Исследование оптимального трехимпульсного перехода на высокую орбиту искусственного спутника Луны

23 / 24

Исследование оптимального трехимпульсного перехода на высокую орбиту…
Инженерный журнал: наука и инновации # 9·2017 23
Investigating the optimal three-impulse transfer into
the high orbit of the artificial lunar satellite
© E.S. Gordienko
Lavochkin Science and Production Association,
Khimki town, Moscow region, 141402, Russia
Bauman Moscow State Technical University, Moscow, 105005, Russia
The article examines the problem of optimum spacecraft ascent from the Earth into the
high circular polar orbit of the artificial lunar satellite with the radius of 6000 km using
the three-impulse bi-elliptic scheme of the so called “Shternfeld” transfer. We have car-
ried out this analysis by taking into account the disturbances from the lunar field noncen-
trality, the Earth and the Sun gravitational fields as well as the engine thrust finiteness.
The optimum trajectory is defined by varying both impulse control parameters and their
application points. The analysis consists of two stages. At the first stage we consider two
options of impulse orientation in the ideal impulse occurrence: at first the impulse orien-
tation is set in the osculating plane by “pitch” γ and “attack” ߙ
angles; then the “yaw”
angle ψ (angle of departure from the plane) is added. It is shown that with the increase of
the maximum distance r ߙ
the optimal points of the impulses application are shifted from
the apsidal points of the orbits. At the same time the second, intermediate impulse is not
directed along the current velocity vector. In the first, “flat” option of the impulse orien-
tation we get characteristics which are similar to the ideal impulse apsidal case. In the
second, “spatial” variant the spacecraft remaining mass slightly increases. At the second
stage we take into account the engine thrust finiteness. The results obtained are very
close to the impulse case.
Keywords: spacecraft, trajectory optimization, artificial lunar satellite, three-impulse
bi-elliptic scheme, “Sternfeld” transfer, Powell’s quasi-Newton method
REFERENCES
[1] Gordienko E.S., Ivashkin V.V. Inzhenernyy zhurnal: nauka i innovatsii — Engi-
neering Journal: Science and Innovation, 2016, iss. 3.
DOI: 10.18698/2308-6033-2016-3-1472
[2] Gordienko E.S., Ivashkin V.V. Kosmicheskiye issledovaniya — Cosmic Re-
search, 2017 (is to be published).
[3] Okhotsimskiy D.E., Eneyev T.M. Uspekhi fizicheskikh nauk — Advances
in Physical Sciences, 1957, vol. 63, no. 1a, pp. 5–32.
[4] Gordienko E.S., Ivashkin V.V., Lyu V. Kosmonavtika i raketostroenye — Cos-
monautics and Rocket Engineering, 2015, no. 1, pp. 37–47.
[5] Attetkov A.V., Galkin S.V., Zarubin V.S. Metody optimizatsii [Optimization
procedure]. V.S. Zarubin, A.P. Krischenko, eds. Moscow, BMSTU Publ., 2003,
440 p.
[6] Standish E.M. JPL Planetary and Lunar Ephemerides. Interoffice memorandum,
1998. Available at:
(accessed February 27, 2017).
[7] Tsiolkovskiy K.E. Trudy po raketnoy tekhnike [Works on rocket technology].
Moscow, Oborongiz Publ., 1947, 368 p.
[8] Tsander F.A. Problemy poleta pri pomoshchi reaktivnykh apparatov [Problems
of flight by jet propulsion: interplanetary flights]. Moscow, Gos. Aviats i avto-
trakt Publ., 1932, 75 p.
[9] Lawden D.F. Optimal Trajectories for Space Navigation. London, Butterworths,
1963. [In Russ.: Louden D.F. Optimalnye traektorii dlya kosmicheskoy navi-
gatsii. Moscow, Mir Publ., 1966, 152 p.]