Harold L. Hallock Gary Welter (cid:129) David G. Simpson Christopher Rouff (cid:129) ACS Without an Attitude 123 Harold L.Hallock DavidG.Simpson Bowie,MD Software Engineering Division USA GoddardSpace Flight Center NASA Gary Welter Greenbelt, MD Software Engineering Division USA GoddardSpace Flight Center NASA Christopher Rouff Greenbelt, MD AppliedPhysics Laboratory USA Johns HopkinsUniversity Laurel, MD USA ISSN 1860-0131 ISSN 2197-6597 (electronic) NASA Monographsin Systems andSoftware Engineering ISBN978-1-4471-7324-3 ISBN978-1-4471-7325-0 (eBook) DOI 10.1007/978-1-4471-7325-0 LibraryofCongressControlNumber:2017934637 ©Springer-VerlagLondonLtd.2017 ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagLondonLtd. Theregisteredcompanyaddressis: 236Gray’sInnRoad,LondonWC1X8HB,UnitedKingdom Preface Ifyoudoacasualsearchforbooksthatcontainthewordattitudeintheirtitle,you’ll find yourself drowning in a sea of over 500 volumes. Of course, most of those books relate more to personal self-improvement than to spacecraft dynamics, but evenwhenthesubjectislimitedtotheareasofaerospaceandastrodynamicsyou’ll still find a fair number of items from which to choose. So, what distinguishes this text from those many other candidates? This book attempts to de-emphasize the formal mathematical description of spacecraft onboard attitude and orbit applica- tions in favor of a more qualitative, concept-oriented presentation of these topics (whetherornotweultimatelyachievedthatgoalissomethingwe’llhavetoleaveto you, dear reader, to decide). As such, it would most likely be described by an attitude control analyst as a (hopefully) amusing light read rather than an essential reference bible. And it certainly would not be the first text an Attitude Control Subsystem (ACS) flight software (FSW) designer would grab if he needed a specification of a Kalman filter algorithm. ACS Without an Attitude is instead intended for programmers and testers new to the field who are seeking a com- monsenseunderstandingofthesubjectmatterthey’recodingandtestinginthehope that they’ll reduce their risk of introducing or missing the key software bug that causes an abrupt termination in their spacecraft’s mission and their careers. ACS Without an Attitude is organized in four major sections. Section One (Chaps. 1–3) contains the attitude, orbit, and dynamics background material required to understand the downstream spacecraft applications. Section Two (Chaps.4and5)isasurveyofthespacecraftsensorsandactuatorsusedtomeasure andcontrolthespacecraftattitudeandorbit.SectionThree(Chaps.6–11)examines howsensordataiscombinedwithreferencedatatomeasureattitudeandorbit,and how desired or commanded attitude parameters are compared with measured atti- tudeparameterstodeterminewhatshouldbedonetomaintainthecurrentpointing, ormodifyittosatisfyfutureneeds.Finally,SectionFour(Chap.12)isasurveyof mission characteristics and how attitude and orbit geometries are selected to accomplish mission objectives. The information presented in these sections was originally collected to support aninformalsetoflecturesin1999and2000instigatedbymyBranchChief,Elaine Shell (Flight Software Branch, NASA Goddard Space Flight Center), who also realized that bulletcharts arean ineffective means to documentinformation, hence thisbook.ThefollowingisalistoftextbooksanddocumentsIdrewon(hopefully nottooblatantly)whilepreparingformytalks,aswellasadditionalreferencesused when writing this book: 1. Spacecraft Attitude Determination and Control, edited by James R. Wertz, Kluwer Academic Publishers (1978). 2. Space Mission Analysis and Design, edited by Wiley J. Larson and James R. Wertz, Microcosm, Inc. and Kluwer Academic Publishers (1992). 3. ReducingSpaceMissionCost,editedbyJamesR.WertzandWileyJ.Larson, Microcosm Press and Kluwer Academic Publishers (1996). 4. SatelliteTechnologyandItsApplications,byP.R.K.Chetty,TABProfessional and Reference Books (1991). 5. An Introduction to the Mathematics and Methods of Astrodynamics, by Richard H. Battin, AIAA Education Series (1987). 6. Modern Inertial Technology Navigation, Guidance, and Control, by Anthony Lawrence, Springer (1998). 7. Modern Control Systems, by Richard C. Dorf and Robert H. Bishop, Addison-Wesley Publishing Company (1995). 8. FeedbackControlofDynamicSystems,byGeneF.Franklin,J.DavidPowell, and Abbas Emami-Naeini, Addison-Wesley Publishing Company (1991). 9. Modern Control Engineering, by Katsuhiko Ogato, Prentice-Hall (1997). 10. Goddard Trajectory Determination System (GTDS) Mathematical Theory, Revision 1, edited by A.C. Long, J.O. Cappellari, Jr., C.E. Velez, and A.J. Fuchs, NASA/GSFC Flight Dynamics Division Code 550 (1989). 11. Fundamentals of Astrodynamics, by Roger R. Bate, Donald D. Mueller, and Jerry E. White, Dover Publications, Inc. (1971). 12. Classical Mechanics, by Herbert Goldstein, Addison-Wesley Publishing Company (1950). 13. Fundamentals of Spacecraft Attitude Determination and Control, by LandisF.MarkleyandJohnL.Crassidis,SpringerScienceandBusinessMedia (2014). (An excellent book for both subject history and in-depth mathematical analysis.) Inaddition, herearesome references targetedtospecifictopicsdiscussed inthe later chapters: 1. Hermite Polynomials, Legendre polynomials, and spherical harmonics: MathematicalMethodsforPhysicists(seventhedition),byG.Arfkin,H.Weber and F. Harris, Academic Press, Inc. (2013), Sect. 18.1. 2. Runge-Kutta integrator: Numerical Methods for Scientists and Engineers by R.W. Hamming, McGraw-Hill (1962). Oneofthestrengthsofmyoriginalsetoflectureswastheparticipationofseveral of my NASA/GSFC Guidance, Navigation, and Control (GN&C) colleagues who supplemented(andoften,andgraciously,corrected)mypresentationswithmaterial drawnfromtheirextensiveexperiencehereatGSFC.Andwhenmypitchesstarted todragabit,theyalsohelpedliventhingsupbyrecountingsomeoftheirmanywar stories accumulated during their years of applying clean-cut mathematical algo- rithms to uncooperative real-life spacecraft. My crew of semi-regular experts included 1. Gary Welter 2. Landis Markley 3. Dave Quinn 4. Dave McGlew 5. Bruce Bromberg Astheyearshaverolledbysincethefirstversionofthemanuscript,thematerial in the book has been updated many times, stimulated by new missions and new ACStechnologies,aswellasnewteachingexperiencesgainedrepeatingthecourse. Finally,asIreachedthepointIcouldnolongerbeartoeditthematerialagain,Gary Welter,DaveSimpson,andChrisRouffhaveriddentotherescuetoco-authorwith me this final version. Lou Hallock—2010 Well … life goes on, including delays from other obligations. Lou passed the torchtous,histhreeamigos,whenheretiredin2011,alongwithencouragementto put our own stamp on the book (sometimes with a “you broke it, you bought it” response to editorial suggestions). We’ve also corralled a couple of colleagues (Scott Starin and Tim McGee) to provide some review and feedback on the near-finaltext.(Ourthankstoyouboth.)So,hereisthemulti-chefresult,wethink well-flavored—though some of you may find certain sections more “in your face” than “without an attitude”. Time to let it go. Enjoy. Gary Welter, Dave Simpson and Chris Rouff—2016 Contents 1 Attitude Conventions and Definitions . .... .... .... .... ..... .. 1 1.1 Definition of the Inertial Reference Frame.. .... .... ..... .. 2 1.2 Defining Attitude via Euler Angles (Right Ascension, Declination, and Roll) ..... .... .... .... .... .... ..... .. 5 1.3 Defining Attitude via Euler Angles (Roll, Pitch, and Yaw) .. .. 8 1.4 Defining Attitude via the Direction Cosine Matrix.... ..... .. 10 1.5 Defining Attitude via the Eigenvector and Rotation Angle... .. 12 1.6 Defining Attitude via Quarternions.... .... .... .... ..... .. 13 1.7 Attitude Format Applications.... .... .... .... .... ..... .. 18 2 General Orbit Background ..... .... .... .... .... .... ..... .. 21 2.1 Historical Perspective. ..... .... .... .... .... .... ..... .. 21 2.2 Orbital Shapes .. .... ..... .... .... .... .... .... ..... .. 24 2.3 Specifying the Orbit’s Orientation in Inertial Space... ..... .. 27 2.4 The Location of the Spacecraft in the Orbit. .... .... ..... .. 29 2.5 Keplerian Element Types... .... .... .... .... .... ..... .. 30 2.6 Orbit Perturbations - Oblate Earth .... .... .... .... ..... .. 31 2.7 Orbit Perturbations - Aerodynamic Drag ... .... .... ..... .. 35 2.8 Orbit Perturbations - Solar Radiation Pressure... .... ..... .. 36 2.9 Orbit Perturbations - Orbit Maneuvers with Thrusters . ..... .. 37 3 Angular Momentum and Torque. .... .... .... .... .... ..... .. 39 3.1 Historical Digression . ..... .... .... .... .... .... ..... .. 39 3.2 Translational Motion . ..... .... .... .... .... .... ..... .. 39 3.3 Rotational Motion ... ..... .... .... .... .... .... ..... .. 41 3.4 Motion of the Center of Mass Versus Motion About the Center of Mass... ..... .... .... .... .... .... ..... .. 46 3.5 How the Moment of Inertia Tensor Describes the Object’s Nature .... .... .... ..... .... .... .... .... .... ..... .. 47 3.6 Types of Torque-Free Rotational Motion... .... .... ..... .. 54 3.7 How Torques Can Influence an Object’s Rotational Motion . .. 57 3.8 Attitude Control Torques... .... .... .... .... .... ..... .. 61 3.9 Environmental Torques .... .... .... .... .... .... ..... .. 63 4 Attitude Measurement Sensors .. .... .... .... .... .... ..... .. 67 4.1 Sun Sensors.... .... ..... .... .... .... .... .... ..... .. 70 4.2 Earth Sensors... .... ..... .... .... .... .... .... ..... .. 75 4.3 Magnetometers.. .... ..... .... .... .... .... .... ..... .. 79 4.4 Star Sensors.... .... ..... .... .... .... .... .... ..... .. 82 4.5 Gyros. .... .... .... ..... .... .... .... .... .... ..... .. 86 5 Attitude Actuators ... .... ..... .... .... .... .... .... ..... .. 95 5.1 Reaction Wheels .... ..... .... .... .... .... .... ..... .. 97 5.2 Magnetic Torquer Bars (MTBs).. .... .... .... .... ..... .. 101 5.3 Thrusters .. .... .... ..... .... .... .... .... .... ..... .. 104 6 Reference Models .... .... ..... .... .... .... .... .... ..... .. 109 6.1 Modeling the Earth’s Gravitational Field... .... .... ..... .. 111 6.2 Modeling the Spacecraft’s Ephemeris . .... .... .... ..... .. 112 6.3 Modeling Solar, Lunar, and Planetary Ephemerides... ..... .. 113 6.4 Modeling the Geomagnetic Field. .... .... .... .... ..... .. 114 6.5 Star Catalogs ... .... ..... .... .... .... .... .... ..... .. 117 6.6 Velocity Aberration .. ..... .... .... .... .... .... ..... .. 118 6.7 Parallax ... .... .... ..... .... .... .... .... .... ..... .. 121 6.8 Stellar Magnitude.... ..... .... .... .... .... .... ..... .. 122 6.9 Star Catalog Examples..... .... .... .... .... .... ..... .. 123 7 Onboard Attitude Determination. .... .... .... .... .... ..... .. 125 7.1 Attitude Propagation with Gyroscope Data . .... .... ..... .. 126 7.2 Reference Attitude... ..... .... .... .... .... .... ..... .. 128 7.3 Minimum Data Attitude Determination .... .... .... ..... .. 130 7.4 Batch Attitude Determination with Vector Observations .... .. 132 7.5 Attitude Uncertainty: The Covariance Matrix.... .... ..... .. 138 7.6 Combining Multiple Attitude Solutions .... .... .... ..... .. 140 7.7 Combining an Attitude Solution with a Vector Measurement ... .... ..... .... .... .... .... .... ..... .. 143 7.8 Measurement Propagation and De-Weighting.... .... ..... .. 144 7.9 Recursive Attitude Estimation ... .... .... .... .... ..... .. 147 7.10 Recursive Attitude Plus Gyro Bias Estimation... .... ..... .. 148 7.11 The Kalman Filter for Recursive Least Squares.. .... ..... .. 152 7.12 Synopsis... .... .... ..... .... .... .... .... .... ..... .. 154 7.13 Mathematics to English Translation of Kalman Filtering.... .. 155 8 Spacecraft State Estimation More Broadly. .... .... .... ..... .. 159 8.1 Attitude-Related Least Squares Problems... .... .... ..... .. 160 8.1.1 Star Tracker Relative Alignments.. .... .... ..... .. 160 8.1.2 Star Tracker Internal Calibrations.. .... .... ..... .. 161 8.1.3 Gyroscope Calibration .. .... .... .... .... ..... .. 161 8.1.4 Sun Sensor Calibration.. .... .... .... .... ..... .. 162 8.1.5 Magnetometer Calibration.... .... .... .... ..... .. 162 8.1.6 Wavefront Calibration... .... .... .... .... ..... .. 163 8.2 General Issues .. .... ..... .... .... .... .... .... ..... .. 163 8.2.1 Observability. ..... .... .... .... .... .... ..... .. 164 8.2.2 State Vector Selection... .... .... .... .... ..... .. 165 8.2.3 Observation Model. .... .... .... .... .... ..... .. 166 8.2.4 Least Squares Filters.... .... .... .... .... ..... .. 167 9 Onboard Orbit Computations ... .... .... .... .... .... ..... .. 169 9.1 CGRO Onboard Orbit Models... .... .... .... .... ..... .. 171 9.2 HST Onboard Orbit Models. .... .... .... .... .... ..... .. 172 9.3 Landsat Orbit Model . ..... .... .... .... .... .... ..... .. 173 9.4 RXTE Orbit Models.. ..... .... .... .... .... .... ..... .. 173 9.5 WMAP Orbit Models. ..... .... .... .... .... .... ..... .. 174 9.6 Onboard Orbit Measurement with GPS .... .... .... ..... .. 175 9.7 Onboard Orbit Measurement with TONS... .... .... ..... .. 176 10 Control Laws: General Qualities. .... .... .... .... .... ..... .. 177 10.1 Definition of Control Law Terms. .... .... .... .... ..... .. 179 10.2 Closed-Loop Control Laws . .... .... .... .... .... ..... .. 180 10.3 Laplace Transforms and Transfer Functions. .... .... ..... .. 182 10.4 Control System Response and Behavior.... .... .... ..... .. 187 10.5 The Harmonic Oscillator in Detail.... .... .... .... ..... .. 190 10.6 Adjusting Gains: The Root Locus Diagram. .... .... ..... .. 193 10.7 Discrete Systems and the Z Transform. .... .... .... ..... .. 195 11 Control Laws: Attitude Applications.. .... .... .... .... ..... .. 197 11.1 Equivalence of L-R-C Circuits and Harmonic Oscillators.... .. 198 11.2 PID Control Laws ... ..... .... .... .... .... .... ..... .. 199 11.2.1 Bang-Bang Control. .... .... .... .... .... ..... .. 202 11.2.2 Proportional Control.... .... .... .... .... ..... .. 202 11.2.3 Proportional-Derivative Control ... .... .... ..... .. 203 11.2.4 Proportional-Integral-Derivative Control. .... ..... .. 203 11.3 Hubble Space Telescope Pointing Control Law.. .... ..... .. 204 11.4 Control Law Tuning.. ..... .... .... .... .... .... ..... .. 207 12 Mission Characteristics ... ..... .... .... .... .... .... ..... .. 213 12.1 Mission Orbit Selection.... .... .... .... .... .... ..... .. 213 12.2 Celestial-Pointers Versus Earth-Pointers.... .... .... ..... .. 217 12.3 Safemodes . .... .... ..... .... .... .... .... .... ..... .. 219 12.4 Mission Attitude Acquisition.... .... .... .... .... ..... .. 222 12.5 Spacecraft Comparisons.... .... .... .... .... .... ..... .. 223 Appendix A: Time Measurement Systems . .... .... .... .... ..... .. 227 Appendix B: Variation on Deriving the Kalman Gain ... .... ..... .. 231 Glossary ... .... .... .... .... ..... .... .... .... .... .... ..... .. 235 References.. .... .... .... .... ..... .... .... .... .... .... ..... .. 265 Index.. .... .... .... .... .... ..... .... .... .... .... .... ..... .. 267 Acronyms ACE Attitude Control Electronics ACS Attitude Control Subsystem AOS Acquisition of Signal AST Advanced Star Tracker AU Astronomical Unit CCD Charge-Coupled Device CDR Critical Design Review CGRO Compton Gamma Ray Observatory CPU Central Processing Unit CSA Celestial Sensor Assembly CSS Coarse Sun Sensor CXO Chandra X-Ray Observatory DCM Direction Cosin Matrix DSCOVR Deep Space Climate Observatory DoD Department of Defense DOY Day of Year DSS Digital Sun Sensor ECI Earth-centered Inertial emf Electromotive force EOS Earth Observing System ESA European Space Agency ESA Earth Scanner Assembly FDF Flight Dynamics Facility FEEP Field Emission Electric Propulsion FES Fine Error Sensor FGS Fine Guidance Sensor FHST Fixed-Head Star Tracker FOG Fiber Optic Gyro FOV Field of View FSS Fine Sun Sensor