In this work, we develop a new method to design energy minimum low-thrust missions (-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach.
Accepté le :
DOI : 10.1051/m2an/2016044
Keywords: Three body problem, optimal control, low-thrust transfer, Lyapunov orbit, Halo orbit, continuation method
Chupin, Maxime 1, 2 ; Haberkorn, Thomas 3 ; Trélat, Emmanuel 1
@article{M2AN_2017__51_3_965_0,
author = {Chupin, Maxime and Haberkorn, Thomas and Tr\'elat, Emmanuel},
title = {Low-thrust {Lyapunov} to {Lyapunov} and {Halo} to {Halo} missions with $L^{2}$-minimization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {965--996},
year = {2017},
publisher = {EDP Sciences},
volume = {51},
number = {3},
doi = {10.1051/m2an/2016044},
mrnumber = {3666653},
zbl = {1370.49024},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2016044/}
}
TY - JOUR
AU - Chupin, Maxime
AU - Haberkorn, Thomas
AU - Trélat, Emmanuel
TI - Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with $L^{2}$-minimization
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2017
SP - 965
EP - 996
VL - 51
IS - 3
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/m2an/2016044/
DO - 10.1051/m2an/2016044
LA - en
ID - M2AN_2017__51_3_965_0
ER -
%0 Journal Article
%A Chupin, Maxime
%A Haberkorn, Thomas
%A Trélat, Emmanuel
%T Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with $L^{2}$-minimization
%J ESAIM: Mathematical Modelling and Numerical Analysis
%D 2017
%P 965-996
%V 51
%N 3
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2016044/
%R 10.1051/m2an/2016044
%G en
%F M2AN_2017__51_3_965_0
Chupin, Maxime; Haberkorn, Thomas; Trélat, Emmanuel. Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with $L^{2}$-minimization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 965-996. doi: 10.1051/m2an/2016044
E.L. Allgower and K. Georg, Introduction to numerical continuation methods, vol. 45 of Classics in Applied Mathematics. Reprint of the 1990 edition [Springer-Verlag, Berlin; MR1059455 (92a:65165)]. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003). | MR | Zbl
, and , Eight-shaped Lissajous orbits in the Earth-Moon system. MathS in Action 4 (2011) 1–23. | MR | Zbl | DOI
and , New smoothing techniques for solving bang–bang optimal control problems–numerical results and statistical interpretation. Optim. Control Appl. Methods 23 (2002) 171–197. | MR | Zbl | DOI
B. Bonnard, L. Faubourg and E. Trélat, Mécanique céleste et contrôle des véhicules spatiaux. Vol. 51 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006). | MR | Zbl
J.-B. Caillau, Contribution àl’étude du contrle en temps minimal des transferts orbitaux. Ph.D. thesis, Institut National Polytechnique de Toulouse, Toulouse, France (2000).
and , Minimum time control of the restricted three-body problem. SIAM J. Control Optim. 50 (2012) 3178–3202. | MR | Zbl | DOI
Z. Chen, J.-B. Caillau and Y. Chitour, -minimization for mechanical systems. Available at: (2015). | HAL | MR
, and , Differential pathfollowing for regular optimal control problems. Optim. Methods Software 27 (2012) 177–196. | MR | Zbl | DOI
B. Daoud, Contribution au contrle optimal du problème circulaire restreint des trois corps. Ph.D. thesis, Université de Bourgogne (2011).
R. Epenoy, Optimal long-duration low-thrust transfers between libration point orbits. In Proc. of the 63rd International Astronautical Congress, in vol. 7. Naples, Italy (2012).
L. Euler, De motu rectilineo trium corpörum se mutuo attrahentium. Oeuvres, Seria Secunda tome XXv Commentationes Astronomicae (1767) 144–151.
and , Quasi-periodic orbits about the translunar libration point. Celestial Mechanics 7 (1973) 458–473. | Zbl | DOI
, , and , Trajectories and Orbital Maneuvers for the First Libration-Point Satellite. J. Guid. Control Dynam. 3 (1980) 549–554. | DOI
and , Homotopy method for minimum consumption orbit transfer problem. ESAIM: COCV 12 (2006) 294–310. | MR | Zbl | Numdam
G. Gomez and J. Masdemont, Some zero cost transfers between libration point orbits. In Point Orbits, AAS paper 00-177, AAS/AIAA Astrodynamics Specialist Conference (2000).
G. Gómez, J. Masdemont, C. Simó and A. Jorba, Study refinement of semi-analytical halo orbit theory: Executive summary.
G. Gómez, J. Masdemont and C. Simó, Lissajous orbits around halo orbits. Adv. Astronaut. Sci. (1997). ESOC Contract No.: 8625/89/D/MD (SC) (1991).
, and , Low thrust minimum-fuel orbital transfer: a homotopic approach. J. Guid. Control Dynam. 27 (2004) 1046–1060. | DOI
, and , Practical techniques for low-thrust trajectory optimization with homotopic approach. J. Guid. Control Dynam. 35 (2012) 245–258. | DOI
and , Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D 132 (1999) 189–213. | MR | Zbl | DOI
W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross, Dynamical systems, the three-body problem and space mission design. In International Conference on Differential Equations, Vols. 1, 2 (Berlin, 1999). World Sci. Publ., River Edge, NJ (2000) 1167–1181. | MR | Zbl
, , and , Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10 (2000) 427–469. | MR | Zbl | DOI
J.-L. Lagrange, Essai sur le problème des trois corps. Prix de l’académie royale des Sciences de paris, tome IX, Oeuvres de Lagrange 6, Gauthier-Villars (1772) 272–282.
C. Martin and B.A. Conway, Optimal low-thrust trajectories using stable manifolds. In Spacecraft Trajectory Optimization, edited by B.A. Conway. Cambridge University Press (2010) 238–262.
K.R. Meyer, G.R. Hall and D. Offin, Introduction to Hamiltonian dynamical systems and the -body problem. Vol. 90 of Appl. Math. Sci., 2nd edition. Springer, New York (2009). | MR | Zbl
, and , Combined optimal low-thrust and stable-manifold trajectories to the Earth-Moon halo orbits. AIP Conf. Proc. 886 (2007) 100–112. | MR | DOI
, and , Low-energy, low-thrust transfers to the Moon. Celestial Mech. Dyn. Astron. 105 (2009) 61–74. | MR | Zbl | DOI
, and , Optimal low-thrust invariant manifold trajectories via attainable sets. J. Guid. Control Dynam. 34 (2011) 1644–1655. | DOI
and , Low-thrust transfers in the Earth-Moon system, including applications to libration point orbits. J. Guid. Control Dynam. 33 (2010) 533–549. | DOI
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. Translated by D.E. Brown. A Pergamon Press Book. The Macmillan Co., New York (1964). | MR | Zbl
, Analytic construction of periodic orbits about the collinear points. Celestial Mech. 22 (1980) 241–253. | MR | Zbl | DOI
, and , Low-thrust variable-specific-impulse transfers and guidance to unstable periodic orbits. J. Guid. Control Dynam. 28 (2005) 280–290. | DOI
T. Starchville and R. Melton, Optimal low-thrust trajectories to Earth-Moon L2 Halo orbits (circular problem). In Proc. of the AAS/AIAA Astrodynamics Specialists Conference. American Astro- nomical Soc. (1997) 97–714.
V.G. Szebehely, Theory of Orbits – The restricted problem of three bodies. Academic Press (1967).
, Fast numerical approximation of invariant manifolds in the circular restricted three-body problem. Commun. Nonlinear Sci. Numer. Simulat. 32 (2016) 89–98. | MR | Zbl | DOI
E. Trélat, Contrôle optimal. In Mathématiques Concrètes. [Concrete Mathematics]. Théorie & applications. [Theory and applications]. Vuibert, Paris (2005). | MR | Zbl
, Optimal control and applications to aerospace: some results and challenges. J. Optim. Theory Appl. 154 (2012) 713–758. | MR | Zbl | DOI
and , The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Eq. 258 (2015) 81–114. | MR | Zbl | DOI
L.T. Watson, HOMPACK90: FORTRAN 90 Codes for Globally Convergent Homotopy Algorithms. Department of Computer Science, Virginia Polytechnic Institute and State University (1996).
F.B. Zazzera, F. Topputo and M. Massari, Assessment of mission design including utilisation of libration points and weak stability boundaries. Technical Report 03-4103b, European Space Agency, the Advanced Concepts Team. Available on line at: www.esa.int/act (2004).
, , and , Low-thrust minimum-fuel optimization in the circular restricted three-body problem. J. Guid. Control Dynam. 38 (2015) 1501–1510. | DOI
Cité par Sources :
