Orbital transfers: optimization methods      and recent results

Mauro Pontani

doi:10.3934/naco.2011.1.435

Article Contents

2011, Volume 1, Issue 3: 435-485. Doi: 10.3934/naco.2011.1.435

This issue Previous Article Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization Next Article A note on monotone approximations of minimum and maximum functions and multi-objective problems

Orbital transfers: optimization methods and recent results

Mauro Pontani^1,

1.
Scuola di Ingegneria Aerospaziale -- University of Rome "La Sapienza", via Salaria 851, 00138 Rome

Received: April 2011

Revised: July 2011

Published: September 2011

Abstract / Introduction Related Papers Cited by

Abstract

A wide variety of techniques have been employed in the past for optimizing orbital transfers, which represent the trajectories that lead a spacecraft from a given initial orbit to a specified final orbit. This paper describes several original approaches to optimizing impulsive and finite--thrust orbital transfers, and presents some very recent results. First, impulsive transfers between Keplerian trajectories are considered. A new, analytical optimization method applied to these transfers leads to conclusions of a global nature for transfers involving both ellipses and escape trajectories, without any limitation on the number of impulses, and with possible constraints on the radius of closest approach and greatest recession from the attracting body. A direct optimization technique, termed direct collocation with nonlinear programming algorithm, is then applied to finite--thrust transfers between circular orbits. Lastly, low--thrust orbital transfers are optimized through the joint use of the necessary conditions for optimality and of the recently introduced heuristic method referred to as particle swarm optimization. This work offers a complete description and demonstrates the effectiveness of the distinct techniques applied to optimizing orbital transfer problems of different nature.

Keywords:

Orbital Transfers,
direct collocation with nonlinear programming,
particle swarm optimization.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	P. J. Angeline, Evolutionary optimization versus particle swarm optimization: philosophy and performance Differences, Evolutionary programming VII, Lecture Notes in Computer Science, 1447, Springer (1998), 601-610.
[2]	R. B. Barrar, An analytic proof that the Hohmann-type transfer is the true minimum two-impulse transfer, Astronautica Acta, IX (1963), 1-11.
[3]	R. H. Battin, An introduction to the mathematics and methods of astrodynamics, AIAA Education Series, AIAA, New York (1987), 529-530.
[4]	D. J. Bell and D. H. Jacobson, "Singular Optimal Control Problems," Academic Press, Inc., London, 1975.
[5]	R. Bellman, "Dynamic Programming," Princeton University Press, Princeton, 1957.
[6]	C. R. Bessette and D. B. Spencer, Optimal space trajectory design: A heuristic-based approach, Advances in the Astronautical Sciences, 124 (2006), 1611-1628.
[7]	C. R. Bessette and D. B. Spencer, Identifying optimal interplanetary trajectories through a genetic approach, Paper AIAA 2006-6306, (2006).
[8]	J. T. Betts, Optimal interplanetary orbit transfers by direct transcription, Journal of the Astronautical Sciences, 42 (1994), 247-326.
[9]	G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago (1946), 108-112.
[10]	K. R. Brown, E. F. Harrold and G. W. Johnson, Rapid optimization of multiple-burn rocket flights, NASA CR-1430 (1969).
[11]	R. G. Brusch and T. L. Vincent, Numerical implementation of a second-order variational endpoint condition, AIAA Journal, 8 (1970), 2230-2235.doi: 10.2514/3.6092.
[12]	A. E. Bryson and Y. C. Ho, "Applied Optimal Control," Ginn and Company, Waltham, 1969.
[13]	A. Carlisle and G. Dozier, An Off-The-Shelf PSO, Proceedings of the Workshop on Particle Swarm Optimization, Indianapolis (2001).
[14]	P. Cicala, "An Engineering Approach to the Calculus of Variations," Levrotto & Bella, Torino, 1957.
[15]	M. Clerc, The swarm and the queen: Towards a deterministic and adaptive particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999), Washington (1999).doi: 10.1109/CEC.1999.785513.
[16]	A. R. Cockshott and B. E. Hartman, Improving the Fermentation medium for Echinocandin B production part II: particle swarm optimization, Process Biochemistry, 36 (2001), 661-669.doi: 10.1016/S0032-9592(00)00261-2.
[17]	B. A. Conway, Optimal low-thrust interception of earth-crossing asteroids, Journal of Guidance, Control, and Dynamics, 20 (1997), 995-1002.doi: 10.2514/2.4146.
[18]	V. Coverstone-Carroll and S. N. Williams, Optimal low thrust trajectories using differential inclusion concepts, Journal of the Astronautical Sciences, 42 (1994), 379-393.
[19]	R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, (1995).doi: 10.1109/MHS.1995.494215.
[20]	R. C. Eberhart and Y. Shi, Comparison between genetic algorithms and particle swarm optimization, Evolutionary programming VII, Lecture Notes in Computer Science, Springer 1447 (1998), 611-616.doi: 10.1007/BFb0040812.
[21]	R. C. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2000), La Jolla (2000).
[22]	R. C. Eberhart and Y. Shi, Particle swarm optimization: developments, applications, and resources, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2001), Seoul (2001).
[23]	T. N. Edelbaum, Some extensions of the Hohmann transfer maneuver, ARS Journal, 29 (1959), 864-865.
[24]	T. N. Edelbaum, Propulsion requirements for controllable satellites, ARS Journal, 31 (1961), 1079-1089.
[25]	A. P. Engelbrecht, "Computational Intelligence. An Introduction," Wiley, Chichester, 2007.
[26]	P. J. Enright and B. A. Conway, Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming, Journal of Guidance, Control, and Dynamics, 14 (1991), 981-985.doi: 10.2514/3.20739.
[27]	P. J. Enright and B. A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal of Guidance, Control, and Dynamics, 15 (1992), 994-1002.doi: 10.2514/3.20934.
[28]	P. C. Fourie and A. A. Groenwold, Particle swarms in topology optimization, Proceedings of the Fourth World Congress of Structural and Multidisciplinary Optimization, Liaoning Electronic Press, (2001), 1771-1776.
[29]	P. C. Fourie and A. A. Groenwold, The particle swarm optimization algorithm in size and shape optimization, Structural and Multidisciplinary Optimization, 23 (2002), 259-267.
[30]	Y. Gao and and C. Kluever, Low-thrust interplanetary orbit transfer using hybrid trajectory optimization method with multiple shooting, Paper AIAA 2004-5088 (2004).
[31]	D. E. Goldberg, "Genetic Algorithms in Search, Optimization, and Machine Learning," Addison Wesley, Boston, 1989.
[32]	C. R. Hargraves and S. W. Paris, Direct trajectory optimization using nonlinear programming and collocation, Journal of Guidance, Control, and Dynamics, 10 (1987), 338-342.doi: 10.2514/3.20223.
[33]	R. Hassan, B. Cohanim and O. de Weck, Comparison of particle swarm optimization and the genetic algorithm, Paper AIAA 2005-1897, (2005).
[34]	G. A. Hazelrigg, Globally optimal impulsive transfers via Green's Theorem, Journal of Guidance, Control, and Dynamics, 7 (1983), 462-470.doi: 10.2514/3.19879.
[35]	A. L. Herman and B. A. Conway, Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592-599.doi: 10.2514/3.21662.
[36]	A. L. Herman and B. A. Conway, Optimal low-thrust, earth-moon orbit transfer, Journal of Guidance, Control, and Dynamics, 21 (1998), 141-147.doi: 10.2514/2.4210.
[37]	N. Higashi and H. Iba, Particle swarm optimization with Gaussian mutation, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003).
[38]	F. B. Hildebrand, "Introduction to Numerical Analysis," Dover, New York, 1987.
[39]	R. F. Hoelker and R. Silber, The bi-elliptical transfer between coplanar circular orbits, Proceedings of the 4th Symposium on Ballistic Missiles and Space Technology, Pergamon Press, New York, 3 (1961), 164-175.
[40]	W. Hohmann, Die Erreichbarkeit der Himmelskoerper, Oldenbourg, Munich (1925); also
[41]	X. Hu and R. Eberhart, Solving constrained nonlinear optimization problems with particle swarm optimization, Proceedings of the Sixth World Multiconference on Systemics, Cybernetics and Informatics (SCI 2002), Orlando (2002).
[42]	X. Hu, R. Eberhart and Y. Shi, Engineering optimization with particle swarm, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003).
[43]	X. Hu, Y. Shi and R. Eberhart, Recent advances in particle swarm, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2004), Portland (2004).
[44]	M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1994), 941-958.
[45]	M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1999), 941-958.
[46]	A. B. Jenkin, Representative mission trade studies for low-thrust transfers to geosynchronous orbits, paper AIAA 2004-5086 (2004).
[47]	V. Kalivarapu and E. Winer, Implementation of digital pheromones in particle swarm optimization for constrained optimization problems, Paper AIAA 2008-1974 (2008).
[48]	J. A. Kechichian, Reformulation of Edelbaum's low-thrust transfer problem using optimal control theory, Journal of Guidance, Control, and Dynamics, 20 (1997), 988-994.doi: 10.2514/2.4145.
[49]	J. A. Kechichian, Low-thrust eccentricity-constrained orbit raising, Journal of Spacecraft and Rockets, 35 (1998), 327-335.doi: 10.2514/2.3330.
[50]	J. A. Kechichian, Optimal altitude-constrained low-thrust transfer between inclined circular orbits, Journal of the Astronautical Sciences, 54 (2006), 485-503.
[51]	J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of the IEEE International Conference on Neural Networks, Piscataway, (1995).doi: 10.1109/ICNN.1995.488968.
[52]	J. Kennedy and R. Eberhart, "Swarm Intelligence," Academic Press, San Diego, 2001.
[53]	M. S. Khurana, H. Winarto and A. K. Sinha, Application of swarm approach and artificial neural networks for airfoil shape optimization, Paper AIAA 2008-5954, (2008).
[54]	S. Kitayama, K. Yamazaki and M. Arakawa, Adaptive range particle swarm optimization, Paper AIAA 2006-6912, (2006).
[55]	C. A. Kluever and B. L. Pierson, Optimal low-thrust, three-dimensional earth-moon trajectories, Journal of Guidance, Control, and Dynamics, 18 (1995), 830-837.doi: 10.2514/3.21466.
[56]	C. A. Kluever and S. R. Oleson, Direct approach for computing near-optimal low-thrust earth-orbit transfers, Journal of Spacecraft and Rockets, 35 (1998), 509-515.doi: 10.2514/2.3360.
[57]	R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444.doi: 10.2514/3.3165.
[58]	S. Koziel and Z. Michalewicz, Evolutionary algorithms, homorphous mappings, and constrained parameter optimization, Evolutionary Computation, 7 (1999), 19-44.doi: 10.1162/evco.1999.7.1.19.
[59]	D. F. Lawden, "Optimal Trajectories for Space Navigation," Butterworths, London, 1963.
[60]	D. F. Lawden, Optimal intermediate-thrust arcs in a gravitational field, Astronautica Acta, 8 (1962), 106-123.
[61]	G. Leitmann, A calculus of variations solution of Goddard's problem, Astronautica Acta, 2 (1956), 55-62.
[62]	G. Leitmann (Ed.), "Optimization Techniques," Academic Press, New York, 1962.
[63]	J. P. Marec, "Optimal Space Trajectories," Elsevier, Amsterdam, 1979.
[64]	S. McAdoo, D. J. Jezewski and G. S. Dawkins, Development of a method for optimal maneuver analysis of complex space missions, NASA TN D-7882 (1975).
[65]	R. Mendes, R., J. Kennedy and J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on Evolutionary Computation, 8 (2004), 204-210.doi: 10.1109/TEVC.2004.826074.
[66]	A. Miele, General variational theory of the flight paths of Rocket-Powered aircraft, missiles, and satellite carriers, Astronautica Acta, 4 (1958), 11-21.
[67]	A. Miele, V. K. Basapur and W. Y. Lee, Optimal trajectories for aeroassisted, noncoplanar orbital transfer, Acta Astronautica, 15 (1987), 399-411.doi: 10.1016/0094-5765(87)90176-7.
[68]	H. G. Moyer, Minimum impulse coplanar circle-ellipse transfer, AIAA Journal, 3 (1965), 723-726.doi: 10.2514/3.2954.
[69]	C. O. Ourique, E. C. Biscaia and J. C. Pinto, The use of particle swarm optimization for dynamical analysis in chemical processes, Computers & Chemical Engineering, 26 (2002), 1783-1793.doi: 10.1016/S0098-1354(02)00153-9.
[70]	J. I. Palmore, An elementary proof of the optimality of Hohmann transfers, Journal of Guidance, Control, and Dynamics, 7 (1984), 629-630.doi: 10.2514/3.56375.
[71]	K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization method for constrained optimization problems, in "Intelligent Technologies - Theory and Applications: New Trends in Intelligent Technologies, Frontiers in Artificial Intelligence and Applications series"(eds. P. Sincak, J. Vascak, V. Kvasnicka, J. Pospichal) 76 (2002), 214-220.
[72]	K. E. Parsopoulos and M. N. Vrahatis, On the computation of all global minimizers through particle swarm optimization, IEEE Transactions on Evolutionary Computation, 8 (2004), 211-224.doi: 10.1109/TEVC.2004.826076.
[73]	M. Pontani, Simple method to determine globally optimal orbital transfers, Journal of Guidance, Control, and Dynamics, 32 (2009), 899-914.doi: 10.2514/1.38143.
[74]	M. Pontani and B. A. Conway, Particle swarm optimization applied to space trajectories, Journal of Guidance, Control, and Dynamics, 33 (2010), 1429-1441.doi: 10.2514/1.48475.
[75]	M. Pontani and B. A. Conway, Swarming theory applied to space trajectory optimization, Spacecraft Trajectory Optimization, Cambridge University Press, 2010, 1429-1441.
[76]	M. Pontani, C. Martin and B. A. Conway, New numerical methods for determining periodic orbits in the circular restricted three-body problem, Proceedings of the 61st International Astronautical Congress, Prague, paper IAC-10-C1.1.2, (2010).
[77]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Princeton University Press, New York, 1962.
[78]	J. E. Prussing, Simple proof of the global optimality of the Hohmann transfer, Journal of Guidance, Control, and Dynamics, 15 (1992), 1037-1038.doi: 10.2514/3.20941.
[79]	D. C. Redding, Optimal low-thrust transfers to geosynchronous orbit, Stanford University Guidance and Control Lab, report SUDAAR 539, Stanford (1983).
[80]	H. M. Robbins, Optimality of intermediate thrust arcs of rocket trajectories, AIAA Journal, 3 (1965), 1094-1098.doi: 10.2514/3.3060.
[81]	H. M. Robbins, Analytical study of the impulsive approximation, AIAA Journal, 4 (1966), 1417-1423.doi: 10.2514/3.3687.
[82]	M. Rosa Sentinella and L. Casalino, Cooperative evolutionary algorithm for space trajectory optimization, Celestial Mechanics and Dynamical Astronomy, 105 (2009), 211-227.doi: 10.1007/s10569-009-9223-4.
[83]	W. A. Scheel and B. A. Conway, Optimization of very-low-thrust, many-revolution spacecraft trajectories, Journal of Guidance, Control, and Dynamics, 17 (1994), 1182-1192.doi: 10.2514/3.21331.
[84]	H. Seywald, Trajectory optimization based on differential inclusion, Journal of Guidance, Control, and Dynamics, 17 (1994), 480-487.doi: 10.2514/3.21224.
[85]	A. Shternfeld, "Soviet Space Science," Basic Books, Inc., New York, 1959.
[86]	C. J. Spaans and E. Mooij, Performance evaluation of global trajectory optimization methods for a solar polar sail mission, Paper AIAA 2009-5666, (2009).
[87]	S. Tang and B. A. Conway, Optimization of low-thrust interplanetary trajectories using collocation and nonlinear programming, Journal of Guidance, Control, and Dynamics, 18 (1995), 599-604.doi: 10.2514/3.21429.
[88]	L. Ting, Optimum orbital transfer by impulses, ARS Journal, 30 (1960), 1013-1018.
[89]	L. Ting, Optimum orbital transfer by several impulses, Astronautica Acta, 6 (1960), 256-265.
[90]	M. Vasile, E. Minisci and M. Locatelli, On testing global optimization algorithms for space trajectory design, Paper AIAA 2008-6277, (2008).
[91]	G. Venter and J. Sobieszczanski-Sobieski, Particle swarm optimization, AIAA Journal, 41 (2003), 1583-1589.doi: 10.2514/2.2111.
[92]	N. X. Vinh, General theory of optimal trajectory for rocket flight in a resisting medium, Journal of Optimization Theory and Applications, 11 (1973), 189-202.doi: 10.1007/BF00935883.
[93]	C. B. Winn, Minimum fuel transfers between coaxial orbits, both coplanar and noncoplanar, Paper AAS 66-119 (1966).
[94]	C. Zee, Effects of finite thrusting time in orbit maneuvers, AIAA Journal, 1 (1963), 60-64.doi: 10.2514/3.1469.
[95]	K. Zhu, J. Li and H. Baoyin, Trajectory optimization of multi-asteroids exploration with low thrust, Transactions of the Japan Society for Aeronautical and Space Sciences, 52 (2009), 47-54.
[96]	K. Zhu, J. Li and H. Baoyin, Satellite scheduling considering maximum observation coverage time and minimum orbital transfer fuel cost, Acta Astronautica, 66 (2010), 220-229.doi: 10.1016/j.actaastro.2009.05.029.
[97]	K. P. Zondervan, L. J. Wood and T. K. Caughey, Optimal low-thrust, three-burn orbit transfers with large plane changes, Journal of the Astronautical Sciences, 32 (1984), 407-427.