SPACE TRAJECTORY OPTIMISATION IN HIGH FIDELITY MODELS Advisor

139 Pages·2017·8.91 MB·Italian

SPACE TRAJECTORY OPTIMISATION IN HIGH
FIDELITY MODELS
Industrial Engineering Faculty
Department of Aerospace Science and Technology
Master of Science in Space Engineering
Advisor:
Prof. Francesco Topputo Graduation Thesis of:
Erind Veruari
Co-advisor:
837650
Ph.D. Diogene A. Dei Tos
Academic Year 2015-2016

Sommario
Il campo della progettazione ed ottimizzazione di traiettorie spaziali procede di pari
passo con l’evoluzione del mondo scientifico e tecnologico. Le richieste in questo ambito
prevedono trasferimenti che abbiano un alto livello di accuratezza e, al contempo, un
basso costo in termini di propellente a bordo. Un esempio esplicativo è rappresentato dal
numero crescente di satelliti a bassissima autorità di controllo in orbita (cubesats), il cui
studio per missioni interplanetarie si sta intensificando.
Tra le varie strategie di progettazione di missione, quelle che sfruttano la dinamica
del problema dei tre corpi offrono una serie di soluzioni a basso costo con caratteristiche
stimolanti. Tuttavia, il loro utilizzo in modelli reali del sistema solare presenta grandi
discrepanze. Il seguente lavoro prende spunto da questa divergenza, muovendosi in due
direzioni, una teorica ed una pratica.
Quella teorica prevede la riscrittura delle equazioni del moto del problema a tre corpi
inserendo le perturbazioni dovute alle azioni gravitazionali degli altri pianeti, così come
l’effetto della pressione della radiazione solare. Le equazioni, ottenute a partire dal for-
malismo Lagrangiano, vengono poi ruotate in un sistema di riferimento roto-pulsante, nel
quale si mantengono le caratteristiche delle orbite progettate in un modello a tre corpi.
Esso permette, inoltre, un facile confronto tra le orbite rifinite e quelle di partenza.
Dal punto di vista pratico, questo lavoro si occupa di creare alcuni algoritmi che siano
in grado di propagare le traiettorie e poi ottimizzarle. Innanzitutto, vengono sviluppati
alcunistrumentinumericichepermettanolasoluzionedelproblemaadn-corpi. Perquesti
viene presentata una strategia di validazione tramite software open-source, i quali sono
stati utilizzati nell’ambito della progettazione di missioni già in volo. In secondo luogo,
si sfruttano orbite progettate in modelli meno accurati come soluzioni di partenza per la
soluzione di un problema non lineare vincolato al contorno, che rappresenta il metodo di
ottimizzazione della soluzione.
Questi strumenti vengono, infine, applicati al caso del satellite LISA Path Finder
(LPF)nell’ambitodell’estensionediduratadellamissione. Conquestospecificoesempiosi
èvolutotestarelacapacitàdell’algoritmodiotteneresoluzioniconvergentiancheinquella
regione altamente instabile rappresentata dai punti Lagrangiani del sistema dinamico.
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Abstract
The design and the optimization of space trajectories goes side by side with the evolution
of the scientific and technological world. In this field, there are requirements of transfers
with a high-level of accuracy and, at the same time, of low costs in terms of on-board
propellant. The high number of ultra-low thrust orbiting satellites (cubesats), whose
study for interplanetary mission is growing, gives an instructive example of this situation.
Among the different mission design strategies, the ones that exploit restricted three-
bodyproblem(R3BP)dynamicsprovidemanylowcostsolutionswithchallengingcharac-
teristics. However,theirinclusioninrealsolarsystemmodelsresultsinhighdiscrepancies.
This work starts from this incongruence, and takes a twin-track approach, analyzing it
from a theoretical and a practical point of view.
From a theoretical point of view,taking into consideration the gravitational effects of
a set of n-bodies and adding the perturbing effect of the solar radiation pressure. The
equations have been computed exploiting the Lagrangian formalism. Afterwards, they
have been rotated in a roto-pulsating reference frame (RPF), where the features of the
orbits designed in R3BP are preserved. Moreover, RPF allows to easily compare guess
and refined solutions.
This work aims at creating some algorithms, which can be used for trajectory propa-
gation and optimization in practical terms. First of all, numerical tools which solve the
n-body problem are established. Then, they are validated using open-source softwares,
which have been adopted in the design process of already flown missions. In the sec-
ond place, formerly designed orbits in less accurate models are considered as guesses for
the solution of the non-linear constrained boundary value problem, which represents the
optimization strategy for the trajectory.
Finally, these tools are applied to the case of the mission extension for the LISA Path
Finder (LPF) satellite. Using this practical example, the converging properties of the
algorithm have been tested, specifically in the highly unstable region that is represented
by the Lagrangian points of the dynamic system.
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Contents
1 Introduction 1
1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Research question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Models in inertial frames 7
2.1 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Inertial reference frames . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 J2000 and EME2000 . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 n-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 The general n-body problem . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 The restricted n-body problem and the Lagrangian formulation . 12
2.2.3 Planeto-centred equations of motion . . . . . . . . . . . . . . . . 15
2.3 Gravitational perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Non-gravitational perturbations . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 SRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Models in rotating frames 27
3.1 Definition of RPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Rotation of the equations into RPF . . . . . . . . . . . . . . . . . . . . . 33
3.3 Alternative derivation of the equation of motion . . . . . . . . . . . . . . 35
3.4 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 Gravitational effects . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.2 Non-gravitational effects . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Logics of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.1 The Restricted 2 Body Problem . . . . . . . . . . . . . . . . . . . 44
3.6.2 Restricted 3 Body Problem . . . . . . . . . . . . . . . . . . . . . 46
4 Validation of the models 51
4.1 Integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Runge–Kutta–Fehlberg methods and ODE78 . . . . . . . . . . . . 52
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4.2 JPL’s SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 GMAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Validation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.1 Validation with SPICE . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.2 Validation with GMAT . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Impulsive trajectory optimization 81
5.1 Introduction to a multiple shooting strategy . . . . . . . . . . . . . . . . 81
5.2 The variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.2 Equality Linear Constraints . . . . . . . . . . . . . . . . . . . . . 89
5.3.3 Inequality Linear Constraints . . . . . . . . . . . . . . . . . . . . 89
5.3.4 Equality Non-Linear Constraints . . . . . . . . . . . . . . . . . . 91
5.3.5 Inequality Non-Linear Constraints . . . . . . . . . . . . . . . . . . 96
6 Application to the LISA PathFinder mission extension 99
6.1 Presentation of the LPF mission . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 Saddle Point and LPF’s EoL disposal . . . . . . . . . . . . . . . . 102
6.2 Guess Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Optimisation in RPF n-body problem . . . . . . . . . . . . . . . . . . . . 111
6.4 Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.1 Direct solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.2 Indirect solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4.3 Divergence and change of class . . . . . . . . . . . . . . . . . . . . 118
7 Conclusions 123
7.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography 127
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List of Figures
2.1 Representation of J2000 and EME2000 reference frame . . . . . . . . . . 9
2.2 Geometrical representation of forces on general n-body model . . . . . . 11
2.3 Change of origin: from SSB to Planet . . . . . . . . . . . . . . . . . . . . 16
2.4 Change of origin: from SSB to Planet . . . . . . . . . . . . . . . . . . . . 18
3.1 Logics for writing the equations of motion in RPF . . . . . . . . . . . . . 44
4.1 GMAT interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 RPF trajectory for astroids . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 RPF trajectory for astroids . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Trend of the relative error for 1862 Apollo in long integration period . . . 69
4.5 Trend of the relative error for the first 6 asteroids in long integration period 70
4.6 Trend of the relative error for the second 6 asteroids in long integration
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.7 RPF trajectory for astroids . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8 Errors on different primaries for long integration period . . . . . . . . . . 75
4.9 Trajectory evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.10 Partiular of error trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.11 Trajectory evolution for J002E3 . . . . . . . . . . . . . . . . . . . . . . . 78
5.1 Multiple shooting strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Sparsity pattern of equality linear contraint matrix . . . . . . . . . . . . 89
5.3 Sparsity pattern of equality linear contraint matrix - particular . . . . . . 90
5.4 Sparsity pattern of equality non-linear constraint matrix . . . . . . . . . 97
6.1 LPF technolgy, courtesy of ESA . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 xy motion of LPF in RPF . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 LPF manifold extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 SP trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5 Intersection of LPF and SP trajectory . . . . . . . . . . . . . . . . . . . 104
6.6 Miss distance depending on departure date . . . . . . . . . . . . . . . . . 105
6.7 Synodic trajectory for minimum and maximum miss distance cases . . . 106
6.8 Time evolution for miss distance . . . . . . . . . . . . . . . . . . . . . . . 106
6.9 Shooting from 4-body to n-body: 2 manoeuvre, 5 segments . . . . . . . . 108
6.10 Shooting from 4-body to n-body in different cases . . . . . . . . . . . . . 109
6.11 Shooting from 4-body to n-body in different cases . . . . . . . . . . . . . 109
6.12 Periodic solution for the phasing problem . . . . . . . . . . . . . . . . . . 110
6.13 Shooting from 4-body to n-body with different number of segments . . . 113
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6.14 Shooting from 4-body to n-body: 3 manouevres, 98 segments . . . . . . . 113
6.15 Converged direct solution: 6 manoeuvres, 9 segments . . . . . . . . . . . 116
6.16 Converged indirect solution: 2 manoeuvres, 21 segments . . . . . . . . . 117
6.17 Converged indirect solution: 2 manoeuvres, 19 segments . . . . . . . . . 119
6.18 Converged indirect solution: 2 manoeuvres, 19 segments . . . . . . . . . 120
6.19 Converged indirect solution: 2 manoeuvres, 19 segments . . . . . . . . . 121
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List of Tables
4.1 Butcher’s tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Name and properties of chosen asteroids . . . . . . . . . . . . . . . . . . 62
4.3 Validation periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Name and properties of chosen dwarf planet . . . . . . . . . . . . . . . . 62
4.5 Attractros and their gravitational parameters . . . . . . . . . . . . . . . 64
4.6 Absolute and relative maximum positions errors . . . . . . . . . . . . . . 67
4.7 Absolute and relative maximum velocity errors . . . . . . . . . . . . . . . 68
4.8 Absolute errors for the position in Earth RPF . . . . . . . . . . . . . . . 73
4.9 Postion absolute errors in Jupiter RPF . . . . . . . . . . . . . . . . . . . 74
4.10 Mininum max. absolute errors related to the primary . . . . . . . . . . . 77
4.11 Physical and integration properties . . . . . . . . . . . . . . . . . . . . . 77
4.12 Initial conditions for the integrators . . . . . . . . . . . . . . . . . . . . . 78
4.13 Max relative and absolute errors after 100 days integration . . . . . . . . 78
4.14 Initial conditions for the integrators . . . . . . . . . . . . . . . . . . . . . 79
4.15 Max relative and absolute errors after 100 days integration . . . . . . . . 79
4.16 Comparison the integrators . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 Percentage of trajectories with miss distance smaller than the tolerance . 104
6.2 Minimum and maximum miss distance . . . . . . . . . . . . . . . . . . . 105
6.3 Values for the main parameters . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Direct trajectory properties . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Comparison between different refined solutions . . . . . . . . . . . . . . . 118
6.6 Indirect trajectory properties . . . . . . . . . . . . . . . . . . . . . . . . 118
6.7 Properties of non-converging solution . . . . . . . . . . . . . . . . . . . . 120
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Chapter 1
Introduction
1.1 Context
The evolution of space exploration has highly changed throughout the centuries. It
began as a fascinating study of the stars and their motion. It then developed as a
philosophicalquestiondethroningthehumankindasthecentreoftheuniverse. Nowadays,
it goes on in the search of new frontiers to reach, and new forms of life to unveil.
Science has played the main role in these discoveries. Astronomy led the knowledge
of the surrounding universe to enter in the history, whereas the orbital mechanics gave
a physical meaning to the motion of celestial bodies. Astrodynamics extended those
notions to man-made objects, whose presence has become a natural part even of daily
life, through communication satellites and navigation systems.
Space trajectory design is one of the branches of the astrodynamics that has mostly
evolved in the last twenty years, due to an outstanding interest in space mission and their
implication to scientific progress. The design of a trajectory considers the gravitational
effects of celestial bodies on orbiting vehicles. The aim is to achieve a transfer whose
requirements are met. However, new scientific frontiers need more demanding ones, and
this translates into the development of higher fidelity models.
The classic approach to the trajectory design has consolidate bases. The formulation
of the problem dates back to Sir Isaac Newton and his universal gravitational law [33].
The approximated formulation, called Two-Body Problem (2BP), has an analytical so-
lution. It can be efficiently used for the trajectory design, where only one main body is
considered and the artificial object describes conics around it. In the case of a transfer
between different primaries the conics can be patched together. The simplification of
the model permits the generation of meaningful solutions. Nevertheless, these solutions
must be corrected because of the gravitational effects of other influencing celestial bodies.
The perturbation of these attractors are accounted in more complex models, where the
solutions are refined.
New possibilities emerge as the trajectories are directly designed in more refined mod-
els, like the Restricted Three Body Problem (R3BP). It takes advantage of the idea of
co-rotating bodies with respect to a common barycentre. Then it considers the spacecraft
as a point mass in relative motion. In this way it is easier to understand the mutual in-
teraction between celestial bodies and the probe. Mathematically speaking the problem
has not an analytical solution, but it is possible to find one integral of motion. In this
1