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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.

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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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.

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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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.

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-

!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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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70th International Astronautical Congress, Washington D.C., United States, 21-25 October 2019.

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.

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