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B-plane coordinates at Mars sphere-of-influence(Mars-centered mean equator and IAU node of epoch)--------------------------------------------------b-magnitude 9131.22236299615 kilometersb dot r -7884.79362085750 kilometersb dot t 4605.35030143918 kilometerstheta 300.288346415312 degreesv-infinity 2706.43936253276 meters/secondr-periapsis 4995.80284573990 kilometersdecl-asymptote 7.47149194895923 degreesrasc-asymptote 281.319000130777 degreesflight path angle -86.6386680328833 degreestime and conditions at Mars entry interface(Mars-centered mean equator and IAU node of epoch)--------------------------------------------------calendar date December 24, 2003TDB time 02:07:38.158TDB Julian date 2452997.58863608sma (km) eccentricity inclination (deg) argper (deg)-.584304937602D+04 0.185515028038D+01 0.599817961400D+02 0.113992445290D+03raan (deg) true anomaly (deg) arglat (deg)0.105639013468D+03 0.356950775849D+03 0.110943221140D+03rx (km) ry (km) rz (km) rmag (km)-.176830316788D+04 -.235141535790D+04 0.404435562654D+04 0.500128610588D+04vx (kps) vy (kps) vz (kps) vmag (kps)0.215587049620D+01 -.412637403497D+01 -.166793392649D+01 0.494537600754D+01B-plane coordinates at Mars entry interface(Mars-centered mean equator and IAU node of epoch)--------------------------------------------------b-magnitude 9130.09430947819 kilometersb dot r -7882.77552850412 kilometersb dot t 4606.56836128398 kilometerstheta 300.301334295321 degreesv-infinity 2707.36011349352 meters/secondr-periapsis 4996.68531218008 kilometersdecl-asymptote 7.46173444525707 degreesrasc-asymptote 281.299129226053 degreesflight path angle -1.98134902376417 degreestime and conditions of Mars at entry interface(heliocentric EME2000)----------------------calendar date December 24, 2003TDB time 02:07:38.158TDB Julian date 2452997.58863608page 12
sma (au) eccentricity inclination (deg) argper (deg)0.152368043785D+01 0.935420837075D-01 0.246772248956D+02 0.332979294518D+03raan (deg) true anomaly (deg) arglat (deg) period (days)0.337165819738D+01 0.704475973849D+02 0.434268919026D+02 0.686972361210D+03rx (km) ry (km) rz (km) rmag (km)0.150787801633D+09 0.145972773409D+09 0.628789377224D+08 0.219086221628D+09vx (kps) vy (kps) vz (kps) vmag (kps)-.166518297346D+02 0.168787718592D+02 0.819178961924D+01 0.250854896335D+02transfer time 198.324237048160 daysProblem setupThis section provides additional details about the software implementation. For good scaling during theoptimization, the time unit used in all heliocentric calculations is days, position is expressed inastronomical units, and the velocity and TCM delta-v unit is astronomical units per day. The areocentricor Mars-centered motion is modeled using seconds, kilometers and kilometers per second.During the solution of this optimal control problem, the itcm_ocs software models the heliocentricmotion implicitly using collocation and the Mars-centered orbital motion explicitly using numericalintegration of the equations of motion.(1) Performance index – minimize TCM delta-vThe objective function or performance index J for this simulation is the scalar magnitude of theimpulsive TCM delta-v vector. For this classic trajectory optimization problem, this index is simplyJ VThe value of the maxmin indicator in tells the software whether the user is minimizing or maximizingthe performance index.The components of the impulsive TCM delta-v are the optimization parameters used by the SparseOptimization Suite to solve this trajectory problem.(2) Point functions – position and velocity vector “matching” at the TCMThe optimal solution must satisfy the following state vector boundary conditions (equality constraints) atthe initial time defined by the user:r t rt 0 s / c TCM s / c TCMt t tv v v 0s / c TCM s / c TCM TCMwhere rs/cand vs/care the heliocentric position and velocity vectors of the spacecraft prior to the TCMmaneuver time t TCM, rs/cand vs/care the heliocentric position and velocity vectors of the spacecraftafter the TCM delta-v is applied, and v is the impulsive TCM delta-v vector.page 13
Page 4 and 5: upper bound for TCM delta-v compone
Page 6 and 7: The next input is an integer that d
Page 8 and 9: calendar date June 8, 2003TDB time
Page 10 and 11: sma (au) eccentricity inclination (
Page 14 and 15: (3) Point functions - SOI distance
Page 16 and 17: Heliocentric equations of motionFig
Page 18 and 19: In these equations, and r eqare th
Page 20 and 21: Geocentric-to-areocentric coordinat
Page 22 and 23: Algorithm resources“Update to Mar
Page 24 and 25: s2e-x (au) = x-component of Earth p
Page 26 and 27: time and conditions after TCM maneu
Page 28 and 29: calendar date December 24, 2003TDB
Page 30 and 31: ----------------------calendar date
Page 32 and 33: time and conditions at Mars SOI(Mar
Page 34 and 35: time and conditions at Mars sphere-

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