/~~~~~~~ ~'. ,;/_ ., i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f', k· ' S 7. 0 ~ ~~~~~ ·,. '''':'--" ""-J' x ' ii 5 T/~~~~~~~~~' q z ?p~ ~ ~ ~~~~'-'. -' ' 'L ,'' "' I ci~~~'""7: .... ''''''~ 1(oI~~~~~~~~~~~~~~~~~~~~nci"~~~~~~~~~~~~~' ;R --,'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j. DTERSMINAT ION,SUBSYSTEM I i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, ;' '~/ ',\* i ' /,i~~·EM'AI-At1' - ' ' ' ' . , .. SPECQIFICA~TIONS'` .i ~~~~~~~-- r ~. i .t +-% ~.- V~~~~~~~r~_' " , ' ''~v N .·~~~~~,·' : z' '% - '::' "~'' ' ·~''-,z '""·' -... '"' , ......?.':; ,,<" .~:iI ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-.. L /' ' ) .. d~' 1.- il · ) · 'i. i L i i; j~~~~~~~~~~l^ ' .1) ~~~~~~~~~~~~Y~~~~~~~~~~ .i · ~ .i -~.,,-~.~~ ' / (NASA-TM-X-65984 ) GODDARD TRAJECTORY N 72- 30O816 , .,-Q: ... DETERMlINATION SUBSY STEM: M9ATHEMATICAL k_ % SP E CIFTC AT IONS W.E. Wagner, et al (NASA) -_/3.· ~ar. 1972 333 CSCL 22C Uca i-- ~·--,::--..:- ~~~~~~~~~ ~~~~~~~ : .1... 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'~ ' "- · ,'-, ~- ' ,;""~",._ '/ ~'\'-'.· -~~~~: - , '.,/'",_5L".,-,../REPRODUCED BY ? ,'7, .'. ;,---,'_., c '.~ U.S. DEPARTMENT OF COMMERCE ~ .?.~~,, ~ ' ;~ ~.- ~~"'_7·,... NATIONAL TECHNICAL / :7 'u*;,7 :t - 9 -- .'..:'.~~ .. ,'... ). INFORMATION SERVICE [. ' : ' " ~ _ .'.."-J' .'-'' . _ /:'/:: SPRINGFIELD, VA 22161 ) ,,.; ~':,r./,. ,.-,> ,/ >-t ,'~' '- ' .,'' ~ " . '~ ~ \ .' , . h -. / ' ;: .- ~ , .~~~~~~~~~~.~ ' ... - 'I -.z - - ' L . , , . ) . ~ - ' rj ' 1.- ~ 'w. GODDARD TRAJECTORY DETERMINATION SUBSYSTEM MATHEMATICAL SPECIFICATIONS Edited by W. E. Wagner Computer Sciences Corporation and C. E. Velez Goddard Space Flight Center March 1972 GODDARD SPACE FLIGHT CENTER Greenbelt, Maryland C ACKNOWLEDGEMENT The editors wish to acknowledge the contribution made by many individuals at Goddard Space Flight Center and at Computer Sciences Corporation who participated in the technical preparation of manuscripts to these mathematical specifications. These individuals are: P. D. Argentiero, GSFC R. W. Barbieri, GSFC N. R. Burton, CSC L. H. Carpenter, GSFC J. L. Cooley, GSFC V. T. Laczo, GSFC E. J. Lefferts, GSFC D. H. Novak, CSC C. E. Velez, GSFC W. E. Wagner, CSC D. S. Woolston, GSFC A special acknowledgement is given to J. W. Siry, GSFC E. R. Watkins, Jr. GSFC C. E. Doll, GSFC for helpful technical discussions and consultation. ABSTRACT This document presents the mathematical specifications of the Goddard Trajectory Determination Subsystem (GTDS) of the Flight Dynamics System (FDS). These specifications include the mathe- matical description of the coordinate systems, dynamic and measure- ment model, numerical integration techniques, and statistical estimation concepts. iii TABLE OF CONENTS Page Chapter 1 - Introduction .................. 1-1 1.1 Operating Modes ......... eeeleeele····leee·eeeeleel 1-1 1.2 Near Real Time and Postflight Operation .................... 1-2 1.3 Data Preprocessing ....... e· e e eeeeeeeeeeeeeee. **ee 1-2 1.4 Tracking Systems ....... 1-4 1.5 Spacecraft Dynamics ..... ...........................Ope ra t i o n . e e e * . e . .e 1-4 * - . e · ·. · · · · f·- · · · ·. · · · · 1.6 Estimation Process ....... * - . - . - i, * * . * . * * * . 1-5 1.7 References ............. 1-7 Chapter 2 - Tracking Systems and Preprocessing ................ 2-1 2.1 Tracking System Description ........... 2-1 2.1.1 Goddard Range and Range-Rate (GRARR) UHF System ...... 2-1 2.1.2 C-Band Pulse Radar ............................ 2-3 2.1.3 Minitrack Interferometer ......................... 2-4 2.2 Preprocessing for Option A (Metric) Processing .............. 2-7 2.2.1 GRARR (VHF) System ......... * .e.e.e..eee . .. ee. 2-8 2.2.2 C-Band Radar .............. ..eeeeee. eee. ·.. 2-13 2.2.3 Minitrack Interferometer . ..... .ee.e.e ..eee .e.el . 2-14 2.3 Preprocessing for Option B (Polynomial) Preprocessing.... .... 2-21 2.3.1 GRARR System ............. *...ee........ e.... 2-2$ 2.3.2 C-Band Radar .............. 2-23 2.3.3 Minitrack ................. ·················· 2-24 2.4 References .................... ............. e.e... 2-29 Chapter 3 - Coordinates and Time Systems ........ ............ 3-i 3.1 General Comments and Definitions .... · .. o. ·o.o. .. o ... o. 3-i 3.2 . Coordinate System Descriptions ...... ooo ... JO. .. 0 : 3-4 3.2.1 Body Centered Inertial ....... * a ea ·.eo .. oo 3.2.2 Body Centered Rotating........ .\.IOO·..I . iv TABLE OF CONTENTS-(continued) Page 3.2.3 Local Plane System ............................. 3-6 3.2.4 Topocentric Local Tangent ...................... 3-7 3.2.5 Orbit Plane .............. .... ............ · .... 3-7 3.2.6 Keplerian Elements ............................. 3-9 3.3 Specific Transformations ............. ... ............ 3-9 3.3.1 1950.0 Inertial To Body-Fixed .......... ............ .. 3-9 3.3.2 Selenocentric True of Epoch to Selenographic ...........·. 3-20 3.3.3 Spherical-Cartesian Transformations ..... ........ ... ....3-26 3.3.4 Geocentric to Orbit Plane ............. ............. 3-31 3.3.5 Earth-Fixed Geodetic Transformations .... ......... 3-32 3.3.6 Earth-Fixed to Topocentric Local Tangent .. ........... 3-39 3.3.7 Keplerian-Cartesian Transformations ..... ... ......... .. 3-41 3.4 Time Systems ..................................... 3-51 3.4.1 Ephemeris Time, ET .......... ........ ·........ 3-51 3.4.2 Atomic Time, A.1 . ............ .· . e . e e ee . 3-51 3.4.3 Universal Time, UT ........... ·.... ... 3-51 3.4.4 Uncorrected Universal Time, UTO . .... e... l. ... 3-53 3.4.5 Universal Time, UT1 ........... e e .... ... .. 3-53 3.4.6 Universal Time, UT2 ........... .· . 3-54 3.4.7 Universal Time Coordinated, UTC .. .· . 3-55 3.4.8 Station Time, ST ...... ....... .. 0 . .. .. 3-55 3.5 Transformations Between Time Systems ................... 3-55 3.5.1 Transformations by Standard Formula.. mo.m o ll... 3-55 3.5.2 Transformations by Time Polynomials . *·. el. e e e 3-57 3.6 Trajectory Related Events and Equations .... ............... 3-59 3.6.1 Geomagnetic Latitude and Longitude .... 3-59 3.6.2 Apofocus and Perifocus Altitudes ..... 3-59 3.6.3 Rotation Rates of Node and Perifocus .. .. 3-59 oleomo 3.6.4 Keplerian, Anomalistic, and Nodal Periods . * . 3-60 oleoeel 3.6.5 Apsis Points ......... ...., . .. 3-60 3.6.6 Altitude Extremes ................ .0 3-60 3.6.7 Encounters ..................... eel o l 3-61 v TABLE OF CONTENTS-(continued) Page 3.7 References .................. ..................... 3-62 Chapter 4 - Acceleration Model .................. .. ...... 4-1 4.1 Contribution Due to N-Point Masses ...................... 4-1 4.2 Nonsphericity ...................... ........... 4-4 4.3 Drag and Atmospheric Models .......................... 4-10 4.3.2 Jacchia 1960 ................................... 4-13 4.3.2 Paetzold 1962 ................................. 4-14 4.3.3 Jacchia-Nicolet 1963-1964 ....... .................. 4-16 4.3.4 Harris-Priester 1962 ............................ 4-18 4.3.5 Modified Harris-Priester ........................ 4-19 4.4 Solar Radiation Pressure .............................. 4-23 4.5 Attitude Control .................................. 4-25 4.6 Thrust .................. .................. 4-28 4.7 References ....................................... 4-32 Chapter 5 - Variational Equations........................... 5-1 5.1 Contribution Due to N-Point Masses ............. ...... 5-4 5.2 Nonsphericity ................... ........... ..... 5-5 5.3 Drag ............................... ........... 5-9 5.4 Solar Radiation Pressure ............................. 5-13 5.5 Attitude Control ........................ ........... 5-15 5.6 Thrust .................................... .... 5-17 Chapter 6 - Numerical Integration of the Equations of Motion and Variational Equations ........................ 6-1 6.1 Adams-Cowell Ordinate Second Sum Formulas ............... 6-1 6.2 Predict-Pseudo Correct Algorithm for Equations of Motion....... 6-5 6.3 Corrector-Only Cowell Integration for Linear Systems.......... 6-7 6.4 Corrector-Only Algorithm for Variational Equations ........... 6-9 6.5 Mapping of Position Partials ........................... 6-13 6.6 Starting Procedure ................... ............... 6-14 6.7 Interpolation ...... ................................ 6-16 6.8 Local Error Control ................................. 6-16 vi TABLE OF CONTENTS-(continued) Page 6.9 Time Regularization .. ................... 6-18 6.10 References ...................................... 6-21 Chapter 7 - Observation Models ............................ 7-1 7.1 General Description ................................ 7-2 7.2 Ideal Observations ................................. 7-4 7.2.1 Transformation of Coordinates-Geocentric Inertial to Topo- centric