Attitude Control for the LUMIO CubeSat in Deep Space

Abstract

The Lunar Meteoroid Impact Observer (LUMIO) is a 12U CubeSat designed to observe, quantify, and characterize the impact of meteoroids on the lunar surface. The combination of a highly demanding Concept of Operations (ConOps) and the characteristics of the deep-space environment determine the configuration of the spacecraft. This paper presents the preliminary Attitude Determination and Control System (ADCS) design of LUMIO, the reaction wheels desaturation strategy and Moon tracking control laws. The proposed solution is shown to (a) prevent the saturation of the reaction wheels, (b) minimize propellant consumption, (c) minimize the parasitic ∆V, (d) keep the pointing angle below a certain limit, and e) maximize power generation. Although no attempt is made to optimize the control parameters, the most efficient alternative in terms of propellant consumption is identified. The proposed LUMIO design could lay the foundations for a standardized minimum mass and volume ADCS system for CubeSats operating in deep-space.

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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.
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IAC-19,C1,6,4,x50894
ATTITUDE CONTROL FOR THE LUMIO CUBESAT IN DEEP SPACE
´
A. Romero-Calvo,1J. D. Biggs, 1F. Topputo1
The Lunar Meteoroid Impact Observer (LUMIO) is a 12U CubeSat designed to observe, quantify, and characterize the
impact of meteoroids on the lunar surface. The combination of a highly demanding Concept of Operations (ConOps)
and the characteristics of the deep-space environment determine the configuration of the spacecraft. This paper presents
the preliminary Attitude Determination and Control System (ADCS) design of LUMIO, the reaction wheels desaturation
strategy and Moon tracking control laws. The proposed solution is shown to (a) prevent the saturation of the reaction
wheels, (b) minimize propellant consumption, (c) minimize the parasitic ∆V, (d) keep the pointing angle below a certain
limit, and e) maximize power generation. Although no attempt is made to optimize the control parameters, the most
efficient alternative in terms of propellant consumption is identified. The proposed LUMIO design could lay the foundations
for a standardized minimum mass and volume ADCS system for CubeSats operating in deep-space.
Nomenclature
AActual DCM matrix
AdDesired DCM matrix
AeError DCM matrix
AiArea of surface i
CjJacobi constant
~cpi Surface i position vector
cSpeed of light
ˆ
~
dDisturbance estimation
~
FiSolar pressure force in panel i
γThrusters tilting angle
~
hd
rDesired angular momentum
~
hrAngular momentum of the reaction wheels
∧Inverse hat map
horbit Altitude of the spacecraft
ISolar irradiance
~
Itot Total impulse vector
JGeneric inertia matrix
Jdepl Inertia matrix for LUMIO’s deployed configuration
Jmin Minimization function
Jpack Inertia matrix for LUMIO’s packed configuration
kiControl parameters
lDistance between nozzles and Y-Zplane
mNumber of thrusters
mSC Mass of the spacecraft
NMatrix of thrust directions
1Department of Aerospace Science and Technology, Politecnico di
Milano, Via Giuseppe La Masa, 34, 20156, Milan, Italy;
alvaro.romero.calvo@gmail.com
~nsUnit vector normal to surface
Ω Aperture angle of a regular tetrahedron
Hadamard’s product
PSolar constant
RReaction wheels matrix
RMMoon radius
ρdDiffusely reflected radiation
ρsSpecularly reflected radiation
rDistance to the Sun
~
SUnit vector from Sun to surface
∗Pseudo-inverse operator
sValues after saturation
∆tMI B Minimum Impulse Bit
TTorque matrix
θPositive real number
~
TSRP Solar radiation pressure torque
~
tThrust vector
TTranspose operator
·Time derivative
~uRW Control momentum applied by the reaction wheels
~ucControl input
~udes Desired thrust torque
~ulim Thrust threshold
∆VSpacecraft velocity increment
VSkew-symmetric matrix or hat map
~ω Actual angular velocity
~ωdDesired angular velocity
~ωeAngular velocity error
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XBody frame axis
~xMNormalized Moon pointing vector (J2000)
~xSNormalized Sun pointing vector (J2000)
xDistance between nozzles and Xaxis
xiDCM unitary axis
YBody frame axis
Y0Tilted body frame axis
ZBody frame axis
Z0Tilted body frame axis
Acronyms
ADCS Attitude Determination and Control System.
CDR Concurrent Design Review.
CMG Control Moment Gyroscope.
CoM Center of Mass.
ConOps Concept of Operations.
DCM Direction Cosine Matrix.
ESA European Space Agency.
LUCE LUnar CubeSat for Exploration.
LUMIO Lunar Meteoroid Impact Observer.
MIB Minimum Impulse Bit.
RW Reaction Wheel.
SRP Solar Radiation Pressure.
TLO Top-Level Objectives.
1 Introduction
The last decade has witnessed a paradigm shift in the space
sector due to the popularization of nanosatellites. Their
appearance has democratized access to space, boosted the
development of miniaturized technologies and extended the
possibilities of distributed spacecraft architectures [1]. These
new capabilities, mainly tested in low-Earth orbits, have also
laid the foundations for the development of interplanetary
nanosatellite missions such as Mars Cube One [2].
CubeSats are a standardized class of nanosatellites ini-
tially conceived as educational tools or technology demon-
strators [3]. Their low cost, versatility and fast develop-
ment time have led to their employment for actual scien-
tific projects. Interplanetary missions may benefit from their
scalability, modularity and distributed architecture to obtain
redundant and more detailed scientific information [4]. Some
examples of recently proposed interplanetary CubeSats are
the University of Colorado’s Earth Escape Explorer (CU-
E3), the Cornell University Cislunar Explorers or the Fluid
& Reason-LLC Team Miles [5].
The Lunar Meteoroid Impact Observer (LUMIO) is a
12U CubeSat mission to observe, quantify, and characterize
the meteoroid impacts on the surface of the Moon by detect-
ing their flashes on the lunar far-side. This complements the
knowledge gathered by Earth-based observations of the lunar
nearside, thus synthesizing a global information on the lunar
meteoroid environment. LUMIO envisages a 12U CubeSat
form-factor placed in a halo orbit at Earth-Moon L2 to char-
acterize the lunar meteoroid flux by detecting the impact
flashes produced on the far-side of the Moon. The mission
employs the LUMIO-Cam, an optical instrument capable of
detecting light flashes in the visible spectrum [6]. LUMIO is
one of the two winners of ESA’s LUnar CubeSat for Explo-
ration (LUCE) SysNova competition, and as such it is being
considered by ESA for implementation in the near future.
One of the major challenges of the mission is the strict
pointing budget, which imposes high-precision tracking of
a specific attitude that maximizes power generation. This
is particularly relevant for the Attitude Determination and
Control System (ADCS) due to the limited capacity of
the reaction wheels. In addition, the de-tumbling and
de-saturation maneuvers are undertaken using only four
thrusters, which adds to the complexity of the control de-
sign.
This paper describes the attitude control strategy for the
LUMIO mission focusing on the configuration design of the
reaction wheels and thruster-based de-saturation. Due to
the tight constraint on the maximum momentum storage,
the placement of the reaction wheels significantly affects the
desaturation strategy and requires optimization. Different
desaturation strategies are presented which require an un-
conventional approach to their design due to the employment
of only four thrusters.
The work is organized as follows: Sec. 2 summarizes the
mission and its most relevant characteristics for the ADCS
subsystem, whose configuration is discussed in Sec. 3 and
control laws in Sec. 4. The performance of different reaction
wheel configurations and desaturation strategies is analyzed
in Sec. 5. Finally, the conclusions and potential future de-
velopments are presented in Sec. 6.
2 Mission Overview
2.1 Top-Level Objectives (TLO)
The LUMIO mission aims to characterize the flux, magni-
tude, luminous energy, and size of the meteoroids impacting
the lunar farside. This would help advance the understanding
of how meteoroids evolve in the cislunar space and comple-
ment the existing observations of the lunar nearside. From
the technological perspective, the mission wants to demon-
strate the deployment and autonomous operation of a Cube-
Sat in the lunar environment [7]. Those goals are summarized
in the TLO listed in Tab. 2.
2.2 Concept of Operations (ConOps)
In the Circular Restricted Three-Body Problem, the libration
points are at rest with respect to a frame co-rotating with
the smaller and larger primaries. Consequently, a halo orbit-
ing the Earth–Moon L2 always faces the lunar farside. On
top of this, for a wide range of Jacobi energies, Earth–Moon
L2 halos are almost locked into a 2:1 resonance, that is 2
orbital revolutions in 1 synodic period Tsyn = 29.4873 days.
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Tab. 2: Top-Level Objectives of LUMIO [7].
ID Objective
01 To perform remote sensing of the lunar surface
and measurement of astronomical observations not
achievable by past, current, or planned lunar mis-
sions
02 To demonstrate deployment and autonomous op-
eration of CubeSats in lunar environment, includ-
ing localization and navigation aspects
03 To demonstrate miniaturization of optical instru-
mentation and associate technology in lunar envi-
ronment
04 To perform inter-satellite link to a larger Lunar
Communications Orbiter for relay of data and for
TT&C
05 To demonstrate CubeSat trajectory control capa-
bilities into lunar environment
06 To gain European flight heritage in emplacing and
operating assets at Earth-Moon Lagrange points
The quasi resonance locking, which is also preserved in the
full ephemeris quasi-halos, enables LUMIO operations to be
steady, repetitive, and regular. Within the operative phase,
each synodic month LUMIO moves along a) a Science orbit
(dark solid line in Fig. 1) and b) a Navigation and Engineer-
ing orbit (light colored solid line in Fig. 1). During the Sci-
ence orbit, lasting approximately 14 days, the Moon farside
has optimal illumination conditions to perform flash obser-
vations (i.e., at least half lunar disk is dark). On the other
hand, during the Nav&Eng orbit the Moon farside illumina-
tion conditions are apt to optical navigation routines. In this
way, LUMIO preliminary ConOps is somewhat simplified and
tight to both resonance mechanisms and illumination condi-
tions to properly enable scientific or other operations.
2.3 ADCS requirements
LUMIO ConOps determine the design of the ADCS, whose
high-level requirements are summarized in Tab. 3. The en-
tries 03 to 05 have a special relevance in the context of this
work. During the science and navigation phases the space-
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Earth
Anti-Helion
Helion
Antapex
Apex
Fig. 1: LUMIO Concept of Operations [8].
Tab. 3: LUMIO ADCS high-level requirements [7].
ID Name Requirement
01 De-Tumbling After the separation from the Lu-
nar Orbiter, the ADCS is required
to de-tumble the spacecraft from
tip-off rates of, up to, 30 deg/s in
each axis.
02 Initialization Maneuver the solar panels to a
power safe mode within a time
compatible with the electrical en-
ergy capability.
03 Moon Point-
ing
The ADCS is required to point
with an accuracy of less than 0.1
deg during the science and naviga-
tion phases.
04 Power Maxi-
mization
The attitude is required to max-
imize the power generation capa-
bility of the solar panels given
the Moon pointing (halo and part
of the transfer) constraint and
the Earth pointing (parking) con-
straint.
05 Pointing Sta-
bilization
The ADCS is required to provide a
minimum pointing stabilization of
79.90 arcsec/s during the science
phase.
06 Slew Rate The ADCS shall provide a maxi-
mum slew rate of 1 deg/s.
craft has to point the Moon to enable full disk coverage (03).
In the former case, this requirement arises from the need of
visualizing the impacts of meteoroids, while in the later it is
determined by the visual navigation algorithms. If one axis
is fixed, the second degree of freedom in the attitude is de-
termined by power generation (04). In addition, the input
power is maximized by allowing a rotation of the solar ar-
rays around their axis. A certain pointing stabilization (05)
is required to avoid blurred images and enable flash detection
with the LUMIO Camera. Requirements 01, 02 and 06 are
common in ADCS design, the last arising from operational
range of the star tracker.
2.4 ADCS hardware
The preliminary list of ADCS hardware was defined in the
Phase A design of LUMIO. During the Concurrent Design
Review (CDR) performed at the European Space Agency
(ESA) Concurrent Design Facility [9] it was decided to up-
date the hardware components with the ones listed in Tab. 4.
The sizing and model selection of the reaction wheels and the
orientation of thrusters depend on the desaturation strategy,
and are hence the main outcomes of this work.
3 ADCS Configuration
3.1 Deep space environment
The deep space environment presents unique characteris-
tics that result in specific ADCS architectures. The ab-
sence of relevant magnetic fields in lunar orbit discards any
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Tab. 4: LUMIO ADCS hardware list
Component Qty. Model
Sun Sensor 2 SolarMEMS nanoSSOC-D60
Star Tracker 2 Hyperion ST400
IMU 1 Sensonor STIM 300
Thruster 4 Aerojet MPS-130
RW 3 See Sec. 3.5
magnetorquer-based desaturation procedure. A dedicated
ADCS propulsion system or a careful management of Solar
Radiation Pressure (SRP) torques is then required to desat-
urate the momentum management devices. In the case of
LUMIO, the strict requirement ADCS-03 discards the sec-
ond option, and hence a set of dedicated ADCS thrusters
must be installed.
Given the large orbit altitude, the reflected radiation
coming from the surface is negligible and the major distur-
bance to be considered is the SRP. Internal disturbances such
as the mechanical oscillations of the solar arrays or liquid
sloshing have only a short-term effect and, nevertheless, can-
not be faithfully simulated in such an early design phase.
3.2 Inertial properties
The inertia matrix, Center of Mass (CoM) position and prin-
cipal axes of inertia were computed with a block model de-
veloped in CATIA V5 6R-2018. The mass of each block was
assigned according to the preliminary mass budget (which
totals 22.82 kg) and distribution of hardware components.
Fig. 2 depicts the deployed and packed inertial Catia models
with the corresponding blocks.
Fig. 2: Catia inertial modeling for (a) LUMIO’s deployed
configuration, and (b) LUMIO’s packed configuration.
Fig. 3: LUMIO’s Body Frame and surface numbering;
adapted from [10].
The simulation returned a maximum deviation between
the principal inertial and geometrical axes of symmetry of
11◦. The maximum displacement of the CoM from the geo-
metrical center was of 1.6 cm. Since these computations cor-
respond to a preliminary state of the design, it was decided
to identify the principal axes of inertia with the geometric
ones and estimate the effect of SRP in a worst case scenario
with the CoM displaced 1.6 cm in the axis of the solar arrays
(Y). The body reference system represented in Fig. 3 is con-
sidered in this work. Under these assumptions, the principal
inertia matrices for the packed and deployed configurations
are
Jpack =

30.5 0 0
0 20.9 0
0 0 27.1

×10−2[kg m2],(1)
Jdepl =

100.9 0 0
0 25.1 0
0 0 91.6

×10−2[kg m2].(2)
3.3 Target reference frame
The Earth–Moon L2 quasi-halo orbit with a Cj= 3.09 Jacobi
constant is the baseline operational scenario for LUMIO [11].
The desired attitude is calculated from a predefined SPICE
kernel by applying the reference frame definition
Ad=





~x1=~xM
~x2=~xS×~x1
|~xS×~x1|
~x3=~x1×~x2
|~x1×~x2|





,(3)
where ~xMand ~xSdenote the normalized Moon and Sun
pointing vectors in the inertial reference frame J2000. It can
be shown that this reference frame maximizes power genera-
tion by allowing the solar arrays to be always normal to the
Sun vector [6].
The angular velocity ~wdand its derivative ˙
~wdare com-
puted numerically by making use of the Direction Cosine
Matrix (DCM) Kinematic Differential Equation
~ωV
d=−dAd
dtAT
d,(4)
with Adbeing the desired DCM from the J2000 inertial to
the body reference frames and the superscript “V” denoting
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the skew-symmetric matrix or hat map, defined for the case
of ~w as
~ωV=

0−ω3ω2
ω30−ω1
−ω2ω10

.(5)
The actual DCM matrix is named as Ain this work.
3.4 Solar Radiation Pressure
The intensity of solar radiation at Earth’s orbit is approxi-
mately 1366.1 W/m2. Considering a mean distance to Sun
of 149.6×106km, the inverse power law becomes
I=P
4πr2,(6)
Ibeing the total solar irradiance in W/m2,P≈3.842 ×1026
W the solar constant and rthe distance to Sun. The distance
between LUMIO and the moon oscillates between 4×104km
and 9×104km, so the Lunar radiation pressure can be safely
neglected, as at 100 km altitude it already represents a 20%
of solar radiation pressure [12].
The force exerted by SRP on a flat, Lambertian surface
is given by [13]
~
Fi=I
cAi(~
S·~nsi)n(1 −ρs)~
S+h2ρs(~
S·~nsi) + 2
3ρdi~nsio,
(7)
where cis the speed of light, Ais the surface area, ~
Sis the
unit vector pointing from the Sun to the surface, ~nsis a
unit vector normal to the surface (and directed towards the
interior), and ρs= 0.6 and ρd= 0.1 are the specularly and
diffusely reflected radiation. The vectors are expressed in the
body frame.
The SRP torque is computed with a simplified model that
neglects the interaction between different surfaces (shadows,
reflections). The result is
~
TSRP =
n
X
i=1
~cpi ×~
Fiif ~
S·~nsi >0,~
0 otherwise, (8)
with ~cpi being the position vectors between the center of mass
of the spacecraft and the center of pressure of the surfaces,
assumed to be the same as their geometrical centers. If the
spacecraft is eclipsed, the torque is set to zero. The eclipse,
without making distinctions between umbra and penumbra,
is produced when
~xM·~xS≤RM+horbit
pR2
M+ (RM+horbit)2,(9)
where RMis the radius of the Moon and horbit is the altitude
of the spacecraft. The surfaces of LUMIO are numbered
according to Fig. 3, and correspond to the columns of
ns=

00001−1 sin α−sin αsin α−sin α
001−1 0 0 0 0 0 0
1−1 0 0 0 0 cos α−cos αcos α−cos α

(10)
cp=

0 0 0 0 −10 10 0 0 0 0
0 0 −10 10 0 0 −45 −45 45 45
−10 10 0 0 0 0 0 0 0 0

−

0
1.6
0

[cm],
A=66664412121212×10−2[m2],
where αis the rotation angle of the solar panels, set to max-
imize the product (~
S·~nsi). In the target reference frame,
where the second base vector approximately points to the
Sun, it is possible to align ~
Sand ~nsi with a rotation of the
solar panels around their axis.
3.5 Reaction wheels configuration
A fundamental process when implementing a set of momen-
tum management devices is defining their optimum arrange-
ment. The optimization goal can be the minimization of
the torque requirements (or, equivalently, power demand)
or the optimization of the momentum/torque envelope or
workspace size. This problem has historically raised signifi-
cantly less attention than the development of ADCS control
laws [14]. In [15] it is shown that a tilt angle of arctan(1/2)
in a pyramidal single gimbal Control Moment Gyroscope
(CMG) system generates a spherical momentum workspace.
The maximum momentum and torque envelopes are com-
puted in [16] for four dissimilar reaction wheels. The influ-
ence of the tilt angle of a pyramidal configuration on power
consumption is studied in [14] for a specific maneuver. In
[17] several 3-axis and 4-axis arrangements are simulated in
the same scenario and the ones with a minimum total torque
level are identified. An interesting approach to the problem
is presented in [18], where an optimization procedure is given
to obtain the orientations Rand momentum bias for a set
of three identical reaction wheels subjected to momentum,
torque and power constraints.
The previous works highlight that the optimum arrange-
ment is mission-dependent. In the case of LUMIO, a major
importance has been given to increasing the period between
desaturation maneuvers (i.e., optimizing the angular momen-
tum capacity for the mission profile). This choice aims to
reduce the effects that those maneuvers would cause in the
normal operation of the spacecraft. A more common goal is
the reduction of maximum power consumption (or maximum
torque), but a significant minimum value is required in any
case to compensate the ADCS thrusters torque during de-
saturation. For other situations, the power requirements can
be lowered by performing slower maneuvers. The analysis of
alternatives is given in Sec. 5.
3.6 Reaction wheels control
The equations of attitude motion of a spacecraft with reac-
tion wheels are given by [19]
J˙
~ω +~ω ×J~ω =~uRW +~
TSRP ,(11)
˙
~
hr=−R∗(~uc+~ω ×R~
hr),(12)
~uRW =−A˙
~
hs
r−~ω ×R~
hs
r,(13)
where ~uRW is the control momentum applied by the reaction
wheels, ~ucis the control input, ~
hris the angular momen-
tum of the reaction wheels, Ris the actuators matrix (where
each column represents the axis of rotation of each wheel)
and the superscript “ ∗” denotes the pseudo-inverse. This
model allows imposing a limitation on the maximum angular
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Fig. 4: Aerojet MPS-130 main specifications and 1U version
representation; adapted from Aerojet MPS-130 datasheet.
momentum and torque, with the superscript “s” referring to
the values after saturation.
3.7 Thrusters configuration
Two 2U versions of the Aerojet MPS-130 were selected at
the ESA CDR as the baseline propulsion system [10]. Its
preliminary design and main specifications are given in Fig.
4.
The location of both modules is given in Fig. 5. In or-
der to enable a 3-axis control, the nozzles, directed towards
the −Xaxis, are tilted an angle γ= 10◦in the direction
of the green arrows. A minimum torque of 6.3 mNm in the
Y0and Z0axes is achieved with this configuration. At the
time of writing it is unclear if each thruster is throttleable
in the range 0.0625-0.3125 N or if the thrust value is set be-
fore launch. In the former case, lower torque values would
be achieved by firing the opposite thrusters. If the reaction
wheels provide higher torque values, a static desaturation
strategy can be considered (i.e., the reaction wheels could be
desaturated without producing an off-pointing of the satel-
lite).
The shift of the CoM is not a major concern for this con-
figuration. At the beginning of the science mission a ∆V
of 25.9 m/s will remain from the initially allocated 195.5
m/s [10]. According to the Aerojet MPS-130 datasheet, that
would correspond to approximately 371 g of usable propel-
lant. In a worst case scenario, its consumption would produce
a position shift of +3 mm in the X direction for a initial mass
of 20 kg, which has little effect on the torque matrix. The
Fig. 5: Thrusters configuration. The green arrows represent
the tilting directions of the ADCS thrusters; adapted from
[10].
reader should note that the Aerojet MPS-130 engine stores
its propellant in glass state and employs a piston to remove
sloshing [20]. Once measured, the CoM position would then
be subjected to a small uncertainty solely dependent on fuel
consumption.
3.8 Thrusters control
The ADCS thrusters control strategy aims to map a desired
torque value ~udes to the actual thrust vector ~
t, where each
row corresponds to one thruster. A discrete projective con-
trol is firstly evaluated. It imposes a constant thrust level
(t= 0.0625 N) and calculates the combination of on/off
states that best follows the desired control for each time step.
This combination minimizes the function
Jn
min =||~udes −~un||,with ~un=T~
t(14)
where nidentifies one of the 16 possible combinations of 4
thrusters, Tis the torque matrix and each element of ~
tis
either 0 or 0.0625 N. The torque matrix for a tilting of γ≈
10◦is computed easily after noticing that the nozzles rest
approximately in the X-Yand X-Zplanes. Therefore:
T=

xsin γ−xsin γ x sin γ−xsin γ
−xcos γ l sin γ x cos γ−lsin γ
−lsin γ x cos γ l sin γ−xcos γ

,(15)
where x= 0.09 m is the distance between the nozzles and
the Xaxis and l= 0.15 m is the distance to the Y-Zplane.
A thrust threshold is set as ~ulim = [2,3.9,3.9]TmNm, cor-
responding to the minimum possible compensated torque on
each axis. If |udes,i|is below k4~ulim, it is set to 0, while if
it is between k4~ulim and ~ulim it is set to ~ulim .k4is a tun-
ing parameter that controls the sensitivity of the propulsion
system. If k4is close to 0, a higher pointing accuracy will
be achieved with a higher propellant consumption, while if
it is close to 1 the second is minimized at the expenses of
pointing accuracy. It should be noted that the axes Yand Z
are coupled since the torque uncoupling is produced in the
Y0and Z0axes. Also, that the generation of torque will lead
in any case to a parasitic ∆Vin the Xaxis.
The projective control assumes that a single thrust level
is available for each thruster. This strategy may be improved
if a range of thrust values is allowed for each thruster. Un-
fortunately, it was not possible to determine if the Aerojet
MPS-130 thrusters have such capability. If that was the case,
a constrained optimal control problem could be formulated
with the objective of minimizing the total thrust function
Jmin =
m=4
X
i
ti(16)
subjected to T~
t=~udes and ti>0 N. In this case, tiranges
from 0.0625 to 0.3125 N. The output should then be corrected
to impose ti∈[0.0625,0.3125] N, in such a way that the
control value is limited to 0.3125 N, while if it drops below
k4umin it is set to 0. Again, the range between k4umin and
umin is set to umin.
The previous strategy can be implemented through a
standard linear solver or, more efficiently, by following the
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procedure described in [21]. According to its Theorem 1,
given any targeted ~udes torque, the vector of thrust values
can be computed as
~
t=T+~udes +θ~ω, (17)
with T+=TT(T T T)−1and ~ω > ~
0 being the Moore-Penrose
pseudoinverse and the kernel (null-space) of T, respectively,
and θ > maxi=1,...,m(−A+T)i/wibeing a positive real num-
ber that ensures that ti≥~
0. In order to activate the
thrusters when a minimum thrust limit is set, the minimum
thrust value 0.0625 N can be added to θ.
The Minimum Impulse Bit (MIB) of the thrusters is
∆tMI B = 0.0064. This lower limit is respected in the simu-
lations carried out in Sec. 5 by imposing a lower time step
limit for the variable time solver.
3.9 Parasitic ∆V
The parasitic vectorial ~
∆Vis given by
~
∆Vpar =N
mSC
~
Itot,(18)
where ~
Itot is the total impulse produced by each thruster, N
the matrix of thrust directions and mSC = 22.82 kg is the
mass of the spacecraft. The matrix Nis given by
N=

cos γcos γcos γcos γ
sin γ0−sin γ0
0 sin γ0−sin γ

.(19)
This vectorial formulation has the advantage of showing
specific information in the body axes, an useful knowledge
for optimizing the station-keeping maneuver.
4 Control laws
The attitude control laws for Moon tracking and simultane-
ous desaturation and tracking are given in this section, while
the stability proofs for both are detailed in the Appendix A.
4.1 Moon tracking control law
A possible Lyapunov-stable tracking control law in the pres-
ence of SRP disturbances would be given by [22, 23]
(20)
~u =−k1~ωe−k2(AT
e−Ae)V+~ω ×J ~ω
+J(Ae˙
~ωd−[~ωe]∧Ae~ωd)−ˆ
~
d,
with kibeing the control tuning parameters, ~ωe=~ω −Ae~ωd
the angular velocity error, Ae=AAT
dthe error DCM matrix
and ˆ
~
dthe disturbances estimation, equal in this case to the
SRP disturbance described in Sec. 3.4. The superscript ∧
denotes the inverse hat map. The application of this control
law consequently ensures the satisfaction of the requirement
ADCS-03 during the science and navigation phases.
4.2 Desaturation and tracking control law
In order to desaturate the reaction wheels, a control algo-
rithm must a) reduce their angular momentum to a safe
value, and b) simultaneously fire the spacecraft thrusters
to continue tracking the Moon and satisfy the requirement
ADCS-03 during desaturation. The ideal tracking law given
in Eq. (20) is then augmented and linked to the ADCS
thrusters, while a new law of the form
~uc=k3R(~
hr−sign(~
hT
r)~
hd
r)−~ω ×R~
hr,(21)
where k3is the control parameter, hd
ris the desired angular
momentum, and “ ” refers to Hadamard’s product, asymp-
totically stabilizes the reaction wheels spin. The multiplica-
tion of the desired angular momentum vector by the sign of
the actual momentum of the reaction wheels aims to avoid
an unwinding behavior. This expression can also be used to
control the thrusters in a zero-sum strategy, but as their op-
eration is not continuous that would derive in an off-pointing
of the satellite.
The tracking control law must finally be followed by the
ADCS thrusters and include the control torque given by Eq.
(21) as an external disturbance, resulting in
(22)
~u =−k1~ωe−k2(AT
e−Ae)V+~ω ×J ~ω
+J(Ae˙
~ωd−[~ωe]∧Ae~ωd)−ˆ
~
d−~uc.
5 Results and discussion
5.1 Reaction wheels configuration
The Moon tracking has to be ensured by the ADCS while the
spacecraft is subjected to external disturbance torques. The
most important of them is the SRP, which follows a repetitive
oscillatory track in the X−Zbody axes throughout the
mission. This behavior is depicted in Fig. 6 and is consistent
with the L2 halo 2:1 resonance (see Sec. 2.2).
The control load should be equally distributed among
the reaction wheels in order to extend the period between
desaturation maneuvers. This discards any configuration on
which a single reaction wheel is aligned with the directions
[±1,±1,0], that correspond to the maximum control torque
requirement. Two options are here studied. The first is
the basic 3-axis configuration, while the second is a Regu-
lar Tetrahedron around the Xaxis where one of the wheels
lies in the X-Zplane. The aperture of the tetrahedron with
respect to the symmetry axis is set to Ω = 60◦. The corre-
sponding Rmatrices are
Tab. 5: Proposed reaction wheels for LUMIO. Last column
reports the mass saving with respect to the heaviest option.
ID ~
hmax
r
˙
~
hmax
rm∆m
(mNms) (mNm) (g) (g)
Hyperion RW400 1 15 8 155 -675
GOMspace GSW600 19 2 180 -600
Hyperion RW400 2 30 8 210 -510
Hyperion RW400 3 50 8 380 0
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Copyright 2019 by the authors. Published by the IAF, with permission and released to the IAF to publish in all forms.
Fig. 6: SRP torque in the body frame during the first 30
days of mission.
Rb=

100
010
001

, Rt=






cos Ω cos Ω
√1 + cos2Ω
cos Ω
√1 + cos2Ω
0−√3/2
√1 + cos2Ω
√3/2
√1 + cos2Ω
sin Ω −1/2
√1 + cos2Ω−1/2
√1 + cos2Ω






.
(23)
The reaction wheels listed in Tab. 5 are subsequently con-
sidered. The objective is to select the technology that en-
sures a desaturation-free period of at least 15 days without
increasing unnecessarily the mass of the vehicle. The first
30 days of mission are simulated in Figs. 7 and 8 with the
two proposed RWs configurations. The torque requirements
are extremely low in both cases, although slightly higher
in the basic configuration. The angular momentum evolu-
tion is qualitatively similar with only small differences in the
maximum values. Both configurations can avoid the desat-
uration maneuver during 15 days with almost all the RW
models. However, since the wheels are close to saturation
after two weeks and a simplified SRP model was employed,
it was decided to implement the Hyperion RW400 2 set with
respective maximum angular momentum and torque of 30
mNms and 8 mNm. The 30 days evolution of the angular
momentum is smoother with the 60◦tetrahedron configura-
tion, which will be subsequently considered.
Fig. 7: a) Torque and b) angular momentum for the Carte-
sian RWs configuration during the first 30 days in the oper-
ational orbit.
5.2 Rigid desaturation schedule
The science phase of LUMIO lasts approximately 14 days.
A rigid desaturation schedule is here considered by imposing
desaturation maneuvers at days -1 and 15 of the resonant
orbit. This choice leaves the science and engineering orbits
completely free of disturbances, but does not minimize the
number of maneuvers.
Simulations cover the first year of the mission and im-
plement the models described in Secs. 3 and 4 with the
deployed matrix configuration given in Sec. 3.2. The pro-
jective thrusters control is first evaluated with parameters
k1= 0.0005, k2= 0.0005, k3= 0.01 and k4= 0.5. The
tracking algorithm was assumed by the propulsion system
during desaturation, imposing k1= 100 and k2= 100. The
RWs desaturation control k3cannot be increased without
obtaining a larger pointing deviation since the thrusters are
Fig. 8: a) Torque and b) angular momentum for the 60◦
tetrahedron RWs configuration during the first 30 days in
the operational orbit.
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Copyright 2019 by the authors. Published by the IAF, with permission and released to the IAF to publish in all forms.
Fig. 9: Angular momentum evolution for a rigid desatura-
tion schedule employing a) projective thrusters control, and
b) optimal thrusters control.
operating at a fixed thrust level. A maximum torque of 0.25
mNm and angular velocity of 2 ×10−4rad/s was produced
with this configuration, satisfying both the torque capabili-
ties of the RWs and the requirement ADCS-05. The optimal
thrusters control is compared with the previous by placing
the same parameters with a larger k3= 0.3, which according
to Eq. (21) results in a faster desaturation. The maximum
torque for this simulation is 7 mNm with a maximum an-
gular velocity of 3 ×10−4rad/s, satisfying again the torque
capabilities of the RWs and the requirement ADCS-05. This
time, however, the increase in k3brings the maximum torque
closer to the limit of the RWs.
Angular momentum, pointing angle, total impulse and
parasitic ∆Vplots are given in Figs. 9, 10, 11 and 12, re-
spectively. The saturation limits are always respected, but
an off-pointing of up to 0.2◦, slightly larger than the 0.1◦
Fig. 10: Pointing angle evolution for a rigid desaturation
schedule employing a) projective thrusters control, and b)
optimal thrusters control.
Fig. 11: Thrusters impulse evolution for a rigid desaturation
schedule employing a) projective thrusters control, and b)
optimal thrusters control.
limit imposed by the requirement ADCS-03, is produced in
some maneuvers. The steeper desaturation control law of the
optimal thrusters control scenario increases the off-pointing
in most cases. In terms of performance, the optimal control
achieves a significantly lower propellant consumption with
total impulse reductions of a 45% with respect to the projec-
tive control. This implies a significantly lower parasitic ∆V,
which is mainly directed in the Xaxis.
5.3 Flexible desaturation schedule
In the flexible desaturation scheme, the desaturation routine
is activated when 90% of the maximum momentum is reached
and stopped when a 66% is achieved. In this way, the pre-
viously observed oscillatory angular momentum evolution of
the reaction wheels are employed to save propellant. The
Fig. 12: Parasitic ∆Vevolution for a rigid desaturation
schedule employing a) projective thrusters control, and b)
optimal thrusters control.
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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.
Copyright 2019 by the authors. Published by the IAF, with permission and released to the IAF to publish in all forms.
Fig. 13: Angular momentum evolution for a flexible desat-
uration schedule employing a) projective thrusters control,
and b) optimal thrusters control.
activation and stopping thresholds have been chosen arbi-
trarily, and indeed power consumption may be reduced with
lower values.
The control parameters are set to k1= 0.0005, k2=
0.0005, k3= 0.0005 and k4= 0.5 for the projective thrusters
control case. As before, k1= 100 and k2= 100 during de-
saturation. The maximum simulated torque is 0.15 mNm,
while the angular velocity reached 2 ×10−4rad/s. These
values satisfy again the torque capabilities of the RWs and
the requirement ADCS-05. The optimal thrusters control pa-
rameters are the same but implementing a more demanding
RWs desaturation law with k3= 0.3. A maximum torque of
8 mNm and angular velocity of 4 ×10−4rad/s was produced
with this configuration, saturating the RWs and failing the
requirement ADCS-05 on a single maneuver.
Angular momentum, pointing angle, total impulse and
Fig. 14: Pointing angle evolution for a flexible desaturation
schedule employing a) projective thrusters control, and b)
optimal thrusters control.
Fig. 15: Thrusters impulse evolution for a flexible desatura-
tion schedule employing a) projective thrusters control, and
b) optimal thrusters control.
parasitic ∆Vplots are depicted for the new desaturation
strategy in Figs. 13, 14, 15 and 16, respectively. As in the
rigid desaturation case, the saturation limits are always re-
spected, but an off-pointing of up to 0.2◦is again produced
in some maneuvers and becomes slightly larger when k3ac-
quires a larger value. A lower propellant consumption is
observed with the optimal control, achieving total impulse
and parasitic ∆Vreductions of a 45% with respect to the
projective control.
5.4 Summary of desaturation results
The main outputs of the previously discussed desaturation
strategies have been summarized in Tab. 6. Although no
attempt was made to optimize the set of parameters ki, the
flexible-optimal desaturation arises as the most efficient in
Fig. 16: Parasitic ∆Vevolution for a flexible desaturation
schedule employing a) projective thrusters control, and b)
optimal thrusters control.
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Copyright 2019 by the authors. Published by the IAF, with permission and released to the IAF to publish in all forms.
Tab. 6: Summary of performances for different desaturation
strategies and a 1-year mission.
Rigid
Proj.
Rigid
Opt.
Flex.
Proj.
Flex.
Opt.
k15·10−45·10−45·10−45·10−4
k25·10−45·10−45·10−45·10−4
kdes
1100 100 100 100
kdes
2100 100 100 100
k30.01 0.3 5 ·10−45·10−4
k40.5 0.5 0.5 0.5
T.I. (Ns) 35.134 19.333 17.973 9.822
∆V (m/s) 1.573 0.865 0.805 0.440
terms of propellant consumption. In all cases, the parasitic
∆Vis shown to be negligible in comparison with the station-
keeping ∆V, which for LUMIO’s orbit was estimated to be
18.3 m/s with a 1σinterval.
6 Conclusions and future work
The post-CDR ADCS design of LUMIO has been presented
in this paper. Suitable RW models and configurations are
analyzed with the goal of extending the time period between
desaturation maneuvers. A desaturation strategy is devel-
oped on which the off-pointing of the satellite is not neces-
sarily produced, and various numerical implementations are
tested.
A set of three Hyperion RW400 2 (maximum angular
momentum of 30 mNms, maximum torque of 8 mNm) is
selected and oriented in a 60◦tetrahedron configuration. The
simulations, based on a simplified SRP model, show that this
setup is able to remain more than 15 days without reaching
saturation. Mass savings of 510 grams are achieved with
respect to the baseline configuration.
Among the desaturation strategies under study, the flex-
ible schedule with an optimal thrusters control law is identi-
fied as the most efficient in terms of propellant consumption.
The feasibility of this implementation relies on the throt-
tlability of the Aerojet MPS-130 ADCS thrusters and the
proper selection of the control parameters ki. Although less
efficient, a projective control could also be used if a single
thrust value is generated by each thruster. In all cases the
parasitic ∆Vis shown to be negligible in comparison with
the station-keeping costraint.
The ADCS requirements listed in Tab. 3 were generally
satisfied by the proposed configuration. Only small devi-
ations in the pointing angle (ADCS-03) were observed for
certain maneuvers, but these are due to a mismatch between
the torque produced by the ADCS thrusters and the RWs
during desaturation. A proper tuning of the kiparameters
should then be able to overcome this problem.
This study was not aimed to optimize the configuration
and control parameters of the system. Small differences in
performance are observed between the proposed RW arrange-
ments, and hence the selected 60◦tetrahedron has to be
taken just as a suitable solution. Future works should address
the effect of different configurations on power consumption
and angular momentum storage by making use, for instance,
of the procedure described in [18]. The desaturation strat-
egy and control parameters kishould be selected to satisfy
further mission requirements in next phases of development.
Those may include the minimization of power and energy
consumption, the avoidance of structural and liquid sloshing
resonance frequencies or other operational restrictions.
Acknowledgments
The work described in this paper is a spin-off of a bigger
project: LUMIO. For this reason the authors are grateful to
the whole LUMIO Team. The authors would like to acknowl-
edge fruitful discussions with Simone Ceccherini and Palash
Patole, which were greatly appreciated.
Appendix A: Stability proofs
The desaturation law described by Eq. (20) (RW, tracking),
Eq. (21) (RW, desaturation) and Eq. (22) (thrusters, track-
ing) result in asymptotically stable dynamics if (a) the space-
craft is driven to the desired attitude during normal opera-
tion, (b) the reaction wheels reach the desired state ~
hd
rduring
desaturation, and (c) the spacecraft keeps the desired atti-
tude during desaturation.
According to the Lyapunov’s Second Stability Theorem,
an autonomous nonlinear dynamic system described by [19]
˙
~x =f(~x), f (~xe) = 0,(24)
where ~xeis an isolated equilibrium point, is said to be asymp-
totically stable if in some finite neighbourhood Dof ~xethere
exists a scalar function V(~x) with continuous first partial
derivatives in ~x and tthat satisfies
(i) V(~x)>0 for all ~x 6=~x∗in Dand V(~x∗) = 0,(25)
(ii) ˙
V(~x)<0 for all ~x 6=~x∗in Dand ˙
V(~x∗) = 0.(26)
For the aforementioned case (a), the Lyapunov function
V(~x) = 1
2~ωeJ ~ωe+k2tr(I−Ae) (27)
can be differentiated to give
˙
V(~x) = ~ωeJ˙
~ωe−k2tr(˙
Ae).(28)
Implementing the DCM Kinematic Differential Equation
˙
Ae=−~ωV
eAeand considering that
(29)
˙
~ωe=˙
~ω −d
dt(Ae~ωd)
=J−1(J~ω ×~ω +~u +~
d)−d
dt(Ae~ωd)
then the derivative of the Lyapunov function given in Eq.
(27) results
˙
V(~x) = ~ωehJ~ω ×~ω +~u +ˆ
~
d−Jd
dt(Ae~ωd)−k2(Ae−AT
e)Vi,
(30)
where the property tr(~ω×
eAe) = −~ωT
e(Ae−AT
e)Vhas been
used. If the control law represented by Eq. (20) is substi-
tuted, the result is
˙
V(~x) = −k1w2
e(31)
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and hence the conditions given by Eq. (26) and Eq. (26) are
satisfied. In other words, the tracking control law makes the
system asymptotically stable.
For the case (b), the stability condition is demonstrated
by substituting Eq. (21) into Eq. (12), which gives
˙
~
hr=−k3(~
hr−sign(~
hT
r)~
hd
r).(32)
This is a first-order differential equation whose solution
asymptotically tends to the desired values ~
hd
r.
Finally, the procedure described in case (a) can be used
to demonstrate that the control law in case (c) asymptoti-
cally stabilizes the system. This time, the disturbance vector
includes a new term −~uc, that arises from the desaturation
torque of the reaction wheels.
References
[1] E. Hand. “Interplanetary small satellites come of age”.
In: Science 361.6404 (2018), pp. 736–737.
[2] A. Klesh, B. Clement, C. Colley, J. Essmiller, D. For-
gette, J. Krajewski, A. Marinan, T. Martin-Mur, J.
Steinkraus, D. Sternberg, T. Werne, and B. Young.
“MarCO: Early Operations of the First CubeSats to
Mars”. In: Proceedings of the AIAA/USU Conference
on Small Satel lites. Session 9. 2018.
[3] H. Heidt, J. Puig-Suari, A. S. Moore, S. Nakasuka,
and R. J. Twiggs. “CubeSat: A new Generation of Pi-
cosatellite for Education and Industry Low-Cost Space
Experimentation”. In: Proceedings of the AIAA/USU
Conference on Small Satel lites. Session 5. 2000.
[4] G. Benedetti, N. Bloise, D. Boi, F. Caruso, A. Civita,
S. Corpino, E. Garofalo, G. Governale, L. Mascolo, G.
Mazzella, M. Quarata, D. Riccobono, G. Sacchiero, D.
Teodonio, and P. M. Vernicari. “Interplanetary Cube-
Sats for asteroid exploration: Mission analysis and de-
sign”. In: Acta Astronautica 154 (2019), pp. 238–255.
[5] A. Schorr, K. F. Robinson, and D. Hitt. “NASA’s
Space Launch System: Enabling Exploration and Dis-
covery”. In: Proceedings of the 15th Reinventing Space
Conference. BIS-RS-2017-13. Oct. 1, 2017.
[6] S. Speretta, A. Cervone, P. Sundaramoorthy, R.
Noomen, S. Mestry, A. Cipriano, F. Topputo, J. Biggs,
P. Di Lizia, M. Massari, K. V. Mani, D. A. Dei Tos, S.
Ceccherini, V. Franzese, A. Ivanov, D. Labate, L. Tom-
masi, A. Jochemsen, J. Gailis, R. Furfaro, V. Reddy, J.
Vennekens, and R. Walker. “LUMIO: An Autonomous
CubeSat for Lunar Exploration”. In: Space Operations:
Inspiring Humankind’s Future. Ed. by H. Pasquier,
C. A. Cruzen, M. Schmidhuber, and Y. H. Lee. Cham:
Springer International Publishing, 2019, pp. 103–134.
[7] F. Topputo, M. Massari, J. Biggs, P. Di Lizia, D. Dei
Tos, K. Mani, S. Ceccherini, V. Franzese, A. Cer-
vone, R. Noomen, P. Sundaramoorthy, S. Speretta, S.
Mestry, A. Carmo Cipriano, S. Pepper, M. van de Poel,
A. Ivanov, D. Labate, L. Tommasi, A. Jochemsen, Q.
Leroy, R. Furfaro, V. Reddy, and K. Jacquinot. Lunar
Meteoroid Impact Observer: a CubeSat at Earth-Moon
L2. European Space Agency, Nov. 1, 2017.
[8] V. Franzese, P. Di Lizia, and F. Topputo. “Au-
tonomous Optical Navigation for the Lunar Meteoroid
Impacts Observer”. In: Journal of Guidance, Control,
and Dynamics 42.7 (2019), pp. 1579–1586.
[9] ESA-ESTEC. CDF Study Report of Lunar Mete-
oroid Impacts Observer (LUMIO) Mission. Tech. rep.
LUMIO-CDF-R-36. European Space Agency, Noord-
wijk, the Netherlands, May 2018.
[10] European Space Agency. CDF Study Report of LU-
MIO: Review of SysNova Award LUMIO Study. Euro-
pean Space Agency, Feb. 1, 2018.
[11] Ana Cipriano, Diogene Alessandro Dei Tos, and
Francesco Topputo. “Orbit Design for LUMIO: The
Lunar Meteoroid Impacts Observer”. In: Frontiers in
Astronomy and Space Sciences 5 (2018), pp. 1–23.
[12] R. Floberghagen, P. Visser, and F. Weischede. “Lunar
Albedo Force Modeling and its Effect on Low Lunar
Orbit and Gravity Field Determination”. In: Advances
in Space Research 23.4 (1999), pp. 733–788.
[13] B. Wie. Space Vehicle Dynamics and Control. AIAA
Education Series, 2006.
[14] S. Abolfazl and M. Mehran. In: International Jour-
nal of Aeronautical and Space Sciences 15.2 (2014),
pp. 190–198.
[15] H. Kurokawa. “A Geometric Study of Single Gimbal
Control Moment Gyros”. In: 1998.
[16] T. Shengyong, C. Xibin, and Z. Yulin. “Configuration
optimization of four dissimilar redundant flywheels
with application to IPACS”. In: Proceedings of the 31st
Chinese Control Conference. 2012, pp. 4664–4669.
[17] Z. Ismail and R. Varatharajoo. “A study of reaction
wheel configurations for a 3-axis satellite attitude con-
trol”. In: Advances in Space Research 45.6 (2010),
pp. 750–759.
[18] D. S. Bayard. “An optimization result with applica-
tion to optimal spacecraft reaction wheel orientation
design”. In: Proceedings of the 2001 American Control
Conference. (Cat. No.01CH37148). Vol. 2. 2001, 1473–
1478 vol.2.
[19] H. Schaub and J. L. Junkins. Analytical Mechanics of
Space Systems. American Institute of Aeronautics and
Astronautics (AIAA), 2003.
[20] D. Schmuland, C. Carpenter, R. Masse, and J. Overly.
“New insights into additive manufacturing processes:
enabling low-cost, high-impulse propulsion systems”.
In: Proceedings of the 27th Annual AIAA/USU Con-
ference on Small Satel lites (2013).
IAC-19,C1,6,4,x50894 Page 12 of 13

70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.
Copyright 2019 by the authors. Published by the IAF, with permission and released to the IAF to publish in all forms.
[21] R. S. Sanchez Pena, R. Alonso, and P. A. Anigstein.
“Robust optimal solution to the attitude/force control
problem”. In: IEEE Transactions on Aerospace and
Electronic Systems 36.3 (2000), pp. 784–792.
[22] Y. Bai, J. D. Biggs, X. Wang, and N. Cui. “A singu-
lar adaptive attitude control with active disturbance
rejection”. In: European Journal of Control 35 (2017),
pp. 50–56.
[23] Y. Bai, J. D. Biggs, F. Bernelli-Zazzera, and N. Cui.
“Adaptive Attitude Tracking with Active Uncertainty
Rejection”. In: Journal of Guidance, Control, and Dy-
namics 41.2 (2018), pp. 550–558.
IAC-19,C1,6,4,x50894 Page 13 of 13