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Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling

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Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling by Zachary James Folcik B.S. Computer Science Michigan Technological University, 2000 SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2008 © 2008 Massachusetts Institute of Technology. All rights reserved. Signature of Author:_______________________________________________________ Department of Aeronautics and Astronautics May 23, 2008 Certified by:_____________________________________________________________ Dr. Paul J. Cefola Lecturer, Department of Aeronautics and Astronautics Thesis Supervisor Certified by:_____________________________________________________________ Professor Jonathan P. How Professor, Department of Aeronautics and Astronautics Thesis Advisor Accepted by:_____________________________________________________________ Professor David L. Darmofal Associate Department Head Chair, Committee on Graduate Students 1 [This page intentionally left blank.] 2 Orbit Determination Using Modern Filters/Smoothers and Continuous Thrust Modeling by Zachary James Folcik Submitted to the Department of Aeronautics and Astronautics on May 23, 2008 in Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics ABSTRACT The development of electric propulsion technology for spacecraft has led to reduced costs and longer lifespans for certain types of satellites. Because these satellites frequently undergo continuous thrust, predicting their motion and performing orbit determination on them has introduced complications for space surveillance networks. One way to improve orbit determination for these satellites is to make use of new estimation techniques. This has been accomplished by applying the Backward Smoothing Extended Kalman Filter (BSEKF) to the problem of orbit determination. The BSEKF outperforms other nonlinear filters because it treats nonlinearities in both the measurement and dynamic functions. The performance of this filter is evaluated in comparison to an existing Extended Semianalytic Kalman Filter (ESKF). The BSEKF was implemented in the R&D Goddard Trajectory Determination System (GTDS) for this thesis while the ESKF was implemented in 1981 and has been tested extensively since then. Radar and optical satellite tracking observations were simulated using an initial truth orbit and were processed by the ESKF and BSEKF to estimate satellite trajectories. The trajectory estimates from each filter were compared with the initial truth orbit and were evaluated for accuracy and convergence speed. The BSEKF provided substantial improvements in accuracy and convergence over the ESKF for the simulated test cases. Additionally, this study used the solutions offered by optimal thrust trajectory analysis to model the perturbations caused by continuous thrust. Optimal thrust trajectory analysis makes use of Optimal Control Theory and numerical optimization techniques to calculate minimum time and minimum fuel trajectories from one orbit to another. Because satellite operators are motivated to save fuel, it was assumed that optimal thrust trajectories would be useful to predict thrust perturbed satellite motion. Software was developed to calculate the optimal trajectories and associated thrust plans. A new force model was implemented in GTDS to accept externally generated thrust plans and apply them to a given satellite trajectory. Test cases are presented to verify the correctness of the mathematics and software. Also, test cases involving a real satellite using electric propulsion were executed. These tests demonstrated that optimal thrust modeling could provide order of magnitude reductions in orbit determination errors for a satellite with low-thrust electric propulsion. Thesis Supervisor: Dr. Paul J. Cefola Title: Lecturer, Department of Aeronautics and Astronautics 3 DISCLAIMERS The views expressed in this article are those of the author and do not reflect the official policy or position of MIT, MIT Lincoln Laboratory, the United States Air Force, Department of Defense, or the U.S. Government. This work is sponsored by the Air Force under Air Force Contract FA8721-05-C-002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government. 4 Acknowledgements I would first like to thank Paul Cefola for his outstanding advising, teaching and mentoring over the past six years. I have been very lucky to have such a thoughtful and attentive thesis advisor. Through your example, you have inspired me to always strive for excellence in my work. I look forward to many more years of collaboration with you. I thank David Chan, Rick Abbot, Susan Andrews, Forrest Hunsberger, Val Heinz, and Curt von Braun for their support and guidance which allowed me to enter the Lincoln Scholars Program. I am very grateful for the opportunity to go back to school. I also thank Bonnie Tuohy and Deborah Simmons for their gracious help with release review. I also thank Sid Sridharan and Kent Pollock for hiring me at MIT Lincoln Laboratory. Kent, you have taught me so much about software programming and how to do so in a team. Sid, I thank you for guiding my career, giving me interesting projects and showing me how to be a creative engineer. I thank my family, Tom and Peggy Folcik and Jeffrey Dawley, for their constant support, encouragement and love. Your words and understanding helped me stay grounded and secure during my foray into graduate school. I am very grateful to the system administrators in Group 93: Susan Champigny, Douglas Piercey, Ernest Cacciapuoti, John Duncan, and Edward Christiansen. Thank you very much for your diligence and time in resurrecting the data I needed for this thesis. I thank the members of the Lincoln Scholars Committee for giving me the opportunity to join the program and go to graduate school. I thank my academic advisor, Prof. Jonathan How, for his guidance. I also thank Jean Kechichian at the Aerospace Corporation for his advice in trajectory optimization. I also thank Paul Schumacher and Jim Wright for their helpful comments at the American Astronautical Society meeting in Galveston, TX. I appreciate the assistance of Erick Lansard, the Director of Research at Thales Alenia Space, Leonardo Mazzini at Thales Alenia Space, Heiner Klinkrad at the European Space Operations Center, and Andrea Cotellessa at the European Space Agency in being very responsive to my requests for information about the ARTEMIS satellite and its recovery. I thank you for all of the information you provided. I appreciate the efforts of Paul Cefola, Wayne McClain, Ron Proulx, Mark Slutsky, Leo Early, and the many MIT students at Draper Laboratory who have made GTDS and DSST what it is today. Your work has culminated in an orbit determination tool that is accurate, useful, and extensible. My thesis work would have been much more difficult, if not impossible, without standing on your shoulders. 5 [This page intentionally left blank] 6 Table of Contents Chapter 1 Introduction................................................................................................... 15 1.1 Motivation ............................................................................................................. 15 1.2 Overview of Thesis ............................................................................................... 18 Chapter 2 Background in Orbit Propagation, Optimal Thrust Trajectories, and Estimation .................................................................................................... 21 2.1 Satellite Orbit Propagation ................................................................................. 21 2.1.1 Satellite Orbit Propagation Using Cowell’s Method ................................. 23 2.1.2 Analytic Satellite Theory .............................................................................. 24 2.1.3 Draper Semianalytic Satellite Theory ......................................................... 25 2.1.3.1 Perturbing Forces and Variation of Parameters ................................. 29 2.1.3.2 Gaussian VOP Formulation ................................................................... 32 2.1.3.3 Lagrangian VOP Formulation............................................................... 34 2.1.3.4 DSST Formulation for VOP Equations ................................................ 36 2.1.3.5 Equinoctial Orbital Elements and Variational Equations .................. 38 2.1.3.6 DSST Application of Generalized Method of Averaging .................... 42 2.2 Application of Electric Propulsion (EP) to satellites ......................................... 58 2.2.1 Electric propulsion motors ............................................................................ 58 2.2.2 Electric propulsion optimal orbit transfer .................................................. 68 2.2.2.1 Optimal Control Theory Background .................................................. 68 2.2.2.2 Minimum-Time Trajectory Optimization Problem ............................ 72 2.2.2.3 Numerical Solution Method for Trajectory Optimization Problem .. 78 2.2.2.4 Averaged Numerical Solution Method for Trajectory Optimization Problem .................................................................................................... 81 2.2.3 Electric Propulsion for Satellite Station
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