SATELLITE DYNAMICS TOOLBOX FOR PRELIMINARY DESIGN PHASE

Abstract

This paper presents the latest developments of the Satellite Dynamics Toolbox dedicated to the modeling of large flexible space structures. The satellite is considered as a flexible multi-body system with open or closed loop kinematics chains of flexible bodies. Each body (platform, reaction wheels, booms, antenna, solar panels, ...) is modeled as a substructure in which each sizing parameters can be declared as a varying or uncertain parameter. The whole model is thus fully compatible with the MATLAB robust control toolbox to perform sensitivity analysis and pointing performance budget at the preliminary design phase level. INTRODUCTION Attitude control synthesis during preliminary design phase of large spacecraft fitted with various flexible appendages, like large Earth observation satellites, requires accurate model to perform pointing performance budget in response to external and internal disturbances. The later ones, due for instance to local mechanisms like SADM (Solar Array Driving Mechanism) or unbalances in reaction wheel actuators, can excite the flexible modes of the solar panels and induce resonance phenomena in the whole spacecraft as consequence. 1 The resonance frequencies and magnitudes depend on the mass/inertia ratios between the spacecraft main body and the solar array. Finally this mass/inertia ratio depends also on the angular configuration of the solar array with respect to the main body and thus requires to simultaneously consider the 3 attitude degrees-of-freedom. From the modeling point of view, the challenge consists in deriving the simplest model required to capture the 3-axes dynamics interactions between: • the flexible modes of the overall spacecraft structure derived by sub-structured model approach and depending both on the angular configuration of the solar array and the attitude control system, • the local mechanisms with their associated disturbances, while taking into account the parametric uncertainties and variations. Such a model requires a multi-body/flexible-body system approach. Many modeling tools for multi-body systems are available nowadays. They are often based on EULER-LAGRANGE equations and they generally take into account non-linear terms to describe the rigid motion in most of the cases and more rarely to include the non-linear deformations of the flexible bodies. The effect * Prof, ISAE-SUAPERO, 10 av. Edouard 1 of parametric variations can then be captured using several linearizations of the non-linear model on a given grid in the parametric space, followed by an interpolation of the linear models. However this approach results time-consuming and the exact parametric dependency is lost. For space applications , the linear model assumption is commonly adopted during preliminary design phase and is valid considering the very small and slow motions for attitude control studies. The Satellite Dynamic Toolbox (SDT) was developed to efficiently derive an LPV (Linear Parameter Varying) model of the dynamic behavior of flexible multi-body systems with the objective to be fully compatible with the MATLAB robust control toolbox to perform worst-case analyses and control design.

Full text

(Preprint) AAS 20-161
SATELLITE DYNAMICS TOOLBOX FOR PRELIMINARY DESIGN
PHASE
Daniel Alazard∗
, Francesco Sanfedino†
This paper presents the latest developments of the Satellite Dynamics Toolbox
dedicated to the modeling of large flexible space structures. The satellite is con-
sidered as a flexible multi-body system with open or closed loop kinematics chains
of flexible bodies. Each body (platform, reaction wheels, booms, antenna, solar
panels, ...) is modeled as a substructure in which each sizing parameters can be
declared as a varying or uncertain parameter. The whole model is thus fully com-
patible with the MATLAB robust control toolbox to perform sensitivity analysis
and pointing performance budget at the preliminary design phase level.
INTRODUCTION
Attitude control synthesis during preliminary design phase of large spacecraft fitted with vari-
ous flexible appendages, like large Earth observation satellites, requires accurate model to perform
pointing performance budget in response to external and internal disturbances. The later ones, due
for instance to local mechanisms like SADM (Solar Array Driving Mechanism) or unbalances in
reaction wheel actuators, can excite the flexible modes of the solar panels and induce resonance
phenomena in the whole spacecraft as consequence.1The resonance frequencies and magnitudes
depend on the mass/inertia ratios between the spacecraft main body and the solar array. Finally this
mass/inertia ratio depends also on the angular configuration of the solar array with respect to the
main body and thus requires to simultaneously consider the 3 attitude degrees-of-freedom.
From the modeling point of view, the challenge consists in deriving the simplest model required
to capture the 3-axes dynamics interactions between:
•the flexible modes of the overall spacecraft structure derived by sub-structured model ap-
proach and depending both on the angular configuration of the solar array and the attitude
control system,
•the local mechanisms with their associated disturbances,
while taking into account the parametric uncertainties and variations.
Such a model requires a multi-body/flexible-body system approach. Many modeling tools for
multi-body systems are available nowadays. They are often based on EUL ER -L AG RA NG E equa-
tions and they generally take into account non-linear terms to describe the rigid motion in most of
the cases and more rarely to include the non-linear deformations of the flexible bodies. The effect
∗Prof, ISAE-SUAPERO, 10 av. Edouard Belin, 31055 Toulouse, France. daniel.alazard@isae.fr
†Ass. Prof., ISAE-SUAPERO, 10 av. Edouard Belin, 31055 Toulouse, France. francesco.sanfedino@isae.fr
1

of parametric variations can then be captured using several linearizations of the non-linear model
on a given grid in the parametric space, followed by an interpolation of the linear models. However
this approach results time-consuming and the exact parametric dependency is lost. For space ap-
plications, the linear model assumption is commonly adopted during preliminary design phase and
is valid considering the very small and slow motions for attitude control studies. The Satellite Dy-
namic Toolbox (SDT) was developed to efficiently derive an LPV (Linear Parameter Varying) model
of the dynamic behavior of flexible multi-body systems with the objective to be fully compatible
with the MATLA Brrobust control toolbox to perform worst-case analyses and control design.
Starting from the Satellite Dynamics Toolbox (SDT) User Guide,2this paper summarizes the
TITOP (Two-Input Two-Output Ports) framework used to synthesize LPV models for any configura-
tion of open or closed kinematic chain of flexible bodies. An overview on the developed MATL ABr
-SIMULINKrinterface is finally given together with an illustration of its features.
NOMENCLATURE
Rb= (O, xb,yb,zb): Main body (hub B) reference frame. Ois a reference point on the main
body. xb,yb,zbare unit vectors.
Ra= (P, xa,ya,za): Appendage Areference frame. Pis the appendage’s anchoring point on
the hub. xa,ya,zaare unit vectors.
B: Hub’s (B) centre of mass.
A: Appendage’s (A) centre of mass.
G: Overall assembly’s centre of mass B+A.
Pa/b : Direction cosine matrix of the rotation from frame Rbto frame Ra, that
contains the coordinates of vectors xa,ya,zaexpressed in frame Rb.
aP: Inertial acceleration (vector) of body Bat point P.
ω: Angular speed (vector) of Rbwith respect to the inertial frame.
¨xP: acceleration twist at point P:¨xP= [aT
P˙
ωT]T.
Fext : External forces (vector) applied to B.
Text,B : External torques (vector) applied to Bat point B.
FB/A: Force (vector) applied by Bon A.
TB/A,P : Torque (vector) applied by Bon Aat point P.
WB/A,P : Wrench applied by Bon Aat point P:WB/A,P = [FT
B/ATT
B/A,P ]T.
DB
B: Hub’s static dynamics model expressed at point B.
mB: Hub’s Bmass.
IB
B: Inertia tensor 3×3of Bat point B.
MA
P(s): Appendage’s dynamic model at point P.
mA: Appendage’s mass.
IA
A: Inertia tensor 3×3of Aat point A.
τP B : Kinematic model between points Pand B:
:τP B =13(∗−−→
P B)
03×313.
2

(∗−−→
P B): Skew symmetric matrix associated with vector −−→
P B: if −−→
P B =

x
y
z
Ri
then
h(∗−−→
P B)iRi
=

0−z y
z0−x
−y x 0
Ri
.
Na: Number of appendages.
If there is a flexible appendage, we add the following definitions:
N: Number of flexible modes.
η: Modal coordinates’ vector.
ωi:ith flexible mode’s angular frequency.
ξi:ith flexible mode’s damping ratio.
li,P :ith flexible mode’s vector of modal participations (1×6), expressed at point Pin the
frame Ra.
LP: Matrix N×6of the modal participation factors expressed at point P: (LP=
[lT
1,P ,lT
2,P ,· · · ,lT
N,P ]T).
If there is a revolute joint between the appendage and the hub, we add the following definitions:
ra: Unit vector along the revolute joint axis: ra= [xrayrazra]T
Ra.
¨
θ: Revolute joint’s angular acceleration.
Cm: Revolute joint’s torque.
General definitions:
−−→
AB : vector from point Ato point B.
[X]Ri:X(model, vector or tensor) expressed in frame Ri.
dv
dt |Ri= 0 : Derivative of vector vwith respect to frame Ri.
u∧v: Cross product of vector uwith vector v(u∧v= (∗u)v)
u.v: Scalar product of vector uwith vector v(u.v= [u]T
Ri[v]Ri,∀ Ri)
s: LA PL ACE’s variable.
1n: Identity matrix n×n.
0n×m: Zero matrix n×m.
AT: Transpose of A.
diag(ωi): Diagonal matrix N×N: diag(ωi)(i, i) = ωi,i= 1,· · · , N.
G−1I(s) : Inversion of channels numbered in the vector of indices Iin the square system G(s)
(see appendix 2).
GI,J(s) : Sub-system of G(s)from inputs indexed in vector Jto outputs indexed in vector I
BACKGROUND ON FLEXIBLE-MULTI-BODY SYSTEMS DYNAMICS
Summary of the SDT
The objective is to compute the inverse dynamic model, that is the relation between the 6 com-
ponents of the wrench Wext,B = [FT
ext TT
ext,B]Tof the resulting external force and torque ap-
plied to the satellite at its center of mass B(input) and the 6 components of the acceleration twist
3

¨xB= [aT
B˙
ωT]Tof point B(output). The satellite is assumed to be composed of a rigid main body
or hub Bwith one or several flexible appendage Aiconnected to the base at points Pi(see Fig. 1).
The model can also take into account a revolute joint at the point Pibetween the appendage and the
main body. Then the 6×6inverse dynamic model is augmented by a new channel corresponding to
the transfer between Ti, the torque applied around the i-th joint axis by a mechanism (for instance a
SADM) and the angular acceleration ¨
θiinside the i-th joint.
A2
P1
P2
B
B
¨
θ1
T1
−T1
CMG
RWA
~
Ωk
~
Ωj
¨
θj
A1
[¨xB]RbaB
˙
ωRb
[Wext]Rb=Fext
Text,B Rb
Rb
Ra1
Figure 1. Sketch of a Spacecraft composed of a rigid base and several dynamic appendages.
In the case of a single appendage A, the block diagram representation of this model is depicted
in the Figure 2 where (see2for a detailed nomenclature):
•[DB
B]Rb=mB1303×3
03×3[IB
B]Rb,
•MA
PRa(s) = 16
000xrayrazraDA
PRa(s)








16
0
0
0
xra
yra
zra








(Ais the center of mass
of appendage Aand [ra]Ra= [xra, yra, zra,]Tis a unit vector along the revolute joint axis
expressed in the appendage frame Ra),
•DA
PRa(s) = [DA
P]Ra−PN
i=1 lT
i,P li,P s2
s2+2ξiωis+ω2
i
(li,P ,ωi,ξiare, respectively: the modal
participation factors at point P, the frequencies and the damping ratios of the Nflexible
modes of body A),
•[DA
P]Ra= [τAP ]T
RamA1303×3
03×3[IA
A]Ra[τAP ]Ra(τAP is the kimematic model between
points Aand P),
•the notation MA
P−17
Ra(s) stands for the inversion of the 7-th channel in the system MA
PRa(s)
(see also appendix 2),
4

•Pa/b is the Direction Cosine Matrix (DCM) of the rotation from frame Rbto frame Ra.
[Wext,B]Rb[¨xB]Rb
DB
B−1
Rb
¨
θ
T
WB/A,P Ra
WB/A,BRb
[MA
P]−17
Ra(s)
DCM
[¨xP]Ra
−
+
"PT
a/b 03×3
03×3PT
a/b #
τT
P B τP,B
Pa/b 03×3
03×3Pa/b 
Figure 2. The block diagram representation of the inverse dynamic model
[MB+A
B]−1
Rb(s) of the system composed of a main body Bwith a flexible appendage
Aconnected to the main body at the point Pthrough a revolute joint. Cmis the
driving torque around the joint axis. ¨
θis the angular acceleration inside the joint.
Finally,
•the inverse dynamic model [MB+A
B]−1
Rb(s) of the composite B+Acan be transported to any
reference point Oof the main body B(then denoted [MB+A
O]−1
Rb(s)),
•the obtained model can be parameterized according to
σ4= tan θ
4
where θis the angular configuration (in radian) of the revolute joint3,4 (see also appendix
1). Such a model, denoted [MB+A
O]−1
Rb(s, σ4)is an LPV (Linear Paremetric Variant) model
valid for a whole revolution of the appendage (θ∈[−π, +π]or σ4∈[−1,1]) and is fully
compatible with the tools of the MATLA BrRobust Control Toolbox.
Summary of the TITOP model approach
The TITOP approach allows to extend the previous framework to any kind of open kinematics
chains (or tree-like structures) of flexible bodies, i.e.: systems where an intermediate body can be
flexible. In the approach described in the previous section, a flexible body is modeled by a singe port
model and thus can only be located at the ends of the chain (as the leaves of the tree-like structure).
The flexible substructure Aconnected to the parent substructure Pat the point Pand to the child
substructure Cat point Cis depicted in Fig. 3. The double-port or TITOP (Two-Input Two-Output
Port) model DA
P,C (s), proposed in,5, 6 is a linear dynamic model between 12 inputs:
5

•the 6 components in Raof the wrench WC/A,C =FC/A
TC/A,C applied by the child sub-
structure Cto Aat point C,
•the 6 components in Raof the acceleration twist ¨
xP=aP
˙
ωP(time-derivative of the twist)
of point P,
and 12 outputs:
•the 6 components in Raof the acceleration twist ¨
xC=aC
˙
ωCof point C,
•the 6 components in Raof the wrench WA/P,P =FA/P
TA/P,P applied by Ato the parent
substructure Pat point P,
and can be represented by the block-diagram depicted in Fig. 4.
P
P
C
¨
xP=aP
˙
ωP
WA/P,P =FA/P
TA/P,P 
¨
xC=aC
˙
ωCWC/A,C =FC/A
TC/A,C 
A
A
C
Ra
Undeformed link
at equilibrium
Figure 3. The flexible sub-structure Ain the structure.
¨
xC
WA/P,P
DA
P C (s)
¨
xP
WC/A,C
6
6
6
6
Figure 4. Block-diagram of the TITOP model DA
P,C (s) of A.
The state space representation of such a TITOP model DA
P,C (s) is detailed in5and requires data
(frequencies ωj, damping ratios ξj, modal participation factors lj,p at point P, modal shapes ϕj,C
6

¨
xC
WA/P,P
DA
P C (s)
¨
xP
WC/A,C
DP
∗P(s)
DC
C∗(s)
Figure 5. Connection of the substructures P,A,C.
at point Cof the flexible modes and the kinematic model τCP ) which can be directly extracted
from the NASTRANranalysis output file assuming the sub-structure Ais clamped at point Pand
free at point C. The TITOP model embeds, in the same minimal state-space realization, the direct
dynamic model (transfer from acceleration to force) of the substructure Aat the point Pand the
inverse dynamic model (transfer from force to acceleration) of Aat the point C.
The connection of Ato the parent substructure Pat point Pand the child substructure Cat point
Ccan be modeled just by feedback connections of the TITOP models of the various sub-structures
according to Fig. 5.
Furthermore, hDA
P,C i−1I(s)denotes the model where the channels numbered in the index vec-
tor Iare inverted (according to the procedure described in appendix 2). This channel inversion
operation is very useful to analyze the dynamics of the link Afor various boundary conditions:
•clamped at Pand clamped at C:
DA
P C −1[1:6] (s)
WC/A,C
¨
xC
¨
xPWA/P,P
•free at Pand free at C:
DA
P C −1[7:12] (s)
¨
xC
WC/A,C
WA/P,P ¨
xP
•free at Pand clamped at C:
DA
P C −1(s) ¨
xP
WC/A,C
¨
xC
WA/P,P
The analytical TITOP model of a beam was used in7to co-design the 3-axis attitude control laws
and the boom linking the deployable reflector to the main body of an Earth-observation satellite. An
extension to NINOP (N-port model) was proposed in8, 9 to model a solar array composed of several
panels. Channel inversion operation can be used to model closed kinematics chain of flexible bodies
as it was highlighted in6on the four bar mechanism.
7

MATLABr/SIMULINKrLIBRARY
A MATL AB r/SIMULINKrlibrary of elementary components was developed to build the model
of a whole spacecraft using such a substructure approach (Figure 6, left). All the input ports and
outputs of the various blocks take into account the 6 degrees of freedom. The main basic blocks are:
•multi-port rigid body,
•analytical multi-port models for simple bodies like beams, plates, spinning wheels (see Figure
7),
•TITOP model for complex flexible bodies designed using NASTRAN/PATRAN. The names
of the .f06 and .bdf files and the node numbers corresponding to the connection ports can
be specified in the dialog box (see Figure 6, right).
•TITOP model of revolute joints,
•DCM associated to a rotation around a given axis and parameterized for a whole rotation
([−π, +π]).
Figure 6. The SDT library (left) and an example of dialog box (right).
yP
G
O
Ra
Ω
A
z
x
m, Iw, Irw
[¨xP]Ra
WA/.,P Ra
1-port model of a spinning wheel
P
C
l
s
ρ, E, ν, Iz, Iy, ξ
x
y
z
Ra
A
[¨xP]Ra
WA/.,P Ra
[¨xC]Ra
W./A,CRa
2-port model of a flexible beam
W./A,CiRa
A
[¨xCi]Ra
[¨xPi]Ra
WA/.,PiRa
Pi
Ci
ρ, E, ν
Ra
x
z
y
h
O
lx
ly
Multi-port model of a flexible plate
Figure 7. Analytical models of simple bodies.
All the parameters specified in the dialog boxes can be declared as varying parameters (using
ureal syntax) in such a way that an LPV of the whole system can be obtained thank to the function
ulinearize. Then robust control design and parametric sensitivity analysis can be performed.
8

CONTROL/MECHANICAL CO-DESIGN WITH TITOP APPROACH: AN ILLUSTRATION
This example shows how to perform in TITOP framework the co-design of a 3-axis attitude
controller and of a 2-part boom which links a large deployed reflector to the main body of a
large Earth observation satellite (see Figure 8 and7for more details). The TITOP models of the
2 uniform booms can be parameterized according to the thickness tand the perimeter pof the
square cross-section. Thus, these 2 sizing parameters are included in the set of the decision vari-
ables of the optimal control problem. The 3-axis attitude controller is structured and decentralized:
K=diag(Kx(s), Ky(s), Kz(s)). Finally the angular configuration θof the solar array SA w.r.t.
to the main body Bis considered as an uncertain parameter varying in the range [−π, +π]. The
co-design of the booms (tand p) and of the 3-axis attitude controller (K) can be expressed as an
optimization problem:
{b
K,b
t, bp}= arg min
K, t∈[t, t],p∈[p, p]ρ(l1+l2)t p
| {z }
boom mass
such that:
•max
θ∈[−π, π]kW1(s)TText,B →δ θ (s,K, t, p,θ)k∞≤1: worst-case disturbance rejection for a given
frequency-domain template 1/W1(s)on the transfer function from orbital disturbing torques
Text,B to the pointng error δθ,
•max
θ∈[−π, π]kW2(s)Tn→δ θm(s,K, t, p,θ)k∞≤1: worst-case stability margin for a given frequency-
domain template 1/W2(s)on the output sensitivity function.
[¨xB]Rb
Dboom2
C1C2(s,[t, p])
Dboom1
P1C1(s,[t, p])
DB
B−1
DSA
A(s)
03
Text,B Rb
Dantenna
C2(s)
P1
C1
C2
B
BSA
antenna
boom1
boom2
A
θ
r
R(r,tan θ
4)RT(r,tan θ
4)
RW(s)
SST(s)
n
Ky(s)
Kz(s)
Kx(s)
u
[˙
ω]Rbδθ
13
s2
δθm
Figure 8. The structured model of a large Earth observation flexible satellite for
attitude control and boom mechanical co-design.
CONCLUSION
The various tools presented in this paper and some tutorials can be downloaded at:
https://personnel.isae-supaero.fr/daniel-alazard/matlab-packages/
satellite-dynamics-toolbox.html.
Further developments will be focused on the non-linear terms in order to develop a very general tool
which can be used to model aerospace systems where non-linear terms cannot be neglected. In this
category some examples can be cited: spinning-satellites with flexible booms or helicopter rotor
9

with flexible blades (inclusion of centrifugal force terms), pointing systems embedded in strato-
spheric balloons (gravity terms to be taken into account).
ACKNOWLEDGMENT
We thank all the people that have contributed to the development of the Satellite Dynamics Tool-
box, more particularly CHRISTELLE CUMER, from ONERA, Toulouse, France.
APPENDIX 1: LPV MODEL ACCORDING TO THE ANGULAR CONFIGURATION
Let us define Pa+θ/a as the DCM between the appendage frame Ra+θafter a rotation of θaround
ra-axis and the initial appendage frame Ra, i.e. (for a given vector v):
[v]Ra=Pa+θ/a[v]Ra+θ
with:
•Pa+θ/a =Pr/a 

cos θ−sin θ0
sin θcos θ0
0 0 1 
PT
r/a,
•Pr/a =Pra

det(Pra)00
0 1 0
0 0 1 
,
•Pra= [ker([ra]T
Ra) [ra]Ra].
It is possible to define an LPV (Linear Parameter Varying) model of the system valid ∀θ∈
[−π, +π], using the new parameter σ4= tan θ
4∈[−1,1]. Indeed, the trigonometric relations:3
cos θ=(1 + σ2
4)2−8σ2
4
(1 + σ2
4)2,sin θ=4σ4(1 −σ2
4)
(1 + σ2
4)2
define the block diagram representation of Pa+θ/a depicted in Figure 9 where the varying parameter
σ4can be declared using the MATLA B function ureal of the Robust Control Toolbox. Then, the
MATLA B function ulinearize allows to compute an LFT representation of the system and gives
access to all the tools of the Robust Control Toolbox.
APPENDIX 2: CHANNEL INVERSION
Let us consider a square (same number of inputs and outputs) linear system G(s) = Dm×m+
Cm×n(s1n−An×n)−1Bn×mwith order nand mchannels (i.e. minputs and moutputs).
Single channel inversion: the system corresponding to the inversion of the i-th channel of the
system G(s) (i∈[1, m]) is denoted G−1i(s) and can be characterized by the following state-space
realization (equation (1)).
10

σ4
σ4
σ4
σ4
PT
r/a
Pr/a
Figure 9. The block diagram representation of Pa+θ/a parameterized with σ4= tan θ
4.
ui
y1
yi−1
yi+1
ym
G−1i(s)
.
.
.
.
.
.
u1
ui−1
yi
.
.
.
ui+1
um
.
.
.
Figure 10. Inversion of the i-th channel of G.
Let Jbe the vector of indices form 1to mwithout i:J= [1,· · · , i −1, i + 1,· · · , m]. It is assumed
that D(i, i)6= 0. Then let us denote fi=1
D(i,i)and define:
e
G−1i(s) ≡


A−fiB(:, i)C(i, :) [B(:,J)−fiB(:, i)D(i, J)fiB(:, i)]
C(J,:) −fiD(J, i)C(i, :)
−fiC(i, :)  D(J,J)−fiD(J, i)D(i, J)fiD(J, i)
−fiD(i, J)fi

.
In e
G−1i(s), the i-th inverted channel appears on the last channel, it is thus required to re-order the
channels using the vector of indices K= [1,· · · , i −1, m, i + 1,· · · , m −1] and then:
G−1i(s) = e
G−1i
K,K(s) (1)
Let uand ybe the input and output vectors of G, this inversion can be represented by the block
diagram depicted in Figure 10.
Multi channel inversion: Let Ibe the vector (with qcomponents) of indices corresponding to the
channels to be inverted. The successive inversion of the qchannels in G(s) is denoted:
G−1I(s) = hhG−1I(1) −1I(2) i···i−1I(q)(s) .(2)
11

REFERENCES
[1] C. Cumer, D. Alazard, A. Grynadier, and C. Pittet, “Codesign mechanics / attitude control for a simpli-
fied AOCS preliminary synthesis,” ESA GNC 2014 - 9th International ESA Conference on Guidance,
navigation & Control Systems, Porto, PT, 2014, pp. 1–9.
[2] D. Alazard and C. Cumer, “Satellite Dynamics Toolbox,” https://
personnel.isae-supaero.fr/daniel-alazard/matlab-packages/
satellite-dynamics-toolbox.html, 2011. Online.
[3] V. Dubanchet, Modeling and Control of a Flexible Space Robot to Capture a Tumbling Debris. PhD
thesis, Institut Sup´
erieur de l’A´
eronautique et de l’Espace, Polytechnique Montr´
eal, 2016.
[4] N. Guy, D. Alazard, C. Cumer, and C. Charbonnel, “Dynamic modeling and analysis of spacecraft with
variable tilt of flexible appendages,” Journal of Dynamic Systems Measurement and Control, Vol. 136,
January 2014, 10.1115/1.4025998.
[5] D. Alazard, J. A. P. Gonzalez, T. Loquen, and C. Cumer, “Two-input two-output port model
for mechanical systems,” Scitech 2015 - 53rd AIAA Aerospace Sciences Meeting, Kissim-
mee, US, 2015, pp. 1–18. Thanks to AIAA. The definitive version is available at
http://arc.aiaa.org/action/showPublications. The original PDF of the article can be found at ARC AIAA
Scitech web site : http://arc.aiaa.org/series/6.scitech, 6.2015-1778.
[6] J. Chebbi, V. Dubanchet, J. A. P. Gonzalez, and D. Alazard, “Linear dynamics of flexible
multibody systems : a system-based approach,” Multibody System Dynamics, Vol. 41, Septem-
ber 2017, pp. 75–100. Thanks to Springer editor. The definitive version is available at
http://www.springerlink.com The original PDF of the article can be found at Multibody System Dy-
namics website: http://link.springer.com/journal/11044, 10.1007/s11044-016-9559-y.
[7] H. Murali, D. Alazard, L. Massotti, F. Ankersen, and C. Toglia, “Mechanical-Attitude Controller Co-
design of Large Flexible Space Structures,” Advances in Aerospace Guidance, Navigation and Control
(J. Bordeneuve-Guib, A. Drouin, and C. Roos, eds.), pp. 659–678, Springer International Publishing,
2015, 10.1007/978-3-319-17518-8 38.
[8] F. Sanfedino, D. Alazard, V. Pommier-Budinger, F. Boquet, and A. Falcoz, “Dynamic modeling and
analysis of micro-vibration jitter of a spacecraft with solar arrays drive mechanism for control purposes,”
10th International ESA Conference on Guidance, Navigation & Control Systems (GNC 2017), Salzburg,
AT, 2017, pp. 1–15.
[9] F. Sanfedino, D. Alazard, V. Pommier-Budinger, A. Falcoz, and F. Boquet, “Finite element based N-
Port model for preliminary design of multibody systems,” Journal of Sound and Vibration, Vol. 415,
February 2018, pp. 128–146. Thanks to Elsevier editor. The original PDF of the article can be found at
http://www.sciencedirect.com/science/article/pii/S0022460X17307915, 10.1016/j.jsv.2017.11.021.
[10] N. Murdoch, D. Alazard, B. Knapmeyer-Endrun, N. A. Teanby, and R. Myhill, “Flexible Mode Mod-
elling of the InSight Lander and Consequences for the SEIS Instrument,” Space Science Reviews,
Vol. 214, December 2018, pp. 1–24, 10.1007/s11214-018-0553-y.
[11] J. A. P. Gonzalez, D. Alazard, T. Loquen, C. Pittet, and C. Cumer, “Flexible Multibody System Linear
Modeling for Control Using Component Modes Synthesis and Double-Port Approach,” Journal of Dy-
namic Systems, Measurement, and Control, Vol. 138, December 2016, p. 121004, 10.1115/1.4034149.
[12] F. Sanfedino, D. Alazard, V. Pommier-Budinger, F. Boquet, and A. Falcoz, “A novel dynamic model of
a reaction wheel assembly for high accuracy pointing space missions,” ASME 2018 Dynamic Systems
and Control Conference (DSCC2018), Atlanta, US, 2018, pp. 1–10, 10.1115/DSCC2018-8918.
[13] J. A. P. Gonzalez, C. Pittet, D. Alazard, and T. Loquen, “Integrated Control/Structure Design
of a Large Space Structure using Structured Hinfinity Control,” 20th IFAC Symposium on Auto-
matic Control in Aerospace (ACA 2016), Vol. 49, Shrerbrooke, CA, Elsevier, 2016, pp. 302–307,
10.1016/j.ifacol.2016.09.052.
[14] J. A. P. Gonzalez, T. Loquen, D. Alazard, and C. Pittet, “Linear Modeling of a Flexible Substructure
Actuated through Piezoelectric Components for Use in Integrated Control/Structure Design,” 20th IFAC
Symposium on Automatic Control in Aerospace (ACA 2016), Vol. 49, Sherbrooke, CA, Elsevier, 2016,
pp. 296–301, 10.1016/j.ifacol.2016.09.051.
[15] C. Cumer, D. Alazard, A. Grynadier, and C. Pittet, “Codesign mechanics / attitude control for a simpli-
fied AOCS preliminary synthesis,” ESA GNC 2014 - 9th International ESA Conference on Guidance,
navigation & Control Systems, Porto, PT, 2014, pp. 1–9.
[16] D. Alazard, C. Cumer, and K. Tantawi, “Linear dynamic modeling of spacecraft with various flexible
appendages and on-board angular momentums,” 2008.
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