Abstract
We consider optimization problems for spatial trajectories of space flights for space vehicles with an additional fuel tank from a low circular orbit of a man-made Earth satellite to a geotransitional orbit. The motion of the space vehicle is controlled through a jet engine with bounded thrust. To discard the additional tank, the vehicle needs to switch off its engine and spend a certain time. The mass of the discarded tank is assumed to be proportional to the mass of the expended fuel, and the mass of the engine and additional constructions is proportional to the thrust. We minimize the value of the impulse needed to transfer to the geostationary orbit for a given useful mass, or, which is the same, maximize the useful mass for a given value of this impulse. In the first part of the paper, we consider in detail the history of this problem and computational schemes that can be used to solve this kind of problems.
Similar content being viewed by others
References
Tsentr obrabotki i otobrazheniya poletnoi informatsii GKNPTs im. M.V. Khrunicheva. Proshedshie puski (Khrunichev Center of Processing and Visualization of Flight Information. Past Launches). http://coopi.khrunichev.ru/main.php?id=11
Grigor’ev, I.S. and Danilina, I.A., Optimization of Interorbital Three-dimensional Transfer Trajectories for Stage Spacecraft, Autom. Remote Control, 2007, vol. 68, no. 8, pp. 1372–1390.
Zenger-Bredt, I., A Study of Optimal Conditions for Vertical Flight of a Rocket with Arbitrary Number of Stages in a Gravity Field, Vopr. Raketnoi Tekh., 1954, no. 5(23), pp. 3–26.
Okhotsimskii, D.E. and Eneev, T.M., Some Variational Problems Related to Launching a Man-Made Earth Satellite, Usp. Fiz. Nauk, 1957, vol. 63, no. 1a, pp. 5–32.
Troitskii, V.A., Motion Optimization for a Multi-Stage Rocket, Prikl. Mat. Mekh., 1965, vol. 29, no. 4, pp. 745–750.
Troitskii, V.A., Variational Optimization Problems for Control Processes for Equations with Discontinuous Right-Hand Sides, Prikl. Mat. Mekh., 1962, vol. 26, no. 2, pp. 233–246.
Troitskii, V.A., On Optimal Modes of Flight for Multi-Stage Rockets, Kosm. Issled., 1967, vol. 5, no. 2, pp. 176–183.
Bolonkin, A.A., Optimizing Trajectories in Multi-Stage Rockets, in Studies in Flight Dynamics, no. 1, Moscow: Mashinostroenie, 1965.
Kosmodem’yanskii, V.A., On One Kind of Variational Problems, Prikl. Mat. Mekh., 1963, vol. 27, no. 6, pp. 1111–1116.
Kosmodem’yanskii, V.A., On the Theory of Multistage Rockets, Dokl. Akad. Nauk USSR, 1964, vol. 156, no. 2, pp. 286–289.
Kosmodem’yanskii, V.A., On the Computations of Composite Rockets, Inzhenernyi Zh., 1964, vol. 4, no. 2, pp. 219–224.
Kosmodem’yanskii, V.A., Necessary Variational Calculus Conditions for One Problem of Boltz–Meyer Type, Prikl. Mat. Mekh., 1965, vol. 29, no. 2, pp. 368–372.
Kosmodem’yanskii, V.A., Sufficient Absolute Extremum Conditions in One Variational Problem of Boltz–Meyer Type, Prikl. Mat. Mekh., 1966, vol. 30, no. 3, pp. 599–604.
Kosmodem’yanskii, V.A., Optimal Composition of Stages in a Composite Rocket, Mekh. Tverd. Tela, 1972, no. 1, pp. 25–29.
Kosmodem’yanskii, V.A., On Optimization of Parameters of Composite Rockets, Mekh. Tverd. Tela, 1980, no. 5, pp. 36–41.
Kosmodem’yanskii, V.A., On the Method of Computing the Optimal Program of Stage-Wise Thrust, Mekh. Tverd. Tela, 1987, no. 4, pp. 17–22.
Velichenko, V.V., Optimal Control for Composite Systems, Dokl. Akad. Nauk USSR, 1967, vol. 176, no. 4, pp. 754–756.
Velichenko, V.V., Optimal Control Problems for Equations with Discontinuous Right-Hand Sides, Autom. Remote Control, 1966, vol. 27, no. 7, pp. 1153–1164.
Velichenko, V.V., Optimality Conditions in Control Problems with Intermediate Conditions, Dokl. Akad. Nauk USSR, 1967, vol. 174, no. 5, pp. 1011–1013.
Moiseenko, V.P., Optimization of a Multistage Spacecraft, Tr. TsAGI, 1971, no. 1295.
Moiseenko, V.P., On One Approach to Solving Discontinuous Variational Problems, in Issled. Oper., Moscow: VTs AN SSSR, 1974, no. 4, pp. 146–162.
Medvedev, V.A. and Rozova, V.N., Optimal Control of Stepped Systems, Autom. Remote Control, 1972, vol.33, no. 3, part 1, pp. 359–366.
Fedorenko, R.P., Priblizhennoe reshenie zadach optimal’nogo upravleniya (Approximate Solution of Optimal Control Problems), Moscow: Nauka, 1978.
Fedorenko, R.P., Vvedenie v vychislitel’nuyu fiziku (Introduction to Computational Physics), Moscow: MFTI, 1994.
Ashchepkov, L.T., Optimal Control for Systems with Intermediate Conditions, Prikl. Mat. Mekh., 1981, vol. 45, no. 2, pp. 215–222.
Ashchepkov, L.T., Optimal’noe upravlenie razryvnymi sistemami (Optimal Control for Discontinuous Systems), Novosibirsk: Nauka, Siberia Branch, 1987.
Ashchepkov, L.T., Belov, B.I., Bulatov, V.P., et al., Metody resheniya zadach matematicheskogo programmirovaniya i optimal’nogo upravleniya (Methods for Solving Mathematical Programming and Optimal Control Problems), Novosibirsk: Nauka, Siberia Branch, 1984.
Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V., Optimal’noe upravlenie (Optimal Control), Moscow: Nauka, 1979.
Grigor’ev, I.S. and Grigor’ev, K.G., On the Maximum Principle Conditions in Optimal Control Problems for a Collection of Dynamical Systems and Their Applications to Solving Optimal Control Problems for the Motion of Spacecraft, Kosm. Issled., 2003, vol. 41, no. 3, pp. 307–331.
Grigor’ev, K.G., Zapletin, M.P., and Silaev, D.A., Optimal Motion of a Spacecraft from the Moon Surface to Its Satellite’s Circular Orbit, Kosm. Issled., 1991, vol. 29, no. 5, pp. 695–704.
Grigor’ev, K.G., Zapletina, E.V., and Zapletin, M.P., Optimal Soft Landing of a Spacecraft on the Moon Surface from a Circular Orbit of Its Satellite, Kosm. Issled., 1992, vol. 30, no. 2, pp. 203–211.
Grigor’ev, K.G., Zapletina, E.V., and Zapletin, M.P., Optimal Motion of a Spacecraft from the Moon Surface to a Given Point of Its Man-Made Satellite’s Circular Orbit, Kosm. Issled., 1992, vol. 30, no. 3, pp. 321–332.
Grigor’ev, K.G., Zapletina, E.V., and Zapletin, M.P., Optimal Soft Landing of a Spacecraft from a Circular Orbit of a Man-Made Moon Satellite to a Given Point on Its Surface, Kosm. Issled., 1992, vol. 30, no. 4, pp. 483–494.
Grigor’ev, K.G. and Zapletin, M.P., A Numerical Solution for Boundary Problems of the Maximum Principle in Optimization Problems of Space Flight Dynamics, Izv. Ross. Akad. Nauk, Tekh. Kibern., 1993, no. 1, pp. 91–96.
Grigor’ev, K.G., Zapletina, E.V., and Zapletin, M.P., Optimal Spatial Flights of a Spacecraft between Moon Surface and the Orbit of Its Man-Made Satellite, Kosm. Issled., 1993, vol. 31, no. 5, pp. 34–52.
Grigor’ev, K.G., On Spacecraft Maneuvers with Minimal Mass Expenditure and Bounded Time, Kosm. Issled., 1994, vol. 32, no. 2, pp. 45–60.
Grigor’ev, I.S. and Grigor’ev, K.G., Optimal Trajectories of Flights of a Spacecraft with Jet Engines of Large Bounded Thrust between the Orbit of a Man-Made Earth Satellite and the Moon, Kosm. Issled., 1994, vol. 3, no. 6, pp. 108–129.
Grigor’ev, K.G. and Fedyna, A.V., Optimal Flights of Spacecraft with Jet Engines of Large Bounded Thrust between Complanar Circular Orbits, Kosm. Issled., 1995, vol. 33, no. 4, pp. 403–416.
Grigor’ev, I.S. and Grigor’ev, K.G., Optimal Trajectories for the Return of a Spacecraft with Jet Engines of Large Bounded Thrust from the Moon to the Earth, Kosm. Issled., 1995, vol. 33, no. 5, pp. 513–532.
Grigor’ev, K.G. and Grigor’ev, I.S., A Study of Optimal Spatial Trajectories for Flights of Spacecraft with Jet Engines of Large Bounded Thrust between the Orbits of Man-Made Earth and Moon Satellites, Kosm. Issled., 1997, vol. 35, no. 1, pp. 52–75.
Grigor’ev, K.G. and Zapletin, M.P., On Vertical Launch in Optimization Problems of Rocket Dynamics, Kosm. Issled., 1997, vol. 35, no. 4, pp. 363–377.
Conway, B.A., A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems, J. Optim. Theory Appl., 2012, vol. 152, no. 2, pp. 271–306. http://dx.doi.org/10.1007/s10957-011-9918-z
Pontani, M., Simple Method for Performance Evaluation of Multistage Rockets, Acta Astronautica, 2014, vol. 94, no. 2, pp. 852–864. http://dx.doi.org/10.1016/j.actaastro.2013.09.013
Pontani, M. and Cecchetti, G., Ascent Trajectories of Multistage Launch Vehicles: Numerical Optimization with Second-Order Conditions Verification, ISRN Oper. Res., 2013, vol. 2013, Article ID 498765. http://dx.doi.org/10.1155/2013/498765
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.S. Grigoriev, I.A. Danilina, 2017, published in Avtomatika i Telemekhanika, 2017, No. 12, pp. 131–140.
This paper was recommended for publication by A.P. Kurdyukov, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Grigoriev, I.S., Danilina, I.A. Optimizing flight trajectories for space vehicles with an additional fuel tank. I. Autom Remote Control 78, 2203–2210 (2017). https://doi.org/10.1134/S0005117917120086
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917120086