Legendre pseudospectral method

The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross.[1] A basic version of the Legendre pseudospectral method based on Gauss-Lobatto points was originally proposed by Elnagar and coworkers in 1995.[2] Since then, Ross, Fahroo and their co-workers[3][4] have extended, generalized and applied the method for a large range of problems.[5] An application that has received wide publicity[6] is the use of their method for generating real time trajectories for the International Space Station.

Software

The method was first implemented in DIDO in 2001.[7] Today, it is also available in other software packages such as OTIS and PSOPT.

Flight implementations

The Legendre pseudospectral method has been implemented in flight by NASA several times. The first flight implementation was on November 5, 2006, when NASA used the Legendre pseudospectral method to maneuver the International Space Station to perform the Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazereth Bedrossian using DIDO. Watch a video of this historic maneuver.

See also

References

  1. ↑ Ross, I. M.; Karpenko, M. (2012). "A Review of Pseudospectral Optimal Control: From Theory to Flight". Annual Reviews in Control. 36: 182–197. doi:10.1016/j.arcontrol.2012.09.002.
  2. ↑ G. Elnagar, M. A. Kazemi, and M. Razzaghi, "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," IEEE Transactions on Automatic Control, 40:1793–1796, 1995.
  3. ↑ Kang, W.; Gong, Q.; Ross, I. M.; Fahroo, F. "On the Convergence of Nonlinear Optimal Control Using Pseudospectral Methods for Feedback Linearizable Systems". International Journal of Robust and Nonlinear Control. 17 (1251–1277): 2007.
  4. ↑ Ross, I. M.; Fahroo, F. "Pseudospectral Knotting Methods for Solving Nonsmooth Optimal Control Problems". Journal of Guidance Control and Dynamics. 27 (397–405): 2004.
  5. ↑ Q. Gong, W. Kang, N. Bedrossian, F. Fahroo, P. Sekhavat and K. Bollino, "Pseudospectral Optimal Control for Military and Industrial Applications," 46th IEEE Conference on Decision and Control, New Orleans, LA, pp. 4128–4142, Dec. 2007.
  6. ↑ Kang, W.; Bedrossian, N. "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA $1M in Under Three Hours". SIAM News. 40: 2007.
  7. ↑ J. R. Rea, A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories, S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001. http://dspace.mit.edu/handle/1721.1/8608
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