Параметры публикации

Тип публикации: 

Статья в журнале/сборнике


Verification of the second-order optimality сonditions in the modeling of the SC expedition with the returning to the Earth based on two Lambert’s problems solving



Наименование источника: 

  • Advances in the Astronautical Sciences

Обозначение и номер тома: 

vol. 161


  • San Diego


  • Univelt

Год издания: 



The problem of optimization of the interplanetary spacecraft (SC) expedition from the Earth to Mars with a returning to the Earth is considered. The SC starts since 2020 to 2030 years, duration of the expedition is limited by 1500 days. Two trajectories of interplanetary flights of this mission are approximated by Keplerian orbits and obtained as a solution of Lambert's problems. The gravitational field of the Sun attraction forces is considered to be central Newtonian, the remaining bodies are assumed to be non-attracting material points. Ephemerides DE424 from the NASA SPICE toolkit [1] is used for high precision calculation of the position and velocity of the Earth and Mars in the Solar system at any moment. By the way a lot of factors affecting the trajectory of each body, such as: attraction of the 300 largest bodies of the Solar system, relativistic effects, solar wind are taking into account. In this case, each Lambert problem is given by two moments of time - moments of start and finish of the SC. The position and velocity of the SC coincide with the position and velocity of the center of mass of the body: the Earth or Mars at these instants of time. Lambert's problems are solved numerically with the use of universal variable, Shtump’s functions [2] and Newton's method [3]. Optimization by the time of start and finish of the SC is carried out by the gradient method [4], starting from grid nodes of the problem parameters field. This problem is multiextremal. So finding of a global minimum and verifying the fulfillment of the second-order optimality conditions [5] on the founded solutions are interesting both. A matrix of the second derivatives relative to 4 parameters of the problem is written out to verify the second-order conditions. The Sylvester’s criterion is used - the major minors of the calculated matrixes are verified. To calculate the determinants of matrices, the direct part of Gauss method with the choice of the major element is used. The optimal numerical derivative calculation step is chosen with the use of theory of difference schemes of numerical methods during the problem solution constructing. The problem is solved numerically, 29 local minima of the Earth to Mars flights with the launch in considered time were found and examined. The work is an additional research and development of [6]. REFERENCES [1] http://naif.jpl.nasa.gov/naif [2] Sukhanov A.A. Astrodynamics. Moscow, Seriya “Mekhanika, upravleniye, informatika”, Rotaprint IKI RAN, 2000, 202 p. [3] Fedorenko R.P. Introduction to computational physics. Moscow, ed. Mosk. fiz.-tekhn. in-ta, 1994, 526 p. [4] Vasiliev F.P. Methods of optimization. Мoscow, Factorial press, 2002, 824p. [5] Alekseev V.M., Tihomirov V.M., Fomin S.V. Optimal Control. Moscow, ed. Nauka, 1979, 430 p. [6] Samokhin A.S. Optimization of expedition to Phobos using the impulse control and solution to Lambert problems taking into account attraction of the Earth and Mars. Moscow University Mathematics Bulletin, 2014, vol. 69, no. 2, pp. 84-87. doi: 10.3103/S0027132214020089.

Библиографическая ссылка: 

Самохин А.С., Самохина М.А. Verification of the second-order optimality сonditions in the modeling of the SC expedition with the returning to the Earth based on two Lambert’s problems solving // Advances in the Astronautical Sciences. 2018. vol. 161. С. 843-862.