A multi-objective approach to the design of low thrust space trajectories using optimal control

  • Michael Dellnitz
  • Sina Ober-Blöbaum
  • Marcus Post
  • Oliver Schütze
  • Bianca ThiereEmail author
Original Article


In this article, we introduce a novel three-step approach for solving optimal control problems in space mission design. We demonstrate its potential by the example task of sending a group of spacecraft to a specific Earth L 2 halo orbit. In each of the three steps we make use of recently developed optimization methods and the result of one step serves as input data for the subsequent one. Firstly, we perform a global and multi-objective optimization on a restricted class of control functions. The solutions of this problem are (Pareto-)optimal with respect to ΔV and flight time. Based on the solution set, a compromise trajectory can be chosen suited to the mission goals. In the second step, this selected trajectory serves as initial guess for a direct local optimization. We construct a trajectory using a more flexible control law and, hence, the obtained solutions are improved with respect to control effort. Finally, we consider the improved result as a reference trajectory for a formation flight task and compute trajectories for several spacecraft such that these arrive at the halo orbit in a prescribed relative configuration. The strong points of our three-step approach are that the challenging design of good initial guesses is handled numerically by the global optimization tool and afterwards, the last two steps only have to be performed for one reference trajectory.


Space mission design Low thrust propulsion Three body problem Multi-objective optimization Pareto set Optimal control Formation flight 


5.10.Ce 02.60.Pn 45.20.Jj 45.80.+r 45.10.Db 


  1. Abraham R., Marsden J.E.: Foundations of Mechanics. Addison-Wesley, Boston (1978)zbMATHGoogle Scholar
  2. Baig S., McInnes C.R.: Artificial halo orbits for low-thrust propulsion spacecraft. Celest. Mech. Dyn. Astron. 104, 321–335 (2009)CrossRefADSGoogle Scholar
  3. Barden B.T., Howell K.C., Lo M.W.: Applications of dynamical systems theory to trajectory design for a libration point mission. J. Astronaut. Sci. 45(2), 161–178 (1997)MathSciNetGoogle Scholar
  4. Belbruno E.: Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, Princeton (2004)zbMATHGoogle Scholar
  5. Belbruno E., Miller J.: Sun-perturbed Earth-to-Moon transfers with ballistic capture. J. Guid. Control Dyn. 16, 770–775 (1993)CrossRefADSGoogle Scholar
  6. Belbruno E., Marsden B.: Resonance hopping in comets. Astron. J. 113(4), 1433–1444 (1997)CrossRefADSGoogle Scholar
  7. Betts J.T.: Survey of numerical methods for trajectory optimization. AIAA J. Guid. Control Dyn. 21(2), 193–207 (1998)zbMATHCrossRefGoogle Scholar
  8. Betts, J.T.: Practical methods for optimal control using nonlinear programming. In: Advances in Design and Control. 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (2001)Google Scholar
  9. Biegler L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)CrossRefGoogle Scholar
  10. Binder T., Blank L., Bock H.G., Bulirsch R., Dahmen W., Diehl M., Kronseder T., Marquardt W., Schlöder J.P., von Stryk O.: Introduction to model based optimization of chemical processes on moving horizons. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds) Online Optimization of Large Scale Systems: State of the Art, pp. 295–340. Springer, Berlin (2001)Google Scholar
  11. Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of 9th IFAC World Congress, pp. 242–247. Budapest (1984)Google Scholar
  12. Coello Coello C.A., Lamont G., Van Veldhuizen D.: Evolutionary Algorithms for Solving Multi-objective Problems. Springer, New York (2007)zbMATHGoogle Scholar
  13. Colonius F., Kliemann W.: The Dynamics of Control. Birkhäuser, Boston (2000)Google Scholar
  14. Conway B.A., Chilan Ch.M., Wall B.J.: Evolutionary principles applied to mission planning problems. Celest. Mech. Dyn. Astron. 97, 73–86 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. Coverstone-Caroll V., Hartman J.W., Mason W.M.: Optimal multi-objective low-thrust spacecraft trajectories. Comput. Methods Appl. Mech. Eng. 186, 387–402 (2000)CrossRefGoogle Scholar
  16. Das I., Dennis J.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8, 631–657 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Deb K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)zbMATHGoogle Scholar
  18. Deb K., Pratap A., Agarwal S., Meyarivan T.: A fast and elitist multiobjective genetic algorithm: NSGA–II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  19. Dellnitz, M., Junge, O., Lo, M.W., Thiere, B.: On the detection of energetically efficient trajectories for spacecraft. In: AAS/AIAA Astrodynamics Specialist Conference, Quebec City, Paper AAS 01-326 (2001)Google Scholar
  20. Dellnitz M., Schütze O., Sertl St.: Finding zeros by multilevel subdivision techniques. IMA J. Numer. Anal. 22(2), 167–185 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Dellnitz M., Schütze O., Hestermeyer T.: Covering Pareto sets by multilevel subdivision techniques. J. Optim. Theo. Appl. 124, 113–155 (2005)zbMATHCrossRefGoogle Scholar
  22. Dellnitz, M., Junge, O., Krishnamurthy, A., Ober-Blöbaum, S., Padberg, K., Preis, R.: Efficient control of formation flying spacecraft. In: Meyer auf der Heide, F., Monien, B. (eds.) New Trends in Parallel & Distributed Computing, pp. 235–247. Heinz Nixdorf Institut Verlagsschriftreihe (2006a)Google Scholar
  23. Dellnitz M., Junge O., Post M., Thiere B.: On target for Venus—set oriented computation of energy efficient low thrust trajectories. Celest. Mech. Dyn. Astron. 95, 357–370 (2006b)zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. Deuflhard P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numerische Mathematik 22, 289–315 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  25. Dichmann, D.J., Doedel, E.J., Paffenroth, C.: The computation of periodic solutions of the 3-body problem using the numerical continuation software AUTO. In: International Conference on Libration Point Orbits and Applications, pp. 489–528. World Scientific (2003)Google Scholar
  26. Farquhar, R.W.: The control and use of libration-point satellites. NASA TR R-346 (1970)Google Scholar
  27. Fliege J., Svaiter B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  28. Garcia F., Gómez G.: A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100 (2007)zbMATHCrossRefADSGoogle Scholar
  29. Gawlik E.S., Marsden J.E., Du Toit P.C., Campagnola S.: Lagrangian coherent structures in the planar elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 103, 227–249 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. Gerthsen Ch., Vogel H.: Physik. Springer, New York (1993)Google Scholar
  31. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. Report NA 97-2, Department of Mathematics, University of California, San Diego, CA, USA (1997)Google Scholar
  32. Gill P.E., Jay L.O., Leonard M.W., Petzold L.R., Sharma V.: An SQP method for the optimal control of large-scale dynamical systems. J. Comp. Appl. Math. 20, 197–213 (2000)CrossRefMathSciNetGoogle Scholar
  33. Gómez G., Koon W.S., Lo M.W., Marsden J.E., Masdemont J., Ross S.D.: Invariant manifolds, the spatial three-body problem and space mission design. Adv. Astronaut. Sci. 109(1), 3–22 (2001)Google Scholar
  34. Hairer E., Nørsett S.P., Wanner G.: Solving ordinary differential equations I. Springer, Berlin (1993)zbMATHGoogle Scholar
  35. Han S.P.: Superlinearly convergent variable-metric algorithms for general nonlinear programming problems. Math. Program. 11, 263–282 (1976)CrossRefGoogle Scholar
  36. Hicks G., Ray W.: Approximation methods for optimal control systems. Can. J. Chem. Eng. 49, 522–528 (1971)CrossRefGoogle Scholar
  37. Hillermeier C.: Nonlinear Multiobjective Optimization—A Generalized Homotopy Approach. Birkhäuser, Berlin (2001)zbMATHGoogle Scholar
  38. Howell K., Barden B., Lo M.W.: Application of dynamical systems theory to trajectory design for a libration point mission. J. Astronaut. Sci. 45(2), 161–178 (1997)MathSciNetGoogle Scholar
  39. Howell, K.C., Marchand, B.G.: Control strategies for formation flight in the vicinity of the libration points. In: AIAA/AAS Space Flight Mechanics Conference, Ponce, Puerto Rico, AAS Paper 03-113 (2003)Google Scholar
  40. Junge, O., Levenhagen, J., Seifried, A., Dellnitz, M., Astrium, GmbH: Identification of Halo orbits for energy efficient formation flying. In: Proceedings of the International Symposium Formation Flying, Toulouse (2002)Google Scholar
  41. Junge, O., Ober-Blöbaum, S.: Optimal reconfiguration of formation flying satellites. In: Proceedings of the IEEE Conference on Decision and Control and European Control Conference ECC, Seville (2005)Google Scholar
  42. Junge, O., Marsden, J.E., Ober-Blöbaum, S.: Discrete mechanics and optimal control. In: Proceedings of 16th IFAC World Congress, Prague (2005)Google Scholar
  43. Junge, O., Marsden, J.E., Ober-Blöbaum, S.: Optimal reconfiguration of formation flying spacecraft—a decentralized approach. In: Proceedings of IEEE Conference on Decision and Control and European Control Conference ECC, San Diego, California (2006)Google Scholar
  44. Kanso, E., Marsden, J.E.: Optimal motion of an articulated body in a perfect fluid. In: Proceedings of the IEEE Conference on Decision and Control and European Control Conference ECC, Seville (2005)Google Scholar
  45. Kechichian J.A.: Local regularization of the restricted elliptic three-body problem in rotating coordinates. J. Guid. Control Dyn. 25(6), 1064–1072 (2002)CrossRefGoogle Scholar
  46. Kobilarov M., Desbrun M., Marsden J.E., Sukhatme G.S.: A discrete geometric optimal control framework for systems with symmetries. Rob. Sci. Syst. 3, 1–8 (2007)Google Scholar
  47. Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: The genesis trajectory and heteroclinic connections. In: AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, AAS99-451 (1999)Google Scholar
  48. Koon W.S., Lo M.W., Marsden J.E., Ross S.D.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10, 427–469 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  49. Koon W.S., Lo M.W., Marsden J.E., Ross S.D.: Low energy transfer to the Moon. Celest. Mech. Dyn. Astron. 81, 63–73 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  50. Kraft, D.: On converting optimal control problems into nonlinear programming problems. In: Schittkowsky K. (ed.) Compuational Mathematical Programming, pp. 261–280. F15, NATO ASI series, Springer, Berlin (1985)Google Scholar
  51. Lee, S., von Allmen, P., Fink, W., Petropoulos, A.E., Terrile, R.J.: Multi-objective evolutionary algorithms for low-thrust orbit transfer optimization. In: Genetic and Evolutionary Computation Conference (GECCO 2005) (2005)Google Scholar
  52. Leineweber D., Bauer I., Bock H., Schlöder J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part I: theoretical aspects. Comput. Chem. Eng. 27, 157–166 (2003)CrossRefGoogle Scholar
  53. Leiva A.M., Briozzo C.B.: Extension of fast periodic transfer orbits from the Earth–Moon RTBT to the Sun–Earth–Moon quasi-bicircular problem. Celest. Mech. Dyn. Astron. 101, 225–245 (2008)CrossRefMathSciNetADSGoogle Scholar
  54. Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained multibody dynamics. In: Proceedings of the 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, ASME International Design Engineering Technical Conferences, Las Vegas, Nevada (2007)Google Scholar
  55. Maddock C., Vasile M.: Design of optimal spacecraft-asteroid formations through a hybrid global optimization approach. Int. J. Intell. Comput. Cybern. 1(2), 239–268 (2008)zbMATHCrossRefGoogle Scholar
  56. Marchand, B.G., Howell, K.C.: Formation flight near L2 in the Sun–Earth/Moon ephemeris system including solar radiation pressure. In: AIAA/AAS Astrodynamics Specialist Conference, Big Sky, Montana, AAS Paper 03-596 (2003)Google Scholar
  57. Marchand, B.G., Howell, K.C., Betts, J.T.: Discrete optimal control of S/C formations near the L1 and L2 points of the Sun–Earth/Moon system. In: AIAA/AAS Astrodynamics Specialist Conference, Lake Tahoe, California (2005)Google Scholar
  58. Marsden J.E., West M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  59. McGehee, R.P.: Some homoclinic orbits for the restricted 3-body problem. Ph.D. dissertation, University of Wisconsin (1969)Google Scholar
  60. Meyer K.R., Hall R.: Hamiltonian Mechanics and the n-Body Problem. Springer, New York (1992)Google Scholar
  61. Miettinen K.: Nonlinear Multiobjective Optimization. Kluwer, Dordrecht (1999)zbMATHGoogle Scholar
  62. Mingotti, G., Topputo, F., Bernelli-Zazzera, F.: Low-energy, low-thrust transfers to the Moon. Celest. Mech. Dyn. Astron. 105 (2009). doi: 10.1007/s10569-009-9220-7
  63. Ober-Blöbaum, S.: Discrete mechanics and optimal control. Ph.D. dissertation, University of Paderborn, Paderborn (2008)Google Scholar
  64. Pareto, V.: Manual of Political Economy. The MacMillan (Original edition in French in 1917) (1971)Google Scholar
  65. Pergola, P., Geurts, K., Casaregola, C., Andrenucci, M.: Earth–Mars halo to halo low thrust manifold transfers. Celest. Mech. Dyn. Astron. 105 (2009). doi: 10.1007/s10569-009-9205-6
  66. Powell, M.J.D.: A fast algorithm for nonlinearly constrained optimization calculations. In: Watson G.A. (ed.) Numerical Analysis, vol. 630, pp. 261–280. Lecture notes in mathematics, Springer, Berlin (1978)Google Scholar
  67. Richardson D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. Dyn. Astron. 22(3), 241–253 (1980)zbMATHGoogle Scholar
  68. Schütze, O., Mostaghim, S., Dellnitz, M., Teich, J.: Covering Pareto Sets by multilevel evolutionary subdivision techniques. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) Evolutionary Multi-criterion Optimization. Lecture notes in computer science (2003)Google Scholar
  69. Schütze O., Jourdan L., Legrand T., Talbi E.G., Wojkiewicz J.L.: New analysis of the optimization of electromagnetic shielding properties using conducting polymers and a multi-objective approach. Polym. Adv. Technol. 19, 762–769 (2008a)CrossRefGoogle Scholar
  70. Schütze O., Vasile M., Junge O., Dellnitz M., Izzo D.: Designing optimal low thrust gravity assist trajectories using space pruning and a multi-objective approach. Eng. Optim. 41(2), 155–181 (2008b)CrossRefGoogle Scholar
  71. Schütze, O., Vasile, M., Coello Coello, C.A.: Approximate solutions in space mission design. In: Parallel Problem Solving from Nature (PPSN 2008), pp. 805–814 (2008c)Google Scholar
  72. Stoer J., Bulirsch R.: Introduction into Numerical Analysis. Springer, Berlin (1993)Google Scholar
  73. Szebehely V.: Theory of Orbits. Academic, New York, London (1967)Google Scholar
  74. Tang S., Conway B.A.: Optimization of low-thrust interplanetary trajectories using collocation and nonlinear programming. J. Guid. Control Dyn. 18(3), 599–604 (1995)CrossRefGoogle Scholar
  75. Vasile, M., Schütze, O., Junge, O., Radice, G., Dellnitz, M.: Spiral trajectories in global optimisation of interplanetary and orbital transfers. Technical report, Ariadna Study Report AO4919 05/4106, Contract Number 19699/NL/HE, European Space Agency (2006)Google Scholar
  76. Vasile M.: Hybrid behavioral-based multiobjective space trajectory optimization. Object. Memetic Algorithms Ser. Stud. Comput. Intell. 171, 231–254 (2009)CrossRefGoogle Scholar
  77. Vavrina, M.A., Howell, K.C.: Global low-thrust trajectory optimization through hybridization of a genetic algorithm and a direct method. In: AIAA/AAS Astrodynamics Specialist Conference, AIAA 2008-6614 (2008)Google Scholar
  78. von Stryk, O.: Numerical solution of optimal control problems by direct collocation. In: Bulirsch, R., Miele, A., Stoer, J., Well, K.H. (eds.) Optimal Control—Calculus of Variation, Optimal Control Theory and Numerical Methods, vol. 111, pp. 129–143. International series of numerical mathematics, Birkhäuser (1993)Google Scholar
  79. Walther A., Kowarz A., Griewank A.: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. ACM TOMS 22(2), 131–167 (1996)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Michael Dellnitz
    • 1
  • Sina Ober-Blöbaum
    • 2
  • Marcus Post
    • 3
  • Oliver Schütze
    • 4
  • Bianca Thiere
    • 1
    Email author
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.Control and Dynamical SystemsCalifornia Institute of TechnologyPasadenaUSA
  3. 3.TU MünchenZentrum MathematikGarching bei MünchenGermany
  4. 4.Computer Science DepartmentCINVESTAV-IPNMexico CityMexico

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