Guidance Equations
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o Jo o z E (J t-u¡ l! LLO o l¡t l- GUIDANCE, NAVIGATION f AND CONTR t l-= vt Approved: .7/ "tu, /,f_t> G. M. LEVINE, DIREÇTO CE ANALYSIS APOLLO GUIDANCE AND NAVIGATION PROGRAM vl= - F Approvedr oate: ts Ü:l l- t2Z/ l¡t S.L PPS, SKYLARK MANAGER rrt APOLLO GUIDANCE AND NAVIGATION PROGRAM Ð T Approved Date: ,/1-orl7t (J R. H. BATT DEVELOPMENT APOLLO CE ATION PROGRAM rt) vt Approved: oate, / iZ)ci7 I D. G. HOAG, f APOLLO GUTDAN N TION PROGRAM Approved: ø,// Q, ,A-,-**- Date í ô.( R. R. RAGAN, #epurv DIREC{oR INSTRUMENTATION LABORATORY R- 693 GUIDANCE SYSTEM OPERATIONS PLAN FOR MANNED CSM EARTH ORB ITAL MISS IONS USING PROGRAM SKYLARK 1 SECTION 5 GUIDANCE EQUATIONS OCTOBER 19?1 Itl II. cHARrEs STAR,K DR.APER CAMBRIÓGE, MASSACHUSETTS, O2139 LABORATORY INOEXI¡{Ê DATA SIGTIATOR r0c T PCH sus,lfgf'i -ll ¡ r 4., DATE OPR L,, t&-w ,0-3f "11 ¡{ ¡f Ê-åE3 -æLW * Mtr { '-)¡:, ;tÍi't, f*ti')t'/.15'¿¡ ACKNOWLEDGEMENT This |eport was p|epared under DSR Project 55-23890, sponsored by the ìlanned Spacecraft Center of the National Aeronautics and Space Administration thlough Contlact NAS 9-4065. 'ì, 11 R- 693 GUIDANCE SYSTEM OPERATIONS PLAN FOR MANNED CM EARTH ORBITAL MISSIONS USING PROGRAM SKYLARK 1 SECTION 5 GUIDANCE EQUATIONS ) Signatures appearing on this page designate approval of this document by NASA/MSC. Approved Date: '2/ John R. Ga Section Chief, Guidance Program Section Manned Spacecraft Center, NASA Approved: f Date /a/ z ohn E. Williams, J Chief, Simulation and Flight Software Branch Manned Spacecraft Center, N ApproverJ: Date ,{r es C, Stokes, Jr. ief , Flight Support Divi d Spacecraft Center, NASA lll BLANK ) 1V FOREWORD The Guidance System Operations Plan (GSOP) for Program Skylark 1 is published in five sections as separate volumes. 2, Data Links 3. Digital Autopilots 4. Operational Modes 5. Guidance Equations 7. Erasable Programs Since the information in Section 1 of the Colossus 2E GSOP is also appli- cable to the Skylark Program, Section 1 will not be republished for Skytark. The reader is referred, therefore, to R577 Colossus 2E GSOP, Section 1, Revision 2, January 1970. Also, Section 6 will not be published for Skylark. The changes made to this Skylark GSOP Section 5 are enough to warrant its consideration as a ne\¡/ document. For that reason, it is not being treated as a revi- sio¡¡ of Colossus 3 GSOP 5. Major deletions, additions, and editorial changes have been made to reflect the scope of the new program. paragraphs b.1.3, 5.1.5.8, b.1.b.4, 5.2.4, 5.5.L2' 5.6.11, 5.6.16, and 5.6.17 have been deleted and so indicated in the Table of Contents ancl throughout the document. Also deletec.l and so indicated in text are a number of equations that do not pertain to the Skylark Program. Subsectj.ons 5.4 and 5.8 andparagraphs 5.1.1, 5.t.2, 5,6.1 through 5.6.7.S, b.6.14, 5.6.9, and b.9.1 have heen rewritten. Appendix A, containing a list of pcRs and pcNs, whose implementation is reflected in this issue, has been added. The rest of the docu- ment has undergone extensive editorial changes in the hope of making it more use- ful to the reader. At the same time, the basic form of the Colossus GSOp 5 has been retained to facilitate comparison with earlier GSOps. The volume is published as a control document governing the basic design for the guidance and navigation computations of Skylark 1 and supersedes M. I. T. - DL Section-5 memos pubtished earlier. Revisions constituting changes to the Skylark Program require NASA approval. Bi-ANK VI CONTENTS Section Page 5. 1-1 5. 1 INTRODUCTION 5. 1.1 General Comments 5. 1-1 5. t.2 Sections 4 and 5 Cross Reference 5. t-2 5.1.3 (Deleted from Skylark) 5,1.4 Coordinate Systems 5. 1-6 5. 1.4. 1 Basic Reference Coordinate System 5.1-6 5. 1.4. 2 IMU Stabte Member or Platform Coordinate System 5. L-7 5. 1.4.3 Vehicle or Body Coordinate SYstem 5. 1-9 5. L,4.4 The Earth-fixed Coordinate System 5.1-9 5. 1.4. 5 Apollo Telescope Mount Sensor Coordinate System 5. 1-9 5.L.4.6 Navigation Base Côordinate System 5.1-10 5. 1. 5 General Definitions and Conventions 5. 1- 10 5. 1.5. 1 Error Transition Matrix Maintenance ' 5. 1- 10 5.1.5.2 Altítude Parameters Convention 5. L-12 5.1.5.3 (Deleted from SkYlark) 5. 1.5.4 (Deleted from Skylark) 5. 1.5.5 Time Definitions . 5.1-13 5.2 COASTING FLIGHT NAVIGATION 5.2 -l 5.2. I General Comments 5,2-L 5.2.2 Coasting Integration Routine 5. 2-5 5.2,2.1 General Comments . 5.2-5 5,2.2.2 Encke's Method 5.2-b 5,2.2.3 Disturbing Acceleration 5.2-8 s.2.2, 4 Error-Transition Matrix 5,2-9 5.2.2.5 Numerical Integration Method 5.2-12 5,2.2.6 Coasting Integration Logi.c . 5.2-13 vl1 CONTENTS (cont'd) Section Page 5.2.3 Measurement Incorporation Routine 5.2-tB 5.2.4 (Deleted from Skylark) 5.2.5 Universal Tracking and Rendezvous Navigation Program (P20) 5.2-25 5.2.5. L Tracking 5.2-25 5,2.5.2 Rendezvous Tracking Data Processing Routine (R22) . 5.2-35 5. 2.5.3 Automatic W-matrix Reinitiahzation (WRi) 5.2-51 5.3 POWERED F'LIG}IT NAVIGATION AND GUIDANCE 5.3-1 5.3. 1 General Comments 5.3-1 5.3.2 Powered Flight Navigation - Average-G Routine Ð. J.I 5.3. 3 Powered Flight Guidance Using Cross Product Steering . 5.3-5 5. 3. 3. 1 Introduction 5. 3-5 5.3. 3. 2 Powered Fli.ght Guidance Computation Sequencing ..... 5.3-6 ) 5. 3. 3. 3 Prethrust Computations 5. 3- 14 5. 3. 3. 4 Cross Product Steering R,outine 5.3-32 5. 3. 3. 5 Velocity-to-be-Gained Routine 5.3-37 5.3" 3. 6 SPS Thrust Fail R,outine (R40) 5. 3-40 5. 3. 4 Thrust Monitor Programs . 5.3-41 5. 3. 5 Earth Orbit Insertion Monitor Program-P11 5.3-41 5. 3.5. 1 Introduction 5.3-41 5.3.5.2 Nomenclature for P11 5,3-44 5.3.5.3 Time Sequencing of P11 Subroutines 5.3-45 5. 3. 5. 4 Time Subroutine 5.3-46 5.3. 5.5 State Subroutine 5.3-46 5. 3. 5. 5a Attitude Error Subrt-rulirru 5.3-46 5.3.5.6 Display Subroutine . 5.3-46 5,4 TARGETING 5.4-t 5.4. 1 Rendezvous Targeting Constraints 5.4-2 5. 4.2 NC1 Targeting Program (P31) 5.4-3 5.4. 3 NC2 Targeting Program (P32) 5.4-15 5.4.4 NCC Targeting Program (P33) 5. 4- 16 vLll CONTENTS (cont'd) Section Page 5,4,5 NSR Targeting Program (P34) 5.4-17 5.4. 6 TPI Targeting Program (P35) 5.4-20 5. 4,7 TPM Targeting Program (P36) 5.4-28 5.4, B NPC Targeting Program (P3B) 5.4-31 5.4.9 Targeting Program Routine s 5.4-34 5. 4. 9. t REVUP Routine 5.4-34 5.4.9,2 RADUP Routine 5.4-34 5.4. 9. 3 COE Routine 5.4-34 5. 4,9.4 ITER Routine . 5.4-34 5,4. 9.5 QnOf PI Routine 5.4-38 6.5 BASIC SUBROUTINES . 5.5-1 5.5. I General Comments 5.5-1 5.5.1.1 Solar System Subroutines 5.5-1 5. 5. l. 2 Conic Trajectory Subroutines 5.5-1 5. 5. 1. 3 Additional Subroutines . 5.5-B 5.5,2 Planetary Inertial Orientation Subroutine 5.5-8 5.5. 3 Latitude - Longitu de Subroutine 5.5-9 5.5.4 Solar Ephemerides 5,5-13 5.5.5 Kep1er Subroutine 5.5-13 5.5.6 Lambert Subroutine 5.5-17 5.5.7 Time-Theta Subroutine . 5.5-20 5.5. I Time-Radius Subroutine . 5,5-22 5.5. I Apsides Subroutine 5.5-22 5.5. 10 Mis cellaneous Subroutine s 5.5-26 5. 5. 11 Initial Velocity Subroutine 5.5-33 6. 6. 11. 1 Nomcnclature for the Initial Velocity Subroutine 5.5-34 5. 5. 12 (Deleted from Skylark) 5.5. 13 LOCSAM Subroutine 5.5-37 5. 5. 14 Pericenter-Apocenter (Periapo) Subroutine 5.5-38 5.5. 15 Time-of-Longitude Program (P29) 5.5-38 1X CONTENTS (cont'd) Section Page 5.6 GENERAL SERVICE ROUTINES 5.6-1 5.6-1 5 6. 1 General Comments 5 6.2 IMU Alignment Modes . 5.6-1 5. 6.2.1 IMU Orientation Determination Program (P51) 5.6-1 5.6.2.2 IMU R.ealignment Program (P52) 5.6-4 5. 6.2,3 Backup IMU Orienta tion Determination Prograrn (P53) 5.6-9 5.6.2.4 Backup IMU Realignment Program (P54) 5.6-10 5.6.2.5 ATM Orientation Determination Program . 5.6-11 5.6. 2. 6 ATM Star Tracker Gimbal Angle Program 5.6-15 5.6.3 IMIJ R,outines 5.6-19 5.