Design and Analysis of Optimal Operational Orbits around Venus for the EnVision Mission Proposal

Marta Rocha Rodrigues de Oliveira

Thesis to obtain the Master of Science Degree in Aerospace Engineering

Supervisors: Prof. Paulo Jorge Soares Gil Prof. Richard Ghail

Examination Committee Chairperson: Filipe Szolnoky Ramos Pinto Cunha Supervisor: Paulo Jorge Soares Gil Member of the Committee: João Manuel Gonçalves de Sousa Oliveira

November 2015 ii Acknowledgments

I would like to express my sincere gratitude to my supervisors Professor Paulo Gil from Instituto Superior Tecnico´ and Professor Richard Ghail from Imperial College London. Without their help, counsel, and generous transmission of knowledge, this thesis would not have been possible. I must also thank the EnVision team for the unique opportunity of working with such an exciting Venus project and contributing to an outstanding ESA proposal. Furthermore, I would like to express my appreciation to Dr. Edward Wright and to the NAIF team from JPL for the exceptional support provided. For the very welcomed inspiration, I must thank my friends, in particular the ISU community who encouraged me to pursue my work when I was reaching a breaking point.

This thesis is dedicated to my parents and my sister, who give meaning and purpose to all my pursuits.

iii iv Resumo

Na explorac¸ao˜ espacial, as missoes˜ planetarias´ em orbita´ sao˜ essenciais para obter informac¸ao˜ so- bre os planetas como um todo, e ajudar a resolver questoes˜ cient´ıficas pendentes. Em particular, os planetas mais parecidos com a Terra temˆ sido um alvo privilegiado das principais agenciasˆ espaci- ais internacionais. EnVision e´ uma proposta de missao˜ que tem como objectivo justamente estudar um desses planetas. Projectada para Venus´ e concorrente da proxima´ oportunidade de lanc¸amento da ESA, a proposta ja´ passou pela selectiva revisao˜ tecnica´ para a oportunidade de lanc¸amento M4, e sera´ agora apresentada para a M5, incorporando o feedback da ESA. O objectivo principal e´ estudar proces- sos geologicos´ e atmosfericos,´ nomeadamente processos de superf´ıcie, dinamicaˆ interior do planeta e atmosfera, para determinar as razoes˜ pelas quais a Terra e Venus´ evolu´ıram de forma radicalmente diferente apesar das semelhanc¸as dos dois planetas.

Nesta tese, propomos a estudar e melhorar o desenho da orbita´ operacional a` volta de Venus´ para a missao˜ EnVision. As restric¸oes˜ e requisitos cient´ıficos que afectam a orbita´ vao˜ ser examinados a fim de desenvolver um modelo computacional adaptado aos objectivos da missao.˜ Finalmente, a optimizac¸ao˜ da orbita´ operacional e´ feita para os parametrosˆ com maior influenciaˆ no planeamento da missao.˜

Palavras-chave: orbita´ operacional, design de orbitas,´ requisitos cient´ıficos, observac¸ao˜ de alvos, optimizac¸ao˜ de orbita,´ algoritmos geneticos.´

v vi Abstract

In space exploration, planetary orbiter missions are essential to gain insight into planets as a whole, and to help uncover unanswered scientific questions. In particular, the planets closest to the Earth have been a privileged target of the world’s leading space agencies. EnVision is a mission proposal with the objective of studying one of these planets. Designed for Venus and competing for ESA’s next launch opportunity, the proposal already went through the selective technical review for the M4 launch opportunity, and will now be submitted for the M5 call, incorporating feedback from ESA. The main goal is to study geological and atmospheric processes, namely surface processes, interior dynamics and atmosphere, to determine the reasons behind Venus and Earth’s radically different evolution despite the planets’ similarities.

In this thesis, we propose to study and improve the design of the operational orbit around Venus for the EnVision mission proposal. The constraints and scientific requirements that affect the orbit will be examined in order to develop a computational model adapted to the mission objectives. Finally, the orbit optimization is applied for the parameters with more influence in the mission planning.

Keywords: operational orbit, orbit design, scientific requirements, targets coverage, orbit opti- mization, genetic algorithms.

vii viii Contents

Acknowledgments...... iii Resumo...... v Abstract...... vii List of Tables...... xi List of Figures...... xiv List of Acronyms...... xvii List of Symbols...... 1

1 Introduction 1 1.1 Thesis Objective...... 1 1.2 Studying a Planet from Orbit...... 1 1.2.1 Spacecraft Subsystems...... 1 1.2.2 Operational Orbit(s)...... 3 1.2.3 Popular Target Planets: Earth’s Analogs...... 4 1.3 Studying Venus from Orbit...... 4 1.3.1 Venus’ Challenges...... 4 1.3.2 Venus’ Orbiter Missions...... 5 1.3.3 A Key Orbiter Mission to Venus: Venus Express...... 5 1.4 Studying Venus with the EnVision Mission...... 6 1.4.1 In the Footsteps of Venus Express...... 6 1.4.2 Spacecraft, Payload and Mission Scenario...... 6 1.5 Thesis Approach...... 7

2 Operational Orbit Design Fundamentals8 2.1 Design of an Operational Orbit...... 8 2.1.1 Problem Formulation...... 8 2.1.2 Orbit Representation...... 8 2.1.3 Orbit Propagation...... 11 2.1.4 Orbit Propagation with Perturbations...... 12 2.1.5 Ground Tracks...... 14 2.2 Design of an Operational Orbit around Venus...... 17

ix 2.2.1 Venus Fundamentals and Venus Centered Frames...... 17 2.2.2 Venus Specific Dynamics...... 19

3 Orbit Computation for EnVision 21 3.1 EnVision Orbit Requirements and Constraints...... 21 3.1.1 Mission Time Frame...... 21 3.1.2 Mission Constraints...... 23 3.2 Orbit Computation...... 26 3.2.1 Orbit Dynamics with Provisional Parameters...... 26 3.2.2 VenSAR Fundamentals...... 30 3.2.3 Targets Observation Computation...... 33 3.2.4 Observation Computation Test with Provisional Orbit Parameters...... 34

4 Orbit Optimization Method 36 4.1 Orbit Optimization Approach...... 36 4.1.1 Problem Formulation...... 36 4.1.2 Optimization Method Selection...... 37 4.2 Genetic Algorithm Fundamentals...... 38

5 Targets Observation Optimization 41 5.1 Genetic Algorithm Implementation...... 41 5.1.1 Fitness Function...... 41 5.1.2 Implementation Procedure...... 42 5.1.3 Algorithm Tests and Validation...... 44 5.2 Mission Overview with the Optimal Operational Orbit...... 53

6 Achievements and Future Work 59 Bibliography...... 63

x List of Tables

1.1 Successful orbiter missions to Venus...... 5

2.1 Summary of the Classical Orbit Elements...... 10 2.2 Alternate Orbit Elements...... 11 2.3 Venus Facts Summary...... 17

3.1 EnVision Target Sites...... 24 3.2 VenSAR operating modes parameters and coverage...... 25 3.3 Provisional orbit parameters...... 26

5.1 Fitness function test Conditions...... 45

5.2 Fittest solution Fi = −0.653 for fitness function test conditions...... 45 5.3 Test conditions for short durations...... 48

5.4 Fittest solution Fi = −0.727 for short durations test conditions...... 48

5.5 Fittest solution Fi = −0.181 for short durations test conditions corrected for regular nadir geometry...... 48 5.6 Test conditions for EnVision boundaries and equally weighted fitness terms...... 51

5.7 Fittest solution Fi = −0.495 for EnVision boundaries and equally weighted fitness terms.. 51

5.8 Fittest solution Fi = −0.700 for an optimal orbit solution...... 53

xi xii List of Figures

1.1 Example of a Spacecraft Trade Off Tree...... 3 1.2 Preliminary operational configuration of EnVision’s orbiter...... 7

2.1 Classical Orbital Elements...... 9 2.2 Elliptical Orbit Parameters...... 10 2.3 summary of the Alternate Orbit Elements...... 11 2.4 Declination β and geographical longitude λ...... 13 2.5 Ground track of the International Space Station (ISS) - position of the ISS given by Wol- framAlpha at 13:45 of 01/12/2014 computed from orbital elements determined 8.4 hours before...... 15 2.6 Cross section of the orbited planet with sub-point...... 15 2.7 Ground track for a non-rotating planet...... 16 2.8 Ground track for a planet with prograde motion...... 16 2.9 Ground track for a planet with retrograde motion...... 17 2.10 Comparison of the Earth’s axis tilt (23.4◦ ) and Venus’ tilt (177.3◦ ))...... 17 2.11 Venus Centered Frames...... 19 2.12 Earth Centered Frames...... 20

3.1 Transfer to Venus...... 21 3.2 Arrival orbit and final parking orbit...... 22 3.3 Porkchop plot with time of flight and impulse...... 23 3.4 Interferometry used for the North Pole measurements...... 25 3.5 High resolution mode used to detect the targets...... 26 3.6 Orbit simulation visualization for 5 days (∼ 80 orbits) at 100 s step for provisional parameters. 27 3.7 Ground track plot for 5 days (∼ 80 orbits) at 100 s step for provisional parameters..... 29 3.8 VenSAR and ground tracks...... 30 3.9 SAR geometry...... 31 3.10 Swath strip and footprint...... 31 3.11 Projection pattern of SAR antenna [31]...... 32 3.12 Point reflector imaged by SAR...... 33 3.13 High resolution mode corrected swath...... 34

xiii 3.14 Geometry approximation for VenSAR...... 35 3.15 Targets observations test for the EnVision provisional parameters in the interval 1000000 s - 10000000 s...... 35

4.1 Cross-over example...... 39 4.2 Mutation example...... 39

5.1 Stopping criteria defined with Tolfun [39]...... 44 5.2 Minimum, maximum, and mean fitness function values versus generations for fitness func- tion test conditions...... 46 5.3 Genealogy versus generations for fitness function test conditions...... 46

5.4 Fi versus inclination i and longitude of ascending node Ω for the fitness function test conditions...... 47

5.5 Fi versus time window tf and argument of perigee ω for fitness function test conditions... 47 5.6 Best and mean fitness function values versus generation for short durations test conditions. 49

5.7 Fi versus orbital parameters i and Ω for short durations test conditions...... 49 5.8 Best and mean fitness function values versus generation for short durations test condi- tions corrected for regular nadir geometry...... 50 5.9 Best and mean fitness function values versus generation for EnVision boundaries and equally weighted fitness terms...... 52

5.10 Fi versus orbital parameters i and Ω for EnVision boundaries and equally weighted fitness terms...... 52 5.11 Best and mean fitness function values versus generation for an optimal orbit solution... 53 5.12 Observations test for optimal orbit in the interval 0 s - 1000000 s...... 54 5.13 Observations test for optimal orbit in the interval 5000000 s - 6500000 s...... 55 5.14 Orbit simulation visualization for 10 days (∼ 155 Orbits) at 100 s step for optimal orbit... 55 5.15 Ground track plot for 10 days (∼ 155 orbits) at 100 s step for optimal orbit...... 56 5.16 Plot of the condition in which the distance from Venus to Earth is inferior to 1 AU for the mission’s first year...... 57

xiv List of Symbols

Latin Symbols

a Semi-major axis

an Coefficients of the eccentric anomaly power series given by the Lagrange inversion theorem B Radar’s bandwidth c Speed of light

Cn,m Harmonic coefficients (gravitational perturbations)

dj Position vector relative to the spacecraft e Eccentricity E Eccentric anomaly th Fi Genetic algorithm fitness function for the i design point ft Antenna footprint G Gravitational constant of the orbited planet i Inclination

Jn Harmonics (gravitational perturbations) with m = 0 l True longitude L Antenna length lat Latitude of sub satellite point long Longitude of sub satellite point m Mass of the spacecraft M Mean anomaly n Mean motion rate N Target sites th Ni Fitness term for the i design point that corresponds to the total number of covered sites p Semi-latus rectum

Pn,m Legendre polynomials (gravitational perturbations) R Radius of the planet tf Time window to intersect target sites ~r Spacecraft’s position vector ~r¨ Spacecraft’s acceleration vector

xv Ra Radius of farthest approach (apoapsis)

Ra Azimuthal resolution rot Angular distance resulting from a planet’s rotation for a given time interval of the spacecraft’s motion Rp Radius of closest approach (periapsis)

Rr Ground range resolution

Sn,m Harmonic coefficients (gravitational perturbations) T Total number of target sites considered

TV enus Venus’ period tf Observations time window u Argument of latitude ~u Line of sight vector U Gravitational potential

U0 Gravitational potential without perturbations

Up Gravitational potential perturbations term W Antenna width xsc Spacecraft’s coordinate in x axis of the planet centered inertial frame ysc Spacecraft’s coordinate in y axis of the planet centered inertial frame zsc Spacecraft’s coordinate in z axis of the planet centered inertial frame

Greek Symbols

α Fitness function weight parameter β Declination of the orbiting spacecraft

βi Incidence angle ε Depression angle γ Grazing angle λ Antenna wavelength Λ Geographical longitude µ Gravitational parameter of the planet ν True anomaly ω Argument of periapsis (closest point to the orbital path) Ω Longitude of the ascending node Π Longitude of perigee

ρj Look angle τ Radar’s pulse length θ Vector relative to the planet

θx Angular resolution in the x axis

θy Angular resolution in the y axis

xvi xvii List of Acronyms

AOCS Attitude and Orbit Control System ASCII American Standard Code for Information Interchange COE Classical Orbital Elements DVT Total Delta-V EME2000 Earth Mean Equator and Equinox of Julian Date 2451545.0 Frame FTP File Transfer Protocol InSAR Interferometry Synthetic Aperture Radar J2000 Earth Mean Equator and Equinox of Julian Date 2451545.0 Frame LSK Leap Seconds Kernel MAG Venus Express Magnetometer MRO Mars Reconnaissance Orbiter OCSM Onboard Computer System and Memory PCDU Power and Control Distribution Unit PCK Planetary Constants Kernel SAR Synthetic Aperture Radar SPICAV Ultraviolet and Infrared Atmospheric Spectrometer SPK Spacecraft and Planet Kernel SRS Subsurface Radar Sounder SSP Sub Spacecraft Point TDB Barycentric Dynamical Time TOAST Telecom Orbit Analysis and Simulation Tool TOF Time Of Flight TT& C Telemetry, Tracking and Communications VEM Venus Emissivity MappeR VenSAR Venus Synthetic Aperture Radar VeRa Venus Radio Science Experiment VEX Venus Express VIRTIS Visible and Infrared Thermal Imaging Spectrometer VMC Venus Monitoring Camera VME Venus Mean Equator and IAU vector of Date Frame VME2000 Venus Mean Equator of Date J2000 Frame VSSP VenSAR Sub Spacecraft Point xviii xix Chapter 1

Introduction

1.1 Thesis Objective

The main objective of this thesis is to study and improve the design of a science orbit around Venus for EnVision, Europe’s new medium class mission proposal for the European Space Agency’s Cosmic Vision 2015-2025 M5 call. The different constraints and requirements that affect the orbit design will be analyzed in order to select and refine selected mission features, in particular the observation of selected targets and mission duration.

1.2 Studying a Planet from Orbit

The study of a planet from orbit is essential to gain knowledge about the planet as a whole. A typical orbiter mission can provide data on a broad spectrum of elements such as a planet’s atmosphere, weather, surface, gravity field, magnetic fields, elemental composition, and internal structure.

To achieve these goals different spacecraft subsystems are needed and one or more operational orbits must be selected.

1.2.1 Spacecraft Subsystems

The spacecraft subsystems can be divided in: structures and mechanisms, propulsion, attitude and orbit control system (AOCS), thermal control, power, data handling, telemetry, tracking and communications (TT& C), and payload.

The science payload includes instruments such as cameras for high resolution imaging, context and weather; spectrometers for studying spectrums of different natures and to help identify chemical components; magnetometers for magnetic field studies; radiometers as atmospheric profilers; radar for surface data; and accelerometers for gravity and atmospheric studies, among others.

The TT& C package provides communication (via radio frequency - RF or optical link) for command,

1 data download, radio-science and tracking (by location, velocity). For communications with a ground station on Earth an antenna (fixed or steerable) is needed and it is typically working on X-band or Ka- band [1].

The AOCS stabilizes the orbiter against external (and internal) disturbances and provides orbit con- trol and maintenance to help for instance, the spacecraft satisfy the pointing requirements of the payload, antennae, solar panels, etc. In terms of propulsion, the orbiter needs thrusters to provide acceleration and torque for orbit maintenance and attitude correction. The thruster can assist the other actuators of the AOCS, including momentum wheels, control moment gyroscopes, magnetic torquers, and nutation dampers, that work together with the system’s sensors (magnetometers, sun sensors, star trackers, ac- celerometers, gyroscopes, among others). Moreover, the AOCS control system algorithm is processed on the Onboard Computer System and Memory (OCSM), along with the data handling, processing and storage, the command interpretation and execution, the control functions and the failure detection, iso- lation and recovery [1].

Given the adversity of the space environment, the thermal system is crucial to control the spacecraft thermal environment in different modes (internal dissipation and external input) and under various aspect angles for all the mission phases. This system is driven by equipment and payload requirements, and its control components include coatings, paint, radiators, sun shields, foam, heat pipes, optical reflectors, thermal insulators, among others.

Additionally, the spacecraft might need deployment (covers, baffles, sun shields) and protection against heat, radiation, straylight, etc. For instance for maneuvers such as aerobreaking, in which in order to reduce with drag the high point of the elipse (apoapsis) the spacecraft is driven through the planet’s atmosphere at the low point of the orbit (periapsis).

The power system is the system responsible for providing electrical power to the spacecraft bus and payload. It usually consists of solar panels (body mounted, panel type), batteries (for energy storage, safe mode, eclipses, etc.) and the power and control distribution unit (PCDU). For missions to outer planets (in particular for missions beyond Jupiter) it is important to consider alternative energy sources (e.g. nuclear power) [1].

2 Figure 1.1: Example of a Spacecraft Subsystems Trade Off Tree.1

An orbiter mission is not limited to remote sensing instrumentation; it is also essential to assist in situ instrumentation. Orbiters can help identify sites of interest on the surface of a planet and provide necessary topographic context for landing sites or relevant targets, and they can also assist with future spacecraft navigation, for instance, by identifying the position of an approaching spacecraft and helping with the precision orbit insertion [2].

1.2.2 Operational Orbit(s)

The operational or science orbit is the optimized orbit from which the mission’s scientific observations will be made. Given the scientific goals and spacecraft subsystems for the orbiter mission, the oper- ational orbit(s) selection is designed to satisfy instruments and scientific requirements. The operation orbit is selected after an analysis of the the space environment (e.g. third bodies, solar radiation pres- sure, micrometeorites, space debris, the planet’s atmospheric drag, gravity) [3]. In short, the target orbit selection is driven by requirements such as data resolution, coverage, revisit time, link budgets, visibility from ground stations, eclipse duration, cost of orbit acquisition and maintenance. Often these require- ments are contradictory and a prioritization balance must be made, so that the orbit is optimized for the most relevant requirements [1,4]. In an early design phase, such the proposal phase, when the mission time frame is not precisely determined, the orbit is designed for the drivers with more influence to be

1Graphic provided by Dr.Gunter¨ Kargl from the Space Research Institute of the Austrian Academy of Sciences

3 later on adjusted in the mission scenario.

1.2.3 Popular Target Planets: Earth’s Analogs

In recent orbiter missions, Mars has been a popular target [5]. Missions such as Mars Odyssey, Mars Express and Mars Reconnaissance Orbiter (MRO) helped us gain a more knowledge about the planet and left an important heritage for future missions to other planets. For example, the communications section of the EnVision mission proposal is being developed by Thales based on the communications system of the MRO mission [6].

This prominent interest in the Martian environment is linked to Mars being the terrestrial planet clos- est to Earth in terms of known conditions for the existence of life.

Venus has also been a privileged target as Earth’s twin in terms of its size, distance from the Sun and bulk composition; though a great number of fundamental questions, such as the planet’s geology and its correlation to the atmosphere are still unsolved [6]. In particular, Venus’ science has recently regained interest because of new geological evidence, but also due to the relevance of the subject of climate change. It is possible that the greenhouse effect is responsible for the extreme conditions on Venus. There might be interesting connections to be made between evolution of the climate on Venus and the Earth’s [7]. Hence, Venus remains an attractive target for missions in the near future.

1.3 Studying Venus from Orbit

From the Venera series to Venus Express, Venus’ Earth-like features and its different current state and evolution have inspired scientists and engineers to overcome the challenges of Venus’ exploration.

1.3.1 Venus’ Challenges

The Venus environment is characterized by extreme conditions such as a sulfuric acid cloud layer, high altitude winds, surface ambient temperature and pressure of 470oC and 92 atm, respectively [7]. In order to simulate Venus’ extreme environments there are experiments conducted on Earth such as the the Glenn Extreme Environments Rig, or GEER – one of the world’s top test chambers to recreate Venus on Earth [8].

Even though the study of Venus from orbit presents unique technological challenges, an orbiter mis- sion doesn’t have to face the extreme obstacles of a mission involving a probe or a lander, which maybe explains why there hasn’t been a more detailed science surface mission despite the fact that recent findings suggest there might be volcanic activity [7]. The remote sensing observations are particularly difficult due to Venus’ complex cloud cover of, and by its deep atmosphere characteristics (high temper- atures and pressures)[9].

4 1.3.2 Venus’ Orbiter Missions

There have been numerous international orbiters, atmospheric probes, and landers that have explored Venus. Some of the more relevant missions include the Russian Venera series, NASA’s Pioneer-Venus program that mapped the surface and explored the atmospheric features of the planet, and the Magellan mission that provided more detailed radar maps and topographic information [7].

Mission Launch Year Agency Venera 9 1975 RSA Venera 10 1975 RSA Pioneer Venus 1 1978 NASA Venera 15 1983 RSA Venera 16 1983 RSA Magellan 1989 NASA Venus Express 2005 ESA

Table 1.1: Successful orbiter missions to Venus.

The Venera 9 and 10 missions in Table 1.1 also included landers.

JAXA’s Akatsuki mission flew past Venus on 6 December 2010 after orbit insertion failure, but the insertion might be reattempted in late 2015. The mission’s spacecraft is still operational [10].

1.3.3 A Key Orbiter Mission to Venus: Venus Express

Venus Express was the first ESA mission to Venus. The mission aimed at a global investigation of the planet’s atmosphere, plasma environment, and some surface properties. The broad mission goals, the complexity of the payload and the operational difficulties due to the use of the Mars Express spacecraft, hindered the mission planning [11].

The payload inherited from the Mars Express and Rosetta missions consists of:

• Three spectrometers: an imaging and high-resolution spectrometer for the visible through thermal infrared spectral range (VIRTIS), a high-resolution infrared spectrometer (Planetary Fourier Spec- tometer), and a high-resolution UV and near-IR spectrometer for stellar and solar occultation, and nadir observations (SPICAV);

• A miniature camera operating in the visible and near-IR range (VMC);

• An Analyzer of Space Plasmas and Energetic Atoms for exploring in situ plasma and neutral ener- getic atoms (ASPERA-4);

• A magnetometer for magnetic field measurements and to support the ASPERA-4 (MAG); and

• A spacecraft radio system for a radio science experiment (VeRa) [11][12].

For telecommunications, the spacecraft carried two high gain antennae.

5 The mission was launched in 2005 and the orbiter arrived at Venus in April of the following year. The spacecraft’s operational orbit was a 24-hour elliptical, quasi-polar orbit with a periapsis at 250 km and an apoapsis at 66000 km [11]. In December 16 2014 ESA announced that Venus Express had ended its eight-year mission after exceeding its expected life time [13]. Venus Express was the mission to Venus with the longest observation periods and the mission that helped establish Europe as a leader in Venus research. It contributed with findings on Venus’ atmosphere and its dynamics, and established common points between Earth’s and Venus’ climate evolution [14].

The mission’s most important discoveries include VIRTIS measurements revealing that the planet is spinning slower than previously measured, unusually variable and unstable southern polar vortex, the possibility of recent volcanism, faster high level winds from new cloud motion measurements, the possibility that, in the cold region high in the planet’s atmosphere, carbon dioxide might have conditions to freeze out as ice or snow, and evidence o Venus’ magnetotail, among others [15].

1.4 Studying Venus with the EnVision Mission

1.4.1 In the Footsteps of Venus Express

Venus science has regained interest, especially considering the recent context of Earth-like exoplanets discovery and exploration [6].

Venus Express made important discoveries as mentioned in Section 1.3.3 but also raised new chal- lenges, one of the most relevant being the possibility of recent volcanism. The mission revealed signif- icant changes in mesospheric sulphur dioxide indicators, dark lava surrounding volcanoes, and surface temperature variations that suggest volcanic activity [6]. EnVision is a mission proposal that follows Venus Express to pursue its findings and research. This proposal is a response to ESA’s call for a medium-size mission (M5 call). The proposal already went through the technical review for the M4 launch opportunity, and will incorporate feedback from ESA to improve the mission design. Its main goal is to outline the state of geological activity on Venus and its relation to the atmosphere. EnVision will also provide gravity and geoid data, as well as new spin rate measurements, and new insight into the planet’s interior [6].

1.4.2 Spacecraft, Payload and Mission Scenario

The EnVision mission is to be launched on a Soyuz-Fregat on December 27, 2024. Following aerobreak- ing, the orbiter was planned to reach a low circular science orbit of an altitude of 258 km. EnVision’s payload includes [6]:

• A phased array synthetic aperture radar (VenSAR) with different operational modes – high reso- lution mapping, stereo scan, interferometry (InSAR), and polarimetry, for high improved resolution

6 imaging for selected targets such as the Venera landing sites, global topography and rate mea- surements of surface alteration, and analysis of surface processes;

• A subsurface radar sounder (SRS) for the vertical structure investigation and stratigraphy of geo- logical units, but also to explore sedimentary deposits, and structures below the surface;

• An infrared mapper and spectrometer (VEM) to measure variations in surface temperatures and concentrations of volcanic ally emitted gases;

• A 3.2 m X-band steerable antenna.

Figure 1.2: Preliminary operational configuration of EnVision’s orbiter.[6]

To conclude, EnVision will provide global imaging, topographic and subsurface data with a better resolution than previous missions to Venus, and may uncover the reasons for the radically different evolution of Venus and Earth.

1.5 Thesis Approach

In this thesis, the main goal of selecting and analyzing optimal operational orbits for the EnVision mission will be reached with the following procedure: - First, the fundamentals of an orbit’s dynamics and design will be introduced, as well as Venus’ specificities (Chapter 2); - Then, the orbit dynamics and constraints will be identified in order to develop a computational model of the operational orbit with provisional orbit parameters used in the mission proposal (Chapter 3) - Furthermore, the optimization key drivers will be summarized and the genetic algorithm procedure will be introduced (Chapter 4) - Finally, the optimization algorithm will be implemented and the optimal solutions will be analyzed in terms of its benefits to the mission (Chapter 5).

7 8 Chapter 2

Operational Orbit Design Fundamentals

2.1 Design of an Operational Orbit

2.1.1 Problem Formulation

The orbit (or orbits, for missions with different science phases) selection and design is a fundamental el- ement of the mission design process. Essentially, the task of selecting the operational orbit corresponds to obtaining the values for the orbital elements so that the mission goals are achieved optimally.

In the early stages of the mission design the orbit must be selected and optimized. The orbit needs to satisfy the objectives and requirements of the mission. For some missions it might be important to remain geostationary over a region of interest, or to maintain the apparent angle between an orbited planet, a spacecraft and the Sun constant (Sun-synchronous), or even just to have a repeated ground track. These requirements will induce specific boundaries and required values for different orbital parameters: altitude, eccentricity, inclination, among others. These parameters will be described in more detail in Section 2.1.2. An orbit is not only limited by the science phase of the mission, it can also be restricted by other phases of the mission. For instance, in order to maximize the mission performance from the start, the orbit will be determined taking into account the launch vehicle requirements. In brief, the orbit design is an integrative process, involving all stages of a mission [3].

2.1.2 Orbit Representation

To describe the motion of a spacecraft orbiting a main body in a Keplerian orbit, we assume that gravity is the only force, the mass of the spacecraft is negligible when compared to the mass of the orbited planet. Also the planet is spherically symmetric with uniform density, so it can be treated as a point mass, and other perturbations such as gravitational interactions with other bodies or atmospheric drag are neglected. The Keplerian orbit is a solution of the two-body problem which is described by the

9 following equation of motion [3]:

~r ~r¨ + µ = 0 (2.1) r3

where µ = GM is the gravitational parameter with G being the gravitational constant and M the mass of the orbited body [km3/s2], ~r¨ is the spacecraft’s acceleration [magnitude in km/s2 ], ~r is the spacecraft’s position vector [magnitude in km]. The state vector resulting from this equation translates into the six Classical Orbital Elements (COEs) that describe the ideal Kepler orbit [16, 17].

Figure 2.1: Classical Orbital Elements.

The most popular ways of representing an orbit are position(~r) and velocity (~v) in Cartesian or cylin- drical coordinates and the Keplerian elements.

The latter method was developed by Johannes Kepler to describe the orbit’s size, shape, orientation (orbital plane in space and orbit within the plane) and the spacecraft’s location at a given instant [17]. The Keplerian elements are also known as the Classical Orbit Elements (COEs) and they constitute an essential tool to describe the spacecraft’s motion in a given instant. This description requires six orbital elements:

• Semi-major axis a;

• Eccentricity e;

• Inclination i

• Longitude of the ascending node Ω;

• Argument of periapsis (closest point to the orbited body) ω;

• And, true anomaly ν.

10 In alternative to the true anomaly, it is also possible to use the mean anomaly and the eccentric anomaly to define the spacecraft’s position along the orbit at a given instant [3, 18,1]. The geometry of the orbit can first be described by its eccentricity e. The orbit can be elliptical (0 < e < 1), parabolic (e = 1), hyperbolic (e > 1), or circular (e = 0). For the purpose of this thesis we will focus on the parameters for elliptical and circular orbits, the orbits relevant for this work. The diagram 2.1 summarizes parameters that describe an elliptical orbit considering two-dimensional orbital motion.

Figure 2.2: Elliptical Orbit Parameters.

The value of the inclination can determine the type of the orbit as follows: if i = 0◦ or 180◦ , the orbit is equatorial (stays over the equator), and if i = 90◦ the orbit is polar (travels over the poles).

The Classical Orbital Elements are summarized in Table 2.1[17].

Element Description Range Undefined a Size - Never e Shape see 2.1.2 Never i Tilt −90◦ < i < 90◦ Never Ω Swivel 0◦ < Ω < 360◦ When i = 0◦ or 180◦ ω Angle from ascending node to periapsis 0◦ < ω < 360◦ When i = 0◦ or 180◦ or e = 0 ν Angle from periapsis to the spacecraft’s position 0◦ < ν < 360◦ When e = 0

Table 2.1: Summary of the Classical Orbit Elements.

These orbital parameters are not always defined. In the case of a circular orbit, there is no periapsis, and consequently no argument of periapsis, or true anomaly. In order to account for the absence of the periapsis as a reference, we use the argument of latitude u (also often referred to as θ), which can be related to the argument of peripasis and true anomaly through the following expression: u = w + ν. Essentially, the argument of latitude u is measured from the ascending node to the spacecraft’s position in the direction of the spacecraft’s motion .

In the case of an equatorial orbit, the line of nodes is missing, so the longitude of the ascending node does not exist, and the argument of periapsis is not defined. To replace these elements, we use the

11 longitude of periapsis Π, measured from the principal direction to the periapsis in the direction of the spacecraft’s motion.

Furthermore, in the case of a circular equatorial orbit, for which the longitude of the ascending node, the argument of periapsis, and the true anomaly are all undefined, we use the true longitude l, measured from the principal direction to the spacecraft’s position vector in the direction of the spacecraft’s motion [17].

Element Description Range Application u Angle from Ω to spacecraft 0◦ < u < 360◦ No ω (e = 0) Π From ~x to periapsis 0◦ < Π < 360◦ No Ω (i = 0◦ or 180◦ ) l From ~x to spacecraft 0◦ < l < 360◦ No ω and Ω (i = 0◦ or 180◦ and e = 0)

Table 2.2: Alternate Orbit Elements.

Figure 2.3: Summary of the Alternate Orbit Elements.

2.1.3 Orbit Propagation

In the former subsection (2.1.2) we mentioned three angular parameters measured from the periapsis that give us the spacecraft’s position at a given instant: the true anomaly ν, the eccentric anomaly E, and the mean anomaly M. These measurements are essential to determine the orbit dynamics. The true anomaly is related to the eccentricity and eccentric anomaly through the following tangent expression:

√ 1 − e2 sin(E) tan ν = (2.2) cos(E) − e

The mean anomaly is related to the mean orbit rate as follows:

12 M = M0 + n(t − t0) (2.3) r µ where n = is the mean motion rate of the spacecraft orbiting the planet [16]. a3 In turn, the eccentric anomaly is related to the mean anomaly by Kepler’s transcendental equation [3]:

M = E − e sin(E) (2.4)

This expression is the simplest solution to propagate an orbit [3]. This solution doesn’t consider perturbations (gravitational pull of third bodies, thruster forces, solar pressure, planet’s oblateness, etc.). The perturbations are small enough or corrected in a way that makes it possible to propagate the ele- ments directly. E is solved iteratively with numeric methods such as the Newton method presented in the next equation, until a sufficiently accurate value is reached.

M + e sin(Ei) − Ei Ei+1 = Ei + (2.5) 1 − e cos(Ei)

As an alternative to solving it numerically, an approximation can be used for small values of e:

1 E = M + e sin(M) + e2 sin2(2M) (2.6) 2

This is a second order approximation of the following power series [19]:

∞ X n E = M + ane (2.7) n=1 where the coefficients are given by the Lagrange inversion theorem as

| n | 1 X2 n a = (−1)k (n − 2k)n−1 sin[(n − 2k)M]. (2.8) n 2n−1n! k k=0 The series diverges for e > 0.6627... (Laplace limit) [19].

2.1.4 Orbit Propagation with Perturbations

The assumptions made for the orbit propagation can be corrected for different perturbations. These perturbations correspond to any changes to these classical orbital elements due to other forces beyond the simplified gravity model we have considered. For instance, a planet can’t be treated as a point mass because it is not perfectly spherical [3].

To take into account the asymmetries of an oblate planet, the standard method consists of expanding the gravitational potential in spherical harmonics. To start with, we consider the following gravitational potential U of a planet:

13 ∞ ∞ X µ X R U = ( )n(C cos(mλ) + S sin(mλ))P sin(β) (2.9) r r n,m n,m n,m n=2 m=0

where R is the radius of the planet, r is the radius of the spacecraft position, β is the declination of the orbiting spacecraft, λ is the geographical longitude, Cn,m and Sn,m are the harmonic coefficients and

Pn,m are the corresponding Legendre polynomials.

The harmonics (gravitational perturbations) with m = 0 are given as Jn starting with J2, so Jn is the same as Cn,0.

Figure 2.4: Declination β and geographical longitude λ.

If we consider the approximation of a planet with axial symmetry, the potential can be simplified as follows:

∞ µ X R U = [1 − Jn( )Pn sin(β)] (2.10) r r n n=2

To further explore the gravitational potential we can expand equation 2.10:

µ 1 R 2 1 R 3 1 R 4 U = [1 − J (3 sin2(β) − 1) − J (5 sin2(β) − 3) sin(β) − J (3 − 30 sin2(β) + 35 sin4(β)) r 2 2 r 3 2 r 4 8 r 1 R 4 −J (63 sin5(β) − 70 sin3 β + 15 sin(β)) − ...] 5 8 r (2.11)

Where sin(β) = sin(i) sin(u) = sin(i) sin(ν + w).

Considering perturbations to the order of J4 the equation above becomes:

µ µR2 µR3 U = U + U = − J (3 sin2(i) sin2(ν + w) − 1) − J (5 sin3(i) sin3(ν + w) − 3 sin3(i) sin3(ν + w)) 0 p r 2 2r3 3 2r4 µR4 −J (3 − 305 sin2(i) sin2(ν + w) + 35 sin4(i) sin4(ν + w)) 4 8r5 (2.12)

14 Where U0 is the gravitational potential without perturbations, and Up is the perturbations term [5]. The disturbing forces and perturbing accelerations can be derived from the expression above. In alternative to this method it is possible to model a planet’s gravitational field using a set of point masses [3].

As mentioned before, other perturbations beyond a planet’s asymmetries can affect the spacecraft motion. However, for the purpose of this thesis these effects will not be considered. Indeed the pertur- bations are small enough or corrected in a way that makes it possible to propagate the elements directly (Section 3.2.1).

In brief, to add the perturbations elements it would be necessary to integrate numerically the state equations for orbit propagation [3]:

r a =v ˙ = −µ + f(r) + g(θ) + h (2.13) |r3|

x˙ = v (2.14)

r where v˙ is the derivative velocity of the spacecraft, −µ |r3| is the spherical gravity acceleration 2.1.2, f corresponds to the additional accelerations that depend on the position, g translates the accelerations dependent on orientation (aerodynamic drag and solar pressure), and finally, h gives the accelerations independent of position and orientation (external acceleration from thruster for instance) [3].

Finally, to predict the spacecraft position with precision it would be necessary to add integrators with error correction, such as the Runge-Kutta integrators, in which the equation coefficients are selected to compute a n and n + 1 order while reducing computational cost. For instance, the Runge-Kutta fourth order method implies that the local truncation error is on the order of O(h5), while the total accumulated error is on the order of O(h4)[3].

2.1.5 Ground Tracks

As the COEs help us visualize the orbit in space, ground tracks allow us to visualize the spacecraft’s path on the planet’s surface, and check if certain sites are within the field of view of the spacecraft’s sensors [18,1]. Essentially, the ground track is the projection of the spacecraft’s passage onto the surface of the planet it is orbiting. Depending on the COEs of the orbit, the ground track can take many shapes. The intersection point of the spacecraft passing directly overhead from the frame of reference of the surface is called the sub-point. To better visualize this notion, the key elements are represented in the Figure 2.6.

Given the spacecraft’s position, the corresponding latitude and longitude describe the ground track path, and the spacecraft’s footprint corresponds to the ground area coverage from the nearest edge of

15 Figure 2.5: Ground track of the International Space Station (ISS) - position of the ISS given by Wolfra- mAlpha at 13:45 of 01/12/2014 computed from orbital elements determined 8.4 hours before. coverage swath to the far edge. This concept is essential to optimize the intersection computation of the spacecraft’s ground track path and the sites of interest of the orbited planet and will be further developed for the orbit computation.

Figure 2.6: Cross section of the orbited planet with sub-point location for spherical model [20].

Furthermore, to take into account the planet’s rotation it is necessary to apply a rotation matrix to the coordinates of the spacecraft’s trajectory. A planet can rotate counterclockwise (prograde motion) or clockwise (retrograde motion) and the ground track will appear to shift in succeeding orbits to the East or West depending on that rotation and the fixed orbital plane. For a non-rotating planet, the ground track of an orbit would continuously repeat [17].

In brief, the spacecraft’s position in the planet’s centered inertial frame is given by the following relations:

xsc = r cos(u) cos(Ω) − r sin(u) cos(i) sin(Ω) (2.15)

ysc = r cos(u) sin(Ω) + r sin(u) cos(i) cos(Ω) (2.16)

16 zsc = r sin(u) sin(i) (2.17)

Where xsc, ysc and zsc are the spacecraft’s coordinates in the planet centered inertial frame. In the case of a non-rotating planet, we would get the following longitude and latitude expressions [20]:

z lat = tan−1( sc ) (2.18) p 2 2 xsc + ysc

y long = tan−1( sc ) (2.19) xsc However the longitude needs to be corrected for the additional distance resulting from the planet’s rotation i.e., rot corresponds to the distance covered from the initial instant t0 to the instant t of the spacecraft’s motion due to the planet’s rotation with a given angular velocity. So the longitude becomes:

y long = tan−1( sc ) ± rot (2.20) xsc rot is added for a planet with prograde motion and subtracted for retrograde motion. For further theoretical details [20] can be consulted.

Figure 2.7: Ground track for a non-rotating planet.

Figure 2.8: Ground track for a planet with prograde motion.

17 Figure 2.9: Ground track for a planet with retrograde motion.

2.2 Design of an Operational Orbit around Venus

2.2.1 Venus Fundamentals and Venus Centered Frames

To design an operational orbit around Venus, it is essential to look into a few details of the planet.

Mass 4.9.1024 kg Radius 6051.8 km Surface Temperature 462◦ C Revolution Period 224 days Rotation Period 243 days Number of Moons none Atmosphere carbon dioxide, nitrogen (mainly)

Table 2.3: Venus Facts Summary.

Venus has no natural satellite, which immediately simplifies the gravitational effects. The rotation of the planet is retrograde (clockwise and contrary to the rotation direction of the Sun and the Earth). Also, the planet’s path around the Sun takes around 224 days but takes 243 days to complete a full rotation around its axis, which results in a Venus year being shorter than a Venus day. Indeed the planet’s rotation is extremely slow and has the slowest angular velocity in the Solar System (2.99 × 10−7 [rad/sec]) [21].

Figure 2.10: Comparison of the Earth’s axis tilt (23.4◦ ) and Venus’ tilt (177.3◦ )

On Venus the geographic North and South Pole orientation is the same as on Earth, since IAU defines the geographic north pole of a planet as the planetary pole that is in the same celestial hemisphere

18 relative to the invariable plane of the Solar System as Earth’s North pole. This is why Venus rotation is retrograde [22].

To measure the longitude in Venus the standard range goes from 0 to 360. Longitude is always measured from the prime Meridian: Greenwich for Earth and crater Ariadne in Sedna Planitia for Venus. By convention in planets other than Earth, the longitude is measured in a direction opposite to that in which the planet rotates. Because Venus rotates in a clockwise direction, the longitude on Venus increases in toward the east from the planet’s prime meridian. However, we are used to measure the longitude from the prime meridian toward the east and toward the west with increasing values in degrees until 180◦ . We have chosen this range for a more intuitive visualization of the results, as it is often done for this type of analysis [23].

To go into a little more detail about the Venus reference frames used it’s important to define the different types of frames. A planet’s coordinate system can be: planet-fixed rotating, planet-fixed non- rotating, and inertial. A planet-fixed rotating coordinate system is centered on the body and rotates with the planet, and the planet-fixed non-rotating type is centered on the planet but doesn’t rotate with it. Finally, an inertial coordinate system is fixed at some point in space [24].

As we mentioned there are dynamic and inertial frames. The EME2000 or J2000 frame (Earth Mean Equator and Equinox of Julian Date 2451545.0) is the standard inertial reference frame. The spin axes and prime meridians defined relative to the J2000 inertial reference system are the standard for planets as defined by IAU [22]. We have two main Venus centered frames. These frames both have their origin in Venus’ center of mass [25].

The Venus Mean Equatorial of Date frame or Venus Mean Equator and IAU vector of Date frame (VME) is defined by an X-Y plane corresponding to the Venus equatorial plane of date, and a +Z axis is parallel to the Venus’ rotation axis of date, pointing toward the North side of the invariant plane; +X axis is oriented by the intersection of the Venus’ equator of date with the Earth Mean Equator of J2000; and the +Y axis completes the right-handed system [25].

The Venus Mean Equator of Date J2000 is defined by a +Z axis pointing toward Venus North Pole of date J2000; a +X axis points toward the Venus IAU vector of date J2000 (intersection between the Venus equator of date and the J2000 equator) [25]. Essentially the VME2000 frame is the VME frame frozen at J2000 (using the IAU constants for Venus’ North Pole and prime meridian).

In Figure 2.12 we have summarized the Earth Mean Equator and Equinox of Date frame (EME) and J2000 reference frames to compare with the Venus Centered frames.

19 2.2.2 Venus Specific Dynamics

In the gravitational potential equation obtained, we included the gravitational perturbations terms from

J2 to J5. For Venus the terms to the order of J4 are of the same order of magnitude. Even though J2 (4.458 × 10−6) is the largest term its value is still very small [26]: it is only about 0.4 % of Earth’s value [5]. This reduced perturbations effect is related to the fact that Venus’ flattening coefficient is very close to 0 [22]. Essentially, Venus is almost perfectly spherical, it’s the most spherical planet in the Solar System. This is in turn connected to Venus’ extremely low rotation rate we described in the previous section.

Venus is a complex case to apply the familiar natural orbits typically used for remote sensing mis- sions. For instance, the extremely low perturbations don’t provide the torque that the gravity field of more oblate planets to generate Sun-synchronous orbits (orbits maintaining approximately constant angle be- tween the Sun, orbiter and orbited planet) [5]. Venus is the planet for which the spherical approximation is most accurate in the solar system [22].

Figure 2.11: Venus Centered Frames.

20 Figure 2.12: Earth Centered Frames.

21 22 Chapter 3

Orbit Computation for EnVision

3.1 EnVision Orbit Requirements and Constraints

3.1.1 Mission Time Frame

Venus is the closest planet to Earth with launch windows every 19 months [6]. The choice of launch date is essential to the mission planning. All the occultation predictions and other orbit related constraints depend on the porkchop plots initial analysis for mission scheduling. The porkchop plots are a typical phase in preliminary mission analysis, they are essentially interplan- etary fuel efficiency maps and show how much energy it will take to escape Earth’s gravity, place the spacecraft in the right trajectory, and reach Venus in this case. The required energy varies depending on the planets’ ephemeris data [27]. The data required for these porkchop plots is created by solving the heliocentric, two-body “patched-conic” Lambert problem. The gravitational effect of both the launch and arrivals planets on the heliocentric trajectory is ignored, for more details [27] can be consulted.

Figure 3.1: Transfer to Venus.

In the first draft of the EnVision proposal the mission was scheduled as follows: launch date on the

23 27th of December 2024 on a Soyuz-Fregat from Kourou, cruise of 5 months, arrival on the 7th of May 2025, as presented on the Lambert transfer plot [28]. Capture by conventional bipropellant followed with an initial 308 x 50 000 km altitude orbit. In figure 3.2 we can see the insertion and final orbit obtained with the Classical Orbital Elements tool [29].

Figure 3.2: Arrival orbit and final parking orbit.

Following the orbit insertion, there is a six month period of aerobreaking that will be used to reach the operational orbit for science operations. During aerobreaking the apoapsis of the capture orbit is reduced by the drag effect on the spacecraft passing through the atmosphere at the periapsis. The final provisional operational orbit is well controlled with a 258 km altitude, inclination of 88◦ , and the other parameters concerning the orbital orientation with respect to the Sun were left uncontrolled [6]. These options were compromises made as a result of the requirements developed in the next section.

The porkchop plot was obtained with the MATLAB 2013a software with an adaptation of the script from [27]. The result shows contours of time of flight (TOF), and total delta-v (DVT) for different com- binations of launch date and arrival date. The data is created over the range nominal - span and nominal + span, where the time span is defined in order to optimize the plot visualization. All the con- tour levels were defined as input vectors. We observe two possible launch dates (two optimal minima), including the optimal solution of the 27th of December 2024.

The JPL Solar System Ephemeris were used for the ephemeris data needed for the porkchop plots, in particular the DE421 data set, which includes estimates of the orbits of the Moon and planets and

24 Figure 3.3: Porkchop plot with time of flight and impulse. covers the years 1900 to 2050 (released in 2008). To use the DE421 ephemeris in binary format, it was necessary to obtain the header and data ASCII ephemeris files through JPL File Transfer Protocol (FTP), an then used Fortran to convert and concatenate these files into the binary format.

One of the main challenges of this proposal is the time constraint. It is essential to establish an at- tainable time frame to avoid possible over-runs on schedule. For illustrative purposes it can be assumed that the aerobraking ends before January 2026 and that the nominal mission starts at the beginning of February 2026 and the nominal mission lasts 2 years and 8 months [6].

3.1.2 Mission Constraints

The primary goal of this mission is to understand the geological activity on Venus (tectonics, volcanism, surface processes, interior dynamics). The different science goals result in science objectives: sur- face change, geomorphology, specified targets, thermal emissivity, gravity field, spin rate and spin axis, among others.

We will focus on one of the main objectives: observation of specified sites of interest, since they give us the ground truth. In past missions relevant data on surface processes was found. For instance, the surface images captured by Soviet Venera landers revealed data that suggested pyroclastic or sedimen-

25 tary deposits instead of the basaltic lava flow assumed previously [6].

A major problem with the present data is the image resolution. More accurate measurements are needed to distinguish the presence of different surface materials. There are many data issues and missing information [6]. The ground truth will be best given by the Venera landers, the most significant specified targets. These landers were launched by the soviets between 1961 and 1984 and consist essentially of metal spheres with a landing ring and antenna coil [6]. Because of this metallic nature, the landers appear 6dB brighter than the rest of the surface. Given the radar’s sensitivity to metal, the landers willl be easy to spot [6].

The measurement resolution needed for the location and characterization of the Venera landing sites is of 1-10 m. The minimum resolution required to detect the location of the landers is approximately 4-5 m, and a resolution of 1 m is required for imaging the landers. VenSAR meets these requirements [6]. The same procedure applies to the Vega landers from the Vega program that was a development of the earlier Venera series.

In terms of the mission, the absolute key landers to target are the Venera 9, 10, 13 and 14, since these have surface images, followed by Vega 1, 2 and Venera 8, which have surface composition mea- surements but no images. Other targets beyond the Venera and Vega landers are interesting but not critical landslides: canali, coronae in Helen Planitia, landslide in Diana Chiasma, Imdr Regio [6]...

Target Latitude Longitude Priority 1-3 (1,2-High, 3-Low) Vega 1 7.2◦ N 177.8◦ E 2 Vega 2 7.14◦ S 177.67◦ E 2 Venera 5 3◦ S 18◦ E 3 Venera 6 5◦ S 23◦ E 3 Venera 7 5◦ S 351◦ E 3 Venera 8 10.70◦ S 335.25◦ E 2 Venera 9 31.01◦ N 291.64◦ E 1 Venera 10 15.42◦ N 291.51◦ E 1 Venera 11 14◦ S 299◦ E 3 Venera 12 7◦ S 294◦ E 3 Venera 13 7.5◦ S 303◦ E 1 Venera 14 13.25◦ S 310◦ E 1

Table 3.1: EnVision Target Sites.

Beyond the sites of interest, we will also focus on the North Pole interferometry measurements that will be necessary for many science goals (such as spin axis and rate).

The scientific requirements demand a well controlled near circular orbit (with a maximum eccentricity of 0.001) [6]. Also, the altitude should be as low as possible since the resolution of the gravity field declines rapidly with altitude, which also helps save in terms of fuel usage to correct solar perturbations on the orbit, which progressively increase the orbit altitude. In terms of the drag factor, previous missions detected sensible atmosphere below 200 km altitude: an altitude above 230 km is necessary to be above its effect. The inclination of the orbit needs to take into account the necessary geometry for SAR imaging

26 of the North Pole (restrictive).

As was introduced before, given the science objectives it is essential to review the instruments re- quirements. The main instrument carried by EnVision is VenSAR, a synthetic aperture radar (SAR) antenna (5.47 x 0.60 m), operating at 3200 MHz in the S-band. It has five operating modes: stereo, in- terferometry, polarimetry, high resolution strip-mode, and sliding spotlight. This system is programmable for other modes and parameters. The geometries for these modes depend primarily on the swath width and incidence angles [6]. We have summarized in table 3.2 a few constraints for these operating modes in terms of what will be most significant for this thesis.

Parameter Interferometry High Resolution Input Power 660 W 1874 W Data Rate 53 Mbps 856 Mbps Swath Width 43 km 40 km Incidence Angle (near) 38.2◦ 36.3◦ Incidence Angle (far) 44.1◦ 42.2◦

Table 3.2: VenSAR operating modes parameters and coverage.

VenSAR has a fixed axis of maximum radiated power, also known as boresight, of 32◦ (off-nadir an- gle), which provides an angular separation of at least 20◦ for stereo mode (the more angular separation the better). This configuration is related to the fact that very low orbits are needed and that these orbits require a greater bandwidth for interferometry which needs to be compensated for by increasing the look angle above. VenSAR also faces data rate limitations: 362.9 Tbits data volume are expected to be returned during the mission for the interferometry mode and 7.2 Tbits for the high resolution mode [6]. Due to thermal and data rate limitations, VenSAR has a limited time for active mode (approximately 15 minutes per 92 minutes orbit).

Figure 3.4: Interferometry used for the North Pole measurements [6].

27 Figure 3.5: High resolution mode used to detect the targets [6].

3.2 Orbit Computation

3.2.1 Orbit Dynamics with Provisional Parameters

For the first version of the EnVision proposal, a circular orbit was selected with the following character- istics: a convenient point above 230 km (sensible atmosphere) of 258 km for the spacecraft altitude, an inclination of 88◦ for SAR imaging of the North Pole, and the orbit plane orientation with respect to the Sun was left undefined. To test and simulate the orbit dynamics using Matlab we considered the parameters in the table 3.3, which correspond to the provisional EnVison orbit parameters. The provisional orbit has a period of 1h 32min 5s.

Ω e i u a (altitude) 0◦ 0 88◦ 195◦ 6309.8 km (258 km)

Table 3.3: Provisional orbit parameters.

In sections 2.1.2 and 2.13, we mentioned that different perturbations effects were neglected. Indeed the perturbations are either corrected or small enough not to be considered. For solar perturbations corrections will be needed but the fact that we have low orbits (below 350 km) will help reduce the fuel required for theses corrections [6]. Moreover, we only considered altitudes above 230 km as was rec- ommended in the proposal to avoid the sensible atmosphere detected by Magellan and Venus Express below 200 km [6]. Furthermore, Venus has a flattening coefficient of approximately 0 [22], so orbit apse rotation and nodal regression are very small. To verify the effect of an oblate Venus, we considered the approximate effects of the term with the most impact, J2, on the ascending node and argument of perigee rates of change with the following expressions:

3nJ R2 Ω˙ = − 2 cos i (3.1) 2a2(1 − e2)2

28 3nJ (1 − 5 cos i2) R 2 ω˙ = − 2 ( ) (3.2) 4 a

The impact after 500 days is only of 0.3 km on the spacecraft position vector’s magnitude, which will be well within the range of the swath correction performed in the section that follows (3.2.2), so this effect can be neglected. It is important to mention that for the actual mission, the Precise Orbit Determination (POD) method will be used [6].

In brief, we propagated the elements directly from Kepler’s equation. As an alternative to solving it numerically, we chose the approximation given by equation 2.6 in order to optimize the computing time. To validate the approximation we considered a worst case scenario of a 0.01 eccentricity and ran the simulations for 5000 orbits with a 10 s step for both the numeric eccentric anomaly calculation and for the approximation. We got an error of approximately 0.001 km for the spacecraft position, which validates the approximation. However, it is only valid because we are testing for near-circular orbits, for e=0.8 we have an error to the unit, which is related to the fact that for e > 0.6627... the series diverges (Laplace limit). We obtained the 3D orbit visualization adapted from [30] in order to have a more intuitive interpreta- tion of the orbit propagation results.

Figure 3.6: Orbit simulation visualization for 5 days (∼ 80 orbits) at 100 s step for provisional parameters. The orbital path corresponds to the yellow marker and the ground track on the surface of Venus for the successive orbits is represented in green. 2

In the visualization of the spacecraft’s ground track we included the target sites to have a more intu- itive tool regarding their detection. The North Pole reference point is just representative, it corresponds to all longitudes. The sites of interest plotted were the landers Vega 1,2 and Venera 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. The priority order established in the mission constraints section will be used for the

2The Venus background image is from Steve Albers in connection with NOAA’s Science On a Sphere project (”value-added” global planet and satellite images).

29 optimization chapters. The Venus rotation speed is very low as was stated before. Modeling the mission time is rather slow, since the time step shouldn’t be greater than 100 seconds which will be discussed in the next section (3.2.2). The discontinuous appearance in the trajectory lines for the 5 days (∼ 80 Orbits) are due to the fact that the 100 s time step used to show this effect.

30 Figure 3.7: Ground track plot for 5 days (∼ 80 orbits) at 100 s step for provisional parameters.

31 3.2.2 VenSAR Fundamentals

In order to compute the observations of the target sites with VenSAR, it is necessary to examine valid ge- ometry approximations. To begin with, we can shortly review the payload configuration for the elements of interest to this exercise [6]:

• The VenSAR needs to be aligned along track, which is parallel to the solar array;

• The solar panels rotate about their two axes;

• The VenSAR is inclined to provide the off nadir angle;

• The HGA does not interfere with this structure since it is most of the time in the anti-nadir hemi- sphere, albeit it has a number of positions accessible (as pointing to Earth during science phases).

Figure 3.8: VenSAR and ground tracks.

Furthermore, we have summarized the key concepts and elements of the VenSAR geometry:

• The SAR look angle is the angle between boresight and nadir, also referred to as the off-nadir angle θ, and it is represented in figures 3.8 and 3.9;

• The direction of the incoming wave relative to the horizontal plane is called the grazing angle γ,

and on the Venus surface the waves come with incident angles βi;

• The VenSAR beam comes to the surface of Venus with the incident angles of 36.3◦ and 42.2◦ with respect to the vertical axis for the targets detection mode (38.2◦ and 44.1◦ for the North Pole);

• For a flat Venus surface approximation, the grazing angle and depression angle ε represented in figure 3.9 can be assumed to be equal;

32 • The line of sight vector ~u is the vector pointing from the SAR antenna to the footprint on Venus;

• The area covered by the antenna is the footprint, and the swath width refers to the strip of Venus’ surface from which the mission data is being covered, as illustrated in figure 3.10.

Figure 3.9: SAR geometry.

Figure 3.10: Swath strip and footprint.

To go into further detail about the VenSAR observations, we are considering the strip-mode SAR, used to detect the targets, as a 2-D rectangular aperture as further described in [31]. The aperture in the flight direction corresponds to the length L (5.47 m for VenSAR) and the width W (0.6 m for VenSAR) is the aperture in the perpendicular direction to the orbit path. For a given look angle θ and resulting slant range R (hypotenuse of the triangle represented by the altitude H of the spacecraft and the distance between the radar antenna and the ground track), the following relations for the footprint along track and across track can be applied [31]:

2Rλ ft = (3.3) along L

33 Figure 3.11: Projection pattern of SAR antenna [31].

2Rλ ft = (3.4) across W

Considering the initial scenario for VenSAR, the across track footprint is of 40 km, and along track we have 4.4 km.

It is also important to obtain the resolution needed. The range resolution is the pixels separation of the image perpendicular to the direction of the spacecraft orbiting Venus. The nominal slant range resolution is given by [31]:

∆r = cτ/2, (3.5)

where τ is the pulse length (which corresponds to the inverse of the radar’s bandwidth B), and c is the speed of light. The pulse travels from the antenna to the surface and back. The ground range resolution illustrated in figure 3.9 is given by [31]:

cτ R = (3.6) r 2 sin(θ)

To improve this resolution either the look angle or the bandwidth of the radar need to be increased, albeit in many cases the bandwidth of the radar is limited by the data transmission speed [31].

In the other direction (aziumthal), the cross-range resolution is given by [31]:

Hλ R = . (3.7) a L cos(θ)

However, for the case in this thesis, the strip-mode SAR, this relation is improved by the following theoretical expression [31]:

Ra = L/2 (3.8)

34 Figure 3.12: Point reflector imaged by SAR [31].

One of the requirements for the location and characterization of the Venera landers is a cross-range resolution of 1-10 m, with this radar in strip mode we have 2.7 m (5.47/2 m).

Furthermore, the spotlight-mode, that will be used for imaging in the mission after the detection phase, is essentially a longer synthetic aperture, in which the radar beam is directed in order to follow the target as the spacecraft orbits Venus [32].

3.2.3 Targets Observation Computation

For this project, we will consider the SAR antenna operating as a continuous swath (strip-mode) to detect the targets. However, the SAR can also be generated as a series of bursts (scanSAR), which images a rectangular or square patch of ground, or even in spotlight mode, in which a single image of 5 km along track and 10 km across track is produced. The latter is the most stringent targeting constraints in the mission [6]. This mode will be used to image the Venera landers themselves, after the the high resolution detection phase. The targets observations computation was scripted in Matlab and approached as follows:

• We considered an approximation of a flat surface model that comes with a total error of around 4km when compared with the spherical model swath results from the proposal (Figure 3.14);

• The footprint was assumed to be equal to the swath width, i.e. if the target is inside the swath width, we can assume it can be observed;

• In order to take into account the deviations and to make sure that the target site is not just inter- sected at the swath edge we considered a width of 30 km, equivalent to 0.3 ◦ (33 km for the North Pole) for the initial scenario with the provisional parameters;

• Since for testing the observations, the VenSAR with nadir direction is interesting to consider for its simplicity, both nadir and off-nadir geometries were calculated (only the off-nadir geometry represents the operational SAR);

35 Figure 3.13: High resolution mode corrected swath.

• The off-nadir geometry corresponding to the high resolution mode that will be observing the tar- gets was scripted with incidence angles of 36.3◦ and 42.2◦ (44.1◦ and 42.2◦ for the North Pole interferometric measurements);

• Finally, the swath width is given by y − x, and from the altitude and the tangent relations that give us x and y, we can estimate the swath and the distance between the “VenSAR sub-point” (VSSP) swath longitude and the spacecraft sub-point (SSP) longitude d = x + . VSSP 2

The “VenSAR sub-point” (VSSP) represented in Figure 3.14 corresponds to the spacecraft sub-point (SSP) corrected for the VenSAR observations track. In brief, since the strip SAR image dimension is swath swath limited across track, if the VSSP is between [Target’s longitude - ; Target’s longitude + ], 2 2 the target is considered observed.

3.2.4 Observation Computation Test with Provisional Orbit Parameters

With the geometry described scripted in Matlab, we ran the targets observations script for the interval [1000000 s, 10000000 s] with 10 s step for the EnVision provisional orbit parameters described in 3.2.1. For this test we considered the nadir VenSAR direction geometry and the Venera landers and North Pole as targets (11 in total). The 10 s time step was used to make sure that the area of interest was not missed. With a time step of 100 s we may miss some intersections for large mission times. In Figure 3.15 we can observe the number of detected targets in function of time. In particular, we observe that in these conditions only after 1810 orbits all targets are detected. Essentially, with this test we can verify the observations performance of the provisional parameters that will be improved.

36 Figure 3.14: Geometry approximation for VenSAR in strip-mode.

Figure 3.15: Targets observations test for the EnVision provisional parameters in the interval 1000000 s - 10000000 s.

37 38 Chapter 4

Orbit Optimization Method

4.1 Orbit Optimization Approach

4.1.1 Problem Formulation

EnVision’s main objectives depend on the detection and imaging of target sites. Considering this goal the orbit selection has conflicting criteria (resolution, coverage area, mission duration) that need to be balanced in order to achieve the desired features. The orbit has the following requirements:

• Observe the targets with high priority: Venera 8, 9, 10, 13, 14, Vega 1, 2, and additionally Venera 5, 6, 7, 11, 12 (Table 3.1);

• Ensure polar coverage(Section 3.1.2);

• Maintain orbit as low as possible for gravity field measurements resolution below 250 km and above 230 km to avoid the drag effect (Section 3.1.2);

• Finally, ensure a near circular orbit for the spin axis and rate measurements (eccentricity below 0.001 - Section 3.1.2).

In brief, we want to observe the targets with high priority as soon as possible in a minimum time, while satisfying the orbit constraints (near polar, near circular, low altitude). This way we ensure that the main targets are observed at the beginning of each mission cycle (1 cycle corresponds to a Venus rotation period of 243 days) and that the operations planning for VenSAR will be easier. The observations geometry computation that will be used for the optimization was described in Chapter 2. As an additional objective, we want to detect all the targets listed in Table 3.1. The criteria to evaluate our results includes:

• Minimum effective time to detect the targets with high priority;

• Detection of all targets with high priority;

• Maximum number of targets listed in Table 3.1 detected.

39 4.1.2 Optimization Method Selection

To solve our optimization problem, we need global optimization, i.e. a method that seeks to find the global best solution of an objective function that translates our goals while satisfying the orbit constraints. We refer to the optimization as global since our objectives will likely lead to an objective function with multiple local optima.

Formally, in a global optimization probem we assume continuous objective functions f and con- straints g, finite bounds [xl; xu] related to the decision variable vector x, and a feasible nonempty set D [33]. These assumptions guarantee that the global optimization model is appropriate [33].

In the presence of multiple local minima, if we use traditional local scope search methods we will often find locally optimal solutions [33]. To obtain a globally optimal solution, there are exact methods, in particular deterministic methods, which always produce the same output for a given input and don’t in- volve randomness, and stochastic methods, that use randomness. But there are also heuristic methods, which look for solutions among all possible ones, but do not guarantee that the best solution is found [36].

Recently, a particular algorithm of the heuristic type called genetic algorithm has been often cho- sen as a method for optimizing orbit design [34, 35]. These algorithms have been used to find target orbits, in order to reduce for instance the average revisit time over a targeted site for a selected time frame, but also to optimize the fuel consumption of low Earth orbit constellations for temporary recon- naissance missions or to minimize telecommunications coverage blackouts in interplanetary missions [34, 35]. In some cases, the optimization is preceded by a semi-analytical method to reduce the number of unknowns before the optimization, since this type of tool is usually computationally expensive [34, 35].

On a project developed by the Jet Propulsion Laboratory in orbit design and optimization based on global telecommunication performance metrics [35], they applied a genetic algorithm coupled with the Telecom Orbit Analysis and Simulation Tool (TOAST) to find the optimal orbit for Mars orbiter min- imizing the telecommunications gap time. The optimal solutions obtained were different from the Mars Telecommunications Orbiter (MTO) candidate orbits identified based on the mission’s specific criteria and constraints, which revealed the necessity of an MTO specific assessment.

In a different example, the problem of the initial natural orbit design for regional coverage on Earth [34] was addressed by applying a genetic algorithm for optimizing the number of intersected sites while minimizing the time frame needed. In this case, the genetic algorithm was chosen for its flexible nature in comparison to other techniques such as gradient-descent methods or non-linear programming [34]. The method applied provided several solutions to the problem with different interesting characteristics. In brief, genetic algorithms have proven to be a very successful way of getting solutions in orbit design optimization problems similar to the one addressed in this thesis. We have summarized some of the advantages of selecting the genetic algorithm to optimize the orbit design:

40 • The proven success of the method for problems that are not convex, having many local minima [34, 35];

• The possibility of combining code for integer, real values and options [34];

• The fact that the algorithm’s randomness can accelerate the progress of the optimization, and can make the method less sensitive to modeling errors [37];

• Finally, the possibility that the randomness can lead to a global optimum by escaping a local minimum [37].

4.2 Genetic Algorithm Fundamentals

A genetic algorithm can be applied to solve problems that have discontinuous, non-differentiable, stochas- tic, or highly nonlinear objective functions. Essentially, it is applied when a standard classical, derivative- based optimization algorithm is not ideal [36, 38]. The latter generates a single point at each iteration and selects the next with deterministic computation until an optimal solution is reached, while the genetic algorithm generates a population of points at each iteration, and by random generation, the best point in the population approaches an optimal population [36, 38].

The genetic algorithm is a heuristic technique to find globally true or approximate solutions of a given optimization or search problem. It is inspired by our view of a way the nature finds optimal solutions – evolution. Major ideas of evolutionary biology: inheritance, mutation, selection and recombination form the basis of a generic algorithm.

In this type of algorithm, every point in feature space of a problem is treated as an individual, and features of an individual are treated as a genome. There is a defined so called fitness function that evaluates an individual. All optimization problems aim to minimize or maximize an objective function. In the case of genetic algorithms the objective function is called fitness function. Essentially, it is a function that maps variables into a global number representing an associated evaluation value. It is the first step in the optimization procedure. The fitness function is defined based on the optimization goals [36, 38].

In these algoritms, a set of individuals form a population, and the algorithm starts by generating an initial population in some randomized manner. Next, new generations of population are produced iteratively. To produce a new generation some individuals from a current generation are selected based on their fitness result. Couples of individuals from the selected set form their offspring, and the genomes of latter individuals are generated based on the genomes of their ancestors and using a method that models a recombination in real life biological evolution (cross-over). Moreover, random modifications to the newly produced individuals are applied to model mutations [36, 38]. The purpose and the core idea of the genetic algorithm is that the next generation of population is better than the previous one in a sense of a fitness function. The algorithm terminates when a predefined maximum number of generations were produced or when a sufficient level of fitness has been reached

41 Figure 4.1: Cross-over example.

Figure 4.2: Mutation example. in a population or when there is no more significant changes in a fitness of a new generation compared to the previous one [36, 38].

It must be pointed that there is no guarantee that the next generation would be better than the previous one, just like in real life evolution. But again just like in real life, eventually in time some generation will likely obtain a close to optimum fitness value [36, 38].

With a correctly implemented algorithm, the population will evolve over successive generations so that the fitness of the best and mean individual in each generation increases towards the global optimum. A gene is usually considered converged when 95% of the population share the same value, and the population is said to converge when all the genes have converged [36].

So, besides the connections to ideas from the evolution theory, in brief, a genetic algorithm has the following functions:

• Generate a random set of points in feature space (initial population);

• Repeatedly:

– Analyze the fitness of a current set of points - if a satisfactory solution among the current set exists, terminate the process.

– Select a best subset of the current set based on the values of fitness function on them (Se- lection);

– Produce (by recombination) a new set of points (new generation) based on previously se- lected subset and slightly modify new points in randomized manner (mutation), to finally treat produced points as a new current set (population) [36, 38].

42 Generating the initial population is not a problem in practice and usually generated points (individ- uals) are uniformly distributed over the interested area in feature space. If there exists some a priori information about location of an optimal target point in feature space, say, some non-uniform distribu- tion, it can be used to generate the initial distribution. Selecting a best subset of individuals from the current generation can be made in different ways. The most widely used is the method when the nor- malized fitness (divided by the sum of all fitnesses) is calculated for each individual and a predefined number of best ones is selected. Some modifications of this rule proved to be useful. For example, we can randomly choose subset from a population according to the distribution formed by normalized fitness [36, 38].

A very important factor in the use of these algorithms is a definition of a genome. It can be just a set of features – coordinates in feature space, or it can be a features vector encoded to produce a bit sequence – bit string. The latter case is better adapted to the nature of genetic algorithm as it allows to present clear, simple and real life evolution inspired recombination methods that produces an offspring based on a couple of individuals. Using bit string encoding also allows to measure approximately the running time of an algorithm in a more deterministic way.

Using a real valued vector can also be efficient just considering that every real value can be approxi- mated with some discrete values and we come back again to bit strings. The more difficult case appears when we have some features that have a finite number of values. We can encode that finite set with a bit string, but if the volume of the set of values is not a power of two, there will be some bit encodings that do not correspond to any value of feature. Usually this is solved by interpreting those ‘forbidden’ bit sequences as some allowed feature values [36, 38].

On the overall, it is rather difficult to understand why genetic algorithms are so successful at reaching solutions of high fitness in important practical problems. There are different theories behind the algo- rithm’s behavior , such as the building block hypothesis (BBH), which corresponds to the hypothesis that the genetic algorithm performs by implicitly identifying and recombining ”building blocks”, i.e. low order, low defining-length schemata with above average fitness [36].

43 44 Chapter 5

Targets Observation Optimization

5.1 Genetic Algorithm Implementation

5.1.1 Fitness Function

The unknown variables considered for the optimization are the following:

• Longitude of the ascending node Ω;

• Argument of perigee ω;

• Inclination i;

• True anomaly ν0;

• Semi-major axis a;

• Eccentricity e;

• And finally, the time it takes to observe the mission’s targets tf.

The mission requirements were summarized in Section 4.1.1. To fulfill the main objective identified, we need to select the fitness function that will be minimized by the genetic algorithm for the input bound- aries we define. As mentioned in Section 4.1.1, we want to observe the targets with high priority (Venera 8, 9, 10, 13, 14, Vega 1, 2) as soon as possible in a minimum time, while satisfying the orbit constraints (highly elliptical, near circular, low altitude). Additionally, we want to observe the landers Venera 5, 6, 7,

11, 12. To evaluate our main objective we used the following fitness function Fi:

Ni(Ω, ω, i, ν0, tf, a, e) tfi Fi = −α + β (5.1) N TV enus

th Where Ni and tfi are the fitness terms for the i design point, Ni is the total number of observed sites in tfi seconds, α and β are the fitness weight parameters (values between 0 and 1), which translate the relative importance of the fitness terms (maximizing number of covered sites and minimizing the time needed), N is the total number of target sites considered, and finally TV enus is Venus’ period.

45 As was stated before, modeling mission times for Venus is very slow, because the time step has to provide enough time to pass above the area of interest. Due to extremely different scales of number of observations and mission time the genetic algorithm will tend to kill populations with high mission times if the fitness function weights don’t compensate, that is why α and β are necessary. This effect could result in an optimal solution with minimum time window but losing focus of the number of observations, which is why it is necessary to use different combinations of the fitness function.

The total number of observed sites Ni is obtained from the observations computation described in Section 3.2.3, i.e. if the target is inside the swath width, we can assume it was observed. Considering P8 only the sites with priority Ni = n=1 Tn, where Tn = 1 if the target n was intersected and Tn = 0 if the target n was not intersected. Ni is a dependent variable that can be computed given the candidate orbit and the set of sites to select from. The reason why we divide Ni by N and tfi by TV enus is so that we get better scaled fitness function values. If the solution is 8 observed targets and it took 1000000 s to observe them, it is much more functional to consider the small value from our fitness function then the raw sum of the terms. In particular, we chose to divide by Venus’ period since for the simulation considered after one cycle the track is repeated. Furthermore α is proceeded by a negative signal so that the total number of observed sites is maximized, and the positive signal before β guaranteed that the mission time is minimized.

We could have considered other options such as a fraction of the two fitness terms, or a logarithm applied to one of the terms to reduce it’s impact on the fitness function as an alternative to the weight parameters. It would also be interesting to consider a function for which each target priority level would have a correspondent weight and visiting time, and so the orbit would be optimized in a way that the targets with higher priority were visited first, and then the level 2 targets, and finally the level 3 targets (Table 3.1). Such a function would improve the quality of the solution by taking into account the different priority levels of each target, however it would suffer the weakness of being computationally less efficient. We chose the fitness function for its simplicity and consequent computational efficiency.

5.1.2 Implementation Procedure

To implement the genetic algorithm function in Matlab it is fundamental to go over all the options that were made. To begin with, we defined the population size to 100 individuals in each generation, since with a large population size, the algorithm performs a better search and consequently there is a better chance to get a global minimum. The disadvantage of increasing the population size is the computational cost – the script will take more time to run [39]. 100 individuals was a good compromise between convergence efficiency and computational cost.

With the global population parameters defined, it is essential to establish the conditions for the initial population with the creation function. We chose the standard uniform creation function, which generates a random initial population with a uniform distribution [39].

46 The fitness results given by the fitness function need to be scaled to a range that is adapted to the selection function. We used the default fitness scaling function Rank, which scales the raw scores based on the rank r of each individual i.e., an individual with rank r has a scaled score inversely proportional to √ r [39].

Concerning the genetic algorithm operations we used the following functions and values [39]:

• Stochastic uniform for the selection process i.e, for how the algorithm chooses parents for the next generation;

• Ceil(0.05 x PopulationSize) for the elite count in the reproduction options that determines the num- ber of individuals that survive for the next generation;

• 0.8 for the crossover fraction, which specifies the fraction of the next generation that are produced by crossover;

• Finally, the mutation function was left at the default Gaussian function for unconstrained problems.

The crossover is executed by creating a random binary vector and combining the genes for the offspring after selecting the genes 1 from the first parent, and the genes 0 from the second parent. So if we have P arent1 = [a b c d e] and P arent2 = [1 2 3 4 5], and the binary vector is [0 0 0 1 1], the offspring is given by [1 2 3 d e] [39].

Furthermore, the Gaussian mutation function used essentially adds a random number taken from a Gaussian distribution to each element of the parent vector. The standard deviation is controlled by the parameters Scale and Shrink. The Scale parameter determines the standard deviation at the first generation and for the next generations the Shrink parameter is used. For more details [39] can be consulted.

Moreover, the fitness function can be evaluated in serial, parallel, or vectorized manner [39]. With serial, the genetic algorithm calls the fitness function on one individual at a time as it goes through the population, with parallel it calls the fitness function in parallel, and finally, with the vectorized mode, it calls the fitness function on the entire population at once[39] . The latter was the selected user evaluation function selected since Matlab eagerly consumes vectorized functions and operates much faster.

Finally, the algorithm stops if the average relative change in the best fitness function value over stall generations is less than or equal to the function tolerance value. The algorithm runs until the mean relative change in the fitness function value above the minimum stall generations (default value of 50) is less than the function tolerance TolFun (with set value of 1 × 10−4). If |f(xi)–f(xi + 1)| < T olF un, the iterations stop [39]. It is important to specify that setting small tolerances doesn’t guarantee more accurate results. Instead, a solver can continue futile iterations by failing to recognize when it has converged.

47 Figure 5.1: Stopping criteria defined with Tolfun [39].

5.1.3 Algorithm Tests and Validation

We tested the script in order to evaluate the fitness function’s convergence with the conditions in Table 5.1. We plotted the minimum, maximum, and mean fitness function values in each generation for the selected fitness function Fi, and observed the convergence (|f(xi)–f(xi + 1)| < T olF un) after a few more than 50 generations. As expected the Best, Worst values bars decrease progressively until the fittest solution is reached. In order to visualize what is happening at the elite, mutation, and crossover level we plotted the genealogy in figure 5.3. The lines for each generation are color coded as follows:

• Black lines indicate elite individuals;

• Blue lines indicate crossover offspring;

• Red lines indicate mutation offspring.

We also wanted to see how Fi varied with the change in design variables, and for that purpose separate metric studies were performed to investigate the variation of the fitness function with the orbital elements i, Ω, ω and the time variable tf (Figures 5.4 and 5.5). It is very clear from these studies that the objective function is a multi-minima function. It is also interesting to highlight that in figure 5.4, we can observe an expected symmetry in the fitness function dependency of inclination, and no targets are covered for an equatorial orbit. Furthermore, in figure 5.4 we observe that for the near polar inclinations the fitness function value increases. This is related to the fact that not as many targets are covered for the these inclinations for short durations, which is expected since in this case the optimization is taking into account equal weights both fitness terms. Most targets are located near the equator, so for shorter mission times the algorithm should lead to a fittest solution with a lower inclination. As was mentioned for the observations script tests the ground track of an orbit with lower inclination covers a larger area around the equator than near polar orbits. Naturally, with these conditions, we only get 12 out of the 13 targets covered, since the North Pole is not observed. The fittest solution obtained has an inclination of -49.714◦ (Table 5.2). In figure 5.4 we

48 can observe a peak around that inclination value, corresponding as expected to a global minimum peak. The Ω value associated to that peak corresponds to the fittest solution value. In figure 5.5, we can also identify the global minimum peak associated to the fittest solution’s tf and ω values.

Ω bounds [0◦ - 360◦ ] ω bounds [0◦ - 360◦ ] i bounds [-90◦ - 90◦ ] ◦ ◦ ν0 bounds [0 - 360 ] tf bounds [1000000s - 10000000s] a bounds [6300km - 6400km] e bounds [0 - 0.01] α 1 β 1 Targets Venera, Vega landers, North Pole Geometry Nadir

Table 5.1: Fitness function test conditions.

Ω 272.779◦ ω 342.080◦ i -49.714◦ ◦ ν0 126.598 tf 5670319.015s a 6390.834km e 0.001 Targets 12/13

Table 5.2: Fittest solution Fi = −0.653 for fitness function test conditions.

49 Figure 5.2: Minimum, maximum, and mean fitness function values versus generations for fitness function test conditions.

Figure 5.3: Genealogy versus generations for fitness function test conditions.

50 Figure 5.4: Fi versus inclination i and longitude of ascending node Ω for the fitness function test condi- tions.

Figure 5.5: Fi versus time window tf and argument of perigee ω for fitness function test conditions.

51 After observing the fitness function, it is also necessary to further test the algorithm implementation. For that purpose, we used a simple test with a larger and fixed footprint of 5◦ to target just the Venera landers with the nadir geometry for its intuitive nature (Table 5.3). These set conditions provide us the opportunity to quickly test the algorithm for short durations, since it should very quickly find a solution that covers these targets with a low inclination orbit (the same weights were attributed to the fitness terms). These conditions are set just to test the validity of the algorithm implemented, since it does not satisfy many of the mission constraints (polar coverage, e inferior to 0.001). We plotted the best and mean function values in function of the generation to observe the conver- gence, see Figure 5.6. In this case, as expected, due to the large footprint considered, the lower the inclination the faster the Venera landers near the equator are covered, which explains the parametric plot in Figure 5.7 and the fittest solution obtained (Table 5.4). If we alter back to the real nadir geometry with the narrow footprint calculated from the corrected swath, we only get 2 intersected sites due to the short duration considered (Table 5.5). In both cases the fittest solution is rapidly obtained for low inclinations.

Ω bounds [0◦ - 360◦ ] ω bounds [0◦ - 360◦ ] i bounds [-90◦ - 90◦ ] ◦ ◦ ν0 bounds [0 - 360 ] tf bounds [10000s - 100000s] a bounds [6300km - 6400km] e bounds [0 - 0.01] α 1 β 1 Targets Venera landers, North Pole Geometry Nadir altered with fixed 5◦ footprint

Table 5.3: Test conditions for short durations.

Ω 246.470◦ ω 233.501◦ i 9.281◦ ◦ ν0 151.388 tf 10595.253s a 6390.834km e 0.001 Targets 8/11

Table 5.4: Fittest solution Fi = −0.727 for short durations test conditions.

Ω 53.840◦ ω 1.261◦ i -5.875◦ ◦ ν0 172.123 tf 10000.000s a 6300.000km e 0.007 Targets 2/11

Table 5.5: Fittest solution Fi = −0.181 for short durations test conditions corrected for regular nadir geometry.

52 Figure 5.6: Best and mean fitness function values versus generation for short durations test conditions

Figure 5.7: Fi versus orbital parameters i and Ω for short durations test conditions.

53 Figure 5.8: Best and mean fitness function values versus generation for short durations test conditions corrected for regular nadir geometry.

54 The last test performed is closer to the EnVision actual characteristics (bounds and geometry), but we considered equal weights for the fitness function to observe this combination (Table 5.6). The inclination goes towards the lower boundary allowed due to the equal importance attributed to time window and number of targets observed (table 5.7). With these set conditions the fittest solution obtained doesn’t cover all targets, since equal weights were considered for the fitness terms and higher mission times are dismissed by algorithm. As in previous tests, we can clearly observe the peak corresponding to the fittest solution found in Figure 5.10.

Ω bounds [0◦ - 360◦ ] ω bounds [0◦ - 360◦ ] i bounds [87◦ - 90◦ ] ◦ ◦ ν0 bounds [0 - 360 ] tf bounds [1000000s - 10000000s] a bounds [6300km - 6400km] e bounds [0 - 0.001] α 1 β 1 Targets Vega, Venera landers, North Pole Geometry Off-Nadir

Table 5.6: Test conditions for EnVision boundaries and equally weighted fitness terms.

Ω 281.965◦ ω 107.447◦ i 87.984◦ ◦ ν0 240.937 tf 7372545.231s a 6311.406km e 0.001 Targets 11/13

Table 5.7: Fittest solution Fi = −0.495 for EnVision boundaries and equally weighted fitness terms.

55 Figure 5.9: Best and mean fitness function values versus generation for EnVision boundaries and equally weighted fitness terms.

Figure 5.10: Fi versus orbital parameters i and Ω for EnVision boundaries and equally weighted fitness terms.

56 5.2 Mission Overview with the Optimal Operational Orbit

When searching for an optimal orbit solution, we took into account the priority established for the target sites, and only considered Vega 1, 2, Venera 8, 9, 10, 13, 14 and the North Pole observations. In order to ensure that all targets with high priority were covered, the fitness function weight coefficients considered were α = 0.7 and β = 0.3, otherwise the conditions were the same from table 5.6.

Ω 285.789◦ u 351.669◦ i 88.163◦ tf 5571670.038 s a 6310.828 km e 0.000 Targets 8/8

Table 5.8: Fittest solution Fi = −0.700 an optimal orbit solution.

Figure 5.11: Best and mean fitness function values versus generation for an optimal orbit solution.

The script to obtain an optimal solution was set to satisfy all the mission requirements from Section 3.1.2. In a next step, we checked separately with the targets observation script that with this orbit it is possible to cover all 13 targets just after 1000 orbits, so almost 2 times faster than with the provisional parameters considered in the EnVision proposal. Furthermore, the first observation is immediately achieved during the mission’s first orbit. We calculated a few of the main features of this optimal orbit:

57 • Spacecraft altitude: 259.028 km

• Spacecraft velocity: 7.1747 km/s

• Orbital period: 5526.15 s

Even though the spacecraft’s orbital period remained closely the same as in the previous proposal, the interest sites are optimally covered at the start of the first cycle, and it is possible to reduce the total number of cycles needed for the mission and still repeat the coverage of the sites. To visualize the optimal orbit we obtained the 3D orbit (Figure 5.14) and ground track plot (Figure 5.15). After 155 Orbits we can check on the ground track that 4 sites were intersected (Venera 9, 10, 12 and North Pole), as expected from the results of Figure 5.12.

Figure 5.12: Observations test for optimal orbit in the interval 0 s - 1000000 s.

58 Figure 5.13: Observations test for optimal orbit in the interval 5000000 s - 6500000 s.

Figure 5.14: Orbit simulation visualization for 10 days (∼ 155 Orbits) at 100 s step for optimal orbit.

59 Figure 5.15: Ground track plot for 10 days (∼ 155 orbits) at 100 s step for optimal orbit.

60 Finally, we wanted to look into the importance for mission planning of having the targets covered as early as possible, taking into account that the mission profile has been redefined in the statement of interest sent to ESA on September 2015. The launch is now set for March 2028 (with the possibility of an alternative October 2029 launch). Given this new time frame, we simulated the distance variation between Venus and Earth for the first year to check when the distance is inferior to 1 AU (as a reference distance), knowing that globally it varies from around 0.26 AU to 1.74 AU. For this we used the distance event function from NASA’S Spice toolkit software, an ancillary information system that provides different capabilities such as the inclusion of space geometry and event data into mission design, and science data analysis software, among others [40]. We can observe in Figure 5.16 that Venus is closest to Earth at the earlier phases of the mission. This proximity is why it is important to detect the targets and get the necessary data as soon as possible, as can be achieveed with the optimal solution obatined.

Figure 5.16: Plot of the condition in which the distance from Venus to Earth is inferior to 1 AU for the mission’s first year.

On the overall, the optimality of the method was verified in the sense that the heuristic returned an optimal solution. We evaluated the quality of the solutions through numerous algorithm tests. The accuracy and precision of the method is determined by TolFun – we have an established confidence interval for the purported solution. With the optimal solution obtained we observe all targets in an optimal time and as soon possible.

61 62 Chapter 6

Achievements and Future Work

In this thesis, we investigated EnVision’s operational orbit and optimized it for the observation of selected targets. The problem formulation was developed to design an orbit that covers as many of the target sites as possible, while minimizing the time window in which these observations are performed. A genetic algorithm was implemented to evaluate the combinatorial coverage problem.

On the overall, the developed optimization method was a success in finding a fit solution to EnVision’s challenging case study. We also performed separate metrics studies to investigate the dependence of the algorithm’s fitness function and orbital elements and time variable which showed consistency with the main results.

Even though the metric, constraints and priorities considered are specific to the EnVision mission, the main scripts developed in this thesis can be adapted to other orbiter missions to Venus and even to other planets with the right adaptations, as the observations computation and optimization procedure might be similar.

The results of this thesis were first introduced in the EnVison session during the European Planetary Science Congress in September 2015, and are to be included in the mission proposal to ESA. Further research on this subject will be developed as the mission’s science and payload team refine their data. It will be possible to evaluate other objectives such as telecommunications, spotlight mode coverage, and propulsion metrics applied to the natural orbit.

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0Front cover image from [6].

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