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Robust Thruster Fault Diagnosis : Application

to the rendezvous phase of the Mars Sample

Return mission

Robert Fonod, David Henry, Catherine Charbonnel and Eric Bornschlegl

Abstract This paper addresses robust fault diagnosis of the chaser’s thrusters used

for the rendezvous phase of the Mars Sample Return (MSR) mission. The MSR

mission is a future exploration mission undertaken jointly by the National Aero-

nautics and Space Administration (NASA) and the European Space Agency (ESA).

The goal is to return tangible samples from Mars atmosphere and ground to Earth

for analysis. A residual-based scheme is proposed that is robust against the pres-

ence of unknown time-varying delays induced by the thruster modulator unit. The

proposed fault diagnosis design is based on Eigenstructure Assignment (EA) and

ﬁrst-order Pad´e approximation. The resulted method is able to detect quickly any

kind of thruster faults and to isolate them using a cross-correlation based test. Simu-

lation results from the MSR ”high-ﬁdelity” industrial simulator, provided by Thales

Alenia Space, demonstrate that the proposed method is able to detect and isolate

some thruster faults in a reasonable time, despite of delays in the thruster modu-

lator unit, inaccurate navigation unit, and spatial disturbances (i.e. J2gravitational

perturbation, atmospheric drag, and solar radiation pressure).

Robert Fonod

IMS laboratory, University of Bordeaux 1, 351 cours de la lib´eration, 33405 Talence, France

e-mail: robert.fonod@ims-bordeaux.fr

David Henry

IMS laboratory, University of Bordeaux 1, 351 cours de la lib´eration, 33405 Talence, France

e-mail: david.henry@ims-bordeaux.fr

Catherine Charbonnel

Thales Alenia Space, 100 Boulevard du Midi, 06156 Cannes La Bocca, France

e-mail: catherine.charbonnel@thalesaleniaspace.com

Eric Bornschlegl

European Space Research and Technology Centre, Keplerlaan 1, 2200 AG Noordwijk, Netherlands

e-mail: eric.bornschlegl@esa.int

1

Proceedings of the EuroGNC 2013, 2nd CEAS Specialist Conference

on Guidance, Navigation & Control, Delft University of Technology,

Delft, The Netherlands, April 10-12, 2013

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2 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

1 Introduction

Many space exploration missions require critical autonomous proximity operation.

Mission safety is usually guaranteed through a hierarchical implementation of the

fault diagnosis and fault tolerance with several levels of faults containments deﬁned

from local component/equipment up to global system, i.e. through various equip-

ments (sensors like IMUs, thrusters, reaction wheels etc..) redundancy paths and

ground intervention.

Classical Fault Detection Isolation and Recovery (FDIR) hierarchical implemen-

tation approach (see for instance [3, 15]) may be not sufﬁcient in dynamics deviation

in critical Space operations. This is specially the case for thruster faults during ren-

dezvous and docking/capture proximity operations, and this could lead to mission

loss. On-board robustness and fault tolerance/recovery shall prevail in the dynamics

trajectory conditions.

The objective of this study is to develop an advanced model-based Fault Detec-

tion and Isolation (FDI) scheme able to diagnose thrusters’ faults of the Mars Sam-

ple Return (MSR) chaser spacecraft, on-board/on-line and in time within the critical

dynamics and operations constraints of the last terminal translation (last 20m) of the

rendezvous/capture phase. As mission scenario undertaken, the chaser stays in the

rendezvous/capture corridor, such that it is possible to anticipate the necessary re-

covery actions to successfully meet the capture phase, see Fig. 1 for an illustration.

.

Fig. 1 Illustration of the rendezvous phase of the MSR mission

Numerous fault diagnosis methods are applicable to this problem [12, 13]. In

fact, most of the model-based diagnostic techniques reported in the literature have

the potential to be applied, see [2, 6, 9, 18] for good surveys. In recent years, some

effective techniques of the fault detection and diagnosis for satellite attitude control

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Robust Thruster Fault Diagnosis 3

systems based on inertial wheels have been developed, see for instance the books

[1, 10] and the references given therein. The problem of thrusters’ faults is less con-

sidered in the literature. Among the contributions, one can refer to [5] where an

Iterative Learning Observer (ILO) is designed to achieve estimation of time-varying

thruster faults. The method proposed in [16, 17] is based on the so-called unknown

input observer technique and is applied to the Mars Express mission. The work [11]

addressed the problem of thrusters’ faults diagnosis in the Microscope satellite and

[7] considered the problem of faults affecting the micro-Newton colloidal thrust

system of the LISA Pathﬁnder experiment. Both proposed FDI schemes are based

on H∞/H−ﬁlters to generate residuals robust against spatial disturbances (i.e. J2

disturbances, atmospheric drag and solar radiation), measurement noises and sen-

sor misalignment phenomena, whilst guaranteeing fault sensitivity performances.

Additionally, a Kalman-based projected observer scheme is considered in [7].

In this paper, the proposed FDI scheme consists of a residual generator that is

robust against unknown time-varying delays induced by the thruster modulator unit

and uncertainties on the thruster rise times. These uncertainties are transformed us-

ing Pad´e approximation to unknown inputs and decoupled by means of Eigenstruc-

ture Assignment (EA) technique. This detection scheme allows to detect quickly any

kind of thruster faults. The isolation task is solved using a cross-correlation based

test between the residual signal and the associated thruster open rate. For reduced

computational burdens, the isolation test is based on a sliding time window. The key

feature of the proposed method is the use of a judiciously chosen linear model for

the design of the FDI scheme, i.e. a model that consists of a 6-order model taking

into account both the rotational and linear translation of the spacecraft motions.

The paper is organized as follows: section 2 addresses some theoretical base-

ments. The goal is to develop a robust FDI scheme for linear systems with unknown

time-varying delays in the control input. It is shown that this problem can be solved

using the unknown input decoupling approach by means of EA technique. Section

3 is devoted to the application of the proposed method to the problem of fault de-

tection and isolation of the thrusters that equip the chaser spacecraft involved in the

MSR mission.

2 Problem Description and the Theoretical Foundation of the

Selected FDI Technique

Consider a continuous-time system given by

(˙

x(t) = Ax(t) + But−

τ

(t)+Eff(t)

y(t) = Cx(t)(1)

where x(t)∈Rnis the state vector, u(t)∈Rnuis the non-delayed system input

vector, y(t)∈Rnyis the vector of the available measurements and f(t)∈Rnfis the

fault vector. A,B,Cand Efare known matrices of appropriate dimensions. The

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4 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

pair (A,C)is assumed to be observable. The time-varying delay

τ

(t), induced by

the electronic devices, is assumed to be unknown but upper bounded

τ

(t)≤¯

τ

.

Problem 1. Design a residual generator that is robust in the presence of uncertain

time-varying delay

τ

(t).

In order to solve problem 1, a robust residual generator approach is presented in

this paper. The aim is to model the inﬂuence of the uncertain time-varying delay as

an unknown input. This will be done by using a ﬁrst-order Pad´e approximation and

introducing a new augmented state space description. Then, the unknown inputs are

decoupled by means of EA technique.

2.1 Pad´

e Approximation

The transfer function of the time delay is H(s) = e−

τ

(t)s. This transfer is irrational

and it is necessary to substitute e−

τ

(t)swith an approximation in form of a rational

transfer function. The most common approximation is the Pad´e approximation

e−

τ

(t)s.

=1−k1s+k2s2+...±knsp

1+k1s+k2s2+...+knsp(2)

where pis the order of the approximation and the coefﬁcients kiare functions of p.

In this paper, a ﬁrst-order Pad ´e approximation of the time-varying delay

τ

(t)is

used, when k1=

τ

(t)

2and ki=0, i=2,...,p, that is:

e−

τ

(t)s.

=1−

τ

(t)

2s

1+

τ

(t)

2s

(3)

Considering all system inputs, the transfer function (3) is equivalent with the fol-

lowing state space representation

(˙

xd(t) = Ad(t)xd(t) + Bdu(t)

u(t−

τ

(t)) = Cd(t)xd(t) + Ddu(t)(4)

where xd(t)∈Rnuis the delayed state, u(t−

τ

(t)) ∈Rnuis the delayed input, and

Ad(t) = −2

τ

(t)I,Bd=I,Cd(t) = 4

τ

(t)I,Dd=−Iare matrices with appropriate di-

mension. Furthermore, using (1) and (4), and introducing a new augmented state

vector of the form zT(t) = xT(t)xT

d(t), we obtain:

(˙

z(t) = ˆ

A(t)z(t) + ˆ

Bu(t) + ˆ

Eff(t)

y(t) = ˆ

Cz(t)(5)

where

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Robust Thruster Fault Diagnosis 5

ˆ

A(t) = A BCd(t)

0 Ad(t),ˆ

B=BDd

Bd,ˆ

C=C 0 ,ˆ

Ef=Ef

0

It can be seen, that thanks to the chosen state-space representation (4), the uncer-

tainty is present only in ˆ

A(t). The task is to decompose this matrix into the constant

and time-varying part and to model the uncertainty as an unknown input.

2.2 Expressing the Uncertainty as an Unknown Input

Problem 2. Decompose the matrix ˆ

A(t)in two parts:

ˆ

A(t) = ˆ

A0+

∆

ˆ

A(t)(6)

where ˆ

A0is a constant matrix and

∆

ˆ

A(t)is the time-varying part of ˆ

A(t).

Consider, that

τ

(t)can be expressed as

τ

(t) =

τ

0+

δ

(t):|

δ

(t)| ≤ ¯

δ

(7)

where

τ

0is the nominal delay,

δ

(t)is the variation around

τ

0, and ¯

δ

is the upper

bound of the variation part.

Proposition 1. Let a ∈Rand b ∈Rbe two real scalars, where a 6=0and a +b6=0,

then

(a+b)−1=a−1−a−1b

a+b(8)

Proof. Using some basic arithmetic operations, it can be shown, that (8) holds. ⊓⊔

Therefore, using proposition 1, we can write

1

τ

(t)=

τ

0+

δ

(t)−1=1

τ

0

−1

τ

0

δ

∗(t)(9)

where

δ

∗(t) =

δ

(t)

τ

0+

δ

(t). Problem 2 is solved using (9), that is

ˆ

A0="A BC

τ

0

d

0 A

τ

0

d#,

∆

ˆ

A(t) = "0−BC

τ

0

d

0−A

τ

0

d#

δ

∗(t)(10)

where A

τ

0

d=−2

τ

0Iand C

τ

0

d=4

τ

0I.

The time-varying part

∆

ˆ

A(t)can be expressed as an unknowninput d(t), entering

the augmented dynamics (5) through ˆ

Ed, by:

∆

ˆ

A(t)z(t) = 0−BC

τ

0

d

0−A

τ

0

d

δ

∗(t)z(t) = ˆ

Edd(t)(11)

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6 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

where

ˆ

Ed=−BC

τ

0

d

−A

τ

0

d,d(t) =

δ

∗(t)xd(t)(12)

Now, taking the above notation into account, the design model is expressed in terms

of unknown inputs as

(˙

z(t) = ˆ

A0z(t) + ˆ

Bu(t) + ˆ

Eff(t) + ˆ

Edd(t)

y(t) = ˆ

Cz(t)(13)

This model is the augmented representation of the original system (1), which takes

into account uncertainties caused by electronic-induced delays represented as an

additional unknown input d(t).

2.3 Residual Generator Design Using Eigenstructure Assignment

In order to solve problem 1, we deﬁne the following structure of the residual gener-

ator based on full-order observer (see e.g. [4, 14])

(˙

ze(t) = ( ˆ

A0−Lˆ

C)ze(t) + ˆ

Bu(t) + Ly(t)

r(t) = Wy(t)−ˆ

Cze(t)(14)

where r∈Rnpis the residual vector and ze(t)∈Rn+nuis the augmented state esti-

mation. The matrix W∈Rnp×nyis the residual weighting matrix.

The Laplace transformed residual response to faults and unknown inputs is

r(s) = Gr f (s)f(s) + Grd (s)d(s)(15)

where

Gr f (s) = Wˆ

C(sI−ˆ

A0+Lˆ

C)−1ˆ

Ef(16)

Grd (s) = Wˆ

C(sI−ˆ

A0+Lˆ

C)−1ˆ

Ed(17)

Once ˆ

Edis known, the remaining problem is to ﬁnd the matrices Land Wto sat-

isfy Grd (s) = 0. The assignment of the observer’s eigenvectors and eigenvalues is a

direct way to solve this design problem.

2.3.1 Unknown Input Decoupling by Assigning Left Eigenvectors

Lemma 1. The transfer function Grd (s)can be expanded in terms of the eigenstruc-

ture as

Grd (s) = H(sI−ˆ

Ac)−1ˆ

Ed=

n

∑

i=1

Fi

s−

λ

i

=

n

∑

i=1

HvilT

iˆ

Ed

s−

λ

i

(18)

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Robust Thruster Fault Diagnosis 7

where H=Wˆ

C,Fi=HvilT

iˆ

Ed,viand lT

iare the right and left eigenvectors of

ˆ

Ac=ˆ

A0−Lˆ

Cassociated with eigenvalue

λ

i.

Lemma 2. It is well known that, a given left eigenvector lT

iof ˆ

Acis always orthogo-

nal to the right eigenvectors vjcorresponding to the remaining (n−1)eigenvalues

λ

jof ˆ

Ac, where

λ

i6=

λ

j.

Theorem 1 (Chen and Patton, 1999). If Wˆ

Cˆ

Ed=0and all rows of the matrix H

are left eigenvectors of ˆ

Accorresponding to npeigenvalues of ˆ

Ac, then Grd (s) = 0.

Proof. If the rows of Hare npleft eigenvectors (li,i=1,...,np)of ˆ

Ac, i.e.

H=l1l2... lnpT(19)

then Hvi=0and Fi=0for i=np+1,...,n. If further we have Wˆ

Cˆ

Ed=Hˆ

Ed=0,

i.e. lT

iˆ

Ed=0and Fi=0for i=1,2,...,np, thus Grd (s) = 0.⊓⊔

The ﬁrst step for the design of an unknown input decoupled residual generator

(14) is to compute the weighting matrix Wwhich must satisfy the following neces-

sary condition [4]

Wˆ

Cˆ

Ed=Hˆ

Ed=0(20)

The necessary and sufﬁcient condition for solution (20) to exist is rank(ˆ

Cˆ

Ed)<ny.

If ˆ

Cˆ

Ed=0, any weighting matrix can satisfy this necessary condition. A general

solution is

W=W1(I−ˆ

Cˆ

Ed(ˆ

Cˆ

Ed)+)(21)

where W1∈Rnp×nyis an arbitrary matrix and (ˆ

Cˆ

Ed)+is the pseudo-inverse of

(ˆ

Cˆ

Ed), deﬁned as (ˆ

Cˆ

Ed)+= (( ˆ

Cˆ

Ed)T(ˆ

Cˆ

Ed))−1(ˆ

Cˆ

Ed)T.

The second step is to determine the eigenstructure of the observer. The rows of

Hmust be the npleft eigenvectors of ˆ

Ac. The remaining n−npleft eigenvectors can

be chosen without restraint. For the given (stable) eigenvalue spectrum

Λ

(ˆ

Ac) =

{

λ

i,i=1,...,n}, the following relation holds

lT

i(

λ

iI−ˆ

A0) = −lT

iLˆ

C=−mT

iˆ

C,i=1,...,n(22)

where mT

i=lT

iL. The assignability condition says, that for each

λ

i, the cor-

responding left eigenvector lT

ishould lie in the column subspace spanned by

{ˆ

C(

λ

iI−ˆ

A0)−1}, i.e. a vector miexists such that

lT

i=mT

iKi,i=1,...,np(23)

where Ki=−ˆ

C(

λ

iI−ˆ

A0)−1,i=1,...,np. The projection of liin the subspace

span{Ki}is denoted by:

l◦T

i=m◦T

iKi,i=1,...,np(24)

FrBT2.2

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8 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

where m◦T

i=lT

iKT

i(KiKT

i)−1,i=1,...,np. If lT

i=l◦T

i,lT

iis in span{Ki}and is

assignable. Otherwise, an approximative procedure must be considered in order to

replace lT

iby it’s projection l◦T

i.

The remaining n−npeigenvalues and corresponding eigenvectors can be chosen

freely from the assignable subspace and assigned using some EA technique, e.g.

using singular value decomposition (SVD). Then, the observer matrix Lcan be

computed as follows

L=P−1M(25)

where

M=m◦

1... m◦

npmnp+1... mnT

P=l◦

1... l◦

nplnp+1... lnT

It is obvious, that the ﬁrst npeigenvalues corresponding to the required eigenvectors

lT

i,i=1,...,npmust be real because all these eigenvectors are real-valued.

Remark 1. The remaining design freedom, after unknown input de-coupling, can be

used to optimize other performance indices such as fault sensitivity.

3 Application to the MSR Mission

The robust fault detection scheme presented in the above section is now considered

for the detection and isolation of the faults affecting the chaser’s thrusters unit.

3.1 Modeling the Chaser Dynamics During the Rendezvous Phase

In the interest of brevity, from [8, 19, 20, 21] we only consider the modeling of the

relative position of two spacecrafts on a circular orbit around the planet.

The motion of the chaser is derived from the 2nd Newton law. To proceed, let a,

m,Gand mMdenote the orbit of the target, the mass of the chaser, the gravitational

constant and the mass of the planet Mars. Then, the orbit of the rendezvous being

circular, the velocity of any object (e.g. the chaser and the target) is given by the

relation q

µ

awhere

µ

=G.mM. Let Rl:(Otgt ,−→

Xl,−→

Yl,−→

Zl)be the frame attached to

the target and oriented as shown in Fig. 2. Because the linear velocity of the target

is given by the relation a˙

θ

in the inertial frame Ri:(OM,−→

Xi,−→

Yi,−→

Yi)(see Fig. 2), it

follows:

a.˙

θ

=r

µ

a⇒n=r

µ

a3(26)

During the rendezvous phase, it is assumed that the chaser motion is due to the four

forces: Mars attraction force, centripetal force, Coriolis force and forces due to the

thrusters (Fx,Fy,Fz). Then, from the 2nd Newton law, it follows

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Robust Thruster Fault Diagnosis 9

¨x=n2x+2n˙y−

µ

(a+x)2+y2+z23/2(a+x) + Fx

m

¨y=n2y−2n˙x−

µ

(a+x)2+y2+z23/2y+Fy

m(27)

¨z=−

µ

(a+x)2+y2+z23/2z+Fz

m

where x,y,zdenote the three dimensional position of the chaser (assumed to be a

punctual mass) in Rl.

chaser

target

(sample container)

the rendezvous

orbit

a

−→

Xi

−→

Yi

−→

Xl

−→

Yl

−→

Zi=−→

Zl

θ

x

y

Fig. 2 The Mars rendezvous orbit and the associated frames

Because the distance between the target and the chaser is smaller than the orbit a,

it is possible to derive the so called Hill-Clohessy-Wiltshire (HCW) equations from

(27) by means of a ﬁrst order approximation. This boils down to a linear six order

state space model with the input vector u(t) = (FxFyFz)T, output vector y(t) =

(x y z)and state vector x(t) = (x y z ˙x˙y˙z)T, i.e. from (27) it follows

˙

x(t) = Ax(t) + BRˆ

Qtgt (t),ˆ

Qchs(t)Muthr (t) + Eww(t)

y(t) = Cx(t)

ym(t) = −y(t) + v(t)

(28)

where

A=

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 0 2n

0−n20 0 0 0

003n2−2n0 0

,B=Ew=1

m

0 0 0

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

,C=

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

(29)

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10 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

Further, ˆ

Qtgt (t)∈R4and ˆ

Qchs(t)∈R4denote the attitude’s quaternion of the target

and the chaser, respectively. These quaternions are estimates from the navigation

module (NAV). M∈R3×8refers to the thrusters’ conﬁguration (direction) matrix,

uthr (t) = (uthr1(t),..., ut hr8(t))T, 0 ≤uthri(t)≤1 , i=1,..., 8 are the thruster in-

puts, ym(t)∈R3is the three-dimensional position measured by a LIDAR unit that

is corrupted by the measurement noise v(t)∈R3and w(t)∈R3refers to spatial

disturbances. The quaternions dependent rotation matrix R(.)performs the projec-

tion of the three-dimensional thrust forces (due to the eight thrusters that equip the

chaser) from the chaser’s frame on to the target frame Rl. The numerical values of

the parameters are not shown for reasons of conﬁdentiality.

The considered thruster faults can be modeled in a multiplicative manner accord-

ing to (the index ” f” is used to outline the faulty case)

uf

thr (t) = I8−

Ψ

(t)uthr(t),

Ψ

(t) = diag

ψ

1(t),...,

ψ

8(t)(30)

where 0 ≤

ψ

i(t)≤1, i=1,...,8 are unknown.

Ψ

(t)models thruster faults, e.g. a

locked-in-placed fault can be modeled by

Ψ

i(t) = 1−c

uthri(t)where cdenotes a con-

stant value (the particular values c={0,1}allows to consider closed/open faults)

whereas a ﬁx value of

Ψ

i(t)models a loss of efﬁciency of the ith thruster.

During the rendezvous phase, the thruster management algorithm operates in

the 6DOF mode. It means, that both commanded torque and force are achieved by

thrusters only and thus the thruster faults affect the attitude of the chaser spacecraft.

Taking into account some unknown but bounded delays induced by the electronic

devices, and uncertainties on the thruster rise times due to the thruster modulator

unit that is modeled here as an unknown time-varying delay

τ

(t) =

τ

0+

δ

(t)with

a (constant) nominal delay

τ

0and upper bounded variation part |

δ

(t)| ≤ ¯

δ

. The

motion of the chaser during the rendezvous can be modeled in both fault free (i.e.

Ψ

(t) = 0) and faulty (i.e.

Ψ

(t)6=0) situations according to

(˙

x(t) = Ax(t) + BRˆ

Qtgt (t),ˆ

Qchs(t)MI−

Ψ

(t)uthrt−

τ

(t)+Eww(t)

y(t) = Cx(t) = −ym(t) + v(t)(31)

Now considering R(ˆ

Qtgt (t),ˆ

Qchs(t))Muthr (t)as the input vector u(t), and ap-

proximating the fault model −R(ˆ

Qtgt (t),ˆ

Qchs(t))M

Ψ

(t)uthr(t)in terms of addi-

tive faults f(t)∈R3acting on the state via a constant distribution matrix Ef(then

Ef=B), it follows that the overall model of the relative dynamics that takes into

account both, the attitude Qchs(t), and the relative position (x y z)of the chaser and

the target can be written in the form (1).

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Robust Thruster Fault Diagnosis 11

3.2 Design of the FDI Scheme

3.2.1 Design of the Residual Generator

To design a residual generator r(t), the above derived model (31) is used. It is con-

sidered that y(t) = −ym(t). The sampling period Tsof the NAV is 0.1sand a rea-

sonable value of the nominal time delay was determined to be exactly one sampling

period for the input vector u(t). By using Pad´e approximation of the time delay

τ

(t), the uncertainty caused by the unknown time-varying parameter

δ

(t), intro-

duced in (7), has been modeled as an unknown input d(t)entering the augmented

state space dynamic (13) through the matrix ˆ

Edcomputed as in (12), with

τ

0=0.1.

Following the discussion in section 2.3.1, the residual weighting matrix was deter-

mined to be W=I3, thus the dimension of the resulting residual is np=3, i.e.

r(t) = (r1(t),r2(t),r3(t))T. All the assigned eigenvalues were chosen to be close to

−0.5. Finally, the residual generator (14) is converted to discrete-time (t=kTs)us-

ing a Tustin approximation and implemented within the nonlinear simulator of the

MSR mission.

Remark 2. Since the spatial disturbances w(t)have the same directional properties

as the faults, i.e. Ew=Ef, the residual signal r(t)cannot be decoupled from w(t).

3.2.2 Decision Test and the Isolation Strategy

To make a decision about the fault presence, a simple threshold-based decision test

is applied to the residual norm ||r(k)||2as follows:

||r(k)||2<JT; fault-free

||r(k)||2≥JT; fault declared (32)

where JTis a ﬁxed threshold and || · || denote the Euclidean vector norm.

The proposed isolation strategy is based on the following cross-correlation crite-

rion between the jt h residual signal rjand the associated controlled thrusters open

rate uthri, i.e.:

σ

j(k) = arg

i

min

1

N

k

∑

l=k−N

rj(k)uthri(l),i=1...8,j∈ {1,2,3},∀k∈Z+(33)

This cross-correlation function is a statistical quantity that tries to ﬁnd the associ-

ated thruster index that has the smallest impact on the resulting residual signal. For

real-time reason, this criterion is computed on a N-length sliding-window. The re-

sulting index

σ

j(k)∈ {1,2,...,8}refers to the identiﬁed faulty thruster, using the jt h

residual signal. The decision about the identiﬁed faulty thruster can be considered

in three different ways, i.e.

σ

(k)is computed:

1. as the smallest cross-correlation among all the residuals;

FrBT2.2

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12 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

2. using only one residual signal rj(k),j∈ {1,2,3}, e.g. the ﬁrst

σ

(k) =

σ

1(k);

3. using a voting scheme where, for all residuals rj(k),j=1,2,3 a

σ

j(k)is com-

puted separately, and a majority voting rule is implemented, i.e. the resulting

index by the most

σ

j(k)is the identiﬁed faulty thruster.

A key feature of these isolation methods is that they are static and then, have a low

computational burdens.

To make a ﬁnal decision about the identiﬁed thruster index a conﬁrmation win-

dow Nc>1 is considered, i.e. the identiﬁed index is conﬁrmed at time instant k,

if:

σ

(k) =

σ

(k−1) = ... =

σ

(k−Ns+1)(34)

The whole FDI strategy works as follows: as soon as the fault is declared by the

decision test (32), the above described isolation strategy is executed.

Remark 3. It is obvious, that if the ith thruster is not used by the thruster management

unit, i.e. uthri=0, the minimum cross-correlation will result in

σ

(k) = i. This fact

must be taken into account, and the associated thruster rates below some predeﬁned

(small) threshold shall not be taken into account by the isolation strategy.

Remark 4. With the case of a number of thrusters greater than the DOF, some

thruster faults evolve in the same residual sub-space, therefore using sub-space iso-

lation approach, a full coverage of the isolation problematic cannot be guaranteed.

3.3 Simulation Results

As mentioned, the navigation unit is not considered to deliver “perfect” measure-

ments. Due to this fact, the quaternion estimates and the LIDAR signal are addi-

tionally corrupted by noise that we modeled (according to the industrial speciﬁca-

tions) as an uniform distributed noise. We consider delays between the navigation

module and the control block, delays induced by the thruster modulator unit and

spatial disturbances. The considered disturbances w(t)are solar radiations, J2grav-

itational perturbation and atmospheric drag. The simulated faults correspond to a

single thruster opening at 100% during the last 20m of the rendezvous. Analysis of

different fault scenarios are subject of the future research. The isolation strategy is

computed according to (33) using the majority voting rule. The window length Nc

for fault conﬁrmation is taken as 10 sampling instants. Note that the remark 3, stated

in section 3.2.2, was not considered in this simulation study.

Fig.3 illustrate the behavior of kr(t)k2, the decision test, conﬁrmation window

(green area) and the isolation criteria

σ

(k), for some faulty situations. For each

simulation, the fault occurs at t=1200sand is maintained one minute.

As it can be seen from the ﬁgures, all thruster faults are successfully detected and

isolated by the FDI unit with a reasonable detection and isolation time (see Table 1).

Note that such a strategy succeeds since both the rotational Qchs (t)and linear

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Robust Thruster Fault Diagnosis 13

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.1

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.2

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.3

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.4

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.5

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

Thr. No.

isolated thruster

detection signal

time in second

fault occurs in thruster No.6

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

8

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.7

mag

residual norm

1200 1201 1202 1203 1204 1205

0

2

4

6

8

Thr. No.

isolated thruster

decision signal

time in second

fault occurs in thruster No.8

Fig. 3 Behaviour of the residual norm kr(k)k2, isolation criteria

σ

(k), decision signal and conﬁr-

mation window for some faulty situations

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14 R. Fonod, D. Henry, C. Charbonnel and E. Bornschlegl

translation x(t)of the chaser motions have been considered. Thus, the effects that

faults have on both the chaser attitude and translation motion, are taken into account.

Table 1 Detection and isolation times in seconds

Thruster No. 12345678

Detection time 1.1s 0.7s 0.5s 0.4s 0.4s 0.5s 0.7s 1.1s

Isolation time 2.8s 1.8s 2.1s 2.1s 1.3s 4.0s 2.1s 2.1s

4 Concluding Remarks

This paper has proposed an EA-based fault diagnosis approach to detect and isolate

thruster faults subject to time-varying delays induced by the thruster modulator unit.

The resulted residual generator is robust against the uncertain time variations (ap-

proximated in terms of unknown inputs) around the nominal delay. The key feature

of the proposed method is the use of a judiciously chosen linear model for the design

of the residual generator, i.e. a model that takes into account both the rotational and

translation dynamics of the spacecraft. This allows to propose a fault diagnosis solu-

tion with reduced computational burdens, which is a prior condition for an on-board

implementation. Nonlinear simulations from the ”high-ﬁdelity” industrial simulator

show that despite the presence of measurement noises, delays in the thruster modu-

lator unit and spatial disturbances, the faults are successfully detected and isolated

in a reasonable time.

Acknowledgements This research work was supported by the European Space Agency (ESA)

and Thales Alenia Space in the frame of the ESA Networking/Partnering Initiative (NPI) Program.

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