Parity Space based FDI-Scheme for Vehicle Lateral Dynamics

S. Schneider∗, N. Weinhold∗, S.X. Ding∗ and A. Rehm†∗Institute for Automatic Control and Complex Systems, University of Duisburg-Essen,

Bismarckstrasse 81 BB, 47048 Duisburg, Germany.†Robert Bosch GmbH, Stuttgart, Germany.

Abstract— In this paper, a model-based FDI-scheme for thedetection and isolation of sensor faults in the vehicle lateraldynamics control system ESP is presented. The main objectiveis to handle the model uncertainties and to isolate faults. Inorder to reduce false alarm rate on the one hand and to ensurefault detectability on the other hand, a Parity Space basedsolution is presented. To achieve a numerical stable on-linerealization on an electronic control unit (ECU), the FDI systemis realized in an observer structure. The FDI-scheme has beentested using real driving data.

I. INTRODUCTION

The rapid development of electronic control systems suchas ABS (Anti-lock Breaking System), ESP (Electronic Sta-bility Program), ACC (Adaptive Cruise Control) and theirwide integration in cars mark an important technologicalprogress in the automotive industry in the past decade [1],[2], [3]. A central functionality of these control systems isto improve the active safety by stabilizing the vehicle inextreme driving situations [4]. As far as a critical drivingsituation is identified, controllers will be activated until thevehicle returns to normal situation (see Fig. 1). Due to thisfact, failures of components which are integrated into thesecontrol loops may strongly affect the system stability andsafety. High reliability is thus an essential requirement onthese components [5], [6].

In this paper, a fault detection and isolation (FDI) schemewill be presented, which is applied for detecting and isolatingfaults in

• yaw rate sensor as well as• lateral acceleration sensor

and integrated in the ESP control loops. The FDI scheme isdeveloped based on Parity Space Approach and later realizedin an observer structure in order to achieve a numerical stableon-line realization on an electronic control unit. The majorfocus is devoted to the handling of disturbances, especiallythe road bank angle and a fault isolation strategy.

II. PROBLEM FORMULATION

The main requirements on the FDI system to be integratedin the ESP control loops are (a) low false alarm rate (b)early detection [5], [6]. In addition, it is desired that the FDIsystem is modularly structured so that a system extension byadding additional sensors will not lead to a total re-design.

It is well known that the central problem related to thedesign of model-based FDI systems is to find a compromisebetween high fault detectability and low false alarm rate [7],

oversteer

understeer

without dynamicscontrol system

with dynamicscontrol system

Fig. 1. Improved car safety

[8], [9]. The major difficulty for the application of model-based FDI methods to the sensor fault detection and isolationin ESP lies in the fact that the model uncertainties stronglydepend on driving maneuvers. Therefore, handling of modeluncertainties builds the major focus of our study.

In the past years, efforts of applying advanced model-based FDI methods to improve the performance of FDIsystems integrated in the ESP control loops have been re-markably enhanced. For instance, adaptive FDI schemes havebeen developed aiming at an identification of the corneringstiffness [10], [11]. The robust control theory based handlingof model uncertainties marks another research effort in thisfield. Additionally norm-based robust FDI schemes withadaptive thresholds have been reported [12]. Since the modeluncertainties are only temporarily dominant, the norm-basedFDI schemes that are in some sense a worst-case handling ofmodel uncertainties seem to be too conservative to enhancethe fault detectability. For this reason, there exist needs for analternative way for handling of uncertainties. In this paper,an FDI scheme will be presented, which is developed bytaking into account the road bank angle as unknown inputwith significant influence on the driving dynamics in somedriving situations.

A. Vehicle Bicycle Model

There are a number of mathematical models for thedescription of vehicle lateral dynamics [13], [14], [15].In this study the well known bicycle model is used. Thedecision for using the bicycle model has been made basedon a compromise between the needed on-line computationand sufficient description of system dynamics. The primaryassumptions for this model (Fig. 2) are:

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

WB2.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 1409

1) The center of gravity is on road elevation and the caris not rolling.

2) The lateral force is proportional to the tire slip angle,i.e. the cornering stiffness is constant

3) The tires of each axle are combined to one single tire.

v

CG

M

x0

vsp

�

β + ψv2/�

Fig. 2. Kinematics of the bicycle model

On the assumptions mentioned the vehicle bicycle modelis described by

[βr

]=

[YβKφR

mvYrKφR

mv − 1Nβ

Iz

Nr

Iz

] [βr

]

+

[c,

αV KφR

mvlV c,

αV

Iz

]δ∗L +

[ − gv

0

]sin αx (1)

with the notations

Yβ = − (c,αV + cαH) , Yr = (lHcαH − lV c,

αV ) /v

Nβ = lHcαH − lV c,αV , Nr =

(l2V c,

αV − l2HcαH

)/v.

Symbols used in bicycle model

v Longitudinal speeday Lateral accelerationβ Slip angleψ Yaw angler Yaw rate (ψ)αx Road bank angleδ∗L Adopted steering angle (δL/iL)iL Steering transmission ratiog Gravity constantc′αV , cαH Cornering stiffness (front, rear)lV , lH Distance front/rear axle - center of gravityIz Moment of inertia about the z-axism MassKφR Roll stiffness� Distance center of gravity to center of curvature

Following the relationship

KφRay = v(β + r

)+ g sin αx

we get two sensor models for ay, r given by:

ay =[

Yβ

m

Yr

m

] [βr

]+

c′αV

mδ∗L, (2)

r =[

0 1] [

βr

]. (3)

B. FDI system structure and basic ideas

A model-based FDI system consists of two major units:a residual generator and a residual evaluator. Consideringthe demand for a modular system structure, in our studyresidual generation is realized in two parallel running re-sidual generators, as sketched in Fig. 3. Note that thesetwo residual generators are independent in the sense thateach residual generator is driven by δ∗L and one sensor only.Indeed, this structure is the so-called Generalized ObserverScheme (GOS) [16]. As a result, a structural fault isolationis guaranteed.

Remember that our major focus is on the FDI withconsideration of the road bank angle whose influence onthe system dynamics and on the sensor signals is shownin (1). It is evident that a full decoupling of the generatedresidual signals R1,ay

, R2,r in Fig. 3 from the influence ofthe road bank angle is not possible. In order to reduce thefalse alarm rate caused by the unknown road bank angle, anatural way is to establish a threshold. In our early study,it has been found out that in the context of robust FDI thethreshold should be set very high such that false alarms couldbe prevented in most of the driving maneuvers. As a result,the fault detectability may significantly decrease.

In this paper, an alternative solution is presented aiming atensuring a low false alarm rate on the one side and no loss offault detectability on the other side. Therefore, instead of ahigh threshold, an indicator based on a simple logic is built,which is used to identify the presence of the road bank angle.In case of a large αx, the FDI function will be stopped. Theapplied logic is shown in Fig. 4. The basic idea behind thisscheme is that a large αx will cause considerable changesin both of the residual signals R1,ay

, R2,r (see Fig. 3 andeq.(1)). Thus, on the assumption of no simultaneous faultsthe changes in R1,ay , R2,r indicate the road bank angle.Note that in practice the driving situations with a large roadbank angle are only met temporally and it takes generallyvery short time, thus the application of the above-describedindicator is much more effective than a threshold that shouldbe set high for all driving situations.

residual

generator I

residual

generator II

ä*

äL

ä*

äL

ay

r

residual generation

R1,ay

R2,r

Fig. 3. Residual generation in GOS structure

In the sequent section, the development of the FDI systemshown in Fig. 3-4 will be described in details.

C. Parity space based residual generation

Before we start with the development of the FDI system, abrief overview on the parity space approaches is given in thissub-section. Parity space approaches are a well established

1410

threshold comparison

R1,ay

R2,r

|R1,ay

|>Jth,1

|R2,r

|>Jth,2

decision logic

&

&

alarm ay

alarm r

Fay

Fr

residual evaluation

Fig. 4. Residual evaluation

tool for the design of FDI systems [16], [17], [18], [19].A significant advantage of the parity space methods is thatonly computation of some well-defined algebraic equationsis involved for the FDI system design. This is also the reasonwhy this scheme is adopted here for the residual generationpurpose.

Consider the linear discrete time-invariant system

x(k + 1) = Ax(k) + Bu(k) + Edd(k) + Eff(k) (4)

y(k) = Cx(k) + Du(k) + Fdd(k) + Fff(k)

with x(k) ∈ Rn denotes the state vector, u(k) ∈ R

nu theinput vector, y(k) ∈ R

ny the output vector, d(k) ∈ Rnd the

unknown input vector and f(k) ∈ Rnf the vector of faults to

be detected. A, B, C, D, Ed, Ef , Fd, Ff are known matriceswith appropriate dimensions.

A parity relation based residual generator is expressed by

R(k) = vT

p(ys(k) − Hu,sus(k)) (5)

with the so-called parity vector vTp = [vT

p,0, · · · , vTp,s] ∈ Ps

where Ps = {vTp|vT

pH0,s = 0} is the parity space and s iscalled the order of the parity vector. The dynamics of theresidual generator (5) is governed by

R(k) = vT

p(H0,sx(k − s) + Hd,sds(k) + Hf,sfs(k)) (6)

with

Hu,s =

⎡⎢⎢⎢⎣

D 0 . . . 0CB D . . . 0

......

. . ....

CAs−1B CAs−2B . . . D

⎤⎥⎥⎥⎦ ,

us(k) =[

uT(k − s) . . . uT(k)]T

,

ys(k) =[

yT(k − s) . . . yT(k)]T

,

Hd,s =

⎡⎢⎢⎢⎣

Fd 0 . . . 0CEd Fd . . . 0

......

. . ....

CAs−1Ed CAs−2Ed . . . Fd

⎤⎥⎥⎥⎦ ,

ds(k) =[

dT(k − s) . . . dT(k)]T

,

Hf,s =

⎡⎢⎢⎢⎣

Ff 0 . . . 0CEf Ff . . . 0

......

. . ....

CAs−1Ef CAs−2Ef . . . Ff

⎤⎥⎥⎥⎦ ,

fs(k) =[

fT(k − s) . . . fT(k)]T

(7)

III. FDI SCHEME DEVELOPMENT

In this section, the realization of the FDI scheme sketchedin Fig. 3-4 will be described.

A. Re-modeling

The first step to achieve the objectives formulated at thebeginning is a re-modeling which includes a state spacetransformation. Note that in model (1) the system matrixis a function of velocity v. In order to design a residualgenerator whose dynamics are independent from v, themodel (1) will first be transformed into observer canonicalform. Consider that each observer integrated in the FDIsystem will be driven by one output with respect to faultisolation, two different state transformations correspondingto different output signals are carried out as described below.State transformation I with ay as output

zI = T1(v)[

βr

], T1(v) =

[Qβ,r

mIz

−Yβ

mIzYβ

mYr

m

]

Qβ,r = NβYr − YβNr

State transformation II with r as output

zII = T2

[βr

], T2 =

[ Nβ

Iz

−YβKφR

mv

0 1

]

The state transformations lead, after a straightforward calcu-lation, to

zI = A1zI + B1

[ay

δ∗L

]+ E1,αx sin αx

ay = c1zI + d1δ∗L

A1 =[

0 −Nβ

Iz

1 0

], E1,αx = T1

[ − gv

0

]

B1 =

[KφRQβ,r

Izmv −YβlV c,αV

mIzYβKφR

mv + Nr

Iz

c,αV (lV Yr−Nr)

mIz

]

c1 =[

0 1], d1 =

c,αV

m

as well as

zII = A2zII + B2

[rδ∗L

]+ E2,αx

sin αx

r = c2zII

A2 =[

0 −Nβ

Iz

1 0

], E2,αx

= T2

[ − gv

0

]

B2 =

[KφRQβ,r

Izmv

KφRc,αV (Nβ−YβlV )

IzmvYβKφR

mv + Nr

Iz

lV c,αV

Iz

]

c2 =[

0 1].

For the on-line implementation, the above two transformedmodels are now discretised by means of zero-order-hold witha sampling time dt = 10ms:Discrete-time model I

zI(k + 1) = A1zI(k) + B1

[ay(k)δ∗L(k)

]+ E1,αx

sin αx(k)

ay(k) = c1zI(k) + d1δ∗L(k) (8)

1411

Discrete-time model II

zII(k + 1) = A2zII(k) + B2

[r(k)δ∗L(k)

]+ E2,αx sin αx

r(k) = c2zII(k) (9)

B. Residual generation

Given system models (8) and (9), residual generationusing the Parity space method described in the last sec-tion seems trivial. A well-known method to improve theFDI performance which makes use of the additional designfreedom available for s > n, where s and n stand forthe order of the Parity vector and the order of the systemrespectively, is to make the residual signal as robust aspossible to the disturbance and simultaneously to ensure themaximal sensitivity to the fault [16], [17], [18]. By applyingthis method to systems (8) and (9), we met the followingproblems. First, the noises in the residual signals are verystrong. The reason is that the major difference between theinfluences of the disturbance αx and the both sensor faultslies in the high frequency range. As demonstrated in [19], inthis case the optimal Parity vectors achieved by solving thewell-known min-max problem (minimizing the influence ofthe unknown inputs and maximizing the effects of the faults)work like a high pass. The second problem is the high falsealarm rate. Remember that the core of our FDI scheme isthe establishment of an indicator for αx, which is built onthe logic that if both R1,ay , R2,r are large then FDI willbe stopped due to a large αx. However, the high robustnessagainst the unknown disturbance αx lead to the situationthat the influences of αx on residual signals R1,ay

, R2,r

may considerably differ during some driving maneuvers andthus increase the false alarm rate. For these two reasons,an alternative optimization criterion is needed. In our study,the Parity vectors are selected by solving an optimizationproblem which should lead to a minimization of the influenceof noise and at the same time suitable sensitivity to theunknown disturbance αx.Detection of ay-faultsIt follows form (5) and (6) that the residual generator is givenby

R1,ay(k) = vT

1,p(y1,s(k) − H1,u,su1,s(k))

with the dynamics

R1,ay(k) = vT

1,p(H1,αx,sds(k) + H1,ay,sfay,s(k))

where u1,s(k) =[

ay(k) δ∗L(k)]T

, ds(k) = [sinαx(k −s), · · · , sin αx(k)] and fay,s(k) = [fay

(k − s), · · · , fay(k)].

To determine the parity vector vT1,p we first have to

describe the effects of the unknown input and the consideredfault. The effect of the unknown input is given by E1,αx of(8) and the effect of the fault is given by −E1,ay

which isthe left column of B1 in (8):

1dt

E1,αx= −

[gQβ,r

IzmvgYβ

mv

]

1dt

E1,ay= −

[KφRQβ,r

IzmvKφRYβ

mv + Nr

Izv

]. (10)

With (7) we have for s = 4:

H1,αx,s =

⎡⎢⎢⎢⎣

0 0 · · · 0c1E1,αx

0 · · · 0...

.... . .

...c1A

3E1,αx c1A2E1,αx · · · 0

⎤⎥⎥⎥⎦ ,

H1,ay,s =

⎡⎢⎢⎢⎣

1 0 · · · 0c1E1,ay

1 · · · 0...

.... . .

...c1A

3E1,ayc1A

2E1,ay· · · 1

⎤⎥⎥⎥⎦ . (11)

Note that the sensor signal ay can be written as

ay = Eay + εay + fay

with εay denoting the measurement noise, Eay the mean ofay and fay

the sensor fault. Thus, the distribution matrix ofthe measurement noise εay

is identical with the one of thesensor fault. As noted above, the Parity vectors should beselected in such a way that the influence of the noise onthe residual signal is minimized and at the same time thesensitivity to the unknown disturbance αx is ensured. It isreasonable to define the following performance index J1

minwT

1,p

J1 = minwT

1,p

wT1,pNb,sH1,ay,sH

T1,ay,sN

T

b,sw1,p

wT1,pNb,sH1,αx,sHT

1,αx,sNT

b,sw1,p(12)

With Nb,s the basis matrix of the parity space Ps, vT1,p =

wT1,pNb,s and J1,min = λ1,min, problem (12) can be solved

by a simple eigenvalue/-vector calculation [18]

(Nb,sH1,ay,sHT

1,ay,sNT

b,s

− λ1Nb,sH1,αx,sHT

1,αx,sNT

b,s)w1,p = 0 (13)

to get the desired parity vector v1,p.Detection of r-faults

Analog to R1,ay (k) with A2, B2, c2, d2 = 0 we get theresidual generator and his dynamics

R2,r(k) = vT

2,p(y2,s(k) − H2,u,su2,s(k))= vT

2,pH2,αx,sds(k) + vT

2,pH1,ay,sfr,s(k) (14)

with u2,s(k) =[

r(k) δ∗L(k)]T

and fr,s(k) = [fr(k −s), · · · , fr(k)].

The effect of the unknown input is given by E2,αx of (9),the effect of the fault is given by −E2,r the left column ofB2 in (9):

1dt

E2,αx= −

[ gNβ

Izv

0

]

1dt

E2,r = −[

KφRQβ,r

IzmvKφRYβ

mv + Nr

Izv

](15)

And analogous to (12) we get the optimization index

minwT

2,p

J2 = minwT

2,p

wT2,pNb,sH2,r,sH

T2,r,sN

T

b,sw2,p

wT2,pNb,sH2,αx,sHT

2,αx,sNT

b,sw2,p(16)

which leads to the desired parity vector vT2,p = wT

2,pNb,s.The resultant parity vectors vT

1,p, vT2,p are a function of the

velocity, but both can be approximated by constant vectorsfor the relevant interval of the velocity (v � 20km/h) withsufficient precision.

1412

C. Residual evaluation

The scheme for the residual evaluation and decision logiccan be seen in Fig. 3 and 4. For the first residual we caneasily define a threshold

Jth,1(v) = |vT

1,pH1,ay,s(v)fay,s,min| (17)

with fay,s,min = [fay,min, · · · , fay,min] denoting the (given)minimum detectable fault defined according to the technicalrequirements.

The threshold for the second residual will be chosen withrespect to the minimum false alarm rate as follows. For thefault free case without disturbances, except the unknowninput, the residuals R1,ay

(k), R2,r(k) are given by

R1,ay(k) = vT

1,pH1,αx,s(v)ds(k)R2,r(k) = vT

2,pH2,αx,s(v)ds(k)

with ds(k) = [sinαx(k − s), · · · , sin αx(k)]. From this wecan get the ratio between the residuals as a function of thevelocity

p(v) =∣∣∣∣R1,ay (k)R2,r(k)

∣∣∣∣ =

∣∣∣∣∣vT1,pH1,αx,s(v)ds(k)

vT2,pH2,αx,s(v)ds(k)

∣∣∣∣∣With αx(k) constant for the interval [(k − s) k] this leadsto the relation between the two thresholds Jth,1(v), Jth,2(v).Thus we can obtain a threshold

Jth,2(v) = Jth,1(v)1

p(v)(18)

that leads, in combination with the decision logic (TABLE I),to a well performing FDI system in sense of avoiding falsealarms caused by the unknown input. As a result the residualswill exceed the according threshold at the same time in caseof an existing unknown input.

Using the defined thresholds

Fay = |R1,ay (k)| > Jth,1(v)Fr = |R2,r(k)| > Jth,2(v)

and the decision logic in TABLE I we get two alarm signals,one for faults in the lateral acceleration sensor (alarm ay)and another one for the yaw rate sensor (alarm r).

alarmFay Fr ay r

0 0 0 00 1 0 11 0 1 01 1 0 0

TABLE I

DECISION LOGIC

IV. IMPLEMENTATION AND TEST RESULTS

In this section the implementation of the FDI approach ispresented. As mentioned before the Parity Space Approachonly needs knowledge of algebraic operations for the designprocedure, whereas for the design of an observer basedFDI system knowledge in advanced control theory would

be necessary. On the other hand the Parity Space realizationcauses high amount of on-line computation. Hence for theimplementation stage we will transform the residual gene-rator into an observer form using the known one-to-onetransformation [20].

A. Conversion From Parity Space To Observer

The main goal of the conversion of the parity spacesolution into an observer structure is a reduced amount of on-line computation. As shown in the literature [20], a residualgenerator in observer form given by

z(k + 1) = Gz(k) + Hu(k) + Ly(k)r(k) = −wz(k) + vy(k) + qu(k) (19)

can be realized based on a one-to-one transformationbetween parity space and observer. According to (19)T,L,G, v, w are defined as

T =

⎡⎢⎢⎢⎣

vTp,1 vT

p,2 · · · vTp,s−1 vT

p,s

vTp,2 · · · · · · vT

p,s 0...

. . .. . .

......

vTp,s 0 · · · · · · 0

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

CCA

...CAs−1

⎤⎥⎥⎥⎦

L0 =−

⎡⎢⎢⎢⎣

vTp,0

vTp,1...

vTp,s−1

⎤⎥⎥⎥⎦, G0 =

⎡⎢⎢⎢⎢⎢⎣

0 0 · · · 01 0 · · · 0...

. . .. . .

...0 · · · 1 00 · · · 0 1

⎤⎥⎥⎥⎥⎥⎦∈R

s×(s−1)

G = [G0 g], L = L0 − vT

p,sg, g = 0 ∈ Rs×1

w =[

0 · · · 0 1] ∈ R

1×s, v = vT

p,s. (20)

Following the Luenberger conditions the matrix H = TB −LD and the vector q = −vD can be determined directly.It can easily be seen that the system matrix G represents adead beat observer i.e. its eigenvalues are all placed in theorigin of the z-plane.

B. Results

In this section first results based on real driving data arepresented. The order of the implemented observer has beenset to s = 4. In order to further reduce noise and thereforeto improve the residual signal performance, simple low passfilters have been implemented additionally. Fig. 5-8 shownthe test results for different situations, where the first diagramof each figure is related to residual and threshold of residualgenerator I and the second of generator II, respectively. Thethird diagram sketches the vehicle velocity.

figure fault value time5 - -6 - -7 fay 2m/s2 > 40s8 fr 3.5/s > 40s

TABLE II

ADDED FAULTS

1413

Fig. 5 shows the result, where the road is banked in timespaces ≈ 5s − 55s and ≈ 75s − 95s. It is evident that,referring to the decision logic (TABLE I), no fault is detectedexcept some small peaks. The following figures are showingthe test results driving on public road, fault free case (Fig. 6),occurrence of lat. acc. fault (Fig. 7) and yaw rate fault (Fig.8) respectively. The added faults are described in TABLE II.

0 10 20 30 40 50 60 70 80 90 1000

0.02

0.04

0.06

0.08

resid

ual +

thre

shho

ld

0 10 20 30 40 50 60 70 80 90 1000

1

2

3x 10

3

resid

ual +

thre

shho

ld

0 10 20 30 40 50 60 70 80 90 10026

26.5

27

27.5

time [s]

veloc

ity

R_{1,ay}J_{th,1}

R_{2,r}J_{th,2}

Fig. 5. driving on temporarily banked road (no sensor fault)

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

time [s]

resid

ual +

thre

shho

ld R_{1,ay}J_{th,1}

0 20 40 60 80 100 1200

1

2

3x 10

3

time [s]

resid

ual +

thre

shho

ld R_{2,r}J_{th,2}

0 20 40 60 80 100 1205

10

15

20

25

time [s]

veloc

ity

Fig. 6. driving on public road (no sensor fault)

V. CONCLUSION

In this contribution, an alternative model-based FDIsystem design strategy with respect to the problem ofhandling model uncertainties has been presented. The FDIsystem is based on the well known bicycle model, which issuitable for on-line implementation. Based on the definedindicator for model uncertainties during design, a systemproviding a low false alarm rate without decreasing faultdetectability has been realized. First test results using datafrom real driving maneuver have been presented, which

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

time [s]

resid

ual +

thre

shho

ld R_{1,ay}J_{th,1}

0 20 40 60 80 100 1200

1

2

3x 10

3

time [s]

resid

ual +

thre

shho

ld R_{2,r}J_{th,2}

0 20 40 60 80 100 1205

10

15

20

25

time [s]

veloc

ity

Fig. 7. driving on public road (ay-sensor fault)

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

time [s]

resid

ual +

thre

shho

ld R_{1,ay}J_{th,1}

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

time [s]

resid

ual +

thre

shho

ld R_{2,r}J_{th,2}

0 20 40 60 80 100 1205

10

15

20

25

time [s]

velo

city

Fig. 8. driving on public road (r-sensor fault)

demonstrate good FDI-performance.

Acknowledgment : This work was in part supported by theEuropean Commission under grant IST-2001-32122, IFATIS.

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