April  2014, 10(2): 477-501. doi: 10.3934/jimo.2014.10.477

Designing rendezvous missions with mini-moons using geometric optimal control

1. 

Department of Mathematics, University of Hawaii at Manoa, Honolulu, United States, United States, United States

2. 

Department of Physics, University of Helsinki, Helsinki, Finland

3. 

Institute for Astronomy, University of Hawaii at Manoa, Honolulu, United States

4. 

Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, Paris, France

Received  March 2013 Revised  August 2013 Published  October 2013

Temporarily-captured Natural Earth Satellites (NES) are very appealing targets for space missions for many reasons. Indeed, NES get captured by the Earth's gravity for some period of time, making for a more cost-effective and time-effective mission compared to a deep-space mission, such as the 7-year Hayabusa mission. Moreover, their small size introduces the possibility of returning with the entire temporarily-captured orbiter (TCO) to Earth. Additionally, NES can be seen as interesting targets when examining figures of their orbits. It requires to expand the current state-of-art of the techniques in geometric optimal control applied to low-thrust orbital transfers. Based on a catalogue of over sixteen-thousand NES, and assuming ionic propulsion for the spacecraft, we compute time minimal rendezvous missions for more than $96%$ of the NES. The time optimal control transfers are calculated using classical indirect methods of optimal control based on the Pontryagin Maximum Principle. Additionally we verify the local optimality of the transfers using second order conditions.
Citation: Monique Chyba, Geoff Patterson, Gautier Picot, Mikael Granvik, Robert Jedicke, Jeremie Vaubaillon. Designing rendezvous missions with mini-moons using geometric optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 477-501. doi: 10.3934/jimo.2014.10.477
References:
[1]

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

[2]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004).

[3]

E. L. Allgower and K. Georg, Numerical Continuation Methods. An Introduction,, Springer Series in Computational Mathematics, (1990). doi: 10.1007/978-3-642-61257-2.

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics,, 2nd edition, (1989).

[5]

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

[6]

E. Belbruno, Capture Dynamics and Chaotic Motion in Celestial Mechanics. With Applications to the Construction of Low Energy Transfers,, Princeton University Press, (2004).

[7]

E. Belbruno, Fly me to the Moon. An Insider'S Guide to the New Science of Space Travel,, Princeton University Press, (2007).

[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. doi: 10.4310/CIS.2010.v10.n4.a5.

[9]

B. Bonnard, J.-B. Caillau and E. Télat, Geometric optimal control of elliptic Keplerian orbits,, Discrete Cont. Dyn. Syst. Ser. B, 5 (2005), 929. doi: 10.3934/dcdsb.2005.5.929.

[10]

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. doi: 10.1051/cocv:2007012.

[11]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2003).

[12]

B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle des Véhicules Spatiaux,, Mathématiques & Applications (Berlin) [Mathematics & Applications], 51 (2006).

[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], 5 (1998), 111. doi: 10.1515/form.1993.5.111.

[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. doi: 10.1051/cocv/2010004.

[15]

E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control. Optimization, Estimation and Control,, Revised printing, (1975).

[16]

J.-B. Caillau, Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux,, Ph.D thesis, (2000).

[17]

M. Chyba, G. Picot, G. Patterson, R. Jedicke, M. Granvik and J. Vaubaillon, Time-minimal orbital transfers to temporarily-captured natural Earth satellites,, to appear in OCA5 - Advances in Optimization and Control with Applications, (2013).

[18]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems,, Optim. Methods Softw., 27 (2012), 177. doi: 10.1080/10556788.2011.593625.

[19]

B. Daoud, Contribution au Contrôle Optimal du Problème Circulaire Restreint des Trois Corps,, Ph.D thesis, (2011).

[20]

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

[21]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariants manifolds, the spatial three-body problem and space mission design,, in AAS/AIAA Astrodynamics Specialtists Conference, (2001).

[22]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Connecting orbits and invariant manifolds in the spatial three-body problem,, Nonlinearity, 17 (2004), 1571. doi: 10.1088/0951-7715/17/5/002.

[23]

M. Granvik, J. Vaubaillon and R. Jedicke, The population of natural Earth satellites,, Icarus, 218 (2012), 262. doi: 10.1016/j.icarus.2011.12.003.

[24]

V. Jurdjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).

[25]

J. Kawagachi, A. Fujiwara and T. K. Uesugi, The ion engine cruise operation and the Earth swingby of Hayabusa (MUSES-C),, in Proceddings of the 55th International Astronotical Congress, (2004).

[26]

T. Kubota, T. Hashimoto, J. Kawagachi, M. Uo and M. Shirakawa, Guidance and Navigation of Hayabusa spacecraft to asteroid exploration and sample return mission,, in Proceddings of SICE-ICASE, (2006), 2793. doi: 10.1109/SICE.2006.314761.

[27]

D. Liberzon, Calculus of Variations and Optimal Control Theory. A Concise Introduction,, Princeton University Press, (2012).

[28]

H. Mäurer, First and second order sufficient optimality conditions in mathematical programming and optimal control,, in Mathematical Programming at Oberwolfach (Proc. Conf., (1979), 163.

[29]

A. Moore, Discrete Mechanics and Optimal Control for Space Trajectory Design,, Ph.D thesis, (2011).

[30]

I. Newton, Principes Mathématiques de la Philosophie Naturelle. Tome I, II. (French) Traduction de la Marquise du Chastellet, Augmentée des Commentaires de Clairaut,, Librairie Scientifique et Technique Albert Blanchard, (1966).

[31]

G. Picot, Shooting and numerical continuation method for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion,, Discrete Cont. Dyn. Syst. Ser. B, 17 (2012), 245. doi: 10.3934/dcdsb.2012.17.245.

[32]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Interscience Publishers John Wiley & Sons, (1962).

[33]

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

[34]

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

[35]

A. G. Santo, S. C Lee and R. E Gold, NEAR spacecraft and instrumentation,, J. Astronomical Sciences, 43 (1995), 373.

[36]

A. V. Sary\vchev, Index of second variation of a control system,, Mat. Sb. (N.S), 113(155) (1980), 464.

[37]

V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies,, Academic Press, (1967).

[38]

D. A Vallado, Fundamentals of Astrodynamics and Applications,, Springer, (2001).

[39]

V. Zeidan, First and second order sufficient conditions for optimal control and the calculus of variations,, Appl. Math. and Optim., 11 (1984), 209. doi: 10.1007/BF01442179.

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.

[2]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004).

[3]

E. L. Allgower and K. Georg, Numerical Continuation Methods. An Introduction,, Springer Series in Computational Mathematics, (1990). doi: 10.1007/978-3-642-61257-2.

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics,, 2nd edition, (1989).

[5]

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

[6]

E. Belbruno, Capture Dynamics and Chaotic Motion in Celestial Mechanics. With Applications to the Construction of Low Energy Transfers,, Princeton University Press, (2004).

[7]

E. Belbruno, Fly me to the Moon. An Insider'S Guide to the New Science of Space Travel,, Princeton University Press, (2007).

[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. doi: 10.4310/CIS.2010.v10.n4.a5.

[9]

B. Bonnard, J.-B. Caillau and E. Télat, Geometric optimal control of elliptic Keplerian orbits,, Discrete Cont. Dyn. Syst. Ser. B, 5 (2005), 929. doi: 10.3934/dcdsb.2005.5.929.

[10]

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. doi: 10.1051/cocv:2007012.

[11]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory,, Mathématiques & Applications (Berlin) [Mathematics & Applications], (2003).

[12]

B. Bonnard, L. Faubourg and E. Trélat, Mécanique Céleste et Contrôle des Véhicules Spatiaux,, Mathématiques & Applications (Berlin) [Mathematics & Applications], 51 (2006).

[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], 5 (1998), 111. doi: 10.1515/form.1993.5.111.

[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. doi: 10.1051/cocv/2010004.

[15]

E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control. Optimization, Estimation and Control,, Revised printing, (1975).

[16]

J.-B. Caillau, Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux,, Ph.D thesis, (2000).

[17]

M. Chyba, G. Picot, G. Patterson, R. Jedicke, M. Granvik and J. Vaubaillon, Time-minimal orbital transfers to temporarily-captured natural Earth satellites,, to appear in OCA5 - Advances in Optimization and Control with Applications, (2013).

[18]

J.-B. Caillau, O. Cots and J. Gergaud, Differential continuation for regular optimal control problems,, Optim. Methods Softw., 27 (2012), 177. doi: 10.1080/10556788.2011.593625.

[19]

B. Daoud, Contribution au Contrôle Optimal du Problème Circulaire Restreint des Trois Corps,, Ph.D thesis, (2011).

[20]

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

[21]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariants manifolds, the spatial three-body problem and space mission design,, in AAS/AIAA Astrodynamics Specialtists Conference, (2001).

[22]

G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Connecting orbits and invariant manifolds in the spatial three-body problem,, Nonlinearity, 17 (2004), 1571. doi: 10.1088/0951-7715/17/5/002.

[23]

M. Granvik, J. Vaubaillon and R. Jedicke, The population of natural Earth satellites,, Icarus, 218 (2012), 262. doi: 10.1016/j.icarus.2011.12.003.

[24]

V. Jurdjevic, Geometric Control Theory,, Cambridge Studies in Advanced Mathematics, (1997).

[25]

J. Kawagachi, A. Fujiwara and T. K. Uesugi, The ion engine cruise operation and the Earth swingby of Hayabusa (MUSES-C),, in Proceddings of the 55th International Astronotical Congress, (2004).

[26]

T. Kubota, T. Hashimoto, J. Kawagachi, M. Uo and M. Shirakawa, Guidance and Navigation of Hayabusa spacecraft to asteroid exploration and sample return mission,, in Proceddings of SICE-ICASE, (2006), 2793. doi: 10.1109/SICE.2006.314761.

[27]

D. Liberzon, Calculus of Variations and Optimal Control Theory. A Concise Introduction,, Princeton University Press, (2012).

[28]

H. Mäurer, First and second order sufficient optimality conditions in mathematical programming and optimal control,, in Mathematical Programming at Oberwolfach (Proc. Conf., (1979), 163.

[29]

A. Moore, Discrete Mechanics and Optimal Control for Space Trajectory Design,, Ph.D thesis, (2011).

[30]

I. Newton, Principes Mathématiques de la Philosophie Naturelle. Tome I, II. (French) Traduction de la Marquise du Chastellet, Augmentée des Commentaires de Clairaut,, Librairie Scientifique et Technique Albert Blanchard, (1966).

[31]

G. Picot, Shooting and numerical continuation method for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion,, Discrete Cont. Dyn. Syst. Ser. B, 17 (2012), 245. doi: 10.3934/dcdsb.2012.17.245.

[32]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Interscience Publishers John Wiley & Sons, (1962).

[33]

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

[34]

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

[35]

A. G. Santo, S. C Lee and R. E Gold, NEAR spacecraft and instrumentation,, J. Astronomical Sciences, 43 (1995), 373.

[36]

A. V. Sary\vchev, Index of second variation of a control system,, Mat. Sb. (N.S), 113(155) (1980), 464.

[37]

V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies,, Academic Press, (1967).

[38]

D. A Vallado, Fundamentals of Astrodynamics and Applications,, Springer, (2001).

[39]

V. Zeidan, First and second order sufficient conditions for optimal control and the calculus of variations,, Appl. Math. and Optim., 11 (1984), 209. doi: 10.1007/BF01442179.

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