ROBOT COMPUTER PROGRAM · 2020. 3. 23. · The detailed ascent trajectory profile presented is for...
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2 I NASA TECHNICAL MEMORANDUM Report No. 53873 OPTIMAL TRANSFER TRAJECTORIES FROM EARTH PARKING ORBIT TO VARIOUS TERMINAL CONIC CONSTRAINTS AND MODIFICATIONS TO THE ROBOT COMPUTER PROGRAM I I By Rodney Bradford Program Development . NASA George C. Mmh LL Space Flight Center e I Marshall Spme FZi’bt Center, A hbama * https://ntrs.nasa.gov/search.jsp?R=19700005731 2020-03-23T20:55:38+00:00Z
Transcript of ROBOT COMPUTER PROGRAM · 2020. 3. 23. · The detailed ascent trajectory profile presented is for...
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I N A S A TECHNICAL MEMORANDUM
Report No. 53873
OPTIMAL TRANSFER TRAJECTORIES FROM EARTH PARKING ORBIT TO VARIOUS TERMINAL CONIC CONSTRAINTS AND MODIFICATIONS TO THE ROBOT COMPUTER PROGRAM
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I
By Rodney Bradford Program Development .
NASA
George C. M m h LL Space Flight Center e I Marshall Spme FZi’bt Center, A hbama *
https://ntrs.nasa.gov/search.jsp?R=19700005731 2020-03-23T20:55:38+00:00Z
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T E C H N I C A L R E P O R T S T A N D A R D T I T L E P A G E 1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 3. RECIP IENT 'S CATALOG NO.
T'M X-53873 I 1. T I T L E AND SUBTITLE Optimal Transfer Trajectories From Earth Parking Orbit To Various Terminal Conic Constraints and Modifications To The
5. REPORT DATE
October i. 1969 6. PERFORMlNG ORGANIZATION CODE
ROBOT Computer Program I P D - D O - P 7. AUTHOn(a) I PERFORMING ORGANIZATION REPOR r # Rodney Bradford
P e r f o r m a n c e and Fl ight Mechanics Division
P r o g r a m Development
9. PERFORMING ORGANIZATION NAME AND ADDRESS
P r e l i m i n a r y Design Office
Marsha l l Space Fl ight Center , Alabama 35812 12. SPONSORING AGENCY NAME AND ADDRESS
10. WORK UNIT NO.
1 1 . CONTRACT OR GRANT NO,
13. TYPE OF REPOR;. & PERIOD COVEREC
Technical Memorandum
14. SPONSORING AGENCY CODE
16. ABSTRACT
Reference optimal ( minimum fuel consumption) transfer trajectories were computed by the minimum Hamiltonian-steepest ascent method for departure from a circular parking orbit about the earth to various terminal conic constraints (V, y, r) , (V, y, r, i) , (C3, C,) , ( C3, Ci, i) , and ( C ~ , O , 6) .
17. KEY WORDS
Terminal conditions were a circular orbit at synchronous altitude, 35 862 km, an elliptica orbit with apogee at synchronous altitude (in-and-out-of-parking-orbit-plane) , and an injection burn into a typical 1973 Mars mission hyperbolic orbit. The vehicles used to perform these transfers were the S-IVB stage and the Command Service Module. Also included is a typical ascent trajectory profile utilizing a two-stage Saturn IB launch vehicle. Detailed orbital and trajectory profile parameters are presented in tabular form.
18. DlSTRlBUTlON STATEMENT
STAR Announcement h
Modifications made to the ROBOT-Apollo and AAP Preliminary Mission Profile Optimization computer program are also presented.
19. SECURITY CLASSIF . (of thlr rapart) 20. SECURlTY CLASSIF. (of thli pas.) 21. NO. OF PAGES 22. PRICE
I 135 Unclassified I Unclassified $3.00
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ACKNOWLEDGMENT
Appreciation is expressed to M r s . Shirley J. Johnson of the Dynamics Research Corporation for her diligent efforts in programming the modifications to the ROBOT-Apollo and AAP Preliminary Mission Profile Optimization Computer Program.
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TABLE OF CONTENTS
P a g e
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUMMARY 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTRODUCTION 1
. . . . . . . . . . . . . . . COMPUTER PROGRAM MODIFICATIONS 4
TYPICAL ASCENT TRAJECTORY PROFILES 4 . . . . . . . . . . . .
O P T I M A L TRANSFER T O SYNCHRONOUS A L T I T U D E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CIRCULAR ORBITS 9
n - r n r x = A T mn A X T P - ~ ~ m n n r T T ~ ~ T P A T n n n ~ m c T I T T ~ T T UT 1 1 i v u - 1 ~ 1 1 n f i i v o 1 : Ln L V LUJAII L I b n u V I \ . U L I U v v I 111
. . . . . . . . . . . . . . . . A P O G E E A T SYNCHRONOUS ALTITUDE 10 O P T I M A L TRANSFER T O 1973 MARS MISSION HYPERBOLIC O R B I T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
13 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX A: O U T P U T FORMAT F O R EACH P R I N T TIME INTERVAL AND OPTIONAL SUMMARY P R I N T T A B L E DESCRIPTION . . . . . . . . . . . . 95
APPENDIX B: EQUATIONS FOR T H E COORDINATE SYSTEM TRANSFORMATION AND COMPUTATION O F ADDITIONAL ORBITAL P A R A M E T E R S . . . . . . 101
A P P E N D I X C: ADDITIONAL INPUT VARIABLES REQUIRED AND I N P U T DATA LISTINGS . . . . . . . . . . . . . 11 1
A P P E N D I X D: UNIVAC 1108 FORTRAN LISTING OF SUBROUTINE A P R T N . . . . . . . . . . . . . . . . . 117
APPENDIX E: UNIVAC 1108 FORTRAN LISTING O F . . . . . . . . . . . . . . . . . . SUBROUTINE TRASH 125
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
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Figure
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Table
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LIST OF ILLUSTRATIONS
Title
Terminal conic injection constraints . . . . . . . . . . . . . . . . . .
Inertial Cartesian plumbline coordinate system . . . . . . . . . . Geocentric ephemeris coordinate system and orbital geometry and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Saturn V ascent and circular target orbit injection
Typical Saturn IB ascent and elliptical target orbit injection profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circular target orbit velocity increment diagram. . . . . . . . . . Hyperbolic injection geometry ......................
L I S T OF TABLES
Title
Saturn IB Ascent Trajectory Profile for a ii5-Degree
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Launch Azimuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Saturn IB Ascent Trajectory for a 115-Degree Launch Azimuth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Optimal Transfer Trajectory Characteristics for the Terminal Conic Constraint (V, y , r) - Synchronous Altitude Circular Orbit in the Parking Orbit Plane. . . . . . . . . 35
Optimal Transfer Trajectory for the Terminal Conic Constraint ( V , y , r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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. LIST OF TABLES (Concluded)
Table
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Title
Optimal Transfer Trajectory Characteristics for the Terminal Conic Constraint ( V , y , r, i) - Synchronous Altitude Circular Orbit Inclined at 50 Deg . . . . . . . . . . . . .
Optimal Transfer Trajectory for the Terminal Conic Constraint ( V , y , r, i) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal Transfer Trajectory Characteristics for the Terminal Conic Constraint (C3, Ci) - 6588 x 42 240 Ltm Elliptical Orbit in the Parking Orbit Plane . . . . . . . . . . . . .
Optimal Transfer Trajectory for the Terminal Conic Constraint (C3, Ci). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal Transfer Trajectory Characteristics for the Terminal Conic Constraint ( Cs, Ci, i) - 6588 x 42 240 km Elliptical Orbit Inclined at 50 Deg . . . . . . . . . . . . . . . . . . .
Optimal Transfer Trajectory for the Terminal Conic Constraint (C3, C,, i) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimal Transfer Trajectory Characteristics for the Terminal Conic Constraint (C3, a, 6) - Typical 1973 Mars Mission Hyperbolic Orbit Injection . . . . . . . . . . . . . .
Optimal Transfer Trajectory for a Typical 1973 Mars Mission Hvnerbolic Orbit Iniection . . . . . . . . . . . . . . . . . .
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TECHNICAL MEMORANDUM X-53873
OPTIMAL TRANSFER TRAJECTORIES FROM EARTH PARKING ORBITTO VARIOUS TERMINAL CONIC
CONSTRAINTS AND MODIFICATIONS TO THE ROBOT COMPUTER PROGRAM
SUMMARY
Reference optimal (minimum fuel consumption) transfer trajectories were computed by the minimum Hamiltonian-steepest ascent method for departure from a circular parking orbit about the earth to various terminal " W L " U W l l V U L U I l l U V , . A_-. - rb,-.nn+nn:n+n (11 LI , *.\ ( I T *, P i\ , , J , , . , , .. , A J , (Cj , Ci) , ! c:, c;, i! , 2nd (CS, N , f i ! .
Terminal conditions were a circular orbit at synchronous altitude, 35 862 h, an elliptical orbit with apogee at synchronous altitude (in-and-out- of-parking-orbit-plane) , and an injection burn into a typical 1973 Mars mission hyperbolic orbit. The vehicles used to perform these transfers were the S-IVB stage and the Command Service Module. Also included is a typical ascent trajectory profile utilizing a two-stage Saturn IB launch vehicle. Detailed orbital and trajectory profile parameters are presented in tabular form.
Modifications made to the ROBOT-Apollo and AAP Preliminary Mission Profile Optimization computer program are also presented.
I NTRODUCT I ON
Reference optimal ( minimum fuel consumption) transfer trajectories were computed for the transfer of an S-IVB stage and a Command Service Module (CSM) from a 185.2-km altitude, 37-deg inclined, circular parking orbit about the earth to various terminal conic constraints. The selection of the terminal constraints was based on the cri teria of obtaining a reference cross-section of optimally performed orbital operations. The optimization method employed was the minimum Hamiltonian-steepest ascent method as described in Reference 1.
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The terminal conic constraints were defined as (V, y , r) , (V , y , r, i) , ( C3, Cl) , ( C3, Cl, i) , and ( C3, a,6) I. A physical interpretation of these constraints is presented in Figure I. As shown in this figure the quantities V, y and r are used to specify a circular orbit, whereas the quantities C3 and C l are used to specify the shape of an elliptical orbit. This distinction was made to facilitate the calculation of the transfer trajectories by the computer program employed. The terminal condition for the constraint (V, y , r) was an in-parking- orbit-plane circular orbit at synchronous altitude, 35 862 km. The terminal condition for the constraint (V, y , r, i) was an out-of-parking-orbit-plane circular orbit at synchronous altitude with inclination equal to 50 deg. The terminal conditions for the constraints ( C3, C,) and ( C3, C,, i) corresponded to an elliptical orbit with perigee radius equal to r (earth radius) +- 210 km
and apogee radius equal to r
sponded to an orbital inclination of 37 deg while the constraint ( C3, Cl, i) corresponded to an inclination of 50 deg.
E +- 35 862 km. The constraint ( C3, C,) corre- E
The terminal constraint ( C3, a,6) corresponded to the terminal conditions on a hyperbolic orbit for a typical 1973 Mars mission. For this constraint the date and time of departure were specified along with the right ascension and declination of the outgoing hyperbolic asymptote as measured in an inertial ephermeris coordinate system. The optimization method and computer program employed, as described in Reference I, were referenced to an inertial plumbline coordinate system. Thus, this terminal constraint necessitated a coordinate transformation to an inertial ephemeris system in the computer program and modification of the input variables. Also, additional orbital parameters were computed, a new output format was designed, and a summary print table option incorporated. All modifications made to the computer program are documented in the appendices.
The detailed ascent trajectory profile presented is for a two-stage Saturn IB launched from Cape Kennedy at an azimuth of 115 deg. The S-IB and S-IVB stages a re fueled to capacity and burned to depletion. The CSM propulsion system is then ignited and burned until injected into a 185.2-km altitude, 37-deg inclined, circular parking orbit. At this point of parking orbit injection, an initial state vector was obtained which was used as a departure point to the various terminal constraints.
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I. Symbols a r e defined in Appendices A and B.
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F
a y = v =I-&? 0' (path angle-measured A. (V, y, r ) - specifies a c i rcu lar
orbi t lying in the parking orb i t plane
f r o m the local horizontal to the velocity Vector)
B. (V, Y,r, i ) -specif ies a c i rcu lar orbi t lyin out of the parking orbi t plane Plane
C.
c 3 =v2 - p/r
c = r ~ s i n ( 9 0 ' - ~ ) ( C 3 , C l ) - spec i f i e s the shape of an ell iptical orbi t lying in the parking orb i t plane
D. K 3 , C 1 , i ) - shape of a n lying out of orbi t plane
specifies the ell iptical orbi t the parking
Orbi ta l Plane
Equator ia l Plane
E. (C3 , c y , 6 ) -specif ies a c3 = v: heliocentric t ra jec tory
JS(outgoing asymptote vec tor )
- V,(hyperbolic excess velocity)
/
F i g u r e 1. T e r m i n a l conic in jec t ion c o n s t r a i n t s .
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COMPUTER PROGRAM MOD IF1 CATIONS
Various modifications were made to the computer program [Ref. 13 which was used to compute the optimal transfer trajectories. These changes which were made in the program to make it more compatible for orbital operations computations, consisted of the design of a new output format and labeling code, coordinate system transformation, input variable alteration, and the computation and output of additional orbital parameters.
Appendix A gives a description and example of the output format for each print t ime interval, a definition of the output symbols, and a description of the optional summary print tables incorporated and the variables which must be specified for them. Appendix B gives the equations necessary for the transformation from the inertial plumbline (Fig. 2) to the inertial ephemeris coordinate system (Fig. 3) and equations defining the various orbital parameters included in the output format which were not previously computed in the program. The orbital geometry and notation are also shown in Figure 3 .
Appendix C contains a description of the additional input variables necessary and listings of the input data required for computation of the transfer trajectories presented. Appendix D contains a UNIVAC 1108 FORTRAN listing of the subroutine titled APRTN which w a s modified to perform the coordinate system transformation and the computation of the various additional orbital parameters. The output of these parameters and quantities computed in the original program, in the new output format, is also accomplished by this subroutine. The subroutine, entitled TRASH, encompasses the optional summary print tables. A UNIVAC 1108 FORTRAN listing of the TRASH subroutine is presented in Appendix E.
TY P I CAL ASCENT TRAJECTORY PROF I LES
A typical three-stage Saturn V ascent trajectory and circular orbit in- jection profile is presented in Figure 4. As shown, trajectory arc 1 corresponds to the S-IC stage burn; arc 2 to the S-I1 stage burn; a rc 3 to the first S-IVB stage burn to a 185.2-km parking orbit altitude; arc 4 to the parking orbit coast; arc 5 to the second S-IVB stage burn at perigee of the Hohmann transfer ellipse; arc 6 to the transfer coast; and arc 7 to the third S-IVB stage burn at apogee of the transfer ellipse. Detailed tabulations of Saturn V ascent tra- jectories for various launch azimuths can be found in Reference 2 .
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LAUNCH
F i g u r e 2 . I n e r t i a l C a r t e s i a n plumbline coord ina te s y s t e m .
X?
A
A typical two-stage Saturn IB ascent trajectory and elliptical target orbit injection profile is presented in Figure 5. A s shown, trajectory arc i corresponds to the S-IB stage burn; a r c 2 to the S-IVB stage burn; a rc 3 to the first Command Service Module burn to a 185.2-km parking orbit injection; arc 4 to the parking orbit coast; and a rc 5 to the second CSM burn to injection. A typical Saturn IB ascent trajectory profile for a launch azimuth of 115 deg is presented in Table i. A detailed tabulation of this trajectory is presented in Table 2. The last entries in this table give the position and velocity of the CSM at parking orbit injection. These values of position and velocity were used as the initial "state" for transfer to the various terminal conic constraints.
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NORTH
cu A /
' V \
J x
'I,
- =Argument of Perigee N =Node Vector
P =Perigee Vector K =Radius Vector
n ZRight Ascension of
i =Orbit Inclination v =True Anomaly V =Velocity Vector
v =Flight Path Angle
Ascending Node
- (Measured from local Vertical)
t
Figure 3. Geocentric ephemeris coordinate system and orbital geometry and notation.
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r 1. 5 - I C Burn
2. 5-II Burn 3. First 5 - I P B Burn to 185.2-km Circular Orbit Altitude
4. Parking Orbit Coast
5. Second S - n B Burn (Perigee Burn for Transfer Ellipse)
6. Transfer Coast
Z Third S-mB Burn (Apogee Burn for Target Orbit Injection)
Figure 4. Typical Saturn V ascent and circular target orbit injection profile.
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1. S- IB Burn
2. S-IPB Burn
3. First CSM Burn to 185.2-km Circular Orbit Altitude
4. Parking Orbit Coast
5. Second CSM Burn (Injection Burn for Elliptical Target Orbit)
\
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5
Figure 5 . Typical Saturn IB ascent and elliptical target orbit injection profile.
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OPTIMAL TRANSFER TO SYNCHRONOUS ALTITUDE CIRCULAR ORBITS
The optimal' transfer from the initial state in the parking orbit to the in- parking-o rb it-plane synchronous altitude circular orb it c o r responding to the terminal conic constraint ( V , y , r) r = 42 240 km) required no coast in the parking orbit from the injection point. This condition resulted from the fact that any point in the parking orbit was an optimum departure point for this in-plane target orbit case. Therefore, the S-IVB burn at perigee of the transfer ellipse was specified to occur immediately after parking orbit injection. The optimum burn and transfer time computed is shown in the timing analysis of Table 3 . The remaining vehicle mass after satisfaction of the terminal constraint was 50 156.3 kg as compared to an initid vehicle mass of 128 986.8 kg. A detailed tabulation of this trajectory is presented in Table 4.
( V = 3.072 km/sec, y = 0 deg,
'rhe optimai transfer to ine oui-oI"-par~i~--orbii-piarlt: Lermiiid uuiiicr; constraint ( V , y, r , i) ( V = 3.072 km/sec, y = 0 deg, r = 42 240 km, i = 50 deg) required a plane change of approximately 13 deg. For a minimum velocity impulse to perform this plane change, the perigee point of the transfer ellipse is located at one of the nodal points of the parking orbit where only a portion of the total plane change is performed. Therefore, this transfer required a coast in the parking orbit of 268.4 sec.
The parking orbit injection point was located at a latitude (geocentric latitude) of 15.8 deg and a longitude of -55.6 deg. The perigee burn of 263.8 sec duration began at a latitude and longitude of 5.19 deg and - 4 1 . 6 deg, respectively, and ended at a latitude and longitude of -7.18 deg and -26.44 deg, respectively. Thus, the midpoint of the perigee burn occurred approximately at the equator, adding enough velocity increment to perform a plane change maneuver of 1. 16 deg (Ai,) and modify the S-IVB orbit to that of an ellipse with apogee radius equal to 42 240 km (Hohmann transfer ellipse) . The transfer time along this conic was 5.240 h r . With the plane change of 1. 16 deg performed at perigee, the plane change required at apogee (Ai,) was 11. 9 deg. These amounts were computed to be the optimum perigee/apogee plane change ratio. The length of the apogee burn was 101.2 sec. A complete timing analysis for
2. "Optimal" in the following discussion will refer to minimum fuel consumption.
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this transfer is presented in Table 5. The velocity vector diagrams for both the perigee and apogee burns are shown in Figure 6. A detailed transfer trajectory tabulation is presented in Table 6.
OPTIMAL TRANSFER TO ELLIPTICAL ORBITS WITH APOGEE AT SYNCHRONOUS ALTITUDE
The terminal conic constraint ( Cs, C , ) ( C3 -1ci .33 km2/sec2, C , = (37.40 km2/sec) perigee altitude equal to 210 km and apogee altitude equal to 35 862 km. A typical elliptical orbit injection profile was shown in Figure 4. Again, because of the in-plane target orbit, no particular point in the parldng orbit was an optimum point of departure, and thus the parking orbit injection point was chosen as the departure point. A single burn of the S-IVB stage was required to optimally satisfy the terminal constraint giving an injected mass of 71 361.9 lig as tabulated in the timing analysis of this transfer given in Table 7. Presented in Table 8 is a tabulation of this transfer trajectory.
specified an in-parking-orbit-plnne elliptical orbit with
The analogous out-of-plane terminal conic constraint to the above was ( C3, C1, i) ( C3 = -16.33 km2/sec2, Ci = 67.40 km2/sec, i = 50 deg) . The optimum coast t ime in parking orbit was computed to be 216.9 sec, corresponding to a position of 7.26 deg latitude and -44.2 deg longitude, as shown i n the timing analysis of Table 9. At this position the single S-IVB stage burn of 3 16.7 sec duration commenced. The burn ended at the position of -7.14 deg latitude and -29.6 deg longitude, after the energy and momentum constraints were satisfied and the plane change of approximately 13 deg was performed. A final mass of 59 771.5 kg remained at injection. The tabulation of this transfer maneuver is presented in Table 10.
OPTIMAL TRANSFER TO 1973 MARS M I S S ION HYPERBOLIC ORBIT
The terminal conditions for the injection burn into a typical 1973 M a r s mission trajectory were on a specified hyperbolic orbit. The necessary terminal conic constraint consisted of an energy level, C3, right ascension, CY, and declination, 6, of the outgoing hyperbolic asymptote (C, = + 18.0 km2/sec2, CY = 15.55 deg, 6 = 31.59 deg) along with a specified launch date
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APOGEE INJECTION VELOCITY D I A G R A M
Ai2 = 11.90 deg V, = 1.593 kmlsec Vc = 3.072 kmlsec
AV, : 1.479 kmlsec
PERIGEE INJECTION VELOCITY DIAGRAM
Ail = 1.160 deg Vp : 10.25 kmlsec
Vc 7,793 kmlsec
AV, ~ 2 4 6 4 kmlsec
Figure 6 . Circular target orbit velocity increment diagram.
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(August 6 , 1973) and time ( 14 h r , 38 min, 15.85 sec Greenwich Mean Time) . The hyperbolic injection geometry is illustrated in Figure 7. The right ascension and declination of the outgoing asymptote were referenced to an inertial, geocentric ephemeris coordinate system (Fig. 2) while the equations of motion in the computer program of Reference I were referenced to an inertial plumbline coordinate system. Thus a coordinate transformation was performed. The mechanics of this transformation are given in Appendix B.
OUTGOING A S Y M l f OTE
ESCAPE HYPERBOLA
Figure 7. Hyperbolic injection geometry.
A s tabulated in the timing analysis of Table I1 the optimum parking orbit coast time computed for the transfer was 1587.6 sec, which positioned the spacecraft at -36.3 deg latitude and 39.1 deg longitude. From this position a burn of 455. I-sec duration of the CSM propulsion system was required to insert the vehicle into the specified hyperbolic orbit. The vehicle mass injected into this orbit was 4817.3 kg compared with an initial vehicle mass of 18 029.7 kg. A detailed trajectory tabulation is presented in Table 12.
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CONCLUSIONS
The transfer trajectories presented above provide somewhat of a reference cross-section of optimally performed orbital operations. and detailed trajectories were presented for the optimal transfer of (a) an S-IVB stage vehicle from a parking orbit about the earth to circular orbits at synchronous altitude and elliptical orbits with apogee at synchronous altitude (in-and-out-of-parking-orbit-plane) , (b) a Command Service Module from an earth parking orbit to a typical 1973 M a r s mission hyperbolic orbit. The parking orbit about the earth was a 100 -nautical-mile, 37-deg-inclined circular orbit. An example of the utilization of these type transfer trajectories might be in the support of high-energy missions by the ILRV (Integral Launch and Reentry Vehicle). For instance, some of these proposed missions could involve the delivery of propulsive stages and payloads into low earth orbit for subsequent transfer to synchronous orbit.
Profiles
The input data listings presented should serve as examples for similar runs to be made on the ROBOT computer program. Also, the vehicle roll and pitch programs tabulated in the trajectory tables should be of future use as initial estimates for similar runs. The modifications made to the ROBOT computer program (ephemeris coordinate system transformation, computation of additional orbital parameters, new output format, and inclusion of optional, publishable, summary print tables) make it much more compatible for orbital operations computations.
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W 0 + m b W Q, 00 00 CJ d
II E x s s
w 8 E
c- M d
c- W 0
c- CJ dc
d m 0
* M Q, M
W d 00
00 M CJ *
tx P- W
d * Q, tx
dc 00 c\1 M
Lo d W M
00 M M *
0 0 0
0 0 0
0 0 0
0 0 0
Q) N Q)
0 0 0
0 0 d
0 m 0 m d
0 0 CJ
W d N
E c 0 u
F
0 u 3
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
Q,
Q,
0 0 d
0 m d
0 0 CJ
0 m CJ
0 0 M
W d M 8 w m
74
-
d d b
b 0 0
b b 0
z d
03 Q) 0
ED M d
Q) m 0
00 M d
Q) M d
W cu d
cu 2
Q) CJ d
._ d
cu Q) b
b
0 w Q)
P-
M u5 m 00
Q) 0 43
00
Q) d M
Q)
Q) Q) b
Q)
i? d
0 d
m
. U w F9
E w
75
-
0 m cil
d P-
I
W Q, Q,
W 0 0
W m
m W d
cil P- d 2l 00
0 00 W
0 cil d
0 Q, d
d d d
I
0 d d
I
00 6,
W @3
I I
m 00 m
d P- c- 00 * P- *
0 d 00
W * * d
0 Q, Q,
2 W
m cil 0
m * * m
0 m W
* 0 M
c W P- W
* W c- cil
PIE * ? C d hl
Q, * 00
* 2 Q, 0 00 m
Y
d P- O
d c- 0
d P- 0
d c- 0
M 00 cil
m 00 W
0 0 0
d c- 0
d P- 0
CD d M 00
m m
m m d
m 00 d
M m cil
z p;' 5 a
x E 5 a
0 0 0
3 W
Q, cil Q,
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 m m
0 0 *
0 m * 0 0 Ln
W d M
E 0 w a
0 m m 0 0 W
m m W a x w
76
-
n a Q, a 0 c 0 u
2
W
g d
w
8 F
m W m Lo *
d * m b, 6a d
0 b, W
00 W d
2 cu
0 m P- W Q, 00 00 cu d
0 0 0
Q) 6a Q,
W d m
0 W 00
m *
Q3 6a P- d m d
0 * cu * m h3 00 W SI
0 00 0
00 P- W d 6a d
d b 0
M M
0 0 0
0 In m
Lo d hl
CD *
m c- m Lo m d
0 * a m Q,
-M * hl
d
0 Q) d
I? 0 d d
d b 0
m 00
0 0 0
0 0 *
cu u3 * v, *
co cu 0
Q)
E?
0 m cu cu m d 0 cu c\l
d 0 F5
0 m a, Q, Q)
d c- 0
m m rl
0 0 0
0 m *
d P- In
W *
Q) b W
cu 2
0 cu 6a
d b 0 (I, 0, d
d d * W 0) cn 00 00
d I? 0
m 00 d
0 0 0
0 0 m
* P- m W *
* 6a m W 2
0 d h3
0 d 0 C? P- d
d h3 m h3 6a 0 00 P-
d P- O
m m 6a
0 0 0
0 Lo Lo
d W * W *
P- * Q)
Q)
2
0 d h3
Q, * Q) r- 2
m m (D
00 0 d b W
d P- O
m 00 h3
0 0 0
0 0 W
m 6a m a *
W m m w m d
0 00 Lo
m P- P- - 2
0 Q) * d P- P- Q) m
m 00 W W m d
.
77
-
0 0 d
W
4 E
rn k
d w b z w u d w P-i E 4
3
4 P;
H
cn
E
U
78
-
I -
-
.
CD W dc m W W W 0 m m Q, d v3 m W
I
cil W e- m d
b
b 3
W m
I
0 Q, Q,
d m
I
cil Q) b
b
d Q) b
b
CD cil W
d d
m v3 W d
b
W W d
W
m cil *
Q, Q) dc b Q, Q, m CQ
i? 0 cil W 0 d
0 0 e- m m d 0 0 W W W * d d
0 d 0 * 0 * 0 0
e- W m 0
E 3 2i cn u
a" 5 1 cd
0 k
?-I
.rl
0
0 0
$ Q
79
-
5 0 u
cd
z a 5: E
2
3
Pi w E
u E 8
80
-
c3 vlul 4 c E U J U
m 0 + 0
'", - Q t h 0 0 e
ul W Y
m 0 t I- @ r- 0 N 0 (0 - *
c f W
b -
t ul * 0
' W
z I
W w a
Q O f - 0 9 Z W P . C I .
m o m 9 G 9 m m m 9 9 c
O O O ~ O O O G O O C Q O O 0 0 0 0 0 C O ~ O O O Q 0 0 0 0 0 C O @ C C C C G C G , C . C ? C C X C C C C C C C - C 0 C C C C C C O m t O O Q O O ~ O O ~ O ~ ~ O G ~ ~ C G O @ D C O ~ C O ~ C O ~ O O ~ v)
81
-
V w
nv, N \ o x
Y
U w
* \ o x
Y
aul
U w
aLp x \ b L
X
a x N Y
a s * w
a x I %
w I
e u w
t m ul * 0 U
c
w u x w - V I c
N Q - L n m o 0 0 m r - P N - - &
I l l
0 . .
- - -
o o c c c o c o c c c o o c c o ~ c o o c c @ c c ~ ~ ~ ~ ~ ~ o * 3 0 C C C CS 0 C C 0 t C C c) 0 f - C C C 0 C C 0 C C C - D 0 C 9 0 C 0 C C P O O O O Q Q 0 0 3 C 3 3 0 0 O C 3 O O O O O O Q O O ~ ................................
o o a o c ) o o o c o o o o o ~ O O ~ O O O C Q O O U O O ~ O O ~ ' o o c u c c c c ~ c c E u O - c t c c ~ C C c C C C t C ~ O Q O ~ + , c 0 0 0 c 4 0 0 0 E? 0 8 0 0 P cn Ct 0 0 c c t 3 9 0 c t 0 c c c c, rl; M ..................................
82
-
.
- m - N * w * . - a N - C N N N I I
0 0 0 C D O c c 0 c c c c c t c c c 0 C c c c o c c c Cs c c o CI m 0 0 c! G C C c c OCI z c c c ~ C G c c c c t c 3 c a c c c c c c a o o c o o o o o c c o o c 3 o c o o o c o ~ 3 o o o o c G c 3 E l ~ J) _ _ _ ................................. ~ o ~ ~ ~ o o ~ ~ ~ o t o o o o o o o o c o o o c o o o a o ~ ~ - -NN o n o * a h l r m SI e e 0 0 - - N N 3 - t z m m m rp 0 Ul 0 0 m 0 rl) 0 dl 0 C rl, C J, 0 Ip. I2 an0 0 Ul 0 In G L? 0 IT Q
0 0 u 9 0 0 0 0 0 C O O O O O O C @ 0 0 c c o o o c a o o o o o m c c r c c c c c c o c o c c c o o c c c c c C C K O c c c a c t o w +
o O O O O O O O o 0 0 0 0 0 0 0 Q C c C o o o o O o ~ o o o o h 0 m o u l o L n o & n o m o m o m n u m o m 0 I n 0 I n a l n o m O m O l n O m ~ u
C C C l t 3 O C C C o O C j O G C C C C C C C C C C 0 t m O C . C G G C rll VI .................................. - - ~ ~ o n t f ~ ~ ~ ~ h h ~ o , e ~ c c - - ~ ~ ~ m * ~ m m ~
----------I-- c z L
83
-
z a- 3 IC
I:
s W
c
a
w u t u - v i c
C C. s 0
0 N 9 c
C c 0 f .t -
0 C. 0
0 m 9
- -
a c t
0 h 9
e
m
a c C
0 0. 9 -
C c c c
3 C C
0
I-
0
-. -
c C C: c
0 f I- -
C r c
t
C C C
0 h b
0
0 c 3
0 0 h
0
--.
t t C
C c C
0 N CE L.
0 C. c 0 f Q) c
O C c c. t C * ,
G O 0.0 a e - c
-
.
Y
R d
U 3 IC
85
-
86
-
I -
R d w $ F
87
-
N S Y
> I Y
x x Y
u u r w - u l t
-
2
3 ai t m K
LL L.
M W * + t? 0 c- m c*l 0 Go d
89
-
I ! ! I I I 1
(Y 0
9 f r3 m
m
* c)
0 0 0. 0
c
I
-
.
W
-r L.
t 0 m . c " (u 9 C -
3 30 S C E ' C G G 0 C 3 C t c c t: c c 2 c =i c .z c. E c c - c c c f c t L10 c c c t c c r- 2
9 1
-
92
-
n
% a a
v
PN d
w $ E
0 0 0 0 3 3 0 0 0 0 3 0 0 Q C G O C C 0 O O O O C C C O O O b O Q C C s o o c c, C I C C c c 1 1 l 1 l l ~ 0 l l l ~ l 1 * e e , e C e . . e e . . e .
O C G C C 3 C C ; C C O C C C I T , S C C C C Ci C C C C Ci C C C -I c c c c 0 C C t ; O C . C ’ C c c r - 2
.
9 3
-
a 0 3 d
'2 0 7 )3. 3
2 '3 U
0 a d
w
8 F
94
-
.
APPENDIX A
OUTPUT FORMAT FOR EACH PRINT TIME INTERVAL AND OPTIONAL SUMMARY PRINT TABLE DESCRIPTION
95
-
OUTPUT FORMAT FOR EACH PRINT TIME INTERVAL
TIME X P X DXP DX R
MASS Y P Y DY P DY ALT
WTLBS Z P Z D Z P DZ V
DDXP DDXPG W i P i LATR RDOT
DDYP DDYPG W2 P2 LONGR PTH
DDZP DDZPG W3 P3 AZ VPER
LATP CHIP ALPN DCHIP DALPN VAPO
LONGP CHIY ALPW DCHIY DALPW GRADV
SMA c3 ci ECC RCA RAOP
INC RASNOD ARPG TAN0 PERIOD MEOR
si s 2 s3 SD i SD2 SD3
-
DEFINITION O F OUTPUT SYMBOLS FOR EACH PRINT TIME INTERVAL
i. 2. 3. 4.
TIME. . . . . . . . . . . . . . . . . . MASS. . . . . . . . . . . . . . . . . . WTLBS. . . . . . . . . . . . . . . . . XP, YP, Z P . . . . . . . . . . . . .
time (sec) mass (kg) weight (lb) position coordinates in the plumbline system (m) position coordinates in the ephemeris system (m) velocity components in the plumbline system (m/sec) velocity components in the ephemeris system (m/sec) acceleration components in the plumbline system ( m/sec2) gravitational acceleration components in the plumbline system ( m/sec2) unit vector in the angular momentum direction unit vector in the direction of periapsis latitude (geocentric) and longitude of radius vector (deg) latitude (geocentric) and longitude of periapsis (deg) magnitude of radius vector (m) altitude above ellipsoid ( m ) magnitude of velocity vector (m/sec) velocity heading angle (deg) time derivative of radjus vector (m/sec) flight path angle (deg) angle-of-attack measured from the velocity vector to the projection of the thrust direction in the flight plane (deg) out-of-plane angle -of -attack measured from the projection of the thrust vector in the flight plane to the thrust vector positive toward the angular momentum vector (deg)
5. x , y , z . . . . . . . . . . . . . . . . 6 . DXP, DYP, D Z P . . . . . . . . . . 7. DX, DY, D Z . . . . . . . . . . . . . 8. DDXP, DDYP, D D Z P . . . . . . .
DDXPG, DDYPG, D D Z P G . . . . 9. 10. wi, w2 , w 3 . . . . . . . . . . . . . li. 12.
Pi, P2 , P3 . . . . . . . . . . . . . LATR, L O N G R . . . . . . . . . . .
13 I LATP, LONGP . . . . . . . . . . . 14. 15. 16. 17. 1%. 19. 20.
R . . . . . . . . . . . . . . . . . . . . . ALT. . . . . . . . . . . . . . . . . . . v. . . . . . . . . . . . . . . . . . . . . AZ . . . . . . . . . . . . . . . . . . . RDOT . . . . . . . . . . . . . . . . . PTH . . . . . . . . . . . . . . . . . . ALPN . . . . . . . . . . . . . . . . .
21. ALPW . . . . . . . . . . . . . . . . .
22. CHIP, CHIY . . . . . . . . . . . . . control angles CHI and CHI pitch Yaw ( deg)
.
97
-
DEFINITION OF OUTPUT SYMBOLS FOR EACH PRINT TIME INTERVAL
23.
24.
25. 26. 27.
28. 29. 30. 3 1. 32. 33. 34. 35. 36. 37. 38. 39.
DALPN, DALPW.. . . . . . . . . DCHIP, DCHIY. . . . . . . . . . . VPER, VAPO.. . . . . . . . . . . SMA . . . . . . . . . . . . . . . . . . RASNOD . . . . . . . . . . . . . . . ECC . . . . . . . . . . . . . . . . . . RCA. . . . . . . . . . . . . . . . . . RAOP . . . . . . . . . . . . . . . . . GRADV . . . . . . . . . . . . . . . . ARPG . . . . . . . . . . . . . . . . . TAN0 . . . . . . . . . . . . . . . . . PERIOD. . . . . . . . . . . . . . . . MEOR . . . . . . . . . . . . . . . . . si, s2 , s 3 . . . . . . . . . . . . . . SD i, SD2, SD3 . . . . . . . . . . . c3 . . . . . . . . . . . . . . . . . . . ci . . . . . . . . . . . . . . . . . . .
time derivatives of ALPN and ALPW ( rad/sec2)
and CHI pitch Yaw
time derivatives of CHI ( rad/sec2) velocity at periapsis and apoapsis (m/sec) semimajor axis (m) right ascension of the ascending node (de& eccentricity radius at periapsis (m) radius at apoapsis (m) gravity loss ( d s e c ) argument of periapsis (deg) true anomaly (deg) orbital period (sec) mean orbital rate (rad/sec) calculated outgoing asymptote desired outgoing asymptote twice the specific energy (m2/sec2) specific angular momentum ( m2/sec)
DESCRIPTION O F OPTIONAL SUMMARY PRINT TABLES (QUANTITIES REQUIRED TO BE SPECIFIED)
(1) Print interval for parking orbit coast.
(2 ) Print interval for f i r s t burn.
(3) Print interval for transfer coast.
(4) Print interval for second burn.
If the terminal condition is a circular orbit, the following tables are printed:
a. Parking Orbit Coast b. Perigee Burn c . Conic Conditions After Perigee Burn d. Transfer Coast
98
-
e. Apogee Burn f. Conic Conditions After Apogee Burn
If the terminal condition is an elliptical orbit, the following tables a re printed:
a. Parking Orbit Coast b. First Burn c . Conic Conditions After First Burn d. Orbit Coast e. Second Burn f. Conic Conditions After Second Burn
If second burn required
If the terminal condition is an outgoing hyperbolic asymptote, the following tables a re printed:
.
Tables 1-12 a re examples of the different summary print tables available.
99
-
EQUATIONS FOR THE COORDINATE SYSTEM TRANSFORMATION AND COMPUTATION OF ADDITIONAL ORBITAL PARAMETERS
10 1
-
DEFINITION OF SYMBOLS3
r, 3, ii I-r V r i
52 w
I I [ I
h
P e P L
i, 2, 3
Unit vectors along the X, Y, Z (ephemeris) axes, respectively. Gravitational constant Inertial velocity magnitude Magnitude of radius vector Inclination of orbital plane Argument of periapsis Right ascension of ascending node Indicates vector magnitude Matrix o r determinant notation
SUPERSCRIPTS
Time derivative Hour (universal time) Indicates vector quantity
SUBSCRIPTS
Plumbline coordinate system
Transformation from plumbline to ephemeris coordinates
Indicates reference to X, Y, Z (ephemeris) axes, respectively.
( 5)4 Transformation from plumbline to ephemeris coordinate system: X, Y, Z [3, 41
3. Undefined symbols used car ry the same connotation a s in Reference i.
4. These numbers correspond to those numbering the output symbols of Appendix A.
102
-
s i n B cos K + cos 0 sin ,$I sin K
cos 0 cos K + s in 6 sin 9 s in K
- cos @ s in K
cos 0 cos 9
s in 0 cos 9
s i n 9
sin 8 sin K - cos 0 s in $I COB K - cos 0 sin K - sin 0 sin $ cos K
cos 9 cos K
h e = (GHA)O UT w t + A e h
GHA = Greenwich hour angle measured west from Greenwich at 0 UT of the launch day.
0 UT = Zero hour universal time. h = Launch longitude (east through 360') { 279'395645)
h
K = (3/2)n-A
A = Launch azimuth, @ = Geodetic latitude (28f6079927)
t = GMT nf launch. w = Earth's rotation rate
GHA = 100: 07554260 + Of9856473460 Td + 2: 9015 x
h Td = Days past 0 January I, 1950.
. e
T i
deg/sec 360 w = e 86164.09892 + 0.00164 T
T = Number of Julian Centuries of 36 525 days from 1900 Jan. 0 . 5 UT (Julian Date = 24 15020.0)
(7) Velocity components in the ephemeris system: DX, DY, DZ
( I O ) Unit vector in angular momentum direction: W,, W2, W3
Ci = 4 A2 + B2 + C2
103
-
A = e - Z ? , B = Z A - X Z , C = ~ - e
104
L
B C , w - - w3= c, I- c, 2 - c,' A w - -.
(ii) Unit vector in direction of periapsis: Pi, P,, P3
-
(13) Latitude and longitude of periapsis: LATP, LONGP
h -
t
A T P E R l A P S l S
E O U A T O R I A L P L A N E
O R B I T A L P L A N E
X GP
LATP = sin (i) sin (w )
X / GP
105
-
sin 8 = sin (i) sin (0) -1
0 = sin [sin (i) sin ( w ) ] -
RCA= 1~7~1, RCA = ~ R C A I Xy XY
RCA = R C A C O S e Xy
Pi = R C A COS A w
Xy Pz = R C A sin A
Pi = X - component of unit vector in direction of periapsis
Pz = Y - component of unit vector in direction of periapsis
r = r i , E = E o a t t = t t = time at injection inj i nj
r = a( i - e cos E)
a= i - e cos EO a
ri = magnitude of radius vector (E) at injection
E = eccentric anomaly
a - rl ae cos Eo =
-i (rl sine)
a G
6 = true anomaly Eo = sin
M = n( t - T) = E - e sin E M = mean anomaly
-
.
Eo - e sin Eo T = time at periapsis passage (tinj - 7) = n
= Time increment past periapsis passage for r = ri.
2n Period = T = -. n
- T ) = Time at periapsis passage (‘inj t = T - P
The t
last iteration of a converged run.
value used should be that time of injection computed on the inj
LC)C_A TTON O F GREENWICH MERIDIAN:
E Q U A T O R I A L P L A N E
AT P E R l A P S l S P A S S A G E T *
AT L A U N C H h
AT 0 U T OF
L A U N C H D A Y
LONGP = A [GHA+ w ( t + t inj + tdl e
(26) Semimajor axis: SMA
107
-
v2 2 1 P r a - _ - =,-
S M A = a = 21.1 - r V
(27) Right ascension of the ascending node: RASNOD
A W = W i = s i n S l s i n i = - X Ci
wi sin i
cos Sl = - w2 sin i
-1 w, RASNOD = Sl = sin (s in i ) RASNOD = Sl = cos - I ( -2%)
(28) Eccentricity: ECC = e = [C3 (r2 V2 - r2 ?2)]
(29) Radius at periapsis: RCA = a ( I-e)
(30) Radius at apoapsis: RAOP = a ( i + e)
(31) Gravity loss: GRADV = V - AV C
108
-
.
.
W P W f + W
P
W
W = final weight of vehicle
= weight of propellant burned P
f
r. = initial radius
V. = initial velocity
r = final radius
1
1
f
(32) Argument of periapsis: ARPG
P =P3 Z
z P
sin w = - sin i
Z Q
cos w = - sin i
m P G = w = sin-' (L) sin i ARPG = w = cos
(33) True anomaly: TAN0
r 1
109
-
/
(34) Period: Period
27r Period = T =
$/2 a-3/2
(35) Mean orbital rate: MEOR
27r T MEOR = -.
I
(36) Calculated outgoing asymptote: Si, S2, S3
7
110
- P = i (P* ) + 3 P 2 ) + k ( P $
d l e 2 - 11 si= -(f) Pi + e
Je2-i/ (WIP, - W,P,) ss= -($) P3 + e
-
.
APPENDIX C
ADDITIONAL INPUT VARIABLES REQUIRED AND INPUT DATA LISTINGS
111
-
Program Input Description
Input for ROBOT modified for the special printout shown in Appendix A is the same as that in Reference I, with the addition of the following input variables:
Input Internal Symbol Symbol Explanation
AAE T C+ 5 0 E PRNT = 1 i f special printout is desired
= 0 is normal ROBOT printout is desired
AAE T C+ 24 RA Right ascension as measured in ephemeris coordinates (deg)
AAETC+25 DC Declination (deg)
AAE T C+ 5 4 MO Month of launch
AAET C+ 5 5 IDAY Day of launch
AAET C+ 5 6 IYR Year of launch (years since 1900)
AAET C+ 5 7 HOUR Hour and minute of launch (military time)
AAETC+ 5 8 SEC Fractional minutes of launch time expressed in seconds
If summary print tables are desired, the following variables must be set.
JRBETC+7
JRB E TC+ 29
112
The thrust event number where tables begin
= I if the cutoff condition is a circular
= 2 if the cutoff condition is an elliptical
= 3 if the cutoff condition is an outgoing
orbit
orbit
hyperbolic asymptote
-
. Input Data Lists
The following lists give the input data for the terminal conic constraints.
1 1 3
-
INPUT DATA LISTING FOR TERMINAL CONIC CONSTRAINT (V, y , r, i)
.
.
114
-
c INPUT DATA LISTING FOR TERMINAL CONIC CONSTRAINT (Ca, Ci)
INPUT DATA LISTING FOR TERMINAL CONIC CONSTRAINT (c3, ci, i5)
5 . RW as a "stacked case" behind terminal constraint (C3, C,) input data.
115
-
INPUT DATA LISTING FOR TERMINAL CONIC CONSTRAINT (C3, Q! , 6)
116
-
*
APPENDIX D
UNIVAC 1108 FORTRAN LISTING OF SUBROUTINE APRTN
117
-
118
-
119
-
120
-
c***
C***
C***
C * * C -
C***
C***
C***
C***
C***
C***
C***
c***
C***
C**%
12 1
-
12 2
-
c
.
1 2 3
-
APPENDIX E
UNIVAC 1108 FORTRAN LISTING OF SUBROUTINE TRASH
125
-
DATA RLANK/6H / D A T A ( T I T L E ( I ~ r I = l t l 7 ) / 6 1 1 P I ~ H A R K I N G B ~ H O R U I T r 6 H C O A S T B ~ H P E R I G E
1.6HE EURNr6H C O N I C r 6 H C O N D I t 6 H T I O N S t6HAFTER r 6 H F I R S T t6HRLIRN t
GO TO 171 1 2 4C :!R I T E ( 6 I 9 9 9 1 T r TC r’X LlKG r X r 4 L i j r Ct i I P r C h I Y r ALPN r ALPX
GO T 3 1 7 1 1 2 5 3 h ’ R I T E ( 6 r 9 3 7 ) T r T C r ( S ( I ) r I = 1 3 r l 5 )
171 L I N E S = L I N E S - l I C ( L I N E S . E Q . 9 ) ‘10 TO 7 I F ( k r S ( T - T E ( N @ T + l ) ) . j T * S T E ; ’ P + T E ( N 3 T + l ) 150 T 3 11
1 7 3 > ! R I T E ( 6 r 9 7 8 ) A ( 7 ) 1 7 2 IF(EP.EC*EVDPRT)SO TO 1 7 4
IF(NTAB.LE.NTOL)G3 TO 5 3 2 N T A r~ = V T A! + 1
NOT = NOT + 1 GO T O 4 0 0
1 7 4 iJR I T E ( 6 r 996 ) 5 0 2 DO 5 0 3 I = l r I R C K 503 RACKSP9CE 9
I BCK=O .
126
-
I F ( EP EC E N S P ? T i;ETUIIN G O TO 2
GC TO 1 2
GO TO 1 2
GO TO 1 2
GO TO 1 2
GO TO 1 7 2 C 5 8 E L S B N O T T R A N S I T I O N
4 0 0 DO 2400 I = l t 7 2 4 0 0 A ( I ) = R L A N Y
1 1 2 ' ~ , R I T E ( 6 r 1 1 1 2 ) 4 ( 7 1
117 W R I T F ( 6 r 1 1 1 3 ) 9 ( 7 )
1 2 4 Y R I T E ( 6 r 1 1 2 4 ) 4 1 7 )
1 2 5 ' % ' R I T E ( 6 t 1 1 2 5 ) 6 ( 7 )
i 3 n I - I R I T E ( ~ * ~ ~ ~ ~ ) W ~ L
GO TO ( 4 1 0 9 4 2 2 r 4 2 0 ) *MODE 4113 GO TO (411r412~413r414r41~r416)rN3T 411 DQ 401 I = l s 4 4 0 1 A ( I ) = T I T L E ( I ) 4 0 7 A ( 7 ) = A ( 4 )
N T B L = 3 GO TO 5
4 1 2 A ( 3 ) = T I T L E ( 5 ) A ( 4 ) = T I T L E ( 6 )
4 C 5 A ( 7 ) = T I T L E ( 1 6 ) N T B L = 5 GO T O 5
A ( I ) = T I T L E ( I + 6 ) 413 DO 402 1 x 1 9 4
402 A ( I + 2 ) = T I T L E ( I + 2 ) 404 N T B L = l
GO T O 3 414 A ( l ) = T I T L E ( 1 3 )
A ( 2 ) = T I T L E ( 1 4 ) A ( 3 ) = T I T L E ( 4 ) A ( l ) = A ( 3 1 N T B L r 3 GO T O 5
4 1 5 A ( 3 ) = T I T L E ( 1 5 ) 4 ( 4 ) - T I T L E ( 1 6 ) GO TO 405
416 DO 403 1 x 1 9 4 A ( I ) = T I T L E ( 1 + 6 )
403 A 1 1 + 2 ) = T I T L E ( I + l 2 ) GO TO 404
422 A ( 3 ) = T I T L E ( 1 1 ) A ( 4 ) * T I f L E I 1 2 ) GO T O 405
423 DO 406 I = l t 6 406 A ( I ) = T I T L E ( I + 6 )
GO TO 4 C 4 425 A ( 3 ) = T I T L E ( 1 7 )
A ( 4 ) - T l f L E ( 1 6 ) GO T O 4 0 5
420 GO TO (411r422r423,411r425r423)rNOT
1300 F O R M A T ( l H l r 7 8 X 1 2 H T A R L E N U " E E R / ~ X ~ A ~ S ~ ~ X ~ A ~ / ~ H S E G I N r A 6 r 4 X 7 H W E I G H f * l E 1 3 . 8 ~ 4 H K G S I ~ X I E ~ ? * ~ ~ ~ ~ H L q S / / )
111 1 F 0 R : I A T ( 4 X 4 H T I ; l E 9 6 X A 6 9 5 H T I i.iE 9 5 X l r l R r 6 X 1 3 1 - 1 I N E R V E L O C I T Y Z X 9 H A Z I M U T H S 1 5 X 3 H P T n 5 X 8 H L A T I T U D E 3 X 9 H L ~ ' ~ G I T U 3 E / 5 X 3 H S E c l l X 3 H S E C 8 X 2 ~ K ~ l O X 6 H K M / S E C ~ 1 7 X 3 HDEG8 X 3 H D E G 8 X 3 H ? C G 8x3 tic E C /
9 9 6 FORA'AT ( 1 H 1 )
127
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128
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REFERENCES
1. Gottlieb, Robert G.: ROBOT - Apollo and AAP Preliminary Mission Profile Optimization Program. Part I - Mathematical Formulation, NASA CR-61223, May 1968.
2. Ellison, Bobby; Noblitt, Bobby G. ; and Young, Archie C. : Mission Analysis and Trajectories for Three-Zero-Uegree Inclination Earth Synchronous Orbit Missions Using the Saturn V Vehicle With Three Burns of the S-IVB Stage. NASA TM X-53498, August 1966.
3. Cooper, D. L . ; and Ellis, L. V.: Optimization of Transfer Trajectories by Application of the Calculus of Variations. Northrop Space Laboratories Research and Analysis Section Technical Memorandum ##48, January 25, 1965.
4. C.ia~-ke, TJ. C.; I?c!!mzrr, W. R - J Rnth, R. Y. ; and Scholey, W. J.: Design Parameters for Ballistic Interplanetary Trajectories. Part I: One-way Transfers to Mars and Venus, J P L Technical Report No. 32-77, January 1963.
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APPROVAL TM X-53873
OPTIMAL TRANSFER TRAJECTORIES FROM EARTH PARKING ORBIT TO VARIOUS TERMINAL CONIC
CONSTRAINTS AND MODIFICATIONS TO THE ROBOT COMPUTER PROGRAM
By Rodney Bradford
The information in this report has been reviewed for security classifi- cation. Review of any information concerning Department of Defense o r Atomic Energy Commission programs has been made by the MSFC Security Classifica- tion Officer. This report, in its entirety, has been determined to be unclassi- fied.
This document has also been reviewed and approved for t e c n n i d accuracy.
H. F. THOMAE Chief, Performance & Flight Mechanics Division
W
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Director, Prelimynary Design Office
w. R-.-LUCAS Director, Program Development
131 I
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T M X-53873
INTERNAL
D IR D r . von Braun
D E P - T D r . R e e s
AD-S D r . S tuhl inger
PD-DIR D r . L u c a s
PD-SA -DIR M r . Huber
PD-SA-V M r . Or i l l ion
PD-DO-DIR M r . G o e r n e r D r . Thomason M r . H e y e r M r s . Andrews
P D - D O - P D r . Thomae M r . Scot t
P D - D O - P D M r . Schwar tz M r . J u m p M r . L o w e r y M r . S u m r a l l M r . B r a d f o r d ( 2 0 )
PD-DO-E M r . Digesu
132
DISTRIBUTION
PD-DO-ES M r . Cole
P D - D O - P F M r . Goldsby M r . C r o f t M r . F r e n c h M r . F u l t s M r . L e o n a r d M r . P e r k i n s o n M r . S m a r t
P D -DO - P M M r . H i l l M r . W h e e l e r M r . Young M r . Gold M r . Farr
S & E -AERO-DIR D r . G e i s s l e r M r . H o r n M r . von P u t t k a m e r
S&E-AERO-D M r . E. Dea ton M r . C r e m i n M r . Toe l l e M r . A. Deaton M r . P e r r i n e
S&E-AERO-G M r . Chandler M r . B a k e r M r . C a u s e y M r . Lof t in
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T M X-53873 DISTRIBUTION (Continued)
INTERNAL (Concluded) EXTERNAL
S&E-AERO-F Mr . Lindberg M r . Wittenstein M r . Dickerson Mr . Mullins Mr . Little Mr . Te l f e r Mr . Bauer Mr . Mabry M r . Phi l l ips
S&E -AERO-Y M r - W. Vaughn
S&E -AERO-T D r . H. Krause D r . Teuber
A&TS-MS-IP ( 2 )
A&TS-MS-IL (8)
A&TS-PAT Mr. L. D. Wofford, Jr.
PM-PR-M Mr . Goldston
A&TS-MS-H
A&TS-TU ( 6 )
Scientific and Technical Information
P.O. Box 33 College Pa rk , Maryland 20740 Attn: NASA Representat ive (S-AK/RKT)
Facil i ty (25)
NASA -Manned S p a c e c r a f t Cen te r Houston, Texas 77058 Attn: Library (5)
Dynamics Resea rch Corporat ion Holiday Office Center Huntsville, Alabama 35805
h A I .^ . ?2r. G d t l i eb
M r s . Johnson
Northrop Space Laboratories Technology Dr ive Huntsville, Alabama 35805 Attn: M r . W. A. Klabunde
Mr . D. Raney Mr . W. B. Tucker L ib ra ry ( 5 )
Lockheed Miss i les z d Space Company Technology Dr ive Huntsville, Alabama 35805 Attn: L ib ra ry ( 5 )
Mr . J. Winch Mail Stop AC-50 The Boeing Company P.O. Box 1680 Huntsville, Alabama 35807
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1 3 3
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TM X-53873 DISTRIBUTION (Concluded)
EXTERNAL (Concluded)
Mr . John E. Canady NASA-Langley Research Center Langley Station, Virginia 23365 Mail Stop 215
NASA Headquarters Attn: Col. Burke, MTV (5)
Library (5) RMP (5)
NASA - Ames Research Center Mountain View, California 94035 Attn: Library (5)
NASA-Langley Research Center Hampton, Virginia 23365 Attn: Library (5)
NASA-Electronic s Research Center Cambridge, Mass. 02 138 Attn: M r . W. Miner
Library (5)
Jet Propulsion Laboratory Pasadena, California 9 1 103 Attn: Library (5 )
Mr. J. J. Modorelli (5) Tech. Info. Div. NASA-Lewis Research Center 2 1000 Brookpark Road Cleveland, Ohio 44135
MSFC-RSA. Ala
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