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7x 10

4

time (s)

N

from prosperfrom model

Fig. 3. Axle model validation with Prosper (top: left side vertical force,bottom: right side vertical force)

step is to estimate the longitudinal and lateral forces from

the bicycle model and in the second step we determine the

corresponding vertical forces.

III. SLIDING OBSERVER DESIGN

The input signal is composed of steering angle and the

total torques at the front and the rear of the vehicle. From

the model described in section II, we can derive the state-

space model. The system known inputs are respectively the

steering angle δ, the front total torque Tf and the rear total

torque Tr:

u =[

δ Tf Tr

]T(3)

The output signal Y is composed of the yaw rate α, the

front and rear wheels speeds respectively ωfw, ωrw, the

longitudinal and lateral acceleration respectively x, y :

Y =

αωfw

ωrw

x =1

Mt

(Frx + Ffx cos(δ) + Ffy sin(δ))

y =1

Mt

(Fry − Ffx sin(δ) + Ffy sin(δ))

(4)

where Ffx, Frx,Ffy and Fry are the resultant longitudinal

and lateral tyre forces respectively at the front and the rear

of the vehicle. Note that Mt is the total mass and Iz is the

total inertia about the vertical z-axis. Mfw, Mrw, Ifw and

Irw are the masses and rotational inertias of the front and

rear wheels. lf and lr the distances between the center of

gravity of the vehicle and respectively the front and the rear

axles.

1721

We devide the output vector Y =[

Xv Xa

]Tin two

vectors : Xv composed of velocities and Xa composed of

accelerations as follows : Xv =[

α ωfw ωrw

]Tand

Xa =[

x y]T

.

Xv =

−

lr

Iz

Fry −

lf

Iz

sin(u1)Ffx +lf

Iz

cos(u1)Ffy

rfw

Ifw

Ffx +1

Ifw

u2

rrw

Irw

Frx +1

Irw

u3

(5)

Remark 2. In practice, steering angle, yaw rate, wheel

rotational velocity, the lateral acceleration and the driving

torques are signals available on the CAN-bus of the vehicle.

The braking torque can be deduced from the brakes pressure

which is also availble at the CAN-bus, in our application,

we do not consider braking situations.

By considering the unknown forces vector which is com-

posed of the total front and rear longitudinal and lateral

forces.

uF =[

Ffx Frx Ffy Fry

]T(6)

The model can be written in this form

Xv = Bu + W1(u)uF

Y =

[

Xv

W2(u)uF

] (7)

with,

B =

0 0 0

01

Ifw

0

0 01

Irw

(8)

W1(u) =

−

lf

Iz

sin(u1) 0lf

Iz

cos(u1) −

lr

Iz

rfw

Ifw

0 0 0

0rrw

Irw

0 0

(9)

and

W2(u) =

1

Mt

cos(u1)1

Mt

1

Mt

sin(u1) 0

−

1

Mt

sin(u1) 01

Mt

cos(u1)1

Mt

(10)

u1 is the first component of the known input vector

u, namely the steering angle δ. It is well-known that the

acceleration of mechanical systems is directly correlated to

the forces applied to this system. We can notice it from

the equations describing vehicle dynamics (5). The idea

is to use the availble velocity measurments and to derive

the acceleration using a sliding mode observer. Once the

derivation achieved, we use an estimator derived from the

equations of the model to deduce the unknown forces.

A. First order observer

We begin by developping a first order observer to estimate

the accelerations. From equation (5), we write :

Xv = f(u, uF ) (11)

f ∈ R3 is a vector-field function assumed to be bounded.

The sliding observer is given by

ˆXv = f(u, uF ) + Ksign(Xv − Xv) (12)

K ∈ R3×3 is the observer diagonal positive matrix gain,

we define the estimation error of the velocity by Xv = Xv−

Xv Let V the Lyapunov candidate function of the velocity

error V =1

2XT

v Xv. The time derivative of V is given by

V = XTv

˙Xv. The error dynamics can be obtained by the

formula˜Xv = Xv −

ˆXv , so˜Xv = f(u, uF ) − (f(u, uF ) +

Ksign(Xv− Xv)), then˜Xv = f(u, uF , uF )−Ksign(Xv−

Xv) with f(u, uF , uF ) = f(u, uF ) − f(u, uF ). Finally,

V = XTv (f(u, uF , uF ) − Ksign(Xv)) (13)

To ensure the convergence of the observer, the condition

below must be satisfied.

Kii ≫

∣

∣

∣fi(u, uF , uF )

∣

∣

∣, i = 1, 2, 3 (14)

In this case, Xv converges to 0 in finite time and then˙Xv

converges also to 0 in finite time t0 > 0.

Simulation results : We have simulated a curve trajectory

with a constant streering angle. We injected longitudinal and

lateral forces in the model as unknown inputs. First, we

illustrate the estimation of the accelerations in the figures

4 and 5, (we only represent front wheel acceleration as the

rear wheel is a similar figure) then we present the estimation

of the forces. The results we obtain with this method are

illustrated in figure 6. We have made a zooming in the figure

7 to see more clearly the predominance of chattering. This

result is confirmed in figure 8 where the estimation error

is shown. Note that in the zooming figures, the thick line

corresponds to the model and the thin line to the estimation.

The estimation of the vertical forces is given in figure 9

Discussion : As we can see, the first order observer

introduces a large chattering in the estimates of longitudinal

and lateral forces. Globally, the vertical forces are tracked

with the presence of chattering as well. This is why we

use a second order sliding mode wich normally reduces the

chattering phenomenon.

1722

0 5 10 15−15

−10

−5

0

5

10

15

time (s)

rd/s

²

estimationmodel

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8−2

−1.5

−1

−0.5

0

0.5

1

time (s)

rd/s

²

estimationmodel

Fig. 4. Estimation with first odrer observer of the yaw acceleration (top)and zooming of the figure (bottom)

0 5 10 15−200

−150

−100

−50

0

50

100

150

200

time (s)

rd/s

²

estimationmodel

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8−30

−20

−10

0

10

20

30

time (s)

rd/s

²

estimationmodel

Fig. 5. Estimation with first order observer of wheel acceleration (top) andzooming of the figure (bottom)

B. Second order observer

We use in this section a robust second order observer [3].

In our case, we use the differentiator to estimate the accel-

erations from the velocity vector Xv which is measured. We

assume that Xv is bounded and differentiable. The robust

observer is given by :

{

ˆXv = Z + λ

∣

∣

∣Xv − Xv

∣

∣

∣

1

2

sign(Xv − Xv)

Z = αsign(Xv − Xv)(15)

0 5 10 15−2

−1

0

1

2x 10

4

N

Front longitudinal force

0 5 10 15−2

−1

0

1

2

3x 10

4

N

Rear longitudinal force

0 5 10 15−3

−2

−1

0

1

2

3x 10

4

time (s)

N

Front lateral force

0 5 10 15−3

−2

−1

0

1

2

3x 10

4

time (s)

N

Real lateral force

estimationmodel

Fig. 6. Estimation of longitudinal and lateral forces with first order observer

7 7.2 7.4 7.6 7.8 8−4000

−3000

−2000

−1000

0

1000

2000

N

Front longitudinal force

7 7.2 7.4 7.6 7.8 82000

3000

4000

5000

6000

7000

8000

N

Rear longitudinal force

7 7.2 7.4 7.6 7.8 80.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4

time (s)

N

Front lateral force

7 7.2 7.4 7.6 7.8 86000

7000

8000

9000

10000

11000

12000

time (s)

N

Real lateral force

estimationmodel

Fig. 7. Estimation of longitudinal and lateral forces with first order observer: zooming

Considering the known velocities to be locally bounded

and the observer based on the differentiator (15), for any

initial conditions Z(0), ˆXv(0) there exists a choice of α and

λ such that the observer error e = ˆXv−Xv tends to 0 in finite

time. The complete proof of this theorem is given in [8].

Hense, to get an estimation of the acceleration signal from

the velocities, we have to verify the sufficient conditions for

convergence of the differentiator. These conditions are upon

the gain diagonal matrices α ∈ R3×3 and λ ∈ R

3×3 which

are chosen such that for i = 1, 2, 3, we have :

αii > σ (16)

and

λii > (σ + αii)

√

2

αii − σ(17)

1723

5 6 7 8 9 10−3000

−2000

−1000

0

1000

2000

3000

NFront longitudinal force error

5 6 7 8 9 10−3000

−2000

−1000

0

1000

2000

3000

N

Rear longitudinal force error

5 6 7 8 9 10−5000

0

5000

time (s)

N

Front lateral force error

5 6 7 8 9 10−5000

0

5000

time (s)

N

Real lateral force error

Fig. 8. Error of estimation of longitudinal and lateral forces with first orderobserver

0 5 10 151.4

1.5

1.6

1.7

1.8

1.9

2x 10

5

N

Front left vertical force

0 5 10 151.6

1.7

1.8

1.9

2

2.1x 10

5

N

Front right longitudinal force

0 5 10 151.4

1.6

1.8

2

2.2x 10

5

time (s)

N

Rear left vertical force

0 5 10 151.9

2

2.1

2.2

2.3

2.4

2.5x 10

5

time (s)

N

Rear right vertical force

estimationmodel

Fig. 9. Estimation of vertical forces with first order observer

σ is the an upper bound of the jerks. By using the

differentiator, we want to derive numerically the velocity

vector, we are not interested in jerk estimation. Here we take

σ = 10. To estimate contact forces, we use the estimator

derived from model equations. For the longitudinal forces

we have the expressions

Ffx =1

rfw

(IfwˆXv(2) − u2)

Frx =1

rrw

(IrwˆXv(3) − u3)

(18)

and the lateral forces

Ffy =1

sin(u1)(MtXa(1) − Frx − Ffx cos(u1))

Fry =1

lr(−Iz

ˆXv(1) −

lf

sin(u1)Ffx +

lf

cos(u1)Ffy)

(19)

Note that if the steering angle u1 = 0, we suppose that

the lateral forces are negligible (straight line manoeuvre).

Simulation results : The estimated accelerations are il-

lustrated in figures 10 and 11. We have chosen α = 600and λ = 40 so that the conditions of convergence of the

differentiator are achieved, the results of longitudinal and

lateral force estimation are presented in figure 12. We used

the same conditions of simulation as for the first order

observer. We represent also the estimation error in figure

13.

0 5 10 15−15

−10

−5

0

5

10

15

time (s)

rd/s

²

estimationmodel

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8−2

−1.5

−1

−0.5

0

0.5

1

time (s)

rd/s

²

estimationmodel

Fig. 10. Estimation with second order observer of the yaw acceleration(top) and zooming of the figure (bottom)

0 5 10 15−50

0

50

100

150

200

250

time (s)

rd/s

²

estimationmodel

7.9 7.91 7.92 7.93 7.94 7.95 7.96 7.97 7.98 7.99 8

−0.1

−0.05

0

0.05

0.1

0.15

time (s)

rd/s

²

estimationmodel

Fig. 11. Estimation with second order observer of wheel acceleration (top)and zooming of the figure (bottom)

The estimation of the vertical forces is given in figure 14.

Discussion : We can note that the chattering in the esti-

mated forces is considerably reduced. The estimation error

of the longitudinal and lateral forces is represented in figure

13.

1724

0 5 10 15−0.5

0

0.5

1

1.5

2

2.5x 10

4N

Front longitudinal force

0 5 10 150

0.5

1

1.5

2

2.5

3x 10

4

N

Rear longitudinal force

0 5 10 15−4

−3

−2

−1

0

1

2x 10

4

time (s)

N

Front lateral force

0 5 10 15−1

0

1

2

3

4x 10

4

time (s)

N

Real lateral force

estimationmodel

Fig. 12. Estimation of longitudinal and lateral forces with second orderobserver

9.9 9.92 9.94 9.96 9.98 10−20

−10

0

10

20

N

Front longitudinal force error

9.9 9.92 9.94 9.96 9.98 10−20

−10

0

10

20

N

Rear longitudinal force error

9.9 9.92 9.94 9.96 9.98 10−20

−10

0

10

20

time (s)

N

Front lateral force error

9.9 9.92 9.94 9.96 9.98 10−20

−10

0

10

20

time (s)

N

Real lateral force error

Fig. 13. Error of estimation of longitudinal and lateral forces with secondorder observer

The longitudinal forces errors have an absolute maximum

less than 10 N and lateral forces errors have an absolute max-

imum of 15 N which is insignificant compared to magnitude

of the forces, this is not the case for the first order estimator.

The chattering in the vertical forces is due to the estimation

error in lateral forces, we can notice that the chattering is

less important than the first order observer but it does not

modify considerably the shape of the force. Nevertheless, it is

preferable to use the second order for detecting peak vertical

forces, these peaks can be confused with the chattering when

we use the first order observer.

IV. CONCLUSION

In this paper, we have investigated a new way to de-

termine axle vertical forces by the use of unknown input

observers. We have used both a first and a second order

observer to estimate the longitudinal and lateral forces. We

have noted that the estimation quality is better by using a

0 5 10 151.4

1.5

1.6

1.7

1.8

1.9

2x 10

5

N

Front left vertical force

0 5 10 151.5

1.6

1.7

1.8

1.9

2

2.1x 10

5

N

Front right longitudinal force

0 5 10 151.4

1.6

1.8

2

2.2x 10

5

time (s)

N

Rear left vertical force

0 5 10 151.9

2

2.1

2.2

2.3

2.4

2.5x 10

5

time (s)

N

Rear right vertical force

estimationmodel

Fig. 14. Estimation of vertical forces with second order observer

second order observer. We have also proposed to use an axle

model to derive the vertical forces. Some perpectives of our

investigation is to develop a more sofisticated axle-model, to

extend the approach to other vehicle configurations (tri-axle,

semi-trailer) and to evaluate the performance of other kind

of observers such as higher order observers and high gain

observers.

V. ACKNOWLEDGMENTS

The present research work has been in part supported

by International Campus on Safety and Intermodality in

Transportation the Nord-Pas-de-Calais Region, the European

Community, the Regional Delegation for Research and Tech-

nology, the Ministry of Higher Education and Research, and

the National Center for Scientific Research.

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