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Optimization of Multi-revolution Limited Power Trajectories Using Angular Independent Variable

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Abstract

Optimization of low-thrust trajectories is necessary in the design of space missions using electric propulsion systems. We consider the problem of limited power trajectory optimization, which is a well-known case of the low-thrust optimization problem. In the article, we present an indirect approach to trajectory optimization based on the use of the maximum principle and the continuation method. We introduce the concept of auxiliary longitude and use it as a new independent variable instead of time. The use of equations of motion in the equinoctial elements and a new independent variable allowed us to simplify the optimization of limited power trajectories with a fixed angular distance and free transfer duration. The article presents a new form of necessary optimality conditions for this problem and describes an efficient new numerical method to solve the limited power trajectory optimization problem. We show the existence of several trajectories with a fixed transfer duration and free angular distance that satisfy the necessary optimality conditions. Using numerical examples, we confirm the existence of the limiting values of the characteristic velocity and the product of the cost function value and the transfer duration as the angular distance increases. The high computational performance of the developed technique makes it possible to carry out and present an analysis of the angular flight range and initial true longitude impact on the cost function, transfer duration, and characteristic velocity.

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References

  1. Akhmetshin, R.Z.: Planar problem of an optimal transfer of a low-thrust spacecraft from high-elliptic to geosynchronous orbit. Cosm. Res. 42(3), 238–249 (2004)

    Article  Google Scholar 

  2. Betts, J.T.: Optimal low-thrust orbit transfers with eclipsing. Optim. Control Appl. Methods 36(2), 218–240 (2015)

    Article  MathSciNet  Google Scholar 

  3. Blagonravov A.A.: Collected Works of K.E. Tsiolkovskiy, vol. 2 Reactive Flying Machines. NASA TT F-237 (1965)

  4. Caillau, J.-B., Cots, O., Gergaud, J.: Differential pathfollowing for regular optimal control problems. Optim. Methods Softw. 27(2), 177–196 (2012). (Special issue dedicated to 60th birthday of Andreas Griewank)

    Article  MathSciNet  Google Scholar 

  5. Casalino, L., Colasurdo, G.: Optimization of variable-specific-impulse interplanetary trajectories. J. Guid. Control Dyn. 27(4), 678–684 (2004)

    Article  Google Scholar 

  6. Davidenko, D.F.: On a new numerical solution method for system of nonlinear equations. Dokl. Akad. Nauk SSSR 88(4), 601–601 (1953). (in Russian)

    MathSciNet  Google Scholar 

  7. Edelbaum, T.N.: Optimum low-thrust rendezvous and station keeping. AIAA J. 2(7), 1196–1201 (1964)

    Article  MathSciNet  Google Scholar 

  8. Edelbaum, T.N.: Optimum power-limited orbit transfer in strong gravity fields. AIAA J. 3(5), 921–925 (1965)

    Article  MathSciNet  Google Scholar 

  9. Edelbaum, T.N.: An asymptotic solution for optimum power limited orbit transfer. AIAA J. 4(8), 1491–1494 (1966)

    Article  Google Scholar 

  10. Edelbaum, T.N., Pines, S.: Fifth and sixth integrals for optimum rocket trajectories in a central field. AIAA J. 8(7), 1201–1204 (1970)

    Article  Google Scholar 

  11. Gavurin, M.K.: Nonlinear functional equations and continuous analogues of iteration methods. Izv. Vuz Mat. 5, 18–31 (1958)

    MathSciNet  Google Scholar 

  12. Geffroy, S., Epenoy, R.: Optimal low-thrust transfers with constraints-generalization of averaging techniques. Acta Astronaut. 41(3), 133–149 (1997)

    Article  Google Scholar 

  13. Geffroy, S.: Generalisation des techniques de moyennation en contrle optimal. Application aux problmes de transfert et rendez-vous orbitaux pousse faible. These de doctorat, Institut National Polytechnique de Toulouse (1997)

  14. Gergaud, J., Haberkorn, T.: Homotopy method for minimum consumption orbit transfer problem. ESAIM: control, optimisation and calculus of variations. EDP Sci. 12(2), 294–310 (2006)

    MATH  Google Scholar 

  15. Gobetz, F.W.: Optimal variable-thrust transfer of a power-limited rocket between neighboring circular orbits. AIAA J. 2(2), 339–343 (1964)

    Article  Google Scholar 

  16. Golan, O.M., Breakwell, J.V.: Minimum fuel lunar trajectories for a low-thrust power-limited spacecraft. Dyn. Control 4(4), 383–394 (1994)

    Article  MathSciNet  Google Scholar 

  17. Graham, K.F., Rao, A.V.: Minimum-time trajectory optimization of multiple revolution low-thrust earth-orbit transfers. J. Spacecr. Rocket 52(3), 711–727 (2015)

    Article  Google Scholar 

  18. Graham, K.F., Rao, A.V.: Minimum-time trajectory optimization of low-thrust earth-orbit transfers with eclipsing. J. Spacecr. Rocket 53(2), 289–303 (2016)

    Article  Google Scholar 

  19. Haberkorn, T., Martinon, P., Gergaud, J.: Low-thrust minimum-fuel orbital transfer: a homotopic approach. J. Guid. Control Dyn. 27(6), 1046–1060 (2004)

    Article  Google Scholar 

  20. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations: Nonstiff problems. Springer Series in Computational Mathematics. Springer, Berlin (1993)

    MATH  Google Scholar 

  21. Haissig, C.M., Mease, K.D., Vinh, N.X.: Minimum-fuel, power-limited transfers between coplanar elliptical orbits. Acta Astronat. 29(1), 1–15 (1993)

    Article  Google Scholar 

  22. Irving, J.H.: Low thrust flight: variable exhaust velocity in gravitational fields. In: Seifert, H.S. (ed.) Space Technology. Wiley, New York (1959)

    Google Scholar 

  23. Jiang, F., Baoyin, H., Li, J.: Practical techniques for low-thrust trajectory optimization with homotopic approach. J. Guid. Control Dyn. 35(1), 245–258 (2012)

    Article  Google Scholar 

  24. Junkins, J.L., Taheri, E.: State vector representations for low-thrust trajectory optimization. AAS 18–385, 1–20 (2018)

    Google Scholar 

  25. Kechichian, J.A.: Applied Nonsingular Astrodynamics: Optimal Low-Thrust Orbit Transfer. Cambridge University Press, Cambridge (2018)

    Book  Google Scholar 

  26. Kiforenko, B.N., Vasil’ev, IYu.: On optimization of many-revolution low-thrust orbit transfers: part 1. Int. Appl. Mech. 44(7), 810–817 (2008)

    Article  MathSciNet  Google Scholar 

  27. Kiforenko, B.N., Vasil’ev, IYu.: On optimization of many-revolution low-thrust orbit transfers: part 2. Int. Appl. Mech. 44(9), 1050–1055 (2008)

    Article  MathSciNet  Google Scholar 

  28. Lyness, J.N., Moller, C.B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal. 4(2), 202–210 (1967)

    Article  MathSciNet  Google Scholar 

  29. Marec, J.-P., Vinh, N.X.: Optimal low-thrust, limited power transfers between arbitrary elliptical orbits. Acta Astronat. 4(5–6), 511–540 (1977)

    Article  Google Scholar 

  30. Martins, J.R.R.A., Sturdza, P., Alonso, J.J.: The complex-step derivative approximation. ACM Trans. Math. Softw. 29(3), 245–262 (2003)

    Article  MathSciNet  Google Scholar 

  31. Melbourne, W.G., Sauer, C.G.: Optimum interplanetary rendezvous with power-limited vehicles. AIAA J. 1(1), 54–60 (1963)

    Article  Google Scholar 

  32. Petukhov V.G.: One numerical method to calculate optimal power- limited trajectories. IEPC-95-221, 1-8 (1995)

  33. Petukhov, V.G.: Optimization of multi-orbit transfers between noncoplanar elliptic orbits. Cosm. Res. 42(3), 250–268 (2004)

    Article  Google Scholar 

  34. Petukhov, V.G.: Optimization of interplanetary trajectories for spacecraft with ideally regulated engines using the continuation method. Cosm. Res. 46(3), 219–232 (2008)

    Article  Google Scholar 

  35. Petukhov, V.G.: Method of continuation for optimization of interplanetary low-thrust trajectories. Cosm. Res. 50(3), 249–260 (2012)

    Article  Google Scholar 

  36. Petukhov, V.G.: Application of the angular independent variable and its regularizing transformation in the problems of optimizing low-thrust trajectories. Cosm. Res. 57(5), 351–363 (2019)

    Article  Google Scholar 

  37. Sengupta, P., Vadali, S.R.: Analytical solution for power-limited optimal rendezvous near an elliptic orbit. J. Optim. Theory Appl. 138, 115–137 (2008)

    Article  MathSciNet  Google Scholar 

  38. Shirazi, A., Ceberio, J., Lozano, J.: Spacecraft trajectory optimization: a review of models, objectives, approaches and solutions. Prog. Aerosp. Sci. 102, 76–98 (2018)

    Article  Google Scholar 

  39. Tsiolkovsky, K.E.: Exploration of the universe with reaction machines. Sci. Rev. 5 (1903) (in Russian)

  40. Walker, M.J.H., Ireland, B., Owens, J.: A set modified equinoctial orbit elements. Celest. Mech. 36(4), 409–419 (1985). (ERRATA 38:391–392 (1986))

    Article  Google Scholar 

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Acknowledgements

This study was supported by the Grant in form of subsidies from the federal budget, allocated for state support of scientific research under supervision of leading scientists in Russian institutions of higher education, scientific foundations and state research centers of the Russian Federation (7th stage, Decree of the Government of the Russian Federation No. 220 of 09 April 2010), Project No. 075-15-2019-1894.

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Authors and Affiliations

  1. Research Institute of Applied Mechanics and Electrodynamics of Moscow Aviation Institute (RIAME MAI), Moscow Aviation Institute (MAI), Moscow, Russia

    Alexey Ivanyukhin & Viacheslav Petukhov

  2. Peoples Friendship University of Russia (RUDN University), Moscow, Russia

    Alexey Ivanyukhin

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  1. Alexey Ivanyukhin
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  2. Viacheslav Petukhov
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Correspondence to Alexey Ivanyukhin.

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Communicated by Mauro Pontani.

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Ivanyukhin, A., Petukhov, V. Optimization of Multi-revolution Limited Power Trajectories Using Angular Independent Variable. J Optim Theory Appl 191, 575–599 (2021). https://doi.org/10.1007/s10957-021-01853-8

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