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 was originally proposed by Elnagar and his coworkers in 1995.[2] Since then, Ross, Fahroo and their coworkers[3][4][5][6][7] have extended, generalized and applied the method for a large range of problems.[8] An application that has received wide publicity[9][10] is the use of their method for generating real time trajectories for the International Space Station.

## Fundamentals

There are three basic types of Legendre pseudospectral methods:[1]

1. One based on Gauss-Lobatto points
1. First proposed by Elnagar et al[2] and subsequently extended by Fahroo and Ross[4] to incorporate the covector mapping theorem.
2. Forms the basis for solving general nonlinear finite-horizon optimal control problems.[1][11][12]
3. Incorporated in several software products
2. One based on Gauss-Radau points
1. First proposed by Fahroo and Ross[13] and subsequently extended (by Fahroo and Ross) to incorporate a covector mapping theorem.[5]
2. Forms the basis for solving general nonlinear infinite-horizon optimal control problems.[1][12]
3. Forms the basis for solving general nonlinear finite-horizon problems with one free endpoint.[1][11][12]
3. One based on Gauss points
1. First proposed by Reddien[14]
2. Forms the basis for solving finite-horizon problems with free endpoints[11][12]
3. Incorporated in several software products

## Software

The first software to implement the Legendre pseudospectral method was DIDO in 2001.[12][15] Subsequently, the method was incorporated in the NASA code OTIS.[16] Years later, many other software products emerged at an increasing pace, such as PSOPT, PROPT and GPOPS.

## Flight implementations

The Legendre pseudospectral method (based on Gauss-Lobatto points) has been implemented in flight[1] by NASA on several spacecraft through the use of the software, DIDO. The first flight implementation was on November 5, 2006, when NASA used DIDO to maneuver the International Space Station to perform the Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazareth Bedrossian using DIDO. Watch a video of this historic maneuver.

## References

1. Ross, I. M.; Karpenko, M. (2012). "A Review of Pseudospectral Optimal Control: From Theory to Flight". Annual Reviews in Control. 36 (2): 182–197. doi:10.1016/j.arcontrol.2012.09.002.
2. ^ a b 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. ^ Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327-342
4. ^ a b Fahroo, F. and Ross, I. M., “Costate Estimation by a Legendre Pseudospectral Method,” Journal of Guidance, Control and Dynamics, Vol.24, No.2, March–April 2001, pp.270-277.
5. ^ a b Fahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite-Horizon Optimal Control Problems,” Journal of Guidance, Control and Dynamics, Vol. 31, No. 4, pp. 927-936, 2008.
6. ^ 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.
7. ^ Ross, I. M.; Fahroo, F. (2004). "Pseudospectral Knotting Methods for Solving Nonsmooth Optimal Control Problems". Journal of Guidance Control and Dynamics. 27 (397–405): 2004. Bibcode:2004JGCD...27..397R. doi:10.2514/1.3426.
8. ^ 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.
9. ^ Kang, W.; Bedrossian, N. "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA \$1M in Under Three Hours". SIAM News. 40: 2007.
10. ^ Bedrossian, N. S., Bhatt, S., Kang, W. and Ross, I. M., “Zero-Propellant Maneuver Guidance,” IEEE Control Systems Magazine, Vol.29, No.5, October 2009, pp 53-73; Cover Story.
11. ^ a b c Fahroo F., and Ross, I. M., "Advances in Pseudospectral Methods for Optimal Control," AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2008-7309, Honolulu, Hawaii, August 2008.
12. Ross, Isaac (2015). A Primer on Pontryagin's Principle in Optimal Control. San Francisco: Collegiate Publishers.
13. ^ Fahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems,” AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA
14. ^ Reddien, G.W., "Collocation at Gauss Points as a Discretization in Optimal Control," SIAM Journal on Control and Optimization, Vol. 17, No. 2, March 1979.
15. ^ 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
16. ^ "[ OTIS ] Optimal Trajectories by Implicit Simulation". otis.grc.nasa.gov. Archived from the original on 2016-11-18. Retrieved 2016-12-08.