Preliminary Lander Cubesat Design for Small Asteroid Detumbling Mission

Home , 2003 YN107, 2013 BS45, Asteroid capture, MINERVA (spacecraft)

DEGREE PROJECT IN VEHICLE ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2018

Preliminary Lander CubeSat Design for Small Asteroid Detumbling Mission

AGNE PASKEVICIUTE

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Preliminary Lander CubeSat Design for Small Asteroid Detumbling Mission

AgnPaökeviit

Department of Aeronautical and Vehicle Engineering KTH Royal Institute of Technology

This thesis is submitted for the degree of Master of Science

October 2018 Skiriu ö˛ darbπsavo mamai Graûinai ir tiui Algirdui. Acknowledgements

This thesis would have not been possible without the encouragement and support of numerous people in my life.

I would especially like to thank Dr Michael C. F. Bazzocchi for his support and constructive advice throughout my research. In addition, I would like to express my gratitude to researchers, professors and staat Luleå University of Technology, Space Campus, for their ideas and help in practical matters.

I would like to sincerely thank my family, boyfriend, and friends for believing in me, encour- aging me to reach for my dreams, and loving me without any expectations.

Last but not least, I am truly grateful for the opportunities KTH Royal Institute of Technol- ogy provided. Sammanfattning

Gruvdrift på asteroider förväntas att bli verklighet inom en snar framtid. Det första steget är att omdirigera en asteroid till en stabil omloppsbana runt jorden så att gruvteknik kan demon- streras. Bromsning av asteroidens tumlande är en av de viktigaste stegen i ett rymduppdrag där en asteroid ska omdirigeras. I detta examensarbete föreslås en preliminär asteroidlandare baserad på CubeSat-teknik för ett rymduppdrag där en asteroid ska omdirigeras. En asteroid av Arjuna-typ, 2014 UR, med en diameter på mellan 10.6 och 21.2 m är vald som kandidat för rymduppdraget. På grund av att asteroidens är relativt liten till storlek måste landningen utföras med en aktiv reglermetod och rymdfarkosten måste förankras till asteroiden. Med hjälp av en beslutsmetod utifrån flera mål, PROMETHEE, identifierades förankringsme- toden “mikro-ryggrads-gripare” som den mest lämpliga. Tre huvuduppgifter för rymduppdraget identifierades under designprocessen: dataflöde mellan landaren och moderfarkosten, Delta-V-budgeten och peknoggrannheten. Delta-V som krävs för landning på asteroiden uppskattas att vara högst 10 m/s. Bromsningen av tumlandet kostar högst 15 m/s. Osäkerheten med Delta-V för bromning av tumlandet beror på olika up- pskattningar av asteroidens storlek. Den nödvändiga minsta peknoggrannheten uppskattades vara 6°. Utformningen av landaren, baserade på CubeSat-teknik, använder till största delen kom- ponenter som finns på hyllan, s.k. commercial-o-the-shelf. Det visas att en CubeSat-landare inte kan bromsa tumlandet för en asteroid som roterar snabbt kring flera axlar. Om den valda asteroiden roterar runt en axel med en rotationsperiod på 2.4 timmar, är det möjligt att bromsa tumlandet med endast 1.5 kg drivmedel. Den föreslagna landaren är en 12U CubeSat med en total massa på 15 kg och strömförbrukning på 65 W. Abstract

Asteroid mining is expected to become reality in the near future. The first step is to redirect an asteroid to a stable Earth orbit so that mining technologies can be demonstrated. Detumbling of the asteroid is one of the important steps in asteroid redirection missions. In this thesis, a preliminary lander CubeSat design is suggested for a small asteroid detumbling mission. The candidate asteroid for the detumbling mission is chosen to be 2014 UR, an Arjuna- type asteroid with an estimated diameter ranging from 10.6 to 21.2 m. Due to the small size of the asteroid, the landing must be performed with an active control method after which the spacecraft must be firmly anchored to the asteroid. By using the multi-criteria decision making method PROMETHEE, the microspine gripper is chosen as the most suitable anchoring mechanism. Three main mission drivers are identified during the design process: data-flow between the lander and the mothership, Delta-V budget and pointing accuracy. The Delta-V required for landing on the asteroid and despinning it is estimated to be 10 m/s and 0.15 m/s at most, respectively. The uncertainty with the despinning Delta-V is due to varying estimates of the size of the asteroid. The required minimum pointing accuracy is estimated to be 6. The preliminary lander CubeSat design can be largely realised with commercial o-the- shelf components suggested in this work. Only some of the components have to be custom built or the technologies further developed. It is shown that a CubeSat lander is not able to detumble an asteroid that is rotating fast around multiple axis. However, if the considered asteroid is rotating around a single axis with a rotational period of 2.4 h, it is be possible to despin it by spending just 1.5 kg of propellant. The suggested lander is a 12U CubeSat with an overall mass of 15 kg and power consumption of 65 W. Contents

List of Figures ix

List of Tables xi

Nomenclature xiii

1 Introduction 1 1.1 Literature Review ...... 1 1.2 Motivation ...... 3 1.3 Preliminary Mission Description ...... 4 1.4 Thesis Overview ...... 5

2 Target Asteroid 6 2.1 Spectral Types of Asteroids ...... 6 2.2 Near-Earth Asteroids ...... 7 2.3 Asteroid 2014 UR ...... 10 2.3.1 Spectral Type of 2014 UR ...... 11 2.3.2 Surface Properties ...... 12

3 Forces Acting on The Landed Spacecraft 13 3.1 Introduction ...... 13 3.2 Gravitational Force ...... 16 3.3 Solar Radiation Pressure ...... 16 3.4 Electrostatic Force ...... 16 3.5 Centrifugal Force ...... 17 3.6 Despinning Force ...... 19 3.7 Net Force Acting on The Spacecraft ...... 21 Contents vii

4 Landing Systems Review and Choice 23 4.1 Literature Review ...... 23 4.1.1 Active Descent ...... 24 4.1.2 Passive Descent ...... 26 4.2 Landing Mechanism Selection ...... 27

5 Anchoring Systems Review 29 5.1 Literature Review ...... 29 5.1.1 Slow Anchoring Methods ...... 30 5.1.2 High-Speed Anchoring Methods ...... 32 5.2 Criteria for Successful Anchoring ...... 37 5.3 Discussion of Suitable Anchoring Systems ...... 37 5.3.1 Summary of Anchoring Systems Candidates ...... 38

6 Anchoring System Choice Using MCDM 41 6.1 Introduction ...... 41 6.2 Criteria Description ...... 43 6.3 Criteria Weights ...... 46 6.3.1 Optimal Pairwise Comparison ...... 47 6.3.2 Weights of Criteria ...... 50 6.4 Method of Aggregation ...... 51 6.4.1 PROMETHEE Method ...... 52 6.4.2 Discussion ...... 56

7 Primary Mission Drivers 57 7.1 Mission Data-Flow ...... 57 7.2 Delta-V Budget ...... 59 7.2.1 Sphere of Inﬂuence ...... 59 7.2.2 Delta-V Estimation: Docking ...... 60 7.2.3 Delta-V Estimation: Despinning ...... 62 7.3 Pointing accuracy ...... 62

8 Preliminary Spacecraft Design 64 8.1 Attitude and Orbit Control Subsystem ...... 64 8.1.1 Actuators ...... 65 8.1.2 Sensors ...... 67 8.2 Propulsion Subsystem ...... 69 Contents viii

8.2.1 Propellant Budget ...... 69 8.2.2 Subsystem Choice ...... 71 8.3 Anchoring Subsystem ...... 74 8.4 Communications Subsystem ...... 75 8.5 Other Subsystems ...... 76 8.5.1 Command and Data Handling Subsystem ...... 76 8.5.2 Thermal Subsystem ...... 77 8.5.3 Structure ...... 77 8.6 Power Subsystem ...... 77 8.7 Mass and Power Budget ...... 79

9 Conclusion 81

Bibliography 84

Appendix A Technology Readiness Levels 91

Appendix B Multi-Criteria Decision Making 92 B.1 Pairwise Comparison of Anchoring Criteria ...... 92 B.2 MATLAB Script for Pairwise Dierence Comparison ...... 97 B.3 PROMETHEE Preference Functions and Parameters ...... 100

Appendix C Clohessy-Wiltshire Equations 103 C.1 Coordinate System ...... 103 C.2 Equations of Motion ...... 104

Appendix D Preliminary Design & Subsystems 106 D.1 Moment Arm & Moment of Inertia ...... 106 List of Figures

1.1 Operational concept sketch. Adapted from Probst and Förstner [1]...... 5

2.1 Three subgroups of NEAs and their orbits with respect to Earth’s orbit around the Sun [2]...... 8

3.1 Schematic diagram showing the lander on the surface of asteroid with the main natural forces acting on it...... 14 3.2 Resultant centrifugal force distribution on asteroid 2014 UR for the worst case scenario...... 18 3.3 Best case scenario force and time required for despinning the asteroid. . . . . 19 3.4 Landing location and direction of forces required for despinning the asteroid in the worst case scenario...... 20 3.5 Worst case scenario force and time required for despinning the asteroid. . . . 21

4.1 GRASP lander developed by SFL [3]...... 24 4.2 Spinning lander concepts suggested by SCSG...... 25 4.3 Surface mobility mechanisms patented by University of California [4]. . . . . 26 4.4 MASCOT mobility concept - eccentric arm concept [5]...... 27

5.1 Sample self-opposing drill systems [6]...... 31 5.2 Experimental setup of robotic arm presenting anchoring technology based on sawing method [7]...... 32 5.3 Left: NASA JPL microspine grippers being tested for anchoring strength at

45. Right: CAD view of microspine gripper cross-section [8, 9]...... 33 5.4 Telescoping spike anchoring system [10]...... 34 5.5 Telescoping spike system for ST4/Champollion mission [10]...... 35 5.6 Tethered spike anchoring system [10]...... 36 5.7 Philae harpoon anchoring system mounted in the landing gear [11]...... 36 5.8 Magnetic anchoring concept “Harvestor” by Deep Space Industries [6]. . . . 37 List of Figures x

6.1 Flowchart for the MCDM methodology...... 42

7.1 Data-ﬂow diagram for asteroid detumbling mission. Round shapes denote data source, rectangles denote tasks, hexagon denotes data end user...... 58 7.2 Delta-V required to dock with asteroid 2014 UR versus rendezvous time. . . . 61 7.3 Scheme for estimating required maximum pointing angle to the asteroid 2014 UR, just after spacecraft release from the mothership...... 63

8.1 Attitude control motion in two dimensions [12]...... 66 8.2 Propellant required for docking with the asteroid 2014 UR for dierent spacecraft sizes and dierent speciﬁc impulses...... 70 8.3 Comparison of propellant required for despinning fast spinning and slow spinning asteroid 2014 UR for dierent speciﬁc impulses...... 71

C.1 Coordinate-System Geometry for Relative Motion [13]...... 103 List of Tables

2.1 Orbital parameters of NEA groups [2, 14]...... 7 2.2 List of identiﬁed Arjuna-type asteroids and their parameters (May, 2018) [15]. 9 2.3 Dierent materials representation of S-type, V-type, Q-type, and C-type asteroids...... 12

3.1 Net force and list of separate forces acting on the landed spacecraft in the normal direction to the asteroid surface...... 22

5.1 Criteria for selecting suitable anchoring system...... 39 5.2 Parameters of anchoring sub-systems...... 40

6.1 Maximum power available for dierent CubeSat platforms...... 43 6.2 Points corresponding to volume, mass and power that anchoring system requires. 44 6.3 Numerical criteria for each alternative...... 46 6.4 Similarity Scale proposed by Triantaphyllou...... 47 6.5 Matrix of pairwise dierence comparisons of criteria...... 48 6.6 Closest Discrete Pairwise (CDP) matrix (after re-arrangement)...... 48 6.7 Real Discrete Pairwise (RDP) matrix...... 49 6.8 Weights of each criterion...... 50 6.9 Information about each criteria in order to use PROMETHEE outranking method. 53 6.10 Global preference matrix for all alternatives...... 54 6.11 Positive and negative outranking ﬂows for all alternative anchoring systems. . 55 6.12 Complete ranking of anchoring systems alternatives...... 56

8.1 Mass and power budget for AOC subsystem...... 65 8.2 Mass and power budget for propulsion subsystem (considering best case spinning asteroid scenario)...... 69 8.3 Comparison of dierent propulsion systems performance for the best case scenario [16–21]...... 73 List of Tables xii

8.4 Mass and power budget for anchoring subsystem...... 75 8.5 Mass and power budget for communications subsystem...... 75 8.6 Mass and power budget for C&DH subsystem, thermal subsystem and spacecraft structure...... 76 8.7 Mass budget for power subsystem...... 78 8.8 Total power requirement for each of the three mission phases...... 79 8.9 Total mass and power budget...... 79

A.1 Technology readiness levels according to NASA...... 91

B.1 Verbal and numerical pairwise comparison of each criterion...... 92 B.2 Types of generalised criteria...... 101 B.3 Function choice for each criterion...... 102

D.1 Moment arm and moment of inertia for each CubeSat conﬁguration rotating around an axis by ﬁring two thrusters, as denoted in the schemes...... 106 Nomenclature

Roman Symbols

A matrix with coecients from CDP matrix

A area a semi-major axis c speed of light, = 299,792,458 m/s d dierence between two alternatives d diameter e eccentricity

Ek kinetic energy

F force f function

G gravitational constant, 6.674 10 11 m3 kg-1 s-2 ⇡ ⇥ g criteria g gravity

H absolute magnitude

I impulse

I moment of inertia i inclination Nomenclature xiv

Ibit minimum impulse bit k total number of alternatives l moment arm m mass n number of constraints ne electron density

P power

P preference of one alternative over another p preference function threshold parameter of strict preference p pressure

P period of rotation pv albedo c elementary charge, = 1.602 10 19 C ⇥ Q apogee q perigee q preference function threshold parameter of indierence r radius

T thrust t time

V velocity

V volume

W solar constant, = 1362 W/m2

W weight of criteria w predicted value of pairwise comparison of weights between two criteria Nomenclature xv

X error factor

Greek Symbols a angular acceleration a real value of pairwise comparison of weights between two criteria l vector with Lagrangian coecients

µ standard gravitational parameter w angular velocity

F electric potential f outranking ﬂow value p normalised preference of one alternative over another r density t torque q pointing accuracy half angle

Subscripts

0 initial value a asteroid cent centrifugal el electrostatic f ﬁnal grav gravitational i number of criterion/alternative j number of criterion/alternative prop propellant rad solar radiation Nomenclature xvi sc spacecraft sp speciﬁc

Other Symbols

DV change in velocity

Acronyms / Abbreviations

AOC Attitude and Orbit Control

ARM Asteroid Retrieval Mission

AU Astronomical Unit, 150 million km ⇡ C&DH Command and Data Handling

CDP Closest Discrete Pairwise matrix

COTS Commercial O-The-Shelf

ESA European Space Agency

JAXA Japan Aerospace Exploration Agency

LCDB Light Curve Database

MATLAB multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks

MCDM Multi-Criteria Decision Making

MVP Mass, Volume, Power

NASA National Aeronautics and Space Administration

NEA Near-Earth Asteroid

PROMETHEE Preference Ranking Organisation Method for Enrichment of Evaluations

RDP Real Discrete Pairwise matrix

SOI Sphere of Inﬂuence

TRL Technology Readiness Level

U CubeSat Unit, 1 unit measures 10 10 10 cm3 ⇥ ⇥ Chapter 1

Introduction

Already in 1903 Konstantin Tsiolkovsky in his article “Exploration of Cosmic Space by Means of Reaction Devices” recognised asteroids as potential treasures and listed their exploitation as one of the key points for space conquest [22]. As ambitious as it sounds, with the rapidly improving pace of technology development, asteroid mining is soon to be reality. It also might become the biggest game-changer in economic history [23]. For instance, a 500 m diameter platinum-rich asteroid contains about 174 times the yearly output of platinum on Earth [24]. Furthermore, asteroids rich in volatiles can provide water, which is essential for life support systems, especially in long-term human ﬂight missions. Also, water molecules can be split into hydrogen and oxygen which can be used as rocket fuel. Finally, extraction of necessary materials from bodies in space means that less mass has to be launched from the Earth, which reduces the launch costs. Thus, it is clear that asteroids could provide with resources essential for further space exploration, as well as beneﬁt life on Earth.

1.1 Literature Review

Numerous missions have already been planned, and some executed, for studying and charac- terising asteroids by both national space agencies (JAXA, NASA, ESA) and private companies (such as Planetary Resources [25]). Understanding the characteristics and composition of the asteroids, having experience in approaching, landing, and operating on them is extremely valu- able when preparing for future mining missions. So far, already four missions have been executed to asteroids and one to the comet. The first spacecraft to land on an asteroid was NEAR (Near-Earth Asteroid Rendezvous) Shoemaker [26]; in 2001 the 800 kg spacecraft landed on the asteroid Eros, which is 16.8 km in diameter. Originally, the spacecraft was not designed to land on the asteroid, but the decision to land was made after all science objectives were reached, and the spacecraft was still operational with approximately 36 m/s of DV left. One 1.1 Literature Review 2 of the reasons to land was a possibility of getting even higher resolution images of the surface of Eros. In 2005, the Hayabusa spacecraft reached asteroid Itokawa, which is 550 m along its longest axis [27]. The gravity of Itokawa allowed to enable hopping motion of the small 0.5 kg spacecraft lander MINERVA. Touch-and-go sequence was adapted, which meant the lander was supposed to stay on the surface for 1–2 seconds before hopping to another location. However, by mistake MINERVA was released while Hayabusa was ascending for its automatic station keeping manoeuvre. The lander escaped Itokawa’s gravitational pull due to the release at too high altitude. Nonetheless, Hayabusa spacecraft still managed to land successfully and remained on the asteroid’s surface for half an hour [28]. Just recently, Hayabusa2 spacecraft approached the almost 1000-m-in-diameter asteroid Ryugu and in October, 2018, it will release its main lander MASCOT. Asteroid lander MASCOT developed by DLR (German Aerospace Centre) [5, 29] is an 11 kg CubeSat. Hayabusa2 is not orbiting the asteroid, but is hovering above it, always facing the Earth. MASCOT acts as an autonomous spacecraft by: making attitude correction decisions, activating scientific instruments, actuating hopping mechanism to reach dierent asteroid sites, monitoring and managing system energy and failures. In order to increase the success rate of the mission, MASCOT electronics are fully redundant. The design goal for the lander was to enable it to operate on the surface for up to two asteroid days (spinning period of Ryugu is 7.63 h). OSIRIS-REx by NASA is another spacecraft currently on its way to an asteroid. The target asteroid, Bennu, is around 500 m in diameter. The spacecraft does not contain a lander. It is itself responsible for reaching the asteroid, collecting samples and carrying them back to Earth. It adopts touch-and-go mission sequence and will touch the surface of the asteroid in order to collect the sample, and then will move away from it. Finally, only one mission has been executed to a comet: Rosetta, developed by ESA. The first landing on the comet was accomplished in 2014. Rosetta lander Philae landed on the comet 67P/Churyumov-Gerasimenko (which is 4.3 km in its longest dimension) [30]. All five missions were planned to relatively large bodies, which enabled to utilise their gravity for landing procedures. Thus, none of the spacecraft employed thrusters for full attitude control, and landed in hopping motion. Lessons learnt from the missions described above help to prepare for the next step in asteroid mining: redirecting an asteroid to the vicinity of the Earth. Having an asteroid, or a boulder from an asteroid, in Earth’s or Moon’s orbit will allow for cheaper and faster testing of asteroid mining technologies in real environment. Few missions have already been proposed for redirecting an asteroid, for instance the famous Asteroid Redirect Mission (ARM) by NASA. ARM was supposed to be launched in 2021, but was cancelled due to budget constraints. The goal of the mission was to return a boulder from an asteroid to stable orbit around the Moon [31]. A single spacecraft architecture would have been utilised. Five key functions for asteroid capture 1.2 Motivation 3 would have had to be performed: 1) asteroid and boulder mapping and characterisation; 2) onboard asteroid- and boulder-relative navigation; 3) asteroid surface interaction; 4) boulder capture; 5) boulder restraint during redirection. Instead of returning a boulder from an asteroid, several missions were proposed for returning the whole asteroid. However, many asteroids which are close to Earth and are small enough to be potential candidates for redirection missions, have high rotation rates [32]. Be- fore redirecting such asteroids towards the Earth, they have to first be despun (or detumbled), which is one of the critical steps in asteroid retrieval mission. Brophy et al. in 2012 proposed a mission for returning a whole asteroid of 7 m in diameter [33]. Both single and separable spacecraft architectures were suggested in order to return a near-Earth asteroid to the vicinity of the Earth. A deployable bag would be utilised for capturing and detumbling the asteroid. The deployable bag includes arms which are connected by circumferential hoops. After de- ployment, the hoops would provide compressive force to hold the bag open. For a 6 12 m2 ⇥ asteroid, the bag would measure 10 15 m2. In order to capture the asteroid, its spin and tum- ⇥ bling rate would first need to be matched with the spacecraft. After the velocity and spin rate of the asteroid is matched, the asteroid is captured into the bag. Then the spacecraft and asteroid system is detumbled using a reaction control system. In 2014 Grip et al. suggested a spider- web capture mechanism for NASA’s ARM mission [34]. Dierently to the proposed concept by Brophy et al., this concept does not require for the spacecraft to match asteroid’s spinning rate, which results in lower overall fuel requirements. Instead, the spacecraft approaches the asteroid along its inertial vector. Spider-web capture mechanism is based on 6 robotic limbs attached to the spacecraft, which form a large barrel, surrounding the target asteroid. The cinch lines, connecting robotic limbs in dierent planes, can be tightened by a drawstring bagging mechanism. This makes the limbs close around the asteroid, similarly to how a conventional garbage bag is closed. The asteroid will then be despun, by dissipating the energy through the limbs. Finally, Tethers Unlimited developed a tethered system for despining an asteroid called “Weightless Rendezvous And Net Grapple to Limit Excess Rotation” (WRANGLER) [35]. It is a technique for capturing and despinning an asteroid with a lightweight mesh structure. It works by converting asteroid’s angular momentum into spacecraft’s angular momentum, as it revolves around the asteroid. Then spacecraft is released, and detumbled asteroid can be redirected with a mothership.

1.2 Motivation

As described above, one of the main challenges concerning an asteroid redirection mission is asteroid detumbling. Although mission concepts similar to the ARM mission suggest redi- 1.3 Preliminary Mission Description 4 recting only the boulder from a larger asteroid this way avoiding the detumbling problem, in general such missions are not necessarily seen as less complicated than the ones considering redirection of the whole asteroid. Dierent challenges are introduced: more thorough characterisation of the asteroid is required in order to select a boulder of appropriate size and mass; necessity for precise and fully controlled landing on the asteroid; large and complex capture mechanism requiring to safely bring the boulder to the vicinity of the Earth. An alternative to capturing a boulder from a large asteroid is redirecting a whole of a small asteroid, as explained in the previous section. All three: deployable bag, spider-web mechanism, and tethered WRANGLER systems are developed as alternatives to using spacecraft thrusters, in order to minimise fuel consumption when despinning larger asteroids [36]. How- ever, Bazzocchi and Emami showed that detumbling a typical 10 m in diameter near-Earth asteroid would require only 1 kg of fuel, considering an engine with 3000 s of speciﬁc impulse [37]. Thus, fuel requirements for detumbling should be of no major concern, when small asteroids are considered. Moreover, thrusters used for detumbling an asteroid, could later be used for attitude control during the redirection. Single spacecraft and separable spacecraft architectures are both considered for deep space missions when landing on the body is included. Though single spacecraft typically has lower cost since only one spacecraft needs to be developed [33], separable spacecraft architecture would most likely provide easier landing procedure and more robust communication with the Earth. Landing a small lander is easier than a big spacecraft with its large solar panels and powerful propulsion subsystem. Moreover, if the mothership is always facing the Earth, contact with the Earth can always be maintained by using the mothership as a relay spacecraft. Finally, in order to reduce the cost of the double spacecraft architecture, the lander could be designed as a CubeSat by using commercial o-the-shelf components (COTS) [38]. Therefore, the aim of this study is to suggest a preliminary design of a CubeSat lander for small asteroid detumbling mission, assuming the lander is carried to the vicinity of the asteroid inside the mothership.

1.3 Preliminary Mission Description

Asteroid detumbling mission sequence can be divided into three major steps: identification of the asteroid, landing on the asteroid, detumbling the asteroid. First, asteroid is detected by the mothership. Then, at a predefined distance from the asteroid, the lander spacecraft can be released. Either autonomously, or according to the instructions sent by the mothership, the lander lands on the predetermined location of the asteroid, and anchors itself to its surface to avoid drifting away. Before landing, the spacecraft has to match the spinning rate of the 1.4 Thesis Overview 5 asteroid, so that it constantly sees the same side of it. When the lander is securely attached to the asteroid, the detumbling procedure can be started. The steps for the detumbling procedure can either be calculated by the mothership by analysing asteroid’s tumbling rate and knowing the location of the lander, or by the lander itself. Detumbling is performed by firing the thrusters mounted on the lander. Detumbling mission ends when the angular velocity of the asteroid around each axis is zero, or close to zero. An overview of operational concept is given in Fig. 1.1.

1.4 Thesis Overview

First, a suitable asteroid candidate for detumbling mission is chosen in Chapter 2. The force needed for the spacecraft to detumble the asteroid is estimated in Chapter 3. The landing and anchoring mechanisms for the lander spacecraft are reviewed and selected in Chapters 4,5 and 6. Finally, the main drivers for preliminary spacecraft design are presented in Chapter 7, and subsystems selected in Chapter 8.

Figure 1.1 Operational concept sketch. Adapted from Probst and Förstner [1]. Chapter 2

Target Asteroid

Asteroid populations in the inner Solar System can be categorised within the following groups: asteroid belt, trojans, and near-Earth asteroids (NEAs). Most of the asteroids can be found in the asteroid belt, which is between the orbits of Mars and Jupiter. Trojans are asteroids which reside on one of the Lagrangian points (L4 and L5) of larger planet or moon’s orbit. So far only one Earth trojan has been found, oscillating about Sun-Earth L4 Lagrangian point. However, theoretical estimation is that there should be few hundred of Earth trojans [39]. NEAs come into close proximity with Earth, or intersect its orbit. They are categorised by a perihelion distance of less or equal to 1.3 AU. When choosing a suitable asteroid candidate, the requirements for selection go beyond the goal of this study of asteroid detumbling. This mission can only be feasible if asteroid is accessible from the Earth and is suitable for redirection. For both accessibility and redirection purposes, the greatest driver for asteroid selection is velocity increment requirement. The smaller change in velocity is needed for reaching and redirecting the asteroid, the lower mission costs and mission execution time are. Therefore, NEAs are the most attractive group of asteroids for such mission. Main belt asteroids are much further away compared to NEAs, and the only Earth trojan is at a very dierent inclination compared to the Earth’s orbit.

2.1 Spectral Types of Asteroids

Asteroids are classified by several factors, including their composition and assigned spectral types, mainly based on their colour, albedo, and emission spectrum. The main observed asteroid spectral types are S-type and C-type. Around 75% of all known asteroids in the Solar System fall into C-type group [2]. C-type asteroids are carbonaceous dark objects with their albedos ranging from 0.03 to 0.10, they dominate the outer belt population. The second biggest group of asteroids is S-type asteroids, which dominate the inner asteroid belt population. They 2.2 Near-Earth Asteroids 7 mainly consist of iron- and magnesium- silicates, and comprise about 17% of all known asteroids. S-type asteroids are brighter than C-type, with their albedos ranging from 0.10 to 0.22 [2]. Common asteroid spectral type populations are X-type, V-type, and Q-type. X-type asteroids are further classified into the following subtypes, by their albedos: E-type (pv > 0.30), M-type (0.30 p 0.10), and P-type (p < 0.10). M-type is the most commonly found v v group of asteroids within X-type classification. M-type is associated with metallic meteorites, mostly made up from nickel-iron [40]. V-type asteroids are similar to S-type asteroids by having basaltic properties [41]. Spectrum of Q-type asteroids is generally between V-type and S-type. By composition, Q-type asteroids are similar to ordinary chondrite meteorites [42]. For the purpose of asteroid mining, the most interesting asteroids are C-type and M-type. C-type asteroids are very interesting to explore due to the high concentration of volatiles that can be extracted for fuel production, or life support systems [24]. Moreover, due to the lower mechanical strength, such asteroid type is easier to cut or crush, which therefore should make anchoring process easier [33]. M-type asteroid could be used for material extraction which could then be returned to the Earth, or used for building space structures. An average M-type asteroid contains more platinum group metals than the richest known ore bodies in Earth [23].

2.2 Near-Earth Asteroids

Near-Earth asteroids can be further classified into four groups. The first group consists of the asteroids coming in close proximity to the Earth’s orbit (Amor-type). The second and third groups consists of the asteroids that actually cross Earth’s orbit (Apollo-, and Aten-type asteroids). Their orbits relative to Earth’s orbit can be seen in Fig. 2.1 [2]. The fourth group consists of Atira-type asteroids. Atira-type asteroids orbit inside the Earth’s orbit, but do not cross it. Refer to Table 2.1 for a summary of orbital elements. NEAs can also change their classification in time as their orbits get perturbed by other planets.

Table 2.1 Orbital parameters of NEA groups [2, 14].

Semi-Major Axis, a Apogee, Q Perigee, q Amor 1 AU 1.017 q 1.3 AU   Apollo 1AU 1.017 AU  Aten 1AU 0.9833 AU  Atira 1AU 0.9833 AU   2.2 Near-Earth Asteroids 8

Figure 2.1 Three subgroups of NEAs and their orbits with respect to Earth’s orbit around the Sun [2].

Arjuna-type Asteroids

There is also an unocial classiﬁcation of NEAs: Arjuna-type asteroids. Arjuna-type asteroids orbital parameters merge with Aten and Atira groups – their semi major axis is 0.985 a   1.013 AU. Eccentricity, e, and inclination, i, range within 0 < e < 0.1 and 0 < i < 8.56 respectively. Since inclination and eccentricity are low, the semi-major axis is almost the same as Earth’s orbit, DV required for reaching such asteroids is also low. Therefore, Arjuna-type asteroids are seen as the best possible targets for asteroid exploration missions [14]. As of May 2018, approximately 18,000 NEAs have been discovered using various automatic asteroid survey programmes [43]. Out of them, 19 are identiﬁed as Arjuna-type asteroids (the list is provided in Table 2.2).

Orbital and Physical Parameters

The asteroids are described in physical and orbital parameters. Some of the parameters listed in Table 2.2 are semi-major axis (a), eccentricity (e), inclination (i), absolute magnitude (H), and synodic rotation period of the asteroid (Pa). The data in Table 2.2 is collected from the HORIZONS system provided by Solar System Dynamics Group of Jet Propulsion Laboratory [43]. Parameters are automatically updated to the system few times a year, either from the Light Curve Database (LCDB; Warner et al., 2009), or manually from miscellaneous sources. Parameters listed near asteroid names marked with a star in Table 2.2 are provided from LCDB. Absolute magnitude is a parameter associated with an object’s intrinsic brightness. If the object’s albedo is known, its absolute magnitude can be converted to its size using the relation- 2.2 Near-Earth Asteroids 9

Table 2.2 List of identiﬁed Arjuna-type asteroids and their parameters (May, 2018) [15].

a (AU) e (-) i () H (-) da⇤⇤ (m) Pa (h) 2014 EK24* 1.008 0.070 4.81 23.3 48.5 – 96.9 0.0978 2003 YN107 0.989 0.014 4.32 26.5 11.1 – 22.2 2006 JY26 1.010 0.083 1.44 28.4 4.6 – 9.3 2006 RH120* 1.002 0.035 1.09 29.5 2.8 – 5.6 0.0458 (0.0229) 2008 KT 1.011 0.085 1.98 28.2 5.1 – 10.1 2008 UC202 1.011 0.069 7.45 28.3 4.8 – 9.7 2009 BD* 1.010 0.042 0.38 28.1 5.43 – 10.6 2009 SH2* 0.991 0.094 6.81 24.9 23.3 – 46.4 1.26 2010 HW20 1.011 0.050 8.19 26.1 13.3 – 26.7 2012 FC71 0.988 0.088 4.94 25.2 20.2 – 40.4 2012 LA11 0.987 0.096 5.13 26.1 13.3 – 26.7 2013 BS45 0.992 0.084 0.77 25.9 14.6 – 29.3 2014 QD364 0.986 0.041 4.01 27.2 8.0 – 16.1 2014 UR* 0.996 0.016 8.25 26.6 10.6 – 21.2 2.37 2016 GK135 0.988 0.087 3.16 28.1 5.3 – 10.6 2017 UO7 1.011 0.100 7.86 26.8 9.7 – 19.3 2017 YS1 0.994 0.053 5.54 28.9 3.7 – 7.4 2018 ER1 1.007 0.092 6.44 25.6 16.8 – 33.6 2018 FM3 1.012 0.091 4.57 27.2 8.0 – 16.1 * – data provided by LCDB [44]. ** – estimated value (estimation procedure explained in the text). ship in Equation (2.1)[44].

0.2H 10 da = 1329 , (2.1) ⇤ ppv where da is the object’s diameter in km, and pv is the geometric albedo. Geometric albedo is not known for most asteroids, however it is estimated that the majority of NEAs have geometric albedo ranging between 0.09 and 0.36 [45]. Lightcurve data shows measurements of the brightness change over time. From the data the object’s physical parameters can be estimated, such as rotational states of the asteroid. It can also be determined whether the asteroid is tumbling or spinning around a principal axis. In literature tumbling is also sometimes called non-principal axis (NPA) rotational motion. A tumbling asteroid does not return to the same state after a single period. In order to determine whether the asteroid is tumbling or spinning about a principal axis, lightcurves must be suf- 2.3 Asteroid 2014 UR 10

ﬁciently detailed and cover enough cycles, so that it could be concluded whether the body is strictly periodic with a single rotation frequency [46]. None of asteroids with estimated rotation period in Table 2.2 above have been assigned as tumbling or spinning asteroids. The importance of knowing whether the asteroid is spinning or tumbling is further explained in Section 3.6. Data collected of asteroid 2014 EK24 during its close approaches with Earth in 2014, 2015, and 2016 was not enough to determine whether the asteroid is spinning around its principal axis [44]. Asteroid 2006 RH120 was observed during four consecutive nights in 2007, gathering lightcurve data from approximately 4 h of measurements in total. Interestingly, from observational data two dierent solutions, using 4th and 6th order Fourier series for a rotational period, were derived: P = 1.375 0.001 1 ± min and P = 2.750 0.002 min. Again, due to the lack of observational data it was not 2 ± concluded whether the asteroid is tumbling or not. Lightcurve data collected by Ryan regarding asteroid 2009 SH2 was not published, so the asteroid was not assigned to the tumbling or spinning asteroid list [44]. Data collected by Warner in 2014 regarding asteroid 2014 UR allowed estimation of a period of 2.37 hours. Warner claimed it cannot be concluded the result to be ﬁnal, and the result should be considered as doubtful [47]. The body is not assigned to spinning or tumbling asteroids list.

Spectral Types of Arjuna-type Asteroids

The true size distribution and albedo of Arjuna-type asteroids is not known. Lin et al. in 2018 prepared a photometric survey and taxonomic identiﬁcations of 92 NEAs [48]. The ob- servations were performed in Taiwan from 2012 through 2014. The measurements included dierent colour indices of the asteroids which were brighter than 0.19 (albedo), which allowed to classify them within dierent compositional types. Most of the asteroids, 63%, fell into S/Q- type (S-type composition), 13% within X-type. Metallic asteroids (M-type) belong to X-type class. It can be concluded that distribution of various asteroid types within NEA population is fairly wide. Therefore, without enough data, it cannot be statistically predicted what kind of asteroids in terms of compositional distribution are listed in Table 2.2 [44].

2.3 Asteroid 2014 UR

Potentially hazardous asteroids (PHAs) are asteroids with absolute magnitude (H) of 22 or less (around 150 meters in diameter), and minimum orbital intersection distance to Earth of 0.05 AU [49]. In case of a close approach with Earth, PHAs are considered to be a potential threat. Moreover, in case of a collision event, asteroids bigger than 50 meters in diameter would not disintegrate within the atmosphere, thus causing disastrous eects at the surface 2.3 Asteroid 2014 UR 11

[49]. Therefore, asteroid diameter limit of 50 meters is used to classify asteroids unsuitable for redirection (and thus, detumbling) mission to avoid any potential threats in case the mission does not go as planned. Because rotation rate of the asteroid is one of the driving parameters for spacecraft design in this study, asteroids of interest are those which have rotational period observed (Table 2.2): 2014 EK24, 2006 RH120, 2009 SH2, 2014 UR. The diameter of asteroid 2014 EK24 is estimated to be within 48.5 m and 96.9 m. In case the mission would not go as planned, the asteroid could pose a threat of collision with Earth. Therefore, asteroid 2014 EK24 is not suitable for this mission. Asteroid 2006 RH120 is small, therefore would require less propellant and time for redirection. However, its rotation is much faster compared to 2009 SH2 and 2014 UR asteroids. Fast rotation would signiﬁcantly increase the complexity of the spacecraft, as well as landing procedures, therefore this asteroid is also eliminated as a not suitable candidate. Aster- oid 2014 UR is rotating almost two times slower than the asteroid 2009 SH2. It is also expected to be approximately two times smaller in diameter. Moreover, 2014 UR approaches the Earth closely every year until 2025 (the last close approach with Earth is in February, 2025). The next close approach for asteroid 2009 SH2 is only in 2078 [15]. Therefore, asteroid 2014 UR is the chosen candidate target for the spacecraft lander design. As mentioned before, it has been impossible to do the full characterisation of the chosen asteroid due to the lack of data. The exact diameter, shape, spin rate, composition of the asteroid can only be predicted within rough margins, which brings a lot of uncertainties into the lander mechanism design.

2.3.1 Spectral Type of 2014 UR The population distribution of spectral asteroid types is dierent within main belt asteroids and NEAs. A study in 2016 by Carry et al., determined the taxonomy of 206 various sizes NEAs observed by Sloan Digital Sky Survey (SDSS) [50]. 36.96% of observed asteroids belong to S-type, 23.48% to C-type, 14.35% to V-type, and 8.69% to Q-type. Surprisingly, the majority of NEAs fall within S-type, while 75% of all known asteroids belong to C-type (as mentioned in the text before). The study in 2018 by Perna et al., for the ﬁrst time focused on classifying small NEAs with absolute magnitude H 20 [51]. The study was performed during the period of 30 nights. 147 objects were observed and taxonomically classiﬁed. It has been shown that 39.73% of observed asteroids belong to S-type, 17.81% belong to X-type, 11.64% belong to Q-type, and only 8.90% belong to C-type. Both surveys by Carry et al. and Perna et al. found that the majority of asteroids belong to S-type. However, the result of distribution of other spectral types dier noticeably. Within the 2.3 Asteroid 2014 UR 12

Table 2.3 Dierent materials representation of S-type, V-type, Q-type, and C-type asteroids.

S-, V-, Q-type C-type Compressive strength 100 – 300 10 – 80 (MPa) Plaster, Representing materials Basalt limestone, kaolinite population of small NEAs the second biggest type observed was X-type, while within NEAs of all sizes the second biggest type is C-type. Therefore, it is dicult to draw accurate conclusions of what type of asteroid 2014 UR could be. The biggest likelihood is for it to be an S-type. The other types including M-type is less likely, since three dierent types belong to X-type, and M-type is just one of them. Since material properties are of high importance for design of anchoring mechanism, it is important to draw a conclusion of what kind of properties asteroid 2014 UR is expected to have. Compressive strength is of particular importance, since it shows how hard it is to break the asteroid in order to anchor on it. Apart from S-type; V-type and Q-type asteroids are also very likely to be discovered within NEAs population. All three types have similar compressive strength values. Another group of asteroids that is likely to be found within NEAs is C-type. Table 2.3 shows what compressive strength each asteroid is expected to have, and what kind of materials here on Earth are good representatives for listed asteroid types. V-type, S-type, and Q-type asteroids have similar bulk density (and thus compressive strength) as basalt [52–54]. C-type asteroids have similar density to plaster, limestone and kaolinite. Such information will be useful in further chapters of this study, especially when analysing anchoring mechanism.

2.3.2 Surface Properties Many asteroids are believed to be rubble piles. Rubble pile asteroid is formed from smaller pieces which are holding together due to self-gravitation forces. However, as can be seen in Table 2.2, all listed small asteroids are fast rotators. Being of low mass the asteroids have almost negligible gravitational force, and quite often the same order or even higher centrifugal force. The spinning rate of such asteroids is typically higher than their “rubble pile limit”, meaning that such objects are monolithic bodies [33, 55]. In order for the body to have a regolith layer, its gravity must be higher than its centrifugal acceleration. Therefore, it can be assumed that it is unlikely for the surface of asteroid 2014 UR to contain a layer of regolith. Chapter 3

Forces Acting on The Landed Spacecraft

In order to design a spacecraft for landing and despinning the asteroid, forces acting on the spacecraft must be taken into consideration. Not only they aect the spacecraft control, but also the requirements for landing and anchoring systems. This chapter describes the main forces that act on the spacecraft, their calculation procedure, and the magnitude range of the net force that can be expected when landed on the asteroid.

3.1 Introduction

Since some of the characteristics and parameters of the asteroid and spacecraft are unknown or vague, it is desirable to estimate a range of the net force acting on the spacecraft, i.e. minimum and maximum possible forces. Natural forces acting on the lander include: gravitational force, centrifugal force, solar radiation pressure, and electrostatic force (Fig. 3.1)[56, 57]. Other forces such as gas drag force, impact of dust particles and possible seismic motion are of very small magnitude or could even be non-existing for the chosen asteroid, thus assumed to be negligible. The main lander induced force is despinning force, i.e. force induced by thrusters during the mission stage of asteroid despinning. The equations and assumptions used for calculating the range of each force are described in the sections below. For calculating the range of the forces the minimum and maximum possible parameters and characteristics of the spacecraft and asteroid are given and explained. Parameters of the asteroid and spacecraft that are used extensively throughout this chapter are described in this section below. Individual parameters applicable only to calculation of a speciﬁc force are provided in the relevant following sections.

Size of the spacecraft. It is not known yet what size the spacecraft will be, but it has been decided that it has to be a CubeSat (refer to Section 1.2). As of 2011, the largest Cube- 3.1 Introduction 14

Figure 3.1 Schematic diagram showing the lander on the surface of asteroid with the main natural forces acting on it.

Sat that a standard dispenser can support is 27U, weighing 54 kg, and having dimensions of 34 35 36 cm3 [38]. Maximum area that the spacecraft can land on is 0.126 m2. ⇥ ⇥ The smallest possible lander is expected to be smaller than 27U CubeSat, but would not be expected to be less than 3U. The limit of 3U is chosen, since 3U CubeSat INSPIRE (designed by NASA) is the smallest CubeSat designed for a deep space mission to date [58]. The weight of 3U is 4 kg, the spacecraft measures 10 10 30 cm3. The smallest area that the spacecraft ⇥ ⇥ can land on is 0.01 m2. Please note that spacecraft height above the surface is not taken into account when performing calculations.

Size of the asteroid. The density of the asteroid depends on its type. As mentioned in Chap- ter 2, the asteroid is most likely to be S- or C-type. C-type asteroids have density of 1.38 g/cm3 and S-type asteroids have density of 2.71 g/cm3 [59]. As noted in Table 2.2, the diameter of the asteroid can vary between 10.6 to 21.2 m. For simplicity, the asteroid is assumed to be a sphere. Its mass then varies between roughly 860,600 kg and 13,520,000 kg.

Rotation of the asteroid. As mentioned before, it is not clear whether the asteroid is spinning or tumbling, since it was observed only for a short period of time. The observational period of 2.37 h shows the change in brightness – it is a period over which the asteroid returns to the same state. For instance, let us say the asteroid is actually tumbling, not spinning, and its 3.1 Introduction 15 observed period is 2 h. In case of a tumbling asteroid, it could be that its rotational period about one axis is 1 hour, and 2 hours around another axis. The lightcurve data would still show that it returns to the same state after two hours. In this case, if the spinning assumption is used, then angular velocity is calculated assuming 2 h spinning period around one axis. The angular momentum of an asteroid spinning around one axis would be much lower than angular momentum of asteroid tumbling around two axes. This aects estimation for spacecraft fuel consumption: less fuel is needed to despin the asteroid which spins about one axis, than the one which has angular velocity around two axes. Asteroid 2014 UR has a rotation period of 2.37 h, and it is not clear whether the asteroid is spinning, or tumbling. The kinetic energy equation for a rotating body is:

1 E = w T Iˆw, (3.1) k 2 where w is angular velocity, Iˆ is inertia tensor. Angular velocity is expressed with respect to inertial reference frame which has its origin ﬁxed to the mass centre of the body. Inertia tensor for a sphere is:

100 2 Iˆ = mr2 010, (3.2) 5 2 3 001 6 7 4 5 where m is mass of the body, and r is radius. It can be seen that the kinetic energy correlates with the angular velocity, meaning the faster the body is spinning, the higher its kinetic energy is. Thus, in order to calculate for the worst case scenario using spherical body assumption, it should be assumed that the asteroid has angular velocity around all three axes. It is typical for such small asteroids to have a tumbling period of just 10 minutes [33], thus it is decided to use this value as rotation period around two axes. Spinning period about the third axis is 2.37 h. The angular velocity can then be expressed as: w =[0.01,0.01,0.00074]T rad/s. In the best case scenario the asteroid is spinning around one axis with 2.37 h period, giving the angular velocity of: w =[0,0,0.00074]T rad/s. 3.2 Gravitational Force 16

3.2 Gravitational Force

According to the Newton’s second law, the force applied on the spacecraft due to gravity is

mscma Fgrav = mscg = 2 G, (3.3) ra where G = 6.674 10 11 m3/kg s2 is gravitational constant, m is mass of the spacecraft. ⇥ sc The gravitational force is acting towards the centre of mass of the asteroid. Maximum and minimum possible values of the gravitational force then are 4.3 10 4 N and 8.2 10 6 N ⇥ ⇥ respectively.

3.3 Solar Radiation Pressure

Solar radiation pressure on Earth can be calculated by dividing the solar constant by the speed of light. Currently, the solar constant is known only at 1 AU. That value can be used for calculating solar radiation pressure on the asteroid, since it has orbital parameters similar to Earth. The equation of solar radiation pressure on the asteroid thus is

W p = , (3.4) rad c where W = 1362 W/m2 is solar constant at 1 AU, and c = 299,792,458 m/s is speed of light. Now, assuming the spacecraft lander surface is non-reﬂective, and the spacecraft is facing the Sun at an angle of 0 degrees, the solar radiation force on the spacecraft can be expressed as

W F = A , (3.5) rad c sc where Asc is the area of spacecraft surface facing the Sun. Thus, maximum and minimum possible estimated values of the solar radiation pressure force on the spacecraft are 5.7 10 7 N ⇥ and 4.5 10 8 N respectively. ⇥

3.4 Electrostatic Force

The electrostatic force on the asteroid surface is generated by solar wind and solar radiation. Solar wind electrons are impacting the asteroid surface, while solar radiation stimulates electron emission by photoelectric eect [60]. The currents generated by solar wind and solar radiation vary with surface location and time (due to asteroid rotation). Sunlit surfaces lose 3.5 Centrifugal Force 17 electrons and generate positive potentials, while shadowed surfaces gather electrons and generate negative potentials. Generated electrostatic force can even be capable of levitating the dust on the asteroid, but that of course depends on the size of the grains, as well as other asteroid physical and orbital parameters. The electrostatic repulsive force acting between the asteroid and the lander can be estimated by knowing electrical potential on the asteroid surface and local electron density. To estimate the force, the following equation is used [61]:

qn F F = A e a , (3.6) el sc 2 where F is electrostatic force, q = 1.602 10 19 C is elementary charge, n is local electron el ⇥ e density, and Fa is electric potential on the asteroid surface. Photoelectrons density on the asteroid surface varies from 7 107 1/m3 to 2 108 1/m3, as taken from Nitter et al. (1998), ⇥ ⇥ Havnes et al. (1987), and Grard and Tunaley (1971) studies [62]. Electric potential on sunlit surface reaches up to +5 V, and on a shadowed surface around 550 V to 2550 V[61]. 8 Maximum and minimum expected electrostatic force on the lander is 1.03 10 N and 6.17 11 ⇥ ⇥ 10 N respectively.

3.5 Centrifugal Force

According to Newton’s second law, the centrifugal force in vector form can be expressed as

F = m w (w r ). (3.7) cent sc ⇥ ⇥ a The centrifugal force is acting away from the centre of the asteroid. As mentioned in the introduction, the best case scenario is one-axis spinning case: the asteroid is rotating only about one axis, and its angular velocity is w =[0,0,0.00074]T rad/s. In the worst case scenario – the asteroid is spinning about all three axis with angular velocity of w =[0.01,0.01,0.00074]T rad/s.

Best Case Scenario

The minimum possible centrifugal force is calculated assuming the smallest possible asteroid size of 5.3 m in radius, and minimum spacecraft weight of 4 kg. If minimum possible angular velocity of the asteroid is assumed to be around z-axis: w =[0,0,0.00074]T rad/s, then the location at which highest centrifugal force is acting must be on XY-plane, in this case chosen T 4 T to be ra =[5.3,0,0] m. The centrifugal force is estimated to be: F cent = 10 [0.115,0,0] N. 3.5 Centrifugal Force 18

Figure 3.2 Resultant centrifugal force distribution on asteroid 2014 UR for the worst case scenario.

Worst Case Scenario

Maximum possible centrifugal force is estimated assuming the largest possible asteroid size of 10.6 m in radius, and maximum spacecraft weight of 54 kg. If maximum possible angular velocity is w =[0.01,0.01,0.00074]T rad/s, then the maximum resultant centrifugal force acting on the spacecraft is 0.126 N. The distribution of centrifugal forces around the asteroid surface is depicted in Fig. 3.2. From the distribution of the forces it can be seen that since the angular velocities about both x- and y-axis are the same, the asteroid is rotating with the same velocity around the resultant axis. The angular velocity around z-axis is much slower compared to x- and y-axis, therefore the centrifugal force due to rotation around this axis is much smaller. 3.6 Despinning Force 19

3.6 Despinning Force

Despinning force is the force required for despinning the asteroid so that its angular velocity around all axes is zero. Such force can be found from the relation describing torque:

t = I a = r F, (3.8) ⇥ a ⇥ where t is torque, and F is force. Angular acceleration a essentially can be described as the change of angular velocity over time:

dw a = . (3.9) dt

Again, angular velocity for the best case scenario is: w =[0,0,0.00074]T rad/s. Angular velocity for the worst case scenario is: w =[0.01,0.01,0.00074]T rad/s.

Best Case Scenario

For the most ecient despinning of the asteroid, the spacecraft should land perpendicularly to the axis of rotation. If not, during the despinning procedure, torque would be introduced around the other axes. Using Equation (3.8), the force can be expressed as:

r t F = a ⇥ . (3.10) ? r 2 || a|| With angular velocity around z-axis of w =[0,0,0.00074]T rad/s, minimum moment of inertia 6 2 T of I = 9.67 10 kg m , and landing location at ra =[5.3,0,0] m, despinning force variation ⇥ a with time is shown in Fig. 3.3.

Figure 3.3 Best case scenario force and time required for despinning the asteroid. 3.6 Despinning Force 20

Worst Case Scenario

In order to estimate the required force for despinning the asteroid in the worst case scenario, the landing location on the asteroid has to be determined. In the worst case, as described in Section 3.5, the asteroid is spinning around resultant xy- axis at 0.01 rad/s. Angular motion around z-axis at 0.00074 rad/s is much smaller than motion around xy-axis, which causes small precession. The most ecient way to despin the asteroid is to land on a location at which r is perpendicular to the xy-axis (refer to Fig. 3.4). Asteroid can then be despun around xy-axis by thrusting tangentially against the direction of angular velocity vector (denoted as F1 in the ﬁgure). This would not induce extra torque around z-axis, since the despin force would be applied parallel to this axis, and precession motion assumed to be negligible. After the asteroid is despun around xy-axis, it can then be despun around z-axis, by applying force against the direction of angular velocity vector about z-axis (denoted as F2 in the ﬁgure). Since the force is applied perpendicularly to the spinning axis, Equation (3.10) can be applied.

Figure 3.4 Landing location and direction of forces required for despinning the asteroid in the worst case scenario. 3.7 Net Force Acting on The Spacecraft 21

Figure 3.5 Worst case scenario force and time required for despinning the asteroid.

Figure 3.5 shows the force required to despin the asteroid within certain time frame. As expected, it can be seen that in order to despin the asteroid around xy-axis, the force required is much higher than despinning the asteroid around z-axis. For the worst case scenario, the total force for despinning the asteroid is the sum of exerted force within certain time frame around xy- and z- axis.

3.7 Net Force Acting on The Spacecraft

As mentioned at the beginning of this chapter, forces acting on the spacecraft will aect the choice of its landing and anchoring systems, as well as other subsystems. All the forces calculated in the sections above can be categorised into forces acting normally to the surface of the asteroid (all natural forces), and force acting tangentially to the surface of the asteroid (thrust force exerted for despinning the asteroid). 3.7 Net Force Acting on The Spacecraft 22

Table 3.1 Net force and list of separate forces acting on the landed spacecraft in the normal direction to the asteroid surface.

Fgrav (N) Frad (N) Fel (N) Fcent (N) Net Force (N) 6 8 11 5 6 min 8.2 10 4.5 10 6.2 10 1.2 10 3.7766 1100 ⇥ ⇥ ⇥ ⇥ ⇥ max 4.3 10 4 5.7 10 7 1 10 8 0.126 0.1122266 ⇥ ⇥ ⇥

The list of natural forces is provided in Table 3.1. It can be seen that the dominating force acting on the spacecraft in normal direction is centrifugal force. Signiﬁcant dierence can be noted between minimum and maximum centrifugal forces, which are mainly aected by the angular velocity of the asteroid. Solar radiation pressure and electrostatic forces can be considered to be negligible because of their low magnitude. It can be concluded that the landed spacecraft must be capable of not drifting away from the asteroid when the maximum possible net force of 0.126 N is acting in the normal direction away from the asteroid surface. It is however more dicult to state what is the maximum possible force in tangential direction that the spacecraft should be able to withstand without drifting away. Considering that the timeframe of asteroid redirection mission is several years, even one month for despinning procedure can be seen as a relatively short time period. Thus, the upper cap is placed by the CubeSat propulsion systems capabilities, which are discussed at the end of this study. How- ever, for now, in order to set a requirement for anchoring system, the limit is set to 50 N. In the worst case scenario, 50 N of force would despin the asteroid around xy-axis in less than 7 h and around z-axis in 0.5 h, which equals to just over 7 h of total despinning time. Finally, the results also show the importance of knowing the parameters of the chosen asteroid as precisely as possible. Having more lightcurve data would help to better estimate the angular velocity of the asteroid, and know for certain whether it is spinning or tumbling. This uncertainty is the main reason why the dierence is so great between minimum and maximum possible natural forces acting on the spacecraft, as well as required force for despinning. Chapter 4

Landing Systems Review and Choice

Successful landing on the surface of the asteroid is one of the key steps that have to be accomplished before attempting to detumble the asteroid. Therefore, selecting a suitable landing mechanism is seen as a task of great importance in this study. In this chapter ﬁrst the literature review of existing landing concepts is provided, followed by a suitable landing technique choice for small asteroid detumbling mission.

4.1 Literature Review

For landing on the asteroid surface, either active or passive descent techniques can be employed. Active descent means that the lander can control its descent trajectory after being released from the mothership. Active descent option provides higher accuracy in landing and mobility, by typically involving propulsion and attitude control systems. Passive landing (also called ballistic landing) means that the lander does not control its descent trajectory after the release. Such lander designs result in low weight and volume requirements, and overall re- duced spacecraft complexity. However, uncertainty regarding landing accuracy and mission success is increased. The spacecraft’s capability to move across the surface of the asteroid is also worth considering. Such possibility could improve the reliability of the mission in case the initial position of the landed spacecraft is not satisfactory. Movement across the asteroid surface can be performed either by propulsion systems used for landing, or by simple mechanical mechanisms. Existing concepts for active and passive descents are summarised below, by also describing what mechanisms are used for moving on the asteroid surface, if any. Some of the landing mechanisms are used for the missions described in the literature review in Section 1.1. 4.1 Literature Review 24

4.1.1 Active Descent As mentioned above, the most common active decent option includes propulsion and attitude control systems. One of such examples is the lander Bode designed for a multiple asteroid characterisation mission. Such mission concept has been suggested by Probst and Forstner [1]. Bode is equipped with propulsion and attitude and orbit control (AOC) systems, which allow to perform controlled manoeuvring and landing. Landing in correct orientation is the main requirement, whereas the landing location on the asteroid for exploration purposes is not of such great importance. Thus, the system does not employ any surface mobility mechanisms. For landing and departure manoeuvres the required change in velocity (DV) is estimated to be 100 m/s per asteroid. The disadvantage of such configuration is its high mass requirement. The weight of required wet propulsion subsystem mass is 42.7 kg, and mass of AOCS is 51.1 kg. Another active descent lander, however, not fully controlled, is being developed by Gedex and Space Flight Laboratory (SFL) [3], depicted in Figure 4.1. GRASP (GRavimetric Aster- oid Surface Probe) is a small scale satellite, weighing less than 20 kg, and fitting within a 12U volume CubeSat. GRASP is targeting to operate on small asteroids between 100 m and 1000 m in diameter. The lander is released with spring ejection mechanism from the mothership. Ide- ally GRASP is expected to land on an asteroid in a hopping motion, but is still equipped with propulsion system in case it bounces othe asteroid (not further than 50 km). This can be done by commands from the ground, since the system is not autonomous. The lander uses 6 deployable legs. They are arranged in a way such that the lander can land in 8 stable landing configurations, each supported by 3 legs. The lander can hop across the asteroid’s surface us-

Figure 4.1 GRASP lander developed by SFL [3]. 4.1 Literature Review 25 ing propulsion system. Also, whenever possible, reaction wheels will be employed to provide torque in order to initiate a hopping motion. It is estimated that the total DV needed is 170 m/s, which includes 100% margin (without including DV needed for recovery in case of a bounce- o). Thrust magnitude needed is 100 mN, which is based on asteroid’s gravity and spacecraft’s mass. Propulsion system’s wet mass is 2.8 kg. A controlled landing concept which also increases landing stability is a spinning lander suggested by Southern California Selene Group (SCSG) [63, 64]. Using three sets of thrusters

(a) Spinning lander concept, where I denotes axial thrusters, J denotes radial thrusters, and K denotes tangential thrusters [63].

(b) 3U spinning CubeSat manoeuvring with axial and radial thrusters [64].

Figure 4.2 Spinning lander concepts suggested by SCSG. 4.1 Literature Review 26

(axial, radial, and tangential), the lander’s velocity, spin rate and attitude can be controlled (Figure 4.2). For landing a 3U 5 kg spacecraft on the Moon, each of the thrusters can exert 22 N of force, which is needed to counteract the gravity force. The lander concept is easily scalable and mass-ecient. Most importantly, due to its gyroscopic stiness, such system cannot tip over during the landing, which ensures correct facing of the lander. The lander moves across the surface of the body in a hopping motion, by employing the same thrusters which were used for landing.

4.1.2 Passive Descent In 2017, Wang et al. from University of Southern California patented an “Instrument lander utilizing a CubeSat platform for in situ exploration of asteroids and comets” [4]. The proposed lander lands in uncontrolled descent, where trajectory is influenced by solar radiation pressure and asteroid’s gravitational force. The suggested lander is a 2U CubeSat structure (Fig. 4.3a). 2U shape provides higher probability of landing on one of 4 rectangular faces in horizontal orientation, rather than one of two square faces in vertical orientation. A motor inside the spacecraft drives a flywheel (Fig. 4.3b). The torque generated provides mobility to turn the lander in one or both directions (for correct re-orientation purposes). However, if the spacecraft lands vertically, a break can be used to provide additional torque for the lander to “hop” to another location in order to successfully settle on a preferred side of the spacecraft (Fig- ure 4.3a). When landed, either of two movements can be generated: (1) a hopping motion, and (2) a pivoting motion. Hopping motion can be achieved with the break, and pivoting motion can be achieved with the flywheel.

(a) Mobility mechanism with a break (b) Flywheel mobility mechanism

Figure 4.3 Surface mobility mechanisms patented by University of California [4]. 4.2 Landing Mechanism Selection 27

Figure 4.4 MASCOT mobility concept - eccentric arm concept [5].

The Hayabusa2 lander MASCOT is ejected by a spring mechanism at 100 m altitude above the asteroid surface, with a very small DV of 5 cm/s. MASCOT is not equipped with any attitude or altitude control systems, thus lands by utilising the asteroid’s gravity field only. The descent takes approximately 20 to 30 minutes. MASCOT lands in a hopping motion until it comes to rest in an a-priori unknown orientation [29]. To move on the surface, MASCOT uses an eccentric arm (Fig. 4.4) which can rotate up to few revolutions by activating a brushless DC motor. It can either flip the spacecraft to dierent orientation, or with more energy – hop from the surface. Rosetta lander Philae was also released using spring ejection mechanism for landing on the comet. The touchdown vertical kinetic energy was absorbed by the landing gear – during the touchdown the lander head was pushed towards the tripod, this way transforming mechanical energy into electrical energy by the use of generator. Also, a cold gas thruster was mounted on the lander, which could be fired at touch to reduce the bouncing, by pushing the lander towards the ground. Finally, lander also contained two harpoons with cords, which were supposed to be used for anchoring. However, during touchdown operation the thruster did not fire, thus firm fixation of the lander depended only on harpoons. But the harpoons did not fire either due to unknown reasons. Philae bounced across the comet’s surface for almost 2 h before coming to rest. The weight of landing legs was 10 kg, weight of cold gas system was 4.1 kg [65]. Zhao, et al. in 2012 proposed an asteroid lander concept, inspired by comet lander Philae design [66].

4.2 Landing Mechanism Selection

Both active and passive descent options were described in the section above. It is clear that passive landing methods utilise the gravity of the body. The gravity of asteroid 2014 UR is 4.2 Landing Mechanism Selection 28 estimated in Chapter 3, and is shown to be of very small magnitude, meaning that it cannot be utilised for neither for landing procedure, nor for surface mobility. Thus, only active landing in this case is possible. Regarding active descent options, two main methods were discussed: traditional controlled descent and controlled spinning descent. Both options use propulsion and AOC systems. The main dierence between the two is that spinning descent achieves gyroscopic stability by axi- ally spinning the spacecraft until the touchdown. The advantage of the spinning control system is that it occupies minimal volume (3U) and does not require much propellant. However, when landing on a spinning (or tumbling) asteroid, the spacecraft has to match the asteroid’s spin rate, so that the spacecraft constantly follows the same part of the asteroid before attempting to land. A spinning lander is limited to motion along one axis. For instance, when landing on the Moon, it would use its radial thrusters to create continuous up and down motion. In case of a mission to a small spinning asteroid body, during the landing phase it would also need to be thrusting along other axes in order to match asteroid’s spinning rate. This would result in destabilising spacecraft’s spinning motion. Therefore, due to the reasons mentioned above, the chosen method for landing on a small spinning asteroid surface is traditional controlled descent, which will employ propulsion and AOC subsystems. Chapter 5

Anchoring Systems Review

The main purpose of the anchoring system is to ensure that the spacecraft remains on the surface in the right orientation, by reacting to all forces or torques that might occur. When landing on a small asteroid with negligible gravity, during the impact with the surface, the spacecraft can gain energy from the asteroid rotation, which could cause the spacecraft to escape. Therefore, it is important to dissipate as much energy as possible during the impact, to ensure that the energy gained from the asteroid rotation would be less than necessary to escape asteroid’s surface [67]. Such energy can be dissipated by employing a suitable anchoring system. Since exact physical properties of asteroid 2014 UR are unknown prior to the mission, the anchoring system must be able to function within a range of dierent possible asteroid parameters.

5.1 Literature Review

Anchoring methods can be divided into two categories: slow anchoring and high-speed anchoring. The main advantage of high-speed anchoring methods is little time required, which means the spacecraft does not need to be held in place during the anchoring process for a long time. However, some slow anchoring systems solve this problem by employing self-opposing mechanism. Also, slow anchoring systems typically oer reusability, while most of the fast anchoring systems can only be used once. Slow anchoring methods include hammering, wetting ﬂuid, melting strategies, drilling, and sawing. High-speed anchoring methods include microspine gripper, tethered spike, and telescoping spike strategies [6, 10]. Each method is described below. Some of the technologies described have technology readiness level (TRL) assigned to them according to system developed by NASA. Please refer to Appendix A for a description of each level. 5.1 Literature Review 30

5.1.1 Slow Anchoring Methods

Hammering. A suitable approach where fast penetration is not possible, is hammering the anchor in the surface [6]. The diculty with this approach is that the nail must be perpendicular to the surface, otherwise the nail gets bent during the process. It is easier to penetrate into plaster or limestone, however it is more dicult with higher strength materials such as basalt.

Wetting fluid. Application of wetting fluid is another slow anchoring method. Fluid such as cement, epoxy, foam, or similar, can be injected on the surface through some hollow tube, and this way anchor spacecraft to the surface. The advantage of this method is that during the process no reaction force is exerted on the spacecraft. Application of various wetting fluids in space environment has already been demonstrated [6]. However this method has stability issues, since the spacecraft is required to stay still during the anchoring process.

Melting. A possible way to anchor to the asteroid is by melting a hole in it, and then placing an expansion device inside that hole [10]. For this technology to work, heated probe would be needed for melting, which has signiﬁcant power consumption requirements. Of course, this method would be highly dependent on the melting point of surface materials of the body, thus is mainly to be used on bodies rich in ice.

Drilling. One of the most commonly known slow anchoring techniques is drilling. However, long time and fairly high drill force required do not make this technique particularly attractive [10]. This method is feasible for penetrating low compressive strength materials such as plaster, limestone, but not that often employed for harder materials such as basalt [6, 68]. Drilling technology is also a popular choice as a sample collection tool for missions to comets or C- type asteroids.

Self-opposing drilling. Force applied towards the surface of the body for a long period of time generates a reaction force on the spacecraft, therefore thrusters (or some other anchoring mechanisms) are required to hold the spacecraft in place. However, self-opposing anchoring systems are being developed. Such systems apply opposing force to the reaction force, thus there is no need to hold spacecraft in place by the use of thrusters. Cadtrak Engineering developed a self-opposing drilling technique. Such technique decreases spacecraft preload requirement (to oppose the reaction force), and thus minimises the propulsion system requirements. It works by using multiple drilling arms in a coordinated manner, as to keep the forces in equi- librium. Such sample drilling system is shown in Fig. 5.1a. The pull-out force of the system is 200 N normal to the surface for basalt rock. To achieve such strength the drills penetrate into 5.1 Literature Review 31

(a) Cadtrak Engineering (b) Honeybee Robotics

Figure 5.1 Sample self-opposing drill systems [6] a rock up to 3 mm depth in roughly 20 s. Required preload from a spacecraft is approximately 10 N. Total power consumption is 10 W. The subsystem weighs 5 kg [69, 68, 6]. The system has also been tested on a lower strength material kaolinite. The pull-out strength was measured to be 100 N normal to the surface, at 3 mm penetration depth. Power, preload and anchoring time requirements for lower compressive strength bodies are expected to be smaller. The system is suitable for rocks, gravel and consolidated soil surfaces, and can penetrate at wide range of angles. The system suggested by Cadtrak uses 4 drills. The drills were tested in laboratory environment, therefore, TRL is 4. Similar approach is also suggested by Honeybee Robotics (Fig. 5.1b)[6]. Net force component is perpendicular to the surface, thus propulsion system of the spacecraft only has to exert force in opposite direction. Speciﬁc power requirements or pull-out strength on dierent rocks is not provided. However, if the same drill bit is used as for Cadtrak Engineering system, the values are expected to be similar. Important to note – because the mechanism of self-opposing system is designed dierently, the preload force requirement during operation might be dierent. Both Cadtrak Engineering and Honeybee Robotics systems are foldable and do not seem to occupy more than 1U of volume (as can be seen in the pictures). Drills are expected to be reusable until degraded.

Self-opposing sawing. A dierent anchoring system for asteroid exploration is presented by Zhang et al., which is based on sawing method (Fig. 5.2)[7]. Three robotic arms of the spacecraft are equipped with three sawing disks. The disks present with the possibility to anchor onto hard rock surfaces, such as plaster, marble, and granite. After disks penetrate into the surface, the self-lock mechanism ﬁxes the robotic arms on that surface. The system provides at least 157 N and 225 N force in tangential and normal directions respectively. It 5.1 Literature Review 32

Figure 5.2 Experimental setup of robotic arm presenting anchoring technology based on sawing method [7]. takes 1 minute to penetrate into plaster, and 3 minutes to penetrate into granite. The system can successfully operate at penetration angles of 45, 60, and 75 degrees. The preload required for anchoring into plaster is only 0.5 N, and 8.6 N for penetrating into granite – which is less than drilling system requirements. The power requirement is 58.5 W and 11.2 W for granite and plaster respectively. The robotic legs are foldable, and as it seems from the pictures, would not take more than 1 U of volume. The weight of the system is 7.8 kg. It can also be reusable, as long as sawing disks are not degraded. Sawing disks have been tested in laboratory environment, therefore TRL is 4.

5.1.2 High-Speed Anchoring Methods

Microspine Gripper. A self-opposing high-speed anchoring system is being developed by Parness et al., at NASA Jet Propulsion Laboratory [70]. Microspine gripper (Fig. 5.3), initially developed for rock climbing robots, can be used for grasping asteroid surface. Microspine gripper plants itself onto the surface by arraying many attachment points (toes of microspines) on the rock. Each microspine drags and stretches relative to the neighbouring microspines in order to ﬁnd suitable point on the surface to grip on. Thus, the lower surface roughness is, the harder it is for the microspines to grip on it – the strength of the device decreases. Poorest performance of grippers was demonstrated on loose materials, such as bonded pumice, loose lava rock, pea pebbles and sand [8]. The pull-out forces were only around 1 N. However, when tested on consolidated rocks, the gripper performed well by achieving more than 120 N of pull out force in both tangential and normal directions. A single microspine gripper only weighs 1.05 kg, and takes less than 1U of space – as can be clearly seen from the pictures. 5.1 Literature Review 33

Figure 5.3 Left: NASA JPL microspine grippers being tested for anchoring strength at 45. Right: CAD view of microspine gripper cross-section [8, 9].

The microspine gripper configuration consists of 16 carriages with 16 microspines in each carriage, and two actuators. The disengage actuator acts as a release mechanism by pulling the carriages away from the rock surface. It is a 2 in (51 mm) custom designed linear actuator. The second, engage actuator, is responsible for securing the anchoring by applying tension to engagement cables, which connect to the carriages. The engage actuator is a 12 V DC brushed motor provided by Maxon Motor USA. Such motors, as stated in Maxon Motor USA website, require 6 W (or less) of power. Since the power requirement of a linear actuator is not specified, by including factor of safety, it can be assumed that the system developed by JPL should not consume more than 10 W of power in total during any stage of gripper activation. The normal position of the gripper is locked, which means that once it holds onto the surface, no extra power is needed for it to stay there. So far, microspine gripper system has been demonstrated in relevant environment, thus TRL is assumed to be 6. The complete tool also includes anchoring drill and drive train, and could then be classified as a slow anchoring tool (due to time required for drilling). It was designed for the NASA ARM mission. The complete tool is referred as a Microspine Tool. After the grippers have a stable grasp with the surface, the anchoring drill is driven into the rock. During the drilling operation, reaction forces and torques are redirected to the rock, thus no thrusting from the spacecraft to counteract the reaction forces is required. After the hole is drilled, a T-shaped anchor is deployed from the tip of the drill, cutting a T-shaped slot at the bottom of the borehole. The tool provides very high anchoring forces and also allows to collect the samples from the surface [70, 8]. 5.1 Literature Review 34

Figure 5.4 Telescoping spike anchoring system [10].

Telescoping spike. The telescoping spike (Fig. 5.4) penetrates the surface with spikes of increasing diameter towards the depth of the body. It was ﬁrst developed by Stelzner and Nasif [10] for ST4/Champollion mission. ST4/Champollion mission was supposed to be a joint NASA and CNES (the French Space Agency) mission to a comet. The mission was cancelled in 1999 due to budget constraints. The telescoping spike concept was chosen for anchoring to the surface of the comet. The spike is released explosively during the touchdown. A gas generator of the anchoring system is triggered by a laser altimeter, which senses the proximity of the surface. Gas generator accelerates telescoping spike which penetrates into the comet. ST4/Champollion anchor is designed to provide at least 450 N pull-out strength in any direction. Because of the uncertainties in comet properties, the anchor is designed to penetrate up to 3 m of comet surface depth. The dissipation of energy during penetration is achieved via plastic deformation of anchoring parts. The telescoping spike system consists of three main parts: spike, tube 1, and tube 2 (Fig. 5.5). During anchoring, a certain length of the spike section is plastically deformed to the tube 1 (which has larger diameter) during coupling. Even though the anchoring process is robust and fast, the main disadvantage is the large lateral loads imparted on spacecraft during the process. For ST4/Champollion mission 100 kg spacecraft the anchor to the surface of the comet, the anchor must be released at 120 m/s velocity. The penetrating spike weighs 2.3 kg. The system has been tested on homogeneous and heterogeneous surfaces. 450 N anchoring strength in any direction has been achieved when penetrating at impact angles of up to 45 into materials of up to 10 MPa of compressive strength. Inspired by ST4/Champollion mission to the comet, Liu et al. in 2013 suggested using telescoping spike concept for anchoring to the asteroid [71]. The tests of their proposed anchoring system were successfully performed on sand and soil, which have densities of 1.71 g/cm3 and 1.89 g/cm3 respectively. The anchoring force for sand and consolidated soil substances is 18 N and 10 N, where initial penetration velocity is 10.2 m/s and 22.6 m/s respectively. The subsys- 5.1 Literature Review 35

Figure 5.5 Telescoping spike system for ST4/Champollion mission [10]. tem weighs around 1 kg, and occupies less than 1 U of CubeSat volume [11, 71]. The telescoping spike system does not provide re-anchoring possibility. Since the telescoping components have been tested in laboratory and relevant environment, TRL is assumed to be 4.

Tethered Spike. Tethered spike and multi-legged tethered spike concepts are depicted in Fig. 5.6a and Fig. 5.6b respectively. The spike is explosively released from the spacecraft and penetrates the surface of the body. The tether is then being pulled until the required tension is reached. However, if the body is rocky and has heterogeneous surface, the spike can deﬂect from its intended path. Therefore, the tethered spike is more suitable for gravel and consolidated soil surfaces [6]. Multi-legged tether spike system works in the same manner, except it uses multiple spikes for penetration. It provides tighter anchoring, but since the interaction with the surface is increased, it poses more risks. Quadrelli et al. demonstrated that a single tether provides more than 450 N of anchoring strength in any direction when anchoring to low compressive strength materials (representing C-type asteroids). System applicability to wide range of angles was also demonstrated. Tethered anchoring system was installed into the Rosetta lander Philae (Fig. 5.7)[11]. Two harpoons were mounted in the landing gear. One of them was supposed to be ﬁred automati- 5.1 Literature Review 36

(a) Tether spike (b) Multi-legged tether spike

Figure 5.6 Tethered spike anchoring system [10]. cally. The firing mechanism would have been activated by firing the projectile, after sending a touchdown signal from the landing gear. Rewind system would have then spooled up the anchor cable until the required tension was reached. However, the signal was not sent, therefore the system failed to fire. Since the system was not demonstrated in the relevant space environment, its TRL is assumed to be 6. The weight of a single harpoon subsystem is 0.9 kg, and it occupies roughly 0.5U of CubeSat volume.

Magnetic Anchoring. For metal-rich, magnetised asteroids, magnetic anchoring is possible. One of the concepts is envelopment – it uses cables to grab the asteroid as shown in Fig. 5.8.

(a) Philae landing gear (b) Anchoring harpoon projectile

Figure 5.7 Philae harpoon anchoring system mounted in the landing gear [11]. 5.2 Criteria for Successful Anchoring 37

Figure 5.8 Magnetic anchoring concept “Harvestor” by Deep Space Industries [6].

5.2 Criteria for Successful Anchoring

Due to the uncertainties with the physical parameters of asteroid 2014 UR (refer to Chapter 2), multiple selection criteria for choosing the most suitable anchoring methods have to be established. Criteria listed in Table 5.1 are used for the selection of the ideal anchoring system candidate. Each criterion is provided with an explanation of why it is important.

5.3 Discussion of Suitable Anchoring Systems

In the literature review above two types of drilling systems were considered: the ones requiring very high preload, and the ones which are self-opposing, thus having less preload requirements. In case of former, the preload force sometimes must be applied for fairly long periods of time, which results in high propellant requirements, and increased mission complexity. Therefore only self-opposed drilling systems will be considered as a potential candidate for asteroid detumbling mission. Another slow anchoring method, hammering, requires very powerful nail-gun to be able to penetrate into harder composition surfaces. Significant preload of about 10 Ns is required. To rebound such energy the spacecraft would need to produce a reaction force, by, for instance, firing thrusters in the opposite direction. Moreover, this method requires the nail to be perpendicular to the surface. Such requirement significantly increases the complexity of the spacecraft, to be able to autonomously correctly position the nail. Therefore, the discussed hammering method is ruled out as unsuitable for such small asteroid mission, where one of the criteria is applicability to wide range of angles. 5.3 Discussion of Suitable Anchoring Systems 38

The method of wetting fluid is also considered as not viable. Such method would be more suitable for large bodies (so that gravity can be utilized), since near-perfect stability during the anchoring process is required. However, in case of small tumbling asteroids of negligible gravitational force, very high requirements on propulsion and control systems would be imposed for longer times. As such, design of the spacecraft would be too complex. The melting method is not viable for S-type asteroids (which are expected to be the majority). Such method could be successfully utilized on asteroids rich in ice, however melting other types of asteroids would impose high power requirements on the spacecraft. Moreover, it would also be a very slow anchoring process. The microspine tool consisting of microspine gripper, drilling mechanism and drivetrain provides pull-out strength of 280 N and 190 N in tangential and normal directions respectively. Such anchoring strength is much greater than needed, therefore it was decided to consider only the microspine gripper system. It is much lighter and takes less space, and also the power requirement is lower since no drilling is required. It satisfies the criteria presented for anchoring systems, therefore the microspine gripper, as a fast anchoring method, will be considered next. Multi-tethered spike anchoring system would be unnecessarily complex for a detumbling mission on small asteroid body. The benefit of higher strength from multiple anchors is not considered to be significant, since anchoring strength provided by one anchor is already enough. Multiple anchors would also significantly increase system complexity. Moreover, the small asteroid diameter would impose limits on tether length. Therefore, a system with one tethered spike is considered as potential candidate, assuming it would be fired when the spacecraft is hovering close to asteroid surface. Finally, the magnetic anchoring method is only suitable for ferromagnetic bodies. There- fore, since composition of the chosen asteroid 2014 UR is unlikely to be M-type, this anchoring technique cannot be selected. After initial overview of anchoring concepts, the remaining methods to select from are self-opposing drilling, sawing, microspine gripper, tethered spike, and telescoping spike.

5.3.1 Summary of Anchoring Systems Candidates Selected anchoring systems will be compared in the next chapter according to the criteria presented in Section 5.2. Summary of parameters of each candidate is presented in Table 5.2. 5.3 Discussion of Suitable Anchoring Systems 39

Table 5.1 Criteria for selecting suitable anchoring system.

Criteria Explanation Unit 1 Small mass In order to minimise the mass of the spacecraft it is Kilograms preferable for each sub-system to have as minimal mass as possible. 2 Small volume In order to minimise the volume of the spacecraft it CubeSat is preferable for each sub-system to occupy as little units space as possible. 3 Low power High power requirements result in heavy batteries oc- Watts requirement cupying large volume of the spacecraft. Moreover, high power requirement on one subsystem would result in compromising the operation of other subsystems. Therefore, subsystems with low power requirements are preferred. 4 Low preload Some anchoring systems might need additional Newtons requirement thrusting during the anchoring process. This compli- cates spacecraft design. Therefore, subsystems with no preload or low preload requirements are preferred. 5 High TRL More developed and tested technologies are pre- TRL level ferred over less advanced ones. (1 – 9) 6 Re-anchoring In case anchoring process is not successful during the Qualitative possibility ﬁrst attempt, re-anchoring capability is highly pre- (no, partial, ferred, which signiﬁcantly increases the robustness yes) of the mission. 7 Anchoring ability In case the landing location is on a steep rock, anchor- Qualitative over wide range ing at an angle might be needed. Therefore, systems (none, of angles able to anchor on wide range of angles are preferred. narrow, moderate, wide) 8 Suitability for Due to possibility of asteroid having C-type or S-type Qualitative wide range of composition, anchoring should be possible on both (gravel, asteroid surfaces types of asteroids. soil, rocks) 9 Short anchoring Shorter anchoring times save fuel needed for hover- Minutes, time ing near the asteroid, or holding onto its surface. It seconds also reduces complexity of the mission. 10 High pull-out Sucient anchoring strength (at least 50 N and Newtons strength 0.126 N in tangential and normal directions respectively) is needed to ensure that the spacecraft will not drift away. 5.3 Discussion of Suitable Anchoring Systems 40 c > 450, > 450 > 450, > 450 Tethered Spike Telescoping NA NA NA NA NA NA < 10 Negligible Negligible Gripper Microspine Rocks Rocks Gravel/soil Gravel/soil Sawing 157, 225 > 140, > 130 c 5 7.844664 1.05 0.9 1 < 1 < 1 < 1 1.5 < 1 Table 5.2 Parameters of anchoring sub-systems. Wide Moderate Wide Wide Wide Partial Partial Yes No No Slow Anchoring Methods Fast Anchoring Methods Drilling Self-Opposing Rocks, gravel/soil S-typeS-type 40 10 58.41 8.637 S-typeS-type 00:20 NF, 200 03:00 NF, NF NF, NF C-typeC-type NF NF 11.22 0.459 C-typeC-type NF NF, 100 01:00 – consolidated. Criteria Subsystem Mass (kg) Subsystem Volume (U) Power (W) Preload required (N) TRL Re-anchoring capability Range of angles for anchoring Applicable surfaces Anchoring time (mm:ss) Pull-out strength (tangent, normal) (N) c NA – not applicable. NF – not found. Chapter 6

Anchoring System Choice Using MCDM

As presented in Section 5.2, there are multiple criteria according to which the best anchoring system is chosen. From Table 5.2 it can be seen that all of the listed anchoring technologies have advantages and disadvantages. All of them show great performance in one set of criteria while signiﬁcantly underperform in another set of criteria. Therefore, decision making for which anchoring system to choose is not straight forward. In order to choose the most suitable system, multi-criteria decision making (MCDM, sometimes abbreviated as multi-criteria decision analysis – MCDA) technique is used.

6.1 Introduction

The purpose of MCDM is to employ decision maker’s preference when ﬁnding a unique optimal solution for problems involving multiple criteria. In any decision making methodology the following steps are involved (as deﬁned by Majumder and Mrinmoy) [72]:

1. Identiﬁcation of the objective/goal of the decision making process.

2. Selection of criteria, parameters, factors and decider.

3. Selection of the alternatives.

4. Selection of the weighing methods to represent importance.

5. Method of aggregation.

6. Decision making based on the aggregation results.

For this study the goal of the multi-criteria decision making process is to choose the most suitable anchoring system for a small asteroid detumbling mission. The ﬂowchart of the process 6.1 Introduction 42 is provided below in Fig. 6.1. The steps of selecting criteria and alternatives (i.e. anchoring systems) were already described in Chapter 5, with criteria provided in Table 5.1, and alternatives with their parameters listed in Table 5.2. Since all criteria are not of equal importance, a method for assigning weights to each criterion has to be selected. After the weights are calculated, method of aggregation for the alternatives is chosen. Finally, the most suitable system can be selected. Exact numerical descriptions of each criteria, procedure of assigning weights, and choice of suitable aggregation method are described in the sections below. The reader is suggested to refer to the ﬂowchart diagram throughout all sections of this chapter for a better understanding of the MCDM process.

Figure 6.1 Flowchart for the MCDM methodology. 6.2 Criteria Description 43

6.2 Criteria Description

In order to have a decision making process as accurate as possible, it is important to ensure that the criteria are not overlapping and also to convert qualitative values into quantitative ones. In this section, the overlapping criteria will be put under one criterion, and qualitative criteria will be given suitable quantitative descriptions. The ﬁnal table of all criteria used for the MCDM process will then be presented. For the interest of the reader, it might be important to note that at this point some of the criteria will have dierent units. The units will however be normalised later from 0 to 1 when aggregation method (PROMETHEE in the ﬂow chart) is assigned, and preference functions selected. This will be explained in Section 6.4. Finally, in the criteria parameters table (Table 5.2), it can be seen that the parameters of some of the alternatives are unknown for both S-type and C-type asteroids. For evaluation purposes, power, preload required, and anchoring time values will be taken for anchoring to S- type asteroid. It is a reasonable choice, not only because all values are known for this asteroid type but also because as described in Section 2.3.1, majority of small NEAs are expected to be S-type.

Mass, Volume and Power (MVP)

Mass, volume and power (MVP) are interdependent parameters of similar importance. For instance, higher power requirements result in larger batteries, which hence add more weight and volume. Therefore, it is decided to put all three parameters under a single criterion, by using points system, as explained below. 1U CubeSat must weigh not more than 1.33 kg, according to the standard CubeSat launch dispenser [73]. Thus, if a system occupies 1U of volume, it is assumed to be of roughly the same importance as if the system weighs 1.33 kg. Comparison of mass and volume with power is not that simple. Knowing that the eective area of body-mounted solar cells, is related to the area of the sides of satellite, it can be said that the maximum power available for the satellite

Table 6.1 Maximum power available for dierent CubeSat platforms.

Volume (U) Surface area (cm2) Power available (W) 1 600 30 2 800 40 3 1400 70 4 1600 80 6 2200 110 6.2 Criteria Description 44

Table 6.2 Points corresponding to volume, mass and power that anchoring system requires.

Volume (U) Points Mass (kg) Points Power (W) Points 0 < V 1 1 0 < m 1.33 1 5 < P 15 1    1 < V 2 2 1.33 < m 2.66 2 15 < P 25 2    2 < V 3 3 2.66 < m 3.99 3 25 < P 55 3    3 < V 4 4 3.99 < m 5.32 4 55 < P 65 4    4 < V 5 5 5.32 < m 6.65 5 65 < P 80 5    5 < V 6 6 6.65 < m 7.98 6 80 < P 95 6    is related to its surface area. For instance, the GomSpace power system for 1U CubeSat can provide 30 W of power (total of 600 cm2 of surface area), and power system for 3U CubeSat can provide 60 W of power (1200 cm2 of surface area if panels are mounted on the sides of the spacecraft, excluding top and bottom). Thus, an assumption is made that 200 cm2 of available spacecraft surface area can provide 10 W of power. Table 6.1 provides estimates of available power for up to 6U CubeSat platforms. Since 5U platforms as such do not exist, it could be assumed that 5U would provide maximum of 95 W of power. During the anchoring, the power has to be provided not only to the anchoring system, but also distributed to propulsion system (if anchoring requires preload), on-board computer for calculations and sending instructions for anchoring, and maybe for communications with mothership if needed. 15 W of power is decided to be left for such operations, meaning that the anchoring mechanism can consume maximum available power minus the margin of 15 W. If less than 5 W of power is required, then it can be assumed that power consumption is negligible. The following point system as presented in Table 6.2 is thus used. For instance, a self-opposing drilling system scores 8 points for MVP criterion: 4 points for the weight (5 kg), 1 point for the volume (< 1U), and 3 points for power consumption (40 W); which in total is 8 points. Using the same principle, sawing, microspine gripper, tethered spike, and telescoping systems score 11, 3, 3, and 2 points respectively.

Re-anchoring

Re-anchoring capability can be described as “none”, “partial”, or “capable”. “None” means that re-anchoring is impossible after failed anchoring attempt. This applies to tethered spike and telescoping systems. “Partial” means that the system is capable to re-anchor for a limited number of times. For instance, drill bits or sawing tool parts are worn out fairly quickly, especially after anchoring to hard rocks such as basalt. Finally, “capable” means a successful re-anchoring many times. This applies to microspine gripper, since it can re-anchor many times before being worn out [8]. The score for each alternative is assigned as follows: 0 points for 6.2 Criteria Description 45 no re-anchoring capability, 0.5 points for partial re-anchoring, and 1 point for a system which is capable to re-anchor.

Anchoring at Wide Range of Angles

Ability of the mechanism to anchor to rocks at wide range of angles is described as “none”, “narrow”, “moderate”, and “wide”. “None” means that the system is capable to anchor only at a certain angle. For instance, a hammering tool described in the anchoring systems literature review is able to anchor only when placed at 90 angle to the rock. “Narrow” means, that system can anchor at a range of 10 angle to the rock. “Moderate” provides range of anchoring at ± 30 to the rock. And ﬁnally, “wide” means that the system is capable of anchoring at 45 ± ± to the rock. The score is distributed as follows: if the system is not capable to anchor at any range of angles, it gets 0 points, if it is capable to anchor at narrow range of angles it gets 1 point, if the range is moderate, the system gets 2 points, and if the range is wide, the system gets 3 points.

Applicability to Dierent Surfaces

The systems selected are applicable to anchor to rock, gravel or consolidated soil surfaces (or a combination of few). As mentioned in Section 2.3.2, the surface of the asteroid most likely will be rocky. Therefore, ability to anchor to a rocky surface is greatly preferred. Thus, the points for dierent surface applicability are distributed as follows: 0 for gravel, soil, or combination of both; 0.9 for rock; 1 for combination of gravel, soil, and rock, or soil and rock.

Sucient Anchoring Strength

As shown in Chapter 3, no more than 0.125 N of anchoring strength in normal direction is required, and no more than 50 N of anchoring strength in tangential direction is required. There- fore, it is decided to assign 0 if sucient anchoring strength of 0.125 N and 50 N in normal and tangential directions respectively is not reached, and 1 if sucient anchoring strength is reached. It is not very important whether the system can achieve 100 N or 200 N, since both of these anchoring strengths would lead to the same asteroid detumbling performance. It can be seen from the criteria table that the tangential anchoring strength value for a drilling system is not provided. However, assuming the drill bit would not break when 50 N force is applied in tangential direction, the drilling system pull-out strength can be considered as sucient. If the self-opposing drilling system gets selected, the required material for a drill bit to withstand such strength, will be looked at. It can also be seen that tethered spike and 6.3 Criteria Weights 46

Table 6.3 Numerical criteria for each alternative.

Self- Microspine Tethered Criteria Opposing Sawing Telescoping Gripper Spike Drilling

g1 MVP (score) 8 11 3 3 2 Preload g 10 8.637 0 0 0 2 Required (N)

g3 TRL (level) 4 4 6 6 4 Re-anchoring g 0.5 0.5 1 0 0 4 (score) Anchoring at g5 Wide Range of 32333 Angles (score) Applicability to g6 Dierent 1 0.9 0.9 0 0 Surfaces (score) Anchoring Time g 0.33(3) 3 0 0 0 7 (minutes) Sucient g8 Anchoring 1 1 1 unknown unknown Strength (1 or 0) telescoping systems strength values for anchoring on S-type asteroid are not provided, therefore the anchoring strength will be left as unknown for these two alternatives.

6.3 Criteria Weights

Not all criteria are of equal importance. Therefore a system of showing how each criterion diers from one another is needed. The importance of criterion can be shown by assigning weight, such that: k Â Wj = 1, (6.1) j=1 where W is weight, subscript j denotes each criteria, and k is number of alternatives. The weights can be found from pairwise comparisons of each two criteria. Triantaphyllou provides a method of deriving relative criteria weights from dierence comparisons [74], by 6.3 Criteria Weights 47 asking a question “How much is criterion 1 more important than criterion 2?”. The proposed similarity scale used to compare two criteria is presented in Table 6.4. For instance, a pairwise comparison of criteria g and g is w = W W , where g 1 2 12 | 1 2| denotes criteria. It can be seen that pairwise comparison does not show which criteria is more important – only how similar the two are. For instance, using similarity scale, value of w12 (dierence of importance between having small MVP and having low preload requirement) is 0.1, because both criteria are seen to be of almost identical importance. In the same way, the rest of the comparisons are made, and are presented in matrix form in Table 6.5. More detailed reasoning behind each pairwise criteria comparison is provided in Appendix B.1. The values of dierence between criteria in Table 6.5 are re-arranged in Table 6.6. The increasing dierence in similarity of each criteria pair shows the order of criteria importance.

As can be seen in the table, the order of criteria from least important to most important is: g3, g7, g2, g1, g4, g5, g6, g8.

6.3.1 Optimal Pairwise Comparison It is important to note, that according to the Human Rationality assumption [74], a decision maker is considered to be a rational person. This means, that decision maker is trying to minimise their regret and maximise the proﬁt by putting eort in trying to minimise the errors involved in pairwise comparisons. However, it can be seen that the values in Table 6.6 are not perfectly consistent, which is due to human error. For instance, criterion g6 is more important than criterion g7, and the dierence between the two is 0.8. Criterion g6 is also more important than criterion g3, and the dierence between the two is 0.7. This should mean that criterion

Table 6.4 Similarity Scale proposed by Triantaphyllou.

Intensity of Similarity Deﬁnition 0 The two entities are identical. 0.10 Almost identical. 0.20 Very similar. 0.30 Almost very similar. 0.40 Almost similar. 0.50 Similar. 0.60 Almost dissimilar. 0.70 Almost very dissimilar. 0.80 Very dissimilar. 0.90 Almost completely dissimilar. 1 Completely dissimilar. 6.3 Criteria Weights 48

Table 6.5 Matrix of pairwise dierence comparisons of criteria.

W1 W2 W3 W4 W5 W6 W7 W8

W1 0 0.1 0.3 0.3 0.5 0.6 0.3 0.6 W2 0.1 0 0.2 0.3 0.5 0.7 0.2 0.7 W3 0.3 0.2 0 0.4 0.6 0.7 0.1 0.8 W4 0.3 0.3 0.4 0 0.2 0.4 0.4 0.5 W5 0.5 0.5 0.6 0.2 0 0.2 0.5 0.4 W6 0.6 0.7 0.7 0.4 0.2 0 0.8 0.2 W7 0.3 0.2 0.1 0.4 0.5 0.8 0 0.8 W8 0.6 0.7 0.8 0.5 0.4 0.2 0.8 0

Table 6.6 Closest Discrete Pairwise (CDP) matrix (after re-arrangement).

W3 W7 W2 W1 W4 W5 W6 W8

W3 0 0.1 0.2 0.3 0.4 0.6 0.7 0.8 W7 0.1 0 0.2 0.3 0.4 0.5 0.8 0.8 W2 0.2 0.2 0 0.1 0.3 0.5 0.7 0.7 W1 0.3 0.3 0.1 0 0.3 0.5 0.6 0.6 W4 0.4 0.4 0.3 0.3 0 0.2 0.4 0.5 W5 0.6 0.5 0.5 0.5 0.2 0 0.2 0.4 W6 0.7 0.8 0.7 0.6 0.4 0.2 0 0.2 W8 0.8 0.8 0.7 0.6 0.5 0.4 0.2 0

g3 should be more important than criterion g7 by 0.1 dierence. However, it is the other-way round. Such inconsistencies could be ﬁxed by changing each value until the matrix is perfectly consistent. However, although done quite often, this is a long process of trial-and-error, after which it is still nearly impossible to reach complete consistency. Triantaphyllou [74] suggested a process for achieving perfectly consistent pairwise criteria comparison matrix by solving a linear least squares problem: the error minimisation is inter- preted as minimisation of the sum of squares of the residual vector, which is expressed as a typical linear least squares problem. Therefore, an error factor Xij (due to human error) is introduced with each comparison:

X w = X w = a = a = W W , (6.2) ij ij ji ji ij ji | i j| where a is the actual value of pairwise comparison. Optimal pairwise comparison matrix (Real

Discrete Pairwise (RDP) matrix) can be found, when all Xij variables are equal to 1. Thus the goal is to ﬁnd Xij values which minimize the sum of squares function, subject to constraints. 6.3 Criteria Weights 49

The objective function of quadratic programming problem with linear constraints, as derived by Triantaphyllou [74], is the following:

n 1 n n 1 n 2 minimize f = Â Â Xij 2 Â Â Xij, i=1 j=i+1 ⇥ i=1 j=i+1 (6.3) subject to X w + X w X w = 0 kj kj ji ji ik ik for any n i > j > k 1 and all X 0, where n is number of constraints. The number of ij constraints is equal to the number of criteria. This quadratic problem can be transformed into equivalent system of linear equations (general form):

I AT X 1˜ m = , (6.4) ˜ " AAOON # ⇥ "l # "0# where I is identity matrix of order m (m = n(n 1)/2), A is N m matrix with coecients m ⇥ of CDP matrix in Table 6.6 (where N = n(n 1)(n 2)/6), AT is transpose of A, O is square N matrix of order N (all entries are zeros), X is a vector of size m with Xij variables, l is a vector of size N with Lagrangian coecients li, 1˜ is a vector of size m (all entries are ones), 0˜ is a vector of size N (all entries are zeros). Any column (or row) of matrix AAAAT is linearly dependent on the remaining columns (or rows), and the remaining columns (or rows) are linearly interdependent. Therefore, one of any li values can be set to any arbitrary value, in order to solve the system of linear equations (Equation 6.4) for determining error factors [74].

The RDP matrix can then be found by multiplying error factors by actual values (Xijwij), and is presented in Table 6.7.

Table 6.7 Real Discrete Pairwise (RDP) matrix.

W3 W7 W2 W1 W4 W5 W6 W8

W3 0 0.0812 0.2155 0.3065 0.4940 0.6688 0.8490 1.0162 W7 0.0812 0 0.1343 0.2253 0.4128 0.5876 0.7678 0.9350 W2 0.2155 0.1343 0 0.0910 0.2785 0.4533 0.6335 0.8007 W1 0.3065 0.2253 0.0910 0 0.1875 0.3623 0.5425 0.7097 W4 0.4940 0.4128 0.2785 0.1875 0 0.1748 0.3550 0.5222 W5 0.6688 0.5876 0.4533 0.3623 0.1748 0 0.1802 0.3474 W6 0.8490 0.7678 0.6335 0.5425 0.3550 0.1802 0 0.1672 W8 1.0162 0.9350 0.8007 0.7097 0.5222 0.3474 0.1672 0 6.3 Criteria Weights 50

Table 6.8 Weights of each criterion.

Criterion g3 g7 g2 g1 g4 g5 g6 g8 Weight 0.0904 0.0991 0.1093 0.1202 0.1332 0.1399 0.1506 0.1574

Some initially estimated CDP values showed inconsistency of more than 25%. Largest errors were found for the following entries w38 (X38 = 1.27), w72 (X72 = 0.67), w14 (X14 = 0.62) and w15 (X15 = 0.72). It is nearly impossible for humans to be perfectly consistent, especially when it involves larger number of criteria. Therefore, it is important to check the consistency and introduce error factor in order to obtain the matrix of optimal pairwise comparison, as presented in Ta- ble 6.7. After error correction, RDP matrix shows pairwise criteria comparisons with perfect consistency. The MATLAB code for calculating error values and RDP matrix is provided in Appendix B.2.

6.3.2 Weights of Criteria Finally, since pairwise weight distribution is known, weights of importance of each criterion can be found from Equation 6.2. Again, the order of signiﬁcance of criteria is known to be increasing (from least important to most important) as follows: g3, g7, g2, g1, g4, g5, g6, g8. MATLAB’s non-linear solver fsolve is used for solving the system of equations. The arbitrary initial input for the solver is given in increasing order for each criteria weight: [W3 W7 W2 W1 W4 W5 W6 W8]. The system of equations is then solved until the solution converges. Each weight is then normalised so that the sum of all weights is equal to one. Since all weights have to be normalised to positive values only, the following equation is used:

exp(Wi) Wi = . (6.5) Âexp(Wj) j

Final weights of each criterion are presented in the Table 6.8. It can be seen that the two most critical criteria of anchoring system are ability to provide sucient anchoring strength

(g8), and ability to anchor to dierent surfaces (g6). The second group of the two most important criteria are ability to anchor at wide range of angles (g5), and ability to re-anchor (g4), which is then followed by low mass, volume and power budgets criterion (g1). The last group of lowest importance criteria is low preload requirement (g2), short anchoring time (g7), and high TRL (g3). 6.4 Method of Aggregation 51

It can be seen that the criteria that are the most critical for the success of the mission have the highest weights of importance. Without sucient anchoring strength the mission is impossible. Without ability to anchor to dierent possible surfaces, and wide range of angles, the mission success is not likely. Without ability to re-anchor the mission is fairly risky. If mass, power and volume of the spacecraft are high, the mission complexity and costs increase signiﬁcantly. Finally, high preload requirement, long anchoring time and low TRL would make the spacecraft much more complex, and mission less feasible. It is important to note, however, that some of the criteria are aecting one another. For instance high preload would aect mass and volume, since more fuel would be required. The same applies for anchoring time. Of course, the study would be even more accurate if none of the criteria were overlapping. However, it can be concluded that the weights derived are reasonable, since the most important criteria are not interdependent, and the order of importance of all criteria can be rationally explained as acceptable.

6.4 Method of Aggregation

Correct choice of method for aggregation of all alternatives is of high importance, and thus is seen as another critical step in choosing the best anchoring system. MCDM methods are clas- siﬁed into two main groups: compensatory methods and outranking methods [72]. Compen- satory methods involve trading ogood and bad attributes of each alternative. Bad attributes of an alternative can be outweighed by the good ones. The ﬁnal score of all attributes for each alternative is then calculated in order to determine the most optimal alternative. Outranking methods are rather dierent than compensatory methods. Instead of calculating scores on each alternative, it focuses on eliminating the ones, which are outperformed on enough criteria of sucient importance. One of the most important features of outranking methods is that it is possible to eliminate alternatives under criteria which are dicult to compare (or even incomparable), or information for the alternative is missing. Two most popular outranking methods are ELECTRE (ELimination and Choice Expressing Reality) and PROMETHEE (Preference Ranking Organisation Method for Enrichment of Evaluations) [72]. Both methods perform pair-wise alternatives comparisons, which makes it easier for the decision maker to be involved in the process [75, 76]. One of the main advantage of ELECTRE method is that it considers veto thresholds. Veto threshold is a value decided by a user to a given criterion. If alternative a outranks alternative b to a given criterion, and the dierence between both alternatives is greater than the veto threshold value, then alternative b is omitted from the decision making. However, the major limitation of ELECTRE method is that it provides incomplete ranking of all alternatives. 6.4 Method of Aggregation 52

The PROMETHEE method can provide complete ranking of all alternatives, however its main restriction is that it does not take discordance into account. Concordance and discordance indexes measure satisfaction and dissatisfaction of choosing one alternative over the other [76]. In general, discordance coecient measures the strength of evidence against hypothesis that alternative a is preferred to alternative b. In order to utilise the PROMETHEE method, the decision maker has to select a preference function. It might pose a challenge in case knowledge about the criteria is not sucient.

6.4.1 PROMETHEE Method In the PROMETHEE method, the structure of preference is based on pairwise comparisons between the alternatives. For each criterion, the decision maker assigns a preference function, according to which the alternatives are compared and preference is assigned [75]:

Pj(a,b)=Fj[d j(a,b)], (6.6) where P (a,b) is how much a is preferred over b under a given criterion j, given (a,b) A, j 2 where A is all alternatives. The function assigned to the criterion is Fj, where d j is dierence between a and b: d (a,b)=g (a) g (b), (6.7) j j j where g j(a) and g j(b) are criterion j values of alternatives a and b. Preference value lies between 0 and 1. Values close to 1 show strong preference of one alternative over another, and values close to 0 show weak preference of one alternative over another. As described before, some criteria have to be minimized and some have to be maximized in order to ﬁnd the best solution. Equation (6.6) is applied when criteria has to be maximised. In case a criterion has to be minimized, the preference function is reversed:

P (a,b)=F [ d (a,b)]. (6.8) j j j For each preference function q and/or p parameters have to be identiﬁed. Parameter q is a threshold of indierence; parameter p is a threshold of strict preference. Preference functions and the reasoning for how they are assigned to each criterion are described in Appendix B.3. The summary for each criterion is presented in Table 6.9. 6.4 Method of Aggregation 53 Usual Strength Su cient Anchoring 1 . 0 = q U-shape Surfaces to Di erent Applicability 3 = p Angles at Wide V-shape Range of Anchoring 1 = Re- p V-shape anchoring score (0–1) level score (0–1) 1 or 0 12 = score MVP p (1–14) V-shape 10 = p V-shape quirement Preload Re- 3 = Time p V-shape Anchoring 8 = max min min min max max max max TRL p Table 6.9 Information about each criteria in order to use PROMETHEE outranking method. 0.0904 0.0991 0.1093 0.1202 0.1332 0.1399 0.1506 0.1574 V-shape Criteria Name Criteria Weight Criteria Type Preference Function Unit level minutes Newtons 6.4 Method of Aggregation 54

Table 6.10 Global preference matrix for all alternatives.

Self- Microspine Tethered Opposing Sawing Telescoping Gripper Spike Drilling Self- Opposing 0 0.3154 0.1506 0.2172 0.2172 Drilling Sawing 0.0149 0 0 0.2172 0.2172 Microspine 0.2596 0.4095 0 0.2838 0.3064 Gripper Tethered 0.1930 0.3429 0 0 0.0226 Spike Telescoping 0.1804 0.3303 0.0100 0.0100 0

As discussed before (Table 6.3), anchoring strength criterion values for tethered spike and telescoping system are unknown. For further calculations it is decided to use the value as 1 (the best score). If neither tethered spike nor telescoping spike score as the best alternative, then it does not matter what value is assigned for anchoring strength criterion. In that case these alternatives could not be elected as the best choice in any way. After the preference values are found for each criterion, matrix can be created to show how one alternative is preferred over the other using [75]:

k p(a,b)= Â Pj(a,b)Wj; 8 j=1 (6.9) > k > > The results are presented in: Table 6.10. It can be seen that the strongest preference is of microspine gripper over sawing tool, while the sawing tool over microspine gripper, and tethered spike system over microspine gripper are not preferred at all.

Outranking Flows

Outranking ﬂows can be described as positive or negative. Positive outranking ﬂow value shows how alternative a outranks all other alternatives. The higher value, the better the al- 6.4 Method of Aggregation 55

Table 6.11 Positive and negative outranking ﬂows for all alternative anchoring systems.

+ f f Self-opposing Drilling 0.225 0.162 Sawing 0.112 0.349 Microspine Gripper 0.315 0.040 Tethered Spike 0.140 0.182 Telescoping 0.133 0.191 ternative. Negative outranking flow value shows how alternative a is outranked by all other alternatives. Here, the lower the value, the better the alternative. Positive outranking flow is defined as:

+ 1 f (a)= Â p(a,x). (6.10) n 1 x A 2 Negative outranking ﬂow is deﬁned as:

1 f (a)= Â p(x,a), (6.11) n 1 x A 2 where in both (6.10) and (6.11) f represents value of outranking ﬂow, n is the number of alternatives, x is each alternative a is compared to. The results of positive and negative outranking ﬂows are shown in Table 6.11.

PROMETHEE I and II

The PROMETHEE I partial ranking can be obtained from Table 6.11. It shows whether one of two alternatives is preferred (aPIb), whether two alternatives are indierent (aIIb), or incomparable (aRIb). The rules of partial ranking are the following [75]:

+ + f (a) > f (b) and f (a) < f (b), or 8aPIb if 8f +(a)=f +(b) and f (a) < f (b), or > > > > > f +(b) and f (a)=f (b); > > (6.12) >aIIb if >f +(a)=f +(b) and f (a)=f (b); <> :> f +(a) > f +(b) and f (a) > f (b), or > I >aR b if > 8f +(a) < f +(b) and f (a) < f (b). > < > > :> : 6.4 Method of Aggregation 56

Table 6.12 Complete ranking of anchoring systems alternatives.

Anchoring System f 1 Microspine Gripper 0.275 2 Self-opposing Drill 0.063 3 Tethered Spike 0.042 4 Telescoping 0.058 5 Sawing 0.237

Applying the rules from Equation (6.12) to values in Table 6.11, it can be seen none of the alternatives are incomparable or indierent. PROMETHEE II provides complete ranking, when the net outranking ﬂow is considered:

+ f(a)=f (a) f (a). (6.13) The higher the net ﬂow, the better the alternative. Complete ranking of anchoring systems alternatives is provided in Table 6.12.

6.4.2 Discussion From Table 6.12 it can be seen that the best anchoring system is the microspine gripper. It con- siderably outweighs self-opposing drill, which placed second. Even when the highest score was assigned to sucient anchoring strength criterion for tethered spike and telescoping anchoring systems, in complete ranking they placed only 3rd and 4th. This means that if they had lower value assigned, the net outranking flow value for these two system would have been even worse. Finally, the last in the ranking is sawing system. The main uncertainties in MCDM and particularly in the PROMETHEE aggregation method lie in finding correct criteria weights, and assigning the right preference functions (as well as parameters). Since both steps are based on human decisions, they could not be performed perfectly accurately. However, seeing that first-place alternative has the best result with such high dierence compared to other alternatives, allows to conclude that application of MCDM to this decision making procedure is valid and acceptable, even if some of the weights and functions were to slightly deviate. Chapter 7

Primary Mission Drivers

In order to complete preliminary design of the spacecraft, first the primary drivers of the spacecraft design have to be identified. Primary mission drivers influence subsystem selection and design, which are discussed in the subsequent chapter. The main identified drivers are mission data-flow, total required DV (change in velocity) budget, and spacecraft pointing accuracy. Each driver is discussed in the sections below.

7.1 Mission Data-Flow

Mission data-flow diagram helps to maximise the chances of mission success and minimise system complexity. It shows where the data for reaching mission goals comes from, what processing must be done, and where the results are used [77]. In case of this mission, it is important to see how autonomous the lander is: which data is computed autonomously on-board, and which is communicated from the mothership. The level of lander autonomy will aect the complexity of the lander. The more instruments are installed on-board the mothership, the less complex is the lander. However, it does not mean that the combined result of lander and mothership design is as optimal as possible: communication loads between the two spacecraft become higher since in that case the lander needs to get all the instructions from the mothership; accuracy of the lander operations reduces. Mission data flow distribution mainly aects AOC (attitude and orbit control), communications, and power subsystems. The main tasks that have to be executed in order to achieve the goal of the mission are the following: asteroid identification; landing site location determination; lander-asteroid distance determination; lander attitude determination; thrusting sequence for both landing and detumbling calculation. Mothership is a larger and more powerful spacecraft, since it is responsible for redirecting the asteroid. Therefore, its computational resources are assumed to also be larger than the resources of the lander. To save computational power of a lander spacecraft, the resources 7.1 Mission Data-Flow 58 of the mothership can be utilised for performing asteroid identification and landing site location determination tasks. After the tasks are executed, mothership uploads the asteroid-centric coordinate system together with the landing location coordinates to the lander. Lander attitude and lander-to-asteroid distance determination should be performed by the lander. These tasks are seen as too risky to dedicate them to the mothership: if these tasks were performed by the mothership, they would contain some error due to communication lag, which could result into inaccurate or hard landing, or in the worst case, crash to the asteroid surface. The mothership could, however, also monitor the lander in order to increase the redundancy, but this is outside the scope of this thesis. Asteroid tumbling rate determination task must be performed by the mothership in order to determine the best landing location site. However since detumbling the asteroid is the last critical step for success of this mission, after landing the angular velocity of the asteroid should be measured by the lander again. This needs to be done in case the lander does not land precisely on the assigned landing location.

Figure 7.1 Data-ﬂow diagram for asteroid detumbling mission. Round shapes denote data source, rectangles denote tasks, hexagon denotes data end user. 7.2 Delta-V Budget 59

Thrusting sequences for landing and detumbling are also calculated by the lander spacecraft, since it is equipped with instruments for position determination and asteroid tumbling rate measurement. Figure 7.1 shows the structure of data-flow diagram, where white rectangular boxes indicate the tasks that need to be performed, light grey circles define where the data originates from, dark grey hexagon defines where the data has to go. Instruments are to be defined in the following chapter which completes preliminary spacecraft design.

7.2 Delta-V Budget

DV budget is one of the primary mission drivers. It is used for landing on the asteroid and despinning it. Required change in velocity later translates into propulsion system requirements and amount of propellant needed, which might occupy very high fraction of the total spacecraft mass budget. First, the location where the lander is going to be released from the mothership has to be known. This is aected by the asteroid’s Sphere of Inﬂuence (SOI). Then, DV required for landing on the asteroid can be estimated, which is followed by required DV estimation for asteroid despinning.

7.2.1 Sphere of Influence The sphere of Influence is the region around a celestial body, where the primary gravitational influence on an orbiting object is that celestial body. This means, that the lander must be released further than the SOI radius of the target body, so that the lander is not influenced by the presence of asteroid, until it becomes active. The radius of the sphere of influence of a smaller celestial body orbiting a larger celestial body in circular orbit, can be estimated using the following equation. m 0.4 r = a , (7.1) SOI M ⇣ ⌘ where a is semi-major axis of the smaller body, m is mass of the smaller body, M is mass of the larger body. As presented in the Table 2.2, the parameters of asteroid 2014 UR are unclear – its diameter ranges from 10.6 m to 21.2 m. It is decided to use the upper bound for calculations in order to estimate the highest possible Delta-V requirement. The maximum radius is 10.6 m. As discussed in Chapter 2, the asteroid is most likely to be C-type or S-type. Since the density of S-type asteroids is higher, its value is used for further calculations (r = 2710 kg/m3). Assuming the asteroid to be a spherical body, its mass is estimated to be 1.352 107 kg. ⇥ 7.2 Delta-V Budget 60

The mass of the Sun is 1.9885 1030 kg, semi-major axis of asteroid’s 2014 UR orbit is ⇥ 0.996 AU. Plugging all values in Equation (7.1) gives the value of the maximum possible SOI radius, which is 80.55 m. Therefore, after including some uncertainty margin, the distance for releasing the lander spacecraft from the mothership should be 100 m.

7.2.2 Delta-V Estimation: Docking Knowing that the distance between the asteroid and the lander is 100 m, it is safe to say that the landing manoeuvre can be treated as rendezvous. Because of the small mass (and thus negligible gravity) of asteroid and spacecraft, the problem is treated as two-body problem: spacecraft orbiting the Sun, and asteroid 2014 UR orbiting the Sun. The goal for the spacecraft is to chase and dock with the asteroid. Both the interceptor (spacecraft) and the target (asteroid) are orbiting in the same circular orbit and are very close to each other. The set of linearised dierential equations which solve nearby relative motion problem are called Clohessy-Wiltshire or Hill’s equations of relative motion. The derivation procedure of the equations is described by Vallado [13], and is referred to here. The Satellite Coordinate System, RSW, is used. Such system moves with the satellite: R-axis is collinear with the position vector, S-axis is positive in the direction of the velocity vector aligned with the local horizon, and Y-axis is normal to the plane orbit. In the derivations of the equations xyz notation for RSW components is used. The ﬁgure of the coordinate system can be found in Appendix C. The derivation of equations below and relevant assumptions are also explained in Appendix C. The change in velocity required for the spacecraft to dock with the asteroid is calculated using the following Clohessy-Wiltshire equations:

(6x0 (wt sinwt) y0)w sinwt 2wx0 (4 3coswt)(1 coswt) y˙0 = , (4sinwt 3wt)sinwt + 4(1 coswt)2 8 > wx0 (4 3coswt)+2(1 coswt)y˙0 (7.2) >x˙0 = , <> sinwt z˙ = z w cotwt, > 0 0 > > where:x0,y0,z0 is initial position of the spacecraft, t is rendezvous time, w is angular velocity of the asteroid around the Sun which is calculated as follows:

µ w = 3 , (7.3) srtgt 7.2 Delta-V Budget 61

Figure 7.2 Delta-V required to dock with asteroid 2014 UR versus rendezvous time. where µ is standard gravitational parameter (µ = 1.327 1020 m3s-2 for the Sun), r is the ⇥ tgt semi-major axis of target body’s orbit (rtgt = 0.996 AU for the asteroid 2014 UR). As previously discussed the lander spacecraft is released from the mothership at 100 m distance behind the asteroid. For simplicity, it is assumed that the spacecraft is released in the same orbit as the asteroid. Thus, the initial position of the spacecraft is x = 0, y = 100, 0 0 z0 = 0. Using Equation (7.3), the angular velocity of the asteroid 2014 UR around the Sun is estimated to be w = 2.0035 10 12 rad/s. ⇥ Substituting all values into Equation (7.2), and estimating the initial velocity of the spacecraft, for docking with the asteroid from t = 0 to t = 15 minutes, the graph in Figure 7.2 is obtained. It can be seen that the DV requirement for docking is very small. Docking under one minute can be achieved with DV of slightly more than 1.5 m/s. As the rendezvous time increases, DV requirement decreases, until it becomes almost negligible. However, it is important to note, that in a real case scenario before landing, the spacecraft has to match the rotational rate of the asteroid, i.e. has to orbit the asteroid so that the same spot of the asteroid is always visible. In this case, DV needed just to reach the asteroid was calculated.

Hovering Above The Asteroid

An approximate estimation of how much DV is needed for matching asteroid’s rotational rate and hovering about the surface, can be roughly estimated from DV requirements for docking with the asteroid, which are calculated above. For more accurate estimation, the change in velocity in x- and z-axis should also be included, since the lander travels around the asteroid, rather than in the same direction. However, for simplicity, the value is taken from the calculations made above, since only an approximate estimation of expected DV value is sucient. 7.3 Pointing accuracy 62

The maximum perimeter of the asteroid, assuming its spherical shape, is 66.6 m (when ra = 10.6 m). The maximum possible asteroid’s rotational period about xy-axis (refer to Chapter 3) is 10 min (rotation about z-axis is almost negligible). Thus, in order for the spacecraft to hover 5 m above the same spot of the asteroid, it means that the spacecraft has to travel 97.4 m in 10 minutes. From Figure 7.2 it can be seen that to travel 100 m in 10 min requires DV of roughly 0.2 m/s. Thus it can be concluded that to hover around the asteroid for 5 rotations (50 min) should require approximately 1 m/s of DV budget.

7.2.3 Delta-V Estimation: Despinning The second part for which is necessary to estimate the required change of velocity is despinning the asteroid. Despinning can be seen as changing asteroid’s rotational velocity to 0 rad/s. Rotational velocity of a sphere can be related to linear velocity as follows:

V = w ra, (7.4) l ⇥ a where V l is linear velocity. Same as in Chapter 3, best case and worst case scenarios are calculated. For more details on both scenarios please refer to the aforementioned chapter. Having angular velocity of w =[0,0,0.00074]T rad/s in the best case scenario, and landing T location at ra =[5.3,0,0] m, the linear velocity is 0.0039 m/s. Having angular velocity of w =[0.01,0.01,0.00074]T rad/s in the worst case scenario, and T landing location at ra =[7.4953, 7.4953,0] m, the linear velocity is 0.1501 m/s. a

7.3 Pointing accuracy

Pointing control requirement drives AOC subsystem design, which is responsible for determining the attitude of the spacecraft, and re-orienting (or holding) it to predeﬁned attitude. The more accurate pointing is required, the heavier and more power consuming sensors and actuators have to be selected. In case of this mission, the pointing information is needed in order to land on desired location in desired orientation on the asteroid. Pointing accuracy deﬁnes the range within which the spacecraft must be pointing in order to accomplish the mission goals. Right after the release of the lander spacecraft from the mothership, the lander must be able to track the asteroid and the targeted landing location. Thus, the required minimum pointing accuracy can be estimated as an angle (at a 100 m spacecraft release distance) which ensures the asteroid is always in the range of spacecraft view – as depicted in Figure 7.3. 7.3 Pointing accuracy 63

Figure 7.3 Scheme for estimating required maximum pointing angle to the asteroid 2014 UR, just after spacecraft release from the mothership.

For simplicity the asteroid is assumed to be of a spherical shape. To have a conservative pointing accuracy estimation, its radius is taken as the smallest possible radius of 5.3 m. Thus, the angle qlim is calculated to be 3.03, meaning that the pointing accuracy must be at least 6.07. If the landing sequence was calculated at 5 m above asteroid surface, the spacecraft would land within 0.265 m radius of assigned location. Chapter 8

Preliminary Spacecraft Design

As primary drivers for the spacecraft design have already been identiﬁed in the chapter before, the spacecraft can now be partitioned into its subsystems, each of which are discussed in this chapter in more detail. The propulsion system, acting as a payload since it helps to accomplish the main mission goal, is needed for landing the spacecraft and detumbling the asteroid. In order for the spacecraft to land on the asteroid, it must have some type of attitude control. It can consist of thrusters, or reaction wheels, or both. After landing, the spacecraft has to anchor itself to the surface with the use of a microspine gripper. The spacecraft also needs to be able to communicate with the mothership in order to send data and receive instructions. All the com- putations are to be performed with an on-board computer, which is part of the command and data handling system. Finally, some sort of power system is needed to power all of the processes described above. Thus, the spacecraft shall be partitioned into the following subsystems: propulsion (payload), anchoring (microspine gripper), attitude and orbit control, command and data handling, communications, power, thermal, structure.

8.1 Attitude and Orbit Control Subsystem

Attitude and orbit control subsystem (AOCS) is responsible for determining the attitude of the spacecraft, and re-orienting (or holding) the spacecraft to the predeﬁned attitude. The ﬁnal mass and power budget of AOC subsystem is given in Table 8.1. The explanation of each component and its selection process is provided in the following subsections. 8.1 Attitude and Orbit Control Subsystem 65

Table 8.1 Mass and power budget for AOC subsystem.

Component # Mass, kg Power, W 3-axis star tracker 1 0.9 2.5 Fine sun sensors 4 4 0.035 4 0.130 ⇥ ⇥ IMU 1 0.055 2 Laser altimeter 1 1.5 7 Total 2.595 11.05 Based on dimensions of the components, AOC subsystem occupies 1.5U of volume.

8.1.1 Actuators For attitude control two main techniques are typically used [77]: spinner, and three-axis stabilised. Spinner is an old stabilisation technique, working by the same principle as spinning landers already described in Section 4.1.1. Three-axis stabilisation is a modern, most commonly used technique. It takes four parameters to describe the attitude: 3 deﬁning a vector in inertial space, and 1 for rotation about that vector. Attitude control can be performed by reaction wheels, thrusters, or a combination of both. The highest precision pointing is achieved by using reaction wheels, while thruster type control method is typically enough for lower accuracy pointing applications. The disadvantage is that the momentum collected by the reaction wheels has to be removed. This can destabilise the spacecraft or thrusters have to be used in order to counteract the torque. Since the thrusters are already needed for asteroid detumbling, the goal is to also utilise them for spacecraft attitude control. This simpliﬁes not only the spacecraft design, but also saves mass, which is one of the primary objectives of this study. Reaction wheels shall be included in the design, only if the precision provided by the propulsion system is not sucient. The required pointing accuracy can typically be achieved with thrusters for attitude control or reaction wheels. The pointing accuracy achievable with thrusters only (designed for attitude control) ranges between 0.1 and 1 [77], if reaction wheels are employed spacecraft pointing can be even more accurate. It is important to note, however, that the aforementioned pointing range achievable with thrusters only depends on the selected propulsion system. Therefore, it is necessary to know how the parameters of the propulsion system relate to the required pointing accuracy. 8.1 Attitude and Orbit Control Subsystem 66

Figure 8.1 Attitude control motion in two dimensions [12].

Minimum Impulse Bit

Pointing accuracy of thrusters for attitude control can be estimated by knowing their minimum impulse bit. Minimum impulse bit is the impulse achieved by ﬁring the thruster for the shortest possible time. Typically, the time mainly depends on the shortest valve opening time. Minimum impulse bit can be calculated as follows [78]:

to Ibit = F(t)dt, (8.1) Zton where ton and to are valve opening and closing times, and F is force generated by propulsion system. The smaller Ibit value is, the more accurate pointing can be achieved. The attitude control process in two dimensions is illustrated in Fig. 8.1 (with negligible external torques). When the vehicle reaches maximum allowed rotational displacement qlim, thrusters are fired in opposite direction, which reverses the spacecraft rotation. The burn time for which the thrusters are fired is denoted as t (= t t ). With constant angular rate vehicle b o on then rolls until qlim is reached again. The time during which the spacecraft rolls with zero acceleration is called dead band time. The relationship between dead band time and burn time can be expressed as follows [12]: 4qlimIv tb = , (8.2) tdb nFl where Iv is spacecraft moment of inertia about the rotational axis, n is number of thrusters firing, F is thrust provided by one thruster, l is moment arm from thruster to centre of mass. Minimum impulse bit varies with dead band time, depending on final spacecraft size, and force provided by the propulsion system. If the estimated required minimum impulse bit is smaller than what the propulsion system can provide, reaction wheels must be employed in that case. 8.1 Attitude and Orbit Control Subsystem 67

The moment arm and moment of inertia for each of 3U, 6U, 12U, and 27U CubeSat configurations are listed and their calculation procedure explained in Appendix D.1, when two thrusters are fired simultaneously. As estimated in Section 7.3, the maximum allowed rotational displacement is 3. Finally, commercially available thruster valves for miniature propulsion systems have shortest opening and closing times of roughly 10 ms in total (one of the leading manufacturers for space-graded fluid control components is VACCO). Thus, the minimum dead band time, with some delay included, could be 20 ms. Equation (8.2) can then be rearranged to express maximum thrust value, with which the required minimum impulse bit can be achieved. For all configurations ranging from 3U to 27U CubeSats, the maximum allowed force for sucient attitude control is estimated to be above 70 N, which shows that reaction wheels are not needed to achieve the required pointing accuracy of 6, since propulsion systems for CubeSats most certainly have lower thrust available. Therefore, actuators for AOC subsystem are not needed, as the propulsion subsystem thrusters can be utilised for attitude control. In total, 12 thrusters are needed for full attitude control: 4 thrusters for rotation around a single axis (2 for rotation in one direction).

8.1.2 Sensors The AOCS hardware is completed with a set of sensors required to determine the attitude and help to navigate the spacecraft to the landing location. Traditionally, the sensor selection mainly depends on whether the spacecraft is Earth-pointing, Sun-pointing, or inertial-pointing, and on the required accuracy. For deep space missions the most commonly used sensors are gyroscopes, sun sensors, and star trackers [77]. Gyroscopes, combined with external star or sun sensors, are used for precision attitude determination. Without any knowledge of external reference, gyroscopes measure angle or rotation of the spacecraft with respect to initial reference. Star or sun sensors can provide additional information, which helps to counteract gyroscope errors such as drift bias. Full attitude knowledge can also be attained by using a single piece of hardware – 3-axis star trackers. Solar sensors can then be added for redundancy. The distance between the asteroid and the lander has to be tracked. This is especially important for hovering and landing phases. Knowing that the asteroid might be of irregular shape and is likely to be tumbling, safe hovering above the selected landing location, and then accurate landing are of crucial importance for successful mission execution. Laser or radar altimeters are typically used for distance measuring in space missions.

3-axis star tracker. Even though star trackers are more expensive than sun sensors, they provide much better accuracy. As explained in the subsection before, a 3-axis stabilised system 8.1 Attitude and Orbit Control Subsystem 68 is chosen for this mission. Therefore, at least two external non-parallel vector measurements are required. Each identiﬁed star acts as a reference vector in 3-axis star tracker system, which means that a single component can determine full 3-axis attitude. VECTRONIC Aerospace oers a 3-axis star trackers for space applications [79]. Their product “Star Tracker VST-41M” is suitable for deep space missions and already has ﬂight heritage from previous small satellite missions. The mass of the tracker ranges from 7 to 9 kg, and maximum power consumption is 2.5 W. Achieved accuracy for x-,y- and z-axis is 18, 18 and 122 arc-seconds respectively. Operating temperature ranges from 20 to +65 C. The star tracker occupies 80 100 180 mm3 of volume space. ⇥ ⇥

Fine sun sensors. Knowing the attitude of the spacecraft at all times is crucial for the success of the mission, therefore it is decided to also have redundant sensors. Sun sensors are chosen since they occupy small mass and volume and their power requirements are low. Sun sensor works by determining the spacecraft’s orientation with respect to the Sun. NewSpace Systems oers sun sensors with ﬂight heritage since 2007. Their product “Fine

(digital) Sun-Sensor NFSS-411” oers higher than 0.1 accuracy [80]. The ﬁeld of view of 140 means that only 4 sensors are enough to have full sky coverage. Each sensor weighs 35 g and occupies 34 32 20 mm3 of volume space. Average power consumption is 37.5 mW, ⇥ ⇥ while peaks measure up to 130 mW. Operational temperature range extends from 25 to +70 C.

Inertial Measurement Unit (IMU). In order to ﬁnish the subsystem for precise attitude determination, gyroscopes have to be added. A single gyroscope provides angular velocity around one or two axes. Quite often they are grouped together for full three axes measurements as an inertial reference unit (IRU). IRUs with added acclerometers and inclinometers for position and velocity sensing are called IMUs. Sensor product “STIM300” is an IMU consisting of 3 gyros, 3 accelerometers, and 3 inclinometers [81]. The product weighs less than 55 g and consumes maximum of 2 W of power. The operating temperature ranges from 40 to +85 C. It occupies 35 cm3 of volume space. Even though it is a space-grade product, the information about ﬂight heritage is not provided, nor about the suitability for deep space missions.

Laser altimeter. Currently no o-the-shelf laser altimeters are known to exist for CubeSat missions. Lidars ﬂown on previous space missions typically occupy large volume space and have high power requirements. Thus, a solution of a compact and accurate altimeter is needed. 8.2 Propulsion Subsystem 69

Table 8.2 Mass and power budget for propulsion subsystem (considering best case spinning asteroid scenario).

Component # Mass, kg Power, W PM200 thruster 4 4 0.35 4 6 ⇥ ⇥ Propellant - 1.5 - Propellant tank and other equipment - 3 - Total 5.9 24 Based on the dimensions of the components, propulsion subsystem occupies 4U of volume.

Compact laser altimeter (CLA) is being developed at Johns Hopkins University Applied Physics Laboratory [82]. CLA is expected to weigh less than 1.5 kg, consume less than 7 W of power. It occupies around 1U of volume. The range resolution is less than 0.03 m (at closest approach), and horizontal accuracy of 1 cm at 1 m distance. The maximum operational range is 1–2 km. The technology is indeed impressive and ﬁlls the gap in the market of small spacecraft technologies. The latest information regarding the project dates back to 2012, however its current progress is unclear.

8.2 Propulsion Subsystem

The propulsion subsystem is used for docking with the asteroid, as well as for despinning the asteroid, which is the main goal of the mission. Therefore, the propulsion subsystem can also be seen as the payload of the spacecraft. The biggest contributor to the mass and volume of the propulsion system is the propellant needed. The total change needed in velocity dominates the propellant budget. Mass and power budgets are provided in Table 8.2. The estimation procedure and selection of each component is provided in the text below.

8.2.1 Propellant Budget The required propellant budget estimation is performed in two steps: (1) propellant required for docking with the asteroid; (2) propellant required for despinning the asteroid. The relationship that links DV with propellant and initial mass is [77]:

m0 DV = Isp gln , (8.3) m f 8.2 Propulsion Subsystem 70

2 where Isp is specific impulse of the propulsion system, g is gravity of the Earth (9.81 m/s ), m0 is initial mass, and m f is final mass. Propellant can be estimated as the dierence between initial and final mass: m = m m . Specific impulse of the propulsion systems can be prop 0 f up to around 8000 s, considering the most modern electric propulsion technologies which are currently under development [83].

Propellant Required for Docking With The Asteroid

As explained in Chapter 7, the DV required for docking with the asteroid depends on how fast the procedure has to be performed. 1.5 m/s is required for reaching the asteroid within 1 minute, and 1 m/s is required for hovering above the asteroid for 50 minutes (taken the worst case scenario, when asteroid is tumbling fast). Thus, including some safety factor and taking into account that the both steps of the landing procedure might be needed to be performed slower, the total change in velocity for docking then is assumed to be DV = 10 m/s. The initial spacecraft weight is taken for 27U, 12U, 6U, and 3U CubeSats to be 54 kg, 24 kg, 8 kg, and 4 kg respectively. Please note, that using dierent CubeSat dispensers the weight per unit can either be 1.33 kg or 2 kg. Typically, CubeSats up to 6U weigh 1.33 kg per unit, while larger CubeSats can weigh up to 2 kg per unit. It can be seen from Fig. 8.2 that propellant weight required decreases signiﬁcantly with high

Isp. Thus, required propellant budget strongly depends on what propulsion system is chosen.

Propellant Required for Despinning The Asteroid

In order to ﬁnd propellant required for despinning the asteroid, despinning procedure is treated as change of linear velocity to 0 m/s. The linear velocities for best and the worst cases are

Figure 8.2 Propellant required for docking with the asteroid 2014 UR for dierent spacecraft sizes and dierent speciﬁc impulses. 8.2 Propulsion Subsystem 71

0.0039 m/s and 0.1501 m/s respectively (refer to Chapter 7). However, in this case, the weight of overall mass that has to be despun is m0 = ma + msc. Compared to the possible asteroid mass ranging from 861 103 kg to 1.352 107 kg, the spacecraft mass can be considered ⇥ ⇥ to be negligible. As mentioned earlier, the linear velocity for the best and the worst cases is 0.0039 m/s and 0.1501 m/s respectively. Figure 8.3 shows required propellant for dierent speciﬁc impulse propulsion systems. It can be seen that the range of required propellant mass is very wide, which is due to vague knowledge about the asteroid. When comparing both Fig. 8.2 and 8.3, it can be seen that the mass of propellant needed for asteroid despinning is much higher than propellant needed for docking with the asteroid.

8.2.2 Subsystem Choice The propellant required for despinning the asteroid is signiﬁcantly higher, therefore the propellant needed for docking with the asteroid is seen as negligible. Knowing that the maximum spacecraft weight can be 54 kg, the propellant should occupy not more than 40 kg of the total spacecraft mass budget (leaving 14 kg for other subsystems). From Fig. 8.3 it can be seen that I should be higher than 100 s for the best case scenario. I should be higher than 5100 s sp sp ⇠ for the worst case scenario.

Worst Case Scenario

No chemical or cold gas propulsion systems can provide high enough Isp. Speciﬁc impulse of electrical propulsion systems can range up to 8000 s. Electrical propulsion can be further

Figure 8.3 Comparison of propellant required for despinning fast spinning and slow spinning asteroid 2014 UR for dierent specific impulses. 8.2 Propulsion Subsystem 72 separated into following systems: resistojet, arcjet, ion thruster, solid pulsed plasma (PPT), magnetoplasma dynamic (MPD), hall thruster. Out of all listed systems only ion thruster can provide high enough Isp value, which ranges from 1500 s to 8000 s [83]. The thrust ranges typically from 0.01 mN to 500 mN, which means that the shortest possible asteroid despinning time in this case is around 27 Earth days (estimation procedure can be found in Section 3.6). Two propulsion system candidates were found to have specific impulse value higher than 5200 s: Alta FEEP-150 and IFM Nano Thruster [83, 16]. Qualification phase of Alta FEEP- 150 thruster was expected to take place in 2010, however no updates have been found since 2009. The thruster was being developed for LISA Pathfinder mission by ESA. However, since cold gas propulsion system was chosen for the mission instead [84], the development of FEEP- 150 might have been stopped. The IFM Nano Thruster by Enpulsion has been developed for CubeSat missions in order to meet high DV and low available power requirements. 5000 s of specific impulse can be achieved within 25 W to 40 W of system power [17]. Maximum thrust in this case is only around 0.32 mN. With such small thrust the asteroid despinning would take 8 years, which most certainly is too long. Therefore, it can be concluded that it is impossible to design a CubeSat for detumbling the asteroid for the worst case scenario, by using COTS propulsion systems for detumbling.

Best Case Scenario

The list of propulsion systems considered for the best case asteroid despinning scenario are provided in Table 8.3. All considered propulsion systems have Isp higher than 100 s. The maximum time set for asteroid despinning is 100 days. Considering that asteroid redirection missions could last a couple of years, one third of a year for despinning the asteroid seems to be a reasonable time period. As the mission time increases, so does the complexity of the spacecraft, since all the components have to be operational for longer times. To despin the asteroid within 100 days the thrust T has to be higher than 0.155 mN, but lower than 7 N in order to meet the accuracy requirement. The system has to consume less than 80 W of power (P), leaving some margin for other subsystems since 100 W is the maximum that the 27U satellite can provide, refer to Chapter 6). Comparison of systems TRL (technology readiness level) is also provided. Compared to the asteroid, the spacecraft mass is negligible, thus the initial mass is mass of the asteroid only (mast = 860,588 kg). DV is 0.0039 m/s for despinning the asteroid. The equations used for calculating despinning time and propellant required are Equation (3.8) and Equation (8.3). It can be seen from Table 8.3, that as the speciﬁc impulse of the propulsion system decreases, the thrust value increases. Chemical propulsion systems show advantage over elec- 8.2 Propulsion Subsystem 73

Table 8.3 Comparison of dierent propulsion systems performance for the best case scenario [16–21].

System T, mN Isp,s P,W tdespin, days mprop, kg TRL Electrospray propulsion Accion TILE 5000 1.5 1500 30 10 0.228 NF Accion TILE 500 0.4 1250 8 39 0.274 NF Busek BET-1mN 0.7 800 15 22 0.428 5 Ion propulsion Busek BIT-3 1.4 3500 60 11 0.098 5 Enpulsion IFM 0.4 3500 40 39 0.098 NF Chemical propulsion NanoAvionics EPSS 100 225 7.5 3.7 hours 1.521 9 Aerojet Rocketdyne GR-1 1100 235 12 20.35 min 1.456 6 ECAPS HPGP 100mN 100 225 8 3.7 hours 1.521 6/7 ECAPS HPGP 1N 1000 225 10 22.4 min 1.521 9 Tethers HYDROS 1200 310 25 18.7 min 1.104 6/7 Busek BGT-X1 100 214 4.5 3.7 hours 1.6 5 Busek BGT-X5 500 225 20 44.8 min 1.521 5 Hyperion PM200 500 285 6 44.8 min 1.2 NF NF – not found. trical propulsion systems in higher available thrust, and lower power requirement. Previously used hydrazine is currently not favoured anymore – its high toxicity makes it dicult to work with. Recently hydrazine has been replaced by so called “green” propellants, most of which are ammonia- or water-based. Therefore, only “green” propellant systems are considered.

The Isp of chemical propulsion systems ranges from 225 s to 310 s, which can despin the asteroid in as little as 18.7 min to as long as 3.7 hours. It is signiﬁcantly shorter compared to 10 to 52 days of despinning period that electrical propulsion systems can oer. Power requirements for chemical propulsion systems are also lower (4.5 W to 25 W) compared to power requirements for electrical propulsion systems (8 W to 60 W). However, due to their low spe- ciﬁc impulse, chemical propulsion systems need more propellant. 1.1 kg to 1.6 kg of propellant mass is needed for chemical propulsion systems, while only 0.09 kg to 0.34 kg is needed for electrical propulsion systems. Higher requirement for propellant mass for a chemical propulsion system is acceptable, considering much lower power consumption and shorter despinning times. 8.3 Anchoring Subsystem 74

Thruster. PM200 system developed by Hyperion is chosen as the best candidate for asteroid despinning mission (assuming best case scenario), even though it has slightly longer despinning time compared to HYDROS, HPGP 1N and GR-1. Moreover, due to its higher specific impulse of Isp = 285 s, less propellant is needed. The system is sold as a complete propulsion system with thruster, propellant tank, pipes and valves integrated [21]. A standard system, fitting within 1U, can accommodate only 0.31 kg of propellant, thus, for this mission the propellant tank has to be larger. The disadvantage is that the system has not been flight-proven yet, and its TRL is unclear. Hyperion does not provide thruster mass, but considering that HPGP 1N thruster mass is 0.38 kg, slightly lower value can be expected for PM200 0.5 N thruster. Thus, including some safety factor, 0.35 kg is chosen for mass of the thruster.

Propellant. PM200 is a bi-propellant propulsion system, operating on nitrous oxide and propene. The propellant has to be stored at 9 bar and oxidiser at 45 bar. Pressurisation requirement increases lander design complexity, since it must be ensured that the system will not leak for long duration. Propellant mass is 1.5 kg (refer to Table 8.3), with safety factor included.

Propellant tank and other equipment. Hyperion lists that the full PM200 propulsion system dry mass (including thruster) is 1.1 kg. Excluding thruster the mass would be around 0.75 kg. For this mission four times more valves, more complex piping and heavier propellant tank are needed. Thus, the total system dry mass excluding thrusters is expected to be around 3 kg. Power required for operating valves is considered to be negligible. Operating temperature of the whole system ranges from 5 to +35 C. A single-thruster system designed by Hyperion occupies 1U.

8.3 Anchoring Subsystem

The anchoring subsystem is responsible for anchoring the spacecraft to the asteroid surface. As described in Chapter 6, selected system is microspine gripper. Mass and power budgets are provided in Table 8.4. The subsystem is expected to occupy less than 1U of CubeSat volume. Operational temperature requirements for the system were not found. However, since microspine gripper was supposed to be employed for NASA ARM (Asteroid Retrieval Mission) mission, it is expected to be able to operate in suciently wide range of temperatures. 8.4 Communications Subsystem 75

8.4 Communications Subsystem

Communications subsystem provides the link between the lander spacecraft and the mothership, while mothership communicates with the Earth. The choice of communication band depends on data rate requirements, and distance. Mass and power budget for communications system is provided in Table 8.5 For this particular mission, there is no science data to be transmitted to the mothership. Moreover, the distance between the mothership and the lander is small. Therefore, it can be assumed that low gain antennas on both mothership and lander spacecraft will suce. Links with low antennas typically operate at lower frequencies such as VHF and UHF (very high frequency and ultra high frequency) [77]. The band for VHF is 30 MHz to 300 MHz. The band for UHF is 300 MHz to 3 GHz. The lander has to transmit telemetry to the mothership and receive corrections for landing location or detumbling force (magnitude and direction). Telemetry includes lander subsystems temperature, propellant tanks pressure, power consumption and generation, attitude, system checks, battery state. Telemetry is expected to be sent every 5 minutes during landing and detumbling operation, so that relevant corrections can be made if needed. Telemetry downlink data rates for deep space missions typically has ranges from 5 kb/s to 10 kb/s [77]. Since the spacecraft is small, less telemetry is expected, thus lower end data rate requirement of 5 kb/s is selected. Uplink data rates are not expected to be any higher than 5 kb/s, since only land-

Table 8.4 Mass and power budget for anchoring subsystem.

Component # Mass, kg Power, W Microspine gripper 1 1.05 10 Total 1.05 10 Based on dimensions of the components, anchoring subsystem occupies 1U of volume.

Table 8.5 Mass and power budget for communications subsystem.

Component # Mass, kg Power, W Transceiver 1 0.075 4 Low gain antenna 1 0.1 0.6 Total 0.175 4.6 Based on dimensions of the components, communications subsystem occupies less than 0.2U of volume. 8.5 Other Subsystems 76 ing/detumbling corrections are to be received. Thus, it is assumed that no more than 1250 kb is needed to be sent and received every 5 minutes.

Transceiver. UHF downlink/VHF uplink transceiver developed by ISIS meets the data requirement [85]. Its downlink data rate is 9600 bps, which allows to send 2880 kb of data in 5 minutes. Uplink data rate is the same. Power consumption is only 0.48 W for receiving, and 4 W for transmitting. Transceiver weighs only 75 g and measures 90 96 15 mm3 (a single ⇥ ⇥ standard CubeSat size PCB). Operational temperature range is from 20 to 60 C.

Antenna. A CubeSat dipole deployable antenna, also developed by ISIS, is designed to ﬁt the aforementioned transceiver [86]. It has low weight and low power consumption (0.1 kg and 0.6 W), and is compatible with both UHF and VHF. Operational temperature range is from 20 to 60 C.

8.5 Other Subsystems

This section provides an approximate estimation of command and data handling (C&DH) and thermal subsystems requirements based on previous deep space missions. The mass of the CubeSat structure is based on the mass of 1U, as explained below.

8.5.1 Command and Data Handling Subsystem C&DH subsystem is responsible for distributing commands to other subsystems, collecting telemetry from the subsystems, storing and collecting data, and forwarding the data to the mothership via the communications subsystem. The complexity of C&DH depends on the data ﬂow rates requirements.

Table 8.6 Mass and power budget for C&DH subsystem, thermal subsystem and spacecraft structure.

Subsystem # Mass, kg Power, W C&DH 1 0.567 7.48 Thermal 1 0.85 10.72 Structure 12 12 0.0872 - ⇥ Total 2.246 17.68 8.6 Power Subsystem 77

As CubeSat missions are getting more complex, more powerful C&DH systems are available. Traditionally they are developed for the baseline PC/104 board dimension, which can be stacked together in modules if more computational power is needed [16]. It is known, that C&DH for traditional tasks of CubeSats consume less than 1 W of power. However, in this case autonomous navigation is included, thus the higher computational load. Preliminary subsystem estimates can be drawn from similar deep space missions: an average of 11% of total power is used for powering on-board processing system [77]; subsystem weighs an average of 4% of total spacecraft mass.

8.5.2 Thermal Subsystem Most of the components operational temperature lies within 20 to +60 C range, which is a common operational temperatures range of components designed for space missions. The propulsion system has a bit tighter operational temperature requirements: 5 to +35 C. Traditionally, in order to minimise the heating required to keep the components within required temperature range, passive thermal control systems are used [16]. They require no power input and are lightweight. Most commonly used passive thermal control systems include coatings and ﬁlms. It is also important to remember, that thrusters might get hot during the operation, thus the excess heat must be dissipated. For deep space missions an average thermal control system occupies 6% of the total spacecraft mass, and consumes 15% of the total spacecraft power [77].

8.5.3 Structure Dierent size CubeSats are designed by stacking the required amount of cubes (each measures 10 10 10 cm3) together. The largest CubeSat can measure up to 27U. CubeSat manufactur- ⇥ ⇥ ers such as ISIS develop CubeSat structures of dierent sizes. A 1U CubeSat structure weighs 87 g. So far, without C&DH, thermal, and power subsystems, other subsystems occupy less than 8U. Thus, completed satellite should not be larger than 12U. The weight of 12U structure is 1.05 kg.

8.6 Power Subsystem

The power subsystem is responsible for generating, storing and distributing power. The major design driver is power consumption of the spacecraft. Power can be generated during the mission, and spacecraft can use the energy stored in the battery. The mass and power budget for 8.6 Power Subsystem 78 the subsystem is provided in Table 8.7. Explanation of components’ choice is provided in the text below. The mission can be divided into three smaller phases: landing, anchoring, despinning. Not all subsystems need to be activated during each phase. During landing, microspine gripper is not used. During anchoring, since procedure is very short, communications system is not active. During despinning, the microspine gripper is not active since its normal position is locked. The requirement for total power is provided in Table 8.8. It can be seen that the highest power consumption is during the anchoring phase is 63.18 W.

Solar arrays. Preferably, only body-mounted solar arrays should be considered due to complex landing and docking procedures, which could break any deployable structures in case of impact. ISIS produces solar cells of 1367 W/m2 and 30% eciency [87]. A 64 cm2 area of a solar cell is placed on 10 10 cm2 panel. Using 5 sides of 12U CubeSat (leaving the bottom 20 ⇥ ⇥ 20 cm2 side for anchoring system), in total 28 such panels can be accommodated. 1792 cm2 of total solar cell area provides 73.5 W of power. Total mass of 28 panels is 1.4 kg. It is important to note, that as the incident angle of the Sun increases, power generated by the solar panel decreases. Thus, if one side of the CubeSat is facing the Sun at 0, Sun rays do not reach the other sides of the CubeSat. Therefore, the CubeSat should have deployable solar panels, or the use of fuel cells should be considered.

Electrical power system (EPS). Assuming the best case scenario, the asteroid is spinning with a 2.37 h period, meaning that when landed, the spacecraft is in eclipse for 1.19 h. In order to carry on with the despinning phase, the power supply has to continue. 57.8 W of power is needed for despinning, thus required power capacity is 68.5 Wh.

Table 8.7 Mass budget for power subsystem.

Component # Mass, kg Solar panels 28 28 0.05 ⇥ Electrical power system 3 3 0.195 ⇥ Total 1.985 Based on dimensions of the components, communications subsystem occupies 0.2U of volume. 8.7 Mass and Power Budget 79

Table 8.8 Total power requirement for each of the three mission phases.

Subsystem Power, W Landing Anchoring Despinning AOCS 11.5 on on on Propulsion 24 on on on Microspine gripper 10 oon o Communications 4.6 on oon C&DH 7.48 on on on Thermal control 10.2 on on on Total 57.78 W 63.18 W 57.78 W

EPS developed by NanoAvionics has a 2-cell battery conﬁguration, voltage regulator, and power distributor. The battery provides 3.2 Ah battery capacity and 23 Wh power capacity [88]. Voltage per battery cell is 7.2 V, thus voltage of the whole battery is 14.4 V. To meet the requirement of battery voltage, at least 6 solar cells have to be connected in series (knowing that voltage per solar cell is 2.4 V [87]). As mentioned above, a single 10 10 cm2 panel has ⇥ two solar cells. Thus, a custom made 30 10 cm panel could have six solar cells. In order ⇥ to reach the required 68.7 Wh capacity, 3 such 2-cell battery packs need to be used. 1 battery pack weighs 0.195 kg, and measures 92.9 89.3 25 mm3. ⇥ ⇥

8.7 Mass and Power Budget

Total mass and power budget of all the subsystems is provided in Table 8.9. This CubeSat con- ﬁguration is designed to land on and despin an asteroid, assuming the best case scenario (refer

Table 8.9 Total mass and power budget.

Subsystem Mass, kg Power, W AOC 2.595 11.05 Propulsion 5.9 24 Anchoring 1.05 10 Communications 0.175 4.6 C&DH 0.567 7.47 Thermal 0.85 10.72 Power 1.985 - Structure 1.046 - Total 14.148 63.18 8.7 Mass and Power Budget 80 to Chapter 3 for best case scenario deﬁnition). In case the asteroid is of 21.2 m in diameter and is tumbling with high angular velocity, the only CubeSat-compatible propulsion system existing would be able to detumble it in over 8 years. In this case the mission is not only impossible because of the high longevity requirement of all subsystems, but it also is unreasonably long. The volume of the CubeSat is 7,892 cm3, which is just over 8U. However, taking into account the volume required for harness and mechanisms, as well as some safety factor, the size of the spacecraft would easily exceed 8U. Therefore, it can be concluded that the minimum possible smallest size of the CubeSat is 12U. Important to note, however, that not all required components and subsystems are o-the- shelf or even available yet. The laser altimeter required for determination of distance between asteroid and lander is only being developed. The microspine gripper is not designed to be suitable for CubeSats, thus its integration might be more dicult. C&DH o-the-shelf subsystems are oered for more simple missions, which do not require navigation. Thus, it can be summarised that although it is somewhat possible to utilise the advantages provided by CubeSat platforms, still quite few custom made solutions are needed for completing the design. Chapter 9

Conclusion

The work in this study considered the feasibility of using CubeSat platform for a small asteroid despinning mission. It was assumed that the lander is carried to the vicinity of the asteroid inside the mothership. First, in Chapter 2 a near-Earth asteroid (NEA) was chosen for the mission. The chosen asteroid had to be suitable for both despinning and redirection missions, thus its accessibility from the Earth was taken into account. The asteroid redirection mission was not studied in this work. From the list of all known Arjuna-type asteroids, asteroid 2014 UR was selected as a suitable candidate. Due to the lack of lightcurve data, the parameters of the asteroid were vague: the range of diameter was between 10.6 m to 21.2 m, the spectral asteroid type was not known nor whether it was spinning or tumbling. By referring to photometric surveys of small NEAs, it was concluded that asteroid 2014 UR was most likely to be S- or C-type. Due to the vagueness of asteroid parameters, in Chapter 3 the mission was divided into two possible scenarios: best case and worst case. Best case scenario assumed smallest and lightest possible asteroid spinning around one axis at lowest possible angular velocity. For the worst case scenario, the asteroid was assumed to have the largest possible diameter, and highest density. The asteroid was tumbling at the highest possible angular velocity. In order to find the initial requirements for lander design, forces acting on a landed spacecraft were estimated. Both best and worst case scenarios were considered. Natural forces acting on the lander included: gravitational, solar radiation, electrostatic and centrifugal forces. For the worst case scenario the dominant force was centrifugal. For the best case scenario gravitational and centrifugal forces were of a similar magnitude. Tangential force acting on a lander was asteroid despinning force, when thrusters were utilised. In the worst case 50 N tangential force was acting on the spacecraft when despinning time was 7 h. In the best case, only 0.1 N tangential force was acting on the spacecraft when despinning the asteroid in the same 82 timeframe. The results showed the importance of knowing the parameters of the asteroid as precisely as possible. Since the gravitational force was estimated to be almost negligible, landing on the asteroid was only possible in fully controlled (active) descent, as discussed in Chapter 4. Chapter 5 considered various anchoring systems for establish strong contact with the asteroid surface. Slow and fast anchoring methods were reviewed, and criteria for desired anchoring technique were established. Suitable anchoring systems for the asteroid despinning mission were selected to be the following: self-opposing drilling, sawing, microspine gripper, tethered spike, and telescoping spike. In Chapter 6, multi-criteria decision making (MCDM) technique PROMETHEE was used for selecting the most suitable anchoring system according to the criteria. First, all qualitative criteria were converted into quantitative ones, and all overlapping criteria were put under a single criterion. The importance of all criteria was found by assigning weights to each. Human error was included, and accurate estimation of weights was finalised by solving the least squares problem. The PROMETHEE method was chosen for comparing selected anchoring systems under all criteria. All anchoring systems were ranked. The microspine gripper performed the best. The mission drivers for preliminary spacecraft design were identified in Chapter 7. The main drivers were: mission data flow, required Delta-V budget and pointing accuracy. It was concluded that the lander had to have autonomy for navigating in the vicinity of the asteroid and calculating detumbling procedure, while the mothership was responsible for sending the landing location. Required velocity change for landing on the asteroid was found by using Clohessy-Wiltshire equations, where the landing procedure was treated as docking. The Delta- V budget for despinning the asteroid was estimated for both best and worst case scenarios. Pointing accuracy was defined as an angle which ensures that the asteroid always stays in the field of spacecraft view. Spacecraft subsystems were chosen with regard to the main mission drivers in Chapter 8. The biggest emphasis was placed on AOC (attitude and orbit control) and propulsion subsystems. It was estimated that no actuators were needed for attitude control, and the propulsion system was utilised for this purpose. Star trackers, sun sensors, IMU (inertial measurement unit), and compact laser altimeter were chosen as sensors for AOC subsystem. The propulsion subsystem was dominated by the required propellant budget. It was shown that no propulsion system technologies exist for detumbling the asteroid candidate in the worst case scenario. For the best case scenario, chemical propulsion system PM200 by Hyperion was chosen. Transceiver operating on VHF and UHF band and low gain antenna were picked for communications subsystem. An estimate was made for mass and power of C&DH (command and 83 data handling) and thermal subsystems, based on previous deep space missions. The Cube- Sat structure was estimated to be 12U. Body mounted solar panels and electrical propulsion system (including batteries and regulators) were selected for power subsystem. The anchoring subsystem consisted of microspine gripper. The total mass budget was estimated to be 14 kg, and total power requirement was estimated to be 63 W for a 12U CubeSat. With some custom made components, the CubeSat design was shown to be feasible assuming the best case scenario. Asteroid despinning time was 45 min. It is concluded that accurate knowledge of the asteroid characteristics is of great importance. Bibliography

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Technology Readiness Levels

The maturity of technology is typically assessed with technology readiness level (TRL) measurement system. The most common system is the one provided by NASA in the table below [89].

Table A.1 Technology readiness levels according to NASA.

TRL Description 1 Basic principles observed and reported. 2 Technology concept and/or application formulated. 3 Analytical and experimental critical function and/or characteristic proof-of- concept. 4 Component and/or breadboard validation in laboratory environment. 5 Component and/or breadboard validation in relevant environment. 6 System/sub-system model or prototype demonstration in a relevant environment (ground or space). 7 System prototype demonstration in space environment. 8 Actual system completed and “flight qualified” through test and demonstration (ground or space). 9 Actual system “flight proven” through successful mission operations. Appendix B

Multi-Criteria Decision Making

B.1 Pairwise Comparison of Anchoring Criteria

The comparison of each criteria pair is made by asking a question: “Is it equally important to have both of the following?”. The description of the scores is provided by Table 6.4. Explana- tion of each pairwise comparison is provided in the table below.

Table B.1 Verbal and numerical pairwise comparison of each criterion.

Pair Explanation & Result Score The importance of both criteria is almost Low MVP budget & Low identical. Low preload results in low mass and w12 0.1 preload requirement volume budgets, since less fuel is required for thrusting during anchoring process.

The importance of both criteria is almost very similar. One of the main goals of this study is Low MVP budget & High w13 to minimise mass and volume for a CubeSat 0.3 TRL mission. Therefore, low MVP budgets are given a slightly higher priority than high TRL.

The importance of both criteria is almost very similar. Re-anchoring capability is slightly Low MVP budget & more critical to the mission success. However, w14 0.3 Re-anchoring capability since low MVP budget is one of the main objectives of the study, it is also of great importance. B.1 Pairwise Comparison of Anchoring Criteria 93

Pair Explanation & Result Score

The importance of both criteria is similar. Low MVP budget & Ability Ability to anchor at wide range of angles is w15 to anchor at wide range of 0.5 seen as more critical to the mission success angles than low MVP budget.

The importance of both criteria is almost Low MVP budget & dissimilar. Applicability to possible surfaces is w16 Applicability to dierent 0.6 seen as more important than low MVP budget, surfaces since this is critical to mission success.

The importance of both criteria is almost very similar. Even though shorter anchoring times Low MVP budget & Short w17 increase the possibility of mission success, low 0.3 anchoring time MVP budget is seen as slightly more important, since it is one of the objectives of this study.

The importance of both criteria is almost dissimilar. Sucient anchoring strength is Low MVP budget & w18 more important than low MVP budget, since it 0.6 Sucient anchoring strength is one of the most critical criteria guaranteeing success of the mission.

The importance of both criteria is very similar. However, because high preload requirement Low preload requirement & would quite signiﬁcantly increase mission w23 0.2 High TRL complexity, and would also contribute to higher mass, it is chosen as slightly more important than high TLR requirement.

The importance of both criteria is almost very similar. Re-anchoring capability is slightly Low preload requirement & more critical to the mission success. However, w24 0.3 Re-anchoring capability since low MVP budget is one of the main objectives of the study, it is also of great importance. B.1 Pairwise Comparison of Anchoring Criteria 94

Pair Explanation & Result Score

The importance of both criteria is similar. Low preload requirement & Even though low preload requirement

w25 Ability to anchor at wide decreases complexity of the mission, ability to 0.5 range of angles anchor at wide range of angles is seen as more critical to the mission success.

The importance of both criteria is almost very Low preload requirement & dissimilar. Applicability to anchor to dierent

w26 Applicability to dierent types of surfaces is seen as much more critical 0.7 surfaces to mission success, than low preload requirement.

The importance of both criteria is very similar. Even though both short anchoring time and low preload requirement decrease complexity of the Low preload requirement & w27 mission, the later is seen as slightly more 0.2 Short anchoring time critical since it could impose more signiﬁcant challenges for propulsion system choice in order to be able to provide enough thrust.

The importance of both criteria is almost very dissimilar. Sucient anchoring strength is Low preload requirement & seen as one of the most critical criteria for the w28 0.7 Sucient anchoring strength mission success, therefore it is considered to be much more important than low preload requirement.

The importance of both criteria is almost High TRL & Re-anchoring similar. Since capability to re-anchor is more w34 0.4 capability critical to mission success, this criteria is considered to be more important.

The importance of both criteria is almost High TRL & Ability to dissimilar. Ability to anchor at wide range w35 anchor at wide range of 0.6 angles is seen to be more critical to mission angles success, and thus much more important. B.1 Pairwise Comparison of Anchoring Criteria 95

Pair Explanation & Result Score

The importance of both criteria is almost very dissimilar. Applicability to anchor to dierent High TRL & Applicability to w36 possible surfaces is seen as much more critical 0.7 dierent surfaces to mission success, and thus much more important.

The importance of both criteria is almost High TRL & Short anchoring identical. Short anchoring time is considered to w37 0.1 time be just slightly more important than high TRL, since it would reduce complexity of the mission The importance of both criteria is very High TRL & Sucient dissimilar. Sucient anchoring strength is w38 0.8 anchoring strength deﬁnitely much more important for the mission to be viable.

The importance of both criteria is very similar. Re-anchoring capability & Ability to anchor at wide range of angles is w45 Ability to anchor at wide 0.2 seen as slightly more critical to mission range of angles success than re-anchoring capability.

The importance of both criteria is almost Re-anchoring capability & similar. Applicability to anchor to dierent

w46 Applicability to dierent possible asteroid surfaces is seen as more 0.4 surfaces critical to mission success than re-anchoring capability.

The importance of both criteria is almost Re-anchoring capability & similar. Re-anchoring capability is more w47 0.4 Short anchoring time important for the mission success than short anchoring time.

The importance of both criteria is similar. Re-anchoring capability & Ability to have sucient anchoring strength is w48 0.5 Sucient anchoring strength of higher importance to mission success than re-anchoring capability. B.1 Pairwise Comparison of Anchoring Criteria 96

Pair Explanation & Result Score

The importance of both criteria is very similar. Ability to anchor at wide Applicability of the anchoring system to range of angles & w56 dierent possible asteroid surfaces is seen as 0.2 Applicability to dierent slightly more important than ability to anchor surfaces at wide range of angles.

The importance of both criteria is similar. Ability to anchor at wide Ability to anchor at wide range of angles is w57 range of angles & Short 0.5 more important for mission success than short anchoring time anchoring time.

The importance of both criteria is almost Ability to anchor at wide similar. Having sucient anchoring strength is w58 range of angles & Sucient 0.4 more important for mission success than ability anchoring strength to anchor at wide range of angles.

The importance of both criteria is very Applicability to dierent dissimilar. Ability to anchor to all possible

w67 surfaces & Short anchoring asteroid surfaces is seen as much more 0.8 time important for the mission success than short anchoring time.

The importance of both criteria is very similar. Applicability to dierent However, ability to have sucient anchoring w68 surface & Sucient 0.2 strength is seen as slightly more important for anchoring strength the mission success. The importance of both criteria is very dissimilar. Ability to provide sucient Short anchoring time & w78 anchoring strength is much more critical and 0.8 Sucient anchoring strength important for the mission success than short anchoring time. B.2 MATLAB Script for Pairwise Dierence Comparison 97

B.2 MATLAB Script for Pairwise Dierence Comparison

This MATLAB script analyses user deﬁned matrix CDP (Table 6.6), ﬁnds inconsistency error factors, and calculates the real matrix RDP (Table 6.7), as described in Section 6.3.

1 %% Agne Paskeviciute , agnep@kth . se , KTH Royal Institute of Technology , Sweden . 2018

3 clear all

4 close all

6 %% Inputs

8 in =[00.1 0.2 0.3 0.4 0.6 0.7 0.8

9 0.1 0 0.2 0.3 0.4 0.5 0.8 0.8

10 0.2 0.2 0 0.1 0.3 0.5 0.7 0.7

11 0.3 0.3 0.1 0 0.3 0.5 0.6 0.6

12 0.4 0.4 0.3 0.3 0 0.2 0.4 0.5

13 0.6 0.5 0.5 0.5 0.2 0 0.2 0.4

14 0.7 0.8 0.7 0.6 0.4 0.2 0 0.2

15 0.8 0.8 0.7 0.6 0.5 0.4 0.2 0]; % CDP matrix . Decided by user

17 n = size ( in ,1) ; % number of criteria ( or entities of interest )

18 X = sym ( ’ x%d%d ’ ,[n n ]) ; % empty matrix of error factors for each comparison

20 %% Quadratic programming

22 k =1;

23 j =2;

24 i =3;

25 counter =1;

26 count =1;

28 % generating equations for minimizing the error

29 while ( i ~= ( n +1) ) B.2 MATLAB Script for Pairwise Dierence Comparison 98

30 eqn ( counter ,:) = in ( j , i ) X( j , i )+in ( k , j ) X( k , j ) in ( k , i ) X( k , i )==0;

32 X_v ( count ,1) = X( j , i ); % error factor

33 X_v ( count ,2) = in ( j , i ); % corresponding CDP value

34 X_v ( count+1,1) = X( k , j );

35 X_v ( count+1,2) = in ( k , j );

36 X_v ( count+2,1) = X( k , i );

37 X_v ( count+2,2) = in ( k , i );

38 count = count+3;

40 if i == ( j +1) && j == ( k +1)

41 k =1;

42 j =2;

43 i = i +1;

44 elseif ( i j )~=1&&j == ( k +1) 45 k =1;

46 j = j +1;

47 elseif ( j k )~=1 48 k = k +1;

49 end

51 counter = counter +1;

52 end

54 %% Creating system of linear equations

56 [A, zeros ]=equationsToMatrix ( eqn ); % A is matrix with coefficients from above generating equations expressions

57 A_t = A.’;

59 % rearranging the vector

60 [~ ,idu ]=unique ( X_v (: ,1) );

61 X_v = X_v ( idu ,:) ;

63 % generating a vector of all ones B.2 MATLAB Script for Pairwise Dierence Comparison 99

64 m = size ( X_v ,1) ;

65 ones = ones (1 ,m, ’ uint32 ’ ).’;

67 % generating a vector of Lagrangian multipliers

68 z = size (A,1) ;

69 lambda_s = sym ( ’ lambda%d ’ ,[z 1]) ;

70 lambda_s ( z ,1) = 0; % arbitrary value

73 %% Solving the linear equations

75 eqn2 =[X_v (: ,1) ( A_t lambda_s )==ones , 76 A X_v (: ,1) == zeros ];

77 [ X_s , number ]=equationsToMatrix ( eqn2 ); % X_s is matrix of error factors coefficients

78 solving = linsolve ( X_s , number );

79 lambda_x = double ( solving );

81 r = size ( lambda_s );

82 error = lambda_x ( r : end ); % Separating vector of only error factors

83 final = double ( error . X_v (: ,2) ); % Separating error of only corrected CDP entries

84 sizefinal = length ( final );

86 % Printing error factors for each CDP entry , together with corrected actual values

87 fprintf ( ’ Cell Error Weight \ n ’ )

88 for i =1:length ( final )

89 fprintf ( ’%s %f %f \ n ’ , X_v ( i ,1) , error ( i ), final ( i ));

90 end

93 %% RDP matrix

95 array = X_v (: ,1) ; B.3 PROMETHEE Preference Functions and Parameters 100

96 string = char ( array );

97 cells = str2double ( regexp ( string , ’ \ d ’ , ’ Match ’ ));

99 i =1;

100 size = size ( cells );

101 W = sym ( ’w%d ’ ,[n 1]) ;

102 for c = i : size (1 ,2)

103 no = cells ( c );

104 no_str = str2double ( regexp ( num2str ( no ),’ \ d ’ , ’ match ’ ));

105

106 a = no_str (1) ;

107 b = no_str (2) ;

108

109 % Generating RDP matrix

110 matrix ( a , b )=final ( c );

111 matrix ( b , a )=final ( c );

112 eqn3 ( c ,1) = [abs (W( a ) W( b )) == final ( c )]; % non linear equations for finding weights for each criteria

113

114 c = c +1;

115 end

B.3 PROMETHEE Preference Functions and Parameters

PROMETHEE functions used for the analysis are listed in Table B.2. The functions are used as described by Greco, et al. [75], however some deﬁnitions are changed to suit better for the purpose of this study. In total, Greco et al. suggests 6 dierent preference functions to choose from, however only the three functions depicted below are used. Table B.3 is provided below, including descriptions of how function is chosen and parameters are calculated for each criterion. B.3 PROMETHEE Preference Functions and Parameters 101

Table B.2 Types of generalised criteria.

Generalised criterion Deﬁnition Parameters to ﬁx

Usual Criterion

0 d 0 P(d)=  – (1 d > 0

U-shape Criterion

0 d < q P(d)= q 1 d q (

V-shape Criterion

0 d 0 d  p P(d)=8 p 0 < d < p <>1 p :> B.3 PROMETHEE Preference Functions and Parameters 102

Table B.3 Function choice for each criterion.

Criterion Explanation Criterion From the point system described in Table 6.2, V-shape g MVP The maximum is 14, and minimum is 2. The 1 p = 12 preference is given linearly. The maximum is 10 N, and minimum is 0. Linear increase of preload also increases the V-shape g Preload Required complexity of the spacecraft linearly (as fuel 2 p = 10 consumption increases linearly). Thus, the preference is given as a linear function. The highest TRL is 9, and lowest is 1. It is seen V-shape g TRL that it takes equal amount of time to reach each 3 p = 8 level, therefore the function is linear. The only values possible are 1 or 0. Thus the V-shape g Re-anchoring 4 only possible dierences are 1, 0, 1. p = 1 Anchoring at Wide Maximum value is 3, minimum value is 0. V-shape g 5 Range of Angles Linear preference applies. p = 3 Maximum value is 1, minimum value is 0. Since the expectation of the chosen asteroid to Applicability to have rocky surface is very high, it is considered U-shape g 6 Dierent Surfaces to be equally good whether the system can q = 0.1 anchor on rocks, or both rocks and consolidated soil. The maximum anchoring time is 3 minutes, minimum time is 0. As anchoring time increases linearly, the required extra propellant V-shape g Anchoring Time 7 mass is also expected to increase linearly. p = 3 Therefore, preference for anchoring time dierence is seen to change linearly. It matters only whether the system can reach Sucient Anchoring g sucient anchoring strength or not. Therefore Usual 8 Strength the preference given is either 0 or 1. Appendix C

Clohessy-Wiltshire Equations

C.1 Coordinate System

Satellite coordinate System, RSW, applies to analysing relative motion of spacecraft [13]. Such system moves with the satellite. The example, depicted in Figure C.1 shows motion in Earth’s orbit: R-axis points from the Earth’s centre to the orbiting body, as it moves along the orbit;

Figure C.1 Coordinate-System Geometry for Relative Motion [13]. C.2 Equations of Motion 104

S-axis points in the direction of orbiting body’s velocity vector; the W-axis is ﬁxed to the direction normal to the orbital plane. When the orbit is perfectly circular, xyz axes are aligned to the RSW axes, since S can only be aligned with y axis (velocity vector) in circular orbits. In case of spacecraft docking to asteroid mission, the orbit is around the Sun.

C.2 Equations of Motion

Next, the derivation procedure of the Clohessy-Wiltshire equations is summarised, and applied assumptions are explained. Exact derivation can be found D.A. Vallado “Fundamentals of Astrodynamics and Applications”, Section 5.8 [13]. The derived Clohessy-Wiltshire equations for near-circular orbits around a large celestial body are: f = x¨ 2wy˙ 3w2x, x 8 fy = y¨+ 2wx˙, (C.1) > 2 < fz = z¨+ w z, > where f represents other forces (such:> as thrusting, solar radiation pressure, etc.). The terms of the right hand side of the ﬁrst ( fx) equation – from left to right – are total, Coriolis and centripetal acceleration. Finally, w is angular rate of the target around the body which it is orbiting. In order to solve the Clohessy-Wiltshire equations, Eq. (C.1), external forces ( f ) are assumed to be 0. This assumption is only viable, if the thrusting is impulsive, and does not hold for continuous low thrust to achieve rendezvous. The solution can be derived with Laplace transformations [13]. The derived position and velocity equations of the interceptor with respect to target at time t are:

x˙ 2˙y 2˙y x(t)= 0 sinwt 3x + 0 coswt + 4x + 0 , w 0 w 0 w 8 ✓ ◆ ✓ ◆ > 4˙y0 2˙x0 2˙x0 >y(t)= 6x0 + sinwt + coswt (6wx0 + 3˙yo)t + y0 , > w w w > ✓ ◆ ✓ ◆ > z˙0 >z(t)=z0 coswt + sinwt, (C.2) > w <> x˙(t)=x˙0 coswt +(3wx0 + 2˙y0)sinwt, > >y˙(t)=(6wx0 + 4˙y0)coswt 2˙x0 sinwt (6wx0 + 3˙y0), > > >z˙(t)=z0w sinwt + z˙0,coswt > > :> C.2 Equations of Motion 105 where the terms sin and cos indicate some type of oscillatory motion, and subscript 0 indicates initial position and velocity of the interceptor. Writing the ﬁrst three equations in Eq. (C.2) equal to zero will mean that the interceptor has docket with the target. Rearranging the equations in order to express initial velocity

(x˙0,y˙0,z˙0), relative to the target, essentially provides relations needed to estimate the required velocity change in order to rendezvous interceptor with the target. After some manipulation, the expressions to calculate the initial velocity of the interceptor with respect to the target are the following:

(6x0 (wt sinwt) y0)w sinwt 2wx0 (4 3coswt)(1 coswt) y˙0 = , (4sinwt 3wt)sinwt + 4(1 coswt)2 8 > wx0 (4 3coswt)+2(1 coswt)y˙0 (C.3) >x˙0 = , <> sinwt z˙0 = z0w cotwt. > > All:> equations presented here were derived under the following assumptions: (1) both satellites are within few kilometres of each other; (2) the expression for the position of the interceptor is found by using binomial expansions, and assuming ﬁrst order relations; (3) since the satellites are so close, product of relative-range vector and dot product of target and relative vector were negligible; (4) the target is in the circular orbit; (5) no external forces are acting on the interceptor, thrusting is impulsive. According to all the assumptions mentioned, the Equation (C.3) is considered as suitable for calculating the initial velocity of the lander spacecraft at the time of release from mothership, with respect to the small asteroid 2014 UR, orbiting the Sun in circular orbit. Appendix D

Preliminary Design & Subsystems

D.1 Moment Arm & Moment of Inertia

Moment of inertia about rotational axis for a cuboid is calculated as follows: 1 I = m (a2 + b2), (D.1) 12 sc where msc is mass of the spacecraft, a and b are width and depth. Mass of 3U and 6U spacecraft is 4 kg and 8 kg respectively, while mass of 12U and 27U spacecraft is 24 kg and 54 kg respectively. Size of 1U is 10 10 10 cm3. ⇥ ⇥ The list of 4 dierent spacecraft conﬁgurations rotating around an axis when two thrusters are ﬁred simultaneously as denoted in the schemes, is given in the table below.

Table D.1 Moment arm and moment of inertia for each CubeSat conﬁguration rotating around an axis by ﬁring two thrusters, as denoted in the schemes.

CubeSat Conﬁguration Moment arm, m Moment of inertia, kg m2

3U l = 0.15 I = 0.033 D.1 Moment Arm & Moment of Inertia 107

CubeSat Conﬁguration Moment arm, m Moment of inertia, kg m2

3U l = 0.05 I = 0.0067

6U l = 0.15 I = 0.087

6U l = 0.1 I = 0.033

6U l = 0.15 I = 0.066

12U l = 0.15 I = 0.26 D.1 Moment Arm & Moment of Inertia 108

CubeSat Conﬁguration Moment arm, m Moment of inertia, kg m2

12U l = 0.1 I = 0.16

27U l = 0.15 I = 0.81 www.kth.se