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Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2005 Maneuver Estimation Model for Relative Orbit Determination Tara R. Storch Follow this and additional works at: https://scholar.afit.edu/etd Part of the Astrodynamics Commons Recommended Citation Storch, Tara R., "Maneuver Estimation Model for Relative Orbit Determination" (2005). Theses and Dissertations. 3698. https://scholar.afit.edu/etd/3698 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Maneuver Estimation Model for Relative Orbit Determination THESIS Tara R. Storch, Capt, USAF AFIT/GA/ENY/05-M011 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. AFIT/GA/ENY/05-M011 Maneuver Estimation Model for Relative Orbit Determination THESIS Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science in Astronautical Engineering Tara R. Storch, B.S. Capt, USAF March, 2005 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. AFIT/GA/ENY/05-M011 Maneuver Estimation Model for Relative Orbit Determination Tara R. Storch, B.S. Capt, USAF Approved: /signed/ 21 Mar 2005 Dr. William E. Wiesel date Advisor /signed/ 21 Mar 2005 Lt Col Kerry D. Hicks date Committee Member /signed/ 21 Mar 2005 Lt Col Nathan A. Titus date Committee Member AFIT/GA/ENY/05-M011 Abstract While the use of relative orbit determination has made in reducing minimizing the difficulties inherent in tracking geostationary satellites that are in close proximity, the problem is often compounded by stationkeeping operations or unexpected maneu- vers. If a maneuver occurs, observations will no longer fit predicted data, increasing the risk of misidentification and cross-tagging. The goal of this research was to develop a model that will estimate the mag- nitude, direction, and time of a suspected maneuver performed by a collocated geo- stationary satellite. Relative motion was modelled using Hill’s equations, and least squares estimation was employed to create both a linear non-maneuver model and non-linear maneuver model. Two sets of data for an actual satellite collocation were obtained from the Air Force Maui Optical and Supercomputing (AMOS) site, con- sisting of differential right ascension and declination. Studies conducted with these observations, along with simulation studies, indicate that it is indeed possible to per- form maneuver estimation. It was found, however, that the amount of data required for successful convergence is much greater than that typically obtained for tracking purposes. iv Acknowledgements All praise and glory to my God and King. I thank You, Lord, for all the blessings You have given me. Thank You for this opportunity, for the wonderful classmates, instructors, and friends to help me along, for bringing me through this program, and most of all for using this time at AFIT to teach me about so much more than astro. I would like to thank my advisor, Dr. William Wiesel, for his willingness to impart knowledge and for his enduring patience in the process. I am truly thankful to have had the opportunity to work with him. Thanks to my sponsor, Dr. Chris Sabol, for not only his time and efforts, but also for his intensity and high expectations. I would also like to thank my church family for giving me a home away from home, for their prayers and for challenging me to grow in my faith. And as always, I am indebted to my wonderful family. Thanks to my sister and her family, and my brother for all the laughter, prayers, and support. And finally, special thanks goes to my parents for their unyielding love, encouragement, and faithful prayers during this season of my life. Tara R. Storch v Table of Contents Page Abstract..................................... iv Acknowledgements ............................... v ListofFigures ................................. viii ListofTables.................................. ix ListofSymbols................................. x ListofAbbreviations.............................. xi I. Introduction ............................. 1-1 1.1 Background......................... 1-1 1.2 ProblemStatement..................... 1-2 1.3 MethodofSolution..................... 1-2 II. LiteratureReview .......................... 2-1 2.1 OpticalObservations. 2-1 2.1.1 Fundamentals - Celestial Sphere Geometry . 2-1 2.1.2 The Raven Telescope . 2-2 2.1.3 Topocentric to Geocentric Conversion . 2-3 2.2 LeastSquaresEstimation . 2-4 2.2.1 The Principle of Maximum Likelihood . 2-5 2.2.2 Linearized Dynamics and the State Transition Ma- trix......................... 2-5 2.2.3 Linear Least Squares . 2-7 2.2.4 Nonlinear Least Squares . 2-9 2.3 RelativeMotion....................... 2-11 2.3.1 Equations of Motion . 2-11 2.3.2 Differential Orbital Element Effects . 2-11 2.4 Relative Motion Applications in GEO . 2-12 2.4.1 GEOClusters ................... 2-12 2.4.2 Close Approaches . 2-14 2.5 ChapterOutline ...................... 2-15 vi Page III. Methodology ............................. 3-1 3.1 Coordinate Transformation . 3-1 3.2 The Observation Function and Its Linearization . 3-2 3.3 LinearSystemDynamics. 3-7 3.4 The Maneuver Model: Non-linear System Dynamics . 3-9 3.5 Algorithm.......................... 3-12 3.5.1 DataFiles ..................... 3-12 3.5.2 Initializing the State Vector . 3-13 3.5.3 Reference Orbit . 3-14 3.5.4 Least Squares Algorithm . 3-14 IV. SimulationsandRealData ..................... 4-1 4.1 Relative Orbit Determination Experiment . 4-1 4.2 SimulationStudy...................... 4-2 4.2.1 Non-maneuver Model Initial Simulation . 4-2 4.2.2 Maneuver Model Initial Simulation . 4-3 4.3 Raven Data – 2003 . 4-10 4.3.1 Data Set 1: 29 – 31 July . 4-11 4.3.2 Data Set 2: 30 July - 1 August . 4-17 4.4 Raven Data - 2004 . 4-19 4.4.1 DataSet1:3–4June .............. 4-20 4.4.2 DataSet2:4–5June .............. 4-24 V. Conclusions.............................. 5-1 5.1 Summary .......................... 5-1 5.2 Conclusions......................... 5-1 5.2.1 Observability . 5-1 5.2.2 Higher Order Error Sources . 5-2 5.2.3 Sequential Maneuvers . 5-2 5.3 FutureWork ........................ 5-2 Bibliography .................................. 4 vii List of Figures Figure Page 2.1. GeometryoftheCelestialSphere. 2-1 2.2. RavenImage............................ 2-3 3.1. GeocentricInertialFrame ..................... 3-2 3.2. BodyFixedFrame......................... 3-3 3.3. ManeuverDynamics........................ 3-10 4.1. COWPOKE and TLE predictions for observations on 30,31 July 4-1 4.2. Fit to 30th and 31st, Predict to 1st ................ 4-2 4.3. Estimated and Simulated Observations . 4-4 4.4. Non-maneuver Fit to 29 31July ................ 4-11 − 4.5. Non-maneuver Fit to 29 31July ................ 4-12 − 4.6. Maneuver Fit to 29 31 July, tm =79477............ 4-14 − 4.7. RA vs Dec for 29 31 July, tm =79477 ............. 4-15 − 4.8. Maneuver Fit to 29 31 July, tm = 125340 . 4-15 − 4.9. RA vs Dec for 29 31 July, tm = 125340 . 4-16 − 4.10. Maneuver Fit to 29 31 July, tm = 168380 . 4-17 − 4.11. RA vs Dec for 29 31 July, tm = 168380 . 4-18 − 4.12. Non-maneuver Fit to 31 July - 1 August . 4-19 4.13. Non-maneuverFitto3-4June . 4-22 4.14. Non-maneuver RA vs Dec for 3-4 June . 4-22 4.15. Maneuver Fit to 3 4 June, tm =80008............. 4-23 − 4.16. RA vs Dec for 3 4 June, tm =80008.............. 4-23 − 4.17. RA vs Dec Solution for 3 4 June Propagated to 5 June . 4-24 − 4.18. Maneuver Fit to 4 5 June, tm =87311............. 4-25 − 4.19. RA vs Dec for 4 5 June, tm =87311.............. 4-26 − viii List of Tables Table Page 4.1. Data Arc Time and Length for 29 July - 1 August . 4-5 4.2. Simulated Data Arc Time and Length . 4-6 4.3. Convergence Times and Values for Simulated Data Arcs . 4-7 4.4. State Vector Solution for Convergence Times . 4-8 4.5. State Vector Solution for Convergence Times . 4-9 4.6. State Vector Solution for No Maneuver Data Set in Maneuver Model................................ 4-10 4.7. State Vector Solution for Non-maneuver Fit to 29 31 July . 4-11 − 4.8. Convergence Times and Values for 29-31 July . 4-13 4.9. State Vector Solution for Maneuver Fit to 29 31 July . 4-13 − 4.10. State Vector Solution for Non-maneuver Fit to 30 July - 1 August 4-18 4.11. Data Arc Time and Length for 3 June . 4-20 4.12. State Vector Solution for Maneuver Fit to 3-4 June . 4-21 4.13. State Vector Solution for Maneuver Fit to 4-5 June . 4-25 ix List of Symbols Symbol Page α rightascension ........................ 2-4 d declination .......................... 2-4 a semimajoraxis ........................ 2-12 i inclination........................... 2-12 Ω right ascension of the ascending node . 2-12 M meananomaly ........................ 2-12 ω argumentofperigee ..................... 2-12 u argumentoflatitude ..................... 3-1 rrms rootmeansquareoftheresiduals . 4-3 x List of Abbreviations Abbreviation Page GEO GeosynchronousOrbit .................... 1-1 AMOS AirForceMauiOpticalandSupercomputingsite . 1-2 CCD Charge-Coupled Device . 2-2 COWPOKE Cluster Orbits With Perturbations Of Keplerian Elements 2-11 TLE Two-LineElementset .................... 3-14 xi Maneuver Estimation Model for Relative Orbit Determination I. Introduction 1.1 Background The concept of the geosynchronous orbit, and it’s more specific counterpart, the geostationary orbit, has been around for more than a century.
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