Skip to main content
SpringerLink
Log in

A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The paper describes a continuous second-variation method to solve optimal control problems with terminal constraints where the control is defined on a closed set. The integration of matrix differential equations based on a second-order expansion of a Lagrangian provides linear updates of the control and a locally optimal feedback controller. The process involves a backward and a forward integration stage, which require storing trajectories. A method has been devised to store continuous solutions of ordinary differential equations and compute accurately the continuous expansion of the Lagrangian around a nominal trajectory. Thanks to the continuous approach, the method adapts implicitly the numerical time mesh and provides precise gradient iterates to find an optimal control. The method represents an evolution to the continuous case of discrete second-order techniques of optimal control. The novel method is demonstrated on bang–bang optimal control problems, showing its suitability to identify automatically optimal switching points in the control without insight into the switching structure or a choice of the time mesh. A complex space trajectory problem is tackled to demonstrate the numerical robustness of the method to problems with different time scales.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Betts, J.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–207 (1998). doi:10.2514/2.4231

    Article  MATH  Google Scholar 

  2. Ross, I.M., Fahroo, F.: Legendre pseudospectral approximations of optimal control problems. In: Lecture Notes in Control and Information Sciences, vol. 295, pp. 327–342. Springer, Berlin (2003)

    Google Scholar 

  3. Pontryaguin, L.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)

    Google Scholar 

  4. Speyer, J.L., Jacobson, D.H.: Primer on Optimal Control Theory. Advances in Design and Control. SIAM, Philadelphia (2010)

    Book  MATH  Google Scholar 

  5. Fisher, M.E.: Neighboring extremals for nonlinear systems with control constraints. Dyn. Control 5, 225–240 (1995)

    Article  MATH  Google Scholar 

  6. Mc Reynolds, S.R., Bryson, A.E.J.: A successive sweep method for solving optimal programming problems. Tech. Rep. AD0459518, Harvard University (1965)

  7. Mitter, S.K.: Successive approximation methods for the solution of optimal control problems. Automatica 3, 135–149 (1966)

    Article  MATH  Google Scholar 

  8. Jacobson, D., Mayne, D.: Differential Dynamic Programming. Elsevier, New York (1969)

    Google Scholar 

  9. Mayne, D.: A second order gradient method for determining optimal trajectories for nonlinear discrete-time systems. Int. J. Control 3, 85–95 (1966)

    Article  Google Scholar 

  10. Gershwin, S., Jacobson, D.: A discrete-time differential dynamic programming with application to optimal orbit transfer. AIAA J. 8(9), 1616–1626 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dyer, P., McReynolds, S.R.: Optimization of control systems with discontinuities and terminal constraints. IEEE Trans. Autom. Control 14(3), 223–229 (1969). doi:10.1109/TAC.1969.1099173

    Article  MathSciNet  Google Scholar 

  12. Whiffen, G., Sims, J.: Application of a novel optimal control algorithm to low-thrust trajectory optimization. In: AAS/AIAA Space Fligth Mechanics Meeting (2001). AAS 01-209

    Google Scholar 

  13. Whiffen, G., Sims, J.: Application of the sdc optimal control algorithm to low-thrust escape and capture including fourth-body effects. In: 2nd International Symposium on Low-Thrust Trajectories, Toulouse, pp. 18–20 (2002)

    Google Scholar 

  14. Lantoine, G., Russell, R.: A hybrid differential dynamic programming algorithm for constrained optimal control problems, Part 1: Theory. J. Optim. Theory Appl. 154(2), 382–417 (2012). doi:10.1007/s10957-012-0039-0

    Article  MathSciNet  MATH  Google Scholar 

  15. Lantoine, G., Russell, R.: A hybrid differential dynamic programming algorithm for constrained optimal control problems, Part 2: Application. J. Optim. Theory Appl. 154(2), 418–442 (2012). doi:10.1007/s10957-012-0038-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Bertrand, R., Epenoy, R.: New smoothing techniques for solving bang–bang optimal control problems—numerical results and statistical interpretation. Optim. Control Appl. Methods 23(4), 171–197 (2002). doi:10.1002/oca.709

    Article  MathSciNet  MATH  Google Scholar 

  17. Shamsi, M.: A modified pseudospectral scheme for accurate solution of bang–bang optimal control problems. Optim. Control Appl. Methods 32, 668–680 (2011)

    Article  MathSciNet  Google Scholar 

  18. Jacobson, D.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7(4), 578–595 (1969). doi:10.1137/0307042

    Article  MathSciNet  MATH  Google Scholar 

  19. Colombo, C., Vasile, M., Radice, G.: Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming. Celest. Mech. Dyn. Astron. 105(1), 75–112 (2009). doi:10.1007/s10569-009-9224-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Pantoja, J.F.A.D.O.: Differential dynamic programming and newton’s method. Int. J. Control 47, 1539–1553 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Murray, D., Yakowitz, S.: Differential dynamic programming and Newton’s method for discrete optimal control problems. J. Optim. Theory Appl. 43, 395–414 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liao, L.A., Liao, L.: Global convergence of differential dynamic programming and newton’s method for discrete-time optimal control (1996)

  23. Bullock, T.: Computation of optimal controls by a method based on second variations. Tech. Rep., Stanford University (1966)

  24. Bullock, T., Franklin, G.: A second-order feedback method for optimal control computations. IEEE Trans. Autom. Control 12(6), 666–673 (1967)

    Article  Google Scholar 

  25. Olympio, J.: Optimisation and optimal control methods for planet sequence design of low-thrust interplanetary transfer problems with gravity-assists. Ph.D. thesis, Ecole des Mines de Paris (2008)

  26. Olympio, J.: Algorithm for low thrust optimal interplanetary transfers with escape and capture phases. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, pp. 2008–7363 (2008)

    Google Scholar 

  27. Lantoine, G., Russell, R.: A hybrid differential dynamic programming algorithm for low-thrust optimization. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, pp. 18–21 (2008)

    Google Scholar 

  28. Whiffen, G.: Static dynamic control for optimizing a useful objective. Patent US 6 496 741 B1 (2002)

  29. Jain, S.: Multiresolution strategies for the numerical solution of optimal control problems. Ph.D. thesis, School of Aerospace Engineering, Georgia Institute of Technology (2008)

  30. Izzo, D., Bevilacqua, R., Valente, C.: Internal mesh optimisation and Runge–Kutta collocation in a direct transcription method applied to interplanetary missions. In: IAC (2004)

    Google Scholar 

  31. Ranieri, C., Ocampo, C.: Indirect optimization of three-dimensional finite-burning interplanetary transfers including spiral dynamics. J. Guid. Control Dyn. 32(2), 445–455 (2009). doi:10.2514/1.38170

    Article  Google Scholar 

  32. Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4(5), 303–320 (1969). doi:10.1007/BF00927673

    Article  MathSciNet  MATH  Google Scholar 

  33. Powell, M.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)

    Google Scholar 

  34. Conn, A., Gould, N., Toint, P.: Trust-region methods. MPS-SIAM Series on Optimization, vol. 1. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  35. Rodriguez, J., Renaud, J., Watson, L.: Trust region augmented Lagrangian methods for sequential response surface approximation and optimization. J. Mech. Des. 15, 58–66 (1997). doi:10.1115/1.2826677

    Google Scholar 

  36. Coleman, T., Liao, A.: An efficient trust region method for unconstrained discrete-time optimal control problems. Comput. Optim. Appl. 4, 47–66 (1993)

    Article  MathSciNet  Google Scholar 

  37. Kalman, R.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5(102), 102–119 (1960)

    MathSciNet  Google Scholar 

  38. Bucy, R.: Global theory of the Riccati equation. J. Comput. Syst. Sci. 1(4) (1967). doi:10.1016/S0022-0000(67)80025-4

    Google Scholar 

  39. Jacobson, D.: New conditions for boundedness of the solution of a matrix Riccati differential equation. J. Differ. Equ. 8(2), 258–263 (1970). doi:10.1016/0022-0396(70)90005-7

    Article  MATH  Google Scholar 

  40. Miele, A., Moseley, P., Levy, A., Coggins, G.: On the method of multipliers for mathematical programming problems. J. Optim. Theory Appl. 10(1), 1–33 (1972). doi:10.1007/BF00934960

    Article  MathSciNet  MATH  Google Scholar 

  41. Stoffer, D.: Variable steps for reversible integration methods. Computing 55(1), 1–22 (1995). doi:10.1007/BF02238234

    Article  MathSciNet  MATH  Google Scholar 

  42. Laurent-Varin, J., Bonnans, J., Berend, N., Haddou, M., Talbot, C.: Interior-point approach to trajectory optimization. J. Guid. Control Dyn. 30(5) (2007). doi:10.2514/1.18196

    Google Scholar 

  43. Betts, J., Huffman, W.: Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Methods 19(1), 1–21 (1998). doi:10.2514/2.4231

    Article  MathSciNet  Google Scholar 

  44. Hairer, E., Noersett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd rev. edn. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (2000)

    Google Scholar 

  45. Shampine, L.F.: Interpolation for variable order, Runge–Kutta methods. Comput. Math. Appl. 14(4), 255–260 (1987). doi:10.1007/s10957-013-0274-z

    Article  MathSciNet  MATH  Google Scholar 

  46. Strange, J., Landau, D., Yam, C., Biscani, F., Izzo, D.: Near earth asteroids accessible to human exploration in 2020-2035. In: 42nd Annual Meeting of the Division for Planetary Sciences of the American Astronomical Society (2010)

  47. Petropoulos, A., Whiffen, G.J., Sims, J.A.: Simple control laws for continuous-thrust escape or capture and their use in optimisation. In: AIAA/AAS Astrodynamics Conference, Monterey, CA, USA (2002)

    Google Scholar 

Download references

Acknowledgements

Part of this work has been conducted during the Ph.D. studies of the author, under the sponsorship of Centre National d’Etudes Spatiales (CNES, France) and Thales Alenia Space (France).

Author information

Authors and Affiliations

  1. Aerospace Engineer, Toulouse, France

    Joris T. Olympio

Authors
  1. Joris T. Olympio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joris T. Olympio.

Additional information

Communicated by Ryan P. Russell.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Olympio, J.T. A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory Problems. J Optim Theory Appl 158, 687–716 (2013). https://doi.org/10.1007/s10957-013-0274-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-013-0274-z

Keywords

Search

Navigation