Advertisement

The Journal of the Astronautical Sciences

, Volume 58, Issue 4, pp 615–642 | Cite as

Modified Chebyshev-Picard Iteration Methods for Solution of Boundary Value Problems

  • Xiaoli Bai
  • John L. Junkins
Article

Abstract

Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw-Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation of the entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin’s principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin’s method for a classical Lambert’s problem and with a Chebyshev pseudospectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited.

Keywords

Optimal Control Problem Chebyshev Polynomial Pseudospectral Method Picard Iteration Chebyshev Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    BETTS, J.T. “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 193–20.MATHCrossRefGoogle Scholar
  2. [2]
    LEWIS, F.L. and SYRMOS, V.L. Optimal Control. Wiley-Interscience, New York, NY, 1995.Google Scholar
  3. [3]
    MIGDALASA, A., TORALDO, G., and KUMAR, V. “Nonlinear Optimization and Parallel Computing,” Parallel Computing, Vol. 29, Apr. 2003, pp. 375–391.CrossRefGoogle Scholar
  4. [4]
    TRAVASSOS, R. and KAUFMAN, H. “Parallel Algorithms for Solving Nonlinear Two-Point Boundary-Value Problems which Arise in Optimal Control,” Journal of Optimization Theory and Applications, Vol. 30, No. 1, 1980, pp. 53–71.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    MIRANKER, W.L. and LINIGER, W. “Parallel Methods for the Numerical Integration of Ordinary Differential Equations,” Mathematics of Computation, Vol. 21, Jul. 1969, pp. 303–320.MathSciNetCrossRefGoogle Scholar
  6. [6]
    BETTS, J.T. and HUFFMAN, W. “Trajectory Optimization on a Parallel Processor,” Journal of Guidance, Control, and Dynamics, Vol. 14, Mar.–Apr. 1991, pp. 431–439.MATHCrossRefGoogle Scholar
  7. [7]
    BAI, X. and JUNKINS, J.L. “Solving Initial Value Problems by the Picard-Chebyshev Method with NVIDIA GPUS,” presented at 20th Spaceflight Mechanics Meeting, San Diego, CA, Feb., 2010.Google Scholar
  8. [8]
    BAI, X. and JUNKINS, J.L. “Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value Problems,” presented at Kyle T. Alfriend Astrodynamics Symposium, Monterey, CA, May, 2010.Google Scholar
  9. [9]
    BAI, X. Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems. Ph.D. Dissertation, Texas A&M University, College Station, TX, 2010.Google Scholar
  10. [10]
    PICARD, E. “Sur l’Application des Methodes d’approximations successives à l’Etude de certaines Equatioins differentielles ordinaires,” Journal de Mathematiques, Vol. 9, 1893, pp. 217–271.Google Scholar
  11. [11]
    PICARD, E. Traité d’analyse, Vol. 1, ch. 4.7. Gauthier-Villars, Paris, France, Third Ed., 1922.MATHGoogle Scholar
  12. [12]
    CRAATS, J.V. “On the Region of Convergence of Picard’s Iteration,” ZAMM-Journal of Applied Mathematics and Mechanics, Vol. 52, Dec. 1971, pp. 487–491.CrossRefGoogle Scholar
  13. [13]
    LETTENMEYER, F. “Ober die von einem Punkt ausgehenden Integralkurven einer Differentialgleichung 2. Ordnung,” Deutsche Math, Vol. 7, 1944, pp. 56–74.MathSciNetGoogle Scholar
  14. [14]
    AGARWAL, R.P. “Nonlinear Two-Point Boundary Value Problems,” Indian Journal of Pure and Applied Mathematics, Vol. 4, 1973, pp. 757–769.MathSciNetGoogle Scholar
  15. [15]
    COLES, W.J. and SHERMAN, T.L. “Convergence of Successive Approximations for Nonlinear Two-Point Boundary Value Problems,” SIAM Journal on Applied Mathematics, Vol. 15, Mar. 1967, pp. 426–433.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    BAILEY, P.B. “On the Interval of Convergence of Picard’s Iteration,” ZAMM-Journal of Applied Mathematics and Mechanics, Vol. 48, No. 2, 1968, pp. 127–128.MATHCrossRefGoogle Scholar
  17. [17]
    BAILEY, P., SHAMPINE, L.F., and WALTMAN, P. “Existence and Uniqueness of Solutions of the Second Order Boundary Value Problem,” Bulletin of the American Mathematical Society, Vol. 72, No. 1, 1966, pp. 96–98.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    FOX, L. and PARKER, I.B. Chebyshev Polynomials in Numerical Analysis. London, UK: Oxford University Press, 1972.Google Scholar
  19. [19]
    URABE, M. “An Existence Theorem for Multi-Point Boundary Value Problems,” Funkcialaj Ekvacioj, Vol. 9, 1966, pp. 43–60.MathSciNetMATHGoogle Scholar
  20. [20]
    NORTON, H.J. “The Iterative Solution of Non-Linear Ordinary Differential Equations in Chebyshev Series,” The Computer Journal, Vol. 7, No. 2, 1964, pp. 76–85.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    WRIGHT, K. “Chebyshev Collocation Methods for Ordinary Differential Equations,” The Computer Journal, Vol. 6, No. 4, 1964, pp. 358–365.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    CLENSHAW, C.W. and NORTON, H.J. “The Solution of Nonlinear Ordinary Differential Equations in Chebyshev Series,” The Computer Journal, Vol. 6, No. 1, 1963, pp. 88–92.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    FEAGIN, T. The Numerical Solution of Two Point Boundary Value Problems Using Chebyshev Series. Ph.D. Dissertation, The Universtiy of Texas at Austin, Austin, TX, 1973.Google Scholar
  24. [24]
    INCE, E.L. Ordinary Differential Equations. Dover Publications, Inc, New York, NY, 1956.Google Scholar
  25. [25]
    BATTIN, R. An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics and Astronautics, Inc, Reston, VA, Revised Ed., 1999.MATHCrossRefGoogle Scholar
  26. [26]
    SCHAUB, H. and JUNKINS, J.L. Analytical Mechanics of Space Systems. American Institute of Aeronautics and Astronautics, Inc, Reston, VA, First Ed., 2003.CrossRefGoogle Scholar
  27. [27]
    FAHROO, F. and ROSS, I.M. “Direct Trajectory Optimization by a Chebyshev Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol. 25, Jan.–Feb. 2002, pp. 160–166.CrossRefGoogle Scholar
  28. [28]
    FAHROO, F. and ROSS, I.M. “A Spectral Patching Method for Direct Trajectory Optimization,” Journal of the Astronautical Sciences, Vol. 48, No. 2/3, 2000, pp. 269–286.Google Scholar
  29. [29]
    GONG, Q., FAHROO, F., and ROSS, I.M. “Spectral Algorithm for Pseudospectral Methods in Optimal Control,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 3, 2008, pp. 460–471.MathSciNetCrossRefGoogle Scholar
  30. [30]
    OBERLE, H.J. and TAUBERT, K. “Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem,” Journal of Optimization Theory and Applications, Vol. 95, Nov. 1997, pp. 243–262.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    BAI, X., TURNER, J.D., and JUNKINS, J.L. “Optimal Thrust Design of a Mission to Apophis Based on a Homotopy Method,” presented at the AAS/AIAA Spaceflight Mechanics Meeting, Savannah, GA, Feb. 2009.Google Scholar

Copyright information

© American Astronautical Society, Inc. 2011

Authors and Affiliations

  • Xiaoli Bai
    • 1
  • John L. Junkins
    • 1
  1. 1.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations