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Generalized Hohmann, Bi-Parabolic and Bi-Elliptic Planar Impulsive Transfer Using Transformed Variables

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Abstract

The problem of planar optimal impulsive transfer between ellipses in a Newtonian gravitational field with the final time- free is approached through a transformation of variables. New necessary conditions for optimal Bi-Elliptic Transfers are presented in terms of these transformed variables. The work is applied to examples in which the apses of the ellipses are aligned. The Generalized Hohmann, Bi-Elliptic and Bi-Parabolic Transfers are discussed. An example is presented that shows that Bi-Elliptic Transfer cannot be optimal if the final time is free. The approach can also be applied to determine optimality of transfers for other aligned configurations. This project is then changed to a fixed final time minimization problem. For this problem, it is found that there is a one-to-one correspondence between the final time and the apogee of the transfer ellipse. It is shown from this fact that there can be optimal Bi-Elliptic Transfers if the final time is fixed.

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References

  1. Hohmann, W.: Die Erreichbarkeit der Himmelskoerper, Oldenbourg, Munich (1925), The Attainability of Heavenly bodies NASA Tech. Translation F-44 (1960)

  2. Oberth, H.: Wege Zur Raumschiffahrt. R. Oldenbourg, Munich (1929)

    Book  Google Scholar 

  3. Lawden, D.F.: Minimal rocket trajectories. J. Am. Rocket Soc. 23, 360–382 (1953)

    Article  Google Scholar 

  4. Lawden, D.F.: Stationary rocket trajectories. Quart. J. Mech. 7, 488–504 (1954)

    Article  MathSciNet  Google Scholar 

  5. Lawden, D.F.: Optimal Trajectories for Space Navigation. Butterworths, London (1963)

    MATH  Google Scholar 

  6. Shternfeld, A.: Soviet Space Sciences, pp. 109–111. Basic Books Inc, New York (1959)

    Google Scholar 

  7. Hoelker, R. F., and Silber, R.: The bi-elliptical transfer between co-planar circular orbits. In: Proceedings of the 4th symposium on ballistic missile and space technology, Los Angeles (1959)

  8. Edelbaum, T.N.: Some extensions of the Hohmann transfer maneuver. J. Am. Rocket Soc. 29, 864–865 (1959)

    Google Scholar 

  9. Edelbaum, T.N.: How many impulses? Astronaut. Aeronaut. 5, 64–69 (1967)

    Google Scholar 

  10. Bell, D.J.: Optimal space trajectories. A review of published work. Aeronaut. J. R. Aeronaut. Soc. 72, 141–146 (1968)

    Google Scholar 

  11. Robinson, A.C.: A survey of methods and results in the determination for fuel-optimal space maneuvers. In: A.A.S. Paper 68-091, AAS/AIAA ... Specialist Conference (1968)

  12. Gobetz, F.W., Doll, J.R.: A survey of impulse trajectories. AIAA J. 7, 801–834 (1969)

    Article  Google Scholar 

  13. Breakwell, J.V.: Minimum impulse transfer, Preprint 63-416, AIAA Astrodynamics Conference, New Haven, Aug 19–23 (1963)

  14. Kechician, J.A.: Optimal transfer between close near-circular orbits. Ph.D Thesis, Stanford University (1977)

  15. Marec, J.P.: Optimal Space Trajectories. Elsevier, New York (1979)

    MATH  Google Scholar 

  16. Avandario, P., Mortari, D.: A closed-form solution to the minimum \(\Delta V_{tot}^2\) Lambert’s problem. Celest. Mech. Dyn. Astron. 106, 25–37 (2010)

    Article  Google Scholar 

  17. Zhang, G., Zhou, D., Mortari, D.: Optimal two-impulse rendezvous using constrained multiple-revolution lambert solutions. Celest. Mech. Dyn. Astron. 110, 199–215 (2011)

    Article  MathSciNet  Google Scholar 

  18. Carter, T., Humi, M.: A new approach to optimal impulsive rendezvous near circular orbit. Celest. Mech. Dyn. Astron. 112, 385–426 (2012)

    Article  Google Scholar 

  19. Carter, T., Brient, J.: Linearized impulsive rendezvous problem. J. Optim. Theory Appl. 86, 553–584 (1995)

    Article  MathSciNet  Google Scholar 

  20. Carter, T.: Optimal impulsive space trajectories based on linear equations. J. Optim. Theory Appl. 70, 277–297 (1991)

    Article  MathSciNet  Google Scholar 

  21. Neustadt, L.: Optimization, a moment problem, and nonlinear programming. SIAM J. Control 2, 33–53 (1964)

    MathSciNet  MATH  Google Scholar 

  22. Prussing, J.: Optimal impulsive linear systems: sufficient conditions and maximum number of impulses. J. Astronaut. Sci. 43, 195–206 (1995)

    MathSciNet  Google Scholar 

  23. Ting, L.: Optimum orbital transfers by several impulses. Asronaut. Acta 6, 256–261 (1960)

    Google Scholar 

  24. Kirilink, E., Zaborsky, S.: Optimal bi-elliptic transfer between two generic coplanar elliptical orbits. Acta Astronaut. 139, 996–1000 (2017)

    Google Scholar 

  25. Carter, T., Humi, M.: Two-impulse, bi-parabolic, and bi-elliptical transfers between coplanar elliptical orbits. Acta Astronaut. 177, 158–171 (2020)

    Article  Google Scholar 

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Authors and Affiliations

  1. Eastern Connecticut State University, Willimantic, CT, 06226, USA

    Thomas Carter

  2. Worcester Polytechnic Institute, Worcester, MA, 01609, USA

    Mayer Humi

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  1. Thomas Carter
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  2. Mayer Humi
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Correspondence to Thomas Carter.

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Communicated by Jason L. Speyer.

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Carter, T., Humi, M. Generalized Hohmann, Bi-Parabolic and Bi-Elliptic Planar Impulsive Transfer Using Transformed Variables. J Optim Theory Appl 189, 117–135 (2021). https://doi.org/10.1007/s10957-021-01824-z

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  • DOI: https://doi.org/10.1007/s10957-021-01824-z

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