The methodology for determination of optimal flight path with low thrust between the low-Earth and low-Lunar orbits
Keldysh Research Centre, 8, Onezhskaya str., Moscow, 125438, Russia
AbstractThe optimization of flight trajectories to the Moon of spacecraft with electric propulsion system is the subject of this study.
The problem of flight of spacecraft with electric propulsion system between the low-Earth and low-Lunar orbit in condition of minimal time with searching for program of optimal thrust vector control in terms of minimizing consumption of characteristic velocity is considered in this work. The main aim of this work was to develop a methodology for determination of optimal trajectory for interorbital flight between low-Earth and low-Lunar orbits with low thrust in condition of restricted problem of three bodies.
Optimal flight trajectory was determined with taking into account the gravitational fields of the Earth and the Moon (restricted problem of three-bodies) and without the division into sections. Direction of thrust vector was defined by the law obtained using the maximum principle of Pontryagin. Thrust and density impulse was assumed as constant for the total trajectory. The continuous working mode of electric propulsions has been considered. Moon ephemerides were determined according to the EPM 2008 model developed by "Institute of Applied Astronomy RAS".
Methodical software for calculation of optimal flight trajectories with low thrust between low-Earth and low-lunar orbits in condition of restricted three-body problem without dividing the trajectory on sections has been developed. The estimation of characteristic velocity consumption for flight from the Moon artificial satellite orbit to the Earth artificial satellite orbit has been received.
Results of this work can be used for effectively solution to the transportation problems connected with payloads delivery into low-lunar orbit in the case of decision about step-by-step deployment and operation of ongoing lunar base (including with astronauts) with the respect to investigation and exploration of the Moon.
Keywords:electric propulsion engine, low thrust, flight Earth - Moon
- Grishin S. D., Zakharov Yu. A., Odelevskii V. K. Proektirovanie kosmicheskikh apparatov s dvigatelyami maloi tyagi (Spacecraft design with low trust), Moscow, Mashinostroenie, 1990, 224 p.
- Ivashkin V. V, Petukhov V. G. Available at: http://library.keldysh.ru/preprint.asp?id=2008-81, 2008. no. 81, 32 p.
- R. Bettin. Navedenie v kosmose (Astronautical guidance), Moscow, Mashinostroenie, 1966, 448 p.
- Casaregola C., Geurts K., Pergola P., Biagioni L., Andrenucci M. Mission Analysis and Architecture Definition for a Small Electric Propulsion Transfer Module to the Moon,
43rd AIAA_ASME_SAE_ASEE Joint Propulsion Conference and Exhibit, 2007, 10 p.
- Kluever C.A., Pierson B.L. Optimal Earth-Moon Trajectories Using Nuclear Electric Propulsion, Journal of Guidance, Control, and Dynamics, 1997, vol. 20, no. 2, pp. 239–245.
- Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E. F. Matematicheskaya teoriya optimal'nykh protsessov (Mathematical theory of optimal process), Moscow, Nauka, 1983, 392 p.
- Braison A., Kho Yu-Shi. Prikladnaya teoriya optimal'nogo upravleniya (Applied theory of optimal control), Moscow, Mir, 1972, 544 p.
- Pit'eva E. V. Efemeridy EPM2008. Institute of Applied Astronomy RAS, available at: ftp://quasar.ipa.nw.ru/incoming/EPM2008/.
- Petukhov V. G. Kosmicheskie issledovaniya, 2011, vol. 49, no. 2, pp. 128–137
- Fehlberg E. Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize Control, NASA TR R–287, 1968, 88 p.
- Khairer E., Nersett S., Vanner G. Reshenie obyknovennykh differentsial'nykh uravnenii. Nezhestkie zadachi (Decision of regular differential equation. Nonrigid problem), Moscow, Mir, 1990, 512 p.
- Butcher J.C. Numerical Methods for Ordinary Differential Equations, England, John Wiley&Sons, 2003, 425 p.
- Shamanskii E.V. Metody chislennogo resheniya kraevykh zadach na ETsVM. Nelineinye kraevye zadachi i zadachi na sobstvennye znacheniya dlya differentsial'nykh uravnenii (Numerical methods of boundary problems on EDC. Part II Nonlinear boundary problems and eigenvalue problems for differential equations), Kiev, Naukova dumka, 1966, 244 p.
- Aoki M. Vvedenie v metody optimizatsii. Osnovy i prilozheniya nelineinogo programmirovaniya (Introduction to optimization techniques. Fundamentals and applications of nonlinear programming), Moscow, Nauka, 1977, 344 p.
- Forsait Dzh., Mal'kol'm M., Mouler K. Mashinnye metody matematicheskikh vychislenii (Computer methods for mathematical computations), Moscow, Mir, 1980, 279 p.
- Akhmetshin R. Z. Kosmicheskie issledovaniya, 2004, vol. 42, no. 3, pp. 248–259.