On one Approach to the Solution of the Lambert Problem using the Decompositional Method of Modal Control

Abstract

A new approach to the solution of the Lambert’s problem in spaceflight mechanics is proposed for elliptical orbits. The system of four transcendental algebraic equations is solved using the method of modal synthesis which is based on multilevel decomposition of discrete dynamic system and applied to solve the problem of identification of parameters of discrete system by a state observer. The solution algorithm is as follows: conditional and identification discrete models (systems) are built for the specified system of equations; initial values of estimates are given; initial conditions in the equations of residuals are formed. Using the method of modal synthesis, the problem of search for control of the auxiliary system is solved, as a result of which the matrix of state observer feedback coefficients is calculated. This matrix is used to predict the state vector and to obtain refined estimates --- parameters of the planar orbit. A numerical example of the Lambert’s problem solution using the proposed algorithm is given. In essence, an approach to the solution of nonlinear algebraic systems of the fourth order, which can be extended to systems of any observable order, is proposed. The peculiarity of the proposed algorithm is that the convergence of the iterative process of finding a solution can have a different "adjustable" speed using the control law

Please cite this article as:

Zubov N.E., Ryabchenko V.N., Proletarsky A.V., et al. On one approach to the solution of the Lambert problem using the decompositional method of modal control. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 6 (105), pp. 77--89. DOI: https://doi.org/10.18698/1812-3368-2022-6-77-89

References

[1] Sukhanov A.A. Astrodinamika [Astrodynamics]. Moscow, IKI Publ., 2010.

[2] Izzo D. Revisiting Lambert’s problem. Celest. Mech. Dyn. Astr., 2015, vol. 121, no. 1, pp. 1--15. DOI: https://doi.org/10.1007/s10569-014-9587-y

[3] Sokolov N.L., Zakharov P.A. Autonomous identification of orbit parameters of potentially. Lesnoy vestnik [Forestry Bulletin], 2016, vol. 20, no. 2, pp. 214--224 (in Russ.).

[4] Sangra D., Fantino E. Review of Lambert’s problem. ISSFD, 2015. DOI: https://doi.org/10.48550/arXiv.2104.05283

[5] Bando M., Yamakawa H. New Lambert algorithm using the Hamilton --- Jacobi --- Bellman equation. J. Guid. Control Dyn., 2010, vol. 33, no. 3, pp. 1000--1008. DOI: https://doi.org/10.2514/1.46751

[6] Arora N., Russell R.P. A fast and robust multiple revolution Lambert algorithm using a cosine transformation. Adv. Astronaut. Sci., 2013, vol. 150, art. AAS 13-728.

[7] Bombardelli C., Roa J., Gonzalo J. Approximate analytical solution of the multiple revolution Lambert’s targeting problem Adv. Astronaut. Sci., 2016, vol. 158, art. AAS 16-212.

[8] Sokolov N.L., Sokolov A.P. On one analytic method for defining elements of Kepler orbit by two aircraft locations. Kosmicheskie issledovaniya, 1989, vol. 27, no. 6, pp. 803--807 (in Russ.).

[9] Burdaev M.N. The repositioning maneuver of artificial Earth satellite in a circular orbit with the phasing coils trajectory. Programmnye sistemy: teoriya i prilozheniya [Program Systems: Theory and Applications], 2012, vol. 3, no. 3, pp. 71--78 (in Russ.).

[10] Lysenko L.N. Navedenie i navigatsiya ballisticheskikh raket [Guidance and navigation of ballistic rockets]. Moscow, BMSTU Publ., 2007.

[11] Sikharulidze Yu.G. Ballistika i navedenie letatelnykh apparatov [Guidance and navigation of aircraft]. Moscow, Binom Publ., 2011.

[12] Zubov N.E., Mikrin E.A., Ryabchenko V.N. Matrichnye metody v teorii i praktike sistem avtomaticheskogo upravleniya letatelnykh apparatov [Matrix methods in theory and practice of automatic control systems for aircraft]. Moscow, BMSTU Publ., 2016.

[13] Zubov N.E., Mikrin E.A., Ryabchenko V.N., et al. Analytical synthesis of control laws for lateral motion of aircraft. Russ. Aeronaut., 2015, vol. 58, no. 3, pp. 263--270. DOI: https://doi.org/10.3103/S1068799815030034

[14] Zubov N.E., Mikrin E.A., Misrikhanov M.Sh., et al. Output control of the spectrum of a descriptive dynamical system. Doklady Akademii nauk, 2016, vol. 468, no. 2, pp. 134--136 (in Russ.). DOI: https://doi.org/10.7868/S0869565216140073

[15] Zubov N.E., Lapin A.V., Mikrin E.A., et al. Output control of the spectrum of a linear dynamic system in terms of the Van der Woude method. Dokl. Math., 2017, vol. 96, no. 2, pp. 457--460. DOI: https://doi.org/10.1134/S1064562417050179

[16] Elyasberg P.E. Vvedenie v teoriyu poleta iskusstvennogo sputnika Zemli [Introduction into flight theory of artificial Earth satellites] Moscow, Nauka Publ., 1965.

[17] Mikrin E.A., Zubov N.E., Efanov D., et al. Superhigh-speed iterative solvers of linear matrix. Doklady Akademii nauk, 2018, vol. 482, no. 3, pp. 250--253 (in Russ.). DOI: https://doi.org/10.31857/S086956520003124-9

[18] Gadzhiev M.G., Zhgun K.V., Zubov N.E., et al. Synthesis of fast and superfast solvers of large systems of linear algebraic equations using control theory methods. J. Comput. Syst. Sci. Int., 2019, vol. 58, no. 4, pp. 560--570. DOI: https://doi.org/10.1134/S1064230719020084