The translation of the vehicle with a controlled thrust vector to a given landing location with minimal fuel consumption

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Abstract

The problem of vehicle’s translation to a certain landing location above the surface of the planet is considered. Using the Pontryagin maximum principle, the optimal control problem is reduced to a boundary value problem for a system of nonlinear differential equations. A qualitative analysis of the optimal phase trajectories of the system is carried out, their properties are established, illustrated by the results of numerical modeling. The domains in the plane of phase variables are analytically described, from which it is possible to achieve a terminal set. A synthesis of optimal control is constructed.

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About the authors

N. A. Oryol

Lomonosov Moscow State University

Author for correspondence.
Email: nikita.orel@math.msu.ru
Russian Federation, Moscow

O. Yu. Cherkasov

Shenzhen MSU-BIT University; Lomonosov Moscow State University

Email: oyuche@yandex.ru
China, Shenzhen; Moscow, Russia

References

  1. Goddard R.H. A Method or Reaching Extreme Altitudes// Smithsonian Institute Misc. Collections. 1919. V. 71. № 2. P. 2–80.
  2. Hamel G. Über Eine mit dem Problem der Rakete Zusammenhängende Aufgabe der Variationsrechnung // ZAMM. 1927. Bd 7. H. 6. S. 451–452.
  3. Охоцимский Д.Е. К теории движения ракет // ПММ. 1946. Т. 10. № 2. С. 251–272.
  4. Исаев В.К. Принцип максимума Л. С. Понтрягина и оптимальное программирование тяги ракет // АиТ. 1961. Т. 22. Вып. 8. С. 986–1001.
  5. Охоцимский Д.Е., Энеев Т.М. Некоторые вариационные задачи, связанные с запуском искусственного спутника Земли // УФН. 1957. № 1а. С. 5–32.
  6. Tsien H.S., Evans R.C. Optimum Thrust Programming for a Sounding Rocket // J. American Rocket Society. 1951. V. 21. № 5. P. 99–107.
  7. Tsiotras P., Kelley H.J. Drag-law Effects in the Goddard Problem // J. Automatica. 1991. V. 27. № 3. P. 481–490. https://doi.org/10.23919/ACC.1988.4789942
  8. Miele A. An Extension of the Theory of the Optimum Burning Program for the Level Flight of a Rocket-Powered Aircraft // J. Aeronautical Science. 1957. V. 24. № 12. P. 874–884.
  9. Дмитрук А.В., Самыловский И.А. Исследование оптимальных траекторий в некоторых модификациях простейшей задачи о движении материальной точки с нелинейным сопротивлением и ограниченным расходом топлива // XII Всероссийск. cовещ. по проблемам управления (ВСПУ-2014). М.: Тр. ИПУ РАН, 2014. С. 629–632.
  10. Obert H. Die Rakete zu den Planetenräumen // R. Oldenburg. 1923. S. 1–92.
  11. Indig N., Ben-Asher J.Z., Sigal E. Singular Control for Two-Dimensional Goddard Problems Under Various Trajectory Bending Laws // J. Guidance, Control and Dynamics. 2018. V. 42. № 3. P. 1–15. https://doi.org/10.2514/1.G003670
  12. Малых Е.В., Черкасов О.Ю. Максимизация дальности полета для упрощенной модели летательного аппарата // Изв. РАН. ТиСУ. 2024. № 6. С. 28–40.
  13. Cheng R.K., Conrad D.A. Optimal Translation and Brachistochrone // J. AIAA. 1963. V. 1. № 12. P. 2845–2847.
  14. Keller W.F. Study of Spacecraft Hover and Translation Modes Above the Lunar Surface // J. of Spacecraft and Rockets. 1965. V. 5. № 2. P. 426–430.
  15. Speyer J.L., Bryson A.J. Explicit Guidance Law for Minimum Fuel Horizontal Translation with Bounded Control // Journal AAIA. 1967. V. 5. № 2. P. 340–342.
  16. Понтрягин Л.С., Болтянский В.Г., Гамкрелидзе Р.В., Мищенко Е.Ф. Математическая теория оптимальных процессов. М.: Наука, 1983.
  17. Cherkasov O.Yu., Smirnova N.V. On the Brachistochrone Problem with State Constraints on the Slope Angle // Intern. J. Non-Linear Mech. 2022. V. 139.
  18. Смирнова Н.В. Модифицированная задача о брахистохроне с фазовыми ограничениями и тягой // Вестн. МГУ. Сер. 15. Вычислительная математика и кибернетика. 2023. № 4. С. 54–60.

Supplementary files

Supplementary Files
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1. JATS XML
2. Fig. 1. Setting the task of selecting a landing site.

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3. Fig. 2. Phase portraits of the system (2.11): a - a = 0.5, b - a = 1.

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4. Fig. 3. Mutual arrangement of the terminal set and constraints u = ±u in the plane (u, v) at a = 1 in the case of free v (T).

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5. Fig. 4. In support of statements 1 and 2.

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6. Fig. 5. Results of numerical modelling at x (0) = 0, v (0) = 0, T = 0.5; 1; 1.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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7. Fig. 6. Results of numerical modelling at x (0) = 0, v (0) = 1.5, T = 0.5; 1; 1.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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8. Fig. 7. Results of numerical modelling at x (0) = 0, v (0) = 0.3, T = 2, u = 0.5; 1: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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9. Fig. 8. Results of numerical modelling at x (0) = 0, v (0) = 1.5, T = 1.44, u = 0.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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10. Fig. 9. Results of numerical modelling at x (0) = 0, v (0) = v (T) = 0, T = 1; 1.5; 2; 3: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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11. Fig. 10. Results of numerical modelling at x (0) = 0, v (0) = 0.3, v (T) = 0, T = 1; 1.5; 2: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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12. Fig. 11. Results of numerical modelling at x (0) = 0, v (0) = 1.5, v (T) = 0.7, T = 1; 1.5: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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13. Fig. 12. Mutual arrangement of the terminal set and constraints u = ±u in the plane (u, v) at a = 1 in the case of fixed v (T).

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14. Fig. 13. Results of numerical modelling at x (0) = 0, v (0) = v (T) = 0, T = 2: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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