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

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Abstract

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.

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References

  • Abraham R., Marsden J.E.: Foundations of Mechanics. Addison-Wesley, Boston (1978)

    MATH  Google Scholar 

  • Baig S., McInnes C.R.: Artificial halo orbits for low-thrust propulsion spacecraft. Celest. Mech. Dyn. Astron. 104, 321–335 (2009)

    Article  ADS  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • Belbruno E.: Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, Princeton (2004)

    MATH  Google Scholar 

  • Belbruno E., Miller J.: Sun-perturbed Earth-to-Moon transfers with ballistic capture. J. Guid. Control Dyn. 16, 770–775 (1993)

    Article  ADS  Google Scholar 

  • Belbruno E., Marsden B.: Resonance hopping in comets. Astron. J. 113(4), 1433–1444 (1997)

    Article  ADS  Google Scholar 

  • Betts J.T.: Survey of numerical methods for trajectory optimization. AIAA J. Guid. Control Dyn. 21(2), 193–207 (1998)

    Article  MATH  Google Scholar 

  • 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)

  • Biegler L.T.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)

    Article  Google Scholar 

  • 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 

  • 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)

  • Coello Coello C.A., Lamont G., Van Veldhuizen D.: Evolutionary Algorithms for Solving Multi-objective Problems. Springer, New York (2007)

    MATH  Google Scholar 

  • Colonius F., Kliemann W.: The Dynamics of Control. Birkhäuser, Boston (2000)

    Google Scholar 

  • Conway B.A., Chilan Ch.M., Wall B.J.: Evolutionary principles applied to mission planning problems. Celest. Mech. Dyn. Astron. 97, 73–86 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Deb K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)

    MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

  • Dellnitz M., Schütze O., Sertl St.: Finding zeros by multilevel subdivision techniques. IMA J. Numer. Anal. 22(2), 167–185 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Dellnitz M., Schütze O., Hestermeyer T.: Covering Pareto sets by multilevel subdivision techniques. J. Optim. Theo. Appl. 124, 113–155 (2005)

    Article  MATH  Google Scholar 

  • 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)

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

  • Farquhar, R.W.: The control and use of libration-point satellites. NASA TR R-346 (1970)

  • Fliege J., Svaiter B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Garcia F., Gómez G.: A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100 (2007)

    Article  MATH  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Gerthsen Ch., Vogel H.: Physik. Springer, New York (1993)

    Google Scholar 

  • 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)

  • 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)

    Article  MathSciNet  Google Scholar 

  • 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 

  • Hairer E., Nørsett S.P., Wanner G.: Solving ordinary differential equations I. Springer, Berlin (1993)

    MATH  Google Scholar 

  • Han S.P.: Superlinearly convergent variable-metric algorithms for general nonlinear programming problems. Math. Program. 11, 263–282 (1976)

    Article  Google Scholar 

  • Hicks G., Ray W.: Approximation methods for optimal control systems. Can. J. Chem. Eng. 49, 522–528 (1971)

    Article  Google Scholar 

  • Hillermeier C.: Nonlinear Multiobjective Optimization—A Generalized Homotopy Approach. Birkhäuser, Berlin (2001)

    MATH  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • 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)

  • 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)

  • 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)

  • Junge, O., Marsden, J.E., Ober-Blöbaum, S.: Discrete mechanics and optimal control. In: Proceedings of 16th IFAC World Congress, Prague (2005)

  • 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)

  • 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)

  • Kechichian J.A.: Local regularization of the restricted elliptic three-body problem in rotating coordinates. J. Guid. Control Dyn. 25(6), 1064–1072 (2002)

    Article  Google Scholar 

  • 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 

  • 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)

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • 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)

  • 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)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  Google Scholar 

  • 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)

  • 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)

    Article  MATH  Google Scholar 

  • 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)

  • 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)

  • Marsden J.E., West M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • McGehee, R.P.: Some homoclinic orbits for the restricted 3-body problem. Ph.D. dissertation, University of Wisconsin (1969)

  • Meyer K.R., Hall R.: Hamiltonian Mechanics and the n-Body Problem. Springer, New York (1992)

    Google Scholar 

  • Miettinen K.: Nonlinear Multiobjective Optimization. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  • 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

  • Ober-Blöbaum, S.: Discrete mechanics and optimal control. Ph.D. dissertation, University of Paderborn, Paderborn (2008)

  • Pareto, V.: Manual of Political Economy. The MacMillan (Original edition in French in 1917) (1971)

  • 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

  • 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)

  • Richardson D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. Dyn. Astron. 22(3), 241–253 (1980)

    MATH  Google Scholar 

  • 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)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

  • Stoer J., Bulirsch R.: Introduction into Numerical Analysis. Springer, Berlin (1993)

    Google Scholar 

  • Szebehely V.: Theory of Orbits. Academic, New York, London (1967)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

  • Vasile M.: Hybrid behavioral-based multiobjective space trajectory optimization. Object. Memetic Algorithms Ser. Stud. Comput. Intell. 171, 231–254 (2009)

    Article  Google Scholar 

  • 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)

  • 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)

  • 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)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, 33095, Paderborn, Germany

    Michael Dellnitz & Bianca Thiere

  2. Control and Dynamical Systems, California Institute of Technology, MC 107-81, Pasadena, CA, 91125, USA

    Sina Ober-Blöbaum

  3. TU München, Zentrum Mathematik, Boltzmannstr. 3, 85747, Garching bei München, Germany

    Marcus Post

  4. Computer Science Department, CINVESTAV-IPN, Mexico City, Mexico

    Oliver Schütze

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  1. Michael Dellnitz
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  2. Sina Ober-Blöbaum
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  3. Marcus Post
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  4. Oliver Schütze
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  5. Bianca Thiere
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Correspondence to Bianca Thiere.

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Dellnitz, M., Ober-Blöbaum, S., Post, M. et al. A multi-objective approach to the design of low thrust space trajectories using optimal control. Celest Mech Dyn Astr 105, 33–59 (2009). https://doi.org/10.1007/s10569-009-9229-y

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