Robust Thruster Fault Diagnosis: Application to the rendezvous phase of the Mars Sample Return mission

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 Aeronautics 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 presence 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 Pade approximation. The resulted method is able to detect quickly any kind of thruster faults and to isolate them using a cross-correlation based test. Simulation 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 modulator unit, inaccurate navigation unit, and spatial disturbances (i.e. J2 gravitational perturbation, atmospheric drag, and solar radiation pressure).

Full text

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) + But−
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) = Wy(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... lnpT(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)
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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... mnT
P=l◦
1... l◦
nplnp+1... lnT
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+z23/2(a+x) + Fx
m
¨y=n2y−2n˙x−
µ
(a+x)2+y2+z23/2y+Fy
m(27)
¨z=−
µ
(a+x)2+y2+z23/2z+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) + BRˆ
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) + BRˆ
Qtgt (t),ˆ
Qchs(t)MI−
Ψ
(t)uthrt−
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;
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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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