Optimal Collision Avoidance of Operational Spacecraft in Near-Real Time Final Report of the Spring 2011 Mathematics Clinic Sponsored by SpaceNav, LLC Department of Mathematical and Statistical Sciences University of Colorado Denver The Clinic Team Student Participants Kannanut Chamsri • Jeremy Doan • Rebeccah Dutcher • Yonas Getachew • Pamela Hunter • Michael Kasko • Volodymyr Kondratenko • Anna Le • Danielle Matazzoni • Austin Minney • Mark Mueller • Brian Wainwright • Faculty Advisor Alexander Engau • Sponsor Advisors Matthew Duncan • Joshua Wysack • ii Sponsor: SpaceNav SpaceNav LLC is a Colorado-based aerospace engineering firm providing technical solutions in the areas of Space Situational Awareness, Systems Engineering, and Mission Operations. Their process-driven approach to problem-solving enables to deliver on-time, error free products and services in a variety of functional areas including Systems Engineering and Integra- tion, Space Situational Awareness, Orbit Analysis Mission Operations, and Advanced Concepts Research and Development Cell. The SpaceNav team brings technical expertise and experience from a vast number of programs spanning the U.S. Department of Defense (DoD), Na- tional Aeronautics and Space Administration (NASA) and National Oceanic and Atmospheric Administration (NOAA) customer base. Leveraging a pas- sion for space combined with in-depth knowledge of space operations, Space- Nav provides cost-effective solutions to a vast array of technical problems. Figure 1: SpaceNav web site at http://www.space-nav.com iii Acknowledgments The math clinic is a key element in the mathematics education at UC Denver and offers our students a unique experience to collaboratively learn, advance, and apply their knowledge and skills to a challenging, real-life problem. How- ever, we could not do it without the exceptional support by our sponsors who commit resources and often dedicate a tremendous amount of their own time to help us, and to help our students. Participating in a clinic for the first time myself, I am most thankful to Matthew Duncan, President of SpaceNav, and his colleague Joshua Wysack not only for making this clinic possible, but also for taking an outstanding role in getting involved, showing students di- rections, and spurring motivation by demanding results. Like many of the students, I could learn a lot myself and stay indebted to Matt's and Josh's patience, their encouragement, and our shared leadership throughout the full semester. Needless to say, the problem offered is fantastic and remains an active stimulation for further work or collaboration. The ability to work with our students in such an open and interactive learning environment is easily one of the best aspects that a clinic offers also to faculty. I therefore like to recognize the wide variety of contributions made by the twelve participating students ranging from literature reviews and presentations to the class over independent studies and mutual tutoring to the development or implementation of software and the compilation of this final report. Despite regular guidance by faculty and sponsor advisers, it is ultimately the students' initiative and decision to take responsibility that make a clinic successful, and I congratulate those for whom meeting expectations was just the beginning but never enough. Finally, I thank my colleague Stephen Billups for his service as Clinic Director and his substantial contributions to the Math Clinic Program in our department. While Steve's repeated recruitment of partners and sponsors keeps this program alive, his genuine interest and habitual willingness to iv share his experiences and offer advice truly fills it with life. I also like to thank our Department Chair Michael Jacobson for trusting me with this clinic and allowing me to learn along, and to Angela Beale, Lindsay Hiatt, and Russel Boice for their continuous logistical and technical support. Alexander Engau Denver, June 2011 Student Acknowledgments to Sponsor \Thank you for your organization of such an interesting class. It was a great topic and a good experience of teamwork for me." Volodymyr Kondratenko \I would like to thank you for your time and all the support for the math clinic class. Showing up every single class was very impres- sive; giving us advice, comments and explanations was creating excellent working conditions. Your insight and your thoughtful explanation made a complex problem easy to understand. With- out your guidance and mentorship this class would have been impossible. For me, working on this problem gave me the opportunity to prepare myself for conducting research and strengthening my ex- perience, accumulating more professional knowledge, proficiency and skills. I wish you and your families comfort on difficult days, faith so that you can believe, confidence for when you doubt, patience to accept the truth, and I wish your company having excellent dis- tribution to provide great profits to the company and the world." Kannanut (Mon) Chamsri v Table of Contents The Clinic Team . ii Sponsor: SpaceNav . iii Acknowledgments . iv Table of Contents . vi List of Figures . viii 1 Introduction 1 1.1 Overview and Conduct of Clinic . 5 1.1.1 Project Warmup (January-February) . 6 1.1.2 Phase 1 Projects (February-March) . 6 1.1.3 Phase 2 Projects (March-April) . 8 1.1.4 Project Closeout (May) . 9 1.2 Outline of Report . 9 2 Background 10 2.1 Satellites and Orbits . 10 2.2 Fundamental Orbital Mechanics . 13 2.2.1 Planetary Motion and Two-Body Equation . 15 2.2.2 Motion in Space: Spacecrafts and Rockets . 16 2.2.3 Trajectory Changes by Hohmann Transfers . 18 2.3 Coordinate Frames and Analytic Representations . 23 2.3.1 From Cartesian Coordinates to Keplerian Elements . 24 2.3.2 From Keplerian Elements to Cartesian Coordinates . 27 2.3.3 Closed-Form Analytic Representation . 28 2.3.4 Discussion of Implementation . 35 2.4 Error Propagation and Collision Probabilities . 37 2.4.1 Representation of Uncertainty Errors . 38 2.4.2 Calculation of Collision Probabilities . 39 2.4.3 Discussion of Implementation . 43 vi 3 Optimization 45 3.1 Terminology and Model Assumptions . 45 3.2 Strategies and Closed-Form Solutions . 49 3.2.1 Miss Vector Approach . 49 3.2.2 Probability Gradient Approach . 52 3.3 Discussion of Implementation . 54 4 Implementation 55 4.1 General Assumptions . 56 4.2 The Trade Space Tool . 56 4.2.1 Data Flow . 57 4.2.2 Inputs and Outputs . 59 4.2.3 Propagation Versus Interpolation . 59 4.3 The Collision Module . 62 4.3.1 Data Flow . 63 4.3.2 Inputs and Outputs . 63 4.3.3 Calculation of TCA and Collision Probabilities . 64 5 Experiment 68 5.1 Trade Space Comparison . 69 5.2 Discussion of Results . 70 6 Conclusion 72 List of References 75 A Other Web Resources 83 B Homework Assignments 85 C MATLAB Implementation 87 C.1 Running The Code . 87 C.2 List of Subfunctions . 89 C.2.1 Main Code: Satellite Model . 91 C.2.2 Supplemental Code: Optimization . 104 C.2.3 Supplemental Code: Analytic Form . 113 C.2.4 Supplemental Code: Homework Solutions . 119 C.3 Ephemeris Data Files . 126 vii List of Figures 1 SpaceNav web site at http://www.space-nav.com . iii 1.1 Artwork showing space debris in low and geostationary Earth orbit (based on density data, but not to scale) . 2 1.2 Generic decision flow chart for collision risk management . 4 2.1 Spatial density of equivalent satellite objects in low earth orbits 11 2.2 Spatial density of cataloged space objects in low earth orbits . 12 2.3 Spatial density of cataloged space objects in total . 12 2.4 Illustration of a single-burn maneuver at t = tb and the cor- responding trajectory change to avoid an original conjunction at t = tc .............................. 19 2.5 Illustration of a Hohmann transfer to change satellite trajectories 21 2.6 Output of the Kepler function showing the Keplerian elements as a function of time in minutes, before pre-processing . 31 2.7 Output of the Periodic Test function showing the power spectrum of the Keplerian elements in the frequency domain . 32 2.8 Output of the Kepler function showing the Keplerian elements as a function of time in minutes, after pre-processing . 34 2.9 Two objects with low covariance interaction and collision risk 40 2.10 Two objects with larger covariance interactions and significant ellipsoidal overlap resulting in a larger risk of collision . 40 2.11 Illustration of the encounter plane and its basis vectors . 41 2.12 Illustration of a probability surface with a secondary object . 42 3.1 Encounter plane with the primary object at its origin and two equiprobability curves for current risk and target risk . 47 3.2 Two maneuver strategies for lowering the collision probability and equiprobability curves from current to target risk . 47 viii 3.3 Illustration of the miss vector approach: the primary moves directly along the miss vector in direction opposite to the in- tersection of secondary and the physical encounter plane . 48 3.4 Illustration of the probability gradient approach: the primary moves along the negative gradient in the probability space, resulting in the shortest path out and away from the predicted collision ellipsoid . 48 3.5 Magnitude of the optimal delta-v maneuver versus time of burn 54 3.6 Angle of the optimal delta-v maneuver versus time of burn . 54 3.7 Reduced collision probability after the optimal delta-v maneuver 54 4.1 Generic data flow within the collision module . 63 4.2 Illustration of miss distances: two object trajectories . 66 4.3 Illustration of the distances between two objects over time . 67 4.4 Illustration of the miss distances between two objects over time 67 5.1 2D contours of aggregate probabilities in full ∆v trade space . 69 5.2 2D contours of aggregate probabilities in negative ∆v space .