Model for Touchdown Dynamics of a Lander on the Solar Power Sail Mission

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Lander on the Solar Power Sail Mission

Roger Gutierrez Ramon

Space Engineering, masters level 2016

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

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M ODEL FOR T OUCHDOWN D YNAMICS OF A L ANDER ON THE S ÔLAR P ÔWER S ÂIL M ÎSSION

Roger Gutierrez Ramon

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

S

UPERVISORS

:

Dr. Jun’ichiro Kawaguchi

The University of Tokyo / ISAS/JAXA Sagamihara, Japan

Dr. Osamu Mori

The University of Tokyo / ISAS/JAXA Sagamihara, Japan

Dr. Anita Enmark Luleå University of Technology Kiruna, Sweden

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Model for Touchdown Dynamics of a Lander on the Solar Power Sail Mission

Roger Gutierrez Ramon

Abstract

The ISAS/JAXA Solar Power Sail mission, bound to explore the Jupiter trojans, will face many challenges during its journey. The landing manoeuvre is one of the most critical parts of any space mission that plans to investigate the surface of celestial bodies. Asteroids are mostly unknown bodies and in order to plan a successful landing on their surface, a great number of landing scenarios need to be taken into account. For the future mission to the Jupiter trojans, a study of the landing dynamics and their effects on the lander has to be done. A simple model of a lander has been created based on a design for the ISAS/JAXA Solar Power Sail mission, and the possible landing scenarios have been simulated. For this case, only the last part of the landing, which will be a free-fall has been taken into account. The lander is modelled as a rigid structure with a landing gear composed of four legs. The surface has been modelled as a flat plane with different inclinations and the possibility of including small obstacles or terrain roughness has been implemented. In the model, the lander is allowed 6 degrees of freedom. Several landing possibilities are tested with residual velocities and deviations in the starting point, and the stability of the lander is evaluated respect its geometry. Damping strategies have been considered to protect the instruments and reduce the impact, allowing for a safer landing.

The effect of including crushable honeycomb dampers in the legs is also implemented, simulated and evaluated, by using a model of crushable honeycombs with different characteristics. In addition, the model includes also the position, direction and characteristics of the thrusters. Thus, it could be used to study other phases of the landing sequence where active control of the lander is needed, and evaluate the behaviour and response of different control-loop algorithms for attitude and position control of the lander.

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Acknowledgements

I would like to thank all the people who have helped me during my studies.

I want to thank specially my advisors in ISAS/JAXA, Professor Jun’ichiro Kawaguchi and Professor Osamu Mori, for giving me the chance of working in such an interesting project. I also want to thank Ralf Boden and Javier Hernando-Ayuso for their ideas and contributions to the work.

I finally want to thank my family.

This thesis has been co-funded by the Erasmus Programme of the European Union.

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List of Figures

1.1 Euler diagram showing the types of bodies in the Solar System. . . 2

1.2 Hayabusa bottom/side panel view and instruments. . . 4

1.3 Images of the Itokawa asteroid from (A) Eastern side and (B) Western side. . . 4

1.4 Philae theoretical landing scenario. . . 5

1.5 Deployed solar sail of the IKAROS spacecraft seen from one of the deployable cameras. . . . 6

1.6 Mission sequence of the IKAROS spacecraft. . . 7

1.7 External view of the Hayabusa 2 spacecraft and its instruments. . . 8

1.8 Picture of the IKAROS solar sail (front right) and the new solar sail (back). . . 8

1.9 Mission overview of the SPS. . . 9

1.10 Lander operation sequence. . . 11

1.11 Proposed power descent scheme for the landing scenario. . . 12

2.1 Inertial and body coordinate system position and orientation. . . 13

2.2 Derived CG and principal moments of inertia directions of the Lander. . . 16

2.3 Simplified lander model used for the simulations. . . 16

2.4 Theoretical static stability diagram. . . 18

2.5 Position of the honeycomb dampers in the lander’s legs. . . 21

2.6 Crush strength vs Displacement graph from a single-stage honeycomb crushable test. . . 21

2.7 Strain vs Crush Strength graph showing the removal of the initial peak. . . 22

2.8 Photograph of a honeycomb cartridge from the Apollo missions. . . 23

2.9 Comparison of typical strut stroking-force time histories during landings. . . 23

2.10 Comparison of the honeycomb crushable behaviour. . . 24

3.1 Block diagram of the implementation of the general algorithm. . . 26

3.2 Lander instability condition. . . 28

3.3 Implementation of the roughness of the terrain. . . 29

3.4 Angle of the slope of the terrain. . . 30

3.5 Angle of the direction of the slope of the terrain. . . 30

3.6 Gaussian kernel used to soften the height distribution. . . 31

3.7 Histogram of the height values of the nodes of the terrain for one example. . . 32

3.8 Composition of the complete asteroid surface. . . 33

3.9 Simplified normal force calculations model of the lander-surface interaction. . . 34

3.10 Simplified friction force calculations model of the lander-surface interaction. . . 34

3.11 Block diagram of the algorithm for the honeycomb crushable behaviour. . . 35

3.12 Sketch of the behaviour of the honeycomb model. . . 36

3.13 Time evolution of the different characteristics of the CG. . . 37

3.14 One example frame of the output animations of the time evolution of the lander. . . 37

4.1 Monte Carlo method results with no roughness. . . 39

4.2 Monte Carlo method results with no roughness and residual ang. vel. in Y axis. . . 40

4.3 Monte Carlo method results with no roughness and residual ang. vel. in Z axis. . . 40

4.4 Monte Carlo method results with roughness. . . 41

4.5 Monte Carlo method results with roughness and residual ang. vel. in Y axis. . . 41

4.6 Monte Carlo method results with no roughness and a 140 psi crushable. . . 42

4.7 Monte Carlo method results with no roughness and a 50 psi crushable. . . 42

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4.8 Monte Carlo method results with no roughness and no damping surface. . . 42

4.9 Monte Carlo method results with no roughness, no damping surface and 90 psi crushable. . . . 42

4.10 Monte Carlo method results with no roughness, no damping surface and 210 psi crushable. . . 43

4.11 Comparison of the effect of the different crushables on the lander. . . 44

5.1 Correlation of footpad radius vs. allowable slope. . . 46

5.2 Footpad radius from the lander. . . 47

5.3 Monte Carlo method results with no roughness. . . 47

5.4 3 and 4 leg configurations and comparison between footpad radius and stability radius. . . 48

5.5 Mars Engineering Model surface slope distribution. . . 49

A.1 Time evolution of the different characteristics of the CG. . . 58

A.2 Time evolution of the different characteristics of the CG vs each leg of the lander. . . 59

A.3 3D plot of the lander trajectory (CG and legs). . . 60

A.4 Time evolution of the different characteristics of the legs of the lander. . . 61

A.5 Time evolution of the acceleration in the Z body axis direction. . . 61

A.6 One example frame of the output animations of the time evolution of the lander. . . 62

B.1 Monte Carlo method results with no roughness. . . 64

B.2 Monte Carlo method results with no roughness and residual ang. vel. in Y axis. . . 64

B.3 Monte Carlo method results with no roughness and residual ang. vel. in Z axis. . . 65

B.4 Monte Carlo method results with roughness. . . 65

B.5 Monte Carlo method results with roughness and residual ang. vel. in Y axis. . . 66

B.6 Monte Carlo method results with no roughness and a 140 psi crushable. . . 66

B.7 Monte Carlo method results with no roughness and a 50 psi crushable. . . 67

B.8 Monte Carlo method results with no roughness and no damping surface. . . 69

B.9 Monte Carlo method results with no roughness, no damping surface and 90 psi crushable. . . . 69

B.10 Monte Carlo method results with no roughness, no damping surface and 210 psi crushable. . . 70

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List of Tables

1.1 Rosetta spacecraft characteristics. . . 5

1.2 Parameters of the asteroid Ryugu. . . 7

1.3 Point of departure parameters for the Solar Power Sail lander. . . 10

2.1 Properties and characteristics of the lander used for the model. . . 15

3.1 Asteroid surface properties for the Philae lander study. . . 33

4.1 Parameters and initial conditions for the Monte Carlo method. . . 39

4.2 Parameters and initial conditions with flat terrain and no damping surface. . . 43

4.3 Parameters and initial conditions for the honeycomb simulations. . . 45

B.1 Parameters and initial conditions for the Monte Carlo method. . . 63

B.2 Parameters and initial conditions with flat terrain and no damping surface. . . 68

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Acronyms

ALDN Arrayed Large-area Dust detector for interplanetary space

au astronomical unit

CAD Computer-Aided Design

CE-study Concurrent Engineering study

CG Centre of Gravity

CGT Command Generator Tracker

DLL Design Limit Load

DLR German Aerospace Center or Deutsches Zentrum für Luft- und Raumfahrt

DOF Degrees Of Freedom

DSC Daughter Spacecraft

EDVEGA Electric Delta-V Earth Gravity Assist

ESA European Space Agency

FEA Finite Element Analysis

FOV Field Of View

FreeDyn Free Dynamics Simulation Software

FSA Flexible Solar Array

GAP Gammma-ray burst Polarimeter

GCP Ground Control Point

HP Hovering Position

HST Hubble Space Telescope

IAU International Astronomical Union

IKAROS Interplanetary Kite-craft Accelerated by Radiation Of the Sun

ISAS/JAXA Institute of Space and Astronautical Science/Japanese Aerospace Exploration Agency LabVIEW Laboratory Virtual Instrument Engineering Workbench

LEO Low-Earth Orbit

LIDAR Light Detection And Ranging MASCOT Mobile Asteroid Surface Scout MATLAB Matrix Laboratory by MathWorks

MBDyn Multibody Dynamics Simulation Software

MSC Mother Spacecraft

MSC ADAMS MSC Software Corporation Automated Dynamic Analysis of Mechanical Systems MUSES-C Mu Space Engineering Spacecraft C

NASA National Aeronautics and Space Administration NASTRAN NASA STRucture ANalysis

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ODE Ordinary Differential Equation

ONC Optical Navigation Camera

PID controller Proportional-Integral-Derivative controller RCD Reflectance Control Device

SLIM Smart Lander for Investigation Moon

SPS Solar Power Sail

SRP Solar Radiation Pressure

VLBI Very Long Baseline Interferometry

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1 Introduction

1.1 Outline of the Thesis

The thesis is divided in different chapters in a way I think makes the reading of it easier. The first chapter is an introduction to the work previously done in the Institute of Space and Astronautical Science/Japanese Aerospace Exploration Agency (ISAS/JAXA) and other space agencies regarding the exploration of the Solar system, more precisely asteroids, in order to provide a background on the topic and a clear picture of the objective. The second chapter tackles the theory behind the problem, presenting the physics and the mathematics used to solve it. The third chapter takes the formulation developed in the second chapter and describes the implementation into a usable algorithm, explaining how it works and the possible limitations it has.

The fourth chapter summarizes the results obtained with the implementation for each of the cases and the fifth chapter discusses the results, putting them more into context and explaining how they can affect the mission.

In addition, it shows alternatives to bypass some of the possible problems found on the current design. Finally, the sixth chapter summarizes the extra work done in parallel with the thesis (which will be helpful to continue developing the topic), including the next steps that I think should be taken regarding the final design.

1.2 Background

Efforts to observe, study and understand the universe started thousands of years ago when the first civilisations began looking at the sky and cataloguing the different bright points using stories, legends and tales to give them a meaning. With each scientific and technological new discovery the knowledge was broadened, but also the necessity to further develop the techniques to allow more and better breakthroughs.

With the 20th century, the era of space exploration began, as the first human-made objects left the Earth. The so called "Space Race" saw the rapid succession of several milestones in the first years of the second half of the century. Low-Earth Orbit (LEO) missions led to the first living beings in space, human spaceflight, the first Moon landing of a manned mission and the development of the first space stations.

In the following years, Solar System exploration continued being developed. Historically, the main targets have been the Sun and the different Solar System planets and their moons. It has not been until the last 30 years that the knowledge and the technology available have made possible the exploration of other smaller Solar System bodies, such as asteroids or comets.

ISAS/JAXA has a long story of space observation and exploration missions, focusing mainly on X-ray astron- omy and technology development missions. Many of these technology development missions or experiments had as objectives small bodies of the Solar System, which can give a meaningful insight on the early years of the Universe and the formation of planets and other celestial bodies. I will outline some of these missions in order to provide background information on current mission objectives and development, along with the mission from the European Space Agency (ESA) to the Comet 67P/Churyumov-Gerasimenko, Rosetta-Philae, to provide a comparison with a current comparable mission. In the end, I will add a short description of the current studies and mission status and the specific problem I will try to work on in this thesis.

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1.2.1 Solar System Bodies

Under the hood of the ’Solar System Objects’ name there are many different type of celestial bodies with different characteristics. The International Astronomical Union (IAU), during its XXVIth General Assembly held in in Prague in 2006, approved the current definition of a planet, while in the same way refining the classification of celestial bodies in different classes [IAU 2006 General Assembly, 2006]. Fig. 1.1 shows an Euler graph with the classification of the Solar System bodies.

Planets Satellites (natural)

Dwarf planets

Minor planets Trans-Neptunian objects Plutoids

Small Solar System

bodies Comets

Centaurs Centaurs

Figure 1.1: Euler diagram showing the types of bodies in the Solar System¹.

In [Dunlop et al., 2013] there is a concise and simple description of the main Solar System bodies that suits the needs of this document. I will reproduce it in the next lines with some notes and additions.

The Solar System can be defined as consisting of all the bodies governed by the Sun’s gravitational field. The well known Solar System bodies lie at distances of less than a hundred astronomical unit (au), representing only a small part of the central region of the Solar System. Most of these bodies orbit around the Sun in orbits close to the plane of the solar equator (the ecliptic), thus creating the well-known disk shape of the Solar System.

These bodies are:

Planets are the most massive bodies in the Solar System. A planet orbits around the Sun and has sufficient mass for its self-gravity to overcome rigid body forces. This makes it assume a hydrostatic equilibrium and a (nearly round) shape. Above of that, it has cleared the neighbourhood around its orbit [IAU 2006 General Assembly, 2006]. As of 2016, there are 8 recognized planets in the Solar System: the inner planets Mercury, Venus, Earth, Mars, and the outer planets Jupiter, Saturn, Uranus and Neptune.

Dwarf Planets share most of the characteristics with planets except that they have not cleared the neigh- bourhood around its orbit. Dwarf planets are also not the same as satellites. Pluto falls currently into this division and it is also the prototype of a class of objects called trans-Neptunian Objects, which populate the Kuiper Belt (also known as the Edgeworth-Kuiper Belt).

Asteroids mainly lie between the orbits of Mars and Jupiter (the asteroid belt). They are a family of smaller bodies which have diameters that can range from a few metres to a few hundred kilometres. There are some other asteroids in the inner Solar System, which cross orbits with the inner planets. If they are small, sometimes they can be called meteoroids or micro-meteoroids [Carr, 1984]. Finally, another class of asteroids are the Jupiter trojans, located in either of Jupiter’s Lagrange points, L₄or L₅.

1Tahc (2013). Euler diagram of the main types of Solar System bodies, other than stars, 2013. Wikipedia (Website), https://

commons.wikimedia.org/wiki/File:Euler_diagram_of_solar_system_bodies.svg Date consulted: 2016-06-20.

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1.2 Background

Comets have a nuclei that does not exceed a few kilometres in radius. They usually move in elliptical orbits with high inclinations relative to the ecliptic. They are composed of icy crusts, which sublimates when they get close to the inner Solar System, creating the typical tail of gas (coma).

Satellites are bodies primarily subject the gravitational fields of the planets. They vary greatly in size, being sometimes as big as the smallest planets. Alternatively, the outer planets have systems of rings, consisting of particles of very different sizes, up to a few meters, that also orbit around them.

There are several more groups and sub-groups for classifying the Solar System bodies, like the Centaurs, minor planets or protoplanets, but most of them are still being investigated and there is no consensus for an exact definition. That aside, most of the objects likely to be seen or encountered in the Solar System can be classified in one of the previous described groups.

1.2.2 Previous and Current Missions

In this subsection I list and explain in chronological order (by launch) previous and current similar missions to the one being currently under development by ISAS/JAXA.

1.2.2.1 Hayabusa (ISAS/JAXA, Launch 2003)

Hayabusa (formerly known as Mu Space Engineering Spacecraft C (MUSES-C)) launched in May 2003 and arrived at the target asteroid Itokawa in September 2005. Incidentally, the original target was another and had to be changed after a delay in the launch. It was the first spacecraft to take off from an extra-terrestrial body’s surface (other than the Moon) and the first asteroid sample return mission [Kawaguchi et al., 2008].

The main objective of the Hayabusa mission was the scientific research of an asteroid, but as with many future missions from ISAS/JAXA, there was also a strong technological demonstration effort for new techniques.

These objectives are cited as follows [Yoshikawa et al., 2010]:

1. Use of ion engines as the primary propulsion means for interplanetary cruise.

2. Use of optical means for autonomous navigation and guidance.

3. Collection of asteroid surface sample under micro gravity.

4. Direct re-entry for sample recovery from interplanetary orbit

5. Low thrust and gravity assist manoeuvre combination for orbit change.

Hayabusa was a 510 kg launching wet mass spacecraft (Fig. 1.2 shows a schematic of its components) that had as a target body the near Earth asteroid Itokawa. To reach it, Hayabusa performed various manoeuvres such as an Electric Delta-V Earth Gravity Assist (EDVEGA), during an Earth swing-by [Kawaguchi et al., 2006]. The travel to Itokawa took a little bit more than 2 years. Once the spacecraft arrived at the asteroid, it performed scientific observations and attempted several descents, actually touching down twice. It was planned to stay in the vicinity of Itokawa for 3 months, but due to a malfunction during one of the touchdowns, the fuel chemical thruster leaked and the communication was lost for a month and a half. Finally, the team re-established the communication channel and it was able to recover the spacecraft. They had to re-calculate all the return orbital manoeuvres due to the malfunction of several control systems (fuel leakage, chemical thrusters were broken, as well as two out of three reaction wheels and the ion engines). The samples would return safely to Earth in 2010, roughly 3 years later than the expected return date.

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Figure 1.2: Hayabusa bottom/side panel view and instruments [Kawaguchi et al., 2006].

The Hayabusa mission was a success, retrieving samples from the Itokawa asteroid. It also served as a technological demonstration for different novel techniques. It has given a lot of information on the characteristics and formation of asteroids (Fig. 1.3 shows detailed photos from Itokawa). Successive ISAS/JAXA missions to other small Solar System bodies take knowledge from the conclusions obtained by the Hayabusa mission team and build on top of it, like the follow-up Hayabusa 2 and the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) missions.

Figure 1.3: Images of the Itokawa asteroid from (A) Eastern side and (B) Western side. The orography of the asteroid was believed to have plenty of craters, but instead consisted mainly of large boulders [Saito et al., 2006].

1.2.2.2 Rosetta-Philae (ESA, Launch 2004)

The Rosetta mission is an ESA mission launched in 2004 with an Ariane 5 rocket that had as a target the comet 67P/Churyumov-Gerasimenko. It reached its destination in 2014, releasing a Lander called Philae that was planned to make a soft landing on the surface of the comet [Biele and Ulamec, 2008].

Rosetta had originally a different target, comet 46P/Wirtanen, but like in the case of Hayabusa, a delay in the launch forced the team to change the objective and thus, the whole operation had to be re-arranged [Ulamec et al., 2006]. On its way to 67P/Churyumov-Gerasimenko, Rosetta employed four planetary gravity assist manoeuvres (Earth-Mars-Earth-Earth) to acquire sufficient energy to reach the comet [Glassmeier et al., 2007], while doing two flybys around the main belt asteroid 2867 Steins and 21 Lutetia.

Upon arriving, Rosetta orbited around the asteroid and manoeuvred to eject Philae in the best way possible [Jurado et al., 2014]. Philae’s descent took 7 h and the touchdown was problematic. The lander had several problems (mainly with the hold-down thrusters and the harpoons), which did not allow it to stay on the surface (Fig. 1.4 shows the sequence of how the landing was designed). During the touchdown, it bounced back twice and ended up tipped over on its side and in the shadows of a cliff wall [Hand, 2014]. This made difficult the charging of the batteries using the solar arrays, so the instruments were only usable for roughly three days.

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1.2 Background

Figure 1.4: Philae theoretical landing scenario. During the mission, the hold-downd thrust and the harpoons malfunc- tioned so the lander bounced twice before landing under the shadow of a cliff wall [Biele and Ulamec, 2008].

Table 1.1 shows the Rosetta orbiter characteristics. The spacecraft is three-axis stabilized and the orientation is controlled by 24 thrusters. It equips four reaction wheels, two star trackers, Sun sensors, navigation cameras, three laser gyro packages, solar arrays and several high and low gain antennas [Glassmeier et al., 2007].

The scientific payload of Rosetta included 12 instruments. Amongst these instruments, there were an ultraviolet imaging spectrometer, a cometary secondary ion mass analyser, a grain impact analyser and dust accumulator, a micro-imaging dust analysis system, a microwave instrument, optical, spectroscopic and infra-red remote imaging systems, a spectrometer for ion and neutral species analysis, instruments for plasma observations, a radio science investigation, a visible and infra-red mapping spectrometer and a standard radiation environment monitor [Arnold, 2015].

Table 1.1: Rosetta spacecraft characteristics [Glassmeier et al., 2007].

Physical Size Main Body m 2.8 × 2.1 × 2.0

Span of solar arrays m 32

Launch Mass

Propellant kg 1720

Science Payload kg 165

Philae Lander kg 100

Total kg 2900

Solar Array Output at 3.40 au W 850

at 5.25 au W 395

Propulsion System Number of thrusters 24 bi-propellant

Net thrust N 10

Prime Contractor EADS Astrium, Friedrichshafen

Apart from all the previous characteristics, Rosetta’s lander Philae still had 10 more scientific instruments.

Although not all of the experiments from Philae were successful due to the limitations that the landing imposed, a considerable amount of scientific data was acquired. Combined data from Philae and Rosetta’s instruments has been providing a new insight in the comet’s characteristics [Arnold, 2015].

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1.2.2.3 IKAROS (ISAS/JAXA, Launch 2010)

IKAROS is the first successful interplanetary Solar Power Sail (SPS) technology demonstration in the world.

It is operated by ISAS/JAXA and was launched together with the Akatsuki (Planet-C) Venus climate orbiter, another spacecraft from the Japanese agency.

IKAROS is basically a technology demonstration and validation mission. It performed interplanetary solar- sailing from the Earth to Venus. The main objectives and its implementations to the spacecraft system are as follows [Tsuda et al., 2013a]:

1. Solar sail deployment in space.

2. Use of thin film solar cells attached on the sail for solar power generation.

3. Verify Solar Radiation Pressure (SRP) acting on the solar sail.

4. Demonstrate solar sailing guidance and navigation techniques.

The body of IKAROS is a cylinder surrounded by the sail storage and deployment mechanisms with an initial wet mass of 307 kg. The spacecraft uses a passive spin-stabilising method that is also used to deploy and keep the shape of the sail.

Figure 1.5: Deployed solar sail of the IKAROS spacecraft seen from one of the deployable cameras [Tsuda et al., 2013a].

The solar sail membrane consists of four trapezoidal petals connected to the hub to provide electrical con- nectivity and with a mass at the four tips to support the deployment. Each side of the sail is a bit more than 14 m long. Fig. 1.5 shows a picture of the deployed solar sail in space, taken by the IKAROS own deployable cameras. On the sail there are several instruments, such as the Flexible Solar Array (FSA), a cell for power generation, the Reflectance Control Device (RCD), which encapsulates liquid crystal that changes the optical reflectance of the sheet when electrical voltage is applied, and the Arrayed Large-area Dust detector for interplanetary space (ALDN), a sensor to detect particles impacting on the sail. Other instruments are also present in the main body of the spacecraft. Amongst the usual sensors and devices for interplanetary missions, IKAROS includes a Gammma-ray burst Polarimeter (GAP) to observe gamma-ray bursts from other galaxies, and a Very Long Baseline Interferometry (VLBI) transmitter to perform and test the Delta-Differential One-way Ranging technique [Tsuda et al., 2013a].

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1.2 Background

IKAROS successfully met the four principal objectives at the end of 2010, when the spacecraft passed by the night side of Venus at the closest distance of 80 800 kg. During this time, the mechanism and operation of the deployment of the sail was completed, along with the generation of power using the thin film solar cells attached to the sail and the use of SRP to provide thrust for trajectory manoeuvres (instead of thrusters), reducing the usage of fuel [Tsuda et al., 2013a]. Fig. 1.6 summarizes the mission sequence of IKAROS.

Figure 1.6: Mission sequence of the IKAROS spacecraft [Tsuda et al., 2013a].

IKAROS had an extended mission phase until March 2012. The last contact with the spacecraft, as of the date of this thesis publication, had been in May 21st 2015. The spacecraft entered hibernation mode for the 5th time, as expected, at a distance of about 110 × 10⁶km from Earth².

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1.2.2.4 Hayabusa 2 (ISAS/JAXA, Launch 2014)

Hayabusa 2 is the follow-up from the first Hayabusa mission, and the second asteroid sample return mission from ISAS/JAXA. It is bound to reach the asteroid 1999JU3 (or Ryugu), perform scientific operations and in-situ analysis and come back to Earth with samples collected from the surface.

The Hayabusa 2 baseline design and operation is strongly influenced by the technological heritage of the first Hayabusa mission. This allowed an extremely short development time, taking in total for design, production and test around 4.5 years. The components were improved and upgraded to current state-of-the-art alternatives and the methodologies and operations refined thanks to the experience of the previous Hayabusa and IKAROS missions [Tsuda et al., 2013b].

Table 1.2: Parameters of the asteroid Ryugu [Tsuda et al., 2013b].

Rotation period h 7.6

Diameter kg 0.922 ± 0.048

Aspect ratio 1.3:1.1:1.0

Geometric Albedo 0.063 ± 0.006

Magnitude (H) 18.820 ± 0.021

Slope parameter (G) 0.110 ± 0.007 Perihelion/Aphelion au 0.85/1.4

Spectrum type Cg

The target of Hayabusa 2 is the asteroid Ryugu. Table 1.2 shows its main characteristics.

2JAXA (2016). Small Solar Power Sail Demonstrator ’IKAROS’ (2016). (Website) http://global.jaxa.jp/projects/sat/

ikaros/topics.html#topics4743 Date consulted: 2016-06-21

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Hayabusa 2, as Hayabusa did, used an Earth fly-by to obtain a low-thrust trajectory to reach its destination.

Upon arriving to the asteroid Ryugu, and after 4 years of space travel, Hayabusa 2 will remain in the asteroid’s vicinity for around 1.5 years in order to collect data and to properly execute the manoeuvres necessary to successfully obtain the samples of the asteroid. Hayabusa 2 will take samples from the asteroid’s surface by releasing a projectile in the asteroid’s surface direction and creating a crater. After shielding itself from the explosion by going to the other side of the asteroid, the spacecraft will descend to the surface and collect part of the ejecta. The mission also includes the deployment of a small lander named Mobile Asteroid Surface Scout (MASCOT) developed by the German Aerospace Center or Deutsches Zentrum für Luft- und Raumfahrt (DLR) [Lange et al., 2010]. These manoeuvres use the equipment and sensors of the spacecraft: optical navigation cameras, a Light Detection And Ranging (LIDAR), a laser range finder, target markers, a Flash lamp, ion engines, bi-propellant hydrazine system for the reaction control system, star trackers, inertial reference units, accelerometers, Sun sensors and reaction wheels [Tsuda et al., 2013b]. Fig. 1.7 shows an external view of Hayabusa 2, the equipment and sensors.

Figure 1.7: External view of the Hayabusa 2 spacecraft and its instruments [Tsuda et al., 2013b].

1.2.2.5 Solar Power Sail (SPS) (ISAS/JAXA, Launch early 2020s)

The Solar Power Sail (SPS) mission is the direct follow-up of the space exploration missions ISAS/JAXA has been doing. It aims to combine the know-how and technologies of the Hayabusa 1 & 2 with the new solar sailing technology validated with the IKAROS spacecraft. The ion engine that will be used has an Ispof 7000 s (2 − 3 times larger than the one used by Hayabusa). The solar sail will be several times larger than the one of IKAROS, with a length of 50 m for each side [Mori, 2016]. Fig. 1.8 shows a picture of the deployment test of one of the petals.

Figure 1.8: Picture of the IKAROS solar sail (front right) and the new solar sail (back) during the 50 m solar sail deployment test in Sagamihara, Japan (July 2016).

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1.3 Problem

The main objective of the SPS is to make a rendezvous with a Jupiter trojan asteroid. Since the expected timeline of the mission takes into account a 12 − 13 years cruise phase, it is designed to also perform science operations during this time. It will take advantage of the long journey and the different Solar System locations it will visit during the Earth and Jupiter swing-bys [Mori, 2016]. On arrival, the spacecraft will perform remote science of the asteroid and will release a lander bound to land on the asteroid’s surface and perform surface and sub-surface science operations. An optional objective of the mission is for the lander to take samples from the asteroid, take-off and rendezvous with the Mother Spacecraft (MSC) to transfer these samples. Then, the MSC would make the journey back to Earth to bring back the samples [Mori, 2016]. Fig. 1.9 shows an overview of the first part of the mission.

Some of the instruments the spacecraft will carry will be an exo-zodiacal infra-red telescope, a gamma-ray burst polarimeter, a magnetic field experiment and arrayed large-area dust detectors in interplanetary space. During the cruise, it will also validate new technologies relative to the solar sail and the ion thrusters. For the former, an evolution of the spin deployment system is being studied to cope with such a large thin membrane. It also includes high power generation using thin-film solar cells placed on the sail, since the spacecraft itself will lack the typical large solar panels. In order to steer the spacecraft, the sail will have attached an RCD. It will be able to change the optical properties of the surface using liquid crystal and thus change the direction of the thrust vector [Mori, 2016].

Figure 1.9: Mission overview of the SPS [Mori, 2016].

Due to the long duration of the mission and the long cruise through interplanetary space (and in the outer part of the Solar System), all the equipment needs to be able to withstand extreme conditions and still be fully operational after the journey. Also due to the distance from Earth, the spacecraft needs to be able to operate almost completely autonomously, a critical task when talking about the landing on the asteroid and the following take-off and rendezvous.

1.3 Problem

1.3.1 Background Information

The SPS will make a rendezvous with the Trojan asteroid in order to study the surface and subsurface of the asteroid. In order to do that, the spacecraft needs to make contact with the asteroid, and this is a dangerous manoeuvre. A landing or any other kind of contact with a celestial body is intrinsically complicated, but in the SPS mission the addition of a spin stabilized 50 m solar sail makes it impracticable. Apart from the size, operating and hovering near a considerably large asteroid has a high fuel consumption. These reasons make the usage of a separate lander a necessity.

The lander (or Daughter Spacecraft (DSC)) takes over the objectives of the mission that involve getting to the surface of the asteroid. These objectives are [Trojan Lander Study Group, 2015]:

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• Collection and in-situ analysis of surface sample from a Trojan asteroid (P/D-type).

• Collection and in-situ analysis of sub-surface (up to 1 m depth) sample (extra success).

• Sample transfer to MSC for Earth sample-return (extra success).

To fulfil the objectives, a series of system requirements are imposed on the design of the lander [Trojan Lander Study Group, 2015]:

• Lander must be operational after 15 year cruise-phase (including Jupiter flyby).

• Operation at Trojan asteroid at 5.2 au distance from the Sun (52 W m⁻², < − 100^◦C).

• Autonomous descent and landing on an asteroid with a surface gravity up to 9 mm s⁻².

• Operation for at least 20 h on the asteroid surface for science operations.

• Surface and subsurface collection of samples and in-situ analysis.

• Autonomous ascent, rendezvous, docking and transfer of samples to MSC (extra success).

Table 1.3 shows the point of departure parameters when operating.

Table 1.3: Point of departure parameters for the Solar Power Sail lander [Ulamec et al., 2016].

Target Trojan Asteroid Diameter km 20–30

Density kg m⁻³ 500–4000

Mass Lander (wet, incl. margins) kg 100

Science Payload (incl. sampling device) kg 20

Science Payload Allocated Energy W h 600

Surface Operation Period h 20

A landing scenario derived from the parameters in Table 1.3 was devised in order to put the lander on the surface of the asteroid. The descent operation is separated in two phases: the rehearsal descent and the actual descent. The description of the operation is taken directly from [Trojan Lander Study Group, 2015].

Fig. 1.10 shows the descent and landing operation sequence, and the description of each point is:

(1) The MSC/DSC effectuate a "rehearsal descent", lowering to the separation altitude and returning to the Hovering Position (HP). This will be the base of the Command Generator Tracker (CGT) mode for the DSC operation. During this manoeuvre, it will capture images of the asteroid and transmit them to the Earth for Ground Control Point (GCP) database generation for the DSC descent operation.

(2) The MSC/DSC descend again to separation altitude. The DSC GCP-matching is assumed to be executed manually by the ground station on Earth.

(3) The DSC separates and descends to an altitude of approximately 100 m autonomously (onboard GCP- navigation (CGT mode)). The GCP database onboard is used for GCP template matching. The relative position is estimated from image data and is used as feedback information to the position controller.

(4) The DSC descends to the surface of the asteroid and touches down. The autonomous/onboard 6 Degrees Of Freedom (DOF) control uses Flash LIDAR (vertical and/or horizontal relative position measurement with obstacle recognition) and optical flow techniques for this last part.

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1.3 Problem

Figure 1.10: Lander operation sequence. Includes rehearseal manoeuvre by the MSC and descent, landing and ascent phases by the DSC [Ulamec et al., 2016].

1.3.1.1 Rehearsal Descent Operation

The rehearsal descent operation is the manoeuvre used to inspect the asteroid surface closely and to create a database of GCP, which will be used later on during the actual landing. Since the Optical Navigation Camera (ONC) of the lander and the MSC have different characteristics, the resolution of the images taken will also be different. Particularly, the Field Of View (FOV) of the MSC is 10 times narrower than the one of the lander (6 deg vs 60 deg), making the resolution of the MSC 10 times higher than the one of the lander [Trojan Lander Study Group, 2015].

This means the GCP obtained by the MSC at one altitude will have the same resolution (and therefore can be used) at an altitude 10 times lower by the lander [Trojan Lander Study Group, 2015]. The descent operation starts at the HP of the MSC (around 250 km from the asteroid surface) and goes down to the separation altitude of the lander (around 1 km), where the landers takes over the controlled descent operations. The database acquired during the rehearsal manoeuvre by the MSC (250–1 km) can effectively be used by the lander from the separation point (1 km) to a height of 100 m, where other navigation instruments can already be used to aid in the descent.

1.3.1.2 Actual Descent Operation

After the GCP database is populated during the rehearsal descent operation, the MSC goes back to the HP and prepares to execute the actual descent operation. The actual descent manoeuvre is exactly the same as the rehearsal descent manoeuvre down to the separation altitude. At this point, the lander separates from the MSC and starts descending to the surface of the asteroid.

Fig. 1.11 shows the full manoeuvre. The first part of the descent is a powered descent. The lander thrusts to gain velocity downwards (to make the descent time shorter) and sidewards (to synchronize its motion with the tangential velocity of the target point). This state will be kept during the descent until the height with respect to the asteroid surface is between 5–10 m [Trojan Lander Study Group, 2015]. At this point, the lander will thrust again (upwards and sidewards, but on the opposite direction) to try and minimize the spacecraft’s radial and tangential relative velocity with respect to the asteroid surface. From this height, the final manoeuvre is a free-fall down to the asteroid surface.

The free-fall is a fundamental part of the scientific requirements. In order to obtain good surface and subsurface samples, the surface of the asteroid where the spacecraft lands needs not to be polluted. If the final part (5–10 m) of the descent were not to be a free-fall, there could be the possibility of surface contamination that could invalidate the samples collected.

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Figure 1.11: Proposed power descent scheme for the landing scenario [Trojan Lander Study Group, 2015].

1.3.2 Problem Statement

The phase of the descent made by the MSC, and the power descent part of the lander descent manoeuvred are actively controlled by the different sensors and actuators that the two spacecraft have. These kind of descents have already been extensively studied and tested, since similar procedures were used during the operation of Hayabusa and its sample collection manoeuvre, and will be used when Hayabusa 2 reaches the target asteroid Ryugu [Kawaguchi et al., 2008].

In the free-fall phase of the descent, any active attitude control (mainly thrusters) is out of the picture. This leaves the whole success probability of the manoeuvre to the appropriate reduction of residual relative velocities and the passive mechanisms that can improve the chances.

Past (and present) ISAS/JAXA missions do not have a lander with the same concept as the SPS. Hayabusa did not have a lander at all (it had a deployable package, but it never reached the surface). Hayabusa 2 includes MASCOT, the lander developed by DLR, but the concept is completely different. It is a lot smaller and lighter (only 10 kg), and it will use a hopping mechanism to move around the asteroid surface. It uses the same mechanisms to up-right its position, making the attitude on landing not so critical [Lange et al., 2010]. The technology and procedures for such a landing device have to be studied and improved in order to achieve a successful manoeuvre. Landing on small Solar System bodies is not a trivial task, as has been also seen with Philae and the difficulties it had to land on the comet 67P/Churyumov-Gerasimenko.

The problem this thesis will try to solve can be summarised in one sentence:

Study the landing and touchdown dynamics of the lander on the Solar Power Sail Mission during its free-fall phase and try to find mitigation strategies for the unstable situations in order to ensure a safe

landing on the surface.

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2 Theoretical Development

This part of the thesis is separated in two. In the first part, I explain the entire physical and mathematical model I use to calculate the movements, including translations, rotations and bounces of the lander as a rigid body.

In the second part, I make a small comment in the theoretical static stability condition that should define the upper stability limit for any case. Finally, in the last part, I explain in a more detailed way the behaviour of the damping system used on the lander, study the alternatives and explain the reasoning on why I chose one alternative in particular for the simulations.

2.1 Dynamic Landing Model

To have this section well structured, first I set the bases of the model, explaining the different coordinate systems used and the transformations and relations between them. After that, I write and explain the physical equations to obtain the movement of the lander, and in the last section, I detail the model used to implement the interaction between the lander and the terrain.

2.1.1 General Considerations and Coordinate Systems

I consider the surface of the asteroid to be locally flat near the landing point (the asteroid diameter is large enough to not consider the curvature of the surface locally). All coordinate systems used in the model are orthogonal right-hand systems, and Fig. 2.1 shows them. The coordinate systems and the transformation between them are explained in the following subsections.

α

X_B Y_B ZB

gravity

X_I Y_I Z_I

Figure 2.1: Inertial and body coordinate system position and orientation with the surface (slope represented by the angle α ) and the gravity vector.

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2.1.1.1 Inertial Coordinate System

The inertial coordinate system is a fixed orthogonal right-hand system arbitrarily placed on the surface of the asteroid. The only constraint is that the gravity vector is contained in the Z-axis and has the opposite direction.

Since I consider the gravity vector constant in the whole area, the coordinate system is fixed and is the one used to express the translations of the lander.

2.1.1.2 Body Coordinate System

The body coordinate system is fixed on the lander. Its origin coincides with the Centre of Gravity (CG) of the lander and the axes are the principal axes of inertia, where the Z-axis is aligned with the central axis of the octagonal main body. Fig. 2.3 shows a sketch of this simplified model later on.

At the start of the simulation, the position and attitude of the body coordinate system are set as predefined deviations from the inertial coordinate system, and the evolution of the body coordinate system (and thus, the lander) is calculated and represented as deviations from the start position. The relationship between the body coordinate system and the inertial coordinate system (and the start position) are translations in the three directions and a set of Euler angles φ , θ and ψ, which are rotations around the XB, YBand ZBrespectively.

2.1.1.3 Transformation Matrices

To change the state vectors from one coordinate system to the other, a transformation matrix based on the Euler angles described in the previous section multiplies each of the vectors. The rotation sequence used is 1-2-3 (or also expressed as X (φ )Y (θ )Z(ψ)). The transformation matrix from the initial attitude of the lander, that coincides with the orientation of the inertial coordinate system, to the current attitude of the lander (also known as Forward Mapping) is

T_FS=





cos θ cos ψ − cos φ sin ψ + sin φ sin θ cos ψ sin φ sin ψ + cos φ sin θ cos ψ cos θ sin ψ cosφ cos ψ + sin φ sin θ sin ψ sin φ cos ψ + cos φ sin θ sin ψ

− sin θ sinφ cos θ cos φ cos θ



, (2.1)

where S is for ’start’ attitude and F is for ’final’ attitude. Resulting in Eq. (2.2), which can be used for any rotation from the initial attitude:



 XF

Y_F ZF



=T_FS



 XS

Y_S ZS



. (2.2)

Since the transformation matrix is orthogonal, for the opposite transformation, the transpose of the transformation matrix can be used directly:

TSF = TFS

−1

=TFS

T

. (2.3)

The opposite transformation (from "final" attitude to "start" attitude) can be written then without the use of the inverse of the matrix, writing directly



 XS

YS

Z_S



=TFS

T



 XF

YF

Z_F



. (2.4)

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2.1 Dynamic Landing Model

A second transformation matrix is used to change the body angular velocities (represented with the subscript

’B’) to Euler angular rates (represented with the subscript ’E’). The second transformation

TEB=





1 sin φ tan θ cos φ tan θ 0 cos φ − sin φ 0 _{cos θ}^{sin φ} ^{cos φ}_{cos θ}



 (2.5)

is not orthogonal and results in:



 φ˙ θ˙

˙ ψ



=T_EB



 ωX

ωY

ωZ



, (2.6)

where the vector in the left-hand side of the equation is the representation in Euler angular rates and the vector in the right-hand side of the equation is formed with the components of the body angular velocities.

2.1.2 General Equations of Movement

The lander’s CG acts as the origin point of the body axes coordinate system, which is used to determine the attitude of the lander (its evolution as the time advances). The position and characteristics of the lander’s parts are always referred to this body axes coordinate system, if not stated otherwise.

Fig. 2.2 and Table 2.1 show the properties and characteristics of the lander used for the model. Any change done to them during the simulations (to test how they affect the behaviour of the system) is explained in the chapter or section where it happens.

Table 2.1: Properties and characteristics of the lander used for the model. Distances and positions represented in body coordinate system.

Mass kg 100

CG deviation from center plate

x mm 0

y mm 0

z mm 73.42

Principal Moments of Inertia

Ix kg m⁻² 18

I_y kg m⁻² 18

I_z kg m⁻² 15

Position of leg tip [x,y,z]^>^>^>

Leg 1 m [0.424265, 0.424265, −0.428936]^>

Leg 2 m [0.424265, −0.424265, −0.428936]^>

Leg 3 m [−0.424265, −0.424265, −0.428936]^>

Leg 4 m [−0.424265, 0.424265, −0.428936]^>

2.1.2.1 Equations of Movement: Translational Changes

As stated in the previous section, when calculating the movement of the lander, the model uses a point-mass where forces are applied. This point-mass has the total mass of the lander concentrated and is located in the CG of the lander.

The CG position was consulted in the Computer-Aided Design (CAD) model done during the Concurrent Engineering study (CE-study) [Trojan Lander Study Group, 2015]. Fig. 2.2 shows the CAD model with the position of the CG. Since the deviations in the XY plane are small (17.27 and 34.66 mm in X and Y-axis respectively), in order to simplify the model I assumed the CG was contained in the axis of the octagon, so it

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Figure 2.2: Derived CG and principal moments of inertia directions of the Lander with deployed landing legs [Trojan Lander Study Group, 2015].

had no deviations in the XY plane. The deviation in the Z-axis is big enough (and critical for stability analysis) that it should be taken into account. For this reason, the CG of the model has a deviation from the central plane of 73.42 mm.

Fig. 2.3 shows the lander simplified model with the body axes drawn and the Euler angles rotations associated to each axis.

X_B

Mass Ix, Iy, Iz

Z_B

Y_B

Figure 2.3: Simplified lander model used for the simulations.

To calculate the position of the lander (defined as r) at each time step, I integrate the velocity r =

Z

˙r dt , where ˙r = v . (2.7)

To calculate the velocity of the lander, I integrate the acceleration v =

Z

˙v dt , where ˙v = a = 1

m

∑

^{F + g .} ^(2.8)

I calculate the acceleration by taking into account the component added by the gravity of the asteroid and the sum of the forces that are applied by the interaction between the legs and the surface to the CG. In Eq. (2.8), m is the total mass of the lander. I explain the expressions for the forces in the following subsections.

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2.1 Dynamic Landing Model

2.1.2.2 Equations of Movement: Attitude and Rotation Changes

For the attitude changes, I apply an analogous method. The rotations are done around the CG and they are induced by the residual rotations of the initial conditions and by the interaction between the lander and the surface. Eq. (2.9) shows the calculation of the attitude, integrating the rate of change of the Euler angles with time:

ΦΦΦ = Z

Φ˙

ΦΦ dt . (2.9)

I obtain the rate of change of the Euler angles from the angular velocities of the lander around the body axes. I use the transformation matrix [TEB]. Eq. (2.10), Eq. (2.11) and Eq. (2.12) show the expressions:

φ = ω˙ x+ ωysin φ tan θ + ωzcos φ tan θ , (2.10)

θ = ω˙ ycos φ − ωzsin φ , (2.11)

ψ = ω˙ y

sin φ cos θ + ωz

cos φ

cos θ . (2.12)

To obtain the angular velocities, I integrate the angular accelerations as shown in Eq. (2.13), Eq. (2.14) and Eq. (2.15):

ωx= Z

ω˙xdt Where ω˙x= 1

Ix

∑

^M^x^Ext^{+ ω}^y^ω^z^I^y^{− I}_I ^z

x

, (2.13)

ωy= Z

˙

ωydt Where ω˙y= 1

Iy

∑

^M^y^Ext^{+ ω}^z^ω^x^I^z^{− I}_I ^x

y

, (2.14)

ωz= Z

˙

ωzdt Where ω˙z= 1

Iz

∑

^M^z^Ext^{+ ω}^x^ω^y^I^x^{− I}_I ^y

z

. (2.15)

I calculate the angular accelerations by taking into account how the principal moments of inertia affect the rotations and the sum of moments that are applied by the forces obtained due to the interaction between the legs and the surface. I explain the calculation of these moments in the next section.

2.1.3 Model for Forces Interactions

The external forces applied to the lander in the equations of motion are due to the interaction of the lander with the surface of the asteroid. I divided this interaction in two forces, the normal force and the friction force, and the moment they apply. In this section I explain the more theoretical part of the calculations, while in Chapter 3 I go into more detail on how they are implemented into the code.

2.1.3.1 Normal Force

The normal force

FN= k∆x − cv⊥ (2.16)

is the component normal to the surface of the reaction forces. I obtain it by assuming the lander-surface interaction is a spring-damper system where ∆x is the compression of the spring part of the system, v⊥ is the component normal to the surface of the velocity of the specific leg of the lander, and k and c are the spring/rigidity constant and the damping constant respectively. Similar models have been used in the past to model lander-surface interactions [Hilchenbach, 2004,Nohmi and Miyahara, 2005,Pham et al., 2013,Sun et al., 2010].

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2.1.3.2 Friction Force

The friction force

F_F =

(−µFN

v_k 6= 0

0

v_k

= 0 (2.17)

is the component contained in the surface plane of the reaction forces. I obtain it by assuming the friction interaction to follow the simple Coulomb friction model [Giesbers, 2012], where FN is the normal force calculated using Eq. (2.16), and µ is the friction coefficient. It is always opposing the movement, hence the component of the velocity of the tip of the leg contained in the surface plane. Similar models have been used in the past to model lander-surface interactions [Hilchenbach, 2004, Nohmi and Miyahara, 2005, Pham et al., 2013, Sun et al., 2010].

2.1.3.3 Moments

The moments

M = r_leg×

∑

^{F = r}^leg^{× (F}^N^{+ F}^F⁾ ^(2.18)

are used to obtain the rotations and changes in attitude of the lander due to the interaction of the lander with the surface. I obtain the moments with the cross product of the reaction force vector in every leg and the position of that leg respect the CG. Eq. (2.18) shows how the only forces taken into account are the ones expressed in the previous sections, the normal and the friction force.

2.2 Theoretical Static Stability Condition

I do a theoretical static analysis in order to have a theoretical stability limit for comparison purposes with the dynamic models. These conditions will be ideal, with no movement and without taking into account bouncing and/or irregularities in the terrain, so the limit will never be met in reality (or even in the model), but can show where there is a room for improvement.

CG

α R₁

R₂ mgsinα

mgcosα mg

r d

hCG

leg 1

Figure 2.4: Theoretical static stability diagram. h_CGis the height of the CG, r and d are the moment arm for the CG and the reaction force at leg 2, mg is the product of the mass of the lander and the gravity and α is the slope angle.

Fig. 2.4 is the base of the calculation and I will use a method equivalent to the one used in [Pham et al., 2013].

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2.3 Damping Systems

Taking from Fig. 2.4 the momentum equilibrium at leg 1:

R₂d+ mghCGsin α − mgr cos α = 0 , (2.19)

and supposing the lander loses stability when R2= 0, Eq. (2.19) becomes

mghCGsin α − mgr cos α = 0 . (2.20)

The theoretical static stability condition can be written as

mgh_CGsin α − mgr cos α = 0 −→ tan α ≤ r hCG

. (2.21)

For any slope angle larger than the α obtained in Eq. (2.21), the lander will be theoretically unstable. Since the distance r from leg 1 (contact point) to the CG changes depending on the orientation of the lander (attitude respect to the slope), the unstable region will change for each attitude.

2.3 Damping Systems

Engineers have used damping systems in order to make landings on stiff surfaces softer and protect the equipment of aircraft and spacecraft from high shocks that could break them. Other applications might be to absorb energy and help the lander stop completely, or for stability purposes. The addition of damping systems seems even more important in this particular case, where there are no thrusters that could be used to stop the spacecraft before contact, and the whole manoeuvre is a free-fall from some meters height, so there will be vertical velocity and a shock. However, a proper study needs to be done since the specific implementation for this mission has a lot of particularities. The most important one may be the really small expected gravity field (0.0168 m s⁻²) compared to the one from Earth or even the Moon. Because of this reason, all the alternatives should be taken into account and the minimum amount of assumptions that could bias the results should be made.

2.3.1 Damping Systems Alternatives

Since the goal of damping a structure can be achieved through different mechanisms, I find it suitable to first make a description of these main mechanisms ones and a short comparison.

2.3.1.1 Classification

Generally, damping systems change a lot depending on the application. Although there is a strong heritage in space flight coming from the early ages (the first missions in the 60s and the Apollo project), I think it is worth checking the alternatives so as not to discard anything that could work appropriately only based on the argument ’it is how it has always been done’. Different alternatives for damping systems that can be used are:

Structural flexibility If the structure is not completely rigid, energy is dissipated when it moves at touchdown due to the friction between parts and small plastic deformation. Spring systems may also be used, relying on the friction to dissipate energy on every ’bounce’. This method is used when the energy to be damped is small, and it usually is just an extra to the design. Apart from that, a non-rigid structure is less prone to collapse when loads are applied. An example of applications are some small aircraft or gliders, where during normal operation, landings are done with almost no residual vertical velocity.

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Shock absorbers Are mechanical, hydraulic or even electromagnetic devices that transform the kinetic en- ergy of the shock into another type of energy, to then dissipate it (heat) or even store it (electricity). The most typical ones are the pneumatic and hydraulic shock absorbers, that combine the dissipation by friction of their parts (the pistons inside and the springs and cushions) with the transformation of the kinetic energy into heat for the air or fluid in each case. They are used mostly for vehicle suspension purposes, to reduce the effect of rough terrain.

Plastic deformation These dampers transform the kinetic energy of the shock into plastic deformation. In theory, any material that gets deformed can be used like this, but in practice some geometries and designs are always used, since they allow for a controlled deformation and their behaviour can be better predicted.

Some examples are the honeycomb crushable dampers or new investigations like sintered aluminium fibre.

2.3.1.2 Discussion

In order to choose one of the many alternatives for implementation, the advantages and disadvantages have to be taken into account. In this section, I discuss shortly about the reasons one alternative may or may not be more suitable for the specific application of a Trojan lander.

In general, structural flexibility is always used when designing structures. Completely rigid structures may break down more easily, so a bit of movement is allowed. This may be not enough to be used as the only damping strategy for this specific lander, since the velocity and thus the kinetic energy will not be as close to zero as it would be acceptable for such a design.

Shock absorbers have the ability to dissipate quite a high quantity of energy, and are reusable multiple times without losing much performance. The main problem is that they are usually quite heavy and involve many mechanical elements. This increases the complexity of the design and the possibility that something breaks or malfunctions along the way or during operation is higher. Since a malfunction during operation is difficult to counteract and almost impossible to repair, this is a critical factor.

Dampers that use plastic deformation have the advantage that are usually more lightweight than the other alternatives (they were designed for applications that had that requirement), and that they are mechanically simple. They do not have moving parts, liquids inside or pieces and mechanisms that could break and prevent them from working properly. As a clear disadvantage is that they are not reusable. Once they have been deformed, they stop absorbing energy.

The lander will be kept during the voyage from the Earth to the Trojan asteroid attached to the MSC. Both spacecraft need to fit inside the launcher so the size and the space the lander occupies is critical. For this reason, the legs need to be able to fold during the launch and unfold afterwards when the lander starts to operate. This mechanism already gives the bit of structural flexibility that is required in order to avoid breaking up at the minimum load applied. However, since leaving the whole responsibility of damping during the landing to the structural flexibility seems too risky (and even more in an unknown environment), at least one of the other alternatives has to be adopted.

The lander needs to withstand years travelling through space, harsh known environments and possibly unknown environments. There are uncountable ways that something could break, jeopardizing the entire mission, so the less parts that could be subject to breaking, the better. In this case, the simple design with no mechanical or movable parts of the crushable/plastic deformation dampers has a clear advantage. On top of that, weight is one of the restricting design points, and these dampers are really lightweight. The disadvantage of not being reusable is not an important drawback for this lander, since its operation only takes into account one landing, and in the best of cases, taking off again.

Model for Touchdown Dynamics of a Lander on the Solar Power Sail Mission

Lander on the Solar Power Sail Mission

Roger Gutierrez Ramon

M ODEL FOR T OUCHDOWN D YNAMICS OF A L ANDER ON THE S OLAR P OWER S AIL M ISSION

Roger Gutierrez Ramon

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

S

:

Model for Touchdown Dynamics of a Lander on the Solar Power Sail Mission

Acknowledgements

Contents

List of Figures

List of Tables

Acronyms

1 Introduction

1.1 Outline of the Thesis

1.2 Background

1.3 Problem

2 Theoretical Development

2.1 Dynamic Landing Model

∑

∑

∑

∑

∑

2.2 Theoretical Static Stability Condition

2.3 Damping Systems

M ODEL FOR T OUCHDOWN D YNAMICS OF A L ANDER ON THE S ÔLAR P ÔWER S ÂIL M ÎSSION