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Formation Design of Distributed Telescopes in Earth Orbit with Application to High-Contrast Imaging FORMATION DESIGN OF DISTRIBUTED TELESCOPES IN EARTH ORBIT WITH APPLICATION TO HIGH-CONTRAST IMAGING ADISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Adam Wesley Koenig February 2019 © 2019 by Adam Wesley Koenig. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/rz152by6916 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Simone D'Amico, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Bruce Macintosh I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Zachary Manchester Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract This dissertation presents a new formation design that enables large distributed telescopes that must maintain alignment with inertial targets to be deployed in earth orbit. While previous approaches are infeasible for inter-spacecraft separations larger than a few hundred meters due to the large relative accelerations in earth orbit, the design proposed in this work allows separations within an order of magnitude of the orbit radius. This design is based on a two-phase operations concept that includes observation and reconfiguration phases. During observation phases, one spacecraft uses a quasi-continuous control system to ensure that the formation is aligned with the target. During this phase, control is only applied in the plane perpendicular to the line-of-sight to save propellant, allowing the separation to freely drift within a user-specified control window. During reconfiguration phases, one of the spacecraft performs a sequence of maneuvers that ensure that the formation is aligned with the target at the start of the next observation phase. In conjunction with the proposed operations concept, new absolute and relative orbit designs are developed that exploit the drift along the line-of-sight to minimize propellant consumption. This is accomplished by selecting the orbits to ensure that the relative acceleration remains closely aligned with the line-of-sight throughout all observations. Specifically, the delta-v cost of a properly timed observation maneuver is computed in closed-form. Using this formulation, it is demonstrated that the delta-v required to maintain alignment with any target is globally minimized by ensuring that two requirements are met. First, the spacecraft must have equal orbit radii. Second, the formation should be aligned primarily in the cross-track direction. Additionally, it is demonstrated that this orbit design also minimizes the delta-v cost of re-aligning the formation with the same target over consecutive orbits. Finally, optimal initial orbits for a specified observation sequence that minimize the e↵ect of orbit perturbations on the delta-v cost of the mission are derived in closed-form. To enable accurate and efficient control of the formation during reconfiguration phases, this iv dissertation presents a new real-time algorithm for globally optimal impulsive control of linear time- variant systems. The algorithm is more computationally efficient, robust, and can be applied to a broader class of optimal control problems than previous approaches in literature. A particularly novel feature is accommodation of time-varying, norm-like cost functions. This feature allows the algorithm to account for constraints such as asymmetric thruster configurations and time-varying attitude modes on spacecraft. The dynamics model used by this algorithm is a state transition matrix developed using a new methodology that enables simultaneous inclusion of conservative and non-conservative perturbations. This methodology is used to derive, for the first time in literature, a family of state transition matrices that simultaneously includes the e↵ects of earth oblateness and di↵erential drag on spacecraft relative motion in orbits of arbitrary eccentricity. Through comparison to a high-fidelity orbit propagator, it is demonstrated that the developed models are more accurate than all comparable models in literature. The proposed formation design is used to demonstrate the technical feasibility and scientific value of a small-scale starshade formation deployed in a readily accessible earth orbit. Such a mission could retire key optical and formation-flying technology gaps and perform precursor science in service of future flagship missions. The proposed optical design includes a nanosatellite-compatible telescope separated by several hundred kilometers from a starshade with a diameter of several meters. This design is more than ten times smaller than full-scale designs while providing a deep enough shadow to enable imaging of scientifically interesting targets. This miniaturization is accomplished by in- creasing the inner working angle and designing the starshade to block near-ultraviolet wavelengths. The feasibility and value of the mission are demonstrated through simulations of two example mission profiles. In the first mission, the formation is deployed in a geosynchronous transfer orbit and images a single target for tens of hours to validate the optical performance of the starshade and image a bright exoplanet. In the second mission, the formation images a set of nearby sun-like stars to characterize the density of the surrounding debris disks. These missions are simulated using a navigation and control architecture with errors consistent with the performance of current commer- cially available sensors and actuators. The sensitivity of the delta-v cost of the simulated missions agrees with predictions using analytical models. More importantly, these results demonstrate for the first time that the delta-v cost of these missions is within the capabilities of current propulsion systems for small satellites. In summary, this dissertation presents a novel formation design that enables distributed tele- scopes with large inter-spacecraft separations to be deployed in earth orbit, reducing mission costs v by orders of magnitude. The challenges of operating in earth orbit are overcome using a novel operations concept and orbit design that leverages key findings from modern astrodynamics. This design is used to demonstrate the feasibility and value of a small-scale starshade mission in earth orbit that can retire key technology gaps and perform precursor science in preparation for future flagship missions. This work has resulted in one mission proposal that was selected by NASA As- trophysics and a second that was recommended by NASA’s Starshade Readiness Working Group as a complement to ground-based experiment campaigns. Overall, the proposed formation design can be used to enable or improve the scientific return of a broad class of distributed telescope missions. vi Acknowledgments This work would not have been possible without the generous support of mentors, colleagues, friends, and family. First, I would like to thank my advisor, Professor Simone D’Amico, for his guidance for the past five years. Throughout this process, he encouraged me to explore new approaches to old problems. These e↵orts resulted in numerous publications and several core contributions of this dissertation. His advice has made me a better researcher and communicator. I am grateful to have him as a mentor and look forward to our future collaborations. I would also like to thank Professor Bruce Macintosh for his advice regarding the science and optics portions of this work. His insights helped me to understand the trades between the science and engineering drivers for space telescopes. I would also like to thank Andrew Norton and Eric Nielsen for their patience in explaining telescope behaviors and SNR modeling. Next, I would like to thank the members of my reading committee: Professor Simone D’Amico, Professor Bruce Macintosh, and Professor Zachary Manchester, for their time and insight reviewing this dissertation. I would also like to thank the other members of my defense committee including Dr. Larry Dewell and Professor Mark Cappelli. I would also like to acknowledge the financial support of the Department of Aeronautics and Astronautics and the NASA Space Technology Research Fellowship Grant NNX15AP70H. I am grateful to my colleagues at SLAB: Josh Sullivan, Sumant Sharma, Duncan Eddy, Connor Beierle, Vince Giralo, Matthew Willis, Michelle Chernick, Tommaso Gu↵anti, Corinne Lippe, and Nathan Stacey for providing sounding boards for new ideas and making sure that SLAB is a fun place to work. I would also like to thank Dana Parga for her enthusiastic help with administrative aspects of this work. On a more personal note, I would like to thank my friends at Acts 2 Christian Fellowship and
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