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Prof. Francesco Topputo Co-Advisor 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 Diogene A. Dei Tos, MSc Academic Year 2015-2016 To my grandparents: even if fate kept us distant, I know you have me close to your heart, and I have you close to mine. 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 alcuni strumenti numerici che permettano la soluzione del problema ad n-corpi. Per questi 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’ambito dell’estensione di durata della missione. Con questo specifico esempio si è voluto testare la capacità dell’algoritmo di ottenere soluzioni convergenti anche in quella regione altamente instabile rappresentata dai punti Lagrangiani del sistema dinamico. iv 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- body problem (R3BP) dynamics provide many low cost solutions with challenging charac- teristics. However, their inclusion in real solar system models results in high discrepancies. 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. v vi Contents Sommario iv Abstract v 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 . 32 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 vii 4 Validation of the models 51 4.1 Integration scheme . 51 4.1.1 Runge–Kutta–Fehlberg methods and ODE78 . 52 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 . 72 5 Impulsive trajectory optimization 79 5.1 Introduction to a multiple shooting strategy . 79 5.2 The variational equations . 82 5.3 Implementation . 85 5.3.1 Objective Function . 86 5.3.2 Equality Linear Constraints . 87 5.3.3 Inequality Linear Constraints . 87 5.3.4 Equality Non-Linear Constraints . 88 5.3.5 Inequality Non-Linear Constraints . 94 6 Application to the LISA PathFinder mission extension 97 6.1 Presentation of the LPF mission . 97 6.1.1 Saddle Point and LPF mission extension . 100 6.2 Guess Solutions . 104 6.3 Optimisation in RPF n-body problem . 108 6.4 Analysis of the results . 113 6.4.1 Direct solution . 113 6.4.2 Indirect solution . 114 6.4.3 Divergence and change of class . 116 7 Conclusions 119 7.1 Summary of the results . 119 7.2 Future work . 120 Acknowledgements 123 Bibliography 125 viii 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 Representation of the roto-pulsating frame . 28 3.2 Logics for writing the equations of motion in RPF . 44 4.1 GMAT interface . 57 4.2 RPF trajectory for asteroids: Ceres - Fortuna - Hermione - Hektor . 63 4.3 RPF trajectory for asteroids: Amor - Apollo - Einstein - Aten . 66 4.4 Relative error for 1862 Apollo in long integration period . 67 4.5 Relative error for the first 6 asteroids in long integration period . 68 4.6 Relative error for the second 6 asteroids in long integration period . 69 4.7 RPF trajectory for asteroids: Eureka - Damocles - Chaos - Atira . 70 4.8 Errors on different primaries for long integration period . 71 4.9 Trajectory evolution . 75 4.10 Particular of error trend . 76 4.11 Trajectory evolution for J002E3 . 77 5.1 Multiple shooting strategy . 80 5.2 Sparsity pattern of equality linear contraint matrix . 87 5.3 Sparsity pattern of equality linear constraint matrix - particular . 88 5.4 Sparsity pattern of equality non-linear constraint matrix . 94 6.1 LPF technolgy, courtesy of ESA . 98 6.2 xy motion of LPF in RPF . 99 6.3 LPF manifold extension . 99 6.4 SP trajectory . 100 6.5 Intersection of LPF and SP trajectory . 101 6.6 Miss distance depending on departure date . 102 6.7 Time evolution for miss distance - Minimum miss distance . 103 6.8 Time evolution for miss distance - Maximum miss distance . 103 6.9 Synodic trajectory for minimum and maximum miss distance cases . 104 6.10 Shooting from 4-body to n-body: 2 manoeuvre, 5 segments . 105 6.11 Shooting from 4-body to n-body in different cases . 106 6.12 Shooting from 4-body to n-body in different cases . 107 6.13 Periodic solution for the phasing problem . 108 ix 6.14 Shooting from 4-body to n-body with different number of segments . 110 6.15 Shooting from 4-body to n-body: 3 manouevres, 98 segments . 111 6.16 Converged direct solution: 6 manoeuvres, 9 segments . 114 6.17 Converged indirect solution: 2 manoeuvres, 21 segments . 115 6.18 Converged indirect solution: 2 manoeuvres, 19 segments . 116 6.19 Converged indirect solution: 2 manoeuvres, 19 segments . 117 6.20 Converged indirect solution: 2 manoeuvres, 19 segments . 118 x List of Tables 4.1 Butcher’s tableau .
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