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Flower Constellation Optimization and Implementation

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FLOWER CONSTELLATION OPTIMIZATION AND IMPLEMENTATION A Dissertation by CHRISTIAN BRUCCOLERI Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY December 2007 Major Subject: Aerospace Engineering FLOWER CONSTELLATION OPTIMIZATION AND IMPLEMENTATION A Dissertation by CHRISTIAN BRUCCOLERI Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved by: Chair of Committee, Daniele Mortari Committee Members, John L. Junkins Thomas C. Pollock J. Maurice Rojas Head of Department, Helen Reed December 2007 Major Subject: Aerospace Engineering iii ABSTRACT Flower Constellation Optimization and Implementation. (December 2007) Christian Bruccoleri, M.S., Universit´a di Roma - La Sapienza Chair of Advisory Committee: Dr. Daniele Mortari Satellite constellations provide the infrastructure to implement some of the most im- portant global services of our times both in civilian and military applications, ranging from telecommunications to global positioning, and to observation systems. Flower Constellations constitute a set of satellite constellations characterized by periodic dynamics. They have been introduced while trying to augment the existing design methodologies for satellite constellations. The dynamics of a Flower Constellation identify a set of implicit rotating reference frames on which the satellites follow the same closed-loop relative trajectory. In particular, when one of these rotating refer- ence frames is “Planet Centered, Planet Fixed”, then all the orbits become compatible (or resonant) with the planet; consequently, the projection of the relative path on the planet results in a repeating ground track. The satellite constellations design methodology currently most utilized is the Walker Delta Pattern or, more generally, Walker Constellations. The set of orbital planes and initial spacecraft positions are represented by a set of only three integers and two real parameters rather than by all the orbital elements; Flower Constella- tions provide a more general framework in which most of the former restrictions are removed, by allowing the use of resonant elliptical orbits. Flower Constellations can represent hundreds of spacecraft with a set of 6 integers and 5 real parameters only and existing constellations can be easily reproduced. How to design a Flower Constellation to satisfy specific mission requirements iv is an important problem for promoting the acceptance of this novel concept by the space community. Therefore one of the main goals of this work is that of proposing design techniques that can be applied to satisfy practical mission requirements. The results obtained by applying Global optimization techniques, such as Genetic Algorithms, to some relevant navigation and Earth observation space-based systems show that the Flower Constellations not only are as effective as Walker Constellations, but can also be applied to non-traditional constellation problem domains, such as regional coverage and reconnaissance. v ”Fatti non foste a viver come bruti ma per seguire virtute e canoscenza” Dante Alighieri, Divina Commedia, Inferno canto XXVI, 116-120 to Francesco and Orietta vi ACKNOWLEDGMENTS Accomplishing anything non trivial in life is a long, and often hard, journey. One may only thank his good luck if such a journey is blessed by the help and advice of friends and scholars who ventured upon similar paths before. In the years that I spent trying to achieve this important goal in my life, many people sustained this effort in different ways. I believe that gratitude to my advisor and friend, Dr. Daniele Mortari, can only be expressed with a simple ”Thanks!”, for I know that he likes simple things and short speeches, whereas explaining all the ways in which he, together with Andreea and Anna, helped me would take a chapter of its own. Similarly, I know that Dr. John L. Junkins asks no more satisfaction from his students than seeing them going ahead on their own and accomplishing things in life, both professionally and personally. I must therefore thank him for his precious advice, but above all for his unshaken optimism and foresight. To Dr. Thomas C. Pollock I wish to express my gratitude for his technical advice and for the friendly and comfortable way in which he always approaches his students. I would have had to stop my journey much earlier, however, if it were not for the providential help of StarVision Technologies, Inc. which sponsored my Ph.D. for more than three years, in spite of the difficulties that this proposition entailed. I express my deepest gratitude to Mr. Michael Jacox and Dr. James A. Ochoa for their fundamental support and advice during times in which they were heavily involved in the making of a successful private company. Similarly, I wish to thank all the StarVision Technologies employees for their friendliness. I wish to thank Dr. Walter C. Haisler and Karen Knabe for their responsiveness to the many administrative issues that students face during their career; they were always ready to help and point me in the right direction. vii Many others have accompanied me on various parts of this journey; I wish I could mention them all, but it would not only grow tedious to the reader; it would also make too many words of thanks fade in meaning and intensity. So, I will simply mention some of my friends and colleagues who made a difference in my daily life like Randy and Lori South, Dr. Giovanni Giardini, Dr. Anup Katake, Dr. Puneet Singla, Dr. Matthew P. Wilkins, Troy Henderson, Marco De Santis, Alberto Clocchiatti, and this list could go on for much longer. Finally, I shall remind myself, and the unlikely readers, that every accomplishment is just a new beginning and the climb starts anew: ever towards more misty peaks. viii TABLE OF CONTENTS CHAPTER Page I INTRODUCTION .......................... 1 II ORBITAL MECHANICS BACKGROUND ............ 9 A.OrbitalMechanicsOverview................. 9 B.AvoidSolvingKepler’sEquation............... 17 C.OrbitalPerturbations..................... 27 III FLOWER CONSTELLATION BACKGROUND ......... 31 A.IntroductiontoConstellationDesign............ 31 B.ReferenceFrames....................... 36 C.FlowerConstellationTheory................. 37 D.CompatibleOrbits...................... 44 IV OPTIMIZATION TECHNIQUES ................. 47 A.GeneticAlgorithms...................... 49 B.StrengthandIssuesofGeneticAlgorithms......... 58 V FCVAT AND FCTOOLBOX DESIGN AND IMPLEMENTATION 65 A. Flower Constellation Visualization and Analysis Tool . 65 B.TheFlowerConstellationMatlabToolbox......... 76 VI APPLICATIONS .......................... 79 A.GeometryofEarthObservation............... 79 B. Mission Design for Satellite Constellations . ...... 82 C.GlobalCoverage........................ 89 D.RegionalCoverage...................... 106 E.ReconnaissanceMissions................... 117 VII CONCLUSIONS ........................... 142 REFERENCES ................................... 144 APPENDIX A ................................... 152 ix CHAPTER Page VITA ........................................ 161 x LIST OF TABLES TABLE Page I Comparison of capabilities of Flower and Walker Constellation design methodologies. .......................... 44 II Values of the main GA parameters used for global coverage. ..... 94 III Prograde FC configurations for the global coverage problem. A value of ’*’ in the ω column means that the orbit is circular. ..... 94 IV Retrograde FC configurations for the global coverage problem. .... 95 V Original parameters of “Sistema Quadrifoglio” simulated with FCs. 98 VI FC for continuous persistent coverage of the northern hemisphere with 4 satellites. ............................. 99 VII Parameters of existing constellations for global navigation. ...... 103 VIII Best FC found for GDOP optimization with 24 satellites. ....... 106 IX Broglio-type Flower Constellation for regional coverage parameters. 109 X Broglio-type Flower Constellation. ................... 109 XI Parameters of Molniya-type Flower Constellation for regional coverage. 111 XII Molniya-type FC: satellite orbital elements. .............. 112 XIII Landsat multi-spectral bands. ...................... 122 XIV Sites used by the J2Exploitation program. ............... 136 xi LIST OF FIGURES FIGURE Page 1 Hyperbolic and parabolic orbit geometry. ............... 14 2 Elliptical orbit geometry. The satellite S is orbiting a planet on an elliptical orbit O of semi major axis a and eccentricity e.The planet is occupying one of the foci, F1 and the auxiliary circle A of radius a is used to geometrically define the Eccentric Anomaly, E. The position of the satellite at any instant is defined by the radius vector r which is a function of the True Anomaly, ϕ. ..... 15 3 20-point orbit propagation using different angle steps. ......... 21 4 Relative percentage variation of ΔM between two time steps. .... 24 5 Error in eccentric anomaly, E at last step, as function of e and N. .. 25 6 Error in mean anomaly, M at last step, as function of e and N. ... 26 7 Major cost drivers in mission design. .................. 32 8 Relation between ECI {ˆi, ˆj,kˆ} frame and ECEF {eˆ1, eˆ2, eˆ3};the angle between ˆi ande ˆ1, δ(t), is the GAST angle and it is a function of time since the ECEF frame spins with the Earth. .......... 38 9 Relations between the ECI frame {ˆi, ˆj,kˆ} and the Orbital frame O = {ˆie,ˆip,ˆih}; The orbit is rotated from the reference
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