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University of Cincinnati UNIVERSITY OF CINCINNATI _____________ , 20 _____ I,______________________________________________, hereby submit this as part of the requirements for the degree of: ________________________________________________ in: ________________________________________________ It is entitled: ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Approved by: ________________________ ________________________ ________________________ ________________________ ________________________ Use of Near-Frozen Orbits for Satellite Formation Flying A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE (M.S.) in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering 2001 by Heidi L. Davidz B.S., The Ohio State University, 1997 Committee Chair: Dr. Trevor Williams Abstract There is growing interest in flying coordinated clusters of small spacecraft to perform missions once accomplished by single, larger spacecraft. Using these satellite clusters reduces cost, improves survivability, and increases the flexibility of the mission. One challenge in implementing these satellite clusters is maintaining the formation as it experiences orbital perturbations, most notably due to the non-spherical Earth. Certain aspects of the orbital geometry can remain virtually fixed over extended periods of time due to a natural phenomenon called a frozen orbit. Specifically, the elements of the orbital geometry that can remain fixed are the argument of perigee (a measure of where the orbit is closest to the Earth) and the eccentricity (a measure of how circular or elliptical the orbit is). For satellite formations, using this frozen orbit phenomenon results in considerable propellant savings. In this study, a discussion of current literature on this topic is given. Some examples of formations at near-frozen conditions are shown. There is also a discussion of the propellant impact of near-frozen conditions. If two orbits meet a certain set of initial conditions, the orbits will naturally stay in the vicinity of each other. These orbits are sometimes called “Hill’s orbits”. An algorithm is developed here that determines if the Hill’s orbit conditions will be met given the initial differences in eccentricity and argument of perigee for two satellites. Acknowledgments I would like to thank Professor Trevor Williams for his guidance in conducting this research. Also, I would like to thank my husband Michael Cluff for his patience and support as I completed this work. Table of Contents 1 Introduction ............................................................................................................. 9 1.1 Introduction to Satellite Formation Flight.........................................................9 1.2 Propulsion for Formation Flight......................................................................11 1.3 Proposed Work................................................................................................ 13 2 Satellite Formation Flight Dynamics .................................................................... 13 2.1 Hill’s Orbits..................................................................................................... 13 2.2 Perturbations.................................................................................................... 17 3 Frozen Orbits......................................................................................................... 23 3.1 Definition of Frozen Orbit............................................................................... 23 3.2 Equation Development.................................................................................... 27 3.3 Experience with Frozen Orbits........................................................................ 28 4 Near-Frozen Orbit Test Cases............................................................................... 32 4.1 Introduction ..................................................................................................... 32 4.2 Development of Parameter Values.................................................................. 33 4.3 Development of Difference Equations............................................................ 34 4.4 Simulation Results...........................................................................................36 4.5 Case 4 Motion ................................................................................................. 49 5 Application of Cases to the Clohessy-Wiltshire/Hill’s Equations ........................ 50 5.1 Introduction ..................................................................................................... 50 5.2 Review of the CW/Hill’s Equations................................................................ 50 5.3 Coordinate Systems......................................................................................... 51 5.4 “Algorithm 6” from Vallado and McClain ..................................................... 51 Page 1 5.5 Application of “Algorithm 6” to Find PQW Radius and Velocity Vectors.... 52 5.6 Find IJK Radius and Velocity Vectors............................................................ 54 5.7 Find RSW Radius and Velocity Vectors......................................................... 54 5.8 Identify the CW/Hill’s Initial Conditions .......................................................56 5.9 Numerical Example......................................................................................... 57 6 Propellant Calculations......................................................................................... 64 6.1 Derivation of the Tangential Thrust ∆V Equation .......................................... 65 6.2 Propellant Calculation Results ........................................................................70 7 Summary ............................................................................................................... 78 8 Suggestions for Future Work ................................................................................ 79 9 Bibliography.......................................................................................................... 80 10 Appendix ............................................................................................................... 83 10.1 Matlab Code to Generate Frozen Orbit Circulations .................................... 83 10.2 Matlab Code for Running Cases, Pseudo State Space Program ................... 84 10.3 Matlab Code for Running Cases, ODE Program .......................................... 85 10.4 Matlab Code to Generate Propellant Calculation Graphs ............................. 86 Page 2 Table of Figures Figure 1: Zonal Harmonics ............................................................................................... 17 Figure 2: Sectorial Harmonics........................................................................................... 17 Figure 3: Tesseral Harmonics ........................................................................................... 17 Figure 4: Circulations Around the Frozen Orbit Conditions............................................ 26 Figure 5: Case 1 Eccentricity ............................................................................................ 37 Figure 6: Case 2 Eccentricity ............................................................................................ 37 Figure 7: Case 3 Eccentricity ............................................................................................ 38 Figure 8: Case 4 Eccentricity ............................................................................................ 38 Figure 9: Case 1 Argument of Perigee.............................................................................. 39 Figure 10: Case 2 Argument of Perigee............................................................................ 40 Figure 11: Case 3 Argument of Perigee............................................................................ 40 Figure 12: Case 4 Argument of Perigee............................................................................ 41 Figure 13: Case 1 Eccentricity vs. Argument of Perigee .................................................. 42 Figure 14: Case 2 Eccentricity vs. Argument of Perigee .................................................. 42 Figure 15: Case 3 Eccentricity vs. Argument of Perigee .................................................. 43 Figure 16: Case 4 Eccentricity vs. Argument of Perigee .................................................. 43 Figure 17: Case 1 Eccentricity Difference ........................................................................ 44 Figure 18: Case 2 Eccentricity Difference ........................................................................ 45 Figure 19: Case 3 Eccentricity Difference ........................................................................ 45 Figure 20: Case 4 Eccentricity Difference ........................................................................ 46 Figure 21: Case 1 Argument of Perigee Difference.......................................................... 47 Figure 22: Case 2 Argument of Perigee Difference.......................................................... 47 Page 3 Figure 23: Case 3 Argument 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