An analysis of attitude control laws and thruster configurations for a deep space CubeSat

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POLITECNICO DI MILANO

School of Industrial and Information Engineering Department of Aerospace Science and Technology

Master of Science in Space Engineering

An analysis of attitude control laws and thruster

configurations for a deep space CubeSat

Supervisor:

Prof. James Douglas BIGGS

Master's degree thesis of: Matteo Teodoro PINCA - 883246

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Acknowledgements

At the end of this long adventure which my path at Politecnico di Milano was, and at the end of this thesis, which is its nale act, some thanks are more than dutiful.

The rst thanks surely goes to my family, to mum Paola, dad Andrea, my brother Stefano and grandma Enrica, for all the help they gave to me during these years: for having allowed me to undertake this path, for having spurred me during moments of diculty, for having been happy with me during good moments, for having always listened at me when I needed.

A very special thanks goes to my girlfriend Sara, who has borne and supported me since we met 4 years ago, with all her love, the most beautiful thing in the world, capable of instilling me courage to face this adventure, but also the ones which will come.

A grateful thanks to my supervisor, Professor James Douglas Biggs, for his availability and for his skilful guide during these months. I've never given for granted these things, I've truly appreciated them.

Thanks to my travel fellows: Cristian e Marco, who accompanied me during last hard years of bachelor, and Francesco, with whom gladly I've worked as a team during these (not less hard) years of master of science degree.

Thanks to all my other friends, with whom I shared lot of things outside the eld of studies, and with whom I unavoidably had to reduce relationships in last period, in order to focus my attention on exams and thesis.

A thanks goes also to other professors, from the ones sterner to the ones with whom I felt more com-fortable, because unavoidably they were part of my path in this prestigious university and contributed to my education as aspirant engineer.

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One last thought.

The nal writing of this thesis has occurred during the lockdown to contrast the diusion of SARS-Cov2 virus: as the majority of Italians in this delicate moment I stayed at home, following developments of this pandemic, which deeply turned our habits upside-down, a pandemic which hardly will leave our memory too early.

One last thanks, last but not the least, to all the people who are facing this emergency, providing to us a model of stoicism and a source of solace:

thanks to doctors, nurses and health sta (volunteer or not), who are ghting on the front line to save human lives;

thanks to people, whose work guarantees us essential services, rst of all the food on the table;

thanks to the institutions and to the people who, tough with aws and uncertainties, found on them-selves an unprecedented burden, and strives to make 60 million of Italians continue their lives;

and thanks to all the people who set aside politician persuasions and destructive criticisms, thanks to the people who understood that only respecting rules and people, that only helping us out, each one in his/her small way, we can get out of sticky situations as this one...

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Ringraziamenti

Al termine di questa lunga avventura che è stata la mia permanenza al Politecnico di Milano, e al termine di questo lavoro che ne è l'ultimo atto, alcuni ringraziamenti sono più che doverosi.

Il primo grazie va sicuramente alla mia famiglia, a mamma Paola, papà Andrea, mio fratello Stefano e nonna Enrica, per tutto l'aiuto che mi hanno dato in questi anni: per avermi permesso di intraprendere questo cammino, per avermi spronato nei momenti dicili, per aver gioito con me nei momenti belli, per avermi sempre ascoltato quando ce n'era bisogno.

Un grazie molto speciale va a Sara, la mia ragazza, che da 4 anni mi supporta e mi sopporta con tutto il suo amore, la cosa più bella del mondo, capace di infondermi coraggio per arontare questa avventura e che mai mancherà per le altre che ci si presenteranno.

Un grazie riconoscente va al mio relatore, il professor James Douglas Biggs, per la sua disponibilità e per la sua sapiente guida durante questi mesi, cose che non ho mai dato scontate e che ho tanto apprezzato.

Grazie agli amici, miei compagni di avventura: Cristian e Marco che mi hanno accompagnato negli ultimi duri anni della triennale, e Francesco, con cui ho volentieri fatto squadra durante i non meno duri anni di magistrale.

Grazie a tutti gli altri miei amici, con cui ho condiviso tanto al di fuori dell'università, e con cui inevitabilmente ho ridotto i contatti nell'ultimo periodo, anche troppo forse, per concentrare le mie energie sugli studi.

Un grazie va anche agli altri professori, da quelli più severi a quelli con cui mi sono trovato più a mio agio, perché inevitabilmente sono stati parte del mio cammino in questo prestigioso Ateneo e hanno contribuito alla mia formazione come aspirante ingegnere.

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Un ultimo pensiero.

La stesura nale di questa tesi è avvenuta in pieno lockdown per il contrasto alla diusione del virus SARS-Cov2: come la maggior parte degli italiani in questo momento delicato sono rimasto a casa, seguendo gli sviluppi di questa pandemia che ha stravolto profondamente le nostre abitudini e che dicilmente lascerà la nostra memoria a breve.

Un ultimo grazie, ma non per questo meno importante, a tutti coloro che stanno fronteggiando questa emergenza, fornendo a molti di noi un esempio di stoicismo e una fonte di conforto:

grazie a medici, infermieri e personale sanitario, volontario e non, che combattono in prima linea per salvare vite umane;

grazie a chi col suo lavoro ci sta garantendo i servizi essenziali, primo su tutti il cibo in tavola;

grazie alle istituzioni e a chi, seppur con mille difetti e incertezze, si è trovato un fardello senza precedenti sulle spalle, e si sforza per mandare avanti la vita di 60 milioni di italiani;

e grazie a chi ha messo da parte colori politici e critiche distruttive, e invece ha capito che solo rispettando le regole e le persone, che ha capito che solo dandoci tutti una mano, ognuno nel suo piccolo, possiamo uscire da situazioni brutte come questa...

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List of acronyms

It can be userful to have a recap of the acronyms which will be plentifully used in next chapters:

CG Cold Gas (thrusters)

CoG Centre of Gravity (of the Spacecraft) ctrl Control

EL Electric (thrusters) IC Initial Condition

DCM Direct Cosine Matrix (kinematic equations) NMS Nanosatellite Micropropulsion System

PL Proportional Law (ideal control law) PWM Pulse Width Modulation (control law)

RW Reaction Wheel S/C SpaceCraft

SRP Solar Radiation Pressure (disturbance torque) ST Super Twisting (ideal control law)

U Unit (modular cubic unit of a CubeSat) wrt with respect to

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List of Tables

2.1 Thrusters' specics recap . . . 40

2.2 PWM tolerances . . . 40

2.3 PL kj numerical values . . . 41

2.4 Sliding surface and ST kj numerical values . . . 41

3.1 Gravimetric Specic Impulses . . . 64

5.1 CoG shift eect on torques applied to S/C . . . 91

6.1 Comparison among thrusters, detumbling manoeuvre, including CoG shift . . . 125

6.2 Comparison among thrusters, slew manoeuvre, no CoG shift . . . 125

6.3 References of Figures about detumbling manoeuvre . . . 126

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Abstract

CubeSats are a class of miniaturized satellites, originally developed in 1999 for academic purposes. Their importance has grown with time, with several missions involving CubeSats having been developed by space agencies. To control the attitude of a satellite, actuators are needed such as magnetic actuators, but these are not useful in deep-space. Reaction wheels can be used but require an additional actuator for de-saturation. Alternatively, thrusters can be used, but little has been analysed to the most appropriate controls and congurations of thrusters for Cubesats.

This thesis deals with the attitude control of a 12U CubeSat, using thrusters. The aim of this work is to assess the performance during detumbling and slew manoeuvre comparing dierent control laws and thruster congurations.

Two kinds of thrusters were investigated: Cold Gas thrusters and Electric thrusters. Dierent cong-urations were tested, i.e. dierent numbers of actuators and dierent spatial orientations.

Two ideal control laws were investigated, and they belong to what is called control allocation: the idea is to map the ideal control on each actuator. First one is a simple Proportional Law, and second one is a Super-Twisting algorithm based on Sliding Mode control. Furthermore, a main part of real control implemented is a Pulse Width Mode control, which states when convergence is reached and the manoeuvre ends.

The key performance parameters studied are the time to reach convergence, the Overall Total Impulse (i.e. of the whole conguration, not of the single thruster) and the precision of the manoeuvre (i.e. angular velocity or steady state error evaluated at convergence time).

In addition, a robustness analysis of the sensitivity to the positioning of the centre of mass was undertaken.

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Sommario

I CubeSat sono una classe di satelliti miniaturizzati, nati per scopi accademici nel 1999. La loro im-portanza è cresciuta nel tempo, tanto che sono state realizzate molte missioni che implicano i CubeSat dalle agenzie spaziali. Per controllare l'assetto di un satellite, sono necessari degli attuatori come quelli magnetici, inutili nello spazio profondo. Possono essere impiegate le ruote di reazione (reaction wheels), ma richiedono attuatori addizionali per la desaturazione. In alternativa, possono essere usati i propulsori, ma sono state fatte poche investigazioni approfondite su controllo e congurazioni più appropriati sui CubeSat.

Questa tesi prende in esame il controllo dell'assetto di un CubesSat 12, attraverso l'uso di propulsori. Obiettivo è analizzare le prestazioni durante le manovre di stabilizzazione e puntamento de satellite, confrontando diverse leggi di controllo e congurazioni dei propulsori.

Sono stati studiati due tipi di propulsori: a razzo ed elettrici. Sono state analizzate diverse congu-razioni, cioè diverso numero di attuatori e diverse orientazioni spaziali.

Due sono le leggi di controllo ideale studiate, appartenenti al Control Allocation: l'idea è di mappare il controllo ideale su ogni attuatore. La prima legge è una semplice dipendenza proporzionale, mentre la seconda è un algoritmo chiamato Super Twisting, basato sul controllo Sliding Mode. In più, una parte fondamentale del controllo reale è un controllore a modulazione di larghezza di impulso (PWM), che determina quando si raggiunge convergenza e la manovra termina.

I parametri chiave di prestazione calcolati sono il tempo per raggiungere la convergenza, l'impulso totale complessivo (riferito cioè all'intera congurazione e non ai singoli propulsori) e la precisione della manovra (massima velocità angolare o errore sull'assetto).

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Chapter 1

Introduction

Since they were born in 1999 for academic purposes, CubeSats (i.e. modular miniaturized satellites) became a class of Spacecrafts whose importance grew more and more along the years. Space agencies, such as ESA and NASA, performed missions involving CubeSats in the recent past [52, 53, 54, 55], and other missions are planned for the future [59, 60]. As time goes on, also the complexity of those missions does accordingly, such that the need to develop peculiar technologies and strategies increases too, as a natural consequence [1, 2, 3, 4, 5, 6].

One of mandatory needs, is the one dealing with attitude control: since dimensions of CubeSats are very contained (single modular unit, 1U, is a cube of 10 cm side, the maximum number of units is 12), the actuators must satisfy strict volume and mass constraints. There were large investigation and use of magnetic actuators and reaction wheels [45, 46, 47]. In deep space, Earth's magnetic eld is too low and magnetic actuators become useless. Reaction wheels were investigated in detail also for what concerns faults and disturbances rejection [48, 49], and it is not uncommon that they are paired with thrusters: thrusters are more ecient in detumbling manoeuvre and often are used to desaturate RWs. This work refers to deep-space CubeSats, the ones which lie in orbits around our planet higher than LEOs (Low Earth Orbits). ESA yet used deep-space Cubesats [56, 57, 58], but the mission taken as reference for the models which are going to be shown, is the upcoming LUMIO (LUnar Meteoroid Impact Observer) mission: [7, 8, 9, 61].

The research dealing with propulsion used for attitude control is not so deep as the one dealing with RWs: as said so far, there are proposed missions involving CubeSats, but the investigation about attitude control is often minimized, even less about a propulsive attitude control, demanding it to consolidated and yet tested options, when adopted.

Aim of this work is to provide information about the attitude control, using thrusters, of a 12U CubeSat, during detumbling and slew manoeuvres: a comparative analysis on performances during the manoeuvres considered, concerning the kind of propulsion, how many thrusters and how they are oriented, the ideal control laws and the eect of the Centre of Gravity shift. Baseline of this work is what shown in [15]: the idea is to improve the investigation, taking into account only the rotational problem. What is wanted to be provided, are some plots which may be used for design purposes, i.e. to perform a rst trade-o analysis for a deep space CubeSat similar to the one considered, stressing

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The S/C is modeled as a 12U CubeSat, and three kinds of thrusters were considered, taking as reference structural and propulsive components already available on the market.

Dimensions of the S/C considered are evaluated on the base of [62] structures.

Cold Gas thrusters considered are Hybrid ADN Delta-V used in RCS System [63]. Two kinds of electric thrusters were considered: IFM thrusters [64] and NMS thrusters [65].

Kinematic Equations are considered in DCM (Direct Cosine Matrix) form: computationally they are less convenient than other forms (such as quaternions, [14, 33, 43]), but they were preferred since identify the attitude in an univocal way.

The ideal control laws selected belong to what is called control allocation [17, 18, 19, 20, 21, 22]: the idea is to map the ideal control on each one of the actuators. Control allocation is a well investigated type of control, capable of take into account also faults, disturbances and uncertainties [23, 24, 25, 26, 27, 28], and computationally ecient. Two control laws within this family were investigated.

First ideal control law is what was named Proportional Law, since it is the base of P-I-D (proportional-integral-derivative) controllers, widely known and used in dynamic systems (including spacecraft attitude control [31, 32, 33]). Second ideal control law is the so-called Super-Twisting algorithm, based on Sliding Mode Control. Sliding mode control development is more recent than classic P-I-D, and its investigation begins to go deeper in detail, also because of its robustness to disturbances [36, 37, 38, 39, 40, 41, 42, 43, 44].

Once stated the ideal control law, within the macrosphere of allocation control (and on the base of previous studies), the projective control was considered for Cold Gas thrusters, due to their principle of working [12, 13, 14]: Cold Gas Thrusters, which re with an ON/OFF logic, are well known, since historically are the rst ones that were developed. The challenge, nowadays, and in particular dealing with CubeSats, is to miniaturize their dimensions.

For electric thrusters, and adaptive control was chosen [29, 30], since the way of working of this kind of propulsion diers for Cold Gas one (they work in a throttleable way), and it is based on the concept of pseudo-inverse matrix [34, 35]. In particular, the torque provided by electric thrusters is obtained from the same control proposed for LUMIO mission, shown in Eq. 17 in [10]. Its theoretical validation was provided in [11].

An essential part of the real control implementation is a PWM (Pulse Width Modulation) control [16, 14], to state when the manoeuvre may be considered as concluded.

To increase the robustness of the analysis, a shift of the Centre of Gravity of the Spacecraft was also considered: it was done always with design purposes, to better design the mass distribution of Spacecrft's structure or payload placement inside the Cubesat, or to consider a possible attitude control based on moving masses [50].

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The work is organized as follows:

Chapter 2 deals with modelling of S/C's dynamics and kinematics, as well as ideal control laws, thruster congurations and real control implementation, needed to set up Matlab® and Simulink® numerical simulations.

Chapter 3 introduces the methodology adopted while performing the simulations, and introduces the key performance parameters needed to build the plots showing the results of this analysis. Meanwhile, the Chapter presents in detail a simulation with CG thrusters, to show how to arrive to those plots and make a detailed example.

Chapter 4 follows previous chapter, showing main results and considerations about simulations with Electric thrusters (both IFM and NMS).

Chapter 5 deals with the shift of S/C's Centre of Gravity, and the eects which it produces on key performance parameters and relative plots.

Chapter 6 summarizes what done in this work and presents few conclusive considerations.

A list of the Tables containing references to numerical values used in simulations, relevant recaps and Figures concerning main results, can be found on page 9.

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Chapter 2

CubeSat Dynamics and Control using

Thrusters

This chapter presents the modelling part, dealing with all the models needed to build the simulations in Matlab® and Simulink®.

Starting from general parts such as the SpaceCraft, its dynamics and attitude, as well as the ideal control laws (Section 2.1), the presentation of models continues dealing with thruster congurations and their real control application, for both Cold Gas Thrusters (Section 2.2) and Electric Thrusters (Section 2.3). Finally, a recap with numarical values needed in simulations, is presented in Section 2.4.

2.1 Dynamics and Ideal Control

2.1.1 Spacecraft Specics

The spacecraft considered is a 12U cubesat, modeled simply as a cuboid. Sizes and mass used are based on ESA's LUMIO Mission spacecraft. Data are reported referred to Spacecraft Body Reference Frame (the green one in Figure 2.1.1).

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Specics of the Cuboid S/C are the following:              m = 21.08Kg mass a = 0.3405m length (x-axis) b = 0.2263m width (y- axis) c = 0.2000m height (z-axis)

Since the modelling of sizes is s.t. c < b < a, the principal moments of inertia result as:        Ix= 121m b 2_{+ c}2 ≈ 0.16022854Nm2 Iy= ₁₂1m a2+ c2 ≈ 0.27393504Nm2 Iz=₁₂1m a2+ b2 ≈ 0.29363024Nm2

2.1.2 Euler Equations

The model of the dynamics of the spacecraft is based on the Euler Equations: J ˙ω = Jω × ω + M + u

where J = diag(Ix, Iy, Iz) is the matrix of inertia, ω =

h

ωx ωy ωz

iT

the angular velocity, ˙ω = h

˙

ωx ω˙y ω˙z

iT

its derivative in time, u =h ux uy uz

iT

the control torque, M =h Mx My Mz

iT the overall external torque, due to disturbances.

To evaluate the evolution in time of each component of ω, the Euler Equation formula is inverted as:                ˙ ωx= _I1_x[(Iy− Iz) ωyωz+ Mx+ ux] x-component ˙ ωy=_I1_y [(Iz− Ix) ωxωz+ My+ uy] y-component ˙ ωz=_I1_z [(Ix− Iy) ωyωx+ Mz+ uz] z-component

How to evaluate M, u will be explained in sections 2.1.4, 2.2.2 and 2.3.2.

2.1.3 Kinematic Equations

The attitude evolution in time is evaluated using Direct Cosine Matrix (DCM) model: ˙ A = − [ω]∧A where [ω]∧ =    0 −ωz ωy ωz 0 −ωx −ωy ωx 0  

 is the hat map operator applied to angular velocity vector.

If, during simulations, attitude matrix becomes near-orthonormal, an iterative cycle to be sure that A retuns to be strictly orthonormal is added:

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2.1.4 Solar Radiation Pressure

In this work, the only disturbance considered is the one due to Solar Radiation Pressure (SRP). This disturbance torque MSRP is caused by solar photons: even though a photon's mass is zero, its

energy and momentum are not.

The radiation pressure at a distance of 1 A.U. from Sun is evaluated as P = σ1AU

clight ≈

1358W/m2 3·108_m/s .

Then, in the model adopted there is the strong assumption that the radiation is not absorbed by the S/C, while the reection is taken into account using two coecients: ρs = 0.5 and ρd = 0.1,

respectively specular and diffusive reection coecients.

About the geometry, it is necessary to consider the faces fiof the cuboid spacecraft and their respective

normal versors ni (i = 1 : 6), expressed in Body Reference Frame:

The faces are:

f1= f4= b · cwith normals n1= h 1 0 0 i T and n4= −n1 f2= f5= a · cwith normals n1= h 0 1 0 i T and n5= −n2 f3= f6= a · bwith normals n1= h 0 0 1 iT and n6= −n3.

As it can be seen in the Figure 2.1.2, the thrusters are always located on face f1.

The force produced on each face is calculated as: F_i= P fi(SB· ni) (1 − ρs) SB+ [2ρs(SB· ni) + 2 3ρd n_i

To evaluate the torque produced by these forces, it is useful to dene the arms of the resulting moments and the Centre of Pressure of each face of the S/C (cP i):

on face f1 it is cP 1= h a_/₂ ₀ ₀ i T − CoG on face f2 it is cP 2= h 0 b_/₂ ₀ i T − CoG on face f3 it is cP 3= h 0 0 c_/₂ iT − CoG; on face f4 it is cP 4= − h a_/₂ ₀ ₀ iT − CoG on face f5 it iscP 5= − h 0 b_/₂ ₀ iT − CoG on face f6 it is cP 6= − h 0 0 c_/₂ i T − CoG

In the equations above, CoG is the location of the Centre of Gravity of the S/C in Body Frame. It is modeled in such a way that it can be located also not in coincidence with the origin of the axes of Body Reference Frame, but shifted:

CoG =h errx erry errz

i

·h a b c i

T

whereh errx erry errz i is a vector of percentual errors. In this work it will be considered as:

errx= erry = errz=

 



0% when CoG is exactly at the centre of the S/C ±10% when CoG shifts. 10% is a worst case scenario

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Finally, the distrubance torque is evaluated as: M_SRP =        6 X i=1 cP i× Fi if SB· ni> 0 0 if S_B· n_i< 0 where SB is the Sun Vector in Body Frame:

SB= ASN

and SN the onboard model used to evaluate the position of the Sun in Inertial Reference Frame:

S_N =    cos(n t + B)

sin(nt + B) cos(εecl)

sin(n

t + B) sin(εecl)



 

The parameters used in the equation above are:

n=_{365.25·(3600·24)}2π , the mean motion of the Sun with respect to Earth;

εecl= 23.45°, the angle of inclination of Ecliptic Plane with respect to the Equatorial Inertial Plane;

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2.1.5 Ideal Control Laws

The ideal control law expresses the theoretycal torque that should be provided each time step. In this work only detumbling manoeuvre and slew motion are considered: the desired angular velocity in these cases is ωd= 0constant. This implies that it never appears in the equations, since

(ω − ω_d) = (ω − 0) = ω

If the problem of attitude tracking along the orbit was instead considered, ωdmust have been

consid-ered, since in that case ωd= ωd(t) 6= 0; it is not the case of this work.

Two kinds of ideal control laws will be analyzed: a Proportional Law (named shortly PL) and a Super-Twisting algorithm (named shortly ST).

2.1.5.1 Proportional Law (PL)

This kind of ideal control is one of the simplest, since it is a linear law.

About detumbling manoeuvre, the PL ideal control has the following expression: u_id = −k1Jω

About slew motion, in addition to the term referring to angular velocity, a term referring to attitude error must be condiered, as follows:

Ae= A(t)ATd

e = (AT

e − Ae)∨

u_id= −k1Jω− k2Je

hspace5mmwhere the inverse hat operator (X3×3)∨=

h

X32 X13 X21

iT

satises the property: tr [ω]∧X = −ωT _X−XT∨

Numerical values stated for kj coecients are reported in Table 2.3.

2.1.5.2 Sliding Surface and Super-Twisting (ST)

The Super-Twisting ideal control law is a non-linear law, and in this work it is used in association to a Sliding Surface, which will be:

 



S = ω during Detumbling S = ω + kse during Slew Manoeuvre

this Sliding Surface is put into the ST algorithm as follows:              u_id = −k1     p|Sx|sgn(Sx) p|Sy|sgn(Sy) p|Sz|sgn(Sz)     + v ˙v = −k2sgn(S)

Vector ˙v of second row is numerically integrated each time step to obtain the vector v of rst row. Initial condition for numerical integration is alway set as v0=

h

0 0 0 i

T

. Numerical values stated for kj coecients are reported in Table 2.4.

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2.2 Cold Gas Thrusters

2.2.1 Thrusters Specics and Congurations

From Vacco Cold Gas thruster's datasheet, the thrust F and the minimum impulse bit MIB can be recovered as: _   F = 0, 01N M IB = 1 · 10−4 Ns

Since the minimum time interval in which provide thrust results as ∆tmin =M IB_F = 0.01 s, the actual

time step in simulations chosen is tstep= 0.01s = ∆tmin. This is embedded in Simulink model through

the use of a Zero-Order-Hold block.

First thing to be said about congurations is this: aim is nd T matrix, dimension [3 x n], which expresses the torque provided by each of the n thrusters of the conguration. Each column of the matrix T is the torque around x,y,z axis of the thruster, and is the vectorial product between the arm bi(the location of the thruster) and the force provided by the correspondent thruster (the thrust, Fi),

both expressed with respect to the S/C Body Frame.

To reach T matrix, few geometrical parameters need to be introduced. This is done because the Centre of Gravity of the S/C will be shifted (see Chapter 5), so it is mandatory to refer torques to the CoG where it really is, not only in positionh 0 0 0 i

T

. In this way, the moment arms of applied torques result as:

moment arm on x-axis: l = a

2− CoGxm

nominal moment arms on other axes:  



y0= b₄ m y direction

z0= c₄ m z direction

through which real moment arms on y and z axes can be evaluated:

            

Yp= |y0− CoGy| m arm in y positive direction ("p" means "plus")

Ym= − ky0− CoGyk m arm in y negative direction ("m" means "minus")

Zp= |z0− CoGz| m arm in z positive direction

Zm= − kz0− CoGzk m arm in z negative direction

Figure 2.2.1 below is an example for 4 thrusters: thrusters are the arrows in red, nominal moment arms wrt thruster #3 are the blue ones above, while real moment arms are the dark red ones below.

The parameters Yp, Ym, Zp, Zmmust be evaluated for all the thrusters, for all the congurations which

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Figure 2.2.1: Graphical representation of some geometrical parameters.

Moreover, to refer to each conguration in a compact way, the following nomenclature is introduced: one number = number of thrusters

two Capital letters (CG = stand for Cold Gas thrusters)

one small letter (i,p,o) = expresses the peculiarity of the conguration:

i stands for inplane congurations: the force vector provided by each thruster has only two components which can be nonnull, the 2nd and the 3rd. The Force lies in a plane parallel to the yz one.

p stands for parallel congurations: each Force vector is aligned to one axis, which can be x, y or z.

o stands for outplane congurations: each force vector does not lie in a planed parallel to xy, xz or yz ones, and also is not aligned to one of the axes. It means that each Force Vector has three nonnull components.

Few examples:

4CGi = 4 Cold Gas thrusters inplane (plane parallel to yz). 8CGp = 8 Cold Gas thrusters parallel (to x,y,z S/C axes)

6CGo = 6 Cold Gas thrusters outplane (which have a component in each axis)

Thrusters' congurations are going to be presented in following pages: their analytical expressions and a graphical representation (the ones about Cold Gas thrusters represent as well the Electric thrusters' congurations).

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For outplane congurations, it is convenient to express the Force Vector in spherical coordinates: F =radius of the force vector, θ = angle from z axis, ϕ = angle from x axis, in xy plane.

Fcan't be changed, since the Force of each thruster is always the same; what to do is to choose the an-gular coordinates' values, and afterwards make the conversion from spherical to cartesian coordinates, as reported in following gure:

Figure 2.2.2: Spherical and cartesian coordinates

According to the nomenclature introduced in previous page, the congurations which are going to be shown are: 4CGi 4CGo 6CGi 6CGo 8CGi 8CGp 8CGo

Concerning Cold Gas thrusters, no 12CG congurations have been considered (the possible ones are 12CGp and 12CGo): this happened not because they can't be used, but because simulations with such congurations are too much time consuming. Think to the fact that one single simulation takes about 30 minutes to run.

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4CGi conguration

For this conguration, an angle α needs to be introduced: its graphical representation can be visualized in Figure 2.2.1 above. The numerical value used is α = 15°.

Matrix T is evaluated as follows:

T₁= b₁× F₁=    l Ym Zm   ×    0 −F cos(α) −F sin(α)   =    ZmF cos(α) − YmF sin(α) +F l sin(α) −F l cos(α)    T₂= b₂× F₂=    l Yp Zm   ×    0 +F cos(α) −F sin(α)   =    ZmF cos(α) − YpF sin(α) F l sin(α) F l cos(α)    T₃= b₃× F3=    l Yp Zp   ×    0 +F cos(α) +F sin(α)   =    YpF sin(α) − ZpF cos(α) −F l sin(α) +F l cos(α)    T₄= b₄× F4=    l Ym Zp   ×    0 −F cos(α) +F sin(α)   =    ZpF cos(α) + YmF sin(α) −F l sin(α) −F l cos(α)    T =h T₁ T₂ T₃ T₄ i

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4CGo conguration

For this conguration, θ and ϕ are referred to Thruster #3, while other thrusters' angles are evaluated thruogh simple trigonometric rules.

Numerical values used are θ = 30° and ϕ = 60°.

T₁= b₁× F₁=    l Ym Zm   ×    F sin (π − θ) cos (−ϕ) F sin (π − θ) sin (−ϕ) F cos (π − θ)    T₂= b₂× F₂=    l Yp Zm   ×    F sin (π − θ) cos (ϕ) F sin (π − θ) sin (ϕ) F cos (π − θ)    T₃= b₃× F3=    l Yp Zp   ×    F sin (θ) cos (ϕ) F sin (θ) sin (ϕ) F cos (θ)    T₄= b₄× F4=    l Ym Zp   ×    F sin (θ) cos (−ϕ) F sin (θ) sin (−ϕ) F cos (θ)    T =h T₁ T₂ T₃ T₄ i

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6CGi conguration T₁= b₁× F₁=    l −CoGy Zm   ×    0 −F 0    T₂= b₂× F₂=    l −CoGy Zm   ×    0 0 −F    T₃= b₃× F₃=    l −CoGy Zm   ×    0 F 0    T₄= b₄× F₄=    l −CoGy Zp   ×    0 F 0    T₅= b₅× F₅=    l −CoGy Zp   ×    0 0 F    T₆= b₆× F₆=    l −CoGy Zp   ×    0 −F 0    T =h T1 T2 T3 T4 T5 T6 i

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6CGo conguration

For thrusters #1, #3, #4, #6: θH= 90° and ϕH= 60°. These angles are referred to Thruster #3, the

other angles come from trigonometric rules.

For thrusters #2 and #4: θV= 30° and ϕV= 0°. These angles are referred to Thruster #5, and again,

for the other thrusters' angles, trigonometric rules are used.

T₁= b₁× F₁=    l −CoGy Zm   ×    F sin (θH) cos (−ϕH) F sin (θH) sin (−ϕH) F cos (θH)    T₂= b₂× F₂=    l −CoGy Zm   ×    F sin (π − θV) cos (ϕV) F sin (π − θV) sin (ϕV) F cos (π − θV)    T₃= b₃× F3=    l −CoGy Zm   ×    F sin (θH) cos (ϕH) F sin (θH) sin (ϕH) F cos (θH)    T₄= b₄× F4=    l −CoGy Zp   ×    F sin (θH) cos (ϕH) F sin (θH) sin (ϕH) F cos (θH)    T₅= b₅× F5=    l −CoGy Zp   ×    F sin (θV) cos (ϕV) F sin (θV) sin (ϕV) F cos (θV)    T₆= b₆× F6=    l −CoGy Zp   ×    F sin (θH) cos (−ϕH) F sin (θH) sin (−ϕH) F cos (θH)    T =h T₁ T₂ T₃ T₄ T₅ T₆ i

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8CGi conguration T₁= b₁× F₁=    l Ym Zm   ×    0 −F 0    T5= b5× F5=    l Yp Zp   ×    0 F 0    T₂= b₂× F₂=    l Ym Zm   ×    0 0 −F    T6= b6× F6=    l Yp Zp   ×    0 0 F    T₃= b₃× F₃=    l Yp Zm   ×    0 0 −F    T7= b7× F7=    l Ym Zp   ×    0 0 F    T₄= b₄× F4=    l Yp Zm   ×    0 F 0    T8= b8× F8=    l Ym Zp   ×    0 −F 0    T =h T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ i

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8CGp conguration T₁= b₁× F₁=    l −CoGy Zm   ×    0 −F 0    T5= b5× F5=    l −CoGy Zp   ×    0 0 F    T2= b2× F2=    l −CoGy Zm   ×    0 0 −F    T6= b6× F6=    l −CoGy Zp   ×    0 −F 0    T₃= b₃× F₃=    l −CoGy Zm   ×    0 F 0    T7= b7× F7=    l −CoGy Zm   ×    F 0 0    T₄= b₄× F4=    l −CoGy Zp   ×    0 F 0    T8= b8× F8=    l −CoGy Zp   ×    F 0 0    T =h T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ i

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8CGo conguration

For thrusters #1, #4, #5, #8: θH= 90° and ϕH= 60°. Reference is Thruster #4, then trigonometry.

For thrusters #2, #3, #6, #7: θV= 30° and ϕV= ϕH. Reference is Thruster #6, then trigonometriy.

T₁= b₁× F₁= T₅= b₅× F₅= =    l Ym Zm   ×    F sin (θH) cos (−ϕH) F sin (θH) sin (−ϕH) F cos (θH)    =    l Yp Zp   ×    F sin (θH) cos (ϕH) F sin (θH) sin (ϕH) F cos (θH)    T₂= b₂× F2= T6= b6× F6= =    l Ym Zm   ×    F sin (π − θV) cos (−ϕV) F sin (π − θV) sin (−ϕV) F cos (π − θV)    =    l Yp Zp   ×    F sin (θV) cos (ϕV) F sin (θV) sin (ϕV) F cos (θV)    T₃= b₃× F₃= T₇= b₇× F₇= =    l Yp Zm   ×    F sin (π − θV) cos (ϕV) F sin (π − θV) sin (ϕV) F cos (π − θV)    =    l Ym Zp   ×    F sin (θV) cos (−ϕV) F sin (θV) sin (−ϕV) F cos (θV)    T4= b4× F4= T8= b8× F8= =    l Yp Zm   ×    F sin (θH) cos (ϕH) F sin (θH) sin (ϕH) F cos (θH)    =    l Ym Zp   ×    F sin (θH) cos (−ϕH) F sin (θH) sin (−ϕH) F cos (θH)    T =h T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ i

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2.2.2 Real Control for Cold Gas Thrusters

Theoretical formula is:

u_real= min uid− TuON/OFF where uON/OFF is a vector, whose elements can be only 0 or 1.

The number of elements of uON/OFF is the same number n of thrusters: if a thruster is ON, the

correspondent element is 1, meanwhile, if it is OFF, the correspondent element is 0.

Let's make an example: consider n = 4 thrusters; at some time instant uON/OFF= [ 0 1 1 0 ] T_:

it means that, in that moment, thrusters #1 and #4 are OFF while thrusters #2 and #3 are ON.

In practice, the process to implement is the following:

1st _passage:

calculating all possibles uON/OFF, the minimization process min

uid− TuON/OFF

evaluates the best one to be used, at each time instant.

2nd _passage:

numerically, the best uON/OFF could be the one which turns all thrusters on. In such a case it is not

a good solution for what concernes propellant consumption.

To solve this kind of problem, a PWM control - Pulse Width Modulation - was added. Stating a tolarce on angular velocity and on attitude error, under which they are considered almost zero, the PWM control results as:

in detumbling phase:  



u_ON/OFF= 0 if max(|ωi|) < tolω

u_ON/OFF= u_ON/OFF else

in slew manoeuvre:  



u_ON/OFF= 0 if max(|ωi|) < tolω ∧ err < tole

u_ON/OFF= u_ON/OFF else

Moreover, it is necessary to add this kind of control, since this work is interested only in detumbling manoeuvre or in slew motion, not in the problem of tracking the attitude along the orbit: when all the thrusters shut down, it means that the manoeuvre is ended. What could happen after that moment is not treated here.

Numerical values stated for tolerances are reported in Table 2.2.

3rd passage:

eective torque provided becomes:

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2.3 Electric Thrusters

2.3.1 Thusters Specics and Congurations

The eective torque provided by a conguration of n electric thrusters is expressed by the product between a matrix H and a vector F. In fact, purpose of the Real Control is evaluate F, which is time dependent, to provide the control torque:

u_real(t) = HF (t) But what are H matrix and F vector?

Let's star from vector F:

since Electric Thrusters work in a continuos way, the vector F expresses the force provided by each one of the n thrusters:

F (t) =h F1(t) ... Fi(t) ... Fn(t)

iT

Each Fiis a positive number, and varies between a minimum value Fminand a maximum value Fmax.

As repoerted in Table 2.1, the numerical values for the 2 kinds of electric thrusters considered are: IFM :    Fmax= 500 µN Fmin= 10 µN NMS:    Fmax= 10 mN Fmin= 100 µN

The vector F is the real unkown about Electric Thrusters control, and how to evaluate it will be described in detail, in Section 2.3.2.

While F expresses the intensity part about the torque provided by a conguration of electric thrusters, the matrix H, whose dimension is [3 x n], expresses the geometrical part of the toque. Each column of H, Hi, is referred to correspondent i − th thruster, and it is evaluated as the vectorial product of

vector bi (the arm of the torque, the same used for Cold Gas ones) and vector hi (the orientation in

space of the Force vector of each thruster).

To express each bi, same geometrical parameters l, y0, z0, Yp, Ym, Zp, Zm, yet introduced in Section

2.2.1 for Cold Gas thrusters, are needed.

To express each hi, it is used the same procedure adopted for F vectors of Cold Gas thrusters. What

is needed are spherical coordinates: angles θ and ϕ - whose values have to be stated - while this time the radius of the vector is 1, since the eective module of the vector, Fi, is unkown.

Moreover, to identify each conguration, it is used same way of identication introduced for CG thrusters: number of thrusters, two capital letters (EL stands for Electric thrusters), one small letter (i, p, o). See page 21. Few examples are:

4ELi = 4 Electric thrusters inplane (in yz plane) 8ELp = 8 Electric thrusters parallel (to x,y,z S/C axes)

12ELo = 12 Electric thrusters outplane (which have a component in each axis).

Congurations in detail are presented in following pages. To visualize the ones with 4,6 and 8 thrusters, see Section 2.2.1 about Cold Gas thrusters: graphically they are the same.

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4ELi conguration

As yet introduced for 4CGi conguration, α = 15°.

For the graphical representation, see Figure 2.2.3 on page 23.

H₁= b₁× h₁=    l Ym Zm   ×    0 − cos(α) − sin(α)   =    Zmcos(α) − Ymsin(α) +l sin(α) −l cos(α)    H₂= b₂× h₂=    l Yp Zm   ×    0 + cos(α) − sin(α)   =    Zmcos(α) − Ypsin(α) l sin(α) l cos(α)    H₃= b₃× h₃=    l Yp Zp   ×    0 + cos(α) + sin(α)   =    Ypsin(α) − Zpcos(α) −l sin(α) +l cos(α)    H₄= b₄× h₄=    l Ym Zp   ×    0 − cos(α) + sin(α)   =    Zpcos(α) + Ymsin(α) −l sin(α) −l cos(α)    H =h H₁ H₂ H₃ H₄ i 4ELo conguration

Same angular coordinates used for 4CGo conguration, with same numerical values: θ = 30° and ϕ = 60°.

For the graphical representation, see Figure 2.2.4 on page 24.

H₁= b₁× h₁=    l Ym Zm   ×    sin (π − θ) cos (−ϕ) sin (π − θ) sin (−ϕ) cos (π − θ)    H₂= b₂× h₂=    l Yp Zm   ×    sin (π − θ) cos (ϕ) sin (π − θ) sin (ϕ) cos (π − θ)    H₃= b₃× h₃=    l Yp Zp   ×    sin (θ) cos (ϕ) sin (θ) sin (ϕ) cos (θ)    H₄= b₄× h₄=    l Ym Zp   ×    sin (θ) cos (−ϕ) sin (θ) sin (−ϕ) cos (θ)    H =h H₁ H₂ H₃ H₄ i

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6ELi conguration

See Figure 2.2.5 on page 25 for the graphical representation. H1= b1× h1=    l −CoGy Zm   ×    0 −1 0    H4= b4× h4=    l −CoGy Zp   ×    0 1 0    H2= b2× h2=    l −CoGy Zm   ×    0 0 −1    H5= b5× h5=    l −CoGy Zp   ×    0 0 1    H₃= b₃× h₃=    l −CoGy Zm   ×    0 1 0    H6= b6× h6=    l −CoGy Zp   ×    0 −1 0    H =h H1 H2 H3 H4 H5 H6 i 6ELo conguration

For thrusters #1, #3, #4, #6: θH= 90° and ϕH= 60°. Reference is #3, then trigonometry.

For thrusters #2 and #5 (reference): θV= 30° and ϕV= 0°. See Figure 2.2.6 on page 26.

H₁= b₁× h₁=    l −CoGy Zm   ×    sin (θH) cos (−ϕH) sin (θH) sin (−ϕH) cos (θH)    H₂= b₂× h₂=    l −CoGy Zm   ×    sin (π − θV) cos (ϕV) sin (π − θV) sin (ϕV) cos (π − θV)    H₃= b₃× h₃=    l −CoGy Zm   ×    sin (θH) cos (ϕH) sin (θH) sin (ϕH) cos (θH)    H4= b4× h4=    l −CoGy Zp   ×    sin (θH) cos (ϕH) sin (θH) sin (ϕH) cos (θH)    H5= b5× h5=    l −CoGy Zp   ×    sin (θV) cos (ϕV) sin (θV) sin (ϕV) cos (θV)    H6= b6× h6=    l −CoGy Zp   ×    sin (θH) cos (−ϕH) sin (θH) sin (−ϕH) cos (θH)    H =h H₁ H₂ H₃ H₄ H₅ H₆ i

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8ELi conguration

See Figure 2.2.7 on page 27 for the graphical representation.

H₁= b₁× h₁=    l Ym Zm   ×    0 −1 0    H5= b5× h5=    l Yp Zp   ×    0 1 0    H₂= b₂× h₂=    l Ym Zm   ×    0 0 −1    H6= b6× h6=    l Yp Zp   ×    0 0 1    H₃= b₃× h₃=    l Yp Zm   ×    0 0 −1    H7= b7× h7=    l Ym Zp   ×    0 0 1    H₄= b₄× h₄=    l Yp Zm   ×    0 1 0    H8= b8× h8=    l Ym Zp   ×    0 −1 0    H =h H₁ H2 H3 H4 H5 H6 H7 H8 i 8ELp conguration

H₁= b₁× h1=    l −CoGy Zm   ×    0 −1 0    H5= b5× h5=    l −CoGy Zp   ×    0 0 1    H2= b2× h2=    l −CoGy Zm   ×    0 0 −1    H6= b6× h6=    l −CoGy Zp   ×    0 −1 0    H3= b3× h3=    l −CoGy Zm   ×    0 1 0    H7= b7× h7=    l −CoGy Zm   ×    1 0 0    H4= b4× h4=    l −CoGy Zp   ×    0 1 0    H8= b8× h8=    l −CoGy Zp   ×    1 0 0    H =h H₁ H2 H3 H4 H5 H6 H7 H8 i

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8ELo conguration

Considerations done about 8CGo are valid:

For thrusters #1, #4, #5, #8: θH= 90° and ϕH= 60°. Reference is Thruster #4, then trigonometry.

Fore thrusters #2, #3, #6, #7: θV= 30° and ϕV= ϕH. Reference is Thruster #6, then trigonometriy.

See Figure 2.2.9 for the graphical representation.

H₁= b₁× h₁=    l Ym Zm   ×    sin (θH) cos (−ϕH) sin (θH) sin (−ϕH) cos (θH)    H₂= b₂× h₂=    l Ym Zm   ×    sin (π − θV) cos (−ϕV) sin (π − θV) sin (−ϕV) cos (π − θV)    H₃= b₃× h₃=    l Yp Zm   ×    sin (π − θV) cos (ϕV) sin (π − θV) sin (ϕV) cos (π − θV)    H₄= b₄× h₄=    l Yp Zm   ×    sin (θH) cos (ϕH) sin (θH) sin (ϕH) cos (θH)    H₅= b₅× h₅=    l Yp Zp   ×    sin (θH) cos (ϕH) sin (θH) sin (ϕH) cos (θH)    H₆= b₆× h₆=    l Yp Zp   ×    sin (θV) cos (ϕV) sin (θV) sin (ϕV) cos (θV)    H₇= b₇× h₇=    l Ym Zp   ×    sin (θV) cos (−ϕV) sin (θV) sin (−ϕV) cos (θV)    H₈= b₈× h₈=    l Ym Zp   ×    sin (θH) cos (−ϕH) sin (θH) sin (−ϕH) cos (θH)    H =h H1 H2 H3 H4 H5 H6 H7 H8 i

Since Electric Thrusters provide the thrust in continuous way, simulations with 12 thrusters are no more as computationally heavy as the correspondents whit Cold Gas. That's why 12 thrusters will be considered concerning electric IFM and NMS simulations.

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12ELp conguration

H₁= b₁× h₁=    l Ym Zm   ×    0 −1 0    H7= b7× h7=    l Ym Zp   ×    0 0 1    H₂= b₂× h₂=    l Ym Zm   ×    0 0 −1    H8= b8× h8=    l Ym Zp   ×    0 −1 0    H₃= b₃× h₃=    l Yp Zm   ×    0 0 −1    H9= b9× h9=    l Ym Zm   ×    1 0 0    H₄= b₄× h₄=    l Yp Zm   ×    0 1 0    H10= b10× h10=    l Yp Zm   ×    1 0 0    H₅= b₅× h₅=    l Yp Zp   ×    0 1 0    H11= b11× h11=    l Yp Zp   ×    1 0 0    H₆= b₆× h₆=    l Yp Zp   ×    0 0 1    H12= b12× h12=    l Ym Zp   ×    1 0 0    H =h H₁ H₂ H₃ H₄ H₅ H₆ H₇ H₈ H₉ H₁₀ H₁₁ H₁₂ i 12ELo conguration

About this conguration, only two spherical coordinates, θ0and ϕ0 need to be stated.

All the thrusters are positioned according to these two angles, according to relation presented in following page. Numerical values stated are: θ0= 60° and ϕ0= 30°.

For thrusters #9, #10, #11, #12: θG= θ0 and ϕG = ϕ0.

For thrusters #1, #4, #5, #8: θB= θG and ϕB = 2ϕG.

For thrusters #2, #3, #6, #7: θR= θG and ϕR= ϕG+ϕB−ϕ₂ G.

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H₁= b₁× h₁=    l Ym Zm   ×    sin (π − θB) cos (−ϕB) sin (π − θB) sin (−ϕB) cos (π − θB)    H₂= b₂× h₂=    l Ym Zm   ×    sin (π − θR) cos (−ϕR) sin (π − θR) sin (−ϕR) cos (π − θR)    H₃= b₃× h₃=    l Yp Zm   ×    sin (π − θR) cos (ϕR) sin (π − θR) sin (ϕR) cos (π − θR)    H₄= b₄× h₄=    l Yp Zm   ×    sin (π − θB) cos (ϕB) sin (π − θB) sin (ϕB) cos (π − θB)    H₅= b₅× h₅=    l Yp Zp   ×    sin (θB) cos (ϕB) sin (θB) sin (ϕB) cos (θB)    H₆= b₆× h₆=    l Yp Zp   ×    sin (θR) cos (ϕR) sin (θR) sin (ϕR) cos (θR)    H₇= b₇× h₇=    l Ym Zp   ×    sin (θR) cos (−ϕR) sin (θR) sin (−ϕR) cos (θR)    H₈= b₈× h₈=    l Ym Zp   ×    sin (θB) cos (−ϕB) sin (θB) sin (−ϕB) cos (θB)    H₉= b₉× h₉=    l Ym Zm   ×    sin (π − θG) cos (−ϕG) sin (π − θG) sin (−ϕG) cos (π − θG)    H₁₀= b₁₀× h₁₀=    l Yp Zm   ×    sin (π − θG) cos (ϕG) sin (π − θG) sin (ϕG) cos (π − θG)    H₁₁= b₁₁× h₁₁=    l Yp Zp   ×    sin (θG) cos (ϕG) sin (θG) sin (ϕG) cos (θG)    H₁₂= b₁₂× h₁₂=    l Ym Zp   ×    sin (θG) cos (−ϕG) sin (θG) sin (−ϕG) cos (θG)    H =h H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 i

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Graphical representation of congurations involving 12 thrusters presented so far:

Figure 2.3.1: 12ELp conguration

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2.3.2 Real Control for Electric Thrusters

The real unknown to nd is the time-dependent vector F. It has to be a vector of positive values, and it is stricly realted to the matrix H introduced so far.

1. If H is a full rank matrix (rank(H) = 3), its pseudo-inverse H+ _{exists, and can be numerically}

evaluated (this condition is always satised). Since the size of H is [3 x n], and n ≥ 4, the psesudo-inverse formally is:

   H+_{= H}T _HHT−1 theoretical expression HHT= I property satised by H+

2. Then, it is needed to evaluate the w vector, which is a linear combination of the columns of ker(H), the kernel of matrix H. This vector is numericlly evaluated such that each one of its components is positive: wi> 0.

3. The vector w is fundametal to nd a scalar parameter γ, which is evaluated as: γ = max(−H

+_u id)i

wi

with i = 1, 2, ..., n

4. Once got γ, the force vector is:

F = H+u_id+ γw In this way, each component Fiof the vector F will be positive.

5. As said previously, when an electric thruster is switched ON, its force provided can vary between a maximum and a minimum. Practically, in Simulink models, this peculiarity is demanded to a saturator block, which ensures that Fmin≤ Fi≤ Fmax.

6. Furthermore, a PWM control was added (see on page 30 for reference): PWM in detumbling manoeuvre :    F = 0 if max(|ωi|) < tolω F = F else PWM in slew:   

F = 0 if max(|ωi|) < tolω ∧ err < tole

F = F else

When F = 0, it means that the manoeuvre is ended, and all thrusters shut down. 7. The time-dependent eective torque provided by the thrusters during the manoeuvre is:

u_real= HF

It is userful to stress that H+ _{and w can be evaluated once and for all, since they are xed in time;}

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2.4 Specics and Parameters Values Recap

Substantial dierence between Cold Gas and Electrical thrusters is that the rst ones are not throt-tleable, while second ones are. It means that CG can provide only a null or a constant force F , not intermediate values; electric ones may on the contrary provide a force Fi which continuously varies in

time between a maximum and a minimum. Thrust values and way of work of Thrusters which are going to be analyzed are reported in Table 2.1 below:

minimum thrust maximum thrust throttleability

Cold Gas Vacco thruster 0N 1 · 10−2 _N _NO

Electric IFM thruster 1 · 10−5 N 5 · 10−4 N YES Electric NMS thruster 1 · 10−5 N 1 · 10−2 N YES

Table 2.1: Thrusters' specics recap

Tolerances for PWM control part were tuned in order to allow a comparison between simulations with and without CoG shift. Most simulations reach convergence wrt these values, but non all of them, as it will be shown in next chapters: if all thrusters' congurations with all ideal control laws reach converegence, it would be the best situation, but it is not always possible, so a commun reference needed to be introduced. Remember: convergenge means that the manouevre is ended, i.e. that PWM shuts all the thrusters o.

Here below, in Table 2.2, the numerical values stated are reported:

tolω [rad/s] tole [-]

Cold Gas Vacco thruster 4.3 · 10−4 6.0 · 10−4 Electric IFM thruster 2.0 · 10−5 3.0 · 10−5 Electric NMS thruster 1.2 · 10−4 8.0 · 10−5

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Ideal control laws coecients' numercal values are reported in Tables below: Table 2.3 concerns PL ideal control, while Table 2.4 concerns Sliding Surface and ST ideal control law. These values were stated after some simulations, in which the aim was minimizing angular velocity components' magni-tude. As said so far, they are not rened at best, also because the values stated have only the aim of identify and order of magnitude.

Detumbling Slew Cold Gas Vacco thruster k1= 10

( k1= 1

k2= 0.1

Electric IFM thruster k1= 1

( k1= 1 k2= 0.01 Electric NMS thruster k1= 1 ( k1= 1 k2= 0.01

Table 2.3: PL kj numerical values

Detumbling Slew

Cold Gas Vacco thruster ( k1= √ 10 k2= 0.001      ks= 0.1 k1= 1 k2= 0.001

Electric IFM thruster ( k1= √ 10 k2= 0      ks= 0.01 k1= 100 k2= 0 Electric NMS thruster ( k1= 10 k2= 0.001      ks= 0.1 k1= 10 k2= 0.001

Table 2.4: Sliding surface and ST kj numerical values

These Tables conclude the Modelling part: next chapters present the simulations (and relative results) performed through the implementation of what described up to now.

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Chapter 3

Cold Gas Thrusters Simulations

This Chapter has the aim of introducing the methodology followed in all the simulations performed (Section 3.1), as well as of presenting a whole simulation dealing with CG thrusters as an illustrative example: Sections 3.2 and 3.3 presents key performance parameters searched and how to evaluate the nal results from them, respectively after detumbling and slew manoeuvres. Finally, Section 3.4 contains a digression on Linear Regression model (used to achieve slew motion results), while Section 3.5 contains a note about the connection between Overall-Total-Impulse and propellant mass.

3.1 Simulation Methodology

Models descripted in Chapter 2 were implemented in Matlab® and Simulink®. Once assigned all the Initial Conditions needed, simulations were made run to nd out information wanted.

The goals of the simulations performed are some key performance parameters: convergence time and overall total impulse (for both detumbling and slew motion), maximum angular velocity component at convergence (detumbling only) and steady state error (slew only). How to evaluate and manage them will be explained better in a while.

In this chapter - as in the following one about electric thrusters - the CoG is located exactly at the centre of the CubeSat: h 0 0 0

iT

in Body Reference Frame.

Concerning ideal control laws, introduced in sections 2.1.5, numerical values of coecients kj needed

to be stated: within this context of, they are not rened at best, but they were chosen to indentify an order of magnitude: values used are more or less 0.001, 0.01, 0.1, 1, 10, 100. Other peculiar values are only 0 (sometimes needed to exclude a part of ideal control which may cause problems in the real control part) and √10. Few simulations were performed to state these coecients, such that they minimize the convergence time and the magnitude of angular velocity during the manoeuvres. All the numerical values are reported in Tables 2.3 and 2.4 on the preceding page.

Concerning PWM control, introduced in sections 2.2.2 and 2.3.2, it should be said that tolerances can be more shrunk than ones used, but they were intencionally realxed, to be the same when the

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CoG shift will be performed (Chapter 5) and allow the data to be compared with respect to the same reference. All the numerical values are reported in Table 2.2 on page 40.

Concerning Initial Conditions, 10 attitude matrices were stated for detumbling (Figure 3.2.1 on the following page) and 10 rotations were performed for slew manoeuvre (Figure 3.3.1 on page 52).

As said so far, some key performance parameters are the goal of the simulations performed: they will be put into some graphs which link these parameters the one to the other. Since amount of data produced is quite huge, they will be analysed and summarized in such a way to be presented in the most simple and clear way.

Let's see more in detail what was performed: in this Chapter 3 an example of simulation is going to be presented and analysed in detail, for two reasons:

1. to introduce some concepts and the key performance parameters, useful also for next chapters, in order not to repeat their explanation in each time;

2. to make an example of what happens concerning Cold Gas thrusters.

3.2 Detumbling

Initial condition on angular velocity for detumbling is ω0=

h

10 10 10 iT

deg/s, which is a suitable one for a CubeSat just released in deep space space, after the journey from Earth on the vector.

PWM's tolerance, under which angular velocity is considered almost null and detumbling phase is consequentely considered complited, is stated as tolω= 4.3 · 10−4 rad/s.

A graphical representation of initial conditions on attitude is presented in Figure 3.2.1. These ones will always be the same initial conditions for detumbling simulations involving both PL and ST ideal laws, as well as for electric thrusters detumbling simulations analyzed in next chapters.

Aim of the detumbling simulations is to nd out these key performance parameters: tconv = convergence time.

It is the time when max |ωi| < tolω, the instant in which the manoeuvre is considered to brought

to end and all the thrusters are switched OFF by the PWM part of real control; IT OT = overall Total Impulse.

It is the Total Impulse of the whole conguration: it is the sum of the total areas of the F -t plot of each thruster. Its evaluation is performed through numerical integration; IT OT is NOT the

Total Impulse of the single thruster! ωconv= convergence angular velocity.

It is the max |ωi|evaluated at t = tconv

Plots involving these parameters were built to summarize simulations' results: putting two of them on the axes of each graph, the aim is to understand how a conguration of thrusters behaves.

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Let's see a simulation in detail as an example.

Figure 3.2.1: Initial Conditions on attitude matrix for detumbling

3.2.1 PL simulations

Remenering that PL ideal control, presented in Section 2.1.5, for detumbling is: u_id = −k1Jω

numerical value for the tuning parameter is stated as k1= 10(see Table 2.3).

When all the Initial Conditions are stated, the simulation may start.

As an indicative example, a whole simulation is going to be illustrated in detail: the one chosen is 4CGi conguration, using as initial condition on the attitude the rst one in Figure 3.2.1.

During the simulation the evolutions in time of the angular velocity in its components, as well as of the real control torque applied to the S/C by thrusters, can be tracked.

When also the highest component of ω (t) becomes lower than tolω, the control torque becomes 0

since PWM control shuts down the thrusters. That instant is tconv: the manoeuvre is ended, reaching

convergence.

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Figure 3.2.2: ω (t), ureal(t)- detumbling 4CGi, PL ideal control, no CoG shift

Moreover, the time history of each thruster ring can be tracked, plotting the evolution in time of each component of uON/OF F . Remember that 1 means thruster ON and 0 means thruster OFF:

Figure 3.2.3: uON/OF F(t)- detumbling 4CGi, PL ideal control, no CoG shift

It can be seen from Figures 3.2.2 and 3.2.3 that, wheter the PL ideal control law may be continuous, CG thrusters can't behave as continuous, since their technological limit is the lack of throttleability, They can be only ON or OFF. Indeed, during all the manoeuvre, the behaviour of each thruster is more or less the same wrt the others.

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To evaluate Overall Total Impulse for CG thrusters, the procedure followed is this one: 1. through a numerical integration, Rtconv

0 uON/OF F i dtis evaluated for each i-th thruster;

2. the value obtained is multiplied by F , obtaining the Total Impulse of the single thruster; 3. the values obtained for all the n thrusters are summed up, obtaining the overall total impulse of

the conguration, from now on named simply as IT OT.

About this simulation presented so far, taken as example, the key performance parameters are evaluated by the code as:

       tconv= 92.36s IT OT = 1.5437 Ns ωconv= 0.00042077rad/s

3.2.2 ST simulations

Taking in mind that ST algotithm for detumbling, presented in Section 2.1.5, is:                    S = ω Sliding Surface uid= −k1     p|Sx|sgn(Sx) p|Sy|sgn(Sy) p|Sz|sgn(Sz)    

+ v Super-Twisting ideal control

˙v = −k2sgn(S) part numerically integrated

Numerical values of tuning parameters are stated as k1=

√

10and k2= 0.001. (see Table 2.4).

Again, the time evolution of angular velocity and control torque during the manoeuvre can be tracked (Figure 3.2.4), as well as the evolution in time of uON/OF F components (Figure 3.2.5).

The evaluation of the Overall Total Impulse can be performed through the same steps reperted in PL simulation (see on page 3.2.1).

About this simulation, the key performance parameters are evaluated by the code as:        tconv= 108.83 s IT OT = 1.8157 Ns ωconv= 0.00040094rad/s

Note that, orders of magnitude and values of key performance parameters of PL and ST simulations are very similar.

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Figure 3.2.4: ω (t), ureal(t)- detumbling 4CGi, ST ideal control, no CoG shift

Figure 3.2.5: uON/OF F (t)- detumbling 4CGi, ST ideal control, no CoG shift

From Figures above, it can be seen that during all the manoeuvre, the behaviour of thrusters is not the same of PL simulation: at the end of ST simulation the disconntinuous behaviour of majority of thrusters becomes more relevant, i.e. the ON/OFF jumps became very close one to the other. It will be a commun behaviour in ST simulations.

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3.2.3 Results

For each conguration (see Section 2.2.1), a simulation for both PL and ST ideal control laws was performed, for each one of the 10 Initial Conditions (see Figure 3.2.1).

All the results of these simulations are reported in the graphs presented in following pages.

What is wanted to be achieved is an indication of how good is a conguration with respect to the key performance parameters presented on page 3.2):

convergence time, tconv

conguration Overall Total Impulse, IT OT

convergence angular velocity, ωconv

Taking two of the three key performance parameters per time, the resulting plots are:

Figure 3.2.6: tconv- IT OT detumbling data plot, 4CGi, no CoG shift

Consider that, a result may be consider better than another, wrt the two parameters on the axes, if the marker of that simulation lies as near as possible to the origin of axes.

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Figure 3.2.7: ωconv- tconv detumbling data, 4CGi, no CoG shift