#### 519.6 Synthesis of optimum control of vertical landing of returned space units

##### Mozzhorina T. Y. (Bauman Moscow State Technical University), Osipov V. V. (Bauman Moscow State Technical University)

###### MONTE CARLO METHOD, SOFT LANDING PROBLEM, PROBABILISTIC ANALYSIS, SPACECRAFT RETURN STAGE, OPTIMAL CONTROL, FEEDBACK CONTROL

doi: 10.18698/2309-3684-2020-2-8194

In this paper, we consider one of the possible feedback algorithms for the vertical landing of the returned first stage of the spacecraft for its reuse in the future. It is proposed to use for thrust correction not correction engines, but the main engine of the power plant of the spacecraft, which can be throttled to 60% of the maximum thrust value. A numerical experiment using the Monte Carlo method is performed to evaluate the performance of this algorithm. A soft landing is a landing with a speed of zero or no more than a few meters per second. The last section of the vertical landing is subject to investigation. The optimal program control in this problem statement in terms of minimum fuel consumption is free fall, then turning the engine on at full power until the moment of landing. It is assumed that such parameters as the speed and mass of the spacecraft's return module at an altitude of 2000 m, the specific impulse, as well as the air density and the coefficient of aerodynamic drag can be randomly deviated from the calculated values. It is assumed that these random variables are distributed according to the normal law, are independent, and their deviations from the calculated values do not exceed 1% for the engine pulse and 5% for all other variables. The landing speed is a random value for which the distribution parameters are calculated. The results of the calculation are analyzed.

[1] Official website of the company SpaceX [Electronic resource]: URL: http://www.spacex.com/falcon9
[2] Official website of the Blue origin company [Electronic resource]: URL: https://www.blueorigin.com
[3] Mozzhorina T.Yu., Osipov V.V. Numerical solution of the problem of a soft landing by false position method. Innovatsionnoye razvitiye, 2018, no. 8 (25), pp. 11–15.
[4] Fedorenko R.P. Priblizhennoye resheniye zadach optimal'nogo upravleniya [Approximate solution of optimal control problems]. Moscow, Nauka Publ., 1978, 486 p.
[5] 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.
[6] Mozzhorina T.Yu., Osipov V.V. A probable analysis of the possibility of soft landling on the last vertical part of the engine. International Journal of Applied and Fundamental Research, 2019, no. 7, pp. 136–140.
[7] Letov A.M. Matematicheskaya teoriya protsessov upravleniya [Mathematical theory of control processes]. Moscow, Nauka Publ., 1981, 256 p.
[8] Blackmore L. Autonomous Precision Landing of Space Rockets. The Bridge on Frontiers of Engineering, 2016, no. 4(46), pp. 15–20.
[9] Zhukov B.I., Likhachev V.N., Sikharulidze YU.G., Trifonov O.V., Fedotov V.P., Yaroshevskiy V.S. Kombinirovannyy algoritm upravleniya posadkoy kosmicheskogo apparata «Luna-Glob» [The combined control algorithm for the landing of the spacecraft "Luna-Glob"]. ХI Vserossiyskiy s"yezd po fundamental'nym problemam teoreticheskoy i prikladnoy mekhaniki [Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics], 2015, pp. 1395-1398.
[10] Zhukov B.I., Sazonov V.V., Sikharulidze Y.G., Tuchin A.G., Tuchin D.A., Yaroshevskii V.S., Likhachev V.N., Fedotov V.P. Comparative analysis of algorithms for lunar landing control. Cosmic Research, 2015, vol. 53, no. 6, pp. 441–448.
[11] Afanas'yev V.A., Degtyarev G.L., Meshchanov A.S., Sirazetdinov T.K. Development and studies of algorithms for control of vertical landing of fv modules. Vestnik Kazanskogo gosudarstvennogo tekhnicheskogo universiteta im. A.N. Tupoleva, 2007, no. 3, pp. 56–59.
[12] Yemel'yanova N.M., Poroskun V.I. Monte Carlo simulation of dependent random variables. Geology, geophysics and development of oil and gas fields. 2008, no. 9, pp. 45–49.

Мозжорина Т.Ю., Осипов В.В. Моделирование и синтез оптимального управления вертикальной посадкой возвращаемых космических модулей. Математическое моделирование и численные методы. 2020. № 2. с. 81–94.