Analytical solution of the problem of optimal control of reorientation of solid body (spacecraft), in sense of a combined criteria of quality, based on the quaternions

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Resumo

The problem on optimal reorientation of a solid (spacecraft) from an initial position into a prescribed final angular position on the basis of quaternions is solved. A combined criteria of quality is used, combining in a given proportion the contribution of control forces and the duration of maneuver, as well as the integral of the rotational energy. The synthesis of optimal control is based on a differential equation relating the attitude quaternion and angular momentum of a spacecraft. Analytical solution of optimal control problem is obtained using the necessary conditions of optimality in the form of the Pontryagin’s maximum principle. The properties of optimal rotation are studied in detail. Formalized equations and computational formulas are written to construct the optimal rotation program. Analytical equations and relations for finding the optimal control are presented. Key relations that determine the optimal values of the parameters of rotation control algorithm are given. A constructive scheme for solving the boundary-value problem of the maximum principle for arbitrary turning conditions (initial and final positions and moments of inertia of a solid) is given also. The made numerical experiments confirm the analytical conclusions. In the case of a dynamically symmetric solid body, the problem of spatial reorientation with minimum energy and time consumption is completely solved (in closed form). An example and results of mathematical modeling that confirm the practical feasibility of the developed method for orientation control are given.

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Sobre autores

M. Levskii

Maksimov Space System Research and Development Institute as branch of Khrunichev State Research and Production Space Center

Autor responsável pela correspondência
Email: levskii1966@mail.ru
Rússia, Korolev, Moscow oblast, 141091

Bibliografia

  1. Sinitsin L.I., Kramlikh A.V. Synthesis of the optimal control law for the reorientation of a nanosatellite using the procedure of analytical construction of optimal regulators // J. Phys.: Conf. Ser., 2021. V. 1745. P. 012053. https://doi.org/10.1088/1742-6596/1745/1/012053
  2. Velishchanskii M.A., Krishchenko A.P., Tkachev S.B. Synthesis of spacecraft reorientation algorithms using the concept of the inverse dynamic problem // J. Comput. Syst. Sci. Int. 2003. V. 42. № 5. P. 811–818.
  3. Junkins J.L., Turner J.D. Optimal Spacecraft Rotational Maneuvers. Elsevier. USA, 1986. 515 p.
  4. Reshmin S.A. Threshold absolute value of a relay control when time-optimally bringing a satellite to a gravitationally stable position // J. Comput. Syst. Sci. Int. 2018. V. 57. № 5. P. 713–722. https://doi.org/10.1134/S106423071805012X
  5. Scrivener S., Thompson R. Survey of time-optimal attitude maneuvers // J. Guidance, Control and Dynamics. 1994. V. 17. № 2. P. 225–233.
  6. Zhou H., Wang D., Wu B., E.K. Poh. Time-optimal reorientation for rigid satellite with reaction wheels // International Journal of Control. 2012. V. 85. № 10. P. 1–12. https://doi.org/10.1080/00207179.2012.688873
  7. Reshmin S.A. The threshold absolute value of a relay control bringing a satellite to a gravitationally stable position in optimal time // Dokl. Phys. 2018. V. 63. № 6. P. 257–261. https://doi.org/10.1134/S1028335818060101
  8. Levskii M.V. Pontryagin’s maximum principle in optimal control problems of orientation of a spacecraft // J. Comput. Syst. Sci. Int. 2008. V. 47. № 6. P. 974–986. https://doi.org/10.1134/S1064230708060117
  9. Shen H., Tsiotras P. Time-optimal control of axi-symmetric rigid spacecraft with two controls // AIAA J. Guidance, Control and Dynamics. 1999. V. 22. № 5. P. 682–694. https://doi.org/10.2514/2.4436
  10. Molodenkov A.V., Sapunkov Ya.G. Analytical solution of the minimum time slew maneuver problem for an axially symmetric spacecraft in the class of conical motions // J. Comput. Syst. Sci. Int. 2018. V. 57. № 2. P. 302–318. https://doi.org/10.1134/S1064230718020120
  11. Branets V.N., Chertok M.B., Kaznacheev Yu.V. Optimal turn of a rigid body with a single axis of symmetry // Kosm. Issl. 1984. V. 22. № 3. P. 352–360.
  12. Branets V.N., Shmyglevskii I.P. Use of Quaternions in Problems of Rigid Body Attitude. Nauka, Moscow, 1973. 320 p. [in Russian]
  13. Aipanov S.A., Zhakypov A.T. The method of separation of variables and its application to the problem of a spacecraft’s optimal turn // Cosmic Res. 2020. V. 58, № 1. P. 53–63. https://doi.org/10.1134/S0010952520010013
  14. Strelkova N.A. On optimal reorientation of a solid // Problems of Mechanics of Controlled Motion. Nonlinear Dynamical Systems. PGU, Perm, 1990. P. 115–133 [in Russian].
  15. Levskii M.V. Kinematically optimal spacecraft attitude control // J. Comput. Syst. Sci. Int. 2015. V. 54. № 1. P. 116–132. https://doi.org/10.1134/S1064230714050116
  16. Biryukov V.G., Chelnokov Y.N. Construction of optimal laws of variation of the angular momentum vector of a rigid body // Mech. Solids. 2014. V. 49, № 5. P. 479–494. https://doi.org/10.3103/S002565441405001X
  17. Levskii M.V. Optimal spacecraft terminal attitude control synthesis by the quaternion method // Mech. Solids. 2009. V. 44. № 2. P. 169–183. https://doi.org/10.3103/S0025654409020022
  18. Levskii M.V. About method for solving the optimal control problems of spacecraft spatial orientation // Problems of Nonlinear Analysis in Engineering Systems, 2015. V. 21. № 2. P. 61–75.
  19. Zelepukina O.V., Chelnokov Y.N. Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body // Mech. Solids. 2011. V. 46. № 4. P. 519–533. https://doi.org/10.3103/S0025654411040030
  20. Molodenkov A.V., Sapunkov Ya.G. Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions // J. Comput. Syst. Sci. Int. 2016. V. 55. № 6. P. 969–985. https://doi.org/10.1134/S1064230716060095
  21. Molodenkov A.V., Sapunkov Ya.G. Analytical quasi-optimal solution of the slew problem for an axially symmetric rigid body with a combined performance index // J. Comput. Syst. Sci. Int. 2020. V. 59. № 3. P. 347–357. https://doi.org/10.1134/S1064230720030107
  22. Sapunkov Ya.G. Molodenkov A.V. Analytical solution of the problem on an axisymmetric spacecraft attitude maneuver optimal with respect to a combined functional // Autom. Remote Contr. 2021. V. 82. № 7. P. 1183–1200. https://doi.org/10.31857/S0005231021070059
  23. Molodenkov A.V., Sapunkov Ya.G. Analytical approximate solution of the problem of a spacecraft’s optimal turn with arbitrary boundary conditions // J. Comput. Syst. Sci. Int. 2015. V. 54. № 3. P. 458–468. https://doi.org/10.1134/S1064230715030144
  24. Levskii M.V. Control of the rotation of a solid (spacecraft) with a combined optimality criterion based on quaternions // Mech. Solids. 2023. V. 58. № 5. P. 1483–1499. https://doi.org/10.31857/S0572329922600566
  25. Levskii M.V. Optimal Control of the Angular Momentum for a Solid (Spacecraft) Performing a Spatial Turn // Mech. Solids. 2023. V. 58. № 1. P. 63–77. https://doi.org/10.31857/S0572329922060137
  26. Quang M. Lam. Robust and adaptive reconfigurable control for satellite attitude control subject to under-actuated control condition of reaction wheel assembly // Mathematics in Engineering, Science and Aerospace. 2018. V. 9. № 1. P. 47–63.
  27. Levskii M.V. Special aspects in attitude control of a spacecraft, equipped with inertial actuators // Journal of Computer Science Applications and Information Technology. 2017. V. 2. № 4. P. 1–9.
  28. Gorshkov O.A., Murav’ev V.A., Shagaida A.A. Holl’s and Ionic Plasma Engines for Spacecrafts. Mashinostroenie, Moscow, 2008. 280 p. [in Russian]
  29. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The Mathematical Theory of Optimal Processes. Gordon and Breach, New York, 1986. 392 p.
  30. Young L.C. Lectures on the Calculus of Variations and Optimal Control Theory. EA: Saunders, Philadelphia, 1969. 488 p.
  31. Lyubushin А.А. Modifications of the method of successive approximations for solving optimal control problems // USSR Computat. Math. Math. Phys. 1982. V. 22. № 1. P. 29–34.
  32. Levskii M.V. System for control of spacecraft’s spatial turn. RF Patent No. 2006431// Byull. Izobret. 1994. № 2. P. 49–50. Published at 20.01.1994. [in Russian]
  33. Levskii M.V. Method of controlling a spacecraft turn and system for its. RF Patent No. 2114771// Byull. Izobret. 1998. № 19. P. 234–236. Published at 10.07.1998. [in Russian]
  34. Zhuravlev V.Ph., Klimov D.M. Applied Methods in the Vibration Theory. Nauka, Moscow, 1988. 328 p. [in Russian]
  35. Levskii M.V. Device for regular rigid body precession parameters formation. RF Patent No. 21466381// Byull. Izobret. 2000. № 8. P. 148. Published at 20.03.2000. [in Russian]
  36. Kul’kov V.M., Obukhov V.A., Egorov Yu.G., Belik A.A., Krainov A.M. Comparative evaluation of the effectiveness of applying the perspective types of electric-rocket engines in small spacecrafts // Byull. Samara’s State Space University. 2012. V. 34. № 3. P. 187–195. [in Russian]

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2. Fig. 1. The form of optimal functions a(t) and b(t).

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3. Fig. 2. Change in the projections of the spacecraft kinetic moment during the turn.

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4. Fig. 3. Change of the components of the orientation quaternion L(t) during the reversal.

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5. Fig. 4. The type of functions p1(t), p2(t), p3(t) during the optimal reversal.

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6. Fig. 5. Change of the kinetic moment modulus under optimal control.

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