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Contract number
Time span of the project
Head of the laboratory

As of 01.11.2022

Number of staff members
scientific publications
Objects of intellectual property
General information
Name of the project: Space flight mechanics

Goals and objectives
Project objective: Development of theoretical basics and methods of space flight mechanics necessary for modernization of existing and creation of principally new models of rocket and space units for design of prospective space systems

The practical value of the study

Scientific results:

We have developed methodologies and mathematical software for the optimization of perturbed trajectories with the use of complex-dual numbers numbers. As a result of the conducted work, we produced a software library that allows to compute first and mixed second derivatives of real functions of several variables with a precision close to the precision of the functions. Thanks to the employed technology of object-oriented programming, we could minimize the required amount of modifications of the code used for computing the value of some function in the real region to transform it into code that, apart from computing the value of the function, its first and mixed second derivatives can also be computed. The use of complex-dual number with a vector complex-dual part as the main data type allows to compute an array of first derivatives of this function with a single computation of the function in the complex-dual domain from the elements of a specified vector argument and an array of mixed second derivatives from this vector argument and from an additional scalar argument. This feature of the implementation of the algorithm of computation of derivatives and the library based on it allows to reduce the required computational workload in typical problems of optimal control in comparison with methods relying on multicomplex or hyperdual numbers for the computation of second derivatives.

A new method has been proposed for the diagnostics of the existence of  multi-orbits trajectories of spacecraft with limited thrust by solving the problem of thrust minimization. Minimum-thrust trajectories and the value of the minimum thrust is computed by continuing from the optimal trajectory of a spacecraft with an ideally-controlled thruster of limited power output. The proposed approach allows to automate the process of computation of the optimal trajectory with limited thrust with a fixed angular range and the optimal flight time. In the process of computations, the existence of a solution is tested, and if a solution does not exist, it is possible to produce recommendations for changes in source data (thrust and angular range of flight) up to values at which a solution exists. Building a family of minimum-thrust trajectories in some range of change in specific thrust allows to determine the region of the existence of a solution of a problem with limited thrust on the «specific impulse – thrust» plane. Using the developed method, we have conducted a numerical analysis of optimal trajectories with fixed angular ranges and optimal flight times. As a result of this analysis, we discovered a number of new properties of such trajectories.

We have conducted experimental work to determine the ionic impulse accommodation coefficient and the force applied to the target when it is exposed to a beam at various angles. As a result of this work, we determined that the force applied to the target within the limitations of statistical error does not depend on the angle at which the target is installed. This means that for the ionic impulse accommodation coefficient in the direction of the propagation of the ion beam it is needed to apply a value equal to 1,0. As we expected considering the components of the force applied to the target, we could observe a weak, at a level of 0,2-0,5 mN, decrease of the measured force applied to the target depending on the angle of incidence of ions. We have obtained results of an experimental research of a flow of sprayed material of the target when  it is exposed to an ion beam. We have experimentally determined the type of the main characteristics defining the flow of sprayed particles – the indicatrix of diffusion.

We have researched the dynamics of spatial motion of a passive object transported by an ion beam. A mathematical model has been developed that describes the spatial motion of  a passive object under the impact of the ion beam created by the thruster of an active spacecraft. We have developed simplified methods describing motion of an object with low asymmetry and an axisymmetric object on a geostationary orbit. A law has been developed for controlling the thrust of an engine creating the ion flow of the thruster that ensures the stabilization of the spatial motion of a passive object in the regular precession mode. For an axisymmetric object on a geostationary orbit we have numerically defined the zones where the proposed control ensures the asymptotic stability of the equilibrium position in terms of angle. We have conducted an analysis of the impact of the parameters of the system on motion of a passive object. We introduced a dimensionless parameter describing the ratio of the ionic and gravitational moments for the simplified system we have built bifurcation diagrams describing the locations of equilibrium points of an unperturbed system depending on the introduced parameter and the location of the center of mass. We have conducted a numerical modeling of the impact of the oscillation mode of a passive object on the value of the force transferred by the ion beam. The oscillation modes that are most favorable for transpiration have been determined.

The Laboratory has analyzed the peculiarities of mathematical models within which promising types of motion and deployment orbits of near-Moon communication systems and transportation infrastructure are determined. We have developed methods for designing halo orbits as a single-parameter family of trajectories, designing quasi-halo-orbits as a trajectory of everywhere dense torus winding in the circular restricted three body problem and as bounded quasi-periodic orbits in a pertirbed (ephemerides) model generated by halo orbits of the circular restricted three body problem. We have proposed approaches to solving problems of supporting motion along a quasi-halo orbit in the perturbed model of motion in the case of impulse control for spacecraft with an high-thrust engine and continuous control for spacecraft and ideally-controlled low-thrust engine.

Our researchers have developed a method for computing closed trajectories of a reusable space tug with an electric rocket thruster between a circular near-Earth orbit and a round near-Moon orbit or the L1 point of the Earth-Moon system with the use of quasi-optimal control with feedback. The method reduces the problem of computing a direct and reverse trajectory to one- (for a free longitude of the ascending node) or two-parameter (for a fixed longitude of the ascending node) boundary-value problem for a system of ordinary differential equations of disturbed motion with quasi-optimal control.

The Laboratory is conducting research aimed at the development of a theory and methods for computing low-energy trajectories of spacecraft with high or low thrust in the Earth-Moon system. In doing so, we are studying different approaches to the design of low-energy Lunar missions for spacecraft with high or low thrust. The main idea of designing a low-energy trajectory of a mission to the vicinity of the Moon with an insertion of the spacecraft to the low near-Moon orbit for spacecraft with high thrust has been well-known for a long time, and its essence is the use of Solar gravitational disturbances on mission trajectories in the Earth-Moon system. The main idea for designing a low-energy trajectory for a mission to the vicinity of the Moon with an insertion of the low-thrust spacecraft into the near-Moon orbit has been known for even longer time and is related to the fundamental possibility of ensuring temporary capture of the spacecraft in the restricted three body problem. Both mentioned ideas can and should be used for spacecraft with any thrust. However, for low-thrust spacecraft (with an engine that is efficient thanks to high specific impulse) the implementation of the trajectory of the passage of the «bottleneck» in the vicinity of the libration points of the Earth-Moon system and the implementation of the capture of the spacecraft by the Moon is possible even without the «assistance» of the gravitational disturbance of the trajectory by Solar gravitation. For high-thrust spacecraft, the main idea for designing a trajectory of a mission is to use Solar gravitational disturbances to «crawl through the bottleneck» in the vicinity of the L1 and L2 libration points of the Earth-Moon system and than, thanks to the same disturbances, to be temporary captured by the Moon. The developed approach to the design of Lunar trajectories of high-thrust spacecraft is based on the use of  relations of the restricted three body problem, on the ranging of the chain of the selected parameters of the flight scheme and the gradual introduction of limitations for the type of equality ensuring the solution of the transportation problem.

We have studied the problems of inserting low-thrust spacecraft into target orbits in specified orbital position, the problem of changing the orbital plane with regard to minor spacecraft functioning as part of a low-orbit  group on near-circular orbits in several orbital planes. The Laboratory has developed methods allowing to assess energy expenditure for performing typical manoeuvres: assess the expenditure of the working substance, the aggregate impulse thrust, the aggregate operation time of the propulsion unit over the period of active existence. We have conducted a design and ballistic analysis of the configurations of an orbital satellite group of connected spacecraft on near-circular solar-synchronous orbits. The results of the design and ballistic analysis  include estimates of the duration of flight of a spacecraft to operational orbits as well as the mass of the working substance (fuel) required to deploy an orbital group.

We have developed theoretical foundations and a method of optimization of multi-orbit trajectories inter-orbital flights with limitations on the dose of absorbed radiation.

We have developed new efficient numerical methods and specialized mathematical software for computing and optimizing near-Earth, Lunar and inter-planetary trajectories of spacecraft with electric propulsion units. New results have been accumulated concerning the mechanics of low-thrust space flight, including the development of mathematical models, conditions of  optimality and an analysis of the properties of optimal perturbed trajectories. A methodology has been developed for the use of the mathematical apparatus of complex dual numbers to solving problems of the optimization of low-thrust trajectories. The methodology has been implemented in a software package. Theoretical results have been obtained concerning the dynamics  of transportation of space objects in an ion beam and experimental results on the force and erosion impact of ion beams on  space objects that are important for the implementation of the concept of contactless capture and removal of space debris.

Implemented results of research:

We have received orders from the industry to perform design and ballistic analysis of advanced spacecraft and the development of ballistic mathematical software. In collaboration with the space rocket industry, we have developed and implemented schemes of insertion of  spacecraft with electric propulsion units into geostationary orbits that allowed to increase the mass of the spacecraft delivered to the target orbit load by 25 per cent with a forecasted economic effect worth tens of millions of rubles. In particular, the developed scheme has already been used to deliver the spacecraft  «Express-80», «Express-103», «Express AMU3» and «Express AMU7» developed by ISS JSC onto the geostationary orbit. We have substantiated the possibility of further increasing the mass of spacecraft with electric propulsion units launched into the geostationary orbit (by a factor of 1.5-2 when heavy launch vehicles are used and by the factor of 2-3, up to to 2-2.4 tons, when the «Soyuz-2.1B» is used). We have demonstrated the efficiency of the use of electric propulsion units for the deployment and support of the functioning of low-orbit satellite systems, systems of spacecraft on medium circular and high elliptical orbits, solutions of problems of the reserch of the Moon and the implementation of inter-planetary space missions. 

Education and career development:
  • Three Candidate of Science dissertations have been prepared and defended.
  • Employees of the Laboratory have developed and are delivering the following courses at the Moscow Aviation Institute (at the Aerospace faculty) and at the Patrice Lumumba Peoples' Friendship University of Russia (at the Engineering academy and the S.M. Nikol'skii Mathematical Institute at the Faculty Physics, Mathematics and Natural Sciences): «Fundamentals of the theory of spacecraft motion», «Ballistic problems of electric propulsion spacecraft», «Computational mechanics of space flight», «Design of the orbits of inter-planetary space missions», «Theory of motion of spacecraft», «Informatics and basics of programming», «Theory of automated control», «Mathematical theory of control». 

Other results:

The academic staff of the Laboratory features members and corresponding members of the Russian Academy of Sciences, as well as two fellows of the International Astronautical Federation.

Employees of the Laboratory participate in three Russian Science Foundation grants, in two of the grants our employees are lead researchers.

Employees of the Laboratory act as members of the program and organization committees of Russian and international conferences, such as:


Within the implementation of joint research and works, the academic staff of the Laboratory collaborates with leading universities and enterprises of Russian rocket engineering industry.

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Petukhov, V.G., Ivanyukhin, A.V., Sang Wook, W.
Joint Optimization of Control and Main Trajectory and Design Parameters of an Interplanetary Spacecraft with an Electric Propulsion System // Cosmic Research, 2019, Vol. 57, № 3, pp. 188–203, DOI: https://doi.org/10.1134/S0010952519030079
Starchenko A.
Trajectory Optimization of a Low-Thrust Geostationary Orbit Insertion for Total Ionizing Dose Decrease // Cosmic Research, 2019, Vol. 57, № 4, pp. 289-300, DOI: https://doi.org/10.1134/S0010952519040063
Ivanyukhin, A.V., Petukhov, V.G.
Low-Energy Sub-Optimal Low-Thrust Trajectories to Libration Points and Halo-Orbits // Cosmic Research, 2019, Vol. 57, № 5, pp. 378–388, DOI: https://doi.org/10.1134/S0010952519050022
Aslanov V. S., Ledkov A. S., Konstantinov M. S.
Attitude motion of a space object during its contactless ion beam transportation // Acta Astronautica, 2020, Vol. 179, pp. 359-370, DOI: https://doi.org/10.1016/j.actaastro.2020.11.017
Ivanyukhin A., Petukhov V.
Optimization of multi-revolution limited power trajectories using angular independent variable // Journal of Optimization Theory and Applications, 2021, Vol. 191, № 2, pp. 575-599, DOI: https://10.1007/s10957-021-01853-8
Obukhov V.A., Kirillov V.A., Petukhov V.G., Popov G.A., Svotina V.V., Testoyedov N.A., Usovik I.V.
Problematic issues of spacecraft development for contactless removal of space debris by ion beam // Acta Astronautica, 2021, Vol. 181, pp. 569-578, DOI: https://doi.org/10.1016/j.actaastro.2021.01.043
Aslanov V. S., Ledkov A. S., Petukhov V. G.
Spatial dynamics and attitude control during contactless ion beam transportation // Journal of Guidance, Control, and Dynamics, 2021, Vol. 44, №. 8, pp. 1572-1577, DOI: https://doi.org/10.2514/1.G005561
Petukhov V.G., Ivanyukhin A.V., Popov G.A., N. Testoyedov, Sung Wook Yoon.
Optimization of finite-thrust trajectories with fixed angular distance // Acta Astronautica, 2022, Vol. 197, pp. 354-367, DOI: https://doi.org/10.1016/j.actaastro.2021.03.012
Obukhov V.A., Kirillov V.A., Petukhov V.G., Pokryshkin A.I., Popov G.A., Svotina V.V., Testoyedov N.A., Usovik I.V.
Control of a service satellite during its mission on space debris removal from orbits with high inclination by implementation of an ion beam method // Acta Astronautica, 2022, Vol. 194, pp. 390-400, DOI: https://doi.org/10.1016/j.actaastro.2021.09.041
Ivanyukhin A.V., Petukhov V.G., Sung Wook Yoon.
Minimum-Thrust Transfers to the Moon // Cosmic Research, 2022, Vol. 60, № 6, pp. 480-489, DOI: https://doi.org/10.1134/S0010952522050033
Svotina V.V., Cherkasova M.V.
Space debris removal – Review of technologies and techniques. Flexible or virtual connection between space debris and service spacecraft // Acta Astronautica, DOI: https://doi.org/10.1016/j.actaastro.2022.09.027
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