# Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

**Authors:**Shorikov A.F.^{1}, Goranov A.Y.^{2}^{,3}-
**Affiliations:**- N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences (IMM UB RAS)
- Ural Federal University named after the first President of Russia B.N. Yeltsin
- Scientific and Production Association of Automatics named after Academician N.A. Semikhatov

**Issue:**Vol 20, No 1 (2021)**Pages:**46-64**Section:**AIRCRAFT AND SPACE ROCKET ENGINEERING**URL:**https://journals.ssau.ru/vestnik/article/view/8635**DOI:**https://doi.org/10.18287/2541-7533-2021-20-1-46-64

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## Full Text

## Abstract

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

## About the authors

### A. F. Shorikov

N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences (IMM UB RAS)
**Author for correspondence.**

Email: afshorikov@mail.ru

ORCID iD: 0000-0003-1255-0862

Doctor of Science (Phys. & Math.), Professor

Russian Federation### A. Yu. Goranov

Ural Federal University named after the first President of Russia B.N. Yeltsin; Scientific and Production Association of Automatics named after Academician N.A. Semikhatov
Email: goranovayu@mail.ru

ORCID iD: 0000-0002-1911-8012

Design Engineer

Russian Federation## References

- Appazov R.F., Sytin O.G. Metody proektirovaniya traektoriy nositeley i sputnikov Zemli [Methods of designing trajectories of launch vehicles and Earth satellites]. Moscow: Nauka Publ., 1987. 440 p.
- Ermilov Yu.A., Ivanova E.E., Pantyushin S.V. Upravlenie sblizheniem kosmicheskikh apparatov [Spacecraft rendezvous control]. Moscow: Nauka Publ., 1977. 448 p.
- Ivanov N.M., Lysenko L.N., Martynov A.I. Metody teorii sistem v zadachakh upravleniya kosmicheskim apparatom [Systems theory method in spacecraft control problems]. Moscow: Mashinostroenie Publ., 1981. 255 p.
- Lebedev A.A., Sokolov V.B. Vstrecha na orbite [Orbital rendezvous]. Moscow: Mashinostroenie Publ., 1969. 67 p.
- Krasovskiy N.N. Teoriya upravleniya dvizheniem [Motion control theory]. Moscow: Nauka Publ., 1968. 476 p.
- Krasovskiy N.N. Igrovye zadachi o vstreche dvizheniy [Game-theory problems of space traffic rendezvous]. Moscow: Nauka Publ., 1970. 420 p.
- Krasovskiy N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry [Positional differential games]. Moscow: Nauka Publ., 1974. 456 p.
- Chernous'ko F.L., Melikyan A.A. Igrovye zadachi upravleniya i poiska [Game-theoy problems of control and search]. Moscow: Nauka Publ., 1978. 270 p.
- Tyulyukin V.A., Shorikov A.F. Algorithm for solving terminal control-problems for a linear discrete system. Automation and Remote Control. 1993. V. 54, no. 4. P. 632-643.
- Tyulyukin V.A., Shorikov A.F. Ob odnom algoritme postroeniya oblasti dostizhimosti lineynoy upravlyaemoy sistemy. V sb.: «Negladkie Zadachi Optimizatsii i Upravlenie». Sverdlovsk: UrO AN SSSR Publ., 1988. P. 55-61. (In Russ.)
- Shorikov A.F. Minimaksnoe otsenivanie i upravleni v diskretnykh dinamicheskikh sistemakh [Minimax estimation and control in discrete dynamical systems]. Ekaterinburg: Ural University Publ., 1997. 242 p.
- Shorikov A.F. Algoritm resheniya zadachi optimal'nogo terminal'nogo upravleniya v lineynykh diskretnykh dinamicheskikh sistemakh. Sb. nauchnykh trudov «Informatsionnye tekhnologii v ekonomike: teoriya, modeli i metody». Yekaterinburg: Ural State University of Economics Publ, 2005. P. 119-138. (In Russ.)
- Shorikov A.F., Tyulyukin V.A. Description of the package of computer subroutines for simulation of solving the problem of a posterori minimax estimation. Izvestiya Ural'skogo Gosudarstvennogo Ekonomicheskogo Universiteta. 1999. No. 2 P. 36-49. (In Russ.)
- Goranov A.Y., Shorikov A.F. Modifidied general recursion algebraic method of the linear control systems reachable sets computation. Sb. trudov Shestoy Mezhdunarodnoy nauchnoy konferentsii «Informatsionnye tekhnologii i sistemy» (01-05 marta 2017 g., Bannoe). Chelyabinsk: Chelyabinsk State University Publ., 2017. P. 87-92. (In Russ.)
- Chernikov S.N. Lineynye neravenstva [Linear inequalities]. Moscow: Nauka Publ., 1968. 488 p.