Generalized Computational Nodes for Pseudospectral Methods

  • Received : 2014.03.06
  • Accepted : 2014.06.09
  • Published : 2014.06.30


Pseudo-spectral method typically converges at an exponential rate. However, it requires a special set of fixed collocation points (CPs) to get highly accurate formulas for partial integration and differentiation. In this study, computational nodes for defining the discrete variables of states and controls are built independently of the CPs. The state and control variables at each CP, which are required to transcribe an NOCP into the corresponding NLP, are interpolated, using those variables allocated at each node. Additionally, Lagrange interpolation and spline interpolation are investigated, to provide a guideline for selecting a favorable interpolation method. The proposed techniques are applied to the solution of an NOCP using equally spaced nodes, and the computed results are compared to those using the standard PS approach, to validate the usefulness of the proposed methods.


Supported by : Konkuk University


  1. Q. Gong, I. M. Ross, and F. Fahroo, "Pseudospectral Optimal Control on Arbitrary Grids," AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, August 9-13, 2009.
  2. Bryson, A. E., Jr., and Ho, Y. C., Applied Optimal Control, Hemisphere Publishing, Washington D.C., 1975.
  3. Kirk, D. E., Optimal Control Theory; An Introduction, Dover, New York, 1970.
  4. Garg, D., Patterson, M., Hager, W. W., Rao, A. V., Benson, D. A. and Huntington, G. T., "A Unified Framework for the Numerical Solution of Optimal Control Problems Using Pseudospectral methods," Automatica, Vol. 46, No. 11, November 2010, pp. 1843-1851. DOI:10.1016/j.automatica.2010.06.048
  5. Vijaya Bhaskar, N. R. Babu, and K. Varghese, "Spline Based Trajectory Planning for Cooperative Crane Lifts," Proceedings of the 23rd ISARC, Tokyo, 2006, pp. 418-423
  6. Ogundare, B. S. and Okecha, G. E., "A Pseudo Spline Methods for Solving an Initial Value Problem of Ordinary Differential Equation," Journal of Mathematics and Statistics, Vol. 4, No. 2, 2008, pp. 117-121
  7. Akram, G. and Siddiqi, S. S., "End conditions for interpolatory septic splines," International Journal of Computer Mathematics, Vol. 82, No. 12, December 2005, pp. 1525-1540.
  8. Ross, I. M. and Fahroo, F., "Pseudo-spectral Knotting Methods for Solving Optimal Control Problems," Journal of Guidance, Control and Dynamics, Vol. 27, No. 3, May-June 2004, pp. 397-405. DOI: 10.1109/TAC.2005.860248
  9. Williams, P., "A Gauss-Lobatto Quadrature Method for Solving Optimal Control Problems," Australian and New Zealand Industrial and Applied Mathematics Journal, Vol. 47, 2005, pp. 101-115.
  10. Fahroo, F. and Ross, I. M., "Costate Estimation by a Legendre Pseudo-spectral Method," AIAA Journal of Guidance, Control and Dynamics, Vol. 24, No. 2, 2001, pp. 270-277. DOI: 10.2514/2.4709
  11. Gong, Q., Ross, I. M., Kang, W., and Fahroo, F., "Connections between the Covector Mapping Theorem and Convergence of Pseudo-spectral Methods for Optimal Control," Computational Optimization and Applications, Vol. 41, No. 3, 2008, pp. 307-335.
  12. Kim, C.-J., Sung, S., and Shin, K., "Pseudo-spectral Application to Nonlinear Optimal Trajectory Generation of a Rotorcraft," The First International Conference on Engineering and Technology Innovation, Kenting, Taiwan, November 11-15, 2011.
  13. Kim, C.-J., Sung, S., Park, S. H., and Jung, S. N., " Time- Scale Separation for Rotorcraft Nonlinear Optimal Control Analyses, "Journal of Guidance, Control and Dynamics, Vol. 37, No. 2, March-April 2014, pp. 655-673.
  14. D. Benson, "A Gauss Pseudo-spectral Transcription for Optimal Control," MIT, Ph.D. Thesis, Department of Aeronautics and Astronautics, November 2004.
  15. G. T. Huntington, "Advancement and Analysis of a Gauss Pseudo-spectral Transcription for Optimal Control Problems," Ph. D. Thesis, MIT, June 2007.
  16. I. M. Ross and M. Karpenko, "A Review of Pseudospectral Optimal Control: From theory to flight," Annual Reviews in Control, Vol. 36, No. 2, December 2012, pp. 182-197.

Cited by

  1. Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques vol.14, pp.2, 2016,