# Hohmann transfer via constrained optimization

• Li Xie
• Yi-qun Zhang
• Jun-yan Xu
Article

## Abstract

Inspired by the geometric method proposed by Jean-Pierre MAREC, we first consider the Hohmann transfer problem between two coplanar circular orbits as a static nonlinear programming problem with an inequality constraint. By the Kuhn-Tucker theorem and a second-order sufficient condition for minima, we analytically prove the global minimum of the Hohmann transfer. Two sets of feasible solutions are found: one corresponding to the Hohmann transfer is the global minimum and the other is a local minimum. We next formulate the Hohmann transfer problem as boundary value problems, which are solved by the calculus of variations. The two sets of feasible solutions are also found by numerical examples. Via static and dynamic constrained optimizations, the solution to the Hohmann transfer problem is re-discovered, and its global minimum is analytically verified using nonlinear programming.

## Key words

Hohmann transfer Nonlinear programming Constrained optimization Calculus of variations

O232 V412.4

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## References

1. Avendaño M, Martín-Molina V, Martín-Morales J, et al., 2016. Algebraic approach to the minimum-cost multiimpulse orbit-transfer problem. J Guid Contr Dynam, 39(8):1734–1743. https://doi.org/10.2514/1.G001598
2. Avriel M, 2003. Nonlinear Programming: Analysis and Methods. Dover Publications Inc., Mineola, NY, USA.
3. Barrar RB, 1963. An analytic proof that the Hohmann type transfer is the true minimum two-impulse transfer. Acta Astronaut, 9(1):1–11.Google Scholar
4. Battin RH, 1987. An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, New York, USA.
5. Bertsekas DP, 1999. Nonlinear Programming (2nd Ed.). Athena Scientific, Belmont, Egypt.Google Scholar
6. Bryson AE Jr, Ho YC, 1975. Applied Optimal Control. Hemisphere Publishing Corp., Washington, USA.Google Scholar
7. Cornelisse JW, Schöyer HFR, Wakker KF, 1979. Rocket Propulsion and Spaceflight Dynamics. Pitman, London, UK.Google Scholar
8. Curtis HD, 2014. Orbital Mechanics for Engineering Students. Elsevier, Amsterdam, the Netherlands.Google Scholar
9. Guler O, 2010. Foundations of Optimization. Springer, New York, USA. https://doi.org/10.1007/978-0-387-68407-9
10. Gurfil P, Seidelmann PK, 2016. Celestial Mechanics and Astrodynamics: Theory and Practice. Springer Berlin Heidelberg, Germany. https://doi.org/10.1007/978-3-662-50370-6
11. Hazelrigg GA, 1984. Globally optimal impulsive transfers via Green’s theorem. J Guid Contr Dynam, 7(4):462–470. https://doi.org/10.2514/3.19879
12. Hohmann W, 1960. The Attainability of Heavenly Bodies. NASA Technical Translation F-44, Washington, USA.Google Scholar
13. Hull DG, 2003. Optimal Control Theory for Applications. Springer, New York, USA. https://doi.org/10.1007/978-1-4757-4180-3
14. Kierzenka J, 1998. Studies in the Numerical Solution of Ordinary Differential Equations. PhD Thesis, Southern Methodist University, Dallas, USA.Google Scholar
15. Lawden DF, 1963. Optimal Trajectories for Space Navigation. Butterworths, London, UK.
16. Leitmann G, 1981. The Calculus of Variations and Optimal Control: an Introduction. Springer, New York, USA.
17. Li DY, Li DZ, 1991. Further discussion on optimal transfer between two circular orbits by dual impulse. Chin Space Sci Technol, 12(6):1–10 (in Chinese).Google Scholar
18. Longuski JM, Guzmán JJ, Prussing JE, 2014. Optimal Control with Aerospace Applications. Springer, New York, USA. https://doi.org/10.1007/978-1-4614-8945-0
19. Marec JP, 1979. Optimal Space Trajectories. Elsevier, Amsterdam.
20. Mathwig J, 2004. On Properties of the Hohmann Transfer. MS Thesis, Rice University, Houston, Texas, USA.Google Scholar
21. McCormick GP, 1967. Second order conditions for constrained minima. SIAM J Appl Math, 15(3):641–652. https://doi.org/10.1137/0115056
22. Miele A, Ciarcià M, Mathwig J, 2004. Reflections on the Hohmann transfer. J Optim Theory Appl, 123(2): 233–253. https://doi.org/10.1007/s10957-004-5147-z
23. Moyer HG, 1965. Minimum impulse coplanar circle-ellipse transfer. AIAA J, 3(4):723–726. https://doi.org/10.2514/3.2954
24. Palmore J, 1984. An elementary proof of the optimality of Hohmann transfers. J Guid Contr Dynam, 7(5):629–630. https://doi.org/10.2514/3.56375
25. Pontani M, 2009. Simple method to determine globally optimal orbital transfers. J Guid Contr Dynam, 32(3):899–914. https://doi.org/10.2514/1.38143
26. Prussing JE, 1992. Simple proof of the global optimality of the Hohmann transfer. J Guid Contr Dynam, 15(4): 1037–1038. https://doi.org/10.2514/3.20941
27. Prussing JE, 2010. Primer vector theory and applications. In: Conway BA (Ed.), Spacecraft Trajectory Optimization. Cambridge University Press, Cambridge, p.16-36.Google Scholar
28. Prussing JE, Conway BA, 1993. Orbital Mechanics. Oxford University Press, New York, USA.
29. Shampine LF, Gladwell I, Thompson S, 2003. Solving ODEs with Matlab. Cambridge University Press, Cambridge.
30. Ting L, 1960. Optimum orbital transfer by impulses. ARS J, 30(11):1013–1018. https://doi.org/10.2514/8.5305
31. Vertregt M, 1958. Interplanetary orbits. J Br Interplanet Soc, 16:326–354.Google Scholar
32. Yu ML, 1990. Selection of launch trajectory for launching geosynchronous satellite. Chin Space Sci Technol, 2(1):21–27 (in Chinese).Google Scholar
33. Yuan FY, Matsushima K, 1995. Strong Hohmann transfer theorem. J Guid Contr Dynam, 18(2):371–373. https://doi.org/10.2514/3.21394
34. Zefran M, Desai JP, Kumar V, 1996. Continuous motion plans for robotic systems with changing dynamic behavior. Proc 2nd Int Workshop on Algorithmic Foundations of Robotics.Google Scholar
35. Zhang G, Zhang XY, Cao XB, 2014. Tangent-impulse transfer from elliptic orbit to an excess velocity vector. Chin J Aeronaut, 27(3):577–583. https://doi.org/10.1016/j.cja.2014.04.006