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Verification of the optimal control solutionThe optimal control solution determined by the software can be verified by numerically integrating theorbital equations of motion with the optimal control-computed initial park orbit conditions and theoptimal control solution. This is equivalent to solving an initial value problem (IVP) that uses theoptimal unit thrust vector solution. This part of the leo2geo_ocs computer program uses a Runge-Kutta-Fehlberg 7(8) variable step size method to integrate the orbital equations of motion.The following is a summary of the final optimal solution computed using this explicit numericalintegration method.========================================verification of optimal control solution========================================mission orbit state vector and orbital elements-----------------------------------------------sma (km) eccentricity inclination (deg) argper (deg)0.419999999440D+05 0.100000252738D-02 0.999999595528D+00 0.359999478450D+03raan (deg) true anomaly (deg) arglat (deg) period (min)0.359999985471D+03 0.423927660632D+02 0.423922445128D+02 0.142768906488D+04rx (km) ry (km) rz (km) rmag (km)0.309960417074D+05 0.282912586047D+05 0.493825693260D+03 0.419689620884D+05vx (kps) vy (kps) vz (kps) vmag (kps)-.207699129066D+01 0.227794897693D+01 0.397617220168D-01 0.308294102401D+01transfer time 15.0823889600487 hoursspacecraft mass 835.576420851553 kilogramspropellant mass 164.423579148447 kilogramsdelta-v 5813.28840906505 meters/secondCreating an initial guessThe solution to this classic orbit transfer problem requires a good initial guess for the transfer time alongwith a reasonable guess for the dynamic variables during the orbit transfer.Transfer time initial guessThe leo2geo_ocs software implements three options for specifying the transfer time initial guess. Thefirst technique simply integrates the coplanar transfer orbit trajectory using tangential thrusting until thefinal orbit radius is reached. The second method uses Edelbaum’s algorithm. The first method is bestfor coplanar orbit transfers and the second method should be used for non-coplanar orbit transfers. Thethird initial guess option is a user-specified transfer time.For the first technique, the unit thrust vector for tangential steering during the numerical integration isu 0 1 0 T.simply T page 12
The Edelbaum algorithm is described in Chapter 14 of the book Orbital Mechanics by V. Chobotov andthe technical paper, “The Reformulation of Edelbaum's Low-thrust Transfer Problem Using OptimalControl Theory” by J. A. Kechichian, AIAA-92-4576-CP. The original Edelbaum algorithm isdescribed in “Propulsion Requirements for Controllable Satellites”, ARS Journal, Aug. 1961, pp. 1079-1089. This algorithm is valid for total inclination changes i given by 0 i 114.6 and assumes thatthe thrust acceleration magnitude and spacecraft mass are both constant during the orbit transfer.The initial thrust vector yaw angle 0is given by the following expressiontan 0VV0fsin i 2 cosi 2 where the speed on the initial circular orbit is V0 r0and the speed on the final circular orbit isVf rf. In these equations r 0r e h 0is the geocentric radius of the initial orbit, rf re hfis thegeocentric radius of the final orbit, r eis the radius of the Earth and is the gravitational constant of theEarth. The initial altitude is h 0and the final altitude is h f.The total velocity change required for a low-thrust orbit transfer is given byV0sin0V V0cos0tan i 0 2 The total transfer time is given by t V f where f is the thrust acceleration. This is the transfer timeused for the Edelbaum guess option in the leo2geo_ocs software.Dynamic variables initial guessThe dynamic variables at each grid point of the initial guess are determined by setting the initial guessoption INIT(1) = 6 with INIT(2) = 2 within the odeinp subroutine for this aerospace trajectoryoptimization problem. These program options create an initial guess from the numerical integration ofthe equations of motion coded in the oderhs subroutine. The INIT(1) = 6 program option tells theSparse Optimization Suite to construct an initial guess by solving an initial value problem (IVP) with alinear control approximation. The INIT(2) = 2 program option tells the program to use the Dormand-Prince variable step size numerical method to solve the initial value problem.Binary restart file initial guessBinary restart data files can also be used to initialize a leo2geo_ocs simulation. A typical scenario is1. Create a binary restart file from a converged and optimized simulation2. Modify the original input file with slightly different spacecraft characteristics, propulsiveparameters or perhaps final mission targets and/or constraintspage 13
Page 1 and 2: Program leo2geo_ocsContinuous Low-T
Page 3 and 4: Dynamics, Vol. 22, No. 3, May-June
Page 5 and 6: ************************* initial g
Page 7 and 8: The level of output from the differ
Page 9 and 10: This next plot illustrates the beha
Page 11: page 11
Page 16 and 17: wheremsc iis the initial spacecraft
Page 18 and 19: dhh dt2p s 2wncos Ldkk dt2p s 2w
Page 20 and 21: sin 1uTr1tan uT,nThe pitch angle is
Page 22 and 23: 23JR122hsin L k cos LeJ 1242 22r2
Page 24 and 25: APPENDIX AContents of the Simulatio
Page 26 and 27: APPENDIX BTypical Sparse Optimizati

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