519.6 Optimization of low–mass satellites flight from Earth orbit to Mars orbit using ion engines

Mozzhorina T. Y. (Bauman Moscow State Technical University), Чуванова Л. О. (Bauman Moscow State Technical University)

ION THRUSTERS, THE FALSE POSITION METHOD, BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS, OPTIMAL CONTROL, PONTRYAGIN'S MAXIMUM PRINCIPLE, FLIGHT BETWEEN THE ORBITS OF THE EARTH AND MARS


doi: 10.18698/2309-3684-2021-2-5467


In this paper, optimization of the transfer of a low–mass satellite from Earth orbit to Mars orbit using ion thrusters is considered. The ion engine allows you to minimize fuel consumption and accelerate the spacecraft to fairly high speeds far from the planets of the solar system. The heliocentric section of the flight is subject to consideration. The task is to minimize the flight time. The following assumptions are made in the work: the orbits of the Earth and Mars are circular and lying in the same plane. The angle between the tangential velocity of the spacecraft in the heliocentric system and the direction of thrust action is selected as a control. When compiling the optimization algorithm, the Pontryagin maximum principle was used, which leads the optimization problem of a functional to a boundary value problem for a system of ordinary differential equations. The solution to the boundary value problem was found by one of the numerical methods — the false position method, which gives the most accurate results. The analysis of the results obtained is carried out and a comparison with the data obtained earlier in similar calculations by foreign authors by another numerical solution method is carried out. The conclusion is made about the efficiency of the false position method when solving such problems.


Polytech. Electric rocket engine [Politekh. Elektricheskij raketnyj dvigatel'] [Electronic resource]. Access mode: https://polytech.bm.digital/ontology/368214095364644864/elektricheskij-raketnyij-dvigatel (accessed: 05.03.2021).
Hajtek. Ionnaya tyaga: kak chelovechestvo ispol'zuet elektricheskie dvigateli dlya polyotov v kosmos [Hi–tech. Ion thrust: How humanity uses electric motors to fly into space] [Electronic resource]. Access mode: https://hightech.fm/2019/09/24/ion-space (accessed: 20.03.2021).
Zharkov V.N., Moroz V.I. Pochemu Mars? [Why Mars?] Priroda [Nature], 2000, no.6, pp.58–67.
Russian news agency. Issledovanie Marsa kosmicheskimi apparatami. Dos'e [Exploration of Mars by spacecraft. Dossier] [Electronic resource]. Access mode: https://tass.ru/info/5178916 (accessed: 21.03.2021).
Morshneva I.V., Ovchinnikova S.N. Chislennoe reshenie kraevyh zadach dlya obyknovennyh differencial'nyh uravnenij. Metod strel'by. Metodicheskie ukazaniya dlya studentov 3 i 4 kursov mekhmata [Numerical solution of boundary value problems for ordinary differential equations. The method of shooting. Methodological guidelines for students of the 3rd and 4th courses of mehmat]. Rostov–on–Don, UPL RSU Publ., 2003, 29 p.
Nogin V.D. Vvedenie v optimal'noe upravlenie. Uchebno–metodicheskoe posobie [Introduction to optimal control. Educational and methodical manual]. Saint Petersburg, UTAS Publ., 2008, 92 p.
Grigoriev K.G., Grigoriev I.S., Zapletin M.P. Praktikum po chislennym metodam v zadachah optimal'nogo upravleniya (dopolnenie 1) [Workshop on numerical methods in optimal control problems (Supplement 1)]. Moscow, MSU Publ., 2007, 184 p.
Lietmann G. Optimization techniques: with applications to aerospace systems. Academic Press, 1962, 453 p.
Equity.today. Portal o finansovyh rynkah. Skol'ko letet' s Zemli do Marsa [Equity.today. Portal about financial markets. How long to fly from Earth to Mars] [Electronic resource]. Access mode: https://equity.today/polet-na-mars.html (accessed: 21.03.2021).
Krainov A.Yu., Moiseeva K.M. Chislennye metody resheniya kraevyh zadach dlya obyknovennyh differencial'nyh uravnenij. Uchebnoe posobie [Numerical methods for solving boundary value problems for ordinary differential equations. Study guide]. Tomsk, STT, 2016, 44 p.
Akhmerov R.R., Sadovsky B.N. Ocherki po teorii obyknovennyh differencial'nyh uravnenij [Essays on the theory of ordinary differential equations] [Electronic resource]. Access mode: http://www.nsc.ru/rus/textbooks/akhmerov/ode _unicode/index.html (accessed: 21.03.2021).
Mudrov A.E. Chislennye metody dlya PEVM na yazykah Bejsik, Fortran i Paskal' [Numerical methods for PCs in Basic, Fortran and Pascal languages]. Tomsk, MP "RASKO", 1991, 272 p.
Demidovich B.P., Maron I.A., Shuvalova E.Z. Chislennye metody analiza. Priblizhenie funkcij, differencial'nye i integral'nye uravneniya [Numerical methods of analysis. Approximation of functions, differential and integral equations]. Moscow, Nauka Publ., 1967, 368 p.
Aslamova V.S., Kolmogorov A.G., Stupakova N.N. Vychislitel'naya matematika. Chast' pervaya. Uchebnoe posobie dlya studentov dnevnogo i zaochnogo obucheniya tekhnicheskih i himiko–tekhnologicheskih special'nostej [Computational mathematics. Part I. A textbook for full–time and part–time students of technical and chemical–technological specialties]. Angarsk, ASTA Publ., 2003, 82 p.
Fedorenko R.P. Priblizhennoe reshenie zadach optimal'nogo upravleniya [Approximate solution of optimal control problems]. Moscow, Nauka Publ., 1978, 486 p.
Kuksenko S.P., Gazizov T.R. Iteracionnye metody resheniya sistemy linejnyh algebraicheskih uravnenij s plotnoj matricej [Iterative methods for solving a system of linear algebraic equations with a dense matrix]. Tomsk, TSU Publ., 2007, 208 p.
Mozzhorina T.Yu. Numerical solution to problems of optimal control with switching by means of the shooting method. Mathematical modeling and Computational Methods, 2017, no.2 (14), pp.94–106.


Мозжорина Т.Ю., Чуванова Л.О. Моделирование и оптимизация перелета спутников малой массы с Земной орбиты на орбиту Марса с помощью ионных двигателей. Математическое моделирование и численные методы, 2021, № 2, с. 54–67.



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