The mathematical model of the motion of the aircraft and ballistic algorithms for calculating the nominal and perturbed motion parameters ballistic aircraft


Vinogradov A. V.1*, Borukaeva A. O.2**, Berdikov P. G.2***

1. Military Academy of Strategic Rocket Troops named after Peter the Great, SRTMA, 8, Karbysheva str., Balashikha, Moscow region, 143900, Russia
2. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia



With a view to aircraft and rocket engineering rapid development, considerable complication of aircraft design and cost intensity of its developing, the effectiveness significance at the early stages of design, when aircraft basic structural characteristics are being selected and nominal parameters of its movement are determined, increases.

Occurrence of small asymmetries, as well as small thrust eccentricities for aircraft with power pack, due to technological errors while aircraft components fabrication and their assembly, is possible during production process of various types of aircraft.

The article deals with the algorithms of nominal and perturbed motion parameters of ballistic aircraft while operational change of target marking in flight. The presented algorithms are based on an aircraft mathematical model, which, in its turn, allows ensuring achieving the target movable point in condition of operational changes of target marking.

The applied directedness of the guidance theory is confirmed by the fact guidance system, materialized as the result of the guidance theory application to solving specific problems concerning control of the aircraft center of mass, is necessarily a part of the ballistic aircraft control system.

Equations of motion (basically, Cauchy differential equations) and a technique for differential equation solving form the basis of the mathematical model. The mathematical model complements gravitational field characteristics, atmosphere characteristics, the aircraft dynamic characteristics, and its power pack characteristics.

In the general case, the problem of mathematical modelling of the elastic flying machine dynamics is split into, at least, two more problems. The first one is direct modelling of the elastic flying vehicle behavior, determined by the interacting forces, and the second is accounting for action of flowing forces and moments, stipulated by the elastic flying vehicle deformation, and control system operation.

The presented article considers theoretical basics of the approach to the solution of the set problem, based of real aircraft structure substitution by the equivalent scheme and its realization for the rockets as the simplest from the sketchness viewpoint class of aircraft. Accounting for the aerodynamic forces and moments herewith, caused by the aircraft elastic deformation, is realized based on the simplified stationary hypothesis.


ballistic flying vehicle, calculation algorithm, guidance


  1. Titkov O.S. Aviatsionnye sistemy, 2017, no. 7, pp. 32 – 37.

  2. Lysenko L.N. Voenno-promyshlennyi kur’er, 2005, no. 1 (61), pp. 5.

  3. Fedosov E.A. Razrabotka v SShA kontseptsii i elementov sistemy PRO aviatsionnogo bazirovaniya (Development of concept and elements of aviation-based missile defense system in the USA), Moscow, GosNIIAS, 2014, 63 p.

  4. Ashurbeili I.R. Sredstva vozdushno-kosmicheskogo napadeniya i vozdushno-kosmicheskoi oborony. Sostoyanie i razvitie (Facilities of air-space attack and air-space defense. State-of-the-art and Development), Moscow, Planeta, 2017, 336 p.

  5. Gorchenko L.D., Evseev I.V., Mishin A.A. Voprosy elektromekhaniki. Trudy VNIIEM, 2013, vol. 137, no. 6, pp. 23 – 30.

  6. Panov V.V., Gorchitsa G.I., Balyko Yu.P. et al. Formirovanie ratsional’nogo oblika perspektivnykh aviatsionnykh sistem i kompleksov (Formation of rational appearance of prospective aviation systems and complexes), Moscow, Mashinostroenie, 2010, 157 p.

  7. Arkhangel’skii I.I., Afanas’ev P.P., Golubev I.S. et al. Proektirovanie zenitnykh upravlyaemykh raket (Designing anti-aircraft guided missiles), Moscow, Izd-vo MAI, 2001, 730 p.

  8. Neupokoev F.K. Strel’ba zenitnymi raketami (Shooting anti-aircraft missile), Moscow, Voenizdat, 1991, 343 p.

  9. Orkin B.D., Orkin S.D., D’yachuk A.K. Trudy MAI, 2012, no. 62, URL:

  10. Obnosov B.V., Daneko A.I., Zakharov I.V., Trubnikov A.A., Reshetnikov D.A. Trudy MAI, 2012, no. 62, URL:

  11. Aiden K., Fibel’man Kh., Kramer M. Apparatnye sredstva RS (RS Hardware), Saint Petersburg, Izd-vo BHV, 1996, 544 p.

  12. Novye sredstva vedeniya radioelektronnoi bor’by, Zarubezhnoe voennoe obozrenie, 2014, no. 1, URL:

  13. Filatov V.I. 3-i Mezhdunarodnyi nauchnyi simpozium “Cpetsial’naya svyaz’ i bezopasnost’ informatsii: tekhnologii, upravlenie, ekonomika”: sbornik trudov, Moscow, Rusains, 2017, pp. 109 – 111.

  14. Akulichev A.B. Voennaya mysl’, 2006, no. 9, pp. 76 – 80.

  15. Davydov G.B. Informatsiya i seti svyazi (Information and communication networks), Moscow, Nauka, 1984, 128 p.

  16. Krushevskii A.V. Teoriya igr (Theory of Games), Kiev, Vishcha shkola, 1977, 216 p.

  17. Penin P.I. Sistemy peredachi tsifrovoi informatsii (Digital Information Transmission Systems), Moscow, Sovetskoe radio, 1976, 364 p.

  18. Khokhlachev E.N. Vestnik NPO im. S.A. Lavochkina, 2010, no. 1 (3), pp. 36 – 43.

  19. Kuznetsov I.N. Informatsiya: sbor, zashchita, analiz (Information: collection, protection, analysis), Moscow, Izd -vo Yauza, 2001, 107 p.

  20. D’yakonov V.P., Obraztsov A.A., Smerdov V.Yu. Elektronnye sredstva svyazi (Electronic communication means), Moscow, SOLON-Press, 2009, 430 p.

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