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January  2012, 17(1): 245-269. doi: 10.3934/dcdsb.2012.17.245

Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion

Gautier Picot 1,

1. 

Mathematics Institute, Bourgogne University, 9 avenue Savary, 21078 Dijon, France

Received  March 2010 Revised  April 2011 Published  October 2011

In this article we describe the principle of computations of optimal transfers between quasi-Keplerian orbits in the Earth-Moon system using low-propulsion. The spacecraft's motion is modelled by the equations of the control restricted 3-body problem and we base our work on previous studies concerning the orbit transfer in the two-body problem where geometric and numeric methods were developed to compute optimal solutions. Using numerical simple shooting and continuation methods connected with fundamental results from control theory, such as the Pontryagin Maxium Principle and the second order optimality conditions related to the concept of conjugate points, we compute time-minimal and energy-minimal trajectories between the geostationary initial orbit and a final circular orbit around the Moon, passing through the neighborhood of the libration point $L_1$. Our computations give simple trajectories, obtained by referring to numerical values of the SMART-1 mission.
Keywords: 3-body problem, Earth-Moon transfer, time minimization problem, Geometric control theory, numerical analysis method, energy minimization problem..
Mathematics Subject Classification: Primary: 49K15, 49M05; Secondary: 70F0.
Citation: Gautier Picot. Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 245-269. doi: 10.3934/dcdsb.2012.17.245
References:
[1]

A. A. Agrachev and A. V. Sarychev, On abnormals extremals for Lagrange variational problems, J. Math. Systems Estim. Control, 8 (1998), 87-118.  Google Scholar

[2]

E. L Allgower and K. Georg, "Numerical Continuation Methods. An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990.  Google Scholar

[3]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," Second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[4]

J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170. doi: 10.1137/S1111111102409080.  Google Scholar

[5]

C. Bischof, A. Carle, P. Kladem and A. Mauer, Adifor 2.0: Automatic Differentiation of Fortran 77 programs, IEEE Computational Science and Engineering, 3 (1996), 18-32. doi: 10.1109/99.537089.  Google Scholar

[6]

G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago, Ill., 1946.  Google Scholar

[7]

A. Bombrun, J. Chetboun and J.-B. Pomet, Transfert Terre-Lune en poussée faible par contrôle feedback - La mission SMART-1, (French) INRIA Research Report, 5955 (2006), 1-27. Google Scholar

[8]

B. Bonnard, J.-B. Caillau and G. Picot, Geometric and numerical techniques in optimal control of the two and three-body problems, Commun. Inf. Syst., 10 (2010), 239-278.  Google Scholar

[9]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. and Calc. Var., 13 (2007), 207-236.  Google Scholar

[10]

B. Bonnard, J.-B. Caillau and E. Trélat, COTCOT: Short reference manual, ENSEEIHT-IRIT Technical Report RT/APO/05/1, (2005), 1-15. Google Scholar

[11]

B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Math. and Applications, 40, Springer-Verlag, Berlin, 2003.  Google Scholar

[12]

B. Bonnard, L. Faubourg and E. Trélat, "Mécanique Céleste et Contôle des Véhicules Spatiaux," Mathématiques & Applications (Berlin), 51, Springer-Verlag, Berlin, 2006.  Google Scholar

[13]

B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, (French)[Theory of the singularities of the input/output mapping and optimality of singular trajectories in the minimal-time problem], Forum Math., 5 (1993), 111-159. doi: 10.1515/form.1993.5.111.  Google Scholar

[14]

B. Bonnard, N. Shcherbakova and D. Sugny, The smooth continuation method in optimal control with an application to quantum systems, ESAIM Control Optim. and Calc. Var., 17 (2011), 267-292. doi: 10.1051/cocv/2010004.  Google Scholar

[15]

J.-B. Caillau, "Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux," Ph.D thesis, Toulouse University, 2000. Google Scholar

[16]

J.-B. Caillau, B. Daoud and J. Gergaud, On some Riemannian aspects of two and three-body controlled problems, in "Recent Advances in Optimization and its Applications in Engineering," Springer, (2010), 205-224. Google Scholar

[17]

J.-B. Caillau, B. Daoud and J. Gergaud, Discrete and differential homotopy in circular restricted three-body control, to appear in AIMS Proceedings, 2010. Google Scholar

[18]

J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem, ESAIM Control Optim. Calc. Var., 12 (2006), 294-310. doi: 10.1051/cocv:2006003.  Google Scholar

[19]

G. Gómez, S. Koon, M. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem ans space mission design, in "The Proceedings of AIAA/AAS Astrodynamics Specialist Meeting," Quebec City, Quebec, Canada, 2001. Google Scholar

[20]

M. Guerra and A. Sarychev, Existence and Lipschitzian regularity for relaxed minimizers, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 231-250.  Google Scholar

[21]

J. E. Marsden and S. D. Ross, New methods in celestial mechanics and mission design, Bull. Amer. Math. Soc. (N.S), 43 (2006), 43-73.  Google Scholar

[22]

K. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992.  Google Scholar

[23]

H. Poincaré, "Oeuvres," Gauthier-Villars, Paris, 1934. Google Scholar

[24]

H. Pollard, "Mathematical Introduction to Celestial Mechanics," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966.  Google Scholar

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[26]

G. Racca, B. H. Foing and M. Coradini, SMART-1: The first time of Europe to the moon, Earth, Moon and Planets, 85-86 (2001), 379-390. Google Scholar

[27]

G. Racca, et al., SMART-1 mission description and development status, Planetary and Space Science, 50 (2002), 1323-1337. Google Scholar

[28]

A. V. Saryčev, Index of second variation of a control system, Mat. Sb. (N.S.), 113(155) (1980), 464-486, 496.  Google Scholar

[29]

L. F. Shampine, H. A. Watts and S. Davenport, Solving non-stiff ordinary differential equations-the state of the art, SIAM Rev., 18 (1976), 376-411.  Google Scholar

[30]

V. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, 1967. Google Scholar

show all references

References:
[1]

A. A. Agrachev and A. V. Sarychev, On abnormals extremals for Lagrange variational problems, J. Math. Systems Estim. Control, 8 (1998), 87-118.  Google Scholar

[2]

E. L Allgower and K. Georg, "Numerical Continuation Methods. An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990.  Google Scholar

[3]

V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," Second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.  Google Scholar

[4]

J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170. doi: 10.1137/S1111111102409080.  Google Scholar

[5]

C. Bischof, A. Carle, P. Kladem and A. Mauer, Adifor 2.0: Automatic Differentiation of Fortran 77 programs, IEEE Computational Science and Engineering, 3 (1996), 18-32. doi: 10.1109/99.537089.  Google Scholar

[6]

G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago, Ill., 1946.  Google Scholar

[7]

A. Bombrun, J. Chetboun and J.-B. Pomet, Transfert Terre-Lune en poussée faible par contrôle feedback - La mission SMART-1, (French) INRIA Research Report, 5955 (2006), 1-27. Google Scholar

[8]

B. Bonnard, J.-B. Caillau and G. Picot, Geometric and numerical techniques in optimal control of the two and three-body problems, Commun. Inf. Syst., 10 (2010), 239-278.  Google Scholar

[9]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. and Calc. Var., 13 (2007), 207-236.  Google Scholar

[10]

B. Bonnard, J.-B. Caillau and E. Trélat, COTCOT: Short reference manual, ENSEEIHT-IRIT Technical Report RT/APO/05/1, (2005), 1-15. Google Scholar

[11]

B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Math. and Applications, 40, Springer-Verlag, Berlin, 2003.  Google Scholar

[12]

B. Bonnard, L. Faubourg and E. Trélat, "Mécanique Céleste et Contôle des Véhicules Spatiaux," Mathématiques & Applications (Berlin), 51, Springer-Verlag, Berlin, 2006.  Google Scholar

[13]

B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, (French)[Theory of the singularities of the input/output mapping and optimality of singular trajectories in the minimal-time problem], Forum Math., 5 (1993), 111-159. doi: 10.1515/form.1993.5.111.  Google Scholar

[14]

B. Bonnard, N. Shcherbakova and D. Sugny, The smooth continuation method in optimal control with an application to quantum systems, ESAIM Control Optim. and Calc. Var., 17 (2011), 267-292. doi: 10.1051/cocv/2010004.  Google Scholar

[15]

J.-B. Caillau, "Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux," Ph.D thesis, Toulouse University, 2000. Google Scholar

[16]

J.-B. Caillau, B. Daoud and J. Gergaud, On some Riemannian aspects of two and three-body controlled problems, in "Recent Advances in Optimization and its Applications in Engineering," Springer, (2010), 205-224. Google Scholar

[17]

J.-B. Caillau, B. Daoud and J. Gergaud, Discrete and differential homotopy in circular restricted three-body control, to appear in AIMS Proceedings, 2010. Google Scholar

[18]

J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem, ESAIM Control Optim. Calc. Var., 12 (2006), 294-310. doi: 10.1051/cocv:2006003.  Google Scholar

[19]

G. Gómez, S. Koon, M. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem ans space mission design, in "The Proceedings of AIAA/AAS Astrodynamics Specialist Meeting," Quebec City, Quebec, Canada, 2001. Google Scholar

[20]

M. Guerra and A. Sarychev, Existence and Lipschitzian regularity for relaxed minimizers, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 231-250.  Google Scholar

[21]

J. E. Marsden and S. D. Ross, New methods in celestial mechanics and mission design, Bull. Amer. Math. Soc. (N.S), 43 (2006), 43-73.  Google Scholar

[22]

K. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992.  Google Scholar

[23]

H. Poincaré, "Oeuvres," Gauthier-Villars, Paris, 1934. Google Scholar

[24]

H. Pollard, "Mathematical Introduction to Celestial Mechanics," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966.  Google Scholar

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[26]

G. Racca, B. H. Foing and M. Coradini, SMART-1: The first time of Europe to the moon, Earth, Moon and Planets, 85-86 (2001), 379-390. Google Scholar

[27]

G. Racca, et al., SMART-1 mission description and development status, Planetary and Space Science, 50 (2002), 1323-1337. Google Scholar

[28]

A. V. Saryčev, Index of second variation of a control system, Mat. Sb. (N.S.), 113(155) (1980), 464-486, 496.  Google Scholar

[29]

L. F. Shampine, H. A. Watts and S. Davenport, Solving non-stiff ordinary differential equations-the state of the art, SIAM Rev., 18 (1976), 376-411.  Google Scholar

[30]

V. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, 1967. Google Scholar

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