DEGREE PROJECT IN ,SPACE TECHNOLOGY SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016

Attitude Determination and Control of the CubeSat MIST

JIEWEI ZHOU

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Preface The thesis work presented in this paper has been carried out between January and June 2016 under the scheme of the CubeSat Student Satellite Project MIST at KTH. I did it as a member of the 3rd MIST Student Team under the supervision of Dr. Gunnar Tibert, Associate Professor at the Department of Aeronautical and Vehicle Engineering at KTH, and Dr. Sven Grahn, the former Technical Director of the Swedish Space Corporation and the Project Manager of MIST. I am very thankful to Gunnar and Sven for having offered me this great opportunity to work in a real engineering project and contribute to build a satellite that will be launched in the near future. Many thanks to them also for their valuable guidance during my work, their experience in a large number of real satellite missions definitely helped me learn and make a better thesis. I would like to thank Prof. Yifang Ban for helping me to get in contact with Gunnar and Sven when I was looking for a thesis work at KTH. During my everyday work inside the MIST Student Team I was surrounded by great people that are not only quite professional and skillful in their respective fields but very pleasant persons to deal with as well. Many thanks to all of them for their cooperation to my work, specially to the Team Leader Simon G¨orries,Project Assistant Agnes G˚ardeb¨ack and the rest of members working in attitude control Oscar Bylund and Csaba J´eger. Finally, but certainly not least, I would like to say thanks to my family and friends for supporting me during my work. My very special thanks to Shreyas and Rutvika, who were my most close friends during my Erasmus+ stay in Stockholm.

Jiewei Zhou Madrid July, 2016 Abstract The ADCS concept in MIST reflects the limitations of the CubeSat in terms of space, power and on- board computer computational capability. The control is constrained to the use of only magnetic torquers and the determination to magnetometers and Sun sensors in spite of the the under-actuation and under- determination during eclipses. Usually small satellites with a similar ADCS and demanding requirements fail, therefore MIST would be a design reference for this kind of concept in the case it succeeds. The objectives of this thesis work are the feasibility assessment of the concept to meet the nominal requirements in MIST and the consideration of alternatives. Firstly, the importance of gravitational stabilization and different configurations for the inertial properties are analyzed based on the linear stability regions for nadir pointing spacecraft. Besides, extended stability regions are derived for the case when a momentum wheel is used to consider alternative options for passive stabilization in terms of the inertial properties. Then a controller based on the Asymptotic Periodic Linear Quadratic Regulation (AP LQR) theory, the currently most extended and effective for pure magnetic control in small satellites, is assessed. Also a Liner Quadratic Regulator design by means of numerical optimization methods, which has not been used in any real mission, is considered and its performances compared with the AP LQR. Regarding attitude determination a Linear Kalman Filter is designed using the AP LQR theory. Finally, a robustness analysis is conducted via Monte Carlo simulations for those control and determination strategies. Sammanfattning Systemet f¨orattitydstyrning och -best¨amning i nanosatelliten MIST reflekterar sm˚asatelliters begr¨ansningarnai utrymme, elkraft och omborddatorkapacitet. Regleringen ¨arbegr¨ansadtill styrning med magnetspolar som genererar kraftmoment. F¨orattitydbest¨amningenanv¨andsmagnetometrar och solsensorer trots under-man¨ovreringoch -best¨amningvid solf¨orm¨orkelse. Vanligtvis misslyckas sm˚asatel- liter med liknande reglersystem och h¨ogakrav, s˚aom MIST lyckas skulle den bli ett referenskoncept. M˚alenmed detta examensarbete ¨aratt utf¨oraen genomf¨orbarhetsstudieav ett reglerkoncept f¨oratt m¨otade nominella kraven f¨orMIST samt unders¨oka av alternativa reglersystem. Effekten av gravitation- sstabilisering och olika masstr¨oghetskonfigurationer har analyserats med hj¨alpav linj¨ariseradestabilitet- sregioner f¨oren nadirpekande satellit. Stabilitetsregionerna f¨orstorasd˚aett roterande hjul inf¨orsi ett al- ternativt stabiliseringskoncept eftersom det roterande hjulet p˚averkar de effektiva masstr¨oghetsmomentet. Regleringsalgoritmen som utv¨arderatsi detta arbete ¨arbaserad p˚ateorin om Asymptotisk Periodisk Linj¨ar Kvadratisk Regulering (AP LKR), den som ¨armest anv¨andsamt effektiv f¨orren magnetisk styrning av sm˚asatelliter. En utformning av ett koncept baserat p˚aLinj¨arKvadratisk Reglering med numerisk opti- mering, vilket inte tidigare verkar anv¨ants f¨orett riktigt rymduppdrag, har unders¨oktsoch j¨amf¨ortsmed AP LKR-regleringen. N¨ardet g¨allerattitydbest¨amningens˚ahar ett linj¨artKalmanfilter utformats f¨or AP LKR-regleringen. Slutligen s˚ahar en robusthetsanalys gjorts genom Monte Carlo-simuleringar f¨or styrnings- och best¨amningsstrategierna. Resumen El concepto para el ADCS en MIST refleja las limitaciones de los CubeSats en cuanto a espacio, potencia y capacidad computacional del ordenador a bordo. El control est´arestringido al uso de s´olo magnetopares y la determinaci´ona magnet´ometrosy sensores de Sol a pesar de la imposibilidad de actuaci´onseg´untodos los ejes y el conocimiento incompleto en actitud durante eclipses. Normalmente peque˜nossat´elitescon un ADCS similar y exigentes requisitos fallan, por la tanto MIST ser´ıauna referencia de dise˜nopara este tipo de concepto en el caso de que tenga ´exito. Los objetivos de este trabajo fin de m´asterson la evaluaci´onde la viabilidad del concepto para cumplir los requisitos nominales en MIST y la consideraci´onde alternativas. Primero, la importancia de la estabilizaci´ongravitacional y diferentes configuraciones para las propiedades m´asicasson analizadas en base a las regiones de estabilidad lineales para veh´ıculos espaciales apuntando seg´unnadir. Adem´as, regiones de estabilidad extendidas son deducidas para el caso en el que una rueda de momento es usada con el fin de considerar opciones alternativas de estabilizaci´onpasiva en t´erminosde las propiedades m´asicas. Despu´esun controlador basado en la teor´ıa del Asymptotic Periodic Linear Quadratic Regulation, el actualmente m´asextendido y efectivo para control magn´eticopuro en peque˜nossat´elites,es evaluado. Tambi´enun dise˜node LQR por medio de m´etodos de optimizaci´onnum´erica,el cual no ha sido usado en ninguna misi´onreal, es considerado y sus prestaciones comparadas con el AP LQR. En relaci´ona la determinaci´onde actitud un Linear Kalman Filter es dise˜nadousando la teor´ıadel AP LQR. Finalmente, un an´alisisde robustez es llevado a cabo a trav´esde simulaciones de Monte Carlo para esas estrategias de control y determinaci´on. Contents

1 Introduction 1 1.1 Background...... 1 1.2 Problem Statement...... 1 1.3 Previous Work...... 2

2 Literature Study 2

3 Modeling 3 3.1 Generalities...... 3 3.2 Spacecraft...... 5 3.3 Disturbances...... 6 3.4 Actuators...... 7 3.5 Sensors...... 7

4 ADCS Algorithms 8 4.1 Attitude Control...... 8 4.2 Attitude Determination...... 8

5 Disturbances Analysis 9 5.1 Aerodynamic Torque...... 9 5.2 Residual Dipole...... 10

6 Passive Stabilization 10 6.1 Gravity Gradient...... 10 6.2 Momentum Bias...... 12 6.3 Pitch-Orbital Resonance...... 13

7 ADCS Design 14 7.1 Asymptotic Periodic LQR...... 14 7.2 LQR via Numerical Optimization...... 15 7.3 Asymptotic Periodic LQE...... 20

8 Conclusions and Discussions 21

References 22 1 Introduction There is also a prototype of a propulsion module that will be tested in MIST from one of the experi- 1.1 Background ments. However, these thrusters will not be used for active attitude control. Their only mission related to CubeSats were invented in 1999 as an educational pointing is to divert the CubeSat from the nominal tool. They were conceived to be small and simple attitude (described in Sec. 1.2). enough for university students, and with standard In the case of attitude determination the gyro- sizes to accommodate them in the rockets. Launches scopes in the on-board computer could be used to of CubeSats increased significantly in 2013. The deal with the under-determination during eclipses. number of deployments that year was nearly half of However, this proposal has been ruled out to reduce the total until then. Currently not only universi- power consumption and simplify the on-board data ties are investing in them but the governments and handling. private companies as well. This interest is due to its low cost, the availability of standardized parts, To sum up, the ADCS concept in MIST is quite the increased launch opportunities and its versatility challenging and would provide important design rec- [14, 20]. ommendations for small satellites with similar prob- lems in the case it succeeds. It would decrease the MIST is the first student satellite at KTH, a 3U cost and power consumption and save space being CubeSat to be built between during 2015 to 2017 especially attractive for CubeSats, which are a new and launched soon after that. The payload consists revolution in space of increasing interest used for a of 7 technical and scientific experiments. The exper- wide range of applications. iments have been proposed from inside KTH, from two Swedish companies and from the Swedish Insti- tute of Space Physics in Kiruna. 1.2 Problem Statement The Attitude Determination and Control System (ADCS) concept of MIST consists of magnetic tor- The nominal ADCS requirements for MIST are, quers (MTQs) for control whereas magnetometers [23]: (MGMs) and Sun sensors are used for determina- • Nadir pointing with the axis of the minimum tion. This is not the common practice for small moment of inertia along nadir and another satellites with demanding ADCS requirements and body axis parallel to the direction of flight. usually fails. The reasons are the under-actuation of pure magnetic control (PMC) and lack of com- • An accuracy of 15 deg for nadir pointing during plete attitude information associated to magnetome- sunlight. ters during eclipses. Moreover, magnetic cleanli- ness must be assured and a large effort is required • An accuracy of 5 deg for attitude determination [1,2,4,7,8, 13, 14, 48, 49, 50]. during sunlight. A momentum wheel (MW) has been proposed for MIST. It would remove the under-actuation problem The other ADCS requirements, de-tumbling and of PMC and has a higher level of torque that would nadir pointing acquisition, will not be considered in make the spacecraft dynamics faster. Nevertheless, this report. there are also the following downsides: The main objective of this thesis work is to make an assessment of the ADCS concept’s feasibility. • It has a limited angular momentum storage ca- Moreover, alternatives are also considered for either pacity as there is only internal momentum ex- the case where it is not possible to meet the require- change. Because of that momentum dumping ments with the current concept or when there is still with actuators may be needed as well. possibility to have complementary ADCS hardware. • It is prone to failure due to the presence of a Other aim is to compare different attitude control moving part. and determination strategies for the current concept. New algorithms will be also explored to see whether • It occupies more space and it has a higher there is a possibility to improve the performances ob- power consumption. tained with traditional methods. As to the extension of the work less effort will be Currently, there is not a lot of space left inside the put into the study of alternative concepts and those CubeSat and then it is unlikely that this option is related to determination strategies due to the lack of considered. time.

1 These objectives will be accomplished by means puting the control magnetic moment. The values in of literature survey, analytical studies and numeri- BCF could be obtained directly from MGMs. As to cal simulations. The Princeton CubeSat Toolbox is those in OCF precise position knowledge and mag- available for simulations. However, as it is the aca- netic field model are enough. Thus, it is possible to demic version the functionality is limited and it is achieve nadir pointing using only MTQs and MGMs necessary to write complementary codes. in principle.

1.3 Previous Work Table 2.1: Missions studied for attitude control.

During the previous semester de-tumbling analy- Feature Gurwin- PRISMA UWE-3 COMPASS-1 ses, survey for ADCS simulation toolboxes and pre- Techsat (TANGO) Mass (kg) 48 50 1 1 liminary attitude control studies for the nominal Size (cm × cm × cm) 45 × 45 × 45 80 × 75 × 32 10 × 10 × 10 10 × 10 × 10 pointing requirement were conducted. Mean The preliminary work for nominal pointing con- Altitude (km) 810 700 645 635 ditions consisted of literature survey and numerical Orbit simulations where perfect attitude knowledge was as- Inclination (deg) 98 98 98 98 sumed. The objective was to make an initial assess- Control 3 MTQs 3 MTQs 6 MTQs 3 MTQs ment of the ADCS concept’s feasibility and evaluate Hardware 1 MW 1 RW Control the performances when a MW is considered. The Magnetic studies conducted in this report will be a continua- Moment (Am2) 1 2.5 0.05 0.1 tion of these initial analyses. Residual Dipole Moment (Am2) - 0.1 0.05 - 2 Literature Study Determination 3 MGMs 3 MGMs 9 MGMs 3 MGMs Hardware 6 Sun 3 Gyroscopes 5 Sun Sensors 6 Sun Sensors CubeSats and micro-satellites with a mission pro- 1 Fine Sun Sensors file in terms of ADCS similar to MIST are analyzed. Sensor The aim is to know the usual control and determina- tion strategies as well as the achievable performances for PMC and when there is lack of complete attitude The LQR is based on a linearized model around information during eclipses. Moreover, slightly differ- OCF. It minimizes a cost function that considers con- ent alternatives to MIST’s ADCS concept are studied trol accuracy and effort. For that an optimal feed- as well. back matrix is obtained by solving a system of or- dinary differential equations. In the case of time- The basic properties about some of the missions invariant systems the problem is reduced to the Al- studied are summarized in the Table 2.1. Other satel- gebraic Riccati Equation (ARE). The problem with lites have been also analyzed and will be mentioned PMC for satellites is that the system is time-varying later on as references as well. since the magnetic field changes throughout the orbit. There are two commonly used control algorithms However, there are methods to simplify the problem for PMC. One of them is the Linear Quadratic Regu- by recovering the ARE: lator (LQR) used in Gurwin-Techsat, PRISMA, Hok- ieSat and COMPASS-1 [4,6,8, 13]. Other less ex- • A technique is the Asymptotic Periodic LQR tended method is the COMPASS controller and it used in PRISMA and Gurwin-Techsat. It as- is considered in Gurwin-Techsat and UWE-3 [4, 10]. sumes that the magnetic field is periodic, which It is interesting to note that both algorithms were is more or less true with the orbital period, and used in Gurwin-Techsat and it was launched in 1998. the weight of the control effort in the cost func- Then giving that, for example, UWE-3 (2013) and tion is large enough. Under these conditions an PRISMA (2010) are relatively recent missions it average of the magnetic field over an orbit can could be concluded that there have not been signifi- be used for the ARE [11]. cant breakthroughs in PMC for nadir pointing since at least 20 years ago. • A very similar alternative to the Asymptotic The COMPASS controller feedbacks the differ- Periodic LQR is used in HokieSat. The differ- ences in BCF and OCF1 of both Earth magnetic ence is that now the control magnetic moment field and its derivative, which are then used for com- used is assured to be perpendicular to the Earth 1The coordinate systems BCF and OCF are explained in Sec. 3.1.

2 magnetic field by means of a mapping function was supposed to test several PMC algorithms such [9]. as COMPASS for nadir pointing [10]. But finally it ended with a residual magnetic moment which has The most common estimation method is the Ex- the same order of magnitude as that for control. This tended Kalman Filter (EKF), which is an version is possibly due to the magnetization of the antennas, of Kalman filter valid for nonlinear systems. It is made out of stainless steel. Thus the original plan used in PRISMA, HokieSat and UWE-2 [6,8, 49]. was not able to be conducted [1]. Other similar extensions for nonlinear systems are Gurwin-Techsat has only magnetometers and also considered such as Isotropic Kalman Filter (IKF) then it is more restrictive than MIST in terms of in UWE-3 and Unscented Kalman Filter (UKF) in estimation. Simulations suggested that the LQR and SEAM [2, 50]. COMPASS can meet the pointing requirements with The Kalman filter minimizes the uncertainty in an accuracy of 8 deg and 1 deg, respectively. However the estimation of the state vector. It uses a process it was necessary a small momentum bias in the case model to predict and a measurement model to cor- of the COMPASS controller to achieve the stabiliza- rect the initial estimate. The problem is that those tion. From flight results it is known that a failure of models are supposed to be linear and nearly all prac- the LKF (Linear Kalman Filter) used for the LQR, tical applications imply nonlinear equations. Both possibly due to magnetic disturbances of the stopped EKF and IKF are based on linearized models. The MW, impeded the activation of the controller. For difference between them it is that IKF assumes the the COMPASS controller, in-orbit results showed a same covariance in all the directions for the state vec- nadir pointing accuracy of 3 deg as long as there is a tor simplifying the estimation procedure [48]. On the small momentum bias [4]. other hand UKF propagates the mean and covariance Finally the following conclusions should be through the nonlinear models based on the Unscented pointed out: Transformation to reduce linearization errors [50]. It • A determination concept less restrictive than is more accurate than EKF and it requires a similar that in MIST could have a worse performance computational effort [51]. comparing PRISMA and UWE-3. The possi- In addition to the determination hardware in ble reason is the high level of magnetic dis- MIST, UWE-2 and UWE-3 have gyroscopes and turbances in UWE-3, which affects the magne- SEAM has star trackers. Then, they all have sen- tometers. It is likely that with the same mag- sors to complement the magnetometers and avoid the netic cleanliness the results in UWE-3 would be lack of complete attitude information during eclipses. slightly better. From numerical simulations it is known that the esti- mation error is around 0.2 deg for UWE-3, 30 deg in • Similar ADCS concepts to MIST and slightly UWE-2 and 0.02 deg in SEAM [48, 49, 50]. The high different alternatives usually fail with the on- performance of SEAM is because of the star track- board magnetic disturbances being the main ers. In the case of UWE-3 it should be mentioned reason. that there were some errors not taken into account in the modeling of sensors, thus the actual accuracy • One of the few missions analyzed with reported is expected to be worse. Regarding in-orbit results ADCS success is PRISMA. From its flight re- it is known that the EKF has been proved reliable sults it could be inferred that it is possible to for UWE-3 having an accuracy around 5 deg [1]. Be- meet the ADCS requirements in MIST as long sides, SEAM has not been launched yet and there is as the magnetic cleanliness is assured. no published information for UWE-2. • The Gurwin-Techsat also succeeded with its The missions PRISMA and COMPASS-1 have COMPASS controller. It has a higher nadir similar ADCS concepts to MIST: PMC and only pointing accuracy than PRISMA using only magnetometers available during eclipses. In the case magnetometers for estimation. However, MW of PRISMA, simulations showed a nadir pointing ac- was used. curacy of 5 deg and a estimation error around 2 deg [6]. Afterward in-orbit results indicated a pointing error of 15 deg and an accuracy in the estimation of 3 Modeling 3 deg [7]. As to COMPASS-1 it is known from flight 3.1 Generalities results that its ADCS failed due to large errors in the Sun sensors [13]. The coordinate systems that will be considered In spite of having a reaction wheel (RW) UWE-3 for the study of attitude control and determination

3 are: as the algebraic sum of the on-board electronic equip- ment and set in the worst direction: perpendicular to • Earth-Centered Inertial (ECI) frame. the orbital plane. This assumption is since the orbit is polar and then the Earth magnetic field lines are • Body Coordinate Frame (BCF). The z axis has mainly inside the orbital plane. Studies for increasing the minimum moment of inertia and is toward residual dipole will be conducted. the deployable solar panels. Then the x axis is parallel or perpendicular to the panels with the y axis completing a right handed frame. Table 3.1: MIST reference conditions. Element Parameter Value • Orbit Coordinate Frame (OCF). The z axis is Orbit Eccentricity 0.001 pointing toward zenith and the y axis is perpen- Inclination 97.94 deg dicular to the orbital plane. The x axis com- Mean Altitude 645 km pletes a right handed frame. Besides, the direc- Period 97.57 min 2 tion of the y axis is so that the x axis is near Inertial Ix 0.037 kgm 2 the direction of flight. Properties Iy 0.051 kgm I 0.021 kgm2 Under the nominal pointing condition BCF coincides z x 0 mm with OCF [22]. G y 0 mm Table 3.1 contains the reference conditions nec- G z 0 mm essary for attitude studies. Nearly all the results in G Actuators h˙ 0.23 mNm this report are obtained according to these values. wmax h 1.7 mNms Nevertheless, some of them are uncertain and could wmax µ 0.2 Am2 be different in some cases. Those results where other Cmax Disturbances µ 0 mAm2 conditions are considered will be pointed out. Dx µ 5 mAm2 The orbital parameters correspond to the Two- Dy µ 0 mAm2 Line Element Set (TLE) of the Reference Orbit 1 in Dz C 2.1 [22]. The most doubtful value here is for the alti- D Solar Panel ρ 0.75 tude and it is possible to end with a lower orbit up Sa Solar Panel ρ 0.17 to 400 km. Ss Solar Panel ρ 0.08 The moments of inertia (I , I and I ) are esti- Sd x y z Solar Panel ρ 0 mated assuming a cuboid with a plate in the top for St Radiator ρ 0.15 the deployable solar panels. The mass of the pan- Sa Radiator ρ 0.69 els are obtained from the specifications [41]. Then Ss Radiator ρ 0.16 regarding the cuboid a mass is supposed according Sd Radiator ρ 0 to the typical densities of CubeSats. Furthermore, it St ρ 1 has been assumed that the panels are along BCF x Ea ρ 0 axis. As mentioned before there is other configura- Es ρ 0 tion with the panels along BCF y axis. In that case Ed ρEt 0 the moments of inertia Ix and Iy merely switch their Sensors ω 55 √nT current values. m Hz bm 50 nT The values xG, yG and zG represent the position for the center of mass. They are given with respect Mm 0.5 % ω 0.01 √1 to a coordinate system with the same orientation as s Hz BCF and whose origin is the geometric center of the bs 0.015 cuboid above mentioned. Here, it is only an assump- Ms 0.5 % tion that the center of mass coincides with the that geometric center. In the part of actuators the parameters corre- The drag coefficient varies from 2 to 4 as indicated spond to the capabilities of the MTQs and MW. in Sec. 3.3 and the value here is only an initial guess. Specific explanations about them can be found in Higher coefficients will be considered when analyzing Sec. 3.4. The values are from the specifications of the performances. the actuators bought for MIST [38, 40]. The optical coefficients for radiation pressure Regarding the disturbances the parameters are with the subscript S are for Sun and those with E explained in Sec. 3.3. The residual dipole is obtained are for Earth. The default values from the Princeton

4 CubeSat Toolbox are used [27]. The equation of angular momentum is comple- Finally there are errors for the sensors (explained mented by the kinematic relations between angular in Sec. 3.5). In the case of magnetometers the velocity and parameters that defined the attitude. datasheet is used [39]. Regarding the Sun sensors Both 3-2-1 Euler angles and quaternion will be used it is known from the supplier that a reasonable error in these equations. range is 2 to 6 deg and then the parameters are se- lected to reproduce that range2. These errors depend Linear Model on the temperature and this usually changes over a relatively broad range during a mission. Apart from the above a linear model is also de- rived. This will help to get preliminary results with less computational effort and understand better the 3.2 Spacecraft problems studied. Complete Model The linear model is defined considering the fol- lowing assumptions: The CubeSat is modeled as a rigid body with movable parts to take into account the possible addi- • The equations are linearized around the orbital tion of a momentum wheel. On the other hand, the attitude given by OCF. variable inertial properties caused by the use of the • There is no orbital eccentricity e, thus Ω is di- thrusters will not be modeled. This is since it will rectly the mean motion Ω . oblige to also consider the torque thrusters produce o and as it was mentioned in Sec. 1.1 while they are • The only disturbance is gravity gradient. used nadir pointing is not required. Moreover, it is assumed that the motion of the • The momentum wheel is along the BCF y axis. center of mass is uncoupled with respect to the at- Furthermore dimensionless parameters are intro- titude dynamics. The model SGP4 available in the duced considering that: Princeton CubeSat Toolbox is used to propagate the orbit based on the TLE of the Reference Orbit 1 • Time derivatives and angular velocities are mentioned in Sec. 3.1. This is the most common nondimensionalized with Ωo. model for near-Earth orbits, it takes into account the atmospheric drag and uses low-order Earth gravity • Usual values of the magnetic field Bc in the or- [25, 27]. bit of the CubeSat and magnetic moment µc The attitude dynamics of the motion with respect the actuators are able to provide are used for to OCF is given by the angular momentum equation nondimensionalization.

dHG • The control torque given by the momentum = Iω˙ +ω×Iω+ω×hw = TI +TC +TD. (3.1) dt wheel is nondimensionalized with hwyoΩo being hwyo the angular momentum in equilibrium. Here I is the inertia matrix of the entire spacecraft (including the wheel), ω the angular velocity of the Based on those considerations it is obtained the CubeSat and hw the angular momentum of the wheel state-space representation with respect to the CubeSat. x˙ = Ax + Bu, (3.3) Regarding the torques in the Eq. 3.1, TI is the contribution of the inertial forces and it is where the terms represent: ˙ TI = −IΩ − Ω × IΩ + 2ω × (IGU − I) Ω − Ω × hw. • x is the state vector containing the dimension- (3.2) less angular velocity in BCF and 3-2-1 Euler In this expression Ω is the angular velocity of OCF angles. with respect to ECI frame, Ω˙ its derivative in that T  ? ? ?  frame, IG the moment of inertia around the center of x = p q r ψ θ φ . (3.4) mass and U the identity matrix. •A is the system matrix. It is expressed via The other torques in Eq. 3.1 are disturbances TD Iy−Iz Iz−Ix the inertial ratios σ1 = , σ2 = and (described in Sec. 3.3) and control torque TC (in Ix Iy Sec. 3.4). σ = Iy−Ix . 3 Iz 2The worst case is 5 to 7 deg when the Earth albedo is not modeled. Nevertheless, here an in-between situation has been assumed.

5 A = Based on the assumptions stated above the grav-  hwyo hwyo  0 0 σ1 − 1 + 0 0 −4σ1 − IxΩo IxΩo ity gradient torque is  0 0 0 0 3σ 0   2   hwyo hwyo . 3µ  1 − σ3 − I Ω 0 0 −σ3 − I Ω 0 0  E  z o z o  TGG = uG × IuG. (3.8)  0 0 1 0 0 0  3   rG  0 1 0 0 0 0  1 0 0 0 0 0 Where µE is the gravitational parameter of the Earth (3.5) and uG is the unit vector for the center of mass’s po- sition. • u is the input vector. It comprises the dimen- In the case the orbital eccentricity is small the sionless control magnetic moment and torque gravity gradient torque given by Eq. 3.8 can be sim- provided by the MW in BCF. plified as 2 TGG = 3ΩouG × IuG, (3.9) h i T ? ? ? ˙ ? where Ωo is the mean motion. u = µx µy µz hwy . (3.6) Residual Dipole •B is the input matrix. The Earth magnetic field is in this term and it is in OCF. Its variation The magnetic disturbances of the on-board elec- over the orbit makes the system time-varying. tronics are summarized in a total residual dipole. Then using the general expression for the torque a magnetic dipole produces it is obtained that

TRD = µD × B. (3.10)  0 Bz Iy − By Iy 0  Bc Ix Bc Ix In this equation µ is the residual dipole moment  − Bz 0 Bx − hwyoΩo  D  Bc Bc Bcµc  and B is the Earth magnetic field. Bcµc  By Iy − Bx Iy 0 0  B =  Bc Iz Bc Iz  . For the Earth magnetic field the WMM 2015 and 2   IyΩo  0 0 0 0    the tilted dipole model are used. The latter is only  0 0 0 0  for cases where simplifications are required to un- 0 0 0 0 derstand better and quicker the fundamentals of the (3.7) problems analyzed. Similar linear models and their derivations can be found in literature [15, 16, 18, 19]. Aerodynamic Torque This disturbance is due to the interaction between 3.3 Disturbances the atmosphere and the spacecraft surface. Here the models for torques caused by distur- At orbital altitudes the density is low enough to bances are described. Usual contributions of gravity have a collisionless flow. Under this assumption the gradient, residual dipole, aerodynamic torque and ra- air particles impact the surfaces giving rise to a mo- diation pressure are considered. mentum exchange in the direction of the relative mo- Most of the models associated to the disturbances tion, which in turn produces a force in the same di- considered are functions already implemented in the rection. Princeton CubeSat Toolbox [21, 27]. The only model Given flat surfaces the force for the face i is given that is not in the toolbox is the World Magnetic by  1 Model 2015 (WMM 2015) and then the function 2 ρCDAi cos αiVwkVwk if cos αi > 0 FADi = . available in MATLAB is used. 0 if cos αi ≤ 0 (3.11) Gravity Gradient Where Vw is the relative velocity of the flow, ρ its density, CD the drag coefficient and Ai the area of The model considered in this case is the most the face i. Besides, cos α = − Vw · n being n the i kVwk i i commonly used. It corresponds to the gravity field of unit outward normal of surface i. a spherical and homogeneous body. Moreover when The condition in cos αi is to rule out the faces computing the torque only the first-order term of a not being impacted by the incident flow. CD varies Taylor expansion is considered. This expansion is in between 2 and 4 depending on the amounts of air Lc terms of , where Lc is a typical length of the Cube- particles absorbed and reflected. In the case all the rG Sat and rG is the distance of the center of mass to particles are absorbed it is 2 and when they all are the Earth. reflected it is 4.

6 As for ρ the model AtmJ70 is used. This model The torque provided by the MTQs is the same as takes into account the solar radiation and the geo- that for the residual dipole in Sec. 3.3, magnetic storms providing reasonable predictions. Once the forces are computed the torque is ob- TCm = µC × B for kµC k ≤ µCmax. (3.16) tained supposing that the point of application for each face is its geometric center. Denoting ri as the Here µC is the control magnetic moment and µCmax position for the geometric center of the face i with is its the saturation value. respect to the center of mass it is obtained that As to the MW the control torque is given by

X ˙ ˙ ˙ TAD = ri × FADi. (3.12) TCw = −hw for khwk ≤ hwmax and khwk ≤ hwmax. i (3.17) In this equation h˙ wmax and hwmax are the saturation Radiation Pressure values for the control torque and angular momentum, respectively. This disturbance is due to the interaction of the photons emitted by a source and the faces of the spacecraft. A photon striking a surface can be 3.5 Sensors absorbed, specularly reflected, diffusely reflected or A model for the sensors is obtained based on the transmitted. In relation to the fractions of the in- work [24] done for MIST and [28] conducted previ- coming photons for a surface it is true that ously for other project. The inaccuracies considered for the sensors are: ρa + ρs + ρd + ρt = 1, (3.13) • Bias. It is a static offset that could change from where ρ is the fraction absorbed, ρ specularly re- a s one operating cycle to another. flected, ρd diffusely reflected and ρt transmitted. Assuming flat surfaces the force produced in the • Noise. This is a random error and it is usually face i is given by modeled as Gaussian white noise3.

h  ρd  i FRi = −pAis·ni 2 ρss · ni + ni + (ρa + ρd) s . • Scale factor. This is proportional to the actual 3 (3.14) value of what is measured and comes from the Fp sensitivity of the sensors, that is, the minimum In this equation p = c is the radiation pressure be- ing Fp the flux density of the source and c the speed value they are able to distinguish. of light. s is the unit vector for the position of the source with respect to the center of mass • Crossed-coupling. It is due to errors in the As in the case of the aerodynamic torque here is manufacturing and misalignment in the place- ment of the sensors. Those errors cause that also the condition in s · ni = cos γi, that is, when the measurement in one axis is affected propor- a face is not illuminated by the source (cos γi ≤ 0) there is no force. tionally by the other axes. Once the force for each surface is computed the The mathematical expressions based on the torque is obtained as in the case of the aerodynamic model described above are torque with X T = r × F . (3.15) R i Ri B˜ = bm + MmB + ωm (3.18) i The sources considered are the Sun, the Earth for the magnetometers and albedo and the direct radiation from Earth. u˜S = bs + MsuS + ωs (3.19) 3.4 Actuators for the Sun sensors. Here the terms in the left-hand MIST has three single-axis MTQs and a possi- side are the measured Earth magnetic field (Eq. 3.18) ble MW. There are a lot of details to be added to and Sun vector (Eq. 3.19). In the right-hand side the the ideal models, however only the control limits are 1st term is bias, the 2nd represents scale factor and considered for now. crossed-coupling and the 3rd is noise. 3It is a stochastic process where the current instant is uncorrelated with respect to other instants. Thus, it has a constant power spectral density and it is equally important in all the frequencies. Besides, it follows a Gaussian distribution.

7 4 ADCS Algorithms is based on a stochastic linear measurement model given by 4.1 Attitude Control y = Cx + v. (4.6) The LQR computes an input vector for the linear In Eq. 4.6 the terms represent: model in Eq. 3.3. This input vector minimizes for any initial condition the cost function given by • y is the output vector and contains the mea- surements of the magnetometers and Sun sen- Z T sors minus the Earth magnetic field and Sun T T
J = x Qcx + u Rcu dt, (4.1) vector in OCF, respectively. 0 T   where T is time chosen to evaluate the performance, y = ∆Bx ∆By ∆Bz ∆uSx ∆uSy ∆uSz . (4.7) Qc is the weighting matrix for control accuracy and Rc for control effort. •C is the output matrix. It comprises the Earth The general solution of this Calculus of Variations magnetic field and Sun vector in OCF. problem is −1 T   u = −Rc B P(t)x. (4.2) 0 0 0 By −Bz 0  0 0 0 −Bx 0 Bz  Here P(t) is an unknown matrix to be determined    0 0 0 0 Bx −By  through a system of ordinary differential equations. C =   . (4.8)  0 0 0 uSy −uSz 0  In the case of time-invariant systems P is con-    0 0 0 −uSx 0 uSz  stant and the problem is simplified to the ARE as 0 0 0 0 uSx −uSy mentioned in Sec.2 and which is given by • v represents the errors in the sensors. T −1 T PA + A P − PBR B P + Qc = 0. (4.3) c Now returning to the feedback term in Eq. 4.5 L is However, as seen in Sec. 3.2 the input matrix changes the gain matrix and along the orbit and then the system is time-varying. yˆ = Cxˆ (4.9) Thus, different approaches with a constant P are con- sidered in Sec.7 for a practical design. is the estimated output vector. Finally it should be mentioned that despite the The uncertainties w and v are supposed to be fact that this controller is the most extended for PMC Gaussian white noises in the LQE theory but actu- it can be used for any other actuator. ally they are quite complex to modeled. Regarding the errors in the sensors there are several components 4.2 Attitude Determination of different nature as discussed in Sec. 3.5. In the case of w it contains the nonlinear effects and the lack of The attitude determination method that will be complete knowledge for the torques acting over the implemented in the simulation models is the LKF. CubeSat. However, it is well-known that the assump- It is also known as LQE (Linear Quadratic Estima- tion of Gaussian white noise does usually not worsen tor) and is not common for small satellites. The only significantly the performance of the estimator. mission found where it was used is Gurwin-Techsat The gain matrix L is computed by minimizing the [4]. covariance of the state vector’s estimation error being Adding uncertainties w to Eq. 3.3 it is obtained the solution a linear stochastic model for the system given by T −1 L = Pe(t)C Re . (4.10)

x˙ = Ax + Bu + w. (4.4) Here Pe(t) is the covariance of the estimation error and Re is the covariance of the sensors. The only un- The LQE uses it for a replica of the system to esti- known is Pe(t) and it is necessary to solve a system mate the state vector as of ordinary differential equations to obtain this ma- trix. For time-invariant systems P is constant and ˙ e xˆ = Axˆ + Bu + L (y − yˆ) . (4.5) it comes from the ARE

Here xˆ is the estimated state vector and the 3rd term T T −1 APe + PeA − PeC Re CPe + Qe = 0, (4.11) of the right-hand side is the feedback of the measure- ments. This feedback is necessary since the uncer- where Qe is the covariance of the uncertainties in the tainties in Eq. 4.4 make the estimation drift and it system model. The problem is that C changes over

8 the orbit and along the year. When comparing LQE 5.1 Aerodynamic Torque and LQR in Sec. 4.1 there are a lot of similarities. The order of magnitude of the aerodynamic This is since there is a duality and it can be used for torque is a practical design of the LQE based on an approach T ∼ ρ A V 2L. (5.3) for LQR (discussed in Sec. 7.3). AD c c c In this expression ρc is the typical value of the atmo- spheric density, it depends on the altitude and can 5 Disturbances Analysis be obtained from Fig. 5.1. Then r Here the importance of the different disturbances µE considered now for MIST is assessed. The study will Vc = (5.4) rG be not only for the reference conditions but for other conditions as well given the uncertainty of some pa- is the velocity of the CubeSat and it is also a function rameters in Table 3.1. of the altitude. Table 5.1 summarizes the relevance of the dis- Assuming that the deployable solar panels are turbances for the reference conditions. The proce- long BCF x axis they are the main contribution. dures followed to estimate the aerodynamic torque is Moreover the torque they produce is along the BCF in Sec. 5.1 and the residual dipole torque in Sec. 5.2. y axis. In the case of the torque caused by radiation pres- In the case of the reference altitude it is obtained −14 3 sure it has an order of magnitude given by that ρc = 5 · 10 kg/m , Vc = 7 km/s and therefore

TR ∼ pcAcL = 0.03 µNm, (5.1) TAD ∼ 0.02 µNm. (5.5) where pc = 3 µPa is the usual value of radiation 2 pressure, Ac = 0.03 m the characteristic area of the CubeSat and L = 0.3 m its length. 10-12

Table 5.1: Disturbances under reference conditions. -13

Disturbance Order of Magnitude $ 10 3

(µNm) m

/ g

Residual Dipole 0.2 k #

Aerodynamic Torque 0.02 ; 10-14 Radiation Pressure 0.03

10-15 Returning to the linear model described in 400 500 600 700 800 2 Altitude [ km ] Sec. 3.2 the term IyΩo in Eq. 3.7 is a characteristic torque associated to the inertia of the CubeSat. It represents the gravity gradient torque as well as the Figure 5.1: Atmospheric density. motion. Considering the reference altitude it turns out to be As it was mentioned in Sec. 3.1 it is possible 2 µE to end with an altitude as low as 400 km. Under IyΩo = Iy 3 = 0.06 µNm. (5.2) rG that condition the velocity of the CubeSat remains This result indicates that the disturbances prevail with the same order of magnitude after analyzing over the inertia of the CubeSat under the reference Eq. 5.4 whereas the density becomes much larger as −12 3 conditions. Hence, the open-loop system’s motion ρc = 6 · 10 kg/m . Hence, the aerodynamic torque will be determined by the disturbances rather than also has a huge increase being now by its own properties. When talking about a closed-loop system the con- TAD ∼ 2 µNm. (5.6) trol torque conducts the motion since it will have usu- When MTQs are used the control magnetic moment ally the same order of magnitude as the maximum required is disturbance torque given by the residual dipole. The 2 TAD 2 MTQs have a saturation value of 0.2 Am according µC ∼ ∼ 0.05 Am , (5.7) to Table 3.1. Thus, the residual dipole is not a prob- Bc lem. which is reasonable given their saturation value.

9 5.2 Residual Dipole σ1σ3 > 0 (6.2) The Earth magnetic field has an order of magni- and tude given by 2 (1 + 3σ1 + σ1σ3) > 16σ1σ3. (6.3)  3 RE Bc ∼ Bs , (5.8) rG The stability only depends on the inertial ratios σ1 and σ3. Thus it is possible to plot the stability re- where Bs = 50 µT is the typical value on the Earth’s gions in the plane σ1-σ3 shown in the Fig. 6.1. surface and RE is the radius of the Earth. As pointed out before the worst case for MIST is when the residual dipole is normal to the orbital 1 plane. Hence, in that case the torque it generates are Previous Con-guration along BCF x axis and z axis. New Con-guration Under the reference conditions Bc = 40 µT and 0.5

the residual dipole torque is Lagrange Region 3

< 0 TRD ∼ BcµD = 0.2 µNm. (5.9) A Resonance Line

A! DeBra-Delp Region The Earth magnetic field does not change its or- -0.5 der of magnitude for an altitude of 400 km according to Eq. 5.8. Thus, the altitude is not a key factor for -1 this disturbance. -1 -0.5 0 0.5 1 The most important parameter here is the resid- <1 ual dipole moment and as indicated before in Sec.2 it is the main cause of failures in ADCS concepts simi- Figure 6.1: Gravity gradient stability regions. lar to MIST. Although the estimation now is 5 mAm2 after adding algebraically the contribution of all on- board electronic devices, there is a large possibility In order to understand better the results in to end with a higher value due to unidentified mag- Fig. 6.1 it should be explained the contributions to netic effects of some components in the spacecraft. the stability of the torques involved: CubeSats such as UWE-3 and SEAM4 ended with 2 magnetic disturbances of 0.05 Am . Given this µD • Gravity gradient. The axis with the minimum the disturbance torque is similar to the case of the moment of inertia needs to be along the local aerodynamic torque for an altitude of 400 km. vertical for stability.

6 Passive Stabilization • Centrifugal torque. The axis perpendicular to the orbit plane has to have the maximum mo- 6.1 Gravity Gradient ment of inertia for stability. The environmental torques over the spacecraft • Coriolis torque. It provides gyroscopic stability corresponding to the gravity gradient and inertial irrespective of the inertia matrix. forces could contribute to the stabilization of the at- titude around OCF. This type of passive control is called gravitational stabilization and it is a problem Furthermore the six possible moments of inertia’ that has been analyzed thoroughly in the literature orders defined the same number of regions in the σ1- [15, 16, 18, 19]. σ3 plane as shown in Fig. 6.2. Thus based on this The stability analysis is based on the linear model figure it can be concluded that: the Lagrange re- derived before in Sec. 3.2. Assuming no momentum gion is stable due to gravity gradient and centrifu- bias in the system matrix given by Eq. 3.5 and com- gal forces, and the DeBra-Delp region because of the puting its eigenvalues the conditions for stability are Coriolis forces. The latter is not usually used as the gyroscopic stability disappears when there is energy σ1 > σ3, (6.1) dissipation in the spacecraft. 4The information for UWE-3 is from Table 2.1. As to SEAM it is also a KTH project and there are a lot of data available about it.

10 to the under-actuation. Thus, there are periods of 1 time when the system is basically uncontrolled and

Iy > Iz > Ix they can give rise to significant deviations from OCF if they are long enough. The same simulation as in 0.5 Iz > Iy > Ix Fig. 6.3 is run again with a controller now. The re-

Iy > Ix > Iz sults are in Fig. 6.4 and it seems that the pointing 3

< 0 error will always become large at some point due to Iz > Ix > Iy under-actuation.

-0.5 Ix > Iy > Iz

Ix > Iz > Iy 200 ]

-1 /

-1 -0.5 0 0.5 1 [ 0 A <1 -200 0 2 4 6 8 10

100

] /

Figure 6.2: σ1-σ3 plane. [ 0 3 -100 The configuration of MIST at the end of the last 0 2 4 6 8 10 200

semester was with the deployable solar panels along ] /

BCF y axis. This makes it to be in the 4th quad- [ 0 ? rant of the Fig. 6.1, that is, in a unstable region. In -200 0 2 4 6 8 10 Fig. 6.2 it is seen that the rotation is around the axis Orbit No with intermediate moment of inertia while the axis with minimum moment of inertia is along the local vertical. Thus, according to what was discussed be- Figure 6.4: Controlled system in old configuration. fore the instability in this case is due to centrifugal forces. Fig. 6.3 shows the results in open-loop ob- The decision then was to rotate the panels and tained with a more realistic nonlinear model where put them along the direction of flight. Now the new similar assumptions are made as in the case of the configuration is in the stable Lagrange region but linear model. It is possible to observe that the insta- near the line associated to a phenomenon known as bility is in yaw. There are large oscillations around Pitch-Orbital Resonance as seen in Fig. 6.1. As it ψ = −90 deg as expected since for that orientation will be discussed in Sec. 6.3 this phenomenon can be MIST is in the stable Lagrange region. handled when a controller is used. Therefore, no fur- ther modifications are necessary in terms of inertial properties.

0 ]

/ -50 [

-100 Table 6.1: New configuration stability margins. A -150 Parameter σ1 σ3 σ1-σ3 0 2 4 6 8 10

20 Lower Margin 0.14 0.68 - ]

/ Upper Margin 0.18 0.14 -

[ 0

3 Margin - - 0.10 -20 0 2 4 6 8 10

] 20 /

[ 0 Table 6.1 contains the stability margins of the ? -20 new configuration. They represent the allowable vari- 0 2 4 6 8 10 ations of the inertial ratios. The columns σ and Orbit No 1 σ3 are obtained keeping one of the ratio constant and changing the other one. In the case of σ1 the Figure 6.3: Uncontrolled system in old configuration. lower limit is Eq. 6.1 and the upper limit is actually Eq. 6.16, which is a property of the moments of iner- In principle, the instability with the panels along tia and not a stability condition. As to σ3 the lower BCF y axis could be handled using a controller. How- margin corresponds to Eq. 6.2 and the upper mar- ever, in the case of PMC it is not possible to al- gin to Eq. 6.1. Then the column σ1-σ3 is the margin ways assure stability in the closed-loop system due when both ratios are changed at the same time. It

11 has been computed for the worst case and here that is b > 0 (6.12) the displacement that breaches the condition Eq. 6.1. and 6.2 Momentum Bias b2 > 4c. (6.13)

The previous configuration could be stabilized From the conditions in Eq. 6.1, Eq. 6.11, Eq. 6.12 with a momentum bias along BCF y axis. This is and Eq. 6.13 it can be concluded that now there is known as dual-spin stabilization. Analyses about this one parameter more affecting the stability: η as well combined with the gravitational stabilization are less as σ1 and σ3. Hence, once a value for η is set it is common in the literature and will be covered here possible to plot similar stability regions such as in with some extension. Similar studies can be found in Fig. 6.1. [34] and [35]. Before continuing some clarifications should be The characteristic polynomial of the system ma- made about the Lagrange and DeBra-Delp stability trix given by Eq. 3.5 is regions. The Lagrange region is stabilized by torques that depends on the attitude and not on the angular λ4 + bλ2 + c
λ2 − 3σ
= 0 (6.4) 2 velocity, thus it is said to be statically stable. The where stability in the DeBra-Delp region is due to Corio- lis forces and it is said to be statically unstable but   1 − σ3 gyrically stabilized [15]. b = (σ1 + η) σ3 + η + 3σ1 + 1, (6.5) 1 − σ1 Moreover the Lagrange region also turns out to be stable for finite motions when applying the Lya-  1 − σ  c = (4σ + η) σ + η 3 (6.6) punov’s method. On the other hand, up to now it has 1 3 1 − σ 1 not been possible to demonstrate the nonlinear sta- and bility or instability for the DeBra-Delp region [34, 36]. hwyo η = . (6.7) Once the above clarifications have been stated the I Ω x o new stability regions are analyzed. The Fig. 6.5 is It should be mentioned that the term hwyo of the for a positive value of η. The momentum bias in this IzΩo Eq. 3.5 does not appear in these coefficients. This is case increases the stability contributions of the static due to the relations torques. Thus, the Lagrange region broadens while the DeBra-Delp area decreases as it is shown in the hwyo hwyo Ix Ix = = η (6.8) Fig. 6.5. IzΩo IxΩo Iz Iz and Ix 1 − σ3 = . (6.9) 1 Iz 1 − σ1 Previous Con-guration As in the case without momentum bias the eigen- New Con-guration √ 0.5 values λ = ± 3σ2 correspond to pitch and it is un-

coupled with respect to the motion in the other two 3 angles. This is since the momentum wheel is along < 0 Lagrange Region BCF y axis and the torque produced is perpendicular to the axis of rotation. Thus, given that the motion -0.5 in pitch is still independent the condition in Eq. 6.1 DeBra-Delp Region ! or Ix > Iz remains true for stability. -1 Regarding the other angles the roots of the char- -1 -0.5 0 0.5 1 acteristic polynomial are <1 √ −b ± b2 − 4c λ2 = . (6.10) 2 Figure 6.5: Dual-spin stability regions for η = 0.3. After the same analysis as in pure gravitational stabi- lization the conclusion is that for stability λ2 must be On the other hand Fig. 6.6 is for a negative value real and negative. Based on this fact the remaining of η. Now the static terms are weakened. That is why stability conditions are the Lagrange area decreases and the DeBra-Delp re- gion grows. c > 0, (6.11) Furthermore under the assumption η  1 it is

12 obtained that values of ηmax in both configurations are relatively 1 − σ b ≈ η2 3 (6.14) high and there are a lot of possibilities when modify- 1 − σ1 ing the stability regions with the MW in MIST. and 1 − σ c ≈ η2 3 . (6.15) 1 − σ1 Table 6.2: Stability conditions and wheel capability. The conditions in Eq. 6.11 and Eq. 6.12 are verified Configuration Conditions η automatically since it can be demonstrated that max Previous η > 0.27 or η < −1.29 31.06 |σ1| < 1 (6.16) New η > −0.38 or η < −3.30 42.81 and |σ | < 1. (6.17) 3 Finally, it should be mentioned that in spite of be- Then the remaining stability requirement for the mo- ing able to stabilize with a negative momentum bias tions in yaw and roll gives rise to this option should not be considered. This is due to 1 − σ3 4 the problems of the DeBra-Delp region pointed out > 2 ≈ 0, (6.18) 1 − σ1 η before. which is again the condition Eq. 6.1. Therefore, when η  1 the Lagrange (η → ∞) or DeBra-Delp 6.3 Pitch-Orbital Resonance (η → −∞) region occupies the entire lower-right tri- The motion in pitch is triggered when there is angle. eccentricity in the orbit. This is since the rotation of the local vertical is not uniform and at the same time the gravity gradient will try that the axis with 1 minimum moment of inertia follows it. Previous Con-guration New Con-guration If a small eccentricity is taken into account in the 0.5 linear model of Sec. 3.2 then an additional term cor- responding to the Euler forces will appear in pitch. & In Eq. 3.2 the inertial torque for the Euler forces 3 Lagrange Region

< 0 is TE = −IΩ˙ . (6.19) -0.5 A! DeBra-Delp Region As mentioned before in Sec. 3.2 Ω˙ is the derivative of the OCF angular velocity in ECI frame. From -1 Kepler’s Second Law and the orbit equation rG = -1 -0.5 0 0.5 1 2 h /µE <1 1+e cos ν it is obtained that h (1 + e cos ν)2 Ω = = Ωo , (6.20) Figure 6.6: Dual-spin stability regions for η = −0.2. 2 2 3/2 rG (1 − e ) In the case of MIST, Fig. 6.5 shows that for where h is the specific angular momentum associated η = 0.3 it is already possible to stabilize the pre- to the orbit and ν the true anomaly. The derivative vious configuration. However, when using a negative of Eq. 6.20 in time with respect to ECI frame for η with a similar magnitude it seems that the CubeSat e  1 is is still far away from being stable as seen in Fig. 6.6. e sin ν Ω˙ = −2Ω2 ≈ −2Ω2e sin M. (6.21) Then Table 6.2 contains the stability limits for both 1 + e cos ν o configurations and it is observed that it is necessary η < −1.29, which is a order of magnitude larger than In Eq. 6.21 the relation between true anomaly ν the case of a positive momentum bias. and mean anomaly M for small eccentricity ν = M + 2e sin M was used. Then considering until first- Table 6.2 has as well the column ηmax correspond- ing to the maximum angular momentum storage ca- order terms in the Taylor expansion for Eq. 6.19 only pacity of the MW (taken from Table 3.1). It is seen the equation of motion in pitch is modified becoming that it is possible to control the stability in both con- 2 2 θ¨ − 3σ2Ω θ = 2Ω e sin M. (6.22) figurations. Moreover, as it is known η is the ratio be- o o tween a typical angular momentum the MW can pro- Similar approaches for studying the Pitch-Orbital vide and the CubeSat has along the orbit. Hence, the Resonance can be found in [15] and [16].

13 Eq. 6.22 indicates that the additional term has solution would indicate. The nonlinearity decreases the orbital frequency given that M = Ωot + Mo. its growth rate and the maximum allowable eccen- Hence, there is resonance when the frequency of the tricity is actually smaller than what the linear model system in pitch is equal to the mean motion and that would predict. condition is The same simulations are run again for a con- 1 σ2 = − . (6.23) trolled system in Fig. 6.8. Here the amplitude is also 3 larger when the eccentricity increases but now it is Moreover, it is possible to express σ2 as kept at quite a low value even for e = 0.02, which

σ3 − σ1 is relatively high for a LEO orbit. Thus, the closed- σ2 = , (6.24) loop system is able to handle the resonance problem. 1 − σ1σ3 There are 2 reasons that explain this conclusion: that is, a function of σ1 and σ3. Then taking into account Eq. 6.23 and Eq. 6.24 it is obtained the res- • The orbit for MIST is polar and then the under- onance line in Fig. 6.1. actuation is more or less inside the orbital As pointed out in Sec. 6.1 the new configuration plane. Hence, it is always possible to gener- of MIST is near the resonance line. The solution to ate a torque in the pitch axis with the MTQs Eq. 6.22 has a order of magnitude and even higher eccentricities than 0.02 are not a problem in principle. e θ ∼ , (6.25) 3σ2 + 1 • It takes time to accumulate energy in the sys- tem and therefore to have a large pointing error. where it is observed that higher the eccentricity larger Fig. 6.7 shows that it is necessary 10 orbits to the amplitude. MIST has an intended eccentricity of reach the maximum amplitude for e = 0.01. It 0.001 (from Table 3.1) and then θ ∼ 2 deg for the new also seems that higher the eccentricity less time configuration. Thus, in principle there is no problem is required. However, simulations suggest that but as mentioned before in Sec. 3.1 the inertia matrix it is still several orbits even for e = 0.02. More- used is just a rough estimation and it is possible that over, if there was an orbit with less inclination σ is actually even closer to − 1 . Furthermore, it is 2 3 the under-actuation in pitch would last only a difficult to get such a low eccentricity and analysis portion of an orbit. Then the resonance prob- must be conducted about this problem. lem can be also mitigated for relatively large eccentricities in more equatorial orbits. e = 0:001 e = 0:003

20 ]

/ e = 0:001 [ 0 2

3 e = 0:005 -20 e = 0:01 1.5 e = 0:02 0 5 10 15 20 25 30 e = 0:006

e = 0:01 1 ]

20 /

[ 0.5

]

3 /

[ 0

3 0 -20

0 5 10 15 20 25 30 -0.5 Orbit No -1 15 16 17 18 19 20 Orbit No Figure 6.7: Eccentricity in uncontrolled system.

Fig. 6.7 represents the effect of increasing eccen- Figure 6.8: Eccentricity in controlled system. tricity in open-loop. These results are obtained from a nonlinear model where the only disturbance con- sidered is the gravity gradient. It is seen that the 7 ADCS Design amplitude begins to be unacceptable around 0.45 rad 7.1 Asymptotic Periodic LQR or 25 deg for a eccentricity of 0.01. In the case of e = 0.001 the amplitude is more or less 8 deg. Thus, As mentioned in Sec.2 if the Earth magnetic field it has not grown 10 times for e = 0.01 as the linear is periodic and the weighting matrix for the control

14 effort Rc in Eq. 4.1 is large enough the matrix P The cost function considered in Table 7.1 corre- from Eq. 4.2 is constant and can be obtained from sponds to Eq. 7.5. Regarding Jx and Ju they are the average the parts associated to control accuracy and effort in Eq. 7.5, respectively. From these results it can be 1 Z T PA+AT P −P BR−1BT
dtP+Q = 0, (7.1) concluded that the performances obtained are around c c 2 T 0 5 % of the values 0.5 rad and 0.05 Am used for the weighting matrices, that is, 2 deg and 0.003 Am2 re- where T is taken as the orbital period. spectively. The ARE in Eq. 7.1 can be seen as that associated to the time-invariant system given in closed-loop by 7.2 LQR via Numerical Optimization 1 Z T x˙ = A − BR−1BT P
dtx, (7.2) The matrix P is obtained by optimizing the T c 0 cost function directly with a numerical optimization which is the average over an orbit of the CubeSat’s method. It seems that the attitude control feedback linear model. Moreover, the term between parenthe- in satellites designed in this way has never been im- ses is the average of the closed-loop system matrix. plemented in any real mission. Only works at a theo- In the design of a LQR it is possible to adjust retical level have been found when doing a literature the control accuracy and effort via the matrices Qc survey and they all claim that it has potential advan- and Rc, respectively. They need to be defined rela- tages [52, 53, 54]. tively with respect to each other since the cost func- Here it is possible to consider more factors such as tion has the same optimum as it multiplied by a con- nonlinear effects, other disturbances apart from grav- stant. Based on the requirements stated in Sec. 1.2 ity gradient, estimation errors and so on. Neverthe- and the limitation of the MTQs in Table 3.1 reason- less, a linear model (in Sec. 3.2) with residual dipole, able choices seem to be the most important disturbance as seen in Sec.5, will be used to have a balance between fidelity and  2   1  amount of resources available. 0.001 rad/s U 0 Qc =   (7.3) The weighting matrices in Eq. 4.1 are assigned 0 1
2 U 0.5 rad the same values as in the case of AP LQR. Further- more, the integration time is taken as 10 orbits to be and  1 2 representative enough and the objective function is R = U. (7.4) c 0.05 Am2 redefined as Z T These selections have also considered the limited 1 T T
J = x Qcx + u Rcu dt (7.5) computational capability. Demanding performances T 0 imply quick poles in the averaged closed-loop sys- tem and high update rates for digital implementa- to try to have an order of magnitude near one when tion. The maximum update required is given by the the desired performance is achieved. estimator since it needs to have poles around 4 times faster than the controller and a reasonable update Design of Experiments rate for it is 10 times the velocity of its poles. It is Before optimizing it is always recommendable known from the supplier that the MTQs and MGMs to obtain some properties of the objective function. can be configured from 1 to 8 Hz. Then the fastest This stage is called Design of Experiments. Here pole with the above Q and R is 0.05 1 as seen in c c s techniques are used to select representative samples Table 7.1. In this case the execution rate of the esti- of the design space with the minimum computational mator is around 2 Hz and there is still some margin effort and to process the information associated to to the maximum of 8 Hz. them. The Latin Hypercube Sampling (LHS) is used to Table 7.1: AP LQR performances. get samples. This method divides each design param- Performance Index Value eter in N elements with the same length and selects J 0.0282 the points so that each of those elements appear once. Jx 0.0202 Thus, the number of chosen points is N as well and Ju 0.0080 each design parameter is evenly sampled [45]. 1 |λ|max 0.05 s The following details are considered in the sam- pling:

15 • In addition to the standard selection criterion objective function. As to the other directions of LHS it will be imposed that the distance be- at the beginning the system only becomes less tween the points is maximum. In this way not stable increasing slightly Eq. 7.5 and at some only all the influence of all the parameters are point it will get unstable with a huge growth in considered but also it is avoided large regions Eq. 7.5. in the design space without points. Analysis over the results in the Table 7.3 indicates • The solution from AP LQR is used as a refer- that: ence to rescale the design parameters as • There is only 1.98 % of the points stable while

? Pij the rest are unstable. Some of them are even Pij = . (7.6) PAP ij toward the unstable directions but the distance to PAP has not been enough to make the sys- • It is assumed that the solution is near PAP and tem unstable yet. Thus, there are much more then the bounds are defined as unstable directions than stable directions.

5 ≥ P? ≥ −5. (7.7) • Only 2.02 % of the samples are in the range ij 10−105 and then suddenly there are much more points in the range 105 −107. This is due to the • The number of points is taken as N = 5000. outsize increase of the objective function when This is only a first guess but it seems to be the system becomes unstable. enough to characterize the objective function.

Table 7.2 summarizes the properties of the sam- ples obtained with general statistical parameters. Table 7.3: Frequency table. The minimum is lower than what is obtained from Range Relative Cumulative PAP decreasing in a 67 % and then there is possibil- Frequency Relative ity to achieve much better performances. The other [ % ] Frequency parameters suggest that there are a large diversity of [%] values and it is confirmed by Table 7.3. 0 − 10−2 0.16 0.16 10−2 − 10−1 1.44 1.60 10−1 − 1 0.24 1.84 Table 7.2: General statistical parameters. 1 − 10 0.14 1.98 Parameter Value 10 − 102 0.08 2.06 Minimum 9.36 · 10−3 102 − 103 0.12 2.18 Maximum 9.29 · 107 103 − 104 0.22 2.40 Mean 1.79 · 107 104 − 105 1.60 4.00 Standard Deviation 1.87 · 107 105 − 106 8.72 12.72 106 − 107 30.22 42.94 107 − 108 57.06 100 Before explaining the subsequent results in this section some assumptions are made about the objec- tive function: The derivatives with respect to the design param- • There are regions of design space where the sys- eters are obtained in Table 7.4. A important result tem is stable and where it is unstable. is that the rows 4-6 of P do not affect the objective function. This can be also deduced from Eq. 4.2 tak- • It is considered that there is stability when ing into account that B has zeros in the rows 4-6. Furthermore, the six variables in bold are identified J ≤ 10 (7.8) as those that actually influence the objective function due to the following reasons: as a practical approach. ? • All of them except Prψ have a derivative higher • Taking PAP as origin and supposing that it is than 0.5 in magnitude. near the optimum there are directions where the stability increases and where it decreases. • They are the terms that represent the effects of In those where it increases the control effort the motion in a certain axis for the same degree is also larger leading to a higher value of the of freedom.

16 • Two sets of nearby directions in the design been considered and there are several irrelevant de- space toward stable and unstable regions can sign parameters. be found when considering only those terms as it will be explained later on in this section.

1 Stable Unstable Mixed

Table 7.4: Sensitivity of the design parameters. 0.8

s

e c

Row n a

Column p q r ψ θ φ t

s 0.6 n

p −1.86 0.01 0.03 0.20 0.00 −1.37 I f

q 0.10 −0.52 0.00 0.10 1.13 −0.03 o n

o 0.4

r 0.07 0.01 −1.67 0.13 0.04 −0.11 i

t c

ψ 0 0 0 0 0 0 a r θ 0 0 0 0 0 0 F 0.2 φ 0 0 0 0 0 0 0 -1 -0.5 0 0.5 1 Dot Product Points toward the stable directions and unstable directions among the samples are selected with the criteria Figure 7.1: Directions. J ≤ 0.1 and D ≥ 18 (7.9) and Fig. 7.2 contains the same but taking into account 7 J ≥ 10 , (7.10) only the six important variables mentioned before. respectively. In Eq. 7.9 D is the Frobenius norm of Now it is clear the nearness for each set of directions and that they tend to be opposite to each other. It the point with respect to PAP and its condition is to try to rule out samples in stable regions but toward should be also pointed out that there is more vari- unstable directions. Due to the conditions considered ability for unstable directions and this is since this in Eq. 7.7 the maximum value of D is 30 and 18 is in type of directions is much larger in number. These principle a reasonable limit for the distance. results contribute to confirm that the design param- Then scalar products are taken between: eters of Table 7.4 in bold are those really important for the objective function. Furthermore, the assump- • Stable directions. tion made before about the existence of stable and unstable directions seems to be correct5 considering • Unstable directions. their respective nearness and that the points selected • Stable and unstable directions. have very different D.

All the directions are given by the unit vectors from the P to the points. Moreover, the scalar prod- Stable AP 1 ucts between the same vector are not considered as Unstable Mixed

instances to be studied. 0.8

s e

Fig. 7.1 represent the fractions of instances higher c

n

a t

than a certain value of the dot product for each case s 0.6

n I

above. In principle the stable directions should be f

o n

close with respect to each other and the same for the o 0.4

i

t c

unstable directions. Moreover, both set of directions a r should be mainly opposite to each other. However, F 0.2 the fractions of scalar products higher than 0.5 are less than 20 % for either stable and unstable cases. 0 -1 -0.5 0 0.5 1 Furthermore, the line associated to the mixed dot Dot Product products between the two sets of points is nearly the same as the line for unstable directions. This result is since all the variables in the rows 1-3 of P have Figure 7.2: Directions with the key variables. 5It should be reminded that this assumption is always considering the bounds defined before for P?. Otherwise, it is possible that there are problems in the stable directions when breaching those constraints due to the limitation of the MTQs.

17 Optimization relative tolerance for integration have been used the case of SQP. Despite that SQP stopped since the step The optimization is conducted using the following size was smaller than the tolerance value and without methods: meeting the criteria corresponding to optimization • Sequential Quadratic Programming (SQP). It problems solved by gradient-based methods. This combines quadratic approximations and quasi- means that it was not enough and a better solution Newton algorithms. Thus, it is gradient-based would have been to implement the Earth magnetic and a local method [44, 45]. model directly. Furthermore, the time needed to run a simulation for SQP was around the triple for SQP • Nelder-Mead. This method computes a n- and its actual number of function evaluations would dimensional analogue of a triangle, that is, a be even larger than in the case of the GA. geometrical figure enclosed within n + 1 ver- texes in a n-dimensional space. This is called Table 7.6: Results with guidance. simplex and it is updated in each iteration by substituting the worst vertex by a nearby point Method Total Eval. JJx Ju with a lower value of objective function. Hence, Iter. it is a local derivative-free algorithm [44, 45]. GA 79 1600 0.0088 0.0007 0.0081 Simplex 269 531 0.0090 0.0009 0.0081 • Genetic Algorithms (GAs). These are stochas- SQP 12 484 0.0094 0.0011 0.0083 tic methods that mimic the evolution process to find the optimum. It takes a set of points in the design space as a population of individu- Table 7.6 contains the optimization results with als and measures their suitability to a environ- good initial approximations now. Points from the ment given by the objective function. They are conditions Eq. 7.9 are used as initial population of global and derivative-free algorithms [45, 46]. the GA and the solution by AP LQR is considered Some considerations are taken into account when for Nelder-Mead and SQP. Now, all the results are optimizing Eq. 7.5: very similar being that by the GA still the best and that by SQP still the worst. The GA has obtained • The conditions in Eq. 7.7 are used to keep the nearly the same solution as without guidance. velocity of the poles so that similar update rates Regarding the computational effort the GA still as for the AP LQR are obtained. has the most function evaluations and now SQP is • Only the key variables are used for SQP. The nearly similar to it taking into account what was other variables would make the Hessian matrix mentioned above. The number of function evalua- of the quasi-Newton method nearly singular. tions is now smaller than the case without guidance decreasing around 50 % for Nelder-Mead and SQP The results in Table 7.5 are without elaborate and 25 % in the case of the GA. starting points. In the GA the initial population is generated randomly and a vector of zeros is used for Table 7.7: Standard deviation of the solutions. Nelder-Mead and SQP. The best solution has been obtained by the GA and the worst by SQP. Row Column p q r ψ θ φ p 0.33 4.21 1.65 2.74 3.54 0.22 Table 7.5: Results without guidance. q 1.74 0.80 3.45 1.30 0.25 1.83 r 0.23 4.41 0.15 0.79 5.05 4.00

Method Total Eval. JJx Ju Iter. GA 105 2120 0.0089 0.0007 0.0082 The solution by the GA with guidance has only Simplex 682 1040 0.0138 0.0057 0.0081 reduced the minimum obtained in the Design of Ex- SQP 52 982 0.0268 0.0186 0.0082 periments by 6 %. It means that it was possible to find the solution with a number of samples that has the same order of magnitude as the function eval- The Earth magnetic field of the simulation model uations needed for optimization. Therefore, it sug- is obtained by interpolation and this causes errors gests that good optimization methods for the objec- when computing the derivatives numerically. Thus, a tive function are those able to explore different re- larger number of points for interpolation and a lower gions in design space.

18 As conclusion, the GA is the best optimization Robustness Analysis method for this kind of problems. It is more suitable than Nelder-Mead since it is a global method. When The robustness of the AP LQR and Genetic Algo- comparing with SQP it is not necessary to identity rithm LQR are assessed via Monte Carlo simulations. the key variables. Random values are obtained using uniform distribu- tions for the following parameters: Table 7.4 shows the variability of the different P? obtained. It is again clear that the key vari- ables identified before are those that mainly affect • Mean altitude and orbital inclination. They are the cost function since they all have a standard de- modified within a ±3 % with respect to the ref- viation smaller than 0.8. Moreover, the results in erence conditions. Table 7.8 seem to indicate that the exact solution of the problem for the key variables is in the boundary • Right ascension of the ascending node and ar- defined by Eq. 7.7. gument of perigee. Here the ranges are between 0 and 2π rad.

Table 7.8: Mean of the solutions. • Inertial matrix. The elements in the diagonal Row Column p q r ψ θ φ are modified within a ±2 % while the products p 4.64 - - - - 4.78 of inertia in a range defined by the ±2 % of Ix+Iy+Iz q - 3.59 - - 2.73 - 3 . r - - 4.76 4.08 - - • Residual dipole. It varies from the reference condition until the double of that value.

Fig. 7.3 represents the fraction of dot products be- • Drag coefficient. It changes in its natural range, tween the solution by the GA with guidance and the that is, between 2 and 4. points obtained under the conditions Eq. 7.9 higher than a certain value. It is seen that the solution is • Center of mass’s position. In each direction a close to those stable directions as expected. More- maximum displacement of 10 mm is allowed. over, as the solutions by the other methods are simi- lar to the GA this result can be extended to them as well. Two simulation models are considered for this study. Both takes into account all the disturbances Finally, it should be mentioned that the LQR de- but one of them supposes perfect attitude knowledge signed via numerical optimization is stable when con- while the other uses estimator explained in Sec. 7.3. sidering the averaged close-loop system introduced For each case two sets of Monte Carlo simulations are for the AP LQR. Moreover, the solution here is not obtained with a number of samples Nm = 100. a symmetric matrix as in the case of AP LQR.

Table 7.9: Percentage of stable instances without es- timation errors. 1 First Second Execution Execution 0.8

s AP LQR 79 % 82 %

e

c n

a GA LQR 92 % 98 % t

s 0.6

n

I

f

o n

o 0.4

i

t c

a Table 7.9 contains the results without estimation r F 0.2 errors. The percentage of simulations that met the stability criterion given by Eq. 7.8 is used to mea- 0 sure the robustness. In both executions of the Monte -1 -0.5 0 0.5 1 Dot Product Carlo simulations the values obtained are consistent and this indicates that the Nm chosen was enough. The conclusion is that both controllers are reasonably Figure 7.3: GA LQR versus stable directions. robust being the GA LQR a 15 % better on average.

19 inside the orbital plane. Table 7.10: Percentage of stable instances with esti- mation errors. First Second GA LQR 20

Execution Execution ] AP LQR /

AP LQR 80 % 72 % [ 0 A GA LQR 86 % 90 % -20 0 5 10 15 20

20

] /

[ 0

The results with estimation errors are in Ta- 3 -20 ble 7.10. In this case the Nm chosen was also enough. 0 5 10 15 20 20

Both controllers are slightly less robust than before. ] /

Moreover, GA LQR is 12 % better on average now. [ 0 ? -20 0 5 10 15 20 7.3 Asymptotic Periodic LQE Orbit No As seen in Sec. 4.2, the LQE has the same prob- lem as the LQR. In principle it is also necessary to Figure 7.5: Pointing errors. solve ordinary differential equations to get the solu- tion Pe(t). However, it was mentioned that there is a The pointing error using AP LQR and GA LQR duality between LQE and LQR. Thus, it is possible combined with the AP LQR are represented in to consider the AP LQR theory for the estimation Fig. 7.5. Again the accuracy in yaw and roll are problem. worse since the orbit is polar and the under-actuation Now C is the equivalent to B and it needs to be pe- of MTQs does not affect a lot the motion in pitch. riodic with the orbital period. The Earth magnetic field can be supposed periodic as seen before. The Table 7.11: Performance under sunlight. Sun vector in OCF changes over a year but as the orbit in MIST is Sun-synchronous it could be sup- ψ θ φ ∆ψ ∆θ ∆φ posed periodic. Hence, there are enough arguments (deg)(deg)(deg)(deg)(deg)(deg) to apply the AP LQR theory for the LQE design. AP LQR 3.34 2.16 3.76 1.04 0.59 0.68 GA LQR 2.90 2.60 2.79 0.99 0.59 0.68

] 20 /

[ 0 Table 7.11 and Table 7.12 summarize the perfor- A

" mance in terms of the Root Mean Square (RMS). It -20 0 5 10 15 20 is possible to see that the estimation errors are nearly 20

] the same for AP LQR and GA LQR. As to pointing /

[ 0 error the GA LQR is slightly worse in pitch but bet- 3 " -20 ter in roll. Taking into account all the angles GA 0 5 10 15 20 LQR is more accurate than AP LQR.

] 20 /

[ 0 ?

" -20 Table 7.12: Performance during eclipses. 0 5 10 15 20 Orbit No ψ θ φ ∆ψ ∆θ ∆φ (deg)(deg)(deg)(deg)(deg)(deg) AP LQR 3.99 2.84 3.94 1.86 1.54 2.03 Figure 7.4: Estimation errors. GA LQR 4.02 3.24 2.53 1.85 1.59 2.30

Fig. 7.4 represents the performance of the AP LQE. It is seen that there are peaks in each orbit. The requirements stated in Sec. 7.5 are taken as They correspond to eclipses when the Sun sensors are the RMS when considering all the angles. If the norm not available and there is under-determination using of a vector containing all the angles is used the max- magnetometers only. Also it seems that the errors in imum errors allowed under sunlight for each angle yaw and roll are larger than in pitch. This is since the are 8.66 deg for pointing and 2.89 deg for estimation. orbit is polar and the under-determination is mainly Based on the results from Table 7.11 the conclusion

20 is that it is possible to meet the ADCS requirements changed to have this kind of passive stabiliza- with the ADCS concept in MIST under the reference tion. conditions. • An alternative solution to the rotation of the solar panels is to provide a momentum bias per- 102 pendicular to the orbital plane with a MW. In A this case it should be in the direction of the 3 ? orbital rotation to make the CubeSat be in the

100 ]

/ Lagrange region. The nonlinear stability for the [

e DeBra-Delp region has not been proved and it d

u -2

t 10

i disappears when there is energy dissipation.

l

p m

A • The new configuration of MIST is near the 10-4 Pitch-Orbital Resonance. However, the closed- loop system can handle this problem since the 10-6 orbit is polar and the under-actuation is inside 100 102 104 f ? the orbital plane. • An LQR designed via numerical optimization Figure 7.6: Pointing errors in the frequency domain. could improve the pointing accuracy as well as the robustness of the traditionally used AP Fig. 7.6 represents the pointing error in the fre- LQR. Simulations results indicate that these quency domain. A nondimensionalized frequency f ? advantages are more noticeable while the es- is defined with the mean motion Ωo. As expected the timation errors are small. However, this could maximum amplitude is around the orbital frequency. be since the estimator has not been uncoupled Moreover, the amplitude decreases for high frequen- from the controller in the simulation models cies. causing a higher noise in the system then.

• It seems that an optimization method able to 8 Conclusions and Discussions explore different regions of the design space and derivative-free is the most suitable for an LQR. The following points should be remarked in this There is a large number of variables that do not final part of the report: affect the objective function and it is necessary • Identical ADCS concepts to MIST usually fail to identify them if a quasi-Newton method is with the on-board magnetic disturbances being used. Otherwise the Hessian matrix becomes the main reason. singular.

• PRISMA is the only small satellite with the • The simulation results suggest that the ADCS same ADCS concept as MIST found that suc- concept in MIST can meet its requirements un- ceeded. der its reference conditions.

• It is necessary to assure gravitational stabiliza- • The analyses for different conditions in Sec.5 tion for PMC. The under-actuation could make are supposing perfect attitude knowledge. the closed-loop system unstable. In MIST the When taking into account large estimation er- orientation of the deployable solar panels was rors they could be not valid.

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