Оптимизация траекторий с малой тягой в переменных Кустаанхеймо–Штифеля

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Abstract

В работе рассматривается регуляризация уравнений движения космического аппарата преобразованием Кустаанхеймо — Штифеля для координат и Сундмана для времени в задаче поиска оптимальной траектории межпланетного перелета с двигателем малой тяги. Из принципа максимума Понтрягина находится оптимальное управление вектором тяги при условии ограниченной мощности двигателя. Задача перелета Земля — Марс решается в регулярных переменных. Проводится сравнение найденных траекторий с траекториями, полученными методом продолжения по параметру, а также исследуется чувствительность решений краевой задачи принципа максимума в декартовых и регулярных переменных.

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About the authors

К. Р. Корнеев

Институт прикладной математики им. М. В. Келдыша РАН

Author for correspondence.
Email: kirill_rnd@mail.ru
Russian Federation, Москва

С. П. Трофимов

Институт прикладной математики им. М. В. Келдыша РАН

Email: kirill_rnd@mail.ru
Russian Federation, Москва

References

  1. Улыбышев Ю. П. Обзор методов оптимизации траекторий космических аппаратов с использованием дискретных множеств псевдоимпульсов // Космическая техника и технологии. 2016. Т. 15. № 4. С. 67–79.
  2. Gergaud J., Haberkorn T. Homotopy method for minimum consumption orbit transfer problem // ESAIM: Control, Optimisation and Calculus of Variations. 2006. V. 12. Iss. 2. P. 294–310.
  3. Haberkorn T., Martinon P., Gergaud J. Low thrust minimum-fuel orbital transfer: a homotopic approach // J. Guidance, Control, and Dynamics. 2004. V. 27. Iss. 6. P. 1046–1060.
  4. Mingotti G., Topputo F., Bernelli-Zazzera F. A method to design sun-perturbed earth-to-moon low-thrust transfers with ballistic capture // Proc. XIX Congresso nazionale AIDAA. 2007. V. 17. Art.ID. 21.
  5. Pontryagin L.S., Boltyanskii V. G., Gamkrelidze R. V. et al. Mathematical theory of optimal processes. New York–London: Interscience Publishers John Wiley & Sons, Inc., 1962. 360 p.
  6. Petukhov V. G. Optimal multi-orbit trajectories for inserting a low-thrust spacecraft to a high elliptic orbit // Cosmic Research. 2009. V. 47. Iss. 3. P. 243–250.
  7. Petukhov V. G. Optimization of multi-orbit transfers between noncoplanar elliptic orbits // Cosmic Research. 2004. V. 42. Iss. 3. P. 250–268.
  8. Petukhov V. G. Method of continuation for optimization of interplanetary low-thrust trajectories // Cosmic Research. 2012. V. 50. Iss. 3. P. 249–261.
  9. Petukhov V. G. Optimization of interplanetary trajectories for spacecraft with ideally regulated engines using the continuation method // Cosmic Research. 2008. V. 46. Iss. 3. P. 219–232.
  10. Pérez-Palau D., Epenoy R. Fuel optimization for low-thrust Earth–Moon transfer via indirect optimal control // Celestial Mechanics and Dynamical Astronomy. 2018. V. 130. Iss. 2. Art.ID. 21.
  11. Pan B., Pan X., Zhang S. A new probability-one homotopy method for solving minimum-time low-thrust orbital transfer problems // Astrophysics and Space Science. 2018. V. 363. Iss. 9.
  12. Pan B., Lu P., Pan X. et al. Double-homotopy method for solving optimal control problems // Journal of Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 2016. V. 39. Iss. 8. P. 1706–1720.
  13. Jiang F., Baoyin H., Li J. Practical techniques for low-thrust trajectory optimization with homotopic approach // J. Guidance, Control, and Dynamics. 2012. V. 35. Iss. 1. P. 245–258.
  14. Zhang C., Topputo F., Bernelli-Zazzera F. et al. Low-thrust minimum-fuel optimization in the circular restricted three-body problem // J. Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 2015. V. 38. Iss. 8. P. 1501–1510.
  15. Taheri E., Kolmanovsky I., Atkins E. Enhanced smoothing technique for indirect optimization of minimum-fuel low-thrust trajectories // J. Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 2016. V. 39. Iss. 11. P. 2500–2511.
  16. Taheri E., Junkins J. L. Generic smoothing for optimal bang-off-bang spacecraft maneuvers // J. Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 2018. V. 41. Iss. 11. P. 2470–2475.
  17. Taheri E., Junkins J., Kolmanovsky I. et al. A novel approach for optimal trajectory design with multiple operation modes of propulsion system, part 1 // Acta Astronautica. 2020. V. 172. P. 151–165.
  18. Taheri E., Junkins J., Kolmanovsky I. et al. A novel approach for optimal trajectory design with multiple operation modes of propulsion system, part 2 // Acta Astronautica. 2020. V. 172. P. 166–179.
  19. Junkins J.L., Taheri E. Exploration of alternative state vector choices for low-thrust trajectory optimization // J. Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 2019. V. 42. Iss. 1. P. 47–64.
  20. Geffroy S., Epenoy R. Optimal low-thrust transfers with constraints-generalization of averaging techniques // Acta Astronautica. 1997. V. 41. Iss. 3. P. 133–149.
  21. Sundman K. F. Mémoire sur le problème des trois corps // Acta mathematica. Institut Mittag-Leffler. 1913. V. 36. P. 105–179.
  22. Nacozy P. E. Time elements in Keplerian orbital elements // Celestial mechanics. 1981. V. 23. Iss. 2. P. 173–198.
  23. Brumberg E. V. Length of arc as independent argument for highly eccentric orbits // Celestial Mechanics and Dynamical Astronomy. 1992. V. 53. P. 323–328.
  24. Stiefel E.L., Scheifele G. Linear and Regular Celestial Mechanics. Berlin Heidelberg: Springer-Verlag Berlin Heidelberg, 1971. 306 p.
  25. Levi-Civita T. Sur la régularisation du probleme des trois corps // Acta mathematica. Institut Mittag-Leffler. 1920. V. 42. P. 99–144.
  26. Иванов Д.С., Трофимов С. П., Широбоков М. Г. Численное моделирование орбитального и углового движения космических аппаратов / под ред. М. Ю. Овчинникова. М: ИПМ им. М. В. Келдыша РАН, 2016. 118 с.
  27. Иванюхин А. В. Оптимизация траектории космического аппарата с идеально регулируемым двигателем в переменных Кустаанхеймо — Штифеля // Труды МАИ. 2014. № 75.
  28. Chelnokov Yu.N., Loginov M. Yu. Prediction and Correction of Spacecraft Motion Based on the Solutions of Regular Quaternion Equations in KS-Variables and Isochronous Derivatives // Proc. 29th Saint Petersburg International Conference on Integrated Navigation Systems (ICINS). Saint Petersburg, Russia. IEEE, 2022.
  29. Masat A., Romano M., Colombo C. Kustaanheimo — Stiefel Variables for Planetary Protection Compliance Analysis // J. Guidance, Control, and Dynamics. 2022. V. 45. Iss. 7. P. 1286–1298.
  30. Roa J., Urrutxua H., Peláez J. Stability and chaos in Kustaanheimo — Stiefel space induced by the Hopf fibration // Mon. Not. R. Astron. Soc. 2016. V. 459. Iss. 3. P. 2444–2454.
  31. Nocedal J., Wright S. J. Numerical Optimization. Springer New York, 2006.
  32. Roa J. Regularization in Orbital Mechanics. Berlin, Boston: De Gruyter, 2017. 403 p.
  33. Милютин А.А., Дмитрук А. В., Осмоловский Н. П. Принцип максимума в оптимальном управлении. М.: Центр прикладных исследований мехмата МГУ, 2004. 168 p.
  34. Powers W.F., Tapley B. D. Canonical transformation applications to optimal trajectory analysis. // AIAA Journal. 1969. V. 7. Iss. 3. P. 394–399.
  35. Byrd R.H., Hribar M. E., Nocedal J. An Interior Point Algorithm for Large-Scale Nonlinear Programming // SIAM J. Optim. 1999. V. 9. Iss. 4. P. 877–900.
  36. Folkner W.M., Williams J. G., Boggs D. et al. The planetary and lunar ephemerides DE430 and DE431 // Interplanetary Network Progress Report. 2014. V. 196. Iss. 1. P. 42–196.
  37. Schoenmaekers J. Post-launch Optimisation of the SMART-1 Low-thrust Trajectory to the Moon // Proc. 18th International Symposium on Space Flight Dynamics. 2004. P. 505–510.

Supplementary files

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2. Fig. 1. Different families of solutions and Pareto front recovered by two different approaches

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3. Fig. 2. Difference between fictitious flight time and change in eccentric anomaly for different Pareto-optimal solutions

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4. Fig. 3. Computation time of locally optimal solutions by different approaches

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5. Fig. 4. Conditioning numbers of the sensitivity matrix for two different approaches

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