<%BANNER%>

Investigation of Close-Proximity Operations of an Autonomous Robotic On-Orbit Servicer Using Linearized Orbit Mechanics

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INVESTIGATION OF CLOSE-PROXIMITY OPERATIONS OF AN AUTONOMOUS ROBOTIC ON-ORBIT SERVICER USI NG LINEARIZED ORBIT MECHANICS By SVETLANA ANATOLYEVNA GLADUN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Svetlana A. Gladun

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This document is dedicated to my gra ndmother, my mother, and my father.

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ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Norman Fitz-Coy, for his support, dedication, patience, and encour agement throughout my undergraduate and graduate studies. I would like to acknowledge my committee members, Dr. Gloria Wiens and Dr. Rick Lind, for reviewing the thesis and provi ding valuable comments that improved the quality of this work. I would like to acknowledge Mike Loucks from Space Exploration Engineering for his dedication to and his tir elessness and patience in helping me with STK. I would like to acknowledge Guy Savage fr om Orbital Science Corporation for his assistance in acquiring informa tion regarding the DART mission. I would like to acknowledge my colleague s and friends from Space Systems Group, Chun-Haur (Marvin) Chao and Andy Tatsch, for their advice, help and support, and numerous intellectual and not so conversations I would also like to acknowledge Fred Leve for dedicating his time to making an ADAMS model of DART to include in my thesis. Last, but not least, I would like to thank my family because without them none of this would be possible. I would like to thank my grandmother Antonina Petrovna for playing an important role in my upbringi ng and for always showing me unconditional love and encouragement in my journey thr ough life. I would like to thank my mother Svetlana Victorovna for her unconditional love, courage, and sacrifice in coming to this iv

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country so her children are able to have a be tter life and follow their dreams. I would like to thank my father Anatoly Mihaylovich for his unconditional love and support. I would like to thank my brother Dmitry for believing in me and supporting the decisions I make throughout life. I would like to thank my be st friend Cori for showing me unconditional love and support and bringing happiness into my life. I would like to thank little Dusya for bringing love and joy into my life. v

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................................................................iv LIST OF TABLES ...........................................................................................................viii LIST OF FIGURES ...........................................................................................................ix ABSTRACT .....................................................................................................................xiii CHAPTER 1 INTRODUCTION AND BACKGROUND.................................................................1 2 SPACE ROBOTIC SERVICERS: PAST, PRESENT, AND FUTURE....................10 2.1 State-of-the-Art .....................................................................................................10 2.2 Scope of the Thesis ...............................................................................................25 3 LINEARIZED ORBIT MECHANIC S AND PROXIMITY OPERATIONS............27 3.1 Two-body Problem...............................................................................................27 3.2 Relative Motion....................................................................................................29 3.2.1 Clohessy-Wiltshire Equations for Nearly Circular Orbits......................30 3.2.2 Position and Velocity Solutions for Nearly Circular Orbits...................35 3.3 Close-Proximity Operations.............................................................................38 3.3.1 Linear Two-impulse Rendezvous...............................................................38 3.3.2 Different Approaches of the Target Satellite..............................................40 3.4 Perturbations....................................................................................................42 3.4.1 General Effect of Perturbations..................................................................43 3.4.2 Atmospheric Drag Effects..........................................................................44 3.4.3 Oblateness of the Earth Effects..................................................................45 3.4.4 Third-Body Effects.....................................................................................48 3.4.5 Solar Radiation Effects...............................................................................49 3.4.6 Thrust..........................................................................................................50 4 DEMONSTRATION OF AUTONOMOUS RENDEZVOUS TECHNOLOGY (DART) MISSION OVERVIEW...............................................................................51 4.1 DART Spacecraft..................................................................................................51 4.2 MUBLCOM Spacecraft........................................................................................54 vi

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4.3 Mission.................................................................................................................56 4.3.1 Launch and Rendezvous Phase..................................................................56 4.3.2 Close-Proximity Operations Phase.............................................................57 5 SIMULATIONS USING SATELLITE TO OL KIT (STK) SOFTWARE BY ANALYTICAL GRAPHICS, INC. (AGI).................................................................60 5.1 Satellite Tool Kit (STK).......................................................................................60 5.1.1 STK/Advanced Visualization Option Module...........................................60 5.1.2 STK/Astrogator Module.............................................................................61 5.2 Simulation Scenarios............................................................................................63 5.2.1 Scenario 1...................................................................................................65 5.2.1.1 V-Bar maneuvers..............................................................................69 5.2.1.2 R-Bar maneuvers..............................................................................80 5.2.1.3 Fuel usage.........................................................................................86 5.2.2 Scenario 2...................................................................................................88 5.2.2.1 V-Bar maneuvers..............................................................................92 5.2.2.2 Fuel usage......................................................................................98 5.2.3 Scenario 3...................................................................................................99 5.2.3.1 Targeted thrust efficiency...............................................................100 5.2.3.2 Fixed thrust efficiency targeted maneuver duration....................103 5.2.3.3 Fixed thrust efficiency targeted radial position...........................108 5.3 Discussion of the Simulations Results................................................................112 5.3.1 Relative Position.......................................................................................112 5.3.2 Fuel Usage................................................................................................115 6 CONCLUSION AND RECOMMENDATIONS.....................................................117 6.1 Conclusions.........................................................................................................117 6.2 Recommendations...............................................................................................120 APPENDIX A CLASSICAL ORBITAL ELEMENTS....................................................................121 B CLOHESSY-WILTSHIRE EQUATIONS OF MOTION ALGORITHM...............123 LIST OF REFERENCES.................................................................................................125 BIOGRAPHICAL SKETCH...........................................................................................133 vii

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LIST OF TABLES Table page 4-1. DART Vehicle Propulsi on Systems Characteristics.................................................53 4-2. Location of the 16 Cold-Gas Nitrogen Thrusters......................................................53 4-3. Close-Proximity Operations Timeline.......................................................................58 5-1. Initial Impulse Values for VBar Maneuvers from km to m.............................70 5-2. Final Impulse Values for VBar Maneuvers from km to m..............................71 5-3. Relative Position........................................................................................................ 79 5-4. Initial Values for R-Bar Maneuvers from 150m to 300m...................................81 V 5-5. Final Values for R-Bar Maneuvers from 150m to 300m....................................81 V 5-6. Relative Position........................................................................................................ 83 5-7. Close-Proximity Operations Timeline.......................................................................91 5-8. Initial Values for V-Bar Mane uvers from km to -850m.................................93 V 5-9. Final Values for V-Bar Maneuvers from km to -850m..................................93 V 5-10. Relative Position......................................................................................................9 5 5-11. Targeted Efficiency for Finite Thrusting V-Bar Maneuvers.................................101 5-12. Fixed Efficiency (5%) for Fi nite Thrusting V-Bar Maneuvers.............................105 5-13. Fixed Efficiency (5%) for Fi nite Thrusting V-Bar Maneuvers.............................110 viii

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LIST OF FIGURES Figure page 2-1. Ranger Telerobotic Flight Experiment......................................................................12 2-2. MDRobotics Mobile Servicing System (MBS)......................................................12 2-3. European Robotic Arm (ERA)..................................................................................13 2-4. Japanese Experiment Module Remo te Manipulator System (JEMRMS).................13 2-5. Engineering Test Satellite (ETS) VII.........................................................................14 2-6. Langleys Automated St ructural Assembly Robot....................................................15 2-7. CMU Skyworker.....................................................................................................15 2-8. German Robot ROTEX.............................................................................................15 2-9. NASA Robonaut.....................................................................................................16 2-10. University of Maryland Be am Assembly Teleoperator (BAT)............................16 2-11. NASA AERCam......................................................................................................17 2-12. University of Maryland SCAMP..........................................................................18 2-13. Orbital Sciences DART........................................................................................21 2-14. XSS-10.....................................................................................................................22 2-15. XSS-11.....................................................................................................................23 2-16. TECSAS..................................................................................................................24 2-17. Orbital Express ASTRO........................................................................................24 2-18. SUMO Servicing Spacecraft...................................................................................25 3-1. Geometry of Relative Motion of Two Bodies...........................................................29 3-2. Coordinate System for Relative Motion....................................................................30 ix

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3-3. V-Bar Approach Maneuver.....................................................................................40 3-4. R-Bar Approach Maneuver.....................................................................................41 3-5. Circumnavigation Maneuver.....................................................................................42 3-6. General Effect of Pertur bations on Orbital Elements................................................43 3-7. Effects of Atmospheric Drag.....................................................................................44 3-8. Nodal Regression.......................................................................................................46 3-9. Apsidal Regression....................................................................................................47 3-10. Third-Body Effects..................................................................................................48 4-1. DART Spacecraft Configuration...............................................................................51 4-2. DART Spacecraft Com ponent Configuration...........................................................52 4-3. DART Vehicle Thrusters...........................................................................................53 4-4. MUBLCOM Satellite Configuration.........................................................................54 4-5. MUBLCOM Orbit......................................................................................................55 4-6. Two-Line Elements...................................................................................................55 4-7. DART Rendezvous Phase.........................................................................................56 4-8. DART Proximity Operations.....................................................................................58 5-1. Astrogator..................................................................................................................61 5-2. Astrogator Targeter....................................................................................................62 5-3. Scenario 1 Targeted Close-proximity Operations (Starting at km Point)..........66 5-4. Station-keeping Set-up..............................................................................................67 5-5. Station-keeping..........................................................................................................68 5-6. DART at m Relative to MUBLCO M (Point of Closest Approach)......................69 5-7. V-Bar Maneuvers......................................................................................................70 5-8. STK Untargeted Maneuver Set-Up...........................................................................72 5-9. STK Astrogator Targeted Maneuver Set-Up.............................................................73 x

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5-10. Transfer to km V-Bar..........................................................................................75 5-11. Transfer to m V-Bar........................................................................................76 5-12. Transfer to m V-Bar..........................................................................................77 5-13. Transfer to m V-Bar............................................................................................78 5-14. Error in Relative Pos ition Untargeted/Targeted...................................................80 5-15. R-Bar Maneuvers.....................................................................................................80 5-16. Transfer to 100m R-Bar...........................................................................................84 5-17. Transfer to 300m R-Bar...........................................................................................85 5-18. Error in Relative Pos ition Untargeted/Targeted...................................................86 5-19. Fuel Usage for Ma neuvers in Fig. (5-3B)................................................................88 5-20. Scenario 2 Targeted Close-proximity Operations (Starting at km Point)........89 5-21. Finite Motion Along V-Bar (A Nu mber of Small CW Transfers).......................90 5-22. Multiple Approaches of MUBLCOM (-15m to m).............................................90 5-23. V-Bar Maneuver Multiple Two-Impulse Transfers..............................................92 5-24. Transfer to -950m V-Bar.........................................................................................96 5-25. Transfer to m V-Bar........................................................................................97 5-26. Error in Relative Pos ition Untargeted/Targeted...................................................98 5-27. Fuel Usage for Mane uvers in Fig. (5-20B)..............................................................99 5-28. Astrogator Mission Sequence Control (MCS).......................................................101 5-29. Multiple Approaches of MUBLCOM Finite Thrusting (-15m to m).............102 5-30. Fuel Usage Finite Thrusti ng for Maneuvers in Table (5-11)..............................103 5-31. Astrogator Mission Sequence Control (MCS).......................................................105 5-32. Finite Thrusting Tr ansfer to m V-Bar............................................................106 5-33. Finite Thrusting Transfer to m V-Bar..............................................................107 5-34. Fuel Usage Finite Thrusti ng for Maneuvers in Table (5-12)..............................108 xi

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5-35. Astrogator Mission Sequence Control (MCS).......................................................110 5-36. Finite Thrusting Tr ansfer to m V-Bar............................................................111 5-37. Fuel Usage Finite Thrusti ng for Maneuvers in Table (5-13)..............................112 A-1. Classical Orbital Elements......................................................................................121 xii

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Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science INVESTIGATION OF CLOSE-PROXIMITY OPERATIONS OF AN AUTONOMOUS ROBOTIC ON-ORBIT SERVICER USI NG LINEARIZED ORBIT MECHANICS By Svetlana A. Gladun August 2005 Chair: Norman Fitz-Coy Major Department: Mechanic al and Aerospace Engineering Due to the cost and time disadvantages of satellite replacement, on-orbit servicing of disabled or outdated satelli te has become a priority of the space robotics community. This thesis begins with an overview of pa st, present, and future spacecraft robotic servicers including a discussion on the evolution of the develo pments in various enabling technologies such as vision systems, gui dance navigation and control, collision avoidance, grappling and docking mechanisms, and others. This overview is followed by a discussion of the Clohessy-Wiltshire (CW) equation that governs the linearized relative motion between two spacecraft (a robotic servicer and a target satellite in this case). The effects of the orbital pertur bations not included in the CW model are also discussed. The Demonstration of Autonomous Rendezvous Technology (DART) mission by Orbital Sciences Corporation with sponsorship fr om NASA is utilized in this thesis as the mission profile to be investigated. The th esis investigation focuses on the proximity operations phase of the DART mission, which includes V-Bar and RBar approaches and xiii

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circumnavigation maneuvers of the MUltiple paths, Beyond-Line-of-sight COMmunications (MUBLCOM) satellite by the DART spacecraft. The CW methodology is used to develop and analyze thes e maneuvers that are then simulated in a fully nonlinear model available in Satellite Tool Kit (STK) simulation environment. These CW results are compar ed with the two-point bounda ry value (2PBV) solution available from STKs Astrogator module. Three different close-proximity operations s cenarios are simulated in STK. First, the CW approximation is used perform singl e two-impulse transfer maneuvers. Second, to improve the accuracy of the CW approximation, each maneuver is performed using multiple two-impulse transfers. Third, the maneuvers are performed using finite thrusting. In comparing the first two scenar ios with the Astrogators solution, the CW based approximations resulted in significant re lative position errors in both the V-Bar and R-Bar maneuvers due to the neglected orbital perturbation. In addition to positional errors, fuel consumed during the performance of the mission scenarios were also computed. For scenarios 1 and 2, two cas es regarding the propulsion systems were considered: in Case 1 the propulsion syst em was switched at 21 km behind MUBLCOM (as specified in the DART mission profile) and in Case 2 the switch was performed at 3 km behind MUBLCOM on the V-Bar axis. The fu el consumption analysis indicated that the DART spacecraft only had sufficient fuel to perform scenario one using the propulsion switch scheme of Case 2. The result of this thesis indicates that precise autonomous proximity operations based on the linearized CW model are not feasib le and further analysis of close-proximity operations of an autonomous robotic servicer is needed. xiv

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CHAPTER 1 INTRODUCTION AND BACKGROUND Over the years on-orbit servicing of the disabled or outdated spacecraft became a priority aspect of space robo tics community. Due to the cost and time disadvantage of replacing an entire satellite, repairing it or extending its life on orbit became favorable. The research focus of the servicing missions was aimed at accomplishing the same overall goal maintenance, repair, or retr ieval of existing and future satellites. Numerous technological barriers needed to be overcome before autonomous robotic servicing could become a feasib le operation. An autonomous robotic servicer had to be capable of rendezvousing with and engaging in the docking, repair, refueling, or deorbiting of the target satellite. These maneuve rs required significant developments in the various technologies such as vision systems, guidance naviga tion and control, collision avoidance, grappling and docking mechanisms, and others. This chapter provides a background on the evolution of th ese areas over the years. As early as 1960 the question of rendezvous became important due to the interest and desire to build large space structures, bui lding and assembly of which were to be performed in space. It was envisioned for multiple satellites to join together in space to carry out the mission as one large structure ( 1 ). That was when Clohessy and Wiltshire [ 1 ] (CW) presented their work, closed-form so lution to the rendezvous problem, which is in widespread use to this date. They inve stigated a guidance system that would enable the rendezvous of the satellites in circular orbit. The focus of this thesis is on the use of CW equations; therefore the subject will be addressed in detail in Chapter 3. Four years 1

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2 later Edelbaum [ 2 ] presented an analytical solution to optimum low-thrust power-limited trajectories in inverse-square force field, considering rendezvous of spacecraft that completed transfer between elliptical orbits. In addition to traditional rendezvous, evasive maneuvers, where the target spacecraft would detect the pursuing spacecraft a nd try to avoid it, were of interest. The study of the evasive maneuvers gave an insight into the solution to th e obstacle avoidance problem, which was an important aspect of successful rendezvous mission. In 1984 Jensen [ 3 ], using the principle of proportional gui dance, gave insight into the kinematics and requirements, such as required propulsi on force and amount of fuel needed, of evasive rendezvous maneuvers. Cochran and Lahr [ 4 ] followed with investigation of the guidance problem of intercepting a targ et using the proportional navigation. The space robotics community was focuse d on developing free-flying autonomous space robots that could assist in retrieving a disabled satell ite, if not performing the entire mission. In 1992 Hewgill [ 5 ] presented the means for continuously estimating the motion of a freely rotating object to be used for the grasping tasks of the EVA Retriever, a flying autonomous space robot. Since the rescue missions were performed mainly using the Shuttle, Fitz-Coy and Chatterjee [ 6 ] investigated stability duri ng capture when the rescue vehicle is much smaller than the disabled vehi cle. Additionally, the stability of force-free spinning satellite with deployable flexible appendages was investigated by Mierovitch [ 7 ] in 1974. Twenty years late r Fitz-Coy and Fullerton [ 8 ] focused on the effects of the deployment/retraction of the appendages on the st ability of the system (tumbling satellite and appendages), meaning that the detumbling of the satellite could be accomplished by

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3 passive stabilization th rough energy dissipation, achie ved by deploying flexible appendages. Guidance and navigation technologies were also being developed to ensure safe path for the servicing robot to the satellite, whether it was avoiding the obstacles on the path to the disabled vehicle or maneuvering around the satellite appendages for a desired grasp. In 1994 Yuan and Hsu [ 9 ] presented a new guidance scheme for the final stage of the rendezvous maneuver using augmente d proportional navigation by applying the commanded acceleration in the direction with a bias angle to the normal direction of line of sight (LOS) ( 9 p. 410). A year later Fitz-Coy and Liu [ 10 ] extended on the work of Yuan and Hsu [ 9 ] by applying the augmented proportio nal navigation for rendezvous and docking to translating and rota ting rigid bodies instead of point-masses (as was done in 1994). The point-mass approach was found to lead to the possibility of contact between the vehicles before the maneuver was co mplete, and that could have potentially influenced the overall mission in an unfa vorable way. In 2002 Jacobsen et al. [ 11 ] investigated the danger of possible collision of the servicer robot with uncontrolled spinning satellite, proposing a numerical opt imization method for planning the safest kinematic trajectory for the approach pha se of the capture to accomplish grasp and stabilization of the satellite. The guidance and navigation algorithms developed in Rendezvous and Proximity Operations Program (RPOP) for the proximity operations of the Shuttle Orbiter were pr esented by Clark et al. [ 12 ] in 2003. Receiving raw sensor data from a variety of different navigati on measurement sources, RPOP simultaneously displayed relative navigation and guidance tr ajectory information from each navigation source both digitally and graphi cally. The addition of these algorithms to RPOP during

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4 the manually piloted proximity operations impr oved trajectory control, reduce propellant usage, and reduce piloting variability ( 12 p. 3). At the same time Carpenter and Jackson [ 13 ] developed an optimization tool to be used with traditional CW equations, providing ability for planning high-accuracy rendezvous maneuvers in the presence of disturbing forces. While CW equations de veloped in 1960 were useful for calculating either the relative position of two space vehicles as a function of time or the relative velocity to achieve a desired position ( 13 p. 2), the required simplifying assumptions lead to a possibility of errors occurring in actual use. Carpenter and Jackson [ 13 ] used the CW equations as initial estimate for the Genetic Algorithm (GA) to solve Rendezvous Optimization Problem. Advancements were also made in th e docking and grasping phases of the rendezvous maneuver, implementing the use of vision-based systems. The dynamic machine vision, specifically object tracking a nd visual servoing, was investigated by Nicewarner and Kelley [ 14 ] in 1992. He presented a multi-layered vision-guided system for grasping maneuver, which resolved the conflict between rigorous image processing needs and rapid vision updates to the robot ( 14 p. 9). A year later Ho and McClamroch [ 15 ] proposed a completely automatic spacec raft docking system using a computer vision system and a docking control system ( 15 p. 281). In late 1990s emerging interest in the area of microdynamics was found. Microdynamics of the deployed structure is the dynamic response of the structure under forces for which resulting strain is of order microns per meter of size or less ( 16 p. 2890). The microdynamics of the deployable sp ace structures was becoming a beneficial emerging technology, complete understanding of which was important to the ongoing

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5 progress. In 1997 Warren and Peterson [ 16 ] presented an experimental characterization of microdynamics of a prototype deployed reflector structure ( 16 p. 2890), suggesting that the microdynamics could be stabilized to nanometers of precision through the application of transient loads after the deployment of the structure. Levine [ 17 ] in 1999, using the flight experiment IPEX-II perf ormed during the Space Shuttle STS-85 mission, investigated microdynamic behavior of jointdominate structure on-orbit, concerned with the existence of transient impulses from undergoing rapid thermal variations or instabilities due to internal strain energy releas e mechanisms. Tung et al. [ 18 ] focusing his efforts in the microdynamics of the re dundant flexible multibody systems, presented an innovative approach in the solution of tr acking problem where a mechanical system is constrained to move along a given path. Noor and Venneri [ 19 ] published an articl e in 1994 issue of Aerospace America giving a perspective on future space systems, describing the restructure of the U.S. space program due to economic constraints. The focus was placed on building smaller, cheaper, and faster spacecraft oppose to large, expensive, and complex. The small spacecraft was to be designed with the idea of complementing and/or replacing the large complex spacecrafts. Five mission categories for small spacecraft have been identified by NASA: astrophysics, space physics, Eart h observing, commercial communications, and planetary. The spacecraft to be devel oped was going to be used as Earth-to-orbit (ETO) launchers and space tran sfer vehicle (STV). The vehicle was to be made fully returnable, reusable, robust, and safer and more reliable than the existing one. Many technologies had yet to be developed and brought to a level of reliable and safe application. Due to the new focus on building smaller and cheaper satellites the need for

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6 servicing the spacecraft phased out because now it was supposedly again possible to replace a disabled satellite w ithout incurring a significant financial loss. However, the idea of servicing spacecraft resurfaced. Numerous studies were conducted, trying to demonstrate that servicing a disable satellite was economically efficient in comparison to declaring the vehicle a total loss. In 1993 Levin et al. [ 20 ] conducted a study on feasibility of introducing a salvage/repair of civil, defense, and commercial satellites program. The analysis was based on historical failure data as well as conservative estimate of the future potential for servicing missions ( 20 p. 1). However, the results showed that it was not sufficient to suppor t standalone salvage operation, especially without the interest of the governme nt. Sullivan and Akin in 2001 [ 21 ] constructed three databases for all civilian, m ilitary, commercial, and NGO spacecraft launched from 1957 through 2000 in order to provide a survey of serviceable spacecraft failures. Considering four servicer types, inspection only, reboost only, basic dexterous, and complex dexterous, it was found more cost efficient to service the satellite th an declare it a total loss. A year later Horowitz et al. [ 22 ] investigated Advanced Satellite Maintenance (ASM) potential to be cost effective and viable method for improving satellite operations with the example of NewsSat. ASM missi ons either focused on preventing satellite failure by performing scheduled servicing tasks such as component replacement or on satellite recovery afte r failure. ASM was found to extend satellite life and maintain the revenue earning capabilities of the satellite ( 22 p. 124). Different servicing vehicles were being designed, whether to serve as a single robot or a part of a fleet. Several systems reached their hardware state and either had a flight test demonstration or currently awaiting one. In 2001 Ledebuhr et al. [ 23 ] presented

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7 MicroSats technology development, including critical capabiliti es and technologies necessary for proximity-operations and formation-flying of micro-satellites ( 23 p. 52493). The missions in mind for MicroSats included rendezvous, inspection, proximityoperations (formation flying), docking, and robotic servicing functions (refueling, repowering, or repairing) ( 23 p. 5-2493). Geftke et al. [ 24 ] described the Ranger Telerobotic Shuttle Experiment (Ranger TSX) development. Ranger [ 24 27 ] could potentially service attached experiments and, with a free-flying base, stand alone satellites ( 24 p. 1), using both Shuttle and ground-ba sed control stations. Wagenknecht et al. [ 28 ] in 2003 provided an overview of th e Mini AERCam vehicle design and a detailed description of the GN&C system design, development, and testing ( 28 p. 1). The vehicle was an improvement design of the AERCam SPRINT [ 28 29 ], a visual inspection robot. The vehicle was designed for either remotely piloted operations or supervised autonomous operations including au tomatic stationkeeping and point-to-point maneuvering ( 30 p. 1). A year later Fredrickson et al. [ 30 ] followed with presentation of the docking systems technology, developed for the Mini AERCAm free-flyer, to provide automatic deployment and retrieva l. The docking system technology consisted of two primary elements: magnetic docking, fo r aligning and capturing the vehicle, and vehicle retention, for suppor ting the vehicle during hea vy loading conditions, which occur during launch and landing. The same year Hays et al. [ 31 ] described the evolvement process of a simple, low-co st, lightweight flexible-cable docking mechanism that allows for autonomous cap ture and docking of one spacecraft with another ( 31 p. 1). The newest version of the docking mechanism, ASDS-II, was

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8 intended to provide a passively-maintained, rigid interface between two docking spacecraft . . ( 31 p. 7). Some of the recently developed systems were still in the conceptual stage of their development, however, others had alrea dy completed the simulation phase. In 2001 Matsumoto et al. [ 32 33 ] presented the system concept and design of Hyper Orbital Servicing Vehicle (HOSV), which could be us ed for a variety of servicing missions due to the systems reconfigurability. At the same time Akin [ 25 ] described the technologies developed by UMD SSL for on-orbit servicers, concluding with propos al of development of Miniature Orbital Dexterous Servicing System (MODSS), a Responsive Access, Small Cargo, Affordable Launch (RASCAL)-class fr ee-flying robotic serv icing system. The project was steered towards pr oducing a low cost, ultra-lightwe ight system with variety of servicing applications and was based on the technologies developed for the Ranger TSX [ 24 27 ]. Zimpher [ 34 ] in 2003 described the development of an Autonomous Mission Manager (AMM) that accommodate d highly autonomous mission planning, execution, and monitoring of the servicing on-orbit missions. In 2004 Bosse et al. [ 35 ] described SUMO, Spacecraft for the Universal M odification of Orbits, a technology risk reduction program to demonstrate the in tegration of machine vision, robotics, mechanisms, and autonomous control algorithms to accomplish autonomous rendezvous and grapple of a variety of in terfaces traceable to future sp acecraft servicing operations ( 35 p. 1). Also, Rogers et al. [ 36 ] presented SCOUT progra m, which involves the development of the high-capability, flexible, modular spacecraft architecture, enabling multi-mission compatibility, high payload mass and power fractions, long shelf-life, rapid call-up for launch, rapid initialization on orbit, and manufacturability (quantity). One of

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9 the goals of SCOUT program is to develop a modular microsatellite architecture that is compatible with virtually every launch vehicle (LV) in the world inventory ( 36 p. 2). The space robotics community, embarking on the endeavor of development and advancement set forth by necessity for the au tonomous robotic servicers, is still faced with many technical challenges. Majo r areas, including Avionics, Sensors, & Actuators, Strategic Alliance & Coopera tion, Risk Mitigation, Mechanisms & Interconnects, Rendezvous & Capture, S pacecraft Dynamics & Controls, and Machine Vision, require signif icant breakthrough in order to realize the feasibility of autonomous servicing robots.

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CHAPTER 2 SPACE ROBOTIC SERVICERS: PAST, PRESENT, & FUTURE [ Mankind is drawn to the heavens for the same reason we were once drawn into unknown lands and across the open sea. We choose to explore space because doing so improves our lives, and lifts our national spirit. So let us continue the journey. ( 37 p. 1)] President George W. Bush, January 14, 2004 President Bush [ 37 ] announced a New Vision for Space Exploration Program on January 14, 2004. He delegated new goals for the National Aeronautics and Space Administration (NASA), stating that it was tim e for America to take the next steps ( 37 p. 1). President Bush [ 37 ] outlined specific milestones for the country to achieve, ranging from returning to the Moon to ventur ing out to Mars. To achieve the goals set forth, the new space vision encompassed robotics as a key factor to successful progress as implied by the statement Robotic missions wi ll serve as trailblazers the advanced guard to the unknown ( 37 p. 1). Series of robotic missions to the Moon were planned to make the necessary preparations for human pres ence. Robotic explorations of Mars were to be continued to prepare for future hu man expeditions. The new mandate emphasized the importance of robotic explor ation across solar system to search for resources and life and to understand history of universe. 2.1 State-of-the-Art Even before robotics officially was a part of Presidents space mandate, technological advancements in that area were ongoing. In 1994 Lavery [ 38 ] published an article in Aerospace America surveying the developments of space robotics community and predicting its future. The goal of space robotics community was to create fully 10

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11 autonomous, self-contained robotic systems w ith considerable on-boa rd intelligence ( 38 p. 32). However, due to the lack of the tec hnology to develop such systems, the space robotics community had to adopt a new a pproach, focusing its efforts on advancing teleoperator technology that can be operated from the ground. The space vehicles were divided into three main categories, in which developments were to be pursued: extravehicular robotic (EVR) servicers, scie nce payload servicers, and planetary surface rovers. In order to realize the set goals signifi cant developments in many technologies, including enhanced collision detection a nd avoidance, advanced local proximity sensing, task-level control workstations, improved command and control architectures, reduced mass and volume, worksite recogni tion and representation, improved robotic dexterity, advanced supervis ory control, and improved ove rall system robustness ( 38 p. 37), were required. Over the years many countries, includi ng United States (US), Canada, Russia, Japan, and several European countries, we re competing and working together to accomplish the desired results. US developed Ranger [ 24 27 ] Satellite Servicing Vehicle concept in early 1990s. Ranger [ 24 27 ], a dual-arm, free-flyi ng telerobotic flight experiment, shown in Fig. (2-1), was designe d to conduct a simulated satellite servicing exercise to characterize the operational capab ilities of free-flying robotic system ( 38 p. 35).

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12 Figure 2-1. Ranger Telerobot ic Flight Experiment [ 27 ]. Canada contributed the Shuttle Remote Manipulator System (SRMS) [ 39 ], also know as Canadarm, and the Space Station Remote Manipulator System (SSRMS) [ 40 41 ], known as Canadarm 2. Both were mani pulators designed to maneuver and place large loads. Also Canada developed the Sp ecial Purpose Dexterous Manipulator (SPDM) [ 40 42 44 ], know as Canada Hand, which was dua l-arm robotic system designed to attach to the SSRMS [ 40 41 ]. SSRMS [ 40 41 ], its platform the Mobile Base System (MBS) [ 40 ], and SPDM [ 40 42 44 ] would compose the Mobile Servicing System (MBS), shown in Fig. (2-2) for the International Space Station (ISS). Figure 2-2. MDRobotics Mobile Servicing System (MBS) [ 42 ].

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13 The European Space Agency (ESA) and Rosaviakosmos (RAKA), the Russian Space Agency, were cooperatively devel oping the European Robotic Arm (ERA) [ 45 47 ], designed to assemble and service the Russi an segment of the ISS. Fig. (2-3) shows ERA. Similarly, Japan was developing a Japanese Experiment Module Remote Manipulator System (JEMRMS) [ 48 ] for assistance with experiments conducted on and maintenance of the Japanese segment of the ISS. The system is shown in Fig. (2-4). Figure 2-3. European Robotic Arm (ERA) [ 45 ]. Figure 2-4. Japanese Experiment Module Remote Manipulator System (JEMRMS) [ 48 ].

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14 Also, Japan planned to c onduct a free-flying robotic servicing experiment ( 38 p. 35), Engineering Test Satellite (ETS) VII [ 49 50 ], to verify automated rendezvous and docking technologies ( 38 p. 35), shown in Fig. (2-5). Figure 2-5. Engineering Test Satellite (ETS) VII [ 50 ]. In almost a decade later, NAS A Exploration Team (NEXT) [ 51 ] commissioned a report to provide a survey of the existing and future state-of-the-art robotic capabilities based on the functionalities pertaining to two mission types: planetary surface exploration and in-space operations. This work is re stricted to the robotics for the in-space operations. In-space operations functionali ties consisted of assembly, inspection, maintenance, and human Extra Vehicular Activity (EVA) assistance. For in-space assembly, the existing in-space robots we re found to be limited to SRMS [ 39 ] and SSRMS [ 40 41 ], which were used to move larg e objects and were controlled through teleoperation. The SPDM [ 40 42 44 ], capable of delicate maintenance and servicing through teleoperation, was scheduled to launch in 2005. Langleys Automated Structural Assembly Robot testbed [ 52 ] and Carnegie Mellon Universitys (CMU) Skyworker [ 53 55 ], a prototype Assembly, Inspection, and Maintenance (AIM) robot, shown in Fig. (2 -6) and (2-7) respectively, were able to autonomously assemble fixed structures.

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15 The ROTEX [ 56 ] robot, shown in Fig. (2-8), deve loped by Germany, flew in April 1993 as part of STS-55 to demonstrate capture and assembly, performed through teleoperation and some autonomy, as well as connecting components. Figure 2-6. Langleys Automated Structural Assembly Robot [ 52 ]. A) Robot Arm with End-Effector, B) CloseUp of End-Effector. Figure 2-7. CMU Skyworker [ 54 ]. A) Demonstration Prototype, B) Model. Figure 2-8. German Robot ROTEX [ 56 ].

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16 The Robonaut [ 57 ] was designed by Robot System s Technology Branch at NASA's Johnson Space Center (JSC) in a collaborativ e effort with DARPA. The Robonaut [ 57 ] is an anthropomorphic robotic system with the fu nctionalities capable to assist or replace an astronaut during EVA. As a ground demonstration test bed, Robonaut [ 57 ] performed successful assembly tasks and is shown in Fig. (2-9). Also, Ranger [ 24 27 ] conducted assembly, maintenance, and human EVA assist ance in a neutral buoyancy facility of the University of Maryland (UMD). The Beam Assembly Teleoperator (BAT) [ 25 26 ] built by UMD was able to perform space construction tasks and also to repair satellites, se rvice space hardware, and work in cooperation with astronaut s during its flight on STS-61 B ( 26 p. 1). BAT is shown in Fig. 2-10. Figure 2-9. NAS A Robonaut [ 57 ]. Figure 2-10. University of Maryland Beam Assembly Teleoperator (BAT) [ 26 ].

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17 In 1997, ETS-VII [ 49 50 ] performed rendezvous and docking maneuvers followed by assembly of small structures. After r unning into initial probl em of contacting the satellite, the teleoperat ed mission was successful in verify ing the basic level technologies, but did not include all th e scheduled experiments. For in-space inspection there were no operating robots in space. A German inspection robot experiment Inspektor [ 58 ], designed for in-space operations of Russian Space Station MIR, failed while in flight. AERCam SPRINT [ 28 29 ] teleoperated freeflyer, built by NASA Johnson Sp ace Center (JSC) and successf ully flight tested during STS-87, performed maneuvers and observation su ch as video imaging of Orbiter and ISS to evaluate the efficiency of the free-fl yer. AERCam Integrated Ground Demonstration (IGD) [ 27 28 ] was built and tested on the airbea ring table, capable of autonomously scanning a mock-up of a spacecraft. AERCam Engineering for Complex Systems (ECS) [ 27 ], geared towards advancing autonomy, was under development. Fig. (2-11) shows two generations of AERCam. Figure 2-11. NASA AERCam [ 29 ]. A) AERCam SPRINT, B) Mini AERCam. UMD Space Systems Laboratory (SSL) deve loped the Supplemental Camera and Maneuvering Platform (SCAMP) [ 26 27 ], neutral buoyancy flying camera robot

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18 designed for assistance with viewing during EVA, shown in Fig. (2-12). The first and second generations of the camera were tested in the Neutral Buoyancy Research Facility (NBRF), while the third generation is still in the conceptual stage of the development. Figure 2-12. University of Maryland SCAMP [ 27 ]. In-space maintenance included repla cement of the components, which was demonstrated in space by ROTEX [ 56 ], ETS-VII [ 49 50 ], and BAT [ 25 26 ] robots and on the ground by Ranger [ 24 27 ]. Ranger [ 24 27 ] and Robonaut [ 57 ] have also worked on the ability to access obstructed components. In-space human EVA assistance was provided only by SRMS [ 7] and SSRMS [ 40 41 ]. Robonaut [ 57 ] and Ranger [ 24 27 ] were the ground test beds that demonstrated the capability of assisting the astronauts with handing over tools, holding objects, and shining lights through teleopera tion. Later, Robonaut [ 57 ] was able to hand tools autonomously. It was desired for the robots to perfor m assigned tasks like a human, and this was predicted to be fully realiz ed only under teleoperation ( 51 p. 10). The outcome of the report suggested that fully autonomous robot s would require signi ficant technological developments and continuous monitoring from the ground. However, only a sustained engineering effort focused on developing me thodologies and gaining experience in the

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19 role of robots in space exploration ( 51 p. 11-12) was required to achieve the necessary future robotic capabilities. The robotic missions capable of caring out tasks previously described were becoming more necessary. Composing a large monolithic satellite out of smaller ones was expensive because the spacecraft was designed to carry out multiple missions, therefore having complex structure to support those missions. Such satellites were meant to stay in space for prolonged time, so it was advantageous to send a repair mission, if for any reason the satellite became disabled. The capability and difficulty associated with capturing a satellite that had so me angular motion, whether it was initially stable or not, was realized. In late 70s Kaplan and Nadkarni [ 59 ] investigated several possibilities that could be used for passivating the sate llite to assist in safe capture. The capturing missions were usually c onducted by astronauts through EVA, which was extremely dangerous. This was demonstrat ed when the Shuttle crew of flight STS49 was tasked with capturing and reboosting satellite International Telecommunications Satellite Consortium (INTELSAT) VI [ 60 61 ] satellite that was stranded in the incorrect orbit due to failure of the Titan rocket to put the satellite into the correct orbit. Numerous captures of the satellite were attempted, re sulting in the longest spacewalks ever performed. The problems that arose during EVA were poor visibility of the Earths dark side, insufficient positioning of the end of the robot arm in the attempt to even come near the satellite, and unexpected susceptibility of the satellite to wobbling ( 61 p. 1). After several failures it was decided to perfor m a three-man space walk. Three astronauts assembled a bridge that was the centerpie ce of the Assembly of Space Station by EVA Methods (ASEM) experiment, providing a platform for a triangular formation of the three

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20 astronauts. Standing 120 degrees apart, they grabbed the rim of the satellite to slow and eventually stop its rotation ( 61 p. 1). This mission was proved to be difficult, putting the astronauts lives in great danger. Another example was several servicing missions performed by the astronauts on the Hubble Space Telescope (HST) [ 62 ]. The first servicing mission was conducted in 1993 by the crew of STS-61 to install new equi pment and correct the imperfection in the HSTs primary mirror. The STS-82 perf ormed the second servicing mission in 1997, replacing the old instruments with updated equipment that improved the functionality of the HST. In 1999, STS-103 performed part A of the third servicing mission by replacing failed gyroscopes and making several additions Part B of the third mission and the fourth mission have been cancelled as of Marc h 2004 due to risk concerns in the light of the recent shuttle Columbia tragedy. According to the President Bushs [ 37 ] space mandate, all further shuttle missions, hoped to resume in 2005, would focus on finishing the ISS. HST successful functionality was dependable on the regular servicing missions. Suggestions [ 63 ] have been made to put forth a robotic servicing mission sometime by 2009 to save the HST. A group of scien tists and engineers at NASA Goddard Space Flight Center (GSFC) is considering s ubmitted proposals for a possible mission. NASA JSCs Robonaut [ 57 ] and UMDs Ranger [ 24 27 ] are two of the technologies under the consideration. Several autonomous robotic services we re and are under development due to the increasing demand. Orbital Sciences Corpor ation received sponsorship from NASA in 2001 to design, built, and test the Dem onstration of Autonomous Rendezvous Technology (DART) [ 64 65 ], shown in Fig. (2-13).

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21 Figure 2-13. Orbital Sciences DART [ 64 ]. DART [ 64 65 ], initially scheduled for flight in the fall 2004, was to be the first to locate and rendezvous with a sa tellite completely autonomously. Previously, astronauts had to control the vehicle via teleoperation in order to accomplish any rendezvous and servicing operations. Af ter several delays DART [ 64 65 ] was successfully launched on April 15, 2005 using an Orbital Sciences Pe gasus Launch Vehicle and was scheduled to rendezvous and perform close proximity ope rations, including station keeping, docking axis approach, circumnavigation, and a collision avoidance maneuver ( 64 p. 1), via Advanced Video Guidance Sensor (AVG S). As reported by Spaceflight Now [ 66 ], DART suffered from problems with its guida nce system from the start and, coming to approximately within 300 m to the target sate llite, ran out of fuel, causing the autopilot to initiate the retirement segment of the missi on. It was later reported by Space News [ 67 ] that DART has actually advanced further then originally thought, running into the target satellite and then maneuvering into the reti rement orbit. NASA dubbed the mission to be partially successful and formed a board to in vestigate the mishap. This particular mission is the focus of this thesis and therefore will be discussed in detail in Chapter 4.

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22 Experimental Small Satellite-10 (XSS-10) [ 68 69 ] developed by the U.S. Air Force to evaluate future applications of micro-satellite technologies such as rendezvous, inspection, docking, and close-proximity maneuvering around orbiting satellites. Launched on January 29, 2003, the space robotics mission was pronounced a success. The flight experiment verified semiau tonomous on-orbit rendezvous and inspection capabilities. The XSS-10 was the first de monstration of an autonomous inspection of another resident space object using a highly maneuverable mi cro-satellite. The flight experiment validated the design and operati ons of the micro-satellite's autonomous operations algorithms, the integrated optical camera, and the star sensor design. The XSS-10 program team also verified the critic al station keeping, maneuvering control, and logic guidance and control software necessa ry for autonomous navigation. The ground control capability, innovatively develope d for XSS-10, enabled a small team to successfully interpret the real-time data and control the spacecraft during its short mission [ 69 ]. Experimental Small Satellite-11 (XSS-11) [ 70 71 ] managed by the U. S. Air Force Research Laboratory program and build by Lock heed Martin Co. is another mission that was successfully launched on April 11, 2005. XSS-11 [ 70 71 ], shown in Fig. (2-15), is Figure 2-14. XSS-10 [ 69 ].

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23 going to test the autonomous technologies n eeded for the inspection and repair of the disabled satellites, such as approach and rendezvous maneuvers to several nonoperational US satellites. XSS-11 will also demonstrate technologies for military space surveillance ( 71 p. 1). The mission is scheduled to last approximately a year, with the rendezvous stage set to begin approximately si x weeks after launch. Additional details of the XSS-11 mission are not avai lable in the public domain. Another on-going space robotics project is TEChnology SAtellite for demonstration and verification of Space systems (TECSAS) [ 72 73 ] by the European Aeronautic Defense and Space Company (EADS ), Babakin Space Center, and DLR-RM. The mission consists of launching target and ch aser satellites, show n in Fig. (2-16), the former equipped with a robotic arm and a docking mechanism, to verify robots capabilities for rendezvous and clos e-proximity operations on-orbit. Figure 2-15. XSS-11 [ 70 ].

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24 Figure 2-16. TECSAS [ 73 ]. A) Chaser Satellite, B) Chaser Capturing Target. Also, the Phantom Works division of Boei ng was selected to compete second phase of the Orbital Express [ 74 75 ] project. Phase II consists of finalize th e design, develop and fabricate a prototype servicing satellit e, the Autonomous Space Transport Robotic Operations satellite (ASTRO), and a surrogate serviceable satellite, NextSat, and conduct an on-orbit demonstration to validate the technical feasibility and mission utility of autonomous, robotic on-orbit satellite servicing ( 74 p. 1). Fig. (2-17) shows artists rendering of the mission. It is also a goal to develop a standard upgr adeable vehicle that would be able to be used for a variety of satellite servicing missions. The mission is set to launch in September of 2006. Figure 2-17. Orbital Express ASTRO [ 75 ].

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25 Another space robotics project is Spacecr aft for the Universal Modification of Orbits (SUMO) [ 35 ], sponsored by the Defense Advanced Research Projects Agency (DARPA) and implemented by the Naval Cent er for Space Technology. The servicing spacecraft, shown in Fig. (2-18), is going to demonstrate the integration of machine vision, robotics, mechanisms, and autonomous control algorithms to accomplish autonomous rendezvous and grapple of a variety of interfaces traceable to future spacecraft servicing operations ( 35 p. 1). A demonstration of the prototype is set for December, 2005, while the launch of the SUMO spacecraft is set to occur sometime in 2008. Figure 2-18. SUMO Servicing Spacecraft [ 35 ]. To this day a variety of systems have been developed, yet most of the recent progress is either conceptual or awaiting a f light demonstration, with the exception of the DART mission, which was partially successful. So it is time for the space robotics community to continue the journey ( 37 p. 1). 2.2 Scope of the Thesis This thesis will use the DART project as a framework and will focus on the investigation of the close-pr oximity operations phase of th e mission. The Satellite Tool

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26 Kit (STK v.6.2.) software developed by the Analy tical Graphics Inc. will be used for the visualization of the mission in the space environment. The close-proximity operation phase will be simulated using CW relative mo tion equations for the spacecraft in circular orbit. Chapter 3 will present the analytical background of the linear orbit mechanics, and Chapter 4 will describe the DART mission in detail. Different scenarios will be composed in the STK environment to observe the behavior of the DART spacecraft subjected to the full force model. Specific mane uvers will be examined for their accuracy in the relative position with respect to a target vehicle a nd for the amount of fuel and velocity used. The simulation and the results of this effort will be discussed in Chapter 5. Chapter 6 will give conclusions and recommendations for future work in this area.

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CHAPTER 3 LINEARIZED ORBIT MECHANICS AND PROXIMITY OPERATIONS This chapter discusses relative motion of two spacecrafts, orbiting in closeproximity of each other. The equations govern ing the motion of the robotic servicer with respect to the target satellite are presented for the close-proximity maneuvers. General overview of the perturbation and their effect s on the motion of the spacecraft is given. The author follows reference [ 76 ] for the formulation of this chapter. 3.1 Two-body Problem Equation of motions (EOM) for the target satellite and the robotic servicer are defined using the two-body EOM, therefore it will be discussed first. The derivation of the two-body EOM starts with two fundamental laws, Newtons second law of motion and Newtons law of universal gravitation Newtons second law of motion states that the rate of change of momentum is proportional to the force impressed and is in the same direction as that force ( 77 p. 2). () dmv F dt ma (3.1) where are constant mass, velocity vector, and acceleration vector of a body, respectively, that is subjected to the sum of the forces acting on it. ,, mva Newtons law of universal gravitation states any two bodies attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them ( 77 p. 4). 2 gGMmr F rr (3.2) 27

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28 where g F is the force acting on mass due to mass m M ,mand M are the masses of two bodies, is the vector from r M tom, and is the universal gravitational constant. 826.67010/ G 2dynecmgm x In order to derive the two-body EOM the following assumption are necessary: 1. M m 2. According to the Newtonian mechanics, the coordinate system is inertial (unaccelerating and nonrotating). 3. The bodies are spherically homogene ous (symmetric and with uniform density), which allows treating each body as a point mass located at its geometric center. 4. The only forces acting on the system are gravitational forces, which act along the line joining the cen ters of the two bodies. The system of two bodies is presented in Fig. (3-1). Let be an inertial coordinate system. Let XYZ I JK be a coordinate system displaced from but not rotating or accelerating with respect to The position vectors of bodies XYZ XYZ M and with respect to the origin of the reference frame are m XYZ M r and mr respectively. Therefore, the vector from M to is defined in equation (3.3). m mMrrr (3.3) Applying equations (3.1) a nd (3.2) with respect to XYZ 2 mmGMmrGM mr rr rrr 3 (3.4.1) 2 MMGMmrGm 3 M rr rrr r (3.4.2) Equation (3.3) is differentiated to obtain equation (3.5). Because is an inertial reference frame, the derivative of the equa tion (3.3) is found without differentiating the axes of the coordinate system. XYZ mMrrr (3.5)

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29 Equations (3.4.1) and (3.4.2) are grouped using equation (3.5). 3GMm r r r (3.6) Due to earlier assumption of M m a gravitational parameter is defined as GM and equation (3.6) takes the form of equation (3.7). 3r r r (3.7) Equation (3.7) is the relative two-body EOM (second-order, nonlinear, vector, differential equation). Figure 3-1. Geometry of Relative Motion of Two Bodies [ 76 ]. 3.2 Relative Motion This section develops the governing equa tions for the relative motion of two spacecraft in the nearly circular orbits. Th e relative position and velocity solutions are also found.

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30 3.2.1 Clohessy-Wiltshire Equations for Nearly Circular Orbits Figure (3-2) shows the geometry of the ta rget satellite and se rvicer spacecraft, which will be used to derive the EOM. The target satellite motion is the primary motion, while the motion of the servicer spacecraft is analyzed with resp ect to the target satellite. The satellite coordinate system, ( ) R SWxyz is used as the reference frame for the derivation. Figure 3-2. Coordinate Sy stem for Relative Motion. The axis points from the center of the Earth towards the target satellite along the radial vector. The radial positions and displacements along the axis are collinear with the position vector. The axis points in the direction of the velocity vector for the circular orbits and is perpendicular to the axis. The along-track positions and displacements are along the axis and are normal to th e position vector. The axis points in the direction norm al to the orbital plane. R R S R S W The two-body relative motion e quation (3.7) can be used to define the motion of the target satellite. The moti on of the servicer spacecraft is defined also using equation (3.7) with the addition of the force vector pert F which accounts for the disturbing effects such as applied thrust to move relative to th e target satellite, atmo spheric drag, third-body

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31 and oblateness of the earth effects, and solar radiation. These are de scribed in section 3.4 of this chapter. 3 tgt tgt tgtr r r (3.8) 3 s erv servpert servrr r F (3.9) Referencing the geometry of Fig. (3-2), the relative position vector of the servicer with respect to the target is found. Equation (3.10) is then differentiated twice, as was demonstrated in section (3.1). relservtgtrrr (3.10) relservtgtrrr (3.11) Equations (3.8) and (3.9) are substi tuted into equation (3.11) to yield, 3 tgt serv rel pert 3 s erv tgtr r rF rr (3.12) Additional manipulations are required in order to use equation (3.12). Equation (3.10) is rearranged as to solve for the position vector of the servicer, s ervtgtrelrrr (3.13) Equation (3.13) is divided by 3 s ervr The denominator of the right-hand side (RHS) of the equation (3.14) is rewritten using 1/2 222servtgtreltgttgtrelrelrrrrrrr 3 33 22 22tgtrel tgtrel serv servserv tgttgtrelrelrrrr r rr rrrr (3.14) Assumption of is made, andterm is factored out, 2reltgtrr 2 2tgtr

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32 3 33 2 21 2 1tgtrel serv servtgt tgtrel tgtrr r rr rr r (3.15) Using the binomial expansion of the equation (3.16), equation (3.15) can be rewritten as shown in equation (3.17). 21 3 11 ... 2! 2nnnx xnxwithn (3.16) 33 22 3 1 ... 2tgtrel tgtrel serv servtgt tgtrrrr r rrr (3.17) Equation (3.17) is substituted into the equation (3.12) to yield, 322 3 1 ... 2tgtrel tgtrel tgt rel pert tgt tgt tgtrrrr r rF rr 3r (3.18) Equation (3.18) is expanded, keeping only first-order terms. 322322 3 22tgttgtrel tgtrel rel rel rel pert tgt tgt tgtrrrrr r rr rrr F (3.19) An assumption, 2 tgtrel tgtrr r is small, is made based on the assumption of two satellites being sufficiently close to each other, and the term involving this quantity is dropped from equation (3.19), 3232 2tgttgtrel rel rel pert tgt tgtrrr rr rr F (3.20)

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33 It can be noticed that tgt tgtr R r is a unit vector in the directi on of the position vector of the target satellite, and tgt rel tgtr r r is the x component, therefore e quation (3.21) is the inertial acceleration of the servicer with respect to the target reference frame. 3 3rel rel pert tgtrrxR r F (3.21) The target satellite reference frame, R SW rotates with the motion of the spacecraft in orbit and changes with time, so further analysis of equation (3.21) is necessary. The term is with respect to the inertial frame, so it needs to be written with respect to relr R SW frame. Using equation (3.22), equation (3.21) can be rewritten as shown in equation (3.23). ////2IRRIRRIRIRRIRaar rr (3.22) where is acceleration with respect to rotating reference frame, the term is the tangential acceleration (accounts for change in the angular ra te of rotating reference frame), the Ra / RIRr // RIRIRr term is the centripetal acceler ation (accounts for the rate of change of the tangential velocity), the /2RIRr term is the Coriolis acceleration (demonstrates apparent acceleration the obs erver sees when observing spacecrafts motion from the rotation). 2RIrelrelRrelRrelRRrelrrrrr (3.23)

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34 where 3 tgtr is the mean motion of the target satellite and Rrelr term is zero because the orbit are circular. The remaining cross products are evaluated taking the ,, x yz to be the components of the in the target satellit es reference frame. relr 22 00 00 00 0Rrel RRrelRSW ry R x S xyz RSW r xyz RSW xRyS yx y R x S (3.24) Substitute equations (3.23) and (3.24) into equation (3.21), 22 3 2 22 32 322R Rrel rel pert tgt rel pertrrxRFyRxSxRyS r rxRySzWxRFyRxSxRyS (3.25) Equation (3.25) is written in the component fo rm to give the CW equations of relative motion for the nearly circular orbits. (3.26) 2 223 2x y zxyxf yxfnearcircularorbits zzf

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35 3.2.2 Position and Velocity Solutions for Nearly Circular Orbits Because the application of the soluti on will be to the maneuvers employing impulsive thrust, an assumption of 0pertF is made. This doesnt apply to the cases in which a continuous thrusting is desired. The approach of Laplace operators will be used in order to solve for the x component of the position. The x equation in (3.26) is differentiated, 2 232 32 x x y x xy (3.27) The equation in (3.26) is now substituted into equation (3.27), y 22234 xxxxx 0 (3.28) The Laplace transform of equation (3.28) is taken, 232 2() ()0ooo oxxsXssxsxxsXsx (3.29) Equation (3.29) is expanded and like terms are grouped, 32 22 2222() ()0 ()ooo o ooosXssxsxxsXsx sXsssxsxx (3.30) Equation (3.30) is now solved for () Xs 2222()oooxxx Xs s sss (3.31) The last term in equation (3.31) is expanded using partial fractions, 2 221 2 A BsC ss ss (3.32) Coefficient ,, A BC are found as follows,

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36 2 22 0 22 2 2211 11 ,0s sj sjAs ss j BsC s BC ss (3.33) Substitute equation (3.32) into equation (3.31) with coefficients from equation (3.33), 2 22 222()ooooxxxx Xs ss ss (3.34) The inverse Laplace transform is applied to equation (3.34), 22() sin()cos()oo o oxxx x tx tt (3.35) The x equation in (3.26) is substituted into (3.35), 22 223232 () sin() cos()ooo oo oxyxxy x tx t t (3.36) It is then simplified, 22 ()4 sin()3cos()oo o ooyx y x tx tx t (3.37) Equation (3.37) is differentiated to find the x component of the velocity, taking into the account that the initial conditions are constant. ()cos()32sin()oo o x txtxyt (3.38) In order to find the component of the position and velocity, the equation in (3.26) is integrated with respect to time. First, the equation (3.38) is substituted into the equation in (3.26), y y y 2cos()32sin() 2cos()232sin(oo o oo o ) y xtxyt y xtxy t (3.39)

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37 Integrate equation (3.39) twice, sin()2(32)cos() 24 cos()6sin()oo o oo oyxtxytC xx ytxtC tD (3.40) The constants of integration are found by setting 0 t in equation (3.40), 6463 22ooo o oo ooyxyCCx xx yDDy oy (3.41) Substituting the coefficients yields the component of the position to be, y 24 ()cos()6sin()63oo oo oxy yttx txyty 2o ox o o (3.42) and the component of the velocity to be, y sin()(64)cos()63oo o y xtxytx y (3.43) The approach of Laplace operators will be used in order to solve for the z component of the position. The Laplace transform of the z equation in (3.26) is taken, (3.44) 22 2() ()0oozzsZsszzZs Equation (3.44) is solved for () Z s 2222()oszz Zs ss o (3.45) The inverse Laplace transform is applied to equation (3.45), ()cos()sin()o oz ztztt (3.46) Equation (3.46) is differentiated to find the z component of the velocity, taking into the account that the initial conditions are constant.

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38 ()sin()cos()ooztztzt (3.47) The position and velocity solution to the CW equations for nearly circular orbits are written in the matrix form, ()() () ()() 12 s()1c()0 43c()00 24 6s()610 c()1 00c()oo oo oo oo oo ooxx x yy y MtNt zz z t StTt xx x yy y zz z tt t MttNt t s()30 1 00 3s()00c()2s()0 6c()1002s()4c()30 00s()00c() sin,cos tt t tt t StTtt tt sc s () (3.48) 3.3 Close-Proximity Operations This section develops the analysis for the impulsive rendezvous using the linear CW relative motion equations derived in the pr evious section. Three approaches of the target satellite by the servicer spacecraft, including R-Bar, V-Bar, and circumnavigation maneuvers (for definition refer to section 3.3.2), are described. 3.3.1 Linear Two-impulse Rendezvous In order to calculate th e change in velocity, V required for the two-impulse CW transfer, the stat e transition matrix () t in equation (3.48) is eval uated, and then equation (3.48) is solved for the appropriate state. The x yz is the frame used to derive equation

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39 (3.48) as referenced to Fig. (3-2). The servicer spacecraft ma neuvers with respect to x yz frame. The subscript specifying relative po sition (and velocity) is dropped because only the relative motion is discussed from here on. The relative position and veloci ty vectors are given by, ()()(0)()(0) rtMtrNtr (3.49) ()()(0)()(0) rtStrTtr (3.50) where and are six scalar constants that define the initial state of the orbit. (0) r (0) r The final state for the rendezvous is at the origin of the x yz frame, making The initial relative velocity ()0ffrrt or needed for the successful rendezvous is found from equation (3.49), 1()()ofrNtMt f or (3.51) The first impulse of the CW transfer instanta neously changes the initial relative velocity prior to impulse to the initial relative ve locity after the impulse or or The velocity required for the first impulse is ooovrr (3.52) The final relative velocity upon successful rendezvous is found from equation (3.50), ()() f orStrTtr o (3.53) The second impulse of the CW transfer, ac ting as a stop maneuver, instantaneously changes the final relative velocity prior to impulse f r to the final relative velocity after the impulse f r The velocity required for the second impulse is f fvr (3.54)

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40 The presented equations provide the closed form expres sions necessary to perform rendezvous. The first change in velocity positions the servicer space craft onto the path that will lead towards the ta rget satellite. The second velocity change cancels the velocity of the servicer u pon arrival at the origin of x yz frame, completing the transfer. The scheme for the linear two-im pulse rendezvous described here will be used as a basis for the simulation presented in the Chapter 5. 3.3.2 Different Approaches of the Target Satellite This section discusses three common mane uvers performed to approach or move away from a target satellite: V-Bar ( V ) Approach, R-Bar ( R )Approach, and circumnavigation. Figure 3-3. V-Bar Approach Maneuver. The x yz frame (or CW frame) is shown in Fig. (3-3). The servicer spacecraft is located directly behind, as in Fig. (3-3), or in front of the target satellite along the axis, which is in the direction of the velocity vector. Therefore, approaching or receding along this direction is known as the V-Bar Approach. As can be seen from Fig. (3-3), a number of small CW impulsive maneuvers is us ed to represent the motion of the servicer spacecraft directly along the V-Bar. y

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41 Figure 3-4. R-Bar Approach Maneuver. Fig. (3-4) shows the servicer vehicle locat ed directly below the target satellite along x axis, which is in the direction of the ra dius vector. Theref ore, approaching or receding along this direction is known as the R-Bar Approach. Similarly, the motion directly along the R-Bar direction is repres ented with multiple CW impulsive transfers. The circumnavigation maneuver, shown in Fig. (3-5), allows the servicer spacecraft to fly around the target satellite. VBar Approach, R-Bar Approach, and circumnavigation maneuvers are performed in the orbit plane; they can, for example, correspond to an inspection, docking, or collision avoidance phases of a specific mission.

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42 In all the figures a single arc represents a two-impulse CW transfer. The demonstrated maneuvers are used for the simulations presented in Chapter 5. Figure 3-5. Circumnavigation Maneuver. 3.4 Perturbations A satellite in earth orbit experiences sma ll but significant pertur bations (accelerations) due to the lack of spherical symmetry of the earth, the attraction of the moon and sun for the satellite, and, if the satellite is in low orbit, due to atmospheric drag ( 78 p. 155). The equations developed in the preceding sec tions relied on the as sumptions of the twobody problem and did not account for the disturbances affecting the orbit of the spacecraft. The force model used in the simu lations presented in Chapter 5 implements the following disturbances: atmospheric drag gravitational effects due to oblateness of the Earth, third body effects, so lar radiation, and thrust. Th erefore, a general description for each will be presented in th is section. In depth discussi on of all the perturbations and their effects can be found in [ 76 ].

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43 3.4.1 General Effect of Perturbations Disturbances on orbital motion cause secular and peri odic changes in orbital elements, as shown in Fig. (3-6). Secula r changes grow linearly with time; making the errors in these changes grow unbounded. This is demonstrated in the Fig. (3-6) with a straight line. Periodic changes are separate d into the short-peri odic and long-periodic effects. Short-periodic change s tend to repeat in equal to or less time than the satellites orbital period. Long-periodic changes repeat themselves after significantly longer time than the orbital period, usually one or two or ders of magnitude longe r. These effects are also shown in Fig. (3-6). Figure 3-6. General Effect of Pe rturbations on Orbital Elements [ 76 ]. The orbital elements ( ,,,,, aei ), described in Appendix A, are defined as fast and slow variables. Fast vari ables significantly change duri ng an orbital period, even if the perturbations are not present. True, m ean, and eccentric anomalies are examples of the fast variables. The s hort-periodic changes occur wh en the causative perturbation effect has a fast variable present. Slow va riables have slight ch anges during an orbital period, and these changes are due to the presence of disturbances. Semimajor axis, eccentricity, inclination, node, ar gument of perigee are some of the examples of the slow variables.

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44 3.4.2 Atmospheric Drag Effects Drag is a nonconservative for ce that acts in the directi on opposite to the motion of the satellite. Equation (3.55) provides the expression for the atmospheric drag presented as acceleration, 21 2D dragCAr r m r r (3.55) where is the atmospheric density, D C is the coefficient of drag, A is exposed crosssectional area of the spacecraft, mis the mass of the spacecraft, and is the magnitude of the velocity of the spacecraft. r The atmospheric drag has the effect of a ttenuating the orbit and causing a satellite to reenter the atmosphere and crash into the Earth. This is demonstrated in Fig. (3-7). Figure 3-7. Effects of Atmospheric Drag [ 76 ]. A) Lowering of the Orbit, B) Orbital Properties Under the Effect of Drag. Fig. (3-7A) shows the decreasi ng effect of drag on size and shape of the orbit, hence, change in semi-major axis and eccentricity. Fi g. (3-7B) demonstrates the variation in the orbital elements over the lifetime of a satellite. The effects of atmospheric drag are summarized below, Secular changes caused in semi-major axis, eccentricity, and inclination. Periodic changes caused in all the orbital elements.

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45 The simulations in Chapter 5 employ the Jacchia-Roberts [ 76 ] atmospheric density model, which is a high fidelity, but computa tionally expensive, time-varying model. This model is similar to Jacchia 1971, but uses analytical methods to improve on the performance. Jacchia 1971 model computes atmospheric density based on the composition of the atmosphere, which depends on the satellites altitude as well as a divisional and seasonal variation. It has valid range of 100km-2500km [ 79 ]. 3.4.3 Oblateness of the Earth Effects The Earth is not a perfect sphere, as it is often assumed, making the center of gravity (CG) not coincident w ith the center of mass (CM). The effects of the oblateness of the Earth are most prominent for low Earth orbits (LEO). The av eraging effects of the Earths spin result in a gravitational potential of the satellite with respect to the Earth as expressed in equation (3.56) [ 80 ]. 21k e kk kR UJ P RR c o s (3.56) where R is the distance from the center of the Earth to the spacecraft, e R is the equatorial radius of the Earth, is the Legendre polynomial function of the kth order, is the colatitude, and represents zonal harmonics ( is the first zonal harmonic with the most dominant effects). kP kJ 2J The two primary effects, regression of th e line of nodes and rotation of the line of apsides, are secular changes. Nodal regression is a rotation of the pl ane of the orbit about the axis of rotation of the Earth at a rate that depends on inclina tion and altitude. The torque formed is about the line of nodes, and it pulls the orbital plane towards the equator. The perturbation is observed thr ough a change in the angul ar momentum vector,

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46 which signifies the orbital precession. Fig. (3-8) demonstrates the nodal regression. The nodes move westward for direct orbits and eastward for retrograde orbits. Figure 3-8. Nodal Regression [ 76 ]. The rotation of the line of apsides is the second major effect and is applicable only to eccentric orbits. This perturbation will cau se the major axis of an elliptical path to rotate in the direction of the satellites motion if or and opposite of the direction of motion for The rate of rotation depends on inclination and altitude. Fig. (3-9) demonstr ates the apsidal regression. 63.4 i 116.6 i 63.4116.6 i

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47 Figure 3-9. Apsidal Regression [ 76 ]. The discussion of periodic effects of zonal harmonics can be found in [ 76 ]. The effects of oblateness of the Earth are summarized below, Secular changes in th e node and perigee are the primary effect. Even zonal harmonics cause secular perturbations in ,, M ; have no secular effects. ,, aei For orbits with orbital plane remains fixed, indicating no secular and periodic changes in 90 i i The magnitude of perturbations due to the oblateness of the Earth increases when the eccentricity of the orbit is increased and decreases when the semimajor axis of the orbit is increased. The simulations in Chapter 5 employ the Joint Gravity Model version 2 (JGM2) developed by NASA/GSFC Space Geodesy Branch the University of Texas Center for Space Research, and CNES. It describes the gr avity field of the Earth of the degree and order 21 (maximum 70) [ 79 ] using equation (3.56).

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48 3.4.4 Third-Body Effects The effects of third body, Sun or Moon, should be considered, especially for the satellites in orbits with high altitude. Gravitational attrac tion of a third body causes the perturbations, so it is similar to the orbital changes caused by the oblateness of the Earth. Fig. (3-10) demonstrates the phenomenon ar ising from the third body effects. The trajectories of the satellite and suns (for exam ple) orbits are shown as elliptical rings in the Fig. (3-10). The gravitati onal attraction between the rings generates a rotation of the satellite ring toward the ecliptic solar ring about the intersection line of the rings. A gyroscopic precession about the axis normal to the ecliptic ring plane is a result of a torque on the satellite ring. The Moon w ill have a similar influence, causing the regression about the axis normal to the orbital plane of the Moon. The effect of both the Sun and the Moon is a regression of the orbi tal plane about a mean pole (between the Earths and ecliptic poles). Figure 3-10. Third-Body Effects [ 76 ].

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49 The third body, the Sun or Moon, effects are summarized below, Semi-major axis does not have any secular or long-periodic changes. Secular perturbations occur in the no de and perigee. Since the Moon is closer to the Earth than the Sun, the lunar effects are more pronounced. Long-periodic changes occur in ,,, ei The changes in perigee height induc ed by the Sun and the Moon have the effect of increasing or decreasing the lifetime of the satellite for orbits affected by the atmospheric drag perturbations. The force model in simulations of Ch apter 5 takes into account the third body effect of the Sun and the Moon. 3.4.5 Solar Radiation Effects Solar radiation is a nonconservative force, which effects are more pronounced for orbits with higher altitudes. Equation (3.57) provides the expression for the acceleration due to solar pressure, 21 s olradr S AUAK rC mcR r (3.57) where is the coefficient of reflectivity, rC A is area of the spacecraft,mis the mass of the spacecraft, is percentage of the sun (a s seen from the spacecraft), K is solar flux at 1 AU, is the speed of light, c AU R is the distance from the spacecraft to the sun in AU, and is the unit position vector of the sun as seen from the spacecraft [ Sr 79 ]. The solar radiation effects are summarized below, Radiation pressure causes periodic cha nges in all the orbital elements with more pronounced effects for higher altitude orbits.

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50 Changes in perigee height can have a si gnificant effect on the lifetime of the satellite. Solar radiation effects are more significant for satellites with low mass and large surface area. The force model in simulations of Chapter 5 takes into account the effects of solar radiation. 3.4.6 Thrust The perturbations to the orbit by the accelera tion due to thrust is a planned change. Equation (3.58) provides the expression, () ,thrust thrust thrustsp c F t rF md d m I t (3.58) where s p I is specific impulse of the fuel, dm dt is the motors mass flow rate, and is the current mass of the spacecraft. cm Impulsive thrust maneuvers are implemen ted in the simulations of Chapter 5.

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CHAPTER 4 DEMONSTRATION OF AUTONOMOUS RENDEZVOUS TECHNOLOGY (DART) MISSION OVERVIEW The DART, successfully launched on April 15, 2005, was design as the first robotic servicer to verify the capabilities of autonomous rendezvous with and maneuvering in close proximity of another spacecraft, the Multiple Paths, Beyond-Line-of-Sight Communications (MUBLCOM) satellite. This chapter gives a description of the DART and MUBLCOM vehicles and an overview of th e entire mission, particularly focusing on the proximity operations phase. 4.1 DART Spacecraft The DART spacecraft was built by Orbital Sciences Co. The servicer has a total mass of 363kg, length of 6ft, and diameter of 3ft. Figure 4-1. DART Spacecr aft Configuration [ 65 ]. 51

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52 The DART vehicle, as shown in Fig. (4-1), is composed of tw o main parts. The front is the Advanced Video Guidance Sensor (AVGS) bus, and the aft is the Hydrazine Auxiliary Propulsion System ( HAPS). The specific components of the AVGS bus and HAPS can be seen in Fig. (4-2). Figure 4-2. DART Spacecraft Component Configuration [ 65 ]. Three propulsion systems are used for the control of the DART vehicle. The HAPS system has three hydrazine thrusters, which are used for all the maneuvers preceding the close-proximity phase of the mission, as well as to control pitch and yaw of the DART vehicle, and a set of six cold-gas nitrogen th rusters, which are used to provide three-axis attitude control during or bital drifts and roll c ontrol during HAPS burns ( 65 p. 9 ). The DART vehicle is also equipped with a set of 16 cold-gas nitrogen thrusters, which are used during the close-proximity operation phase for the translation and attitude control.

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53 This thesis doesnt focus on the attitude cont rol; however, the characteristics of all the propulsion systems used are given in Table (4-1) for completeness. Table 4-1. DART Ve hicle Propulsion Systems Characteristics. Propulsion System Thrust (N) Specifi c Impulse (sec) Fuel Mass (kg) HAPS Three Hydrazine Thrusters 222 236 56.88 HAPS set of 6 cold-gas nitrogen thrusters 111 NS 5.77 16 cold-gas nitrogen thrusters 3.6 60 22.68 The HAPS three hydrazine thrusters, shown in Fig. (4-3B), are located in the back of the HAPS section of the DART vehicle with the thrust direction aligned with the axis. BodyX Figure 4-3. DART Vehicl e Thrusters. A) Ortho View, B) Back View. Table (4-2) identifies the locations of th e 16 cold-gas nitrogen thrusters on the DART vehicle, also shown in Fig. (4-3). FWD and AFT represent the location of the forward and aft thrusters, respectively. 90/270 specifies the location of the thruster with respect to the circumference of the face. Table 4-2. Location of the 16 Cold-Gas Nitrogen Thrusters. Thruster # Thruster Exhaust Direction Thruster Location 1 45 Deg. +X/+Y FWD 90 2 +Y FWD 90

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54 3 -Z FWD 90 4 +Z FWD 90 5 45 Deg. +X/-Y FWD 270 6 -Y FWD 270 7 -Z FWD 270 8 +Z FWD 270 9 -X AFT 90 10 +Y AFT 90 11 -Z AFT 90 12 +Z AFT 90 13 -X AFT 270 14 -Y AFT 270 15 -Z AFT 270 16 +Z AFT 270 4.2 MUBLCOM Spacecraft The MUBLCOM, shown in Fig. (4-4), is the target satellite, which will be approached by the DART spacecraft. MU BLCOM retroreflectors were designed specifically for this missions application. Figure 4-4. MUBLCOM Sa tellite Configuration [ 65 ].

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55 Figure (4-5) presents the characteristics of the MUBLCOM orbit. This data was accessed from the satellite database of th e STK software, which contains Two-Line Element (TLE) files, regularly upda ted by the U.S. Strategic Command (USSTRATCOM). TLE sets contain orbital da ta for any satellite, and Fig. (4-6) shows the format of TLE. TLE information was us ed to define the initial state of the MUBLCOM satellite in Astroga tor module of STK. The si mulations in STK and their set up will be discussed in Chapter 5. Figure 4-5. MUBLCOM Orbit. Figure 4-6. Two-Line Elements [ 81 ].

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56 4.3 Mission 4.3.1 Launch and Rendezvous Phase As previously mentioned, the DART sp acecraft was successfully launched from Vandenberg Air Force Base (AFB) in Lompoc, California into Phasi ng Orbit 1, as shown in Fig. (4-7) as point A. Figure 4-7. DART Rendezvous Phase [ 65 ]. Phasing Orbit 1 is a circul ar parking orbit, matching the inclination and Right Ascension of the Ascending Node (RAAN) of the MUBLCOMs orbit. The DART spacecraft then performs a Hohm ann orbit transfer, a fuel and velocity efficient transfer between two coplanar circular orbits, to reac h Phasing Orbit 2, indicated as point C in Fig. (4-7). At this point, the DART spacecraft will be approximately 40 km behind and 7.5 km below the MUBLCOM satellite. Af ter allowing the DART vehicle to drift to approximately 21 km behind MUBLCOM (point D in Fig. (4-7)), the final CW transfer is performed, positioning the DART spacecraft into MUBLCOMs orbit, 3 km behind

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57 relative to the satellite. This is indicated as point E in Fig. (4-7). These maneuvers are accomplished using the three hydrazine th rusters of the HAPS propulsion system. 4.3.2 Close-Proximity Operations Phase The close-proximity operations start with a CW transfer from km to km relative to the MUBLCOM satellite. This and th e rest of the maneuvers in this phase are shown in Fig. (4-8). From km point the DART spacecraft performs forced maneuvers to move along V-Bar and R-Bar directions, wi th the exception of th e transfer to R-Bar axis and circumnavigation maneuvers, which ar e accomplished with CW transfers. Most of the close-proximity maneuvers are to be pe rformed twice as can be seen from Fig. (48). DART performs several docking axis approa ches from m to m, as to simulate the docking maneuver. However, in this th esis the DART spacecraft will perform those approaches along V-Bar axis without transfer ring to the vehicles docking axis. The maneuvers in this phase are accomplished usi ng a set of 16 cold-gas nitrogen thrusters. The close-proximity operations phase is the focus of this thesis, therefore the simulation described in Chapter 5 are rest ricted to the maneuvers shown in Fig. (4-8). The timeline for these maneuvers is given in Table (4-3), which is based on ( 65 p.23 Fig. 10). After completing the proximity operations phase, th e DART spacecraft init iates a retirement burn that is to place the vehicle into a parking orbit. Even though the general prof ile of the DART mission is followed, this thesis does not precisely reflect the real mission, and the changes and/ or simplifications made are noted where applicable. Reference [ 65 ] presents the details of the entire mission for the interested reader.

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58 Figure 4-8. DART Proxi mity Operations [ 65 ]. Table 4-3. Close-Proxim ity Operations Timeline Maneuver Duration (sec) Station-keep at -3km V-Bar 300 CW transfer to -1km 4576 Station-keep at -1km V-Bar 300 Forced motion to -200m V-Bar 3094 Station-keep at m V-Bar 300 Forced motion to -15m V-Bar 826 Station-keep at -15m V-Bar 5400 Forced motion to -5m (1) V-Bar 131 Station-keep at -5m V-Bar 600 Forced motion to -15m (1) V-Bar 134 Station-keep at -15m V-Bar 300 Forced motion to -5m (2) V-Bar 132 Station-keep at -5m V-Bar 300 Forced motion to -15m (2) V-Bar 135 Station-keep at -15m V-Bar 300 Forced motion to -100m V-Bar 473 Station-keep at -100m V-Bar 1800 Forced motion to -200m V-Bar 498 Station-keep at -200m V-Bar 300 Depart to lose AVGS tracking (-350m) V-Bar 589

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59 Return to -200m V-Bar 546 Station-keep at -200m V-Bar 1800 CW transfer to R-Bar 1562 Station-keep at 150m on R-Bar 1800 Forced motion to 100m on R-Bar 248 Station-keep at 100m on R-Bar 300 Forced motion to 300m on R-Bar 585 Station-keep at 300m on R-Bar 300 CW transfer circumnavigation to -1km 4561 Station-keep at -1km on V-Bar 3317 Forced motion to m V-Bar 3094 Station-keep at m V-Bar 300 Depart to lose AVGS tracking (-350m) V-Bar 559 Return to -200m V-Bar 528 Station-keep at -200m V-Bar 300 CW transfer to R-Bar 1565 Station-keep at 150m on R-Bar 300 Forced motion to 100m on R-Bar 255 Station-keep at 100m on R-Bar 300 Forced motion to 300m on R-Bar 658 Station-keep at 300m on R-Bar 425

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CHAPTER 5 SIMULATIONS USING SA TELLITE TOOL KIT (STK) SOFTWARE BY ANALYTICAL GRAPHICS, INC. (AGI) 5.1 Satellite Tool Kit (STK) STK is a commercial-off-the-shelf (COTS) an alysis software for land, sea, air, and space. STK has a capability to present results in both graphical and text formats in order to analyze and determine optimal solutions for space scenarios. The software is used to support all phases of a satellite systems life cycle, including: policy, concept, requirements, development, testing, deploym ent, and operations. STK also calculates data and displays multiple 2-D maps to visualize various time-dependent information for satellites and other space-relate d objects. STK is capable of generating orbi t/trajectory ephemeris and providing acquisition times and sensor coverage analysis for any of the objects modeled in the STK environment. STK/PRO, a collection of additional orbit propagators, attitude profiles, coordinate types and system sensor types, in view constraints, and city, facility, and star data bases, is available as add-on collection of modules ( 82 ). 5.1.1 STK/Advanced Visualization Option Module STK/Advanced VO is a dynamic 3-D envir onment that visualizes complex mission and orbit geometries by displaying realistic 3D views of space, airborne, and terrestrial assets; sensor projections; orbit trajectories; along with assort ed visual cues and analysis aids ( 83 ). 60

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61 5.1.2 STK/Astrogator Module STK/Astrogator is an interactive tool us ed for orbit maneuvering and space mission planning ( 84 ). Figure (5-1) displays STKs gr aphical user interface (GUI) for the Astrogator. Mission control seque nces (MCS), shown on the le ft hand side (LHS) of the figure, display the maneuvers performed duri ng the mission. The detailed information regarding each orbit maneuver is displayed in the right hand side (RHS) of the figure. Bulls-eye mark next to each maneuver segment in the MCS implies that the Astrogator targeter was used to obtain the values displayed on the RHS of Fig. (5-1). Figure 5-1. Astrogator Figure (5-2) displays STKs GUI for the As trogator targeter for the maneuver from km to km along the V-Bar. The c ontrol parameters are the values of for the impulsive thrusting needed to meet the requirements of the mission. Targeting process is started with an initial guess at the values of control parameters. The initial guesses V

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62 for have to be reasonably near a valid soluti on in order for the targeter to successfully converge. In this effort, the values for V V obtained from the CW algorithm were used as the initial guesses. Equality constraints represent the requirem ents of the mission. In this case the DART spacecraft has to complete a maneuve r for duration of 4576 seconds, traveling from km to km relative to MUBCLOM sate llite. At the end of this maneuver the DART spacecraft has to be located 1km in-t rack direction and not displaced in the cross-track and radial directions. Figure 5-2. Astrogator Targeter.

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63 The Astrogator targeter uses an iterative technique referr ed to as the differential corrector to solve, in this cas e, a system of four equations (equality constraints) and three unknowns (control parameters). The process is outlined in equation (5.1) below. intrack intrack xx radial radial yy crosstrackcrosstrack zz do dorr rr rr Srr rr rr tt (5.1) Assuming that the functions are linear, a sensitivity matrix, S containing the partial derivatives calculated numerically by r unning the trajectory with perturbed control parameters is created. PoPoPo Po Po Po PoPointrackintrack intrackintrack intrackintrack xyz radialradial radialradial radialradial xxx crosstrackcrosstrackcrosstrackcrosstrack xrrrrrr rrr rrrrrr rrr S rrrr r Pocrosstrackcrosstrack xx po po po xxxrr rr tt tt tt rrr (5.2) Linear solution is obtained by rearranging equation (5.1), 1 intrack intrack xx radial radial yy crosstrackcrosstrack zz do dorr rr rr rrS rr rr tt (5-3) The inverse of sensitivity matrix is f ound using Singular Valu e Decomposition (SVD). 5.2 Simulation Scenarios The scenarios simulated in STK emulate the close proximity operations phase of the DART mission (refer to Fig. (4-7) in Chapte r 4). The simulated scenarios begin with

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64 Phasing Orbit 2, shown as point C in Fig. (4-6), and cl ose proximity operations begin when the DART spacecraft is positioned in MU BLCOMs orbit, 3 km behind relative to the satellite (refer to point E in Fig. (4-6)). Three different types of scenarios are investigated: 1. The close-proximity operations are performed using a single two-impulse transfer for each maneuver in the phase. 2. The close-proximity operations are performed using multiple two-impulse transfers in order to re present each maneuver of the phase as essentially finite. This scenario more closely represents a motion strictly along V/RBar directions. 3. The close-proximity operations are atte mpted using finite thrusting along VBar direction. A CW algorithm based on the developments in Chapter 3 was written in Matlab (refer to Appendix B) in orde r to compute the values for the change in velocity, V for the impulsive thrusting to be used in STK. Scenarios 1 and 2 were performed in two categories: untargeted and Astrogator targeted maneuvers. Untargeted category implies that the values for obtained from the CW algorithm were directly used to implement an impulse. The Astrogator targeted category implies that the values for obtained from the CW algorithm were used as initial values, and the final values for were found by the Astrogator targeter (described in section 5.1.2) via numerical integration using an eighth order Runge-K utta-Verner integrator with ninth order error control (RKV8th9th). The purpose of considering both categories is to observe the discrepancies V V V

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65 in the relative position values that result due to infeasibility of linearized CW model for precise autonomous proximity operations. 5.2.1 Scenario 1 In this scenario the close-proximity oper ations are performed using a single twoimpulse transfer for each maneuver in the phase. This is demonstrated in Fig. (5-3). Figure (5-3A) displays the mission profile of the DART spacecraft for the closeproximity operations phase (as described in Ch apter 4). Figure (5-3B) displays the STK simulated scenario of the close-proxim ity operations phase. The trajectory of MUBLCOMs orbit is shown. The inbound maneuvers and the outbound maneuvers of the DART spacecraft along the V-Bar and R-Ba r directions with respect to the MUBLCOM satellite are shown. The circ umnavigation maneuver that takes the DART spacecraft from R-Bar to V-Bar axis is also displayed. As per the DART mission profile, stati on-keeping segments occur between each maneuver. During each station-keeping maneuver, the DART spacecrafts relative position is expected to stay constant with respect to MUBLCOM satellite. However, because the CW method is not exact, the DART spacecraft reaches the desired position slightly above or below the V-Bar axis. The ra dial offset from the V-Bar axis has a radial rate associated with it. A th reshold based on DARTs relative ra dial rate is defined, and, when exceeded, a station-keeping burn in the radial direction is triggered to correct the orbit. Figure (5-4) demonstrates the STK set up GUI for the station-keeping maneuver. The stopping condition defines the terms under which the mane uver is ended. Stationkeeping maneuvers stop when a required duration is reached. The UserSelect condition triggers the radial or bit correction burn, when the ra dial rate exceeds a specified value. Two conditions are specified; trigge ring the correct one depends on the location of

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66 the DART spacecraft (below or above the V-Ba r). The value of the threshold depends on the duration and the distance of a specific tr ansfer, which dictates the magnitude of the velocity that spacecraft will have in excess of what is required to keep constant with respect to the satellite. Figure 5-3. Scenario 1 Targeted Close-proximity Operations (Starting at km Point).

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67 Figure 5-4. Sta tion-keeping Set-up The station-keeping maneuver to stay on the V-Bar axis is shown in Fig. (5-5A). The DART spacecraft arrives at a specified lo cation (respective of the maneuver at hand) on the V-Bar axis. Then it perf orms a station-keeping maneuver for a required duration. The DART spacecraft performs radial burns in order to force itself to stay on the V-Bar axis. Upon completion of the station-keep ing maneuver, DART initiates the proceeding maneuver by placing itself onto the departure trajectory. Similar type of tolerance is set up for the station-keeping maneuvers to stay on the R-Bar axis, where the spacecraft will move in the radial (R-Bar) direction for the duration of the maneuver, as shown in Fig. (5-5B). This will keep the vehicle from moving too much in the V-Bar direction, therefore positioning DART directly be low (or above) the MUBLCOM satellite.

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68 Figure 5-5. Station-keeping. A) V-Bar, B) R-Bar. Figure (5-6) shows the DART spacecraft in the 5m vicinity of the MUBLCOM satellite, which is the closest point of a pproach in the mission profile. DARTs and MUBLCOMs local coordinate axis, VN, are displayed at their respective locations. In the VN(Velocity Normal Co-normal) coordinate frame: the C C X axis is along the velocity vector (VIn-Track), theY axis is along the orbit normal (YRCrossTrack), and the V Z axis completes the orthogonal triad ( Z XY Radial). The CW algorithm developed in this thesis (refer to Appendix B) takes into account the relative orientation between CW frame de scribed in Chapter 3 and the VNframe used in STK simulations. Velocity vectors for both vehi cles are shown (red for MUBLCOM, and light blue for DART) in the positive V-Bar direct ion. Because MUBLCOM satellites motion is the primary motion, the motion of the DART spacecraft is analyzed with respect to the MUBLCOMs VNC frame. C

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69 Figure 5-6. DART at m Relative to MUBLCOM (Poi nt of Closest Approach). The two categories of untargeted and Astrogator targeted maneuvers were performed in order to investig ate and quantify the errors in the relative position that occur due to the orbital perturbations that are neglected in the untargeted computations. Due to the accumulation of errors that would occur for multiple CW-based (i.e., untargeted) maneuvers the entire mission prof ile of the close-proximity operation phase will not be analyzed, but several V/R-Bar mane uvers in the phase will be investigated. V 5.2.1.1 V-Bar maneuvers Figure (5-7) displays the maneuvers that are considered for the DART spacecraft transferring relative to the MU BLCOM satellite. The DART sp acecrafts trajectory starts at the -3km, point A, and includes four maneuvers, each represented by a single twoimpulse transfer, positioning DART 5m behind the MUBLCO M satellite, point B. Station-keeping, similar to Fig. (5-5A) is performed between each maneuver for a specified duration.

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70 Figure 5-7. V-Bar Maneuvers. Table (5-1) displays the component values in the MU BLCOM satellites frame for the initial impulsive burn to st art each maneuver. At the beginning and the end of each transfer the DART spacecraft is assumed to be in the same orbit as MUBLCOM satellite, making the relative initial and final velocities to be zero. Therefore, the final impulsive burn is impl emented to seize any relative motion between the vehicles. These values are shown in Table (5-2) and were obtained similarly to the values in Table (5-1). VNC Table 5-1. Initial Impulse Values for V-Bar Maneuvers from km to m. Maneuver V Targeted (km/s) V Untargeted (km/s) % Error -3km to km -0.00009700879 -0.00000020840 -0.00017547925 -0.00009670000 0.00000000000 -0.00017637279 .3183 100 .5092 -1km to m -0.00000556245 0.00000000000 -0.00019624144 -0.00000528616 0.00000000000 -0.00019712353 4.9671 0 .4495 -200m to m 0.00017429195 -0.00000014229 -0.00016198627 0.00017619398 0.00000000000 -0.00016317508 1.0913 100 .7339 -15m to m 0.00008131069 0.00000000000 -0.00000985996 0.00007585655 0.00000000000 -0.00001045073 6.7078 0 5.9916

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71 Table 5-2. Final Impulse Values for V-Bar Maneuvers from km to m. Maneuver V Targeted (km/s) V Untargeted (km/s) % Error -3km to km 0.00009728156 0.00000004957 -0.00017266999 0.00009670000 0.00000000000 -0.00017637279 .5978 100 2.1444 -1km to m 0.00000558119 -0.00000029427 -0.00019687285 0.00000528616 0.00000000000 -0.00019712353 5.2861 100 .1217 -200m to m -0.00017520527 0.00000026803 -0.00016236643 -0.00017619398 0.00000000000 -0.00016317508 .5643 100 .4980 -15m to m -0.00008059413 0.00000001479 -0.00001238031 -0.00007585655 0.00000000000 -0.00001045073 5.8783 100 15.5859 Figure (5-8) displays STK Astrogators GUI set up for the maneuver from -3km to -1km and helps to understand how the values in Table (5-1) for th e untargeted category were obtained. An impulsive maneuver (named Go to -1km) is inserted into the MCS on the LHS with defining characteristics on the RH S of Fig. (5-8). Maneuver direction, thrust axes, propulsion system, a nd values of thrust vector for each direction are specified as shown. The values for the impulsive maneuver were obtained from the CW algorithm and are displayed in Table (5-1) under untargeted category. The propagate segment is inserted into the MCS to id entify the duration during which the DART spacecraft will be traveling to the -1km des tination. The motion is ceased upon arrival at the specified location via the second impulse (named Stop at -1km), which is set up similar to the first impulse and whose values are shown in Table (5 -2) in the untargeted category. The station-keeping maneuver is th en inserted into the MCS, and it keeps the DART spacecraft stationary w ith respect to the MUBCLOM satellite, as was described earlier in this section. V

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72 Figure 5-8. STK Untargeted Maneuver Set-Up. Figure (5-9) displays STK Astrogators GUI set up for the maneuver from -3km to -1km and helps to understand how the values in Table (5-1) for the Astrogator targeted category were obtained. Target ed sequence (marked with bulls -eye) is inserted into the MCS on the LHS of Fig. (5-9). An impulsive maneuver (named V-Ba r) is then inserted into the targeted sequence, defining characteri stics of which are set up on the RHS of Fig. (5-9). Maneuver direction, thru st axes, propulsion system, and values of thrust vector for each direction are specified as shown. The bu lls-eye symbol with a check mark indicates that the desired values for the initial impulse will be targeted via the Astrogator targeter (discussed in section 5.1.2). The Astrogator targeter needs an initial guess for the values, which was obtained from the CW al gorithm. The values displayed in Table (5-1) are obtained by the Astrogator targeter as was described in section 5.1.2. V V

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73 Figure 5-9. STK Astrogator Targeted Maneuver Set-Up. The propagate segment is inserted into th e targeted sequence to notify when the DART spacecraft will reach V-Bar axis. At which point, the motion is ceased upon arrival at the specified location via the second impulse (TargetStop), which is set up similar to the first impulse and whose values are shown in Table (5-2) in the Astrogator targeted category. The station-keeping maneuver is then inse rted into the MCS, and it keeps the DART spacecraft stationary with resp ect to the MUBCLOM satellite, as was described earlier in this section. Tables (5-1, 5-2) also display the per cent error in the com ponents between the untargeted and Astrogator targeted categorie s for each impulse. Untargeted category does not consider an out-of-plane motion, so the percent error displa yed in the Tables (51, 5-2) is in some cases 100%, even though th e Astrogator targeter makes very small out

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74 of-plane corrections. This is due to the a ssumption of the spherically homogeneous Earth in the case of the untargeted category, whereas the Astrogator targeted category takes in account oblateness of the Earth effects. Th e error in impulses for the in-plane motion depends on the accuracy of the relative posit ion, which sometimes can be improved by the station-keeping maneuvers in between each transfer (described below). For that reason the error in impulses for the in-plane motion has an oscillatory behavior, however, overall it is increasing. The error in the relative position after each transfer, s hown in Fig. (5-7), is analyzed next. Figures (5-10) through (5-13) show DARTs arrival at the specified point relative to MUBLCOM on the V-Bar for untar geted (A) and Astrogator targeted (B) categories. It can be observed from the numerical data on each figure that the error increases with each successive figure. The error can sometimes be reduced due to the effect of the station-keeping. This can be seen between Figs. (5-12) and (5-13).

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75 Figure 5-10. Transfer to km V-Ba r. A) Untargeted, B) Targeted.

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76 Figure 5-11. Transfer to m V-Ba r. A) Untargeted, B) Targeted. The numerical data in Fig. (5-12A) displays the achieved untargeted relative position of DART from m to m transfer. Then DA RT spacecraft performs a station-keeping maneuver (red trajectory in Fig. (5-12A)), wh ich brings it closer to V-Bar, therefore correcting the existed error in radial direction. This reduces the amount of error added on to the next transfer from m to m, shown in Fig. (5-13).

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77 Figure 5-12. Transfer to m V-Ba r. A) Untargeted, B) Targeted.

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78 Figure 5-13. Transfer to m V-Ba r. A) Untargeted, B) Targeted. Table (5-3) shows the values for the de sired relative position and the values achieved in the two categories for each direction in the MUBLCOM satellites frame; percent error is also included. The relative position values are obtained based on the two-impulse transfers used for each maneuver, and therefore related to the impulse values listed in Tables (5-1, 5-2). The error in the relative position increases for all directions with each maneuver in the untarge ted category. The error in the radial and VNC

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79 cross-track positions for the targeted category is present due to the limitations of numerical integrator based on the tolerances defined fo r the Astrogator targeter. However, the order of magnitude for the erro r in the radial and cross-track positions for the untargeted category is always greater. This error is present due the linear assumptions of the CW model and assumpti on of spherically homogeneous Earth. Table 5-3. Relative Position. Maneuver Desired (km) Targeted (km) % Error Untargeted (km) % Error -3km to km 1 0 0 .999612 .000000 .000001 .0388 0 100 1.003401 .001258 .000198 .3401 100 100 -1km to m .2 0 0 .199656 .000940 .000193 .1720 100 100 .183754 .000027 .000387 8.1230 100 100 -200m to m .015 0 0 .014809 .000332 .000000 1.2733 100 0 .005382 .004494 .000386 64.1200 100 100 -15m to m .005 0 0 .005002 .000000 .000005 .0400 0 100 .003144 .000870 .000457 37.1200 100 100 Figure (5-14A) shows the relative position error for the duration of the maneuvers, shown in Fig. (5-7), along the V-Bar directi on. Figure (5-14B) zooms in to show the error at the m point, which corresponds to the num erical difference seen from Fig. (5-13) and Table (5-3).

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80 Figure 5-14. Error in Relative Position Un targeted/Targeted. A) V-Bar Transfers for Maneuvers in Fig. (5-7), B) At m Point on the V-Bar. 5.2.1.2 R-Bar maneuvers Figure (5-15) displays the maneuvers th at are considered for the DART spacecraft transferring relative to the MU BLCOM satellite. The DART sp acecrafts trajectory starts at the 100m, point A, and includes two ma neuvers, each represented by a single twoimpulse transfer, positioning DART 300m below the MUBLCO M satellite, point B. Figure 5-15. R-Bar Maneuvers.

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81 At the beginning and the end of each tr ansfer the DART spacecraft is positioned directly below the MUBLCOM satellite, maki ng the relative initial and final in-track distance to be zero. Even though the DART spac ecraft is in a lower orbit than that of MUBLCOM, DARTs relative position must remain constant with respect to the satellite. This is enforced by the station-keeping mane uvers (refer to Fig. (5.5B)), preventing the spacecraft from moving ahead of the MUBLCOM satellite. Table (5-4) displays the component values in the MU BLCOM satellites frame for the initial impulsive burn to start each maneuver. The final impulsive burn is implemented to seize the initial mo tion, stopping at the arrival point. These values are shown in Table (5-5). The numerical data in these tables is obtained similarly to the numerical data in Tables (5-1, 5-2), described earlier. VNC Table 5-4. Initial Values for R-Bar Maneuvers from 150m to 300m. V Maneuver V Targeted (km/s) V Untargeted (km/s) % Error 150m to 100m 0.00005459135 -0.00000129335 0.00023411542 0.00005688173 0.00000000000 0.00025088475 4.1955 100 7.1628 100m to 300m -0.00015994949 -0.00000000970 -0.00019664434 -0.00017359000 0.00000000000 -0.00015434256 8.5280 100 21.5118 Table 5-5. Final Values for R-Bar Maneuvers from 150m to 300m. V Maneuver V Targeted (km/s) V Untargeted (km/s) % Error 150m to 100m -0.00005070418 -0.00000107950 -0.00015060231 -0.00005688173 0.00000000000 -0.00025088475 12.1835 100 66.5876 100m to 300m 0.00025899710 -0.00000003061 0.00051574610 0.00017359000 0.00000000000 0.00015434256 32.9761 100 70.0739

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82 Tables (5-4, 5-5) also display the per cent error in the com ponents between the untargeted and Astrogator targeted categorie s for each impulse. Untargeted category does not consider an out-of-plane motion, so the percent error displa yed in the Tables (54, 5-5) is in some cases 100%, even though th e Astrogator targeter makes very small outof-plane corrections. This is due to the a ssumption of the spherically homogeneous Earth in the case of the untargeted category, whereas the Astrogator targeted category takes in account oblateness of the Earth effects. Th e error in impulses for the in-plane motion depends on the accuracy of the relative posit ion, which sometimes can be improved by the station-keeping maneuvers in between each tr ansfer. It is more difficult to maintain a constant relative position with respect to the MUBLCOM satellite while performing the station-keeping on the R-Bar, and the increase in the error can be noticed with time. The error in the relative pos ition after each transfer is an alyzed next. Figures (516) and (5-17) show DARTs arrival at th e specified point rela tive to MUBLCOM on the R-Bar for untargeted (A) and Astrogator targeted (B) categories. It can be observed from the numerical data on each figure that the e rror increases with each successive figure. Table (5-6) shows the values for the de sired relative position and the values achieved in the two categories for each direction in the MUBLCOM satellites frame; percent error is also included. The relative position values are obtained based on the two-impulse transfers used for each maneuver, and therefore related to the impulse values listed in Tables (5-4, 5-5). The error in the relative position increases for all directions with each maneuver in the untar geted category. The error in the in-track and cross-track positions for the targeted ca tegory is present due to the limitations of numerical integrator based on the tolerances defined fo r the Astrogator targeter. VNC

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83 However, the order of magnitude for the error in the in-track and cr oss-track positions for the untargeted category is always significantl y greater. This error is present due the linear assumptions of the CW model and assu mption of spherically homogeneous Earth. Table 5-6. Relative Position. Maneuver Desired (km) Targeted (km) % Error Untargeted (km) % Error 150m to 100m 0 .1 0 .000000 .100150 .000000 0 .1500 0 .000573 .095801 .000316 100 4.1990 100 100m to 300m 0 .3 0 .000003 .299838 .000000 100 .0540 0 .113922 .420756 .000324 100 40.2520 100

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84 Figure 5-16. Transfer to 100m R-Ba r. A) Untargeted, B) Targeted.

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85 Figure 5-17. Transfer to 300m R-Ba r. A) Untargeted, B) Targeted. Figure (5-18A) shows the relative position error for the duration of the maneuvers along the R-Bar direction, shown in Fig. (515). Figure (5-18B) zooms in to show the error at the 300m point, which corresponds to the numerical di fference seen from Fig. (517) and Table (5-6).

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86 Figure 5-18. Error in Relative Position Un targeted/Targeted. A) R-Bar Transfers for Maneuvers in Fig. (5-13), B) At 300m Point on the R-Bar. 5.2.1.3 Fuel usage One of the capabilities of the Astrogator module in STK is to record the amount of fuel used for each burn executed. The fuel us age is recorded for all the maneuvers in the close-proximity operations phase, as shown in Fig. (5-3B). The two propulsion systems used for the proximity operations phase ar e the three hydrazine th rusters of the HAPS propulsion system and the 16 cold-gas nitrogen thrusters, as described in Chapter 4. According to reference [ 65 ], at the beginning of the proximity operations phase the DART spacecraft performs a switch of the propulsion systems, using the 16 cold-gas nitrogen thrusters to control the translati on of the spacecraft during this phase of the mission. However, it will be demonstrated that the first maneuver of the proximity operations, a two-impulse transfer from km to km, uses a significant amount of fuel and should be accomplished with the use of the three hydrazine thrusters of the HAPS propulsion system. Therefore, this thesis also considers a swit ch of thruster systems at 3km point in the mission to compare impact on the fuel usage.

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87 The first case considers a switch when the DART spacecraft is approximately 21 km behind the MUBLCOM satellite, which is the beginning of the proximity operations phase, as stated in Chapter 4. The fuel usage in this case is displayed in Fig. (5-19B). The total amount of fuel used is 30.786 kg, which exceeds the designated amount of 22.68 kg for 16 cold-gas nitrogen thrusters. The fuel runs out during the first maneuver of the phase, a two-impulse transfer fr om km to km. This means the DART spacecraft cannot continue with the rest of the maneuvers using this propulsion system and places itself into a retirement phase of the mission. According to the real mission [ 66 67 ] the DART spacecraft ran out of fu el approximately 100m behind the MUBLCOM satellite. The second case considers a switch when the DART spacecraft is directly 3 km behind MUBLCOM satellite on the V-Bar, whic h is the beginning of the close proximity operations phase. The fuel usage in this case is displayed in Fig. (5-19A). The total amount of fuel used is 13.412 kg (6.2724 kg three hydrazine thrusters and 7.1392 kg 16 cold-gas nitrogen thrusters), which doe s not exceeds the designated amount of 22.68 kg for 16 cold-gas nitrogen thrusters. Figure (5-19) clearly shows that the largest fuel drop is during the transfer from km to km point on the V-Bar, so it should be performed using the thre e hydrazine thrusters of the HAPS propulsion system.

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88 Figure 5-19. Fuel Usage for Maneuvers in Fig. (5-3B). A) Thrusters Switch at km VBar, B) Thrusters Sw itch at km V-Bar. 5.2.2 Scenario 2 In this scenario the close-proximity oper ations along the V-Bar are performed using multiple two-impulse transfers in order to represent each maneuver of the phase as essentially finite. This scenario more closely represents a motion strictly along V/RBar directions. This is demonstrated in Fi g. (5-20). Figure (5-20A) displays the mission profile of the DART spacecraft for the closeproximity operations phase (as described in Chapter 4). Figure (5-20B) di splays the STK simulated scenario of the close-proximity operations phase. The trajectory of MUBL COMs orbit is displayed. The inbound maneuvers and the outbound maneuvers of the DART spacecraft along the V-Bar and RBar directions with respect to the MUBLCOM satellite are shown. The circumnavigation maneuver that takes the DART spacecraft from R-Bar to V-Bar axis is also displayed. As per the DART mission profile, th e station-keeping segments (previously described in section 5.2.1) occur between each maneuver.

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89 Figure 5-20. Scenario 2 Targ eted Close-proximity Operati ons (Starting at km Point). Figure (5-21) more clearly demonstrates the finite V-Bar motion using multiple two-impulse transfers, both inbound and outbound. Figure (5-22) shows the DART spacecrafts multiple approaches of the MUBL COM satellite with the closest point of approach (5m) in the mission.

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90 Figure 5-21. Finite Motion Along V-Bar (A Number of Small CW Transfers). Figure 5-22. Multiple Approaches of MUBLCOM (-15m to m). Table (5-7) lists the timeline for the proxi mity operations phase of the mission. Note that the V-Bar and the R-Bar maneuvers are divided into multiple two-impulse transfers, and the duration for each transfer is included. In this thesis, forced motion implies a finite thrusting maneuver directly along a specified direc tion. Therefore, a reasonable approximation of finite thrusti ng maneuver is a number of multiple two

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91 impulse transfers. This is performed for a ll forced motion maneuvers listed in Table (57). Table 5-7. Close-Proxim ity Operations Timeline Maneuver Duration (sec) Station-keep at -3km V-Bar 300 CW transfer to -1km 4576 Station-keep at -1km V-Bar 300 Forced motion to -200m V-Bar 3094/16 =193 Station-keep at m V-Bar 300 Forced motion to -15m V-Bar 826/5 = 165 Station-keep at -15m V-Bar 5400 Forced motion to -5m (1) V-Bar 131 Station-keep at -5m V-Bar 600 Forced motion to -15m (1) V-Bar 134 Station-keep at -15m V-Bar 300 Forced motion to -5m (2) V-Bar 132 Station-keep at -5m V-Bar 300 Forced motion to -15m (2) V-Bar 135 Station-keep at -15m V-Bar 300 Forced motion to -100m V-Bar 473/5 = 95 Station-keep at -100m V-Bar 1800 Forced motion to -200m V-Bar 498/5 = 100 Station-keep at -200m V-Bar 300 Depart to lose AVGS tracking (-350m) V-Bar 589/3 = 196 Return to -200m V-Bar 546/3 = 182 Station-keep at -200m V-Bar 1800 CW transfer to R-Bar 1562 Station-keep at 150m on R-Bar 1800 Forced motion to 100m on R-Bar 248/4 = 62 Station-keep at 100m on R-Bar 300 Forced motion to 300m on R-Bar 585/10 = 59 Station-keep at 300m on R-Bar 300 CW transfer circumnavigation to -1km 4561 Station-keep at -1km on V-Bar 3317 Forced motion to m V-Bar 3094/16 = 193 Station-keep at m V-Bar 300 Depart to lose AVGS tracking (-350m) V-Bar 559/3 = 186 Return to -200m V-Bar 528/3 = 176 Station-keep at -200m V-Bar 300 CW transfer to R-Bar 1565 Station-keep at 150m on R-Bar 300 Forced motion to 100m on R-Bar 255/5 = 51 Station-keep at 100m on R-Bar 300

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92 Forced motion to 300m on R-Bar 658/10 = 66 Station-keep at 300m on R-Bar 425 The two categories of untargeted and Astrogator targeted maneuvers were performed in order to investig ate and quantify the errors in the relative position that occur due to the orbital perturbations that are neglected in the untargeted computations. Due to the accumulation of errors that would occur for multiple CW-based (i.e., untargeted) maneuvers the entire mission prof ile of the close-proximity operation phase will not be analyzed, but one V-Bar maneuver consisting of multiple transfers will be investigated. V 5.2.2.1 V-Bar maneuvers The V-Bar maneuver from -1km to -200m will be used for demonstration and is represented using 16 two-impulse transfers. Figu re (5-23) displays three of the transfers, from -1km to -850m, relative to the MUBLCOM satellite. Figure 5-23. V-Bar Maneuver Multiple Two-Impulse Transfers. Table (5-8) displays the component values in the MU BLCOM satellites frame for the initial impulsive burn to star t each transfer. At the beginning and the end of each transfer the DAR T spacecraft is assumed to be in the same orbit as VNC

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93 MUBLCOM satellite, making the relative initial and final velocities to be zero. Therefore, the final impulsive burn is impl emented to negate any relative motion between the vehicles. These values are shown in Table (5-9). The numerical data in these tables is obtained similarly to the numer ical data in Tables (5-1, 52), described in scenario 1. Table 5-8. Initial Values for V-Bar Maneuvers from km to -850m. V Maneuver V Targeted (km/s) V Untargeted (km/s) % Error -1km to -950m 0.00025490101 0.00000115820 .00005194577 0.00025555452 0.00000000000 0.00005196699 .2564 100 .0409 -950m to -900m 0.00025592803 0.00000020678 -0.00005189765 0.00025555452 0.00000000000 -0.00005196699 .1459 100 .1336 -900m to m 0.00025543110 0.00000013292 -0.00005173217 0.00025555452 0.00000000000 -0.00005196699 .0483 100 .4539 Table 5-9. Final Values for V-Bar Maneuvers from km to -850m. V Maneuver V Targeted (km/s) V Untargeted (km/s) % Error -1km to -950m -0.00025483049 -0.00000111354 -0.00006100773 -0.00025555452 0.00000000000 -0.00005196699 .2841 100 14.8190 -950m to -900m -0.00025582437 -0.00000015993 -0.00005228603 -0.00025555452 0.00000000000 -0.00005196699 .1055 100 .6102 -900m to m -0.00025533501 -0.00000007915 -0.00005223273 -0.00025555452 0.00000000000 -0.00005196699 .0860 100 .5088 Tables (5-8, 5-9) also display the per cent error in the com ponents between the untargeted and Astrogator targeted categorie s for each impulse. Untargeted category does not consider an out-of-plane motion, so the percent error displa yed in the Tables (58, 5-9) is in some cases 100%, even though th e Astrogator targeter makes very small out

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94 of-plane corrections. This is due to the a ssumption of the spherically homogeneous Earth in the case of the untargeted category, whereas the Astrogator targeted category takes in account oblateness of the Earth effects. Th e error in impulses fo r the in-plane motion depends on the accuracy of the relative posit ion, which sometimes can be improved by the station-keeping maneuvers in between each transfer. However, there are no stationkeeping required between multiple two-impulse transfers, but only between each maneuver. Therefore, there is nothing to corr ect the error in the ra dial direction impulse and it keeps increasing. The error in the in -track impulse is decreasing because values based on CW algorithm are known to be fairly co rrect for transfers with smaller duration. The error in the relative pos ition after each transfer is an alyzed next. Figures (524) and (5-25) show DARTs arrival at th e specified point rela tive to MUBLCOM on the V-Bar for untargeted (A) and Astr ogator targeted (B) categories. It can be observed from the numerical data on each figure that the e rror increases with each successive figure. The error could sometimes be re duced due to the effect of the station-keeping (scenario 1), but there are no such segments here. Table (5-10) shows the values for the desired relative posit ion and the values achieved in the two categories for each direction in the MUBLCOM satellites frame; percent error is also included. The relative position values are obtained based on the two-impulse transfers used for each maneuver, and therefore related to the impulse values listed in Tables (5-8, 5-9). The error in the relative position increases for all directions with each maneuver in the untarge ted category. The error in the radial and cross-track positions for the targeted category is present due to the limitations of numerical integrator based on the tolerances defined fo r the Astrogator targeter. VNC

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95 However, the order of magnitude for the erro r in the radial and cross-track positions for the untargeted category is always greater. This error is present due the linear assumptions of the CW model and assump tion of spherically homogeneous Earth. Table 5-10. Relative Position. Maneuver Desired (km) Targeted (km) % Error Untargeted (km) % Error -1km to m .95 0 0 .950054 .000000 .000002 .0057 0 100 .951264 .006600 .000224 .1331 100 100 -900m to m .85 0 0 .850061 .000000 .000003 .0072 0 100 .863787 .021861 .000287 1.6220 100 100

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96 Figure 5-24. Transfer to -950m V-Ba r. A) Untargeted, B) Targeted.

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97 Figure 5-25. Transfer to m V-Ba r. A) Untargeted, B) Targeted. Figure (5-26A) shows the relative position error for the duration of the transfers along the V-Bar direction, shown in Fig. (523). Figure (5-26B) zooms in to show the error at the m point, which corresponds to the numerical difference seen from Fig. (5-25) and Table (5-10). The DART spacecra ft performs station-keeping between each maneuver, which usually corrects the relative position error (as in scenario 1). However,

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98 there is no station-keeping between each two-impulse transfer composing a single maneuver, so the error is increasi ng, as observed in Fig. (5-26A). Figure 5-26. Error in Relative Position Un targeted/Targeted. A) V-Bar Transfers for Maneuvers in Fig. (5-21), B) At -850m Point on the V-Bar. 5.2.2.2 Fuel usage This scenario also considers a switch of thruster systems at two different points in the mission to compare impact on the fuel usag e. The first case considers a switch when DART spacecraft is approximately 21 km behind MUBLCOM satellite, which is the beginning of the proximity operations phase, as stated in Chapter 4. The fuel usage in this case is displayed in Fig. (5-27B). Th e total amount of fuel us ed is 53.4870 kg, which exceeds the designated amount of 22.68 kg for 16 cold-gas nitrogen thrusters. The fuel runs out during the first maneuver of the pha se, a two-impulse transfer from km to 3km. This means the DART spacecraft cannot co ntinue with the rest of the maneuvers using this propulsion system and places itself into a retirement phase of the mission. The second case considers a switch when DART spacecraft is directly 3 km behind MUBLCOM satellite on the V-Bar, which is the beginning of the close proximity operations phase. The fuel usage in this case is displayed in Fig. (5-27A). The total

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99 amount of fuel used is 37.3420 kg (6.2724 kg three hydrazine thrusters, 31.0696 kg 16 cold-gas nitrogen thrusters), which ex ceeds the designated amount of 22.68 kg for 16 cold-gas nitrogen thrusters. The fuel runs out during the maneuver from km to m (more specifically from m to m transfer) after circumnavigation. According to the real mission [ 66 67 ] the DART spacecraft ran out of fuel approximately 100m behind the MUBLCOM satellite. Figure 5-27. Fuel Usage for Maneuvers in Fig. (5-18). A) Thrust ers Switch at km VBar, B) Thrusters Switch at km V-Bar. 5.2.3 Scenario 3 In this scenario the close-proximity operat ions are attempted using finite thrusting along V-Bar direction. This scenario represents forced motion strictly along V/R-Bar directions. The finite thrusting maneuver depends on the efficiency with which the thruster can fire. Three different cases are analyzed in order to gain a better understanding: Finite thrusting with targeted thrust efficiency. Finite thrusting with fixed thrust effi ciency and targeted maneuver duration. Finite thrusting with fixed thrust efficiency and targeted radi al relative position.

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100 5.2.3.1 Targeted thrust efficiency In this case the thrust efficiency is set at initial value of .05 (or 5%) for the coldgas nitrogen thrusters. This means that the thrusters at maximum produce 3.6N of thrust due to the specifications (refer to Chapter 4) and at minimum 0.18N of thrust due to limitations in firing precision. The final valu e is obtained with the use of the Astrogator targeter, satisfying the specified duration and relative position constraints set forth by a maneuver. Figure (5-28) displays the STK Astrogato rs set up GUI for the maneuver km to m along the path on the V-Bar and helps to understand how the values in Table (511) were obtained. Targeted sequence (Targe t Intrack) is inserted into the MCS on the LHS of Fig. (5-28). An im pulsive maneuver in the in-tr ack direction (named V-Bar Maneuver) is then inserted into the targ eted sequence to get the DART spacecraft moving in the correct direction along V-Bar. Targeted sequence (Target Zero Radial) is inserted into the targeted sequ ence Target Intrack. A continuous finite burn in radial direction is then inserted into the targeted sequence Target Zero Radial to keep the spacecraft on the V-Bar. The finite burn constantly nulls out the velocity that naturally resulted from the initial impulse. The fini te thrusting, using the propulsion systems specified on the RHS of Fig. (5-28), ceases when the duration for the maneuver is satisfied. The bulls-eye symbol with a check mark next to the thrust efficiency (initially set to .05) indicates that the desired thrust efficiency value will be ta rgeted via the Astrogator targeter (discussed in section 5.1.2). The values displayed in Table (5-11) are obta ined by the Astrogator targeter upon convergence.

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101 Figure 5-28. Astrogator Miss ion Sequence Control (MCS). Table (5-11) lists the values of the targ eted thrust efficiency achieved for the maneuvers from km to m. These values imply that the thruster s can fire with the precision of 1.34% or less, meaning the mi nimum thrust achieved would be 0.048N or less, which is unrealistic. Table 5-11. Targeted Efficiency fo r Finite Thrusting V-Bar Maneuvers. Maneuver Fixed Duration (sec) Targeted Efficiency (%) -1km to m 3094 1.261522 -200m to m 826 1.105794 -15m to m (1) 131 .373203 -5m to m (1) 134 .326344 -15m to m (2) 132 .324918 -5m to m (2) 135 .326344 -15m to m 473 .875791 -100m to m 498 .981818 -200m to m 589 1.237266 -350m to m 546 1.342349 Figure (5-29) shows the DART spacecraft multiple approaches of the MUBLCOM satellite with the closest point of approach (5 m) in the mission. The displayed trajectory

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102 was obtained based on the values listed in the Table (5-11) for the -15m to -5m and -5m to -15m maneuvers. It can be seen that th e resulted trajectory is not represented by the two-impulse hops, but an almo st straight path along the V-Bar. This a more accurate representation a forced motion, which implies a finite thrusting mane uver directly along a specified direction. Figure 5-29. Multiple Approaches of MUBL COM Finite Thrusting (-15m to m). The targeter tries to compute the optimal solution of thrust efficiency based on the given constraints and initial conditions and does not know wh ether the computed answer is realistic or not. In theory it would be possible to have a thrust er continuously firing with a high precision for a long time, achieving the desired conditions and using very small amount of fuel. However, in reality it is impossible to make a thruster, with 3.6N of thrust and to fire with such a precision. The total amount of fuel used for the maneuvers in Table (5-11) is 8.4210 kg for 16 cold-gas nitrogen thrusters, as shown in Fig. (5-30). The amount of fuel used for the same maneuvers in targeted scenario 2 is 60secspI

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103 18.6110 kg for 16 cold-gas nitrogen thrusters. Th is demonstrates that if a thruster could fire with a high precision less fuel would be used. Figure 5-30. Fuel Usage Finite Thrusting for Maneuvers in Table (5-11). 5.2.3.2 Fixed thrust efficiency targeted maneuver duration This case is attempted for the first two maneuvers in the phase from -1km to 200m and from -200m to -15m along V-Bar directi on. In this case the thrust efficiency is set at a fixed value of .05 (or 5%) for the cold-gas nitrogen thrusters. This means that the thrusters at maximum produce 3.6N of th rust due to the specifications (refer to Chapter 4) and at minimum 0.18N of thrust due to limitations in firing precision. The initial value for the duration is set to the desire d value (refer to Table (4-3) in Chapter 4). The final value for the duration is obtained with the use of the Astrogator targeter, satisfying the specified thrust efficiency a nd relative position cons traints set forth by a maneuver.

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104 Figure (5-31) displays the STK Astrogators set up GUI for the maneuver km to m along the path on the V-Bar and helps to understand how the values in Table (512) were obtained. Targeted sequence (Targe t Intrack) is inserted into the MCS on the LHS of Fig. (5-31). An im pulsive maneuver in the in-tr ack direction (named V-Bar Maneuver) is then inserted into the targ eted sequence to get the DART spacecraft moving in the correct direction along V-Bar. Targeted sequence (Target Zero Radial) is inserted into the targeted sequ ence Target Intrack. A continuous finite burn in radial direction is then inserted into the targeted sequence Target Zero Radial to keep the spacecraft on the V-Bar. The finite burn constantly nulls out the velocity that naturally resulted from the initial impulse. The fini te thrusting, using the propulsion systems specified on the RHS of Fig. (5-31), ceases when the duration for the maneuver is satisfied. The bulls-eye symbol with a check mark next to the duration (initially set to 3094 s ec) indicates that the desired duration value will be targeted via th e Astrogator targeter (discussed in section 5.1.2). The values displayed in Table (512) are obtained by the Astrogator targeter. Figures (5-32) and (5-33) show DARTs arrival at the specified point relative to MUBLCOM on the V-Bar for fixed thrust effici ency (A) and Astrogator targeted thrust efficiency (B) cases. Figure (5-32A) shows the trajectory that the DART spacecraft has to follow in order to satisfy the relative posit ion constraints and keep the thrust efficiency constant. Duration of 3094 sec is required to complete this ma neuver; however, the targeter converges approximate ly doubling the value, which is shown in Table (5-12) and Fig. (5-31).

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105 Figure 5-31. Astrogator Miss ion Sequence Control (MCS). Table 5-12. Fixed Efficiency (5%) for Finite Thrusting V-Bar Maneuvers. Maneuver Targeted Duration (sec) -1km to m 30946481.6339 -200m to m 82614.8733 Figure (5-33A) shows the trajectory th at the DART spacecraft has to follow in order to satisfy the relative position constraint s and keep the thrust efficiency constant for the maneuver from m to m along the V-Bar. The Astrogator targeter did not converge and could not find an optimal soluti on for the duration, and could only reach the value displayed in Table (5-12). As can be see from Fig. (5-33A) the DART spacecraft attempts to stay on the V-Bar and move forward, however, can not accomplish this maneuver with the current value of the thrust efficiency. Figures (5-32B) and (5-33B) show that the desired relative position was achieved while keeping the DART spacecraft on the V-Bar, however, with unrealistic thrust efficiency that was found by the Astrogator targeter.

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106 Figure 5-32. Finite Thrusting Transfer to m V-Bar. A) Efficiency of 5% Varied Duration, B) Targeted Efficiency Fixed Duration.

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107 Figure 5-33. Finite Thrusting Transfer to m V-Bar. A) Efficiency of 5% Varied Duration, B) Targeted Efficiency Fixed Duration. In this case the total amount of fuel used for the mane uvers in Table (5-12) is 20.2760 kg for 16 cold-gas nitrogen thrusters, as shown in Fig. (5-34). The amount of fuel used for the same maneuvers in targ eted scenario 2 is 13.7330 kg for 16 cold-gas nitrogen thrusters. This demonstrates that using a number of sma ll two-impulse transfers

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108 to represent motion directly al ong V-Bar is more fuel-efficient that using finite thrusting to accomplish the same. Figure 5-34. Fuel Usage Finite Thrusting for Maneuvers in Table (5-12). 5.2.3.3 Fixed thrust efficiency targeted radial position This case is attempted for the first two maneuvers in the phase from -1km to 200m and from -200m to -15m along V-Bar directi on. In this case the thrust efficiency is set at a fixed value of .05 (or 5%) for the cold-gas nitrogen thrusters. This means that the thrusters at maximum produce 3.6N of th rust due to the specifications (refer to Chapter 4) and at minimum 0.18N of thrust due to limitations in firing precision. The initial value for the radial position is set to the desired value of approximately zero (considering tolerance). The final value is obtained with the use of the Astrogator targeter, satisfying the specified thrust effi ciency and duration cons traints set forth by a maneuver.

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109 Figure (5-35) displays the STK Astrogato rs set up GUI for the maneuver km to m along the path on the V-Bar and helps to understand how the values in Table (513) were obtained. Targeted sequence (Targe t Intrack) is inserted into the MCS on the LHS of Fig. (5-35). An im pulsive maneuver in the in-tr ack direction (named V-Bar Maneuver) is then inserted into the targ eted sequence to get the DART spacecraft moving in the correct direction along V-Bar. Targeted sequence (Target Zero Radial) is inserted into the targeted sequ ence Target Intrack. A continuous finite burn in radial direction is then inserted into the targeted sequence Target Zero Radial to keep the spacecraft on the V-Bar. The finite burn constantly nulls out the velocity that naturally resulted from the initial impulse. The fini te thrusting, using the propulsion systems specified on the RHS of Fig. (5-35), ceases when the radial offset for the maneuver is satisfied. The bulls-eye symbol with a check mark next to the radial offset (initially set to 0.1 km) indicates that the allowable radial offset valu e will be targeted via the Astrogator targeter (discussed in section 5.1.2). The values di splayed in Table (5-13) are obtained by the Astrogator targeter.

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110 Figure 5-35. Astrogator Miss ion Sequence Control (MCS). Table 5-13. Fixed Efficiency (5%) for Finite Thrusting V-Bar Maneuvers. Maneuver Targeted Radial Position -1km to m .1.521671 Figure (5-36) shows DARTs arrival at the specified point relative to MUBLCOM on the V-Bar for fixed thrust efficiency (A) and Astrogator targeted thrust efficiency (B). Figure (5-36A) shows the trajectory that th e DART spacecraft has to follow in order to satisfy the relative position constraints and k eep the thrust efficiency constant for the maneuver from km to m along the V-Bar. The Astrogator targeter did not converge and could not find an optimal solu tion for the radial position, and could only reach the values displayed in Table (5-13) This maneuver cannot be accomplished with the current value of the thrust efficiency. Figure (5-36B) shows that the desired relative position was achieved while keeping the DART spacecraft on the V-Ba r, however, with unrealistic thrust efficiency that was found by the Astrogator targeter.

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111 Figure 5-36. Finite Thrusting Transfer to m V-Bar. A) Efficiency of 5% Varied Radial Distance, B) Targeted Efficiency. Because the first maneuver attempted in th is case, from -1km to -200m, did not converge, the second maneuver from -200m to -15m is not attempted. In this case the total amount of fuel used for the mane uvers in Table (5-13) is 10.0730 kg for 16 cold-gas nitrogen thrusters. The amount of fuel used for the same

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112 maneuvers in targeted scenario 2 is 12.2040 kg for 16 cold-gas nitrogen thrusters, as shown in Fig (5-37). This demonstrates that using a number of small impulsive CW transfers to represent motion di rectly along V-Bar is more fuel efficient that using finite thrusting to accomplish the same. Figure 5-37. Fuel Usage Finite Thrusting for Maneuvers in Table (5-13). 5.3 Discussion of the Simulations Results 5.3.1 Relative Position In the first scenario the close-proximity operations were performed using a single two-impulse transfer for each maneuver. The results were given for the untargeted and Astrogator targeted categories. For both V-Bar and R-Bar maneuvers, the error in the relative position increases for all directions with each maneuver in the untargeted category. The error in the radial and cro ss-track positions for the targeted category is present due to the limitations of numerical in tegrator based on the tolerances defined for the Astrogator targeter. However, performing station-keeping maneuvers between each

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113 V-Bar transfer helped to reduce some of th e error in the radial direction. For the maneuvers along the R-bar the DART spacecraft was positioned directly below (on the R-Bar) the MUBLCOM satellite and maintain c onstant relative position with respect to the satellite. This was enforced by the station-keeping maneuv ers, preventing the spacecraft from moving ahead of the MUBLCOM satellite. Thes e were not as successful at correcting the error as th e V-Bar station-keeping maneuve rs. The order of magnitude for the error in the radial a nd cross-track positions for the untargeted category was always greater. This error is present due the linear assumptions of the CW model and assumption of spherically homogeneous Earth. The generated trajectory demonstrated that using a single two-impulse transfer to represent each maneuver is not a reasonable approximation of finite thrusting maneuver, and it does not correctly reflect a motion on the V/R-Bar. In the second scenario the close-pr oximity operations along the V-Bar are performed using multiple two-impulse transfer s in order to represent each maneuver of the phase as essentially finite. The result s were given for the unt argeted and Astrogator targeted categories. For the maneuvers along th e V-bar, the error in the relative position increases for all directions with each maneuve r in the untargeted category. The error in the radial and cross-track positions for the targeted category is present due to the limitations of numerical integrator based on the tolerances defined for the Astrogator targeter. However, the order of magnitude fo r the error in the radial and cross-track positions for the untargeted category is always greater. This error is present due the linear assumptions of the CW model and a ssumption of spherically homogeneous Earth. The error could not be reduced because the station-keeping is pe rformed between each

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114 maneuver, not between each transfer. The gene rated trajectory demonstrated that using a number of smaller multiple two-impulse transfers is a reasonable approximation of finite thrusting maneuver, and it reflect a motion on the V/R-Bar more accurate than the first scenario. In the third scenario the close-proximity operations are attempted using finite thrusting along V-Bar direction. The results for the targeted thrust efficiency proved to be accurate in the sense of the values for th e relative position and staying on the V-Bar. However, this case was theoretical because in reality the given thrusters would not be able to fire with the precision demanded by the Astrogator targeter in order to satisfy duration and position requirements of the mane uver. The results for the fixed thrust efficiency and targeted durat ion were performed for two maneuvers. For the maneuver from km to m the relative position re sults were accurate, however, the duration was doubled and the trajectory di d not stay along the V-Bar. The next maneuver never converged for the specified constraints. The results for the fixed efficiency and targeted radial position were not accurate because the targeter could not find a solution for even the first maneuver. The last two cases demonstrated that using realistic thrust efficiency with the finite burn would not l ead to the desired results. Th is scenario represents forced motion strictly along V/R-Bar directions. Out of the three scenarios, the third scenar io with targeted thrust efficiency most accurately represented the motion along V/RBar, satisfying the duration and relative position requirements of the proximity opera tions phase of the mission. However, it would not be able to be realized due to the propulsion system limitations.

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115 5.3.2 Fuel Usage In addition to positional errors, fuel consumed during the performance of the mission scenarios were also computed. For each scenario, two cases regarding the propulsion systems were considered: in Case 1 the propulsion system was switched at 21 km behind MUBLCOM (as specified in the DART mission profile) and in Case 2 the switch was performed at 3 km behind MUBLCOM on the V-Bar axis. In the first scenario the close-proximity operations were performed using a single two-impulse transfers for each maneuver. Th e results for Case 1 proved to be fuelefficient not exceeding the designated fuel valu e for 16 cold-gas nitrogen thrusters. In Case 2 the results showed that the assigned fu el amount for 16 cold-gas nitrogen thrusters was exceeded. This means the DART spacecraft cannot continue with the rest of the maneuvers using this propulsion system and pl aces itself into a retirement phase of the mission. A significant amount of fuel is dedicated to the maneuver from km to km, so it should be performed using the thr ee hydrazine thrusters of the HAPS propulsion system. In the second scenario the close-pr oximity operations along the V-Bar are performed using mutiple impulsive two-impulse transfers in order to represent each maneuver of the phase as essentially finite. For both Case 1 and Case 2 the amount of fuel used exceeded the allotted value for 16 co ld-gas nitrogen thrusters. This means the DART spacecraft cannot continue with the rest of the mane uvers using this propulsion system and places itself into a retirement phase of the mission. Case 2 used more fuel than Case 1 due to a significant amount of fu el dedicated to the maneuver from km to 1km. So it should be performed using th e three hydrazine thrusters of the HAPS propulsion system. The values of fuel usage also were greater th an the values from

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116 scenario 1, which was expected due a larger number of the impulsive burn performed in the scenario 2. In the third scenario the close-proximity operations are attempted using finite thrusting along V-Bar direction for first two maneuvers in the phase. Only Case 1 (the propulsion system was switched at 21 km behind MUBLCOM) was investigated. The results for the targeted thrust efficiency cat egory showed less fuel consumption than the fuel usage for the same maneuvers in the second scenario. If the thruster firing could be controlled to the precision de manded by the Astrogator targ eter in order to satisfy duration and position requirements of the ma neuver, a slow finite burn could be performed using significantly less fuel. The results for the fixed thrust efficiency and targeted duration demonstrated more fuel c onsumption than the fuel usage for the same maneuvers in the second scenario. This was expected because the correct thrust efficiency value was held. The amount of fuel used just to perform the first two maneuvers of the close-proximity operations pha se was so large that if the entire mission was to be performed, the DART spacecraft w ould exceed the allotted amount of fuel for 16 cold-gas nitrogen thrusters. The results for the fixed efficiency and targeted radial position showed a significant am ount of fuel was used in the attempt to satisfy the requirements of the first maneuver. It is pr edicted that in this case the amount of fuel used would also exceed the allowed va lue for 16 cold-gas nitrogen thrusters. Out of the three scenarios, the third scenar io with targeted thrust efficiency used less fuel. However, it would not be able to be realized due to the propulsion system limitations. The fuel consumption analysis indicated that the DART spacecraft only had sufficient fuel to perform scenario one using the propulsion switch scheme of Case 2.

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CHAPTER 6 CONCLUSION AND RECOMMENDATIONS 6.1 Conclusions In this research, the close-proximity ope rations of an autonomous robotic on-orbit servicer are investigated usi ng linearized orbital mechanics. First, a background on the evolution of the various technologies necessary for the maneuvers performed by an autonomous robotic servicer is given. Those included, but were not limited to, vision systems, guidance navigation and control, collision avoidance, grappling and docking mechanisms. Chapter 2 gave an overview of past, present, and future of robotic servicers. This chapter discussed numerous space robotic projec ts, the focus of which was to transition from teleoperated to fully autonomous missions. Chapter 3 discussed relative motion of tw o spacecrafts, orbiting in close-proximity of each other. The equations governing the moti on of the robotic servicer with respect to the target satellite were discussed and an alyzed for the close-proximity maneuvers. General overview of the perturbation and thei r effects on the motion of the spacecraft was given. Chapter 4 narrowed the focus of this thesis to the DART mission by Orbital Sciences Corporation. The investigation was focused on the proximity operations phase of the mission, where the DART spacecraf t attempts V/R-Bar approaches and circumnavigation maneuvers of the MUBLCOM satellite, also built by Orbital Sciences 117

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118 Corporation, to verify the capabilities of autonomous rendezvous with and maneuvering in close proximity of another spacecraft. Chapter 5 presents the simulations of th e close proximity operations phase of the DART mission in STK environment. Thr ee different types of scenarios were investigated: 1. The close-proximity operations are performed using a single two-impulse transfer for each maneuver in the phase. 2. The close-proximity operations are performed using multiple two-impulse transfers in order to re present each maneuver of the phase as essentially finite. This scenario more closely represents a motion strictly along V/R-Bar directions. 3. The close-proximity operations are atte mpted using finite thrusting along V-Bar direction for first tw o maneuvers in the phase. Scenarios 1 and 2 were performed in two cat egories: untargeted and Astrogator targeted maneuvers with respect to the relative positi on error. Scenario one and two were also separated into two cases in re gards to switching the propulsion systems. Three different cases were analyzed for the third scenario: Finite thrusting with targeted thrust efficiency. Finite thrusting with fixed thrust effi ciency and targeted maneuver duration. Finite thrusting with fixed thrust efficiency and targeted radi al relative position. Out of the three scenarios, the third scenario with targeted thrust efficiency most accurately represented the motion along V/RBar, satisfying the duration and relative position requirements of the proximity operations phase of the mission.

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119 In addition to positional errors, fuel consumed during the performance of the mission scenarios were also computed. For each scenario, two cases regarding the propulsion systems were considered: in Case 1 the propulsion system was switched at 21 km behind MUBLCOM (as specified in the DART mission profile) and in Case 2 the switch was performed at 3 km behind MUBLCOM on the V-Bar axis. Out of the three scenarios, the third scenar io with targeted thrust efficiency used less fuel. However, it would not be able to be realized due to the propulsion system limitations. For the finite maneuvers, the ta rgeter was able to compute an optimal solution based on the given constraints and initial conditions; however, the computed answer was not realistic. Only in theory it would be possible to ha ve a thruster (with specific characteristics) c ontinuously firing with a hi gh precision for along time, achieving the desired conditions and using very small am ount of fuel. The fuel consumption analysis indicated that realis tically the DART spacecraf t only had sufficient fuel to perform scenario one using th e propulsion switch scheme of Case 2. The results of this research clearly show that using CW methodology for the Guidance, Navigation, and Control (GNC ) during rendezvous and close-proximity operations of a fully autonomous robotic se rvicer is not acceptable. The simulation results demonstrated a significant error in the relative position bot h on the V-Bar and RBar due to perturbation effects on the orbit. However, it was acceptable to use CW algorithm to produce initial values for the two-impulse transfers. An observation regarding the DART mission can be made. A higher fidelity GNC methodology should be implemented in order for the fully autonomous mission to be successful. The result of this thesis indicates that autonomous proximity operations are not currently feasible and

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120 provides a foundation for further analysis of close-proximity operations of an autonomous robotic servicer. 6.2 Recommendations The space robotics community, embarking on the endeavor of development and advancement set forth by necessity for the au tonomous robotic servicers, is still faced with many technical challenges. Majo r areas, including Avi onics, Sensors, & Actuators, Strategic Alliance & Cooperation, Risk Mitigati on, Mechanisms & Interconnects, Rendezvous & Capture, S pacecraft Dynamics & Controls, and Machine Vision, require significant breakthrou gh in order to realize the feasibility of autonomous servicing robots.

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APPENDIX A CLASSICAL ORBITAL ELEMENTS Figure A-1. Classical Orbital Elements [ 76 ]. A) 2D view, B) 3D view. Semi-major Axis (a): The semi-major axis defines the size of the orbit. For a circular orbit the semi-major axis is the sum of th e altitude and the radius of the Earth. Eccentricity (e): The eccentricity defines the shape of the orbit. 0 01 1 1 ecircularorbit eellipticalorbit eparabolicorbit ehyperbolicorbit Inclination (i): The inclination defines the orientati on of the orbital plane. It is the angle between the normal of the equatorial plane and the normal of the orbital plane. Right Ascension of the Ascending Node ( ): RAAN defines the orientation of the spacecraft. It is the angle from the Vernal Equinox ( I ) to the ascending node (spacecrafts crossing of the equatorial plane going from south to north). 121

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122 Argument of Perigee ( ): The argument of perigee defines the orientation of the spacecraft. It is the angle from the ascendi ng node to the eccentricity vector. It is undefined for circular orbits. True Anomaly ( ): The true anomaly defines the location of the spacecraft in the orbit. It is the angle from the eccentricity vector to the position of the satellite in orbit.

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APPENDIX B CLOHESSY-WILTSHIRE EQUATIONS OF MOTION ALGORITHM This appendix holds the code written in Matlab that calculates the change in velocity needed for an impulsive CW transfer, taking into account coordinate transformation between the vehicles local reference frames, CW reference frame and reference frame used in STK simulations. VNC % May 5, 2004 % Author: Svetlana A. Gladun % This program uses CW equations to calcu late needed impulsive manuever to thrust DART closer and away relative to MUBLCOM. % Inputs x_o = -.1; % km initial relative posi tion along x-axis (DART wrt MUBLCOM) y_o = 0; % km initial relative posit ion along y-axis (DART wrt MUBLCOM) z_o = 0; %km initial relative posit ion along z-axis (DART wrt MUBLCOM) x = -.3; % km final relative positi on along x-axis (DART wrt MUBLCOM) y = 0; % km final relative posit ion along x-axis (DART wrt MUBLCOM) z = 0; % km final relative positi on along z-axis (DART wrt MUBLCOM) x_o_dot = 0; % km/s initial relative veloc ity along x-axis prior to impuslive burn (DART wrt MUBLCOM) y_o_dot = 0; % km/s initial relative veloc ity along y-axis prior to impuslive burn (DART wrt MUBLCOM) z_o_dot = 0; % km/s initial relative veloci ty along z-axis prior to impuslive burn (DART wrt MUBLCOM) t = 585; % sec duration of the manuev er (in this case also final time) % Given R = 7123.85; % km a = R; % (km) b/c circular orbit mu = 398600.5; % km^3/s^2 gravitational parameter w = sqrt(mu/(a^3)); % a ngular velocity of DART e = 0; % eccentricity of circular orbit % Define matrices from solu tion to CW equations (3D) 123

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124 M = [4-3*cos(w*t) 0 0;6*sin(w*t)-6*w*t 1 0; 0 0 cos(w*t)]; % coefficients of initial relative position for final relative position N = [(1/w)*sin(w*t) (2/w)*(1-cos(w*t)) 0;(2/w )*(cos(w*t)-1) (4/w)*sin(w*t)-3*t 0;0 0 (1/w)*sin(w*t)]; % coefficients of initial relative velocity for final relative position S = [3*w*sin(w*t) 0 0;6*w*(cos(w*t)-1) 0 0;0 0 -w*sin(w*t)]; % coefficients of initial relative position for final relative velocity T = [cos(w*t) 2*sin(w*t) 0;-2*s in(w*t) 4*cos(w*t)-3 0;0 0 cos( w*t)]; % coefficients of initial relative velocity for final relative velocity % Calculate the initial relative velocity v_i v_i = -(N^(-1))*([x;y;z] M*[x_o;y_o;z_o]) % Calculate delta_v for the maneuver delta_v_CW = v_i [x_o_dot;y_o_dot;z_o_dot] % Change the frame from CW to DART's local frame VNC for STK TF =[cos(pi/2) sin(pi/2) 0;-sin(pi/2) cos( pi/2) 0;0 0 1]*[cos(pi/2) 0 -sin(pi/2);0 1 0;sin(pi/2) 0 cos(pi/2)]; delta_v_VNC = TF*delta_v_CW % Calculate the final relative velocity v_f v_f = S*[x_o;y_o;z_o] + T*v_i

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LIST OF REFERENCES 1. Clohessy, W. H. and Wiltshire, R. S., T erminal Guidance System for Satellite Rendezvous, Journal of the Aerospace Sciences Vol. 27, No. 9, September 1960, pp. 653-674. 2. Edelbaum, T. N., Optimum Low-Thru st Rendezvous and Station Keeping, AIAA Journal Vol. 2, No. 7, July 1964, pp. 1196-1201. 3. Jensen, D. L., Kinematics of Rendezvous Maneuvers, Journal of Guidance and Control Vol. 7, No. 3, May-June 1984, pp. 307-314. 4. Cochran, J. E. and Lahr, B. S., Satelli te Attitude Motion Models for Capture and Retrieval Investigations, Aerospace Engineering Department, Auburn University, NAS8-36470, Alabama, October 1986. 5. Hewgill, L., Motion Estimation of a Freely Rotating Body in Earth Orbit, SPIE Vol. 1829 Cooperative Intelligent Robo tics in Space III (1992), pp. 444-457. 6. Fitz-Coy, N. G. and Chatterjee, A., On the Stability of Satellite Capture, International Symposium on Spacecraft Gr ound Control and Flight Dynamics SCD1, Revista Brasileira de Ciencias Mecanicas, Vol. 16, Special Issue, 1994. 7. Mierovitch, L., Bounds on the Extensi on of Antennas for Stable Spinning Satellites, Journal of Spacecraft, Vol. 11, No. 3, March 1974, pp. 202-204. 8. Fitz-Coy, N. G. and Fullerton, W., Attitude Stability of a Spinning Spacecraft with a Slowly Deploying Appendage," Proceedings of the Flig ht Mechanics/Estimation Theory Symposium NASA Goddard Space Flight Center, Greenbelt, MD, May 1994. 9. Yuan, P. and Hsu, S., Rendezvous Guidance with Proportional Navigation, Journal of Guidance, Control, and Dynamics Vol. 17, No. 2: Engineering Notes, 1994, pp. 409-411. 10. Fitz-Coy, N. G. and Liu, M., A Modified Proportional Navigation Scheme for Rendezvous and Docking with Tumbling Targets: The Planar Case, Proceedings of the Flight Mechanics/Es timation Theory Symposium NASA Goddard Space Flight Center, NASA CP-3186, Gr eenbelt, MD, May 1995, pp. 243-253. 125

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BIOGRAPHICAL SKETCH Svetlana Gladun was born in Moscow, Russia, on November 8, 1979. She immigrated to the United States on July 2, 1995. In May 2003 she received her Bachelor of Science degree in aerospace engineering. And in August 2005 she received her Master of Science degree in aerospace engineering wi th concentration in dynamics and controls. 133


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Title: Investigation of Close-Proximity Operations of an Autonomous Robotic On-Orbit Servicer Using Linearized Orbit Mechanics
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Copyright Date: 2008

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INVESTIGATION OF CLOSE-PROXIMITY OPERATIONS OF AN AUTONOMOUS
ROBOTIC ON-ORBIT SERVICE USING LINEARIZED ORBIT MECHANICS
















By

SVETLANA ANATOLYEVNA GLADUN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Svetlana A. Gladun
































This document is dedicated to my grandmother, my mother, and my father.















ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor, Dr. Norman Fitz-Coy, for his

support, dedication, patience, and encouragement throughout my undergraduate and

graduate studies.

I would like to acknowledge my committee members, Dr. Gloria Wiens and Dr.

Rick Lind, for reviewing the thesis and providing valuable comments that improved the

quality of this work.

I would like to acknowledge Mike Loucks from Space Exploration Engineering for

his dedication to and his tirelessness and patience in helping me with STK.

I would like to acknowledge Guy Savage from Orbital Science Corporation for his

assistance in acquiring information regarding the DART mission.

I would like to acknowledge my colleagues and friends from Space Systems Group,

Chun-Haur (Marvin) Chao and Andy Tatsch, for their advice, help and support, and

numerous intellectual and not so conversations. I would also like to acknowledge Fred

Leve for dedicating his time to making an ADAMS model of DART to include in my

thesis.

Last, but not least, I would like to thank my family because without them none of

this would be possible. I would like to thank my grandmother Antonina Petrovna for

playing an important role in my upbringing and for always showing me unconditional

love and encouragement in my journey through life. I would like to thank my mother

Svetlana Victorovna for her unconditional love, courage, and sacrifice in coming to this









country so her children are able to have a better life and follow their dreams. I would like

to thank my father Anatoly Mihaylovich for his unconditional love and support. I would

like to thank my brother Dmitry for believing in me and supporting the decisions I make

throughout life. I would like to thank my best friend Cori for showing me unconditional

love and support and bringing happiness into my life. I would like to thank little Dusya

for bringing love and joy into my life.
















TABLE OF CONTENTS
Page

A C K N O W L E D G M E N T S ......... .................................................................................... iv

LIST OF TABLES ......... ...... ............................... ................... .... viii

L IST O F FIG U R E S .... ...................................................... .. ....... ............... ix

A B S T R A C T .......................................... .................................................. x iii

CHAPTER

1 INTRODUCTION AND BACKGROUND .............................................................1

2 SPACE ROBOTIC SERVICES: PAST, PRESENT, AND FUTURE ................ 10

2 .1 State-of-the-A rt ......... .. .. ....... ............. .......................... .................... 10
2.2 Scope of the Thesis ........ .... .................................. ..... .. ..................... 25

3 LINEARIZED ORBIT MECHANICS AND PROXIMITY OPERATIONS ............27

3 .1 T w o-b ody P rob lem ..................................................................... ................ .. 2 7
3.2 R elative M option .............. .. .. ... .............................. ....... ........ ............ 29
3.2.1 Clohessy-Wiltshire Equations for Nearly Circular Orbits....................30
3.2.2 Position and Velocity Solutions for Nearly Circular Orbits .................35
3.3 Close-Proxim ity O perations........................................ ......................... 38
3.3.1 Linear Two-impulse Rendezvous............................................................38
3.3.2 Different Approaches of the Target Satellite............................................40
3 .4 P ertu rb action s .................. ................................................ .......... ........... 4 2
3.4.1 General Effect of Perturbations..... .......... ....................................... 43
3.4.2 Atm ospheric D rag Effects ........................................ ...................... 44
3.4.3 Oblateness of the Earth Effects .............. .............................................45
3.4 .4 T hird-B ody E effects ......................................................................... ..... 48
3.4.5 Solar R radiation E ffects................................................................... .....49
3 .4 .6 T h ru st.................................................. ................ 5 0

4 DEMONSTRATION OF AUTONOMOUS RENDEZVOUS TECHNOLOGY
(DART) M ISSION OVERVIEW ..................................... .......................... .......... 51

4.1 DART Spacecraft ................................................ .. ........ ...............5 1
4.2 M U BLCOM Spacecraft............................................... ............................. 54









4 .3 M issio n ........................ ...................................................5 6
4.3.1 Launch and Rendezvous Phase ...................................... ............... 56
4.3.2 Close-Proximity Operations Phase..........................................................57

5 SIMULATIONS USING SATELLITE TOOL KIT (STK) SOFTWARE BY
ANALYTICAL GRAPHICS, INC. (AGI) ...................................... ............... 60

5.1 Satellite Tool K it (STK ) ..................................................... ........................... 60
5.1.1 STK/Advanced Visualization Option Module .......................................60
5.1.2 STK /A strogator M odule.................................... .......................... ......... 61
5.2 Simulation Scenarios ................................. .. .... ... ..................63
5 .2 .1 S c en ario 1 .................................................................. 6 5
5.2.1.1 V -B ar m aneuvers......................................... .......... ............... 69
5.2.1.2 R -B ar m aneuvers........................................................... ... .......... 80
5 .2 .1 .3 F u el u sag e ................................................................................... 8 6
5 .2 .2 S c e n a rio 2 ............................................................................................. 8 8
5.2.2.1 V -B ar m aneuvers......................................... .......... ............... 92
5.2 .2 .2 F u el u sage............ ............................ .......... .......... ... ..... 98
5.2 .3 Scenario 3 .................................................. ................. 99
5.2.3.1 Targeted thrust efficiency ............. .................................100
5.2.3.2 Fixed thrust efficiency targeted maneuver duration..................103
5.2.3.3 Fixed thrust efficiency targeted radial position.........................108
5.3 Discussion of the Simulations Results....................................................112
5.3.1 Relative Position ................. .... ........................ .. .. .. ................ 112
5.3.2 Fuel Usage ............ ......................... .................................115

6 CONCLUSION AND RECOMMENDATIONS ...............................................117

6 .1 C o n c lu sio n s ................................................................................................... 1 1 7
6 .2 R ecom m endation s....................................................................... ..................120

APPENDIX

A CLASSICAL ORBITAL ELEM ENTS ........................................ .....................121

B CLOHESSY-WILTSHIRE EQUATIONS OF MOTION ALGORITHM...............123

L IST O F R E F E R E N C E S ......... .. ............... ................. .............................................. 125

BIOGRAPHICAL SKETCH .............. ........... ... ............. 133
















LIST OF TABLES


Table page

4-1. DART Vehicle Propulsion Systems Characteristics. ..............................................53

4-2. Location of the 16 Cold-Gas Nitrogen Thrusters ............. ..... .................53

4-3. Close-Proxim ity Operations Tim eline................................... ........................ 58

5-1. Initial Impulse Values for V-Bar Maneuvers from -3km to -5m...........................70

5-2. Final Impulse Values for V-Bar Maneuvers from -3km to -5m. ..........................71

5-3. Relative Position ........................................ ............... .. .... ...............79

5-4. AVInitial Values for R-Bar Maneuvers from 150m to 300m.............................. 81

5-5. AVFinal Values for R-Bar Maneuvers from 150m to 300m ............... ...............81

5-6. Relative Position................. ................... .. .. .......... ................. 83

5-7. Close-Proxim ity O operations Tim eline............................................ .....................91

5-8. AV Initial Values for V-Bar Maneuvers from -1km to -850m. ............................93

5-9. AVFinal Values for V-Bar Maneuvers from -1km to -850m. ...............................93

5-10. R elative Position .................. ..................................... ................. 95

5-11. Targeted Efficiency for Finite Thrusting V-Bar Maneuvers ............ ....................101

5-12. Fixed Efficiency (5%) for Finite Thrusting V-Bar Maneuvers...........................105

5-13. Fixed Efficiency (5%) for Finite Thrusting V-Bar Maneuvers...........................110
















LIST OF FIGURES

Figure page

2-1. Ranger Telerobotic Flight Experim ent. .................................................................. 12

2-2. MDRobotics- Mobile Servicing System (MBS).................... ........................... 12

2-3. European R obotic A rm (ERA ). ...................................................... .....................13

2-4. Japanese Experiment Module Remote Manipulator System (JEMRMS). ...............13

2-5. Engineering Test Satellite (ETS) V II. ............................................. .....................14

2-6. Langley's Automated Structural Assembly Robot.................................................15

2-7. C M U Skyw orker............. ................................................................ ........ ... .... 15

2-8. G erm an R obot R O TE X .................................................................... ..................15

2-9. N A SA R obonaut ...................... .. .. ......... .. ........................... .......................... 16

2-10. University of Maryland Beam Assembly Teleoperator (BAT). .........................16

2-11. NA SA AERCam .............. .. ................ ............. ............... 17

2-12. University of Maryland SCAMP................................ ..................18

2-13. Orbital Sciences D AR T. .............................................. .............................. 21

2 -14 X S S -10 ..................................................................2 2

2 -1 5 X S S -1 1 ..................................................................2 3

2-16. TE C SA S. ..................................................................24

2-17. Orbital Express A STRO ..................................................... 24

2-18. SUMO Servicing Spacecraft. .................................................25

3-1. Geometry of Relative Motion of Two Bodies ............... ................. ..........29

3-2. Coordinate System for Relative Motion .............................................30









3-3. V -Bar A approach M maneuver ........................................................................ 40

3-4. R -B ar A approach M maneuver. ........................................ ......................................41

3-5. Circum navigation M maneuver. ........................................ ......................................42

3-6. General Effect of Perturbations on Orbital Elements.............................................43

3-7. E effects of A tm ospheric D rag ................................................................... ........... ..44

3-8. N odal R egression ................................................................................. 46

3-9. A psidal R egression. .................................................................. .. ..... 47

3-10. Third-B ody E effects. ....................... .. .. ........................ .. .. ...... ........... 48

4-1. DART Spacecraft Configuration. ........................................ .......................... 51

4-2. DART Spacecraft Component Configuration. .................................. ...............52

4-3. D A R T V vehicle Thrusters.................................................. ............................. 53

4-4. MUBLCOM Satellite Configuration. ........................................ ..................... 54

4-5. M U B LCO M O rbit. ................................................. .. .... ........ .. ...... 55

4-6. Tw o-L ine E lem ents. ........................ .. .......................... ..... ...............55

4-7. D A R T R endezvous Phase. ............................................................. .....................56

4-8. D AR T Proxim ity O perations.......................................................... ............... 58

5 -1. A stro g ato r ...............................................................6 1

5-2 A strogator T argeter........... ...... ............................................................ ........ .. ....... .. 62

5-3. Scenario 1 Targeted Close-proximity Operations (Starting at -3km Point). .........66

5-4 Station-keeping Set-up ..................................................................... ...................67

5-5. Station-keeping........... .... .............. ... ........................... ............. 68

5-6. DART at -5m Relative to MUBLCOM (Point of Closest Approach)...................... 69

5-7. V -B ar M maneuvers. ........................................... ................... .... .... .. 70

5-8. STK Untargeted Maneuver Set-Up. ........................................ ...................... 72

5-9. STK Astrogator Targeted Maneuver Set-Up ....................................................73









5-10. Transfer to km V -B ar. ............................................... .............................. 75

5-11. Transfer to -200m V -Bar. .............................................. ............................. 76

5-12. Transfer to -15m V -B ar. ............................................... .............................. 77

5-13. Transfer to -5m V -B ar. ................................................. .............................. 78

5-14. Error in Relative Position Untargeted/Targeted. ...............................................80

5-15. R -B ar M aneuvers................ .................................................. ..........80

5-16. Transfer to 100m R -B ar .................................................. ............................... 84

5-17. Transfer to 300m R -B ar ...................... ........................ ................. ............... 85

5-18. Error in Relative Position Untargeted/Targeted. ...............................................86

5-19. Fuel U sage for M aneuvers in Fig. (5-3B)..................................... .....................88

5-20. Scenario 2 Targeted Close-proximity Operations (Starting at -3km Point). .......89

5-21. "Finite" Motion Along V-Bar (A Number of Small CW Transfers)...................... 90

5-22. Multiple Approaches of MUBLCOM (-15m to -5m). ........................... 90

5-23. V-Bar Maneuver Multiple Two-Impulse Transfers............................................92

5-24. Transfer to -950m V -B ar. ........................................... ........................................96

5-25. Transfer to -850m V -Bar. .............................................. ............................. 97

5-26. Error in Relative Position Untargeted/Targeted. ...............................................98

5-27. Fuel Usage for Maneuvers in Fig. (5-20B)................................. ...............99

5-28. Astrogator Mission Sequence Control (MCS)...........................................101

5-29. Multiple Approaches of MUBLCOM Finite Thrusting (-15m to -5m). ...........102

5-30. Fuel Usage -Finite Thrusting for Maneuvers in Table (5-11)..............................103

5-31. Astrogator Mission Sequence Control (MCS)............................. ............... 105

5-32. Finite Thrusting Transfer to -200m V-Bar................................. ...............106

5-33. Finite Thrusting Transfer to -15m V-Bar................................... ............... 107

5-34. Fuel Usage -Finite Thrusting for Maneuvers in Table (5-12)............................108









5-35. Astrogator Mission Sequence Control (MCS)......................................................110

5-36. Finite Thrusting Transfer to -200m V-Bar........................................................ 111

5-37. Fuel Usage -Finite Thrusting for Maneuvers in Table (5-13)..............................112

A -1. Classical O rbital Elem ents ........................................................ ............... 121















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

INVESTIGATION OF CLOSE-PROXIMITY OPERATIONS OF AN AUTONOMOUS
ROBOTIC ON-ORBIT SERVICE USING LINEARIZED ORBIT MECHANICS

By

Svetlana A. Gladun

August 2005

Chair: Norman Fitz-Coy
Major Department: Mechanical and Aerospace Engineering

Due to the cost and time disadvantages of satellite replacement, on-orbit servicing

of disabled or outdated satellite has become a priority of the space robotics community.

This thesis begins with an overview of past, present, and future spacecraft robotic

services including a discussion on the evolution of the developments in various enabling

technologies such as vision systems, guidance navigation and control, collision

avoidance, grappling and docking mechanisms, and others. This overview is followed by

a discussion of the Clohessy-Wiltshire (CW) equation that governs the linearized relative

motion between two spacecraft (a robotic service and a target satellite in this case). The

effects of the orbital perturbations not included in the CW model are also discussed.

The Demonstration of Autonomous Rendezvous Technology (DART) mission by

Orbital Sciences Corporation with sponsorship from NASA is utilized in this thesis as the

mission profile to be investigated. The thesis investigation focuses on the proximity

operations phase of the DART mission, which includes V-Bar and R-Bar approaches and









circumnavigation maneuvers of the MUltiple paths, Beyond-Line-of-sight

COMmunications (MUBLCOM) satellite by the DART spacecraft. The CW

methodology is used to develop and analyze these maneuvers that are then simulated in a

fully nonlinear model available in Satellite Tool Kit (STK) simulation environment.

These CW results are compared with the two-point boundary value (2PBV) solution

available from STK's Astrogator module.

Three different close-proximity operations scenarios are simulated in STK. First,

the CW approximation is used perform single two-impulse transfer maneuvers. Second,

to improve the accuracy of the CW approximation, each maneuver is performed using

multiple two-impulse transfers. Third, the maneuvers are performed using finite

thrusting. In comparing the first two scenarios with the Astrogator's solution, the CW

based approximations resulted in significant relative position errors in both the V-Bar and

R-Bar maneuvers due to the neglected orbital perturbation. In addition to positional

errors, fuel consumed during the performance of the mission scenarios were also

computed. For scenarios 1 and 2, two cases regarding the propulsion systems were

considered: in Case 1 the propulsion system was switched at 21 km behind MUBLCOM

(as specified in the DART mission profile) and in Case 2 the switch was performed at 3

km behind MUBLCOM on the V-Bar axis. The fuel consumption analysis indicated that

the DART spacecraft only had sufficient fuel to perform scenario one using the

propulsion switch scheme of Case 2.

The result of this thesis indicates that precise autonomous proximity operations

based on the linearized CW model are not feasible and further analysis of close-proximity

operations of an autonomous robotic service is needed.














CHAPTER 1
INTRODUCTION AND BACKGROUND

Over the years on-orbit servicing of the disabled or outdated spacecraft became a

priority aspect of space robotics community. Due to the cost and time disadvantage of

replacing an entire satellite, repairing it or extending its life on orbit became favorable.

The research focus of the servicing missions was aimed at accomplishing the same

overall goal maintenance, repair, or retrieval of existing and future satellites.

Numerous technological barriers needed to be overcome before autonomous robotic

servicing could become a feasible operation. An autonomous robotic service had to be

capable of rendezvousing with and engaging in the docking, repair, refueling, or de-

orbiting of the target satellite. These maneuvers required significant developments in the

various technologies such as vision systems, guidance navigation and control, collision

avoidance, grappling and docking mechanisms, and others. This chapter provides a

background on the evolution of these areas over the years.

As early as 1960 the question of rendezvous became important due to the interest

and desire to build large space structures, building and assembly of which were to be

performed in space. It was envisioned for multiple satellites to join together in space to

carry out the mission as one large structure (1). That was when Clohessy and Wiltshire

[1] (CW) presented their work, closed-form solution to the rendezvous problem, which is

in widespread use to this date. They investigated a guidance system that would enable

the rendezvous of the satellites in circular orbit. The focus of this thesis is on the use of

CW equations; therefore the subject will be addressed in detail in Chapter 3. Four years









later Edelbaum [2] presented an analytical solution to optimum low-thrust power-limited

trajectories in inverse-square force field, considering rendezvous of spacecraft that

completed transfer between elliptical orbits.

In addition to traditional rendezvous, evasive maneuvers, where the target

spacecraft would detect the pursuing spacecraft and try to avoid it, were of interest. The

study of the evasive maneuvers gave an insight into the solution to the obstacle avoidance

problem, which was an important aspect of successful rendezvous mission. In 1984

Jensen [3], using the principle of proportional guidance, gave insight into the kinematics

and requirements, such as required propulsion force and amount of fuel needed, of

evasive rendezvous maneuvers. Cochran and Lahr [4] followed with investigation of the

guidance problem of intercepting a target using the proportional navigation.

The space robotics community was focused on developing free-flying autonomous

space robots that could assist in retrieving a disabled satellite, if not performing the entire

mission. In 1992 Hewgill [5] presented the means for continuously estimating the motion

of a freely rotating object to be used for the grasping tasks of the EVA Retriever, a flying

autonomous space robot. Since the rescue missions were performed mainly using the

Shuttle, Fitz-Coy and Chatterjee [6] investigated stability during capture when the rescue

vehicle is much smaller than the disabled vehicle. Additionally, the stability of force-free

spinning satellite with deployable flexible appendages was investigated by Mierovitch [7]

in 1974. Twenty years later Fitz-Coy and Fullerton [8] focused on the effects of the

deployment/retraction of the appendages on the stability of the system (tumbling satellite

and appendages), meaning that the detumbling of the satellite could be accomplished by









passive stabilization through energy dissipation, achieved by deploying flexible

appendages.

Guidance and navigation technologies were also being developed to ensure safe

path for the servicing robot to the satellite, whether it was avoiding the obstacles on the

path to the disabled vehicle or maneuvering around the satellite appendages for a desired

grasp. In 1994 Yuan and Hsu [9] presented a new guidance scheme for the final stage of

the rendezvous maneuver using augmented proportional navigation by applying the

commanded acceleration "in the direction with a bias angle to the normal direction of line

of sight (LOS)" (9, p. 410). A year later Fitz-Coy and Liu [10] extended on the work of

Yuan and Hsu [9] by applying the augmented proportional navigation for rendezvous and

docking to translating and rotating rigid bodies instead of point-masses (as was done in

1994). The point-mass approach was found to lead to the possibility of contact between

the vehicles before the maneuver was complete, and that could have potentially

influenced the overall mission in an unfavorable way. In 2002 Jacobsen et al. [11]

investigated the danger of possible collision of the service robot with uncontrolled

spinning satellite, proposing a numerical optimization method for planning the safest

kinematic trajectory for the approach phase of the capture to accomplish grasp and

stabilization of the satellite. The guidance and navigation algorithms developed in

Rendezvous and Proximity Operations Program (RPOP) for the proximity operations of

the Shuttle Orbiter were presented by Clark et al. [12] in 2003. Receiving raw sensor

data from a variety of different navigation measurement sources, RPOP simultaneously

displayed relative navigation and guidance trajectory information from each navigation

source both digitally and graphically. The addition of these algorithms to RPOP during









the manually piloted proximity operations improved "trajectory control, reduce propellant

usage, and reduce piloting variability" (12, p. 3). At the same time Carpenter and

Jackson [13] developed an optimization tool to be used with traditional CW equations,

providing ability for planning high-accuracy rendezvous maneuvers in the presence of

disturbing forces. While CW equations developed in 1960 were "useful for calculating

either the relative position of two space vehicles as a function of time or the relative

velocity to achieve a desired position" (13, p. 2), the required simplifying assumptions

lead to a possibility of errors occurring in actual use. Carpenter and Jackson [13] used

the CW equations as initial estimate for the Genetic Algorithm (GA) to solve Rendezvous

Optimization Problem.

Advancements were also made in the docking and grasping phases of the

rendezvous maneuver, implementing the use of vision-based systems. The dynamic

machine vision, specifically object tracking and visual serving, was investigated by

Nicewarner and Kelley [14] in 1992. He presented a multi-layered vision-guided system

for grasping maneuver, which resolved "the conflict between rigorous image processing

needs and rapid vision updates to the robot" (14, p. 9). A year later Ho and McClamroch

[15] proposed "a completely automatic spacecraft docking system using a computer

vision system and a docking control system" (15, p. 281).

In late 1990s emerging interest in the area of microdynamics was found.

Microdynamics of the deployed structure is "the dynamic response of the structure under

forces for which resulting strain is of order microns per meter of size or less" (16, p.

2890). The microdynamics of the deployable space structures was becoming a beneficial

emerging technology, complete understanding of which was important to the ongoing









progress. In 1997 Warren and Peterson [16] presented "an experimental characterization

of microdynamics of a prototype deployed reflector structure" (16, p. 2890), suggesting

that the microdynamics could be stabilized to nanometers of precision through the

application of transient loads after the deployment of the structure. Levine [17] in 1999,

using the flight experiment IPEX-II performed during the Space Shuttle STS-85 mission,

investigated microdynamic behavior of joint-dominate structure on-orbit, concerned with

the existence of transient impulses from undergoing rapid thermal variations or

instabilities due to internal strain energy release mechanisms. Tung et al. [18] focusing

his efforts in the microdynamics of the redundant flexible multibody systems, presented

an innovative approach in the solution of tracking problem where a mechanical system is

constrained to move along a given path.

Noor and Venneri [19] published an article in 1994 issue of Aerospace America

giving a perspective on future space systems, describing the restructure of the U.S. space

program due to economic constraints. The focus was placed on building smaller,

cheaper, and faster spacecraft oppose to large, expensive, and complex. The small

spacecraft was to be designed with the idea of complementing and/or replacing the large

complex spacecrafts. Five mission categories for small spacecraft have been identified

by NASA: astrophysics, space physics, Earth observing, commercial communications,

and planetary. The spacecraft to be developed was going to be used as Earth-to-orbit

(ETO) launchers and space transfer vehicle (STV). The vehicle was to be made fully

returnable, reusable, robust, and safer and more reliable than the existing one. Many

technologies had yet to be developed and brought to a level of reliable and safe

application. Due to the new focus on building smaller and cheaper satellites the need for









servicing the spacecraft phased out because now it was supposedly again possible to

replace a disabled satellite without incurring a significant financial loss. However, the

idea of servicing spacecraft resurfaced. Numerous studies were conducted, trying to

demonstrate that servicing a disable satellite was economically efficient in comparison to

declaring the vehicle a total loss. In 1993 Levin et al. [20] conducted a study on

feasibility of introducing a salvage/repair of civil, defense, and commercial satellites

program. "The analysis was based on historical failure data as well as conservative

estimate of the future potential for servicing missions" (20, p. 1). However, the results

showed that it was not sufficient to support standalone salvage operation, especially

without the interest of the government. Sullivan and Akin in 2001 [21] constructed three

databases for all civilian, military, commercial, and NGO spacecraft launched from 1957

through 2000 in order to provide a survey of serviceable spacecraft failures. Considering

four service types, inspection only, reboost only, basic dexterous, and complex

dexterous, it was found more cost efficient to service the satellite than declare it a total

loss. A year later Horowitz et al. [22] investigated Advanced Satellite Maintenance

(ASM) potential to be cost effective and viable method for improving satellite operations

with the example ofNewsSat. ASM missions either focused on preventing satellite

failure by performing scheduled servicing tasks such as component replacement or on

satellite recovery after failure. ASM was found to "extend satellite life and maintain the

revenue earning capabilities of the satellite" (22, p. 124).

Different servicing vehicles were being designed, whether to serve as a single robot

or a part of a fleet. Several systems reached their hardware state and either had a flight

test demonstration or currently awaiting one. In 2001 Ledebuhr et al. [23] presented









MicroSats technology development, including "...critical capabilities and technologies

necessary for proximity-operations and formation-flying of micro-satellites" (23, p. 5-

2493). The missions in mind for MicroSats included "rendezvous, inspection, proximity-

operations (formation flying), docking, and robotic servicing functions (refueling,

repowering, or repairing)" (23, p. 5-2493). Geftke et al. [24] described the Ranger

Telerobotic Shuttle Experiment (Ranger TSX) development. Ranger [24-27] "could

potentially service attached experiments and, with a free-flying base, stand alone

satellites" (24, p. 1), using both Shuttle and ground-based control stations. Wagenknecht

et al. [28] in 2003 provided "an overview of the Mini AERCam vehicle design and a

detailed description of the GN&C system design, development, and testing" (28, p. 1).

The vehicle was an improvement design of the AERCam SPRINT [28, 29], a visual

inspection robot. "The vehicle was designed for either remotely piloted operations or

supervised autonomous operations including automatic stationkeeping and point-to-point

maneuvering" (30, p. 1). A year later Fredrickson et al. [30] followed with presentation

of the docking systems technology, developed for the Mini AERCAm free-flyer, to

provide automatic deployment and retrieval. The docking system technology consisted

of two primary elements: magnetic docking, for aligning and capturing the vehicle, and

vehicle retention, for supporting the vehicle during heavy loading conditions, which

occur during launch and landing. The same year Hays et al. [31] described the

evolvement process of "a simple, low-cost, lightweight flexible-cable docking

mechanism that allows for autonomous capture and docking of one spacecraft with

another" (31, p. 1). The newest version of the docking mechanism, ASDS-II, was









"intended to provide a passively-maintained, rigid interface between two docking

spacecraft. ." (31, p. 7).

Some of the recently developed systems were still in the conceptual stage of their

development, however, others had already completed the simulation phase. In 2001

Matsumoto et al. [32, 33] presented the system concept and design of Hyper Orbital

Servicing Vehicle (HOSV), which could be used for a variety of servicing missions due

to the system's reconfigurability. At the same time Akin [25] described the technologies

developed by UMD SSL for on-orbit services, concluding with proposal of development

of Miniature Orbital Dexterous Servicing System (MODSS), a Responsive Access, Small

Cargo, Affordable Launch (RASCAL)-class free-flying robotic servicing system. The

project was steered towards producing a low cost, ultra-lightweight system with variety

of servicing applications and was based on the technologies developed for the Ranger

TSX [24-27]. Zimpher [34] in 2003 described the development of an Autonomous

Mission Manager (AMM) that accommodated highly autonomous mission planning,

execution, and monitoring of the servicing on-orbit missions. In 2004 Bosse et al. [35]

described SUMO, Spacecraft for the Universal Modification of Orbits, a technology risk

reduction program "to demonstrate the integration of machine vision, robotics,

mechanisms, and autonomous control algorithms to accomplish autonomous rendezvous

and grapple of a variety of interfaces traceable to future spacecraft servicing operations"

(35, p. 1). Also, Rogers et al. [36] presented SCOUT program, which involves the

development of the high-capability, flexible, modular spacecraft architecture, enabling

multi-mission compatibility, high payload mass and power fractions, long shelf-life, rapid

call-up for launch, rapid initialization on orbit, and manufacturability (quantity). "One of









the goals of SCOUT program is to develop a modular microsatellite architecture that is

compatible with virtually every launch vehicle (LV) in the world inventory" (36, p. 2).

The space robotics community, embarking on the endeavor of development and

advancement set forth by necessity for the autonomous robotic services, is still faced

with many technical challenges. Major areas, including "Avionics, Sensors, &

Actuators", "Strategic Alliance & Cooperation", "Risk Mitigation", "Mechanisms &

Interconnects", "Rendezvous & Capture", "Spacecraft Dynamics & Controls", and

"Machine Vision", require significant breakthrough in order to realize the feasibility of

autonomous servicing robots.














CHAPTER 2
SPACE ROBOTIC SERVICES: PAST, PRESENT, & FUTURE

[Mankind is drawn to the heavens for the same reason we were once drawn into
unknown lands and across the open sea. We choose to explore space because
doing so improves our lives, and lifts our national spirit. So let us continue the
journey. (37, p. 1)]
President George W. Bush, January 14, 2004

President Bush [37] announced a New Vision for Space Exploration Program on

January 14, 2004. He delegated new goals for the National Aeronautics and Space

Administration (NASA), stating that it was "time for America to take the next steps" (37,

p. 1). President Bush [37] outlined specific milestones for the country to achieve,

ranging from returning to the Moon to venturing out to Mars. To achieve the goals set

forth, the new space vision encompassed robotics as a key factor to successful progress as

implied by the statement "Robotic missions will serve as trailblazers the advanced

guard to the unknown" (37, p. 1). Series of robotic missions to the Moon were planned to

make the necessary preparations for human presence. Robotic explorations of Mars were

to be continued to prepare for future human expeditions. The new mandate emphasized

the importance of robotic exploration across solar system to search for resources and life

and to understand history of universe.

2.1 State-of-the-Art

Even before robotics officially was a part of President's space mandate,

technological advancements in that area were ongoing. In 1994 Lavery [38] published an

article in Aerospace America surveying the developments of space robotics community

and predicting its future. The goal of space robotics community was to create "fully









autonomous, self-contained robotic systems with considerable on-board intelligence" (38,

p. 32). However, due to the lack of the technology to develop such systems, the space

robotics community had to adopt a new approach, focusing its efforts on advancing

teleoperator technology that can be operated from the ground. The space vehicles were

divided into three main categories, in which developments were to be pursued:

extravehicular robotic (EVR) services, science payload services, and planetary surface

rovers. In order to realize the set goals significant developments in many technologies,

including "enhanced collision detection and avoidance, advanced local proximity

sensing, task-level control workstations, improved command and control architectures,

reduced mass and volume, worksite recognition and representation, improved robotic

dexterity, advanced supervisory control, and improved overall system robustness" (38, p.

37), were required.

Over the years many countries, including United States (US), Canada, Russia,

Japan, and several European countries, were competing and working together to

accomplish the desired results. US developed Ranger [24-27] Satellite Servicing Vehicle

concept in early 1990's. Ranger [24-27], a dual-arm, free-flying telerobotic flight

experiment, shown in Fig. (2-1), was designed to "conduct a simulated satellite servicing

exercise to characterize the operational capabilities of free-flying robotic system" (38, p.

35).























Figure 2-1. Ranger Telerobotic Flight Experiment [27].

Canada contributed the Shuttle Remote Manipulator System (SRMS) [39], also

know as Canadarm, and the Space Station Remote Manipulator System (SSRMS) [40,

41], known as Canadarm 2. Both were manipulators designed to maneuver and place

large loads. Also Canada developed the Special Purpose Dexterous Manipulator (SPDM)

[40, 42-44], know as Canada Hand, which was dual-arm robotic system designed to

attach to the SSRMS [40, 41]. SSRMS [40, 41], its platform the Mobile Base System

(MBS) [40], and SPDM [40, 42-44] would compose the Mobile Servicing System

(MBS), shown in Fig. (2-2) for the International Space Station (ISS).


Figure 2-2. MDRobotics Mobile Servicing System (MBS) [42].











The European Space Agency (ESA) and Rosaviakosmos (RAKA), the Russian


Space Agency, were cooperatively developing the European Robotic Arm (ERA) [45-


47], designed to assemble and service the Russian segment of the ISS. Fig. (2-3) shows


ERA.


Similarly, Japan was developing a Japanese Experiment Module Remote


Manipulator System (JEMRMS) [48] for assistance with experiments conducted on and


maintenance of the Japanese segment of the ISS. The system is shown in Fig. (2-4).


Figure 2-3. European Robotic Arm (ERA) [45].


Main arm boom 1


1 arm joint 3


Main arm base


SMain arm joint 4

Main arm joint 6


Small fine arm electronics
SCamera electronics
Wrist roll joint
:r.,.di ...r .-.i .. Torque sensor
Bm : Wn stcamera head
Boom 2- To
Elbow pitch jont:
Wrist pich joint Wrist yaw joint

Figure 2-4. Japanese Experiment Module Remote Manipulator System (JEMRMS) [48].











Also, Japan planned "to conduct a free-flying robotic servicing experiment" (38, p.

35), Engineering Test Satellite (ETS) VII [49, 50], "to verify automated rendezvous and

docking technologies" (38, p. 35), shown in Fig. (2-5).



Earth direction Space Robot Experiment System
Rabot Ar OTbital replacement unim
Rabot A rm / bTkoard
Advamned robotics had (MITI)
MnitSRmq BSweMbhN MKhM&iM TCPl I
ITaret Sate~Itel Chaser Se tqi]E

NI Flight direction
Relative approach
Rendezvous-Docking direction
Experiment System
Docking chanisem
Proximlt sensor S-band High Gain
Re~znMiu radar Antenna
GPS receiver


Figure 2-5. Engineering Test Satellite (ETS) VII [50].

In almost a decade later, NASA Exploration Team (NEXT) [51] commissioned a

report to provide a survey of the existing and future state-of-the-art robotic capabilities

based on the functionalities pertaining to two mission types: planetary surface exploration

and in-space operations. This work is restricted to the robotics for the in-space

operations. In-space operations functionalities consisted of assembly, inspection,

maintenance, and human Extra Vehicular Activity (EVA) assistance. For in-space

assembly, the existing in-space robots were found to be limited to SRMS [39] and

SSRMS [40, 41], which were used to move large objects and were controlled through

teleoperation. The SPDM [40, 42-44], capable of delicate maintenance and servicing

through teleoperation, was scheduled to launch in 2005.

Langley's Automated Structural Assembly Robot testbed [52] and Carnegie Mellon

University's (CMU) Skyworker [53-55], a prototype Assembly, Inspection, and

Maintenance (AIM) robot, shown in Fig. (2-6) and (2-7) respectively, were able to

autonomously assemble fixed structures.









The ROTEX [56] robot, shown in Fig. (2-8), developed by Germany, flew in April

1993 as part of STS-55 to demonstrate capture and assembly, performed through

teleoperation and some autonomy, as well as connecting components.

.....~~~ ~ ~~ V4...1"""'' i ,_


Figure 2-6. Langley's Automated Structural Assembly Robot [52]. A) Robot Arm with
End-Effector, B) Close-Up of End-Effector.

IA Mn


Figure 2-7. CMU Skyworker [54]. A) Demonstration Prototype, B) Model.












1 *


Figure 2-8. German Robot ROTEX [56].









The Robonaut [57] was designed by Robot Systems Technology Branch at NASA's

Johnson Space Center (JSC) in a collaborative effort with DARPA. The Robonaut [57] is

an anthropomorphic robotic system with the functionalities capable to assist or replace an

astronaut during EVA. As a ground demonstration test bed, Robonaut [57] performed

successful assembly tasks and is shown in Fig. (2-9). Also, Ranger [24-27] conducted

assembly, maintenance, and human EVA assistance in a neutral buoyancy facility of the

University of Maryland (UMD).

The Beam Assembly Teleoperator (BAT) [25, 26] built by UMD was able to

"perform space construction tasks and also to repair satellites, service space hardware,

and work in cooperation with astronauts" during it's flight on STS-61 B (26, p. 1). BAT

is shown in Fig. 2-10.
















Figure 2-9. NASA Robonaut [57].









Figure 2-10. University of Maryland Beam Assembly Teleoperator (BAT) [26].









In 1997, ETS-VII [49, 50] performed rendezvous and docking maneuvers followed

by assembly of small structures. After running into initial problem of contacting the

satellite, the teleoperated mission was successful in verifying the basic level technologies,

but did not include all the scheduled experiments.

For in-space inspection there were no operating robots in space. A German

inspection robot experiment Inspektor [58], designed for in-space operations of Russian

Space Station MIR, failed while in flight. AERCam SPRINT [28, 29] teleoperated free-

flyer, built by NASA Johnson Space Center (JSC) and successfully flight tested during

STS-87, performed maneuvers and observation such as video imaging of Orbiter and ISS

to evaluate the efficiency of the free-flyer. AERCam Integrated Ground Demonstration

(IGD) [27, 28] was built and tested on the airbearing table, capable of "autonomously

scanning a mock-up of a spacecraft." AERCam Engineering for Complex Systems

(ECS) [27], geared towards advancing autonomy, was under development. Fig. (2-11)

shows two generations of AERCam.

B













Figure 2-11. NASA AERCam [29]. A) AERCam SPRINT, B) Mini AERCam.

UMD Space Systems Laboratory (SSL) developed the Supplemental Camera and

Maneuvering Platform (SCAMP) [26, 27], neutral buoyancy flying camera robot









designed for assistance with viewing during EVA, shown in Fig. (2-12). The first and

second generations of the camera were tested in the Neutral Buoyancy Research Facility

(NBRF), while the third generation is still in the conceptual stage of the development.







Pitch y -4 Rdal


Figure 2-12. University of Maryland SCAMP [27].

In-space maintenance included replacement of the components, which was

demonstrated in space by ROTEX [56], ETS-VII [49, 50], and BAT [25, 26] robots and

on the ground by Ranger [24-27]. Ranger [24-27] and Robonaut [57] have also worked

on the ability to access obstructed components.

In-space human EVA assistance was provided only by SRMS [7] and SSRMS [40,

41]. Robonaut [57] and Ranger [24-27] were the ground test beds that demonstrated the

capability of assisting the astronauts with "handing over tools, holding objects, and

shining lights" through teleoperation. Later, Robonaut [57] was able to hand tools

autonomously.

It was desired for the robots to perform assigned tasks like a human, and this was

predicted "to be fully realized only under teleoperation" (51, p. 10). The outcome of the

report suggested that fully autonomous robots would require significant technological

developments and continuous monitoring from the ground. However, "only a sustained

engineering effort focused on developing methodologies and gaining experience in the









role of robots in space exploration" (51, p. 11-12) was required to achieve the necessary

future robotic capabilities.

The robotic missions capable of caring out tasks previously described were

becoming more necessary. Composing a large monolithic satellite out of smaller ones

was expensive because the spacecraft was designed to carry out multiple missions,

therefore having complex structure to support those missions. Such satellites were meant

to stay in space for prolonged time, so it was advantageous to send a repair mission, if for

any reason the satellite became disabled. The capability and difficulty associated with

capturing a satellite that had some angular motion, whether it was initially stable or not,

was realized. In late 70s Kaplan and Nadkarni [59] investigated several possibilities that

could be used for "passivating" the satellite to assist in safe capture.

The capturing missions were usually conducted by astronauts through EVA, which

was extremely dangerous. This was demonstrated when the Shuttle crew of flight STS-

49 was tasked with capturing and reboosting satellite International Telecommunications

Satellite Consortium (INTELSAT) VI [60, 61] satellite that was stranded in the incorrect

orbit due to failure of the Titan rocket to put the satellite into the correct orbit. Numerous

captures of the satellite were attempted, resulting in the longest spacewalks ever

performed. The problems that arose during EVA were "poor visibility of the Earth's dark

side, insufficient positioning of the end of the robot arm in the attempt to even come near

the satellite, and unexpected susceptibility of the satellite to wobbling" (61, p. 1). After

several failures it was decided to perform a three-man space walk. Three astronauts

assembled a bridge that was the centerpiece of the Assembly of Space Station by EVA

Methods (ASEM) experiment, providing a platform for a triangular formation of the three









astronauts. Standing 120 degrees apart, they "grabbed the rim of the satellite to slow and

eventually stop its rotation" (61, p. 1). This mission was proved to be difficult, putting

the astronauts' lives in great danger.

Another example was several servicing missions performed by the astronauts on

the Hubble Space Telescope (HST) [62]. The first servicing mission was conducted in

1993 by the crew of STS-61 to install new equipment and correct the imperfection in the

HST's primary mirror. The STS-82 performed the second servicing mission in 1997,

replacing the old instruments with updated equipment that improved the functionality of

the HST. In 1999, STS-103 performed part A of the third servicing mission by replacing

failed gyroscopes and making several additions. Part B of the third mission and the

fourth mission have been cancelled as of March 2004 due to risk concerns in the light of

the recent shuttle Columbia tragedy. According to the President Bush's [37] space

mandate, all further shuttle missions, hoped to resume in 2005, would focus on finishing

the ISS. HST successful functionality was dependable on the regular servicing missions.

Suggestions [63] have been made to put forth a robotic servicing mission sometime by

2009 to save the HST. A group of scientists and engineers at NASA Goddard Space

Flight Center (GSFC) is considering submitted proposals for a possible mission. NASA

JSC's Robonaut [57] and UMD's Ranger [24-27] are two of the technologies under the

consideration.

Several autonomous robotic services were and are under development due to the

increasing demand. Orbital Sciences Corporation received sponsorship from NASA in

2001 to design, built, and test the Demonstration of Autonomous Rendezvous

Technology (DART) [64, 65], shown in Fig. (2-13).

























Figure 2-13. Orbital Sciences DART [64].

DART [64, 65], initially scheduled for flight in the fall 2004, was to be the first to

locate and rendezvous with a satellite completely autonomously. Previously, astronauts

had to control the vehicle via teleoperation in order to accomplish any rendezvous and

servicing operations. After several delays DART [64, 65] was successfully launched on

April 15, 2005 using an Orbital Sciences Pegasus Launch Vehicle and was scheduled to

rendezvous and perform close proximity operations, "including station keeping, docking

axis approach, circumnavigation, and a collision avoidance maneuver" (64, p. 1), via

Advanced Video Guidance Sensor (AVGS). As reported by Spaceflight Now [66],

DART suffered from problems with its guidance system from the start and, coming to

approximately within 300 m to the target satellite, ran out of fuel, causing the autopilot to

initiate the retirement segment of the mission. It was later reported by Space News [67]

that DART has actually advanced further then originally thought, running into the target

satellite and then maneuvering into the retirement orbit. NASA dubbed the mission to be

partially successful and formed a board to investigate the mishap. This particular mission

is the focus of this thesis and therefore will be discussed in detail in Chapter 4.









Experimental Small Satellite-10 (XSS-10) [68, 69] developed by the U.S. Air Force

to evaluate future applications of micro-satellite technologies such as rendezvous,

inspection, docking, and close-proximity maneuvering around orbiting satellites.

Launched on January 29, 2003, the space robotics mission was pronounced a success.

The flight experiment verified semiautonomous on-orbit rendezvous and inspection

capabilities. "The XSS-10 was the first demonstration of an autonomous inspection of

another resident space object using a highly maneuverable micro-satellite. The flight

experiment validated the design and operations of the micro-satellite's autonomous

operations algorithms, the integrated optical camera, and the star sensor design. The

XSS-10 program team also verified the critical station keeping, maneuvering control, and

logic guidance and control software necessary for autonomous navigation. The ground

control capability, innovatively developed for XSS-10, enabled a small team to

successfully interpret the real-time data and control the spacecraft during its short

mission" [69].

Experimental Small Satellite-11 (XSS-11) [70, 71] managed by the U. S. Air Force

Research Laboratory program and build by Lockheed Martin Co. is another mission that

was successfully launched on April 11, 2005. XSS-11 [70, 71], shown in Fig. (2-15), is


Figure 2-14. XSS-10 [69].









going to test the autonomous technologies needed for the inspection and repair of the

disabled satellites, such as approach and rendezvous maneuvers to several non-

operational US satellites. XSS-11 will also "demonstrate technologies for military space

surveillance" (71, p. 1). The mission is scheduled to last approximately a year, with the

rendezvous stage set to begin approximately six weeks after launch. Additional details of

the XSS-11 mission are not available in the public domain.

Another on-going space robotics project is TEChnology SAtellite for

demonstration and verification of Space systems (TECSAS) [72, 73] by the European

Aeronautic Defense and Space Company (EADS), Babakin Space Center, and DLR-RM.

The mission consists of launching target and chaser satellites, shown in Fig. (2-16), the

former equipped with a robotic arm and a docking mechanism, to verify robot's

capabilities for rendezvous and close-proximity operations on-orbit.


Autonomous Proximity
Operations





S Autonomous
Rendezvous








Figure 2-15. XSS-11 [70].









A B













Figure 2-16. TECSAS [73]. A) Chaser Satellite, B) Chaser Capturing Target.

Also, the Phantom Works division of Boeing was selected to compete second phase

of the Orbital Express [74, 75] project. Phase II consists of "finalize the design, develop

and fabricate a prototype servicing satellite, the Autonomous Space Transport Robotic

Operations satellite (ASTRO), and a surrogate serviceable satellite, NextSat, and conduct

an on-orbit demonstration to validate the technical feasibility and mission utility of

autonomous, robotic on-orbit satellite servicing" (74, p. 1). Fig. (2-17) shows artist's

rendering of the mission. It is also a goal to develop a standard upgradeable vehicle that

would be able to be used for a variety of satellite servicing missions. The mission is set

to launch in September of 2006.












Figure 2-17. Orbital Express ASTRO [75].









Another space robotics project is Spacecraft for the Universal Modification of

Orbits (SUMO) [35], sponsored by the Defense Advanced Research Projects Agency

(DARPA) and implemented by the Naval Center for Space Technology. The servicing

spacecraft, shown in Fig. (2-18), is going to demonstrate "the integration of machine

vision, robotics, mechanisms, and autonomous control algorithms to accomplish

autonomous rendezvous and grapple of a variety of interfaces traceable to future

spacecraft servicing operations" (35, p. 1). A demonstration of the prototype is set for

December, 2005, while the launch of the SUMO spacecraft is set to occur sometime in

2008.
















Figure 2-18. SUMO Servicing Spacecraft [35].

To this day a variety of systems have been developed, yet most of the recent

progress is either conceptual or awaiting a flight demonstration, with the exception of the

DART mission, which was partially successful. So it is time for the space robotics

community to "continue the journey" (37, p. 1).

2.2 Scope of the Thesis

This thesis will use the DART project as a framework and will focus on the

investigation of the close-proximity operations phase of the mission. The Satellite Tool









Kit (STK v.6.2.) software developed by the Analytical Graphics Inc. will be used for the

visualization of the mission in the space environment. The close-proximity operation

phase will be simulated using CW relative motion equations for the spacecraft in circular

orbit. Chapter 3 will present the analytical background of the linear orbit mechanics, and

Chapter 4 will describe the DART mission in detail. Different scenarios will be

composed in the STK environment to observe the behavior of the DART spacecraft

subjected to the full force model. Specific maneuvers will be examined for their accuracy

in the relative position with respect to a target vehicle and for the amount of fuel and

velocity used. The simulation and the results of this effort will be discussed in Chapter 5.

Chapter 6 will give conclusions and recommendations for future work in this area.















CHAPTER 3
LINEARIZED ORBIT MECHANICS AND PROXIMITY OPERATIONS

This chapter discusses relative motion of two spacecrafts, orbiting in close-

proximity of each other. The equations governing the motion of the robotic service with

respect to the target satellite are presented for the close-proximity maneuvers. General

overview of the perturbation and their effects on the motion of the spacecraft is given.

The author follows reference [76] for the formulation of this chapter.

3.1 Two-body Problem

Equation of motions (EOM) for the target satellite and the robotic service are

defined using the two-body EOM, therefore it will be discussed first. The derivation of

the two-body EOM starts with two fundamental laws, Newton's second law of motion and

Newton's law of universal gravitation.

* Newton's second law of motion states that "the rate of change of momentum is
proportional to the force impressed and is in the same direction as that force" (77,
p. 2).

d(mv)
SF ma (3.1)
dt

where m, v, a are constant mass, velocity vector, and acceleration vector of a body,
respectively, that is subjected to the sum of the forces acting on it.

* Newton's law of universal gravitation states "any two bodies attract one another
with a force proportional to the product of their masses and inversely proportional
to the square of the distance between them" (77, p. 4).

GMfm r
F 2 (3.2)
r r









where F is the force acting on mass m due to mass M m and M are the masses of
two bodies, F is the vector from M tom and G = 6.670x10 8dyne cm2 / gm2 is the
universal gravitational constant.

In order to derive the two-body EOM the following assumption are necessary:

1. M >>m.
2. According to the Newtonian mechanics, the coordinate system is inertial
(unaccelerating and nonrotating).
3. The bodies are spherically homogeneous (symmetric and with uniform
density), which allows treating each body as a point mass located at its
geometric center.
4. The only forces acting on the system are gravitational forces, which act
along the line joining the centers of the two bodies.

The system of two bodies is presented in Fig. (3-1). Let YZ be an inertial

coordinate system. Let IK be a coordinate system displaced from XYZ, but not rotating

or accelerating with respect to XYZ. The position vectors of bodies M and m with respect

to the origin of the XYZ reference frame are r, and r, respectively. Therefore, the vector

from M tomr is defined in equation (3.3).

F = f f (3.3)

Applying equations (3.1) and (3.2) with respect to XYZ,

GMm f GM_
mr =- ---> r = r (3.4.1)
r2 r r

GMm f Gm
MrM = r 3 r (3.4.2)
r r r

Equation (3.3) is differentiated to obtain equation (3.5). Because XYZ is an inertial

reference frame, the derivative of the equation (3.3) is found without differentiating the

axes of the coordinate system.


r = r r


(3.5)










Equations (3.4.1) and (3.4.2) are grouped using equation (3.5).

G (M +m)
r =-- r
r


(3.6)


Due to earlier assumption ofM > m, a gravitational parameter is defined as p = GM,

and equation (3.6) takes the form of equation (3.7).


r =-r
r


(3.7)


Equation (3.7) is the relative two-body EOM (second-order, nonlinear, vector, differential

equation).


Figure 3-1. Geometry of Relative Motion of Two Bodies [76].

3.2 Relative Motion

This section develops the governing equations for the relative motion of two

spacecraft in the nearly circular orbits. The relative position and velocity solutions are

also found.









3.2.1 Clohessy-Wiltshire Equations for Nearly Circular Orbits

Figure (3-2) shows the geometry of the target satellite and service spacecraft,

which will be used to derive the EOM. The target satellite motion is the primary motion,

while the motion of the service spacecraft is analyzed with respect to the target satellite.

The satellite coordinate system, RSW(.i2), is used as the reference frame for the

derivation.



u. target satellite

r, / \

EA.RTH ., service spacecraft

0) -*


Figure 3-2. Coordinate System for Relative Motion.

The R axis points from the center of the Earth towards the target satellite along the

radial vector. The radial positions and displacements along theR axis are collinear with

the position vector. The S axis points in the direction of the velocity vector for the

circular orbits and is perpendicular to the R axis. The along-track positions and

displacements are along the S axis and are normal to the position vector. The W axis

points in the direction normal to the orbital plane.

The two-body relative motion equation (3.7) can be used to define the motion of

the target satellite. The motion of the service spacecraft is defined also using equation

(3.7) with the addition of the force vectorF which accounts for the disturbing effects

such as applied thrust to move relative to the target satellite, atmospheric drag, third-body









and oblateness of the earth effects, and solar radiation. These are described in section 3.4

of this chapter.


g tgt (3.8)
tgt
r __ r +F (3.9)
serv e 3 servev pert


Referencing the geometry of Fig. (3-2), the relative position vector of the service

with respect to the target is found. Equation (3.10) is then differentiated twice, as was

demonstrated in section (3.1).

el1 = ere tgt (3.10)


rel = rerv -gt (3.11)

Equations (3.8) and (3.9) are substituted into equation (3.11) to yield,
'ervrt gt (3.12)
rrel = 3 + pertF 3 (3.12)
r r
serv tgt

Additional manipulations are required in order to use equation (3.12). Equation

(3.10) is rearranged as to solve for the position vector of the service,

rse, = gt +rel (3.13)

Equation (3.13) is divided by r e~3 The denominator of the right-hand side (RHS) of the


equation (3.14) is rewritten using |ej = Igt + get = (r g2 + 2rgt + Ie + rel2)


serv tgt rel tgt + rel -.
3 3 (3.14)
*serv '3sev r + 2rF *r +rr 2
'tgt tgt rel rel


Assumption ofrre2 << gt is made, and rg,2 term is factored out,












+re,


serv
serv


(3.15)


Using the binomial expansion of the equation (3.16), equation (3.15) can be rewritten as

shown in equation (3.17).


^n n(n-1)x2
(1+x)" =l+nx++ 2 +...ithn
2!


ser 3tgt3 2r r
serv3 23 2 r gt


(3.16)



(3.17)


Equation (3.17) is substituted into the equation (3.12) to yield,


tgt rel I
"tgt


2r tgt e


(3.18)


+ r 3
tgt


Equation (3.18) is expanded, keeping only first-order terms.


= 3 H\rf 2 l 3 r 2 g, -Fr
t g tg tgt e tgt 2 }\
vl tgP3 rel 2 2 2 2rtt + per
r,9, r,9, r,9


(3.19)


An assumption, tt is small, is made based on the assumption of two satellites


being sufficiently close to each other, and the term involving this quantity is dropped

from equation (3.19),


22g~t, r + Fp ,
"tgt


,rel = 3 rel
tgt


(3.20)









r
It can be noticed thatR = -t"' is a unit vector in the direction of the position vector of the
rtgt
r

target satellite, and,, ei- is the x component, therefore equation (3.21) is the inertial
rt,

acceleration of the service with respect to the target reference frame.


r=e { e I 3x{R}++Fp,, (3.21)
tgt

The target satellite reference frame, RSW, rotates with the motion of the spacecraft in

orbit and changes with time, so further analysis of equation (3.21) is necessary. The

term Fre; is with respect to the inertial frame, so it needs to be written with respect

to RSW frame.

Using equation (3.22), equation (3.21) can be rewritten as shown in equation (3.23).

a1 = R + COR, x rR + CORJ x (ORJ x )+ 20R/, x (3.22)

where a iis acceleration with respect to rotating reference frame, the (iRI x f) term is the

tangential acceleration (accounts for change in the angular rate of rotating reference

frame), the (R /I x (bR/1 x R))term is the centripetal acceleration (accounts for the rate of

change of the tangential velocity), the (2oR/ x I )term is the Coriolis acceleration

(demonstrates apparent acceleration the observer sees when observing spacecraft's

motion from the rotation).

F'eR. = I, -R X el 2R Xel R X (- R X l) (3.23)
reiC Xe! Wl xr 2R Xr (6R re/il










where o = / is the mean motion of the target satellite and (cR x el ) term is zero
'tgt

because the orbit are circular.

The remaining cross products are evaluated taking the(x, y, z) to be the components of

the ie in the target satellite's reference frame.



R X Iel = 0 0 = -oyR + )S



cR X(OR )= 0C x 0 0 = c( x -oyR +iox) (3.24)
xy

R SW
S0 0 ) = -o)xR o)2y
-Oy axO 0

Substitute equations (3.23) and (3.24) into equation (3.21),


r '3 f=- -3xR +F,,, -2 (-coR+ x+)+m +xR+ q'yi
ret (3.25)
R =-2 xR + yS + zW- 3xR + F + F 2oy 2oRm + 2xR + o)2y

Equation (3.25) is written in the component form to give the CW equations of relative

motion for the nearly circular orbits.

x-2oy -32x= f}
y + 2ix = f near circular orbits (3.26)
+ 02Z = f-









3.2.2 Position and Velocity Solutions for Nearly Circular Orbits

Because the application of the solution will be to the maneuvers employing

impulsive thrust, an assumption of Frt = 0 is made. This doesn't apply to the cases in

which a continuous thrusting is desired.

The approach of Laplace operators will be used in order to solve for

the x component of the position. The x equation in (3.26) is differentiated,

x = 3cm2x+2cmy
y (3.27)
x = 302+ 2+2)y

The y equation in (3.26) is now substituted into equation (3.27),

Y = 32x 42 2x -> + C2 = 0 (3.28)

The Laplace transform of equation (3.28) is taken,

Z' [xy +C2] {ssX(s)- s2xo S- o}+ )2 x {sX(s)- x} =0 (3.29)

Equation (3.29) is expanded and like terms are grouped,

s3X(s) S2Xo So o, + SX(s)C2 XoD = 0
(3.30)
sX(s) (s2 + 2) = (s2 + m 2)x, + so + 0

Equation (3.30) is now solved forX(s),


X(s) = x+o + (3.31)
S S2 +2) S(S2 +)(3.31)

The last term in equation (3.31) is expanded using partial fractions,

1 A Bs+C
I 2-- -A + 2)2 (3.32)
S(S2 +3 C2 s s +m9


Coefficient (A, B, C) are found as follows,









1 1
A= 2 s

(3.33)
Bs+C = s(s s2+o2) osJ B C=0
s 2+ 2 ) s=o


Substitute equation (3.32) into equation (3.31) with coefficients from equation (3.33),


X(s) X=o a a ( (3.34)
s s + ) so 9 s +2 )

The inverse Laplace transform is applied to equation (3.34),

xo x Xo
x(t) = x, + + sin(omt) _- cos(ot) (3.35)
0) 0) 0)

The x equation in (3.26) is substituted into (3.35),

3oz2X + 2o)y x 3C)2x + 2aO)y 0
x(t) = x) + 2 + sin(o)t) cos()t) (3.36)


It is then simplified,


x(t)=4xo+ 2J xsin(ot)- 3xo+ 2y cost) (3.37)


Equation (3.37) is differentiated to find the x component of the velocity, taking into the

account that the initial conditions are constant.

x(t) = xo cos(c)t) + (3o)xo + 2jo) sin(o)t) (3.38)

In order to find the y component of the position and velocity, the y equation in

(3.26) is integrated with respect to time. First, the equation (3.38) is substituted into

the y equation in (3.26),

y = -2o) [x cos()t) + (3cxo + 2jo) sin()t)]
S2> cos()t) -2) (3ox + 2Ys t)(3.39)
y = -2oxo cos(ct)- 2c (3mxo + 2 Vo) sin(ct)









Integrate equation (3.39) twice,

y = -xo sin(ct)+ 2(3coxo + 2y o)cos(ct)+ C
22x ( 4x (3.40)
y = --cos(cot) + 6x + sin(cot)+Ct+D (3.40)


The constants of integration are found by setting t = 0 in equation (3.40),

y, = 6 xo + 4y, + C C = -6cxo 3yo
2x -2x (3.41)
y + --D-~D =-+ y,


Substituting the coefficients yields the y component of the position to be,


y(t)= cos(ot)+ 6xo+ sin(t)-(6coXo+3jo)t- +o (3.42)
0c 0 c) 0)

and the y component of the velocity to be,

y = -xo sin(c)t) + (6cvxo + 4y3o)cos(c)t) 6ovxo 3y, (3.43)

The approach of Laplace operators will be used in order to solve for

the z component of the position. The Laplace transform of the z equation in (3.26) is

taken,

I[ z+ 2Z] =s2Z(s) szo 2Z(s) = 0 (3.44)

Equation (3.44) is solved forZ(s),

sz z
Z(s) 2 o 0 (3.45)
(s2 +2) S2 + 2)

The inverse Laplace transform is applied to equation (3.45),


z(t) = zo cos(ot) + z- sin(ct) (3.46)


Equation (3.46) is differentiated to find the z component of the velocity, taking into the

account that the initial conditions are constant.









z(t) = -io sin(iot) + io cos(ot) (3.47)

The position and velocity solution to the CW equations for nearly circular orbits are

written in the matrix form,



o _o
SXo Xo
Y Yol YO
FM(t) N(t)

x xo I S(t) T(t) x

Y Yo Yo
O L
Z oo
1 2
-s(t) -(1- c(t)) 0
4 3 c(ot) 0 0 o o
2 4
M= 6s(t)-6t 1 0 N= (c(t)-I1) -s(t)-3t 0
0 0 c(mot) 1
0 0 -s(cmt)

S3o s(cot) 0 0 c(ct) 2s(ot) 0
S 6(c(6t)- 1) 0 0 T -2s(0t) 4c(ot)-3 0
0 0 -cs(ct) 0 0 c(ct)
s =sin,c =cos (3.48)

3.3 Close-Proximity Operations

This section develops the analysis for the impulsive rendezvous using the linear

CW relative motion equations derived in the previous section. Three approaches of the

target satellite by the service spacecraft, including R-Bar, V-Bar, and circumnavigation

maneuvers (for definition refer to section 3.3.2), are described.

3.3.1 Linear Two-impulse Rendezvous

In order to calculate the change in velocity, AV, required for the two-impulse CW

transfer, the state transition matrix D(t) in equation (3.48) is evaluated, and then equation

(3.48) is solved for the appropriate state. The ij is the frame used to derive equation









(3.48) as referenced to Fig. (3-2). The service spacecraft maneuvers with respect

to iN frame. The subscript specifying relative position (and velocity) is dropped because

only the relative motion is discussed from here on.

The relative position and velocity vectors are given by,

F(t) = M(t)(O0) + N(t)F(O) (3.49)


r(t) = S(t)(0)+ T(t)r(0) (3.50)

where F(0) and F(0) are six scalar constants that define the initial state of the orbit.

The final state for the rendezvous is at the origin of the ji2 frame,

making f, = (tf) = 0. The initial relative velocity Fo needed for the successful

rendezvous is found from equation (3.49),

= N(t) 1M(t ) (3.51)

The first impulse of the CW transfer instantaneously changes the initial relative velocity

prior to impulse ro to the initial relative velocity after the impulse Fo The velocity

required for the first impulse is

A = (3.52)

The final relative velocity upon successful rendezvous is found from equation (3.50),

r, = S(t)o + T(t)r+ (3.53)

The second impulse of the CW transfer, acting as a stop maneuver, instantaneously

changes the final relative velocity prior to impulse rf to the final relative velocity after

the impulse rF The velocity required for the second impulse is


AVf = -f,


(3.54)









The presented equations provide the closed form expressions necessary to perform

rendezvous. The first change in velocity positions the service spacecraft onto the path

that will lead towards the target satellite. The second velocity change cancels the

velocity of the service upon arrival at the origin of i52 frame, completing the transfer.

The scheme for the linear two-impulse rendezvous described here will be used as a basis

for the simulation presented in the Chapter 5.

3.3.2 Different Approaches of the Target Satellite

This section discusses three common maneuvers performed to approach or move

away from a target satellite: V-Bar (V) Approach, R-Bar (R)- Approach, and

circumnavigation.



I desired path service
-R I spacecraft
+ @ t=o
target target's
Z satellite ac,, circular
path rbit




Figure 3-3. V-Bar Approach Maneuver.

The S2frame (or CW frame) is shown in Fig. (3-3). The service spacecraft is

located directly behind, as in Fig. (3-3), or in front of the target satellite along the j axis,

which is in the direction of the velocity vector. Therefore, approaching or receding along

this direction is known as the V-Bar Approach. As can be seen from Fig. (3-3), a

number of small CW impulsive maneuvers is used to represent the motion of the service

spacecraft directly along the V-Bar.












x, -R
target's y, +V
circular target
orbit satellite










; .-- desired path







actual
path I
path service
\ spacecraft
@ t=0

Figure 3-4. R-Bar Approach Maneuver.

Fig. (3-4) shows the service vehicle located directly below the target satellite

along x axis, which is in the direction of the radius vector. Therefore, approaching or

receding along this direction is known as the R-Bar Approach. Similarly, the motion

directly along the R-Bar direction is represented with multiple CW impulsive transfers.

The circumnavigation maneuver, shown in Fig. (3-5), allows the service spacecraft

to fly around the target satellite. V-Bar Approach, R-Bar Approach, and

circumnavigation maneuvers are performed in the orbit plane; they can, for example,

correspond to an inspection, docking, or collision avoidance phases of a specific mission.









In all the figures a single arc represents a two-impulse CW transfer. The demonstrated

maneuvers are used for the simulations presented in Chapter 5.









target target's
satellite circular
orbit
rf


service
spacecraft
@ t=0


Figure 3-5. Circumnavigation Maneuver.

3.4 Perturbations

"A satellite in earth orbit experiences small but significant perturbations (accelerations)

due to the lack of spherical symmetry of the earth, the attraction of the moon and sun for

the satellite, and, if the satellite is in low orbit, due to atmospheric drag" (78, p. 155).

The equations developed in the preceding sections relied on the assumptions of the two-

body problem and did not account for the disturbances affecting the orbit of the

spacecraft. The force model used in the simulations presented in Chapter 5 implements

the following disturbances: atmospheric drag, gravitational effects due to oblateness of

the Earth, third body effects, solar radiation, and thrust. Therefore, a general description

for each will be presented in this section. In depth discussion of all the perturbations and

their effects can be found in [76].







43


3.4.1 General Effect of Perturbations

Disturbances on orbital motion cause secular and periodic changes in orbital

elements, as shown in Fig. (3-6). Secular changes grow linearly with time; making the

errors in these changes grow unbounded. This is demonstrated in the Fig. (3-6) with a

straight line. Periodic changes are separated into the short-periodic and long-periodic

effects. Short-periodic changes tend to repeat in equal to or less time than the satellite's

orbital period. Long-periodic changes repeat themselves after significantly longer time

than the orbital period, usually one or two orders of magnitude longer. These effects are

also shown in Fig. (3-6).



S Mean change Short-periodk plus long-periodic, and secular '


:-- -- **< MSecular
y -Mean change
-s- ^_ /-----
SLong-periodic and


1 It 3j 14
Time

Figure 3-6. General Effect of Perturbations on Orbital Elements [76].

The orbital elements (a, e, i, Q, co, v ), described in Appendix A, are defined as fast

and slow variables. Fast variables significantly change during an orbital period, even if

the perturbations are not present. True, mean, and eccentric anomalies are examples of

the fast variables. The short-periodic changes occur when the causative perturbation

effect has a fast variable present. Slow variables have slight changes during an orbital

period, and these changes are due to the presence of disturbances. Semimajor axis,

eccentricity, inclination, node, argument of perigee are some of the examples of the slow

variables.










3.4.2 Atmospheric Drag Effects

Drag is a nonconservative force that acts in the direction opposite to the motion of

the satellite. Equation (3.55) provides the expression for the atmospheric drag presented

as acceleration,

1 C A -2
rdr=-P-- r (3.55)
2 m R


where p is the atmospheric density, CD is the coefficient of drag, A is exposed cross-

sectional area of the spacecraft, m is the mass of the spacecraft, and r is the magnitude of

the velocity of the spacecraft.

The atmospheric drag has the effect of attenuating the orbit and causing a satellite

to reenter the atmosphere and crash into the Earth. This is demonstrated in Fig. (3-7).



.- -- -- t --, -










The effects of atmospheric drag are summarized below,











Secular changes caused in semi-major axis, eccentricity, and inclination.
Periodic changes caud in a ll te oral element
InM t 74k
> .' A w B





Figure 3-7. Effects of Atmospheric Drag [76]. A) Lowering of the Orbit, B) Orbital
Properties Under the Effect of Drag.

Fig. (3-7A) shows the decreasing effect of drag on size and shape of the orbit, hence,

change in semi-major axis and eccentricity. Fig. (3-7B) demonstrates the variation in the

orbital elements over the lifetime of a satellite.

The effects of atmospheric drag are summarized below,

Secular changes caused in semi-major axis, eccentricity, and inclination.

Periodic changes caused in all the orbital elements.









The simulations in Chapter 5 employ the Jacchia-Roberts [76] atmospheric density

model, which is a high fidelity, but computationally expensive, time-varying model. This

model is similar to Jacchia 1971, but uses analytical methods to improve on the

performance. Jacchia 1971 model computes atmospheric density based on the

composition of the atmosphere, which depends on the satellite's altitude as well as a

divisional and seasonal variation. It has valid range of 100km-2500km [79].

3.4.3 Oblateness of the Earth Effects

The Earth is not a perfect sphere, as it is often assumed, making the center of

gravity (CG) not coincident with the center of mass (CM). The effects of the oblateness

of the Earth are most prominent for low Earth orbits (LEO). The averaging effects of the

Earth's spin result in a gravitational potential of the satellite with respect to the Earth as

expressed in equation (3.56) [80].


U= k JkPk (COs') (3.56)


where R is the distance from the center of the Earth to the spacecraft, Re is the equatorial

radius of the Earth, Pk is the Legendre polynomial function of the kth order, D is the

colatitude, and Jk represents zonal harmonics (J2is the first zonal harmonic with the most

dominant effects).

The two primary effects, regression of the line of nodes and rotation of the line of

apsides, are secular changes. Nodal regression is a rotation of the plane of the orbit about

the axis of rotation of the Earth at a rate that depends on inclination and altitude. The

torque formed is about the line of nodes, and it pulls the orbital plane towards the

equator. The perturbation is observed through a change in the angular momentum vector,









which signifies the orbital precession. Fig. (3-8) demonstrates the nodal regression. The

nodes move westward for direct orbits and eastward for retrograde orbits.

AK Preession of angular momentum








-'4L'
,.-l ,, ./--^ I ^ Extra pun
...i i *-



i fc.,L JN J
'' ra.-. -- 7.. -



Figure 3-8. Nodal Regression [76].

The rotation of the line of apsides is the second major effect and is applicable only

to eccentric orbits. This perturbation will cause the major axis of an elliptical path to

rotate in the direction of the satellite's motion if i < 63.4 or i > 116.6 and opposite of the

direction of motion for 63.4" < i < 116.6 The rate of rotation depends on inclination and

altitude. Fig. (3-9) demonstrates the apsidal regression.














ApApoeEnd) irt)





Prrugt (Pod)
t ',, | --I :, ",' 'l -- '" i




""' ~~.... I~, ,mgil i ,,j


Figure 3-9. Apsidal Regression [76].

The discussion of periodic effects of zonal harmonics can be found in [76]. The

effects of oblateness of the Earth are summarized below,

Secular changes in the node and perigee are the primary effect.

Even zonal harmonics cause secular perturbations in Q, A,M ; a,e,i have

no secular effects.

For orbits within = 90 orbital plane remains fixed, indicating no secular and

periodic changes in Q,i.

The magnitude of perturbations due to the oblateness of the Earth increases

when the eccentricity of the orbit is increased and decreases when the semi-

major axis of the orbit is increased.

The simulations in Chapter 5 employ the Joint Gravity Model version 2 (JGM2)

developed by NASA/GSFC Space Geodesy Branch, the University of Texas Center for

Space Research, and CNES. It describes the gravity field of the Earth of the degree and

order 21 (maximum 70) [79] using equation (3.56).









3.4.4 Third-Body Effects

The effects of third body, Sun or Moon, should be considered, especially for the

satellites in orbits with high altitude. Gravitational attraction of a third body causes the

perturbations, so it is similar to the orbital changes caused by the oblateness of the Earth.

Fig. (3-10) demonstrates the phenomenon arising from the third body effects. The

trajectories of the satellite and sun's (for example) orbits are shown as elliptical rings in

the Fig. (3-10). The gravitational attraction between the rings generates a rotation of the

satellite ring toward the ecliptic solar ring about the intersection line of the rings. A

gyroscopic precession about the axis normal to the ecliptic ring plane is a result of a

torque on the satellite ring. The Moon will have a similar influence, causing the

regression about the axis normal to the orbital plane of the Moon. The effect of both the

Sun and the Moon is a regression of the orbital plane about a mean pole (between the

Earth's and ecliptic poles).



Gyroscopic precession

h

S, Satellite ring
.,, : ,\ Torque

S; Ecliptic solar ring


Figure 3-10. Third-Body Effects [76].









The third body, the Sun or Moon, effects are summarized below,

Semi-major axis does not have any secular or long-periodic changes.

Secular perturbations occur in the node and perigee. Since the Moon is

closer to the Earth than the Sun, the lunar effects are more pronounced.

Long-periodic changes occur in e,i,, oo.

The changes in perigee height induced by the Sun and the Moon have the

effect of increasing or decreasing the lifetime of the satellite for orbits

affected by the atmospheric drag perturbations.

The force model in simulations of Chapter 5 takes into account the third body

effect of the Sun and the Moon.

3.4.5 Solar Radiation Effects

Solar radiation is a nonconservative force, which effects are more pronounced for

orbits with higher altitudes. Equation (3.57) provides the expression for the acceleration

due to solar pressure,

Colrad cr (3.57)
m c RAU)


where C, is the coefficient of reflectivity, A is area of the spacecraft, m is the mass of the

spacecraft, K is percentage of the sun (as seen from the spacecraft), ( is solar flux at 1

AU, c is the speed of light, RAU is the distance from the spacecraft to the sun in AU,

and s is the unit position vector of the sun as seen from the spacecraft [79].

The solar radiation effects are summarized below,

Radiation pressure causes periodic changes in all the orbital elements with

more pronounced effects for higher altitude orbits.









Changes in perigee height can have a significant effect on the lifetime of the

satellite.

Solar radiation effects are more significant for satellites with low mass and

large surface area.

The force model in simulations of Chapter 5 takes into account the effects of solar

radiation.

3.4.6 Thrust

The perturbations to the orbit by the acceleration due to thrust is a planned change.

Equation (3.58) provides the expression,

F rus(t) dm
I,(0 -j d(3.58)
thrust F thrust sp d 3.5 )
m dt

dm
where Is is specific impulse of the fuel, is the motor's mass flow rate, and m is the
dt

current mass of the spacecraft.

Impulsive thrust maneuvers are implemented in the simulations of Chapter 5.















CHAPTER 4
DEMONSTRATION OF AUTONOMOUS RENDEZVOUS TECHNOLOGY (DART)
MISSION OVERVIEW

The DART, successfully launched on April 15, 2005, was design as the first robotic

service to verify the capabilities of autonomous rendezvous with and maneuvering in

close proximity of another spacecraft, the Multiple Paths, Beyond-Line-of-Sight

Communications (MUBLCOM) satellite. This chapter gives a description of the DART

and MUBLCOM vehicles and an overview of the entire mission, particularly focusing on

the proximity operations phase.

4.1 DART Spacecraft

The DART spacecraft was built by Orbital Sciences Co. The service has a total

mass of 363kg, length of 6ft, and diameter of 3ft.
















Hydrazine Auxiliary Propulsion System

AVGS Bus

Figure 4-1. DART Spacecraft Configuration [65].









The DART vehicle, as shown in Fig. (4-1), is composed of two main parts. The

front is the Advanced Video Guidance Sensor (AVGS) bus, and the aft is the Hydrazine

Auxiliary Propulsion System hapsS). The specific components of the AVGS bus and

HAPS can be seen in Fig. (4-2).
























Figure 4-2. DART Spacecraft Component Configuration [65].

Three propulsion systems are used for the control of the DART vehicle. The HAPS

system has three hydrazine thrusters, which are used for all the maneuvers preceding the

close-proximity phase of the mission, as well as to control pitch and yaw of the DART

vehicle, and a set of six cold-gas nitrogen thrusters, which are used "to provide three-axis

attitude control during orbital drifts and roll control during HAPS burs" (65, p. 9). The

DART vehicle is also equipped with a set of 16 cold-gas nitrogen thrusters, which are

used during the close-proximity operation phase for the translation and attitude control.









This thesis doesn't focus on the attitude control; however, the characteristics of all the

propulsion systems used are given in Table (4-1) for completeness.

Table 4-1. DART Vehicle Propulsion Systems Characteristics.
Propulsion System Thrust (N) Specific Impulse (sec) Fuel Mass (kg)
HAPS Three Hydrazine 222 236 56.88
Thrusters
HAPS set of 6 cold-gas 111 NS 5.77
nitrogen thrusters
16 cold-gas nitrogen 3.6 60 22.68
thrusters

The HAPS three hydrazine thrusters, shown in Fig. (4-3B), are located in the back

of the HAPS section of the DART vehicle with the thrust direction aligned with the Xody

axis.

A B













Figure 4-3. DART Vehicle Thrusters. A) Ortho View, B) Back View.

Table (4-2) identifies the locations of the 16 cold-gas nitrogen thrusters on the

DART vehicle, also shown in Fig. (4-3). FWD and AFT represent the location of the

forward and aft thrusters, respectively. 90/270 specifies the location of the thruster with

respect to the circumference of the face.

Table 4-2. Location of the 16 Cold-Gas Nitrogen Thrusters.
Thruster # Thruster Exhaust Direction Thruster Location
1 45 Deg. +X/+Y FWD 90
2 +Y FWD 90











3 -Z FWD 90
4 +Z FWD 90
5 45 Deg. +X/-Y FWD 270
6 -Y FWD 270
7 -Z FWD 270
8 +Z FWD 270
9 -X AFT 90
10 +Y AFT 90
11 -Z AFT 90
12 +Z AFT 90
13 -X AFT 270
14 -Y AFT 270
15 -Z AFT 270
16 +Z AFT 270


4.2 MUBLCOM Spacecraft

The MUBLCOM, shown in Fig. (4-4), is the target satellite, which will be


approached by the DART spacecraft. MUBLCOM retroreflectors were designed


specifically for this mission's application.


-2X SOLAR PANELS


TRANSMITTER
RADIATOR


ARRAY DRIVE


UMBILICAL PLUG


ION WHEEL


Tx/Rx FILTER-
ANTENNA HINGE-
RF FILTER TEST
2X SMA
MAGNETOMETER -


RF OUT




,k


IN TEST SMA

COSINE SENSORS


DEFINED
9 PIN TEST


Figure 4-4. MUBLCOM Satellite Configuration [65].










Figure (4-5) presents the characteristics of the MUBLCOM orbit. This data was

accessed from the satellite database of the STK software, which contains Two-Line

Element (TLE) files, regularly updated by the U.S. Strategic Command

(USSTRATCOM). TLE sets contain orbital data for any satellite, and Fig. (4-6) shows

the format of TLE. TLE information was used to define the initial state of the

MUBLCOM satellite in Astrogator module of STK. The simulations in STK and their

set up will be discussed in Chapter 5.


Coord.System: Earth Centered Mean J2000

-Element Type: Keplerian

OrbitEpoch: 115Apr2005C

Semi-major Axis: 7123.8

Eccentricity: 0.001

Inclination: 97.731

RightAsc ofAsc Node: 157.1

Argument of Periapsis: 96.465

True Anomaly: 291.62


Change...


--I
0:00:00.000 UTCG

85 km

75444623

16deg

3 deg

1 deg

26deg


Figure 4-5. MUBLCOM Orbit.

1st derivative of Mean Drag term or
Name of Satellite Motion or Ballistic Coefficient radiation pressure
(11 characters) coefficent Element Nunter
International Epoch Year & 2nd derivative of Mean & Che sum
Designator Day Fraction Motion, usually blank Ephemeris






of the Asoending of Perigee Revolution number
Node at epoch & check sum

Figure 4-6. Two-Line Elements [81].
Figure 4-6. Two-Line Elements [8 1].








4.3 Mission

4.3.1 Launch and Rendezvous Phase

As previously mentioned, the DART spacecraft was successfully launched from

Vandenberg Air Force Base (AFB) in Lompoc, California into Phasing Orbit 1, as shown

in Fig. (4-7) as point A.

A Insertion into Phasing i- k
Orbit 1
B Begin Transfer to C
Phasing Orbit 2 3 km A
C Insertion into Phasing
Orbit 2
D Begin Transfer to Target
Orbit
E Insertion into Target Orbit
Phasing Orbit 1 D
Phasing Orbit 2
Target Orbit B
SDART Trajectory

Figure 4-7. DART Rendezvous Phase [65].

Phasing Orbit 1 is a circular parking orbit, matching the inclination and Right

Ascension of the Ascending Node (RAAN) of the MUBLCOM's orbit. The DART

spacecraft then performs a Hohmann orbit transfer, a fuel and velocity efficient transfer

between two coplanar circular orbits, to reach Phasing Orbit 2, indicated as point C in

Fig. (4-7). At this point, the DART spacecraft will be approximately 40 km behind and

7.5 km below the MUBLCOM satellite. After allowing the DART vehicle to drift to

approximately 21 km behind MUBLCOM (point D in Fig. (4-7)), the final CW transfer is

performed, positioning the DART spacecraft into MUBLCOM's orbit, 3 km behind









relative to the satellite. This is indicated as point E in Fig. (4-7). These maneuvers are

accomplished using the three hydrazine thrusters of the HAPS propulsion system.

4.3.2 Close-Proximity Operations Phase

The close-proximity operations start with a CW transfer from -3 km to -1 km

relative to the MUBLCOM satellite. This and the rest of the maneuvers in this phase are

shown in Fig. (4-8). From -1km point the DART spacecraft performs forced maneuvers

to move along V-Bar and R-Bar directions, with the exception of the transfer to R-Bar

axis and circumnavigation maneuvers, which are accomplished with CW transfers. Most

of the close-proximity maneuvers are to be performed twice as can be seen from Fig. (4-

8). DART performs several docking axis approaches from -15 m to -5 m, as to simulate

the docking maneuver. However, in this thesis the DART spacecraft will perform those

approaches along V-Bar axis without transferring to the vehicle's docking axis. The

maneuvers in this phase are accomplished using a set of 16 cold-gas nitrogen thrusters.

The close-proximity operations phase is the focus of this thesis, therefore the simulation

described in Chapter 5 are restricted to the maneuvers shown in Fig. (4-8). The timeline

for these maneuvers is given in Table (4-3), which is based on (65, p.23 Fig. 10). After

completing the proximity operations phase, the DART spacecraft initiates a retirement

burn that is to place the vehicle into a parking orbit.

Even though the general profile of the DART mission is followed, this thesis does

not precisely reflect the real mission, and the changes and/or simplifications made are

noted where applicable. Reference [65] presents the details of the entire mission for the

interested reader.














* Station Keeping Points


First -V Bar
Approach
with Docking Axis
Approaches to -5 m


-200 m


/ -350 m
CW Transfer


CW Transfer

Start of Prox Ops
-40 km Behind
Phasing Orbit #2
7.5 km Below


Figure 4-8. DART Proximity Operations [65].


Table 4-3. Close-Proximity Operations Timeline
Maneuver Duration (sec)
Station-keep at -3km V-Bar 300
CW transfer to -1km 4576
Station-keep at -1km V-Bar 300
Forced motion to -200m V-Bar 3094
Station-keep at -200m V-Bar 300
Forced motion to -15m V-Bar 826
Station-keep at -15m V-Bar 5400
Forced motion to -5m (1) V-Bar 131
Station-keep at -5m V-Bar 600
Forced motion to -15m (1) V-Bar 134
Station-keep at -15m V-Bar 300
Forced motion to -5m (2) V-Bar 132
Station-keep at -5m V-Bar 300
Forced motion to -15m (2) V-Bar 135
Station-keep at -15m V-Bar 300
Forced motion to -100m V-Bar 473
Station-keep at -100m V-Bar 1800
Forced motion to -200m V-Bar 498
Station-keep at -200m V-Bar 300
Depart to lose AVGS tracking (-350m) V-Bar 589


300 m









Return to -200m V-Bar 546
Station-keep at -200m V-Bar 1800
CW transfer to R-Bar 1562
Station-keep at 150m on R-Bar 1800
Forced motion to 100m on R-Bar 248
Station-keep at 100m on R-Bar 300
Forced motion to 300m on R-Bar 585
Station-keep at 300m on R-Bar 300
CW transfer circumnavigation to -1km 4561
Station-keep at -1km on V-Bar 3317
Forced motion to -200m V-Bar 3094
Station-keep at -200m V-Bar 300
Depart to lose AVGS tracking (-350m) V-Bar 559
Return to -200m V-Bar 528
Station-keep at -200m V-Bar 300
CW transfer to R-Bar 1565
Station-keep at 150m on R-Bar 300
Forced motion to 100m on R-Bar 255
Station-keep at 100m on R-Bar 300
Forced motion to 300m on R-Bar 658
Station-keep at 300m on R-Bar 425














CHAPTER 5
SIMULATIONS USING SATELLITE TOOL KIT (STK) SOFTWARE BY
ANALYTICAL GRAPHICS, INC. (AGI)

5.1 Satellite Tool Kit (STK)

STK is a commercial-off-the-shelf (COTS) analysis software for land, sea, air, and

space. STK has a capability to present results in both graphical and text formats in order

to analyze and determine optimal solutions for space scenarios. The software is used to

support all phases of a satellite system's life cycle, including: policy, concept,

requirements, development, testing, deployment, and operations. STK also calculates

data and displays multiple 2-D maps to visualize various time-dependent information for

satellites and other space-related objects. STK is capable of generating orbit/trajectory

ephemeris and providing acquisition times and sensor coverage analysis for any of the

objects modeled in the STK environment. STK/PRO, a collection of additional orbit

propagators, attitude profiles, coordinate types and system, sensor types, in view

constraints, and city, facility, and star databases, is available as add-on collection of

modules (82).

5.1.1 STK/Advanced Visualization Option Module

STK/Advanced VO is a dynamic 3-D environment that visualizes complex mission

and orbit geometries by displaying realistic 3-D views of space, airborne, and terrestrial

assets; sensor projections; orbit trajectories; along with assorted visual cues and analysis

aids (83).







61



5.1.2 STK/Astrogator Module

STK/Astrogator is an interactive tool used for orbit maneuvering and space mission


planning (84). Figure (5-1) displays STK's graphical user interface (GUI) for the


Astrogator. Mission control sequences (MCS), shown on the left hand side (LHS) of the


figure, display the maneuvers performed during the mission. The detailed information


regarding each orbit maneuver is displayed in the right hand side (RHS) of the figure.


Bulls-eye mark next to each maneuver segment in the MCS implies that the Astrogator


targeter was used to obtain the values displayed on the RHS of Fig. (5-1).


Propagator: Astrogator


- g- CW Close-Proximity Ops A
is -r:;n, 3oni., Kn.

<5 Prop to VbarCross
4J -
F *TargetStop
t5 Station-keep at-1km -
S lFr mn, ll.n, iG t l'Hin,

Station-keep at-200m
i From -200m to -15m
STargetStop
6 Station-keep at-15m
F 'l:rr rn, -15m to -5m (1)
ST argetStop
J Station-keep at-5m
SFrom -5m to -15m (1)
STargetStop


Results...


Maneuver Direction
AnLuas Vector Type: Cartesian

Thrust X VelZ Radial
Axes- Y vIUX UA0n...

Z n.-1H1 -lcAi el.. En 9 51
Attitude Options...


IT iruV~ ir p, I
IAVGS Vbar pos


i- Decrement Mass Based on
Fuel Usage


Initial' 16 Apr 25005:53:26.829 UTCG


SFinal: 16 Apr 20000:53:26.829 UTCG


Figure 5-1. Astrogator

Figure (5-2) displays STK's GUI for the Astrogator targeter for the maneuver from


-3km to -1km along the V-Bar. The control parameters are the values ofAV for the


impulsive thrusting needed to meet the requirements of the mission. Targeting process is


started with an initial guess at the values of control parameters. The initial guesses


I I







62



for AV have to be reasonably near a valid solution in order for the targeter to successfully


converge. In this effort, the values for AV obtained from the CW algorithm were used as


the initial guesses.


Equality constraints represent the requirements of the mission. In this case the


DART spacecraft has to complete a maneuver for duration of 4576 seconds, traveling


from -3km to -1km relative to MUBCLOM satellite. At the end of this maneuver the


DART spacecraft has to be located -1km in-track direction and not displaced in the


cross-track and radial directions.



Variables Convergence Advanced Log
rControl Parameters


InitialValue: -9.7008e-005 km/sec

Correction: 0 km/sec

Tolerance: 1e)009 km/sec

Equality Constraints --------


,f~rprerr.- 10()0210327km LI

Tolerance. 0.001 km 'Pi


SPerturbation: Ile-06 km/sec r Scaling
Method- By initial value ]
Max Step: D001 km/sec
Value: 0.001 km/sec


Scaling
Method: By desiredvalue
Weight 100000000
Value: (3001 km


L= LCancel Appy IHelp


Figure 5-2. Astrogator Targeter.


Use I Name Desired Valuel Current Value I Semn
rv- CrassTrack /0 kin -OOOGO073287 km jProp to Vbar Crossing
j CrossTrackRate 0 km/sec LlIMO100G(N4878 km/sec 1Prop toVbar Crossing
rv IDuration /4576 sec 14576.71000000 sec lProp to Wbar Crossing
11 IRadial 0Dk .~ D964 k Po toy- Crssng
r ~da~t Dkm/sec iO.W017268343 kmlsec jPrap to Vbar~rosig










The Astrogator targeter uses an iterative technique referred to as the differential

corrector to solve, in this case, a system of four equations (equality constraints) and three

unknowns (control parameters). The process is outlined in equation (5.1) below.


n-track n-track
radial radial
+S r r (5.1)
cross-track cross-track
t d t 0d z oj


Assuming that the functions are linear, a sensitivity matrix, S, containing the partial


derivatives calculated numerically by running the trajectory with perturbed control

parameters is created.






tP to t to t t
in-trackp in-track n-trackp in-track o (n-track, in-track0

rcharap rach r
rad alp radal radalp r adral raalp rad5al

S5.2 Simu n (5.2)
ross-track cross-track cross-track cross-trackt cross-trackox y ross-track o


t the t(refer to Fig. (4-7) in Chapter 4). the simulated scenarios begin with



Linear solution is obtained by rearranging equation (5.1),


n-track n-track
Fx rx r
S+ S1 radial radial (5-3)
cross-track cross-track
-d -z-o t t
-d L o

The inverse of sensitivity matrix is found using Singular Value Decomposition (SVD).

5.2 Simulation Scenarios

The scenarios simulated in STK emulate the close proximity operations phase of

the DART mission (refer to Fig. (4-7) in Chapter 4). The simulated scenarios begin with









Phasing Orbit 2, shown as point C in Fig. (4-6), and close proximity operations begin

when the DART spacecraft is positioned in MUBLCOM's orbit, 3 km behind relative to

the satellite (refer to point E in Fig. (4-6)). Three different types of scenarios are

investigated:

1. The close-proximity operations are performed using a single two-impulse

transfer for each maneuver in the phase.

2. The close-proximity operations are performed using multiple two-impulse

transfers in order to represent each maneuver of the phase as essentially

"finite". This scenario more closely represents a motion strictly along V/R-

Bar directions.

3. The close-proximity operations are attempted using finite thrusting along V-

Bar direction.

A CW algorithm based on the developments in Chapter 3 was written in Matlab

(refer to Appendix B) in order to compute the values for the change in velocity, AV, for

the impulsive thrusting to be used in STK. Scenarios 1 and 2 were performed in two

categories: untargeted and Astrogator targeted maneuvers. Untargeted category implies

that the values for AV obtained from the CW algorithm were directly used to implement

an impulse. The Astrogator targeted category implies that the values for AV obtained

from the CW algorithm were used as initial values, and the final values for AV were

found by the Astrogator targeter (described in section 5.1.2) via numerical integration

using an eighth order Runge-Kutta-Verner integrator with ninth order error control

(RKV8th9th). The purpose of considering both categories is to observe the discrepancies









in the relative position values that result due to infeasibility of linearized CW model for

precise autonomous proximity operations.

5.2.1 Scenario 1

In this scenario the close-proximity operations are performed using a single two-

impulse transfer for each maneuver in the phase. This is demonstrated in Fig. (5-3).

Figure (5-3A) displays the mission profile of the DART spacecraft for the close-

proximity operations phase (as described in Chapter 4). Figure (5-3B) displays the STK

simulated scenario of the close-proximity operations phase. The trajectory of

MUBLCOM's orbit is shown. The inbound maneuvers and the outbound maneuvers of

the DART spacecraft along the V-Bar and R-Bar directions with respect to the

MUBLCOM satellite are shown. The circumnavigation maneuver that takes the DART

spacecraft from R-Bar to V-Bar axis is also displayed.

As per the DART mission profile, station-keeping segments occur between each

maneuver. During each station-keeping maneuver, the DART spacecraft's relative

position is expected to stay constant with respect to MUBLCOM satellite. However,

because the CW method is not exact, the DART spacecraft reaches the desired position

slightly above or below the V-Bar axis. The radial offset from the V-Bar axis has a radial

rate associated with it. A threshold based on DART's relative radial rate is defined, and,

when exceeded, a station-keeping bum in the radial direction is triggered to correct the

orbit. Figure (5-4) demonstrates the STK set up GUI for the station-keeping maneuver.

The stopping condition defines the terms under which the maneuver is ended. Station-

keeping maneuvers stop when a required duration is reached. The "UserSelect"

condition triggers the radial orbit correction bum, when the radial rate exceeds a specified

value. Two conditions are specified; triggering the correct one depends on the location of








66



the DART spacecraft (below or above the V-Bar). The value of the threshold depends on


the duration and the distance of a specific transfer, which dictates the magnitude of the


velocity that spacecraft will have in excess of what is required to keep constant with


respect to the satellite.



Circumnavigation

SStaton Keeping Points
/First-VBar
Approach
with Dodong Axis Second -V Bar
Approaches to -5 m Approach
1515 -200m
-I .V Bar-- -- -- -- ---


SFirst and Second 350 m
Br 100 CW Transfer
150 m

30 m A


Figure 5-3. Scenario 1 Targeted Close-proximity Operations (Starting at -3km Point).
















- %V.1 CW loe-Proi.n,.ro Opt ^
From -.,n. I.::.- lln.
TargetStop
Station-keep at-1km
N Front IKni .::- .I'.H)nl
T rg e r 'I l .P
5 Station-keep at-200n
S FrorI -''l'H ni to-15m
TargetStop
Station-keep at-15m
From -15m to -5m (1)
TargetStop
SStaton-keep at-5m
From -5mto-15m (1)
TargetStop
(, Station-keep at-15m
From -15m to-5m (2)
TargetStop v
< in> |


Propagator: IMy Earth Full Advanced...

Stopping Conditions:
Name User Comment
Duration Stop after a specified duration Insert..

UserSelectl Stop on a specified value using a specified calculation obje
; Remove

Trip: "-0 LttJ{KH '.0I .n, .: Criterion: Cross Either (Inc., .

Tolerance: 0.00000003 km/sec j Repeat 1
SVI Count I
Sequence: Burn Up VerySmall J -Constraints:-
Central Earth
Body: I
UserCalc. RacaRajie
Object 9 I


r"""esul. I Initial: 16Apr200502:09:43.539UTCG I Final: 16Apr200502:14:43.539UTCG


Figure 5-4. Station-keeping Set-up


The station-keeping maneuver to stay on the V-Bar axis is shown in Fig. (5-5A).


The DART spacecraft arrives at a specified location (respective of the maneuver at hand)


on the V-Bar axis. Then it performs a station-keeping maneuver for a required duration.


The DART spacecraft performs radial burns in order to force itself to stay on the V-Bar


axis. Upon completion of the station-keeping maneuver, DART initiates the proceeding


maneuver by placing itself onto the departure trajectory. Similar type of tolerance is set


up for the station-keeping maneuvers to stay on the R-Bar axis, where the spacecraft will


move in the radial (R-Bar) direction for the duration of the maneuver, as shown in Fig.


(5-5B). This will keep the vehicle from moving too much in the V-Bar direction,


therefore positioning DART directly below (or above) the MUBLCOM satellite.





























Figure 5-5. Station-keeping. A) V-Bar, B) R-Bar.

Figure (5-6) shows the DART spacecraft in the 5m vicinity of the MUBLCOM

satellite, which is the closest point of approach in the mission profile. DART's and

MUBLCOM's local coordinate axis, VNC, are displayed at their respective locations.

In the VNC (Velocity Normal Co-normal) coordinate frame: the X axis is along the

velocity vector (V In-Track), the Y axis is along the orbit normal (Y = R x V Cross-

Track), and the Z axis completes the orthogonal triad (Z = X x Y Radial). The CW

algorithm developed in this thesis (refer to Appendix B) takes into account the relative

orientation between CW frame described in Chapter 3 and the VNC frame used in STK

simulations. Velocity vectors for both vehicles are shown (red for MUBLCOM, and light

blue for DART) in the positive V-Bar direction. Because MUBLCOM satellite's motion

is the primary motion, the motion of the DART spacecraft is analyzed with respect to the

MUBLCOM's VNC frame.































Figure 5-6. DART at -5m Relative to MUBLCOM (Point of Closest Approach).

The two categories of untargeted and Astrogator targeted maneuvers were

performed in order to investigate and quantify the errors in the relative position that occur

due to the orbital perturbations that are neglected in the untargeted AV computations.

Due to the accumulation of errors that would occur for multiple CW-based (i.e.,

untargeted) maneuvers the entire mission profile of the close-proximity operation phase

will not be analyzed, but several V/R-Bar maneuvers in the phase will be investigated.

5.2.1.1 V-Bar maneuvers

Figure (5-7) displays the maneuvers that are considered for the DART spacecraft

transferring relative to the MUBLCOM satellite. The DART spacecraft's trajectory starts

at the -3km, point A, and includes four maneuvers, each represented by a single two-

impulse transfer, positioning DART 5m behind the MUBLCOM satellite, point B.

Station-keeping, similar to Fig. (5-5A), is performed between each maneuver for a

specified duration.











Educational Use Only

-5m -15m
\ I 200m -Ikm -3km
-200 V-BarAxis
MUBLCOM


DART

Educational Use Only


Figure 5-7. V-Bar Maneuvers.

Table (5-1) displays the component values in the MUBLCOM satellite's

VNC frame for the initial impulsive burn to start each maneuver. At the beginning and

the end of each transfer the DART spacecraft is assumed to be in the same orbit as

MUBLCOM satellite, making the relative initial and final velocities to be zero.

Therefore, the final impulsive burn is implemented to seize any relative motion between

the vehicles. These values are shown in Table (5-2) and were obtained similarly to the

values in Table (5-1).

Table 5-1. Initial Impulse Values for V-Bar Maneuvers from -3km to -5m.
Maneuver AV Targeted (km/s) AVUntargeted (km/s) % Error
-3km to -1km -0.00009700879 -0.00009670000 .3183

-0.00000020840 0.00000000000 100
-0.00017547925 -0.00017637279 .5092
-1km to -200m -0.00000556245 -0.00000528616 4.9671-

0.00000000000 0.00000000000 0
-0.00019624144 -0.00019712353 .4495
-200m to -15m 0.00017429195 0.00017619398 1.0913-

-0.00000014229 0.00000000000 100
-0.00016198627 -0.00016317508 .7339
-15m to -5m 0.00008131069 0.00007585655 6.7078-

0.00000000000 0.00000000000 0
-0.00000985996 -0.00001045073 5.9916









Table 5-2. Final Impulse Values for V-Bar Maneuvers from -3km to -5m.
Maneuver AV Targeted (km/s) AVUntargeted (km/s) % Error
-3km to -1km 0.00009728156 0.00009670000 .5978
0.00000004957 0.00000000000 100
-0.00017266999 -0.00017637279 2.1444
-1km to -200m 0.00000558119 0.00000528616 5.2861
-0.00000029427 0.00000000000 100
-0.00019687285 -0.00019712353 .1217
-200m to -15m --0.00017520527- -0.00017619398- .5643-
0.00000026803 0.00000000000 100
-0.00016236643 -0.00016317508 .4980
-15m to -5m -0.00008059413 -0.00007585655 5.8783
0.00000001479 0.0000000000 100
-0.00001238031 -0.00001045073 15.5859

Figure (5-8) displays STK Astrogator's GUI set up for the maneuver from -3km to

-1km and helps to understand how the values in Table (5-1) for the untargeted category

were obtained. An impulsive maneuver (named Go to -1km) is inserted into the MCS on

the LHS with defining characteristics on the RHS of Fig. (5-8). Maneuver direction,

thrust axes, propulsion system, and values of thrust vector for each direction are specified

as shown. The AVvalues for the impulsive maneuver were obtained from the CW

algorithm and are displayed in Table (5-1) under untargeted category. The propagate

segment is inserted into the MCS to identify the duration during which the DART

spacecraft will be traveling to the -1km destination. The motion is ceased upon arrival at

the specified location via the second impulse (named Stop at -1km), which is set up

similar to the first impulse and whose values are shown in Table (5-2) in the untargeted

category. The station-keeping maneuver is then inserted into the MCS, and it keeps the

DART spacecraft stationary with respect to the MUBCLOM satellite, as was described

earlier in this section.











Propagator: IAstrogator


- g i:W Clo..'- Pr.,,rr,,r Opi.

& Prop 3:31ae- 1.3 -1.ni
3 lo p jl -l.mn
S ._l r.i .r..-l ep. al l1 .n,
Xj Go to -200m
(, Fr.:.p,.,la to i.:.-*il"0j
4 Stop at-200m
$ Station-keep at-200m
i1 Go to -15m
Propagate to -15m
A4 Stop at-15m
& Station-keep at-15m
4 Go to -5m
P Fropg.ie Io .ni
X; Stop at-5m
O STOP


Results...

R results .. t


Maneuver Direction
Attitude hrust eVector Type: ICartesian
Control: IThrust Vector
Trrui VFIZRc X: -.00009670000 km/se
e Y: 10.0000000000 km/se

Z: -0.00017637279 km/se j
Attitude Options..


IThruster Set

IAVGS Vbar pos


- Decrement Mass Based on
Fuel Usage


Initial: 16 Apr2005 00:53:26.829UTCG


EV Final: 116Apr200500:53:26.829 UTCG Id


Figure 5-8. STK Untargeted Maneuver Set-Up.

Figure (5-9) displays STK Astrogator's GUI set up for the maneuver from -3km to


-1km and helps to understand how the values in Table (5-1) for the Astrogator targeted


category were obtained. Targeted sequence (marked with bulls-eye) is inserted into the


MCS on the LHS of Fig. (5-9). An impulsive maneuver (named V-Bar) is then inserted

into the targeted sequence, defining characteristics of which are set up on the RHS of Fig.


(5-9). Maneuver direction, thrust axes, propulsion system, and values of thrust vector for


each direction are specified as shown. The bulls-eye symbol with a check mark indicates


that the desired AVvalues for the initial impulse will be targeted via the Astrogator


targeter (discussed in section 5.1.2). The Astrogator targeter needs an initial guess for the


AVvalues, which was obtained from the CW algorithm. The values displayed in Table


(5-1) are obtained by the Astrogator targeter as was described in section 5.1.2.


=











Propagator: IA. Irjogar "


H d CW Close-Proximity Ops A
I rGr 1p'.m |G -ll.n

Prop to Vbar Cross

TargetStop
4 stop motion
41-
SStation-keep at-lkm
I ltrom 1 imIO *-i-m
STargetStop
< Station-keep at-200m
S 1 rorr -.Irrn to -15m
TargetStop
(' Station-keep at-15m
I rc.nl -1 5mto-5m(1)
STargetStop
(' Station-keep at-5m


Results...


Mareuer Direction -----------
Attitude Th------ Vector Type: ICartesian
Control: Thrust Vector
____x:__-____ X 4) 7O l CH -. *.n -,' e |
Thrust XVelZ Radialxs
Attitu: e r' '' .n. -
Z: -01: ( 11) I-. -4- `4. rr,.- e
Attitude Options...


IThrusterSet

IAVGS Vbar pos


Decrement Mass Based on
Fuel Usage


Initial: 16 Apr2005 00:53:26.829 UTCG


SFinal: 116Apr 20500:53:26.829 UTCG


Figure 5-9. STK Astrogator Targeted Maneuver Set-Up.

The propagate segment is inserted into the targeted sequence to notify when the DART


spacecraft will reach V-Bar axis. At which point, the motion is ceased upon arrival at the


specified location via the second impulse (TargetStop), which is set up similar to the first


impulse and whose values are shown in Table (5-2) in the Astrogator targeted category.


The station-keeping maneuver is then inserted into the MCS, and it keeps the DART


spacecraft stationary with respect to the MUBCLOM satellite, as was described earlier in


this section.


Tables (5-1, 5-2) also display the percent error in the components between the


untargeted and Astrogator targeted categories for each impulse. Untargeted category


does not consider an out-of-plane motion, so the percent error displayed in the Tables (5-


1, 5-2) is in some cases 100%, even though the Astrogator targeter makes very small out-









of-plane corrections. This is due to the assumption of the spherically homogeneous Earth

in the case of the untargeted category, whereas the Astrogator targeted category takes in

account oblateness of the Earth effects. The error in impulses for the in-plane motion

depends on the accuracy of the relative position, which sometimes can be improved by

the station-keeping maneuvers in between each transfer (described below). For that

reason the error in impulses for the in-plane motion has an oscillatory behavior, however,

overall it is increasing.

The error in the relative position after each transfer, shown in Fig. (5-7), is

analyzed next. Figures (5-10) through (5-13) show DART's arrival at the specified point

relative to MUBLCOM on the V-Bar for untargeted (A) and Astrogator targeted (B)

categories. It can be observed from the numerical data on each figure that the error

increases with each successive figure. The error can sometimes be reduced due to the

effect of the station-keeping. This can be seen between Figs. (5-12) and (5-13).













DART Relative Position -km Educational Use Only
Time (UTCG): 16 Apr 2005 02:09:42-829
InTrack (km): -1.003401
Radial (km): 0.001258
CrossTrack (km): -0.000198



DART








Educati nal Use Only A



DART Relative Position -km Educato Use Only
Time(UTCG): 1 Apr 2005 02:09:43.539
InTrack (km): -0.999612
Radial (km): 0.000000
CrossTrack (km): -0.000001



RT







Ed o al UseOny B


1km V-Bar. A) Untargeted, B) Targeted.


Figure 5-10. Transferto












DART Relative Position -km Educational Use Only -
Time (UTCG): 16 Apr 2005 03:06:16.829
InTrack (km): -0.183754
Radial (km): 0.000027
CrossTrack (km): 0.000387


/ Educational Jse Only A



DART Relative Position -km Educational Use Only.
Time (UTCG): 16Apr 2005 03:06:16.015
InTrsx. rmi -0.199656
Radial (km): -0.000940
-rrr w 1 n i 00 0193


Figure 5-11. Transfer to -200m V-Bar. A) Untargeted, B) Targeted.

The numerical data in Fig. (5-12A) displays the achieved untargeted relative position of

DART from -200m to -15m transfer. Then DART spacecraft performs a station-keeping

maneuver (red trajectory in Fig. (5-12A)), which brings it closer to V-Bar, therefore

correcting the existed error in radial direction. This reduces the amount of error added on

to the next transfer from -15m to -5m, shown in Fig. (5-13).














DART Relative Position -km Educational Use Only
Time(UTCG): 16 Apr 2005 03:25:02.829
InTrack (km): -0.005382
Radial (km): 0.004494
CrossTrack (km): 0.000386


DART Relative Position.-km Educational Use Only
Time (UTCG): 16 Apr 2005 03:25:03.104'


InTrack (km):
Radial (km):
CrossTrack (km)i



-mR-A IiP rr'il


-0.014809
-0.000332
-0.000000



- - - - - -


Educational Use Only B



Figure 5-12. Transfer to -15m V-Bar. A) Untargeted, B) Targeted.












DART Relative Position -km Educational Use Only
Time (UTCG): 16 Apr 2005 04:57:13.829
InTrack (km): 0.003144
Radial (km): 0.000870
CrossTrack (km): 0.000457


ART







Educational Use Only A



DART Relative Position -km Educational Use Only
Time (UTCG): 16 Apr 2005 04:57:14.079
InTrack (kn): -0.005002
Radial (km): -0.000000
CrossTrack (km): -0.000005











Educational Use Only B


Figure 5-13. Transfer to -5m V-Bar. A) Untargeted, B) Targeted.

Table (5-3) shows the values for the desired relative position and the values

achieved in the two categories for each direction in the MUBLCOM satellite's

VNC frame; percent error is also included. The relative position values are obtained

based on the two-impulse transfers used for each maneuver, and therefore related to the

impulse values listed in Tables (5-1, 5-2). The error in the relative position increases for

all directions with each maneuver in the untargeted category. The error in the radial and









cross-track positions for the targeted category is present due to the limitations of

numerical integrator based on the tolerances defined for the Astrogator targeter.

However, the order of magnitude for the error in the radial and cross-track positions for

the untargeted category is always greater. This error is present due the linear

assumptions of the CW model and assumption of spherically homogeneous Earth.

Table 5-3. Relative Position.
Maneuver Desired Targeted % Error Untargeted % Error
(km) (km) (km)
-3km to -1km --1 -.999612 .0388 -1.003401 .3401
0 .000000 0 .001258 100
0 -.000001 100 -.000198 100
-1km to -200m -.2 -.199656- .1720- -.183754- 8.1230-
0 -.000940 100 .000027 100
0 .000193 100 .000387 100
-200m to -15m -.015 -.014809 1.2733 -.005382- 64.1200-
0 -.000332 100 .004494 100
0 -.000000 0 .000386 100
-15m to -5m -.005 -.005002 .0400- -.003144- 37.1200-
0 -.000000 0 .000870 100
0 -.000005 100 .000457 100

Figure (5-14A) shows the relative position error for the duration of the maneuvers,

shown in Fig. (5-7), along the V-Bar direction. Figure (5-14B) zooms in to show the error

at the -5m point, which corresponds to the numerical difference seen from Fig. (5-13)

and Table (5-3).







80


Relative Position Error for V-Bar Maneuvers x 10-3 Relative Position Error for V-Bar Maneuvers
0.01 2
-- In-Track t- in-Track
Radal Radial
----- Cross-Track -- Cross-Track



0 -05 -






0-11

-0.02
00 5000 10000 15000 1.38 1.39 1.4 1.41 1.42 1.43 144 1.45 1.46 1.47
Time (sec) A Time (sec) x 104 B


Figure 5-14. Error in Relative Position Untargeted/Targeted. A) V-Bar Transfers for
Maneuvers in Fig. (5-7), B) At -5m Point on the V-Bar.

5.2.1.2 R-Bar maneuvers

Figure (5-15) displays the maneuvers that are considered for the DART spacecraft


transferring relative to the MUBLCOM satellite. The DART spacecraft's trajectory starts


at the 100m, point A, and includes two maneuvers, each represented by a single two-


impulse transfer, positioning DART 300m below the MUBLCOM satellite, point B.


Educational Use Only







MUBt COM
I-
I.Oi -ril- A
---- DART



RB B

R-Bar Axi s


Figure 5-15. R-Bar Maneuvers.









At the beginning and the end of each transfer the DART spacecraft is positioned

directly below the MUBLCOM satellite, making the relative initial and final in-track

distance to be zero. Even though the DART spacecraft is in a lower orbit than that of

MUBLCOM, DART's relative position must remain constant with respect to the satellite.

This is enforced by the station-keeping maneuvers (refer to Fig. (5.5B)), preventing the

spacecraft from moving ahead of the MUBLCOM satellite.

Table (5-4) displays the component values in the MUBLCOM satellite's

VNC frame for the initial impulsive burn to start each maneuver. The final impulsive

burn is implemented to seize the initial motion, stopping at the arrival point. These

values are shown in Table (5-5). The numerical data in these tables is obtained similarly

to the numerical data in Tables (5-1, 5-2), described earlier.

Table 5-4. AVInitial Values for R-Bar Maneuvers from 150m to 300m.
Maneuver AV Targeted (km/s) AVUntargeted (km/s) % Error
150m to 100m 0.00005459135 0.00005688173 4.1955-
-0.00000129335 0.00000000000 100
0.00023411542 0.00025088475 7.1628
100m to 300m -0.00015994949 -0.00017359000- 8.5280
-0.00000000970 0.0000000000 100
-0.00019664434 -0.00015434256 21.5118

Table 5-5. AVFinal Values for R-Bar Maneuvers from 150m to 300m.
Maneuver AV Targeted (km/s) AVUntargeted (km/s) % Error
150m to 100m -0.00005070418 -0.00005688173 12.1835
-0.00000107950 0.00000000000 100
-0.00015060231 -0.00025088475 66.5876
100m to 300m 0.00025899710 0.00017359000- 32.9761
-0.00000003061 0.00000000000 100
0.00051574610 0.00015434256 70.0739









Tables (5-4, 5-5) also display the percent error in the components between the

untargeted and Astrogator targeted categories for each impulse. Untargeted category

does not consider an out-of-plane motion, so the percent error displayed in the Tables (5-

4, 5-5) is in some cases 100%, even though the Astrogator targeter makes very small out-

of-plane corrections. This is due to the assumption of the spherically homogeneous Earth

in the case of the untargeted category, whereas the Astrogator targeted category takes in

account oblateness of the Earth effects. The error in impulses for the in-plane motion

depends on the accuracy of the relative position, which sometimes can be improved by

the station-keeping maneuvers in between each transfer. It is more difficult to maintain a

constant relative position with respect to the MUBLCOM satellite while performing the

station-keeping on the R-Bar, and the increase in the error can be noticed with time.

The error in the relative position after each transfer is analyzed next. Figures (5-

16) and (5-17) show DART's arrival at the specified point relative to MUBLCOM on the

R-Bar for untargeted (A) and Astrogator targeted (B) categories. It can be observed from

the numerical data on each figure that the error increases with each successive figure.

Table (5-6) shows the values for the desired relative position and the values

achieved in the two categories for each direction in the MUBLCOM satellite's

VNC frame; percent error is also included. The relative position values are obtained

based on the two-impulse transfers used for each maneuver, and therefore related to the

impulse values listed in Tables (5-4, 5-5). The error in the relative position increases for

all directions with each maneuver in the untargeted category. The error in the in-track

and cross-track positions for the targeted category is present due to the limitations of

numerical integrator based on the tolerances defined for the Astrogator targeter.









However, the order of magnitude for the error in the in-track and cross-track positions for

the untargeted category is always significantly greater. This error is present due the

linear assumptions of the CW model and assumption of spherically homogeneous Earth.

Table 5-6. Relative Position.
Maneuver Desired (km) Targeted (km) % Error Untargeted (km) % Error
150m to 100m 0 .000000 0 -.000573- 100
-.1 -.100150 .1500 -.095801 4.1990
0 -.000000 0 .000316 100
100m to 300m 0 -.000003 100 .113922 100
-.3 -.299838 .0540 -.420756 40.2520
0 -.000000 0 .000324 100







84




DART Relative Position-km Educational Use Only -
Time (UTCG): 16 Apr 2005 08:09:12.237
InTrack (km): -0.000573
Radial (km): -0.095801
CrossTrack (km): 0.000316



DART





/

Education ny A



DART Relative Position-km Educational Use Only
Time (UTCG): 16 Apr 2005 08:09:11.636
InTrack (km): 0.00000
Radial (km): -0.100150
CrossTrack (km): -0.000000



DAPT







Educational Use Only B


Figure 5-16. Transfer to 100m R-Bar. A) Untargeted, B) Targeted.













DART Relative Position -km Educational Use 0
Time (UTCG): 1Apr 2005 0823-57.237-
InTrack (km): 0.113922
Radial (km): -0.420756
CrossTrack (km): 0.000324












Educational Use Only A

3G h I Ert


DART Relative Position -km Edicational Jse Only
Time (UTCG): 16 Apr 2005 08:23:56.575
InTrack (km): -0.000003
Radial (kn): -0.299838
CrossTrack (km): -0.000000



~4- T


I Educational Use Only B


Figure 5-17. Transfer to 300m R-Bar. A) Untargeted, B) Targeted.

Figure (5-18A) shows the relative position error for the duration of the maneuvers

along the R-Bar direction, shown in Fig. (5-15). Figure (5-18B) zooms in to show the

error at the 300m point, which corresponds to the numerical difference seen from Fig. (5-

17) and Table (5-6).







86


Relative Position Error for R-Bar Maneuvers Relative Position Error for R-Bar Maneuvers
0.15
0.1
01

SB
E 0U5 E
c /- 005
0
0 050





-0.15 In-Track In-Track
Radial -0 1 Radial
----- Cross-Track ----- Cross-Track
-02
0 200 400 600 800 1000 1200 1110 1115 1120 1125 1130 1135 1140 1145
Time (sec) A Time (sec) B

Figure 5-18. Error in Relative Position Untargeted/Targeted. A) R-Bar Transfers for
Maneuvers in Fig. (5-13), B) At 300m Point on the R-Bar.

5.2.1.3 Fuel usage

One of the capabilities of the Astrogator module in STK is to record the amount of


fuel used for each bum executed. The fuel usage is recorded for all the maneuvers in the


close-proximity operations phase, as shown in Fig. (5-3B). The two propulsion systems


used for the proximity operations phase are the three hydrazine thrusters of the HAPS


propulsion system and the 16 cold-gas nitrogen thrusters, as described in Chapter 4.


According to reference [65], at the beginning of the proximity operations phase the


DART spacecraft performs a switch of the propulsion systems, using the 16 cold-gas


nitrogen thrusters to control the translation of the spacecraft during this phase of the


mission. However, it will be demonstrated that the first maneuver of the proximity


operations, a two-impulse transfer from -21km to -3km, uses a significant amount of fuel


and should be accomplished with the use of the three hydrazine thrusters of the HAPS


propulsion system. Therefore, this thesis also considers a switch of thruster systems at -


3km point in the mission to compare impact on the fuel usage.