Relative Orbital Motion Dynamical Models for Orbits about Nonspherical Bodies Item Type text; Electronic Thesis Authors Burnett, Ethan Ryan Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 07:16:25 Link to Item http://hdl.handle.net/10150/628098 Relative Orbital Motion Dynamical Models for Orbits about Nonspherical Bodies by Ethan R. Burnett Copyright c Ethan R. Burnett 2018 A Thesis Submitted to the Faculty of the Department of Aerospace and Mechanical Engineering In Partial Fulfillment of the Requirements For the Degree of Master of Science With a Major in Aerospace Engineering In the Graduate College The University of Arizona 2 0 1 8 2 3 Acknowledgments I would like to thank my advisor, Dr. Eric Butcher, for his encouragement and excellent teaching. I greatly enjoyed working with my research colleagues: Mohammad Maadani, Bharani Malladi, David Yaylali, Jingwei Wang, Morad Nazari, and Arman Dabiri. I thank my parents and my brothers for their steadfast support. 4 Vita Education Master of Science, Aerospace Engineering Emphasis: Dynamics and Control University of Arizona, Tucson, AZ May 2018 Bachelor of Science, Aerospace Engineering Minor, Physics University of Arizona, Tucson, AZ May 2016 Publications Ethan Burnett and Eric Butcher, \Linearized Relative Orbital Motion Dynamics in a Ro- tating Second Degree and Order Gravity Field," AAS 18-232, AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, August 19 { 23, 2018 (submitted) Ethan Burnett, Eric Butcher, and T. Alan Lovell, \Linearized Relative Orbital Motion Model About an Oblate Body Without Averaging," AAS 18-218, AAS/AIAA Astrodynam- ics Specialist Conference, Snowbird, UT, August 19 { 23, 2018 (submitted) Ethan Burnett and Andrew J. Sinclair, \Interpolation on the Unit Sphere in Laplace's Method," AAS 17-793, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 { 24, 2017 Kristofer Drozd, Ethan Burnett, Eric Sahr, Drew McNeely, Vittorio Franzese, and Natividad Ramos Mor´on,\Block-Like Explorer of a Near-Earth Body by achieving Orbital Proximity (BEEBOP)," AAS 17-846, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 { 24, 2017 Eric Butcher, Ethan Burnett, Jingwei Wang, and T. Alan Lovell, \A New Time-Explicit J2-Perturbed Nonlinear Relative Orbit Model with Perturbation Solutions," AAS 17-758, AIAA/AAS Astrodynamics Specialist Conference, Columbia River Gorge, Stevenson, WA, August 20 { 24, 2017 Eric Butcher, Ethan Burnett, and T. Alan Lovell, \Comparison of Relative Orbital Motion Perturbation Solutions in Cartesian and Spherical Coordinates," AAS 17-202, AIAA/AAS Spaceflight Mechanics Meeting, San Antonio, TX, February 5 { 9, 2017 5 Table of Contents List of Figures ................................. 7 List of Tables .................................. 9 Abstract ..................................... 10 1. Introduction ................................ 11 2. Orbital Motion and Relative Motion ................ 15 2.1. Classical Theory of Satellite Orbits . 15 2.1.1. Gravitation . 15 2.1.2. The Two-Body Problem and Elliptic Orbits . 16 2.1.3. Orbital Elements for Elliptic and Circular Orbits . 18 2.2. Analytical Relative Dynamics of Co-orbiting Bodies . 19 2.2.1. Relative Orbital Motion . 20 2.2.2. The Hill-Clohessy-Wiltshire Model . 21 2.2.3. Representing Model Error . 26 2.3. Orbital Perturbations . 28 2.3.1. Perturbed Orbital Motion . 28 2.3.2. Types of Perturbations . 29 2.3.3. Perturbed Relative Orbital Motion . 30 3. Nonspherical Body Gravitation and Orbital Mechanics .... 34 3.1. Gravitational Potential Expressed by Spherical Harmonics . 34 3.2. The Shape of the Earth . 37 3.3. Orbital Motion under the Influence of J2 ................ 37 3.4. Moons, Planetoids, and Asteroids . 38 6 Table of Contents|Continued 3.5. Orbital Motion about Small Asteroids . 39 4. A J2-Perturbed Relative Orbit Model ............... 43 4.1. Formulation . 43 4.1.1. Case of Vanishing Chief Eccentricity . 47 4.2. Model Analysis and Validation . 50 5. A Linearized Relative Orbit Model for Orbits in a Rotating Second Degree and Order Gravity Field .............. 55 5.1. Formulation . 55 5.1.1. Case of Vanishing Chief Eccentricity . 70 5.1.2. Extension to Eccentric Chief Orbits . 74 5.2. Model Analysis . 77 5.2.1. The Case of Equatorial Orbits . 77 5.2.2. A Special Case LTI Model . 78 5.2.3. Analysis of Unstable and Stable Eigenspaces . 82 5.3. Model Validation via Simulation . 87 5.3.1. Hypothetical Asteroid for Case Studies . 88 5.3.2. Variable Chief Inclination with Constant Γ . 88 5.3.3. Variable Γ with Constant Chief Inclination . 95 5.3.4. Variable Chief Semimajor Axis with Constant Body Rotation Rate . 99 5.3.5. Interesting Cases . 101 6. Conclusions and Suggestions for Future Work ......... 107 References .................................... 109 7 List of Figures Figure 2.1. Elliptical Orbit Geometry22 .................... 18 Figure 2.2. The 3-1-3 Euler Angles and the Orbit Plane23 .......... 19 Figure 2.3. Relative Motion Problem Geometry . 20 Figure 2.4. Relative Distance for Unperturbed and Perturbed Cases . 32 Figure 2.5. ∆h(t) for Unperturbed and Perturbed Cases . 32 Figure 3.1. Primary Body Mass Distribution Geometry . 34 21 Figure 3.2. Nodal Regression due to J2 ................... 38 Figure 3.3. Asteroids and Comets Visited by Spacecraft24 .......... 39 Figure 3.4. Spacecraft Orbital Ejection due to SRP Forces . 42 Figure 3.5. Spacecraft Orbital Velocity Before and During Ejection . 42 Figure 4.1. Relative Orbit for 4 Orbits (LEO with J2)............ 51 Figure 4.2. Small Parameter Values for Two Orbits (LEO with J2)..... 52 Figure 4.3. Radial Error (LEO) . 53 Figure 4.4. Along-Track Error (LEO) . 53 Figure 4.5. Cross-Track Error (LEO) . 53 Figure 4.6. Error Norm (LEO) . 53 Figure 4.7. Maximum Error Norm over Two Orbits vs. Chief Orbit Radius 54 Figure 5.1. Problem Geometry for Orbiting a Rotating Ellipsoidal Body . 58 Figure 5.2. Libration Points for Rotating Ellipsoidal Body . 79 Figure 5.3. Superposition of In-Plane Modes at Stable Libration Point . 84 Figure 5.4. Simulated In-Plane Motion at Stable Libration Point . 84 Figure 5.5. Stable and Unstable Manifolds (Unstable Libration Point) . 87 Figure 5.6. Average Modeling Error vs. Inclination, Γ = 2:4 . 90 8 List of Figures|Continued Figure 5.7. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 2:4....................................... 92 Figure 5.8. Average Modeling Error vs. Inclination, Γ = 1:2 . 93 Figure 5.9. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 1:2....................................... 93 Figure 5.10. Average Modeling Error vs. Inclination, Γ = 0:8 . 94 Figure 5.11. Maximum Deviation of Relative Orbit Angular Momentum, Γ = 0:8....................................... 95 Figure 5.12. Average Modeling Error vs. Γ, i = 10◦ .............. 96 Figure 5.13. Maximum Deviation of Relative Orbit Angular Momentum, i = 10◦ 97 Figure 5.14. Average Modeling Error vs. Γ, i = 45◦ .............. 97 Figure 5.15. Maximum Deviation of Relative Orbit Angular Momentum, i = 45◦ 98 Figure 5.16. Average Modeling Error vs. Γ, i = 75◦ .............. 98 Figure 5.17. Maximum Deviation of Relative Orbit Angular Momentum, i = 75◦ 99 Figure 5.18. Average Modeling Error vs. Semimajor Axis, i = 75◦ ...... 100 Figure 5.19. Max Deviation of Relative Angular Momentum vs. Semimajor Axis...................................... 100 Figure 5.20. Relative Distance with Relative Orbit Resonance . 103 Figure 5.21. ∆h(t) with Relative Orbit Resonance . 103 Figure 5.22. Model Error (Relative Orbit Resonance Case) . 104 Figure 5.23. Relative Orbit (Resonance Case) . 104 Figure 5.24. Relative Orbit (Marginal Stability Case) . 106 Figure 5.25. Model Error (Marginal Stability Case) . 106 9 List of Tables Table 2.1. Orbital Elements and Differences . 31 Table 3.1. Asteroid Physical Parameters . 40 Table 4.1. Orbital Elements for LEO Relative Orbit Scenario with J2 ... 51 Table 5.1. Unique O Matrix Terms in 5.31, Oij = Oji ............ 62 Table 5.2. Unique E Matrix Terms in 5.32, Eij = Eji ............ 62 Table 5.3. Unstable and Stable Eigenvalues for Example Case . 83 Table 5.4. Simulated Asteroid Data . 88 Table 5.5. Orbital Elements, Variable Chief Inclination . 88 Table 5.6. Orbital Elements, Variable Chief Semimajor Axis . 99 10 Abstract RELATIVE ORBITAL MOTION DYNAMICAL MODELS FOR ORBITS ABOUT NONSPHERICAL BODIES by Ethan R. Burnett Master of Science University of Arizona Tucson, Arizona, 2018 Dr. Eric A. Butcher, Chair Relative orbital motion dynamical models are presented and discussed. Two types of models are primarily discussed in this work: a linear relative motion model account- ing for J2 (the gravitational parameter associated with the oblateness of the Earth), and a new linear relative motion model accounting for both nonzero second degree and order gravity terms C20 = −J2 and C22. The latter model, referred to as the \second-order model," is useful for simulating and studying spacecraft relative motion in orbits about uniformly rotating asteroids. This model is derived in two alternate forms. The first makes use of averaging in the kinematics and the second avoids any use of averaging. Additional work is devoted to analyzing the stability of relative orbital motion in rotating second degree and order gravity fields. To facilitate this, a parameter called the relative orbit angular momentum is introduced. For commensurate angular rates of primary body rotation and orbital mean mo- tion, a special case linear time-invariant (LTI) model is obtained from the unaveraged second-order model. This is connected to the topic of libration points in the body frame of rotating gravitating triaxial ellipsoids, and shown to successfully predict the instability of libration points collinear with the long axis and the stability of libration points collinear with the short axis.