Abstract
A numerical study of optimal time-fixed low-thrust limited power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is performed by means of two different approaches. The first approach uses a numerical method based on the second variation theory, usually known as neighboring extremals method, to solve the two-point boundary value problem obtained from the application of the Pontryagin Maximum Principle to the optimization problem formulated as a Mayer problem with the radial distance and the components of the velocity vector as state variables. The second approach is based on the solution of the two-point boundary value problem defined by a first-order analytical solution expressed in terms of non-singular orbital elements, which include short periodic terms, and derived through canonical transformations theory in a previous work. For transfers between close orbits, a simplified solution expressed by a linear system of algebraic equations is straightforwardly derived from this analytical first-order solution. In this case, the two-point boundary value problem can be solved by simple techniques. Numerical results are presented for transfers between circular orbits, considering several radius ratios and transfer durations. Some maneuvers involving orbits with arbitrary small eccentricities are also considered. The fuel consumption is taken as the performance criterion in comparison of the results.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig9_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig10_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig11_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig12_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig13_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig14_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig15_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig16_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig17_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig18_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40314-016-0325-9/MediaObjects/40314_2016_325_Fig19_HTML.gif)
Similar content being viewed by others
References
Bertrand R, Bernussou J, Geffroy S et al (2001) Electric transfer optimization for mars sample return mission. Acta Astronaut 48(5):651–660
Bonnard B, Caillau JB, Dujol R (2006) Averaging and optimal control of elliptic Keplerian orbits with low propulsion. Syst Control Lett 55:755–760
Camino O, Alonso M, Blake R et al (2005) SMART-1: Europe’s lunar mission paving the way for new cost effective ground operations (RCSGSO). In: Proceedings of the 6th International Symposium on Reducing the Costs of Spacecraft Ground Systems and Operations (RCSGSO), pp 14–17. ESA/ESOC, Darmstadt
Da Silva Fernandes S, Carvalho FC (2008) A first-order analytical theory for optimal low-thrust limited-power transfers between arbitrary elliptical coplanar orbits. Math Probl Eng 30 (article ID 525930)
Da Silva Fernandes S, Silveira Filho CR, Golfetto WA (2012) A numerical study of low-thrust limited power trajectories between coplanar circular orbits in an inverse-square force field. Math Probl Eng 24 (article ID 168632)
Da Silva Fernandes S, Carvalho FC, Moraes RV (2015) Optimal low-thrust transfers between coplanar orbits with small eccentricities. Comput Appl Math (accepted for publication)
Edelbaum TN (1964) Optimum low-thrust rendezvous and station kee**. AIAA J 2(7):1196–1201
Edelbaum TN (1965) Optimum power-limited orbit transfer in strong gravity fields. AIAA J 3(5):921–925
Edelbaum TN (1966) An asymptotic solution for optimum power limited orbit transfer. AIAA J 4(8):1491–1494
Geffroy S, Epenoy R (1997) Optimal low-thrust transfers with constraints—generalization of averaging techniques. Acta Astronaut 41(3):133–149
Gobetz FW (1964) Optimal variable-thrust transfer of a power-limited rocket between neighboring circular orbits. AIAA J 2(2):339–343
Gobetz FW (1965) A linear theory of optimum low-thrust rendezvous trajectories. J Astronaut Sci 12(3):69–76
Haissig CM, Mease KD, Vinh NX (1993) Minimum-fuel power-limited transfers between coplanar elliptical orbits. Acta Astronaut 29(1):1–15
Huang W (2012) Solving coplanar power-limited orbit transfer problem by primer vector approximation method. Int J Aerosp Eng 9 (article ID 480320)
Jamison BR, Coverstone V (2010) Analytical study of the primer vector and orbit transfer switching function. J Guid Control Dyn 33(1):235–245
Jiang F, Baoyin H, Li J (2012) Practical techniques for low-thrust trajectory optimization with homotopic approach. J Guid Control Dyn 35(1):245–258
Kiforenko BN (2005) Optimal low-thrust orbital transfers in a central gravity field. Int Appl Mech 41(11):1211–1238
Marec JP (1979) Optimal space trajectories. Elsevier, New York
Marec JP, Vinh NX (1980) Étude generale des transferts optimaux a poussee faible et puissance limitee entre orbites elliptiques quelconques. ONERA Publication
Quarta AA, Mengali G (2013) Trajectory approximation for low-performance electric sail with constant thrust angle. J Guid Control Dyn 36(3):884–887
Racca GD, Marini A, Stagnaro L et al (2002) SMART-1 mission description and developments status. Planet Space Sci 50:1323–1337
Racca GD (2003) New challengers to trajectory design by the use of electric propulsion and other new means wandering in the solar system. Celest Mech Dyn Astron 85(1):1–24
Rayman MD, Varghese P, Lehman DH et al (2000) Results from the Deep Space 1 technology validation mission. Acta Astronaut 47(2–9):475–487
Roberts SM, Shipman JS, Roth CV (1968) Continuation in quasilinearization. J Optim Theory Appl 2(3):164–178
Stoer J, Bulirsch R (2002) Introduction to numerical analysis, 3rd edn. Springer, New York
Sukhanov AA (2000) Optimization of interplanetary low-thrust transfers. Cosm Res 38(6):584–587
Vallado DA (2007) Fundamentals of astrodynamics and applications, 3rd edn. Springer, New York
Acknowledgments
This research has been supported by CNPq under contract 304913/2013-8 and FAPESP under contract 2012/21023-6.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Elbert E. N. Macau, Antônio Fernando Bertachini de Almeida Prado and Cristiano Fiorilo de Melo.
Rights and permissions
About this article
Cite this article
das Chagas Carvalho, F., da Silva Fernandes, S. & de Moraes, R.V. A numerical study for optimal low-thrust limited power transfers between coplanar orbits with small eccentricities. Comp. Appl. Math. 35, 907–936 (2016). https://doi.org/10.1007/s40314-016-0325-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-016-0325-9