Low-Impact and Damped State Feedback Control of a Solar Sail on an Optimal Non- Keplerian Planet-Centered Orbit Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Ryan Micah Gero Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2009 Thesis Committee: Dr. Richard J. Freuler, Advisor Dr. Gerald M. Gregorek Copyright by Ryan Micah Gero 2009 Abstract Consider the most fundamental difference between a solar sail and conventional spacecraft: propellant. In order to effect propulsion, solar sails receive a constant supply of massless photons to reflect while conventional spacecraft must carry a limited supply of fuel. At first glance, solar sails are infinitely more efficient than conventional spacecraft simply because of this fact. However, while certainly an advantage to solar sailing, propellant consumption is not the proper metric for spacecraft comparison or the only appealing facet of the solar sail. It can be shown by a reasonable and straightforward analysis that solar sails have the potential to out-perform conventional spacecraft on the basis of effective specific impulse, a parameter that incorporates launch and payload masses as well as total mission duration via an adaptation of the illustrious rocket equation. Pair an aggressive specific impulse with the orbital possibilities that arise when solar sail performance is at a level capable of producing spacecraft accelerations the same order of magnitude as local solar or planetary gravitational acceleration, and the engineer finds significantly fewer constraints limiting the design of future space missions. Imagine a spacecraft for which a Lagrange equilibrium point becomes a large surface, rather than a singular location, on which it is able to remain at rest. Picture a space vehicle hovering high above an ecliptic plane or perhaps racing along some other non-Keplerian orbit taking measurements and relaying signals from positions previously untenable. Solar sails can do all of these ii things, and it is the intent of this body of work to generate a proof of concept for one of the most attainable and pertinent capabilities unique to solar sails mentioned thus far. In the pages that follow it will be shown that a solar sail is inherently stable for some of the optimal non-Keplerian family of planet-centered orbits, and can be stabilized by straightforward control schemes for the rest. Beginning from scratch with a radiation pressure model, gain parameters were developed for low-impact and damped state feedback control via sail pitch attitude variation. Optimal orbits are attainable, as these trajectories were designed to minimize the required spacecraft acceleration and thus lower the solar sail performance requirement. Planet-centered orbits are pertinent, since a solar sail must inevitably begin its journey by escaping from the planet Earth and most of NASA’s recent efforts in space are geared towards the exploration of nearby planets and their moons. Uniqueness stems from the specification of the non-Keplerian family of orbits, since solar sails are capable of sustaining them whereas modern conventional spacecraft are not. In today’s day and age, with payload miniaturization and the ability to manufacture extremely light weight reflective materials, solar sailing has the potential to become reality within the next five to ten years. The concepts highlighted in this thesis have a significant probability of being among the first demonstrated capabilities of solar sail spacecraft once they take flight. iii Dedication Dedicated to the memory of my late grandmother Eleanor R. Gero, who has been there for all of my accomplishments up until this one. iv Acknowledgements To my advisor, Dr. Richard J. Freuler, I wish to extend the deepest sense of gratitude. You have been with me since the very beginning of my college career, and you looked on as my team’s robot that was built for the annual Freshman Engineering Honors program competition could not decide where it wanted to go or what it wanted to do. Several years later you watched as I could not decide where I wanted to go academically or what I wanted to do professionally. I think I’m headed in the right direction now, and you have been a significant source of guidance. To the professor of my first aeronautical and astronautical engineering course, Dr. Gerald M. Gregorek, know that you are regarded by me as a beacon of lucidity. Together, you and Dr. Freuler always managed to keep my attention focused on aerospace as I wandered through many different fields of study as an undergraduate. Know that your signatures of approval concerning my research give me great pride. To the graduate studies committee chaired by Dr. Mo-How Herman Shen, and the graduate program coordinator Ms. Carol Scott, I thank you for the opportunity to finish my degree after many years of absence. Without your help none of this would have been possible. Lastly and most importantly, I am fortunate to have always had the support and encouragement of my family and close friends. For this I am eternally indebted, and promise to return the favor whenever and however I can. v Vita December 9, 1980 . Born - New London, Connecticut, USA 1999 . Floyd E. Kellam High School, Virginia Beach, Virginia 2000 – 2004 . Undergraduate Teaching Assistant, Freshman Engineering Honors Program, The Ohio State University 2004 . B.S. Engineering Physics, B.S. Applied Mathematics, Astronomy, The Ohio State University 2004 – 2006 . Designated Student Naval Aviator, United States Navy 2006 – present . Designated Naval Aviator, United States Navy Fields of Study Major Field: Aeronautical and Astronautical Engineering Other Fields: Engineering Physics Applied Mathematics Astronomy vi Table of Contents Abstract . ii Dedication . iv Acknowledgements . v Vita . vi List of Tables . ix List of Figures . x Nomenclature . xii Chapters: 1 INTRODUCTION . 1 1.1 Hunting Halley . 1 1.2 Keeping the Embers Burning . 3 1.3 Experiments in Space . 4 1.4 Less is More . 6 1.5 Motivation for this Thesis . 8 2 SOLAR SAIL DESIGNS . 11 2.1 Square Sail . 11 2.2 Heliogyro . 13 2.3 Disc Sail . 15 3 ANALYTICAL FOUNDATIONS . 17 3.1 A Quantum Portrayal of Radiation Pressure Physics . 18 3.1.1 Momentum Transport by a Single Photon . 18 3.1.2 Momentum Transport by a Flux of Photons . 18 3.2 Solar Sail Force Model . 19 3.2.1 The Solar Sail as a Perfect Reflector . 20 3.2.2 Effects of Imperfect Reflection on a Solar Sail . 22 vii 3.3 Characterizing Solar Sail Performance . 23 3.3.1 Solar Sail Buoyancy Ratio . 23 3.3.2 Critical Solar Sail Burden . 24 3.3.3 A Realistic Performance Estimate . 24 3.4 The Solar Sail vs. Conventional Spacecraft . 26 3.4.1 Assessing Conventional Rockets . 26 3.4.2 Solar Sail Equivalent Specific Impulse . 27 3.4.3 Substantiating a Worthwhile Venture . 28 3.5 Planet-Centered Non-Keplerian Orbital Dynamics . 29 3.5.1 Equations of Motion . 30 3.5.2 Orbit Optimization . 34 3.5.3 Variational Equations . 36 4 STABILITY . 40 4.1 The Characteristic Polynomial . 40 4.2 Forbidden Regions in the - z Plane . 41 5 CONTROLLABILITY . 43 5.1 Modifying the Variational Equation . 43 5.2 Modeling in State Space . 44 5.3 State Controllability . 45 5.4 Output Controllability . 46 6 OBSERVABILITY . 49 6.1 Observability Matrix . 49 6.2 Displacement Outputs . 50 6.3 Rate Outputs . 50 7 STATE FEEDBACK . 52 7.1 Gain Matrix Definition . 52 7.2 Transfer Function Determination . 53 7.3 Pole-Zero Configuration . 55 7.4 Pole Placement . 60 7.4.1 Gain Determination . 61 7.4.2 Low-Impact Control Method . 63 7.4.3 Testing for Lowest Impact . 64 7.4.4 Damped Control Method . 65 8 SUMMARY AND CONCLUSION . 70 References . 72 viii List of Tables Table 1.1 Chronological History of Solar Sail Missions to Space . ..