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Spline parameterization based nonlinear trajectory optimization along 4D waypoints

  • Ahmed, Kawser (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior) ;
  • Bousson, Kouamana (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior) ;
  • Coelho, Milca de Freitas (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior)
  • Received : 2018.10.02
  • Accepted : 2019.05.08
  • Published : 2019.09.25

Abstract

Flight trajectory optimization has become an important factor not only to reduce the operational costs (e.g.,, fuel and time related costs) of the airliners but also to reduce the environmental impact (e.g.,, emissions, contrails and noise etc.) caused by the airliners. So far, these factors have been dealt with in the context of 2D and 3D trajectory optimization, which are no longer efficient. Presently, the 4D trajectory optimization is required in order to cope with the current air traffic management (ATM). This study deals with a cubic spline approximation method for solving 4D trajectory optimization problem (TOP). The state vector, its time derivative and control vector are parameterized using cubic spline interpolation (CSI). Consequently, the objective function and constraints are expressed as functions of the value of state and control at the temporal nodes, this representation transforms the TOP into nonlinear programming problem (NLP). The proposed method is successfully applied to the generation of a minimum length optimal trajectories along 4D waypoints, where the method generated smooth 4D optimal trajectories with very accurate results.

Keywords

trajectory optimization;4D waypoint navigation;spline parameterization;Nonlinear programming

References

  1. Athans, M.A. and Falb, P.L. (2006), Optimal Control: An Introduction to The Theory and Its Applications, Dover Publications, Mineola, New York, U.S.A.
  2. Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (2006), Nonlinear Programming: Theory and Algorithms, Wiley-Interscience, New Jersey, U.S.A.
  3. Bellman, R.E. (1957), Dynamic Programming, Princeton University Press, Princeton, New Jersey, U.S.A.
  4. Benson, D.A., Huntington, G.T., Thorvaldsen, T.P. and Rao, A.V. (2006), "Direct trajectory optimization and costate estimation via an orthogonal collocation method", J. Guid. Control Dyn., 29(6), 1435-1440. https://doi.org/10.2514/1.20478. https://doi.org/10.2514/1.20478
  5. Betts, J.T. (1998), "Survey of numerical methods for trajectory optimization", J. Guid. Control Dyn., 21(2), 193-207. https://doi.org/10.2514/2.4231. https://doi.org/10.2514/2.4231
  6. Betts, J.T. (2010), Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Ed., SIAM Press, Philadelphia, U.S.A.
  7. Bousson, K. (2003), "Chebyshev pseudospectral trajectory optimization of differential inclusion models", Proceedings of the SAE World Aviation Congress, Montreal, Canada, January.
  8. Bousson, K. (2005), "Single gridpoint dynamic programming for trajectory optimization", Proceedings of the AIAA Atmospheric Flight Mechanics Conference and Exhibit, San Francisco, California, U.S.A., August.
  9. Bousson, K. and Machado, P. (2010), "4D flight trajectory optimization based on pseudospectral methods", World Acad. Sci. Eng. Technol. Int. J. Aerosp. Mech. Eng., 4(9), 879-885. https://doi.org/10.5281/zenodo.1076520.
  10. Bryson, A. E. and Ho, Y. C. (1975), Applied Optimal Control: Optimization, Estimation and, Control, Taylor & Francis, New York, U.S.A.
  11. Burden, R.L. and Faires J.D. (2011), Numerical Analysis (Ninth Ed.), Cengage Learning, Boston, Massachusetts, U.S.A.
  12. Dontchev, A.L., Hager, W.W. and Veliov, V.M. (2000), "Second-order Runge-Kutta approximations in control constrained optimal control", SIAM J. Numer. Anal., 38(1), 202-226. https://doi.org/10.1137/S0036142999351765. https://doi.org/10.1137/S0036142999351765
  13. Enright, P.J. and Conway, B.A. (1992), "Discrete approximations to optimal trajectories using direct transcription and nonlinear programming", J. Guid. Control Dyn., 15(4), 994-1002. https://doi.org/10.2514/3.20934. https://doi.org/10.2514/3.20934
  14. Fahroo, F. and Ross, I.M. (2002), "Direct trajectory optimization by a chebyshev pseudospectral method", J. Guid. Control Dyn., 25(1), 160-166. https://doi.org/10.2514/2.4862. https://doi.org/10.2514/2.4862
  15. Guo, X. and Zhu, M. (2013), "Direct trajectory optimization based on a mapped chebyshev pseudospectral method", Chin. J. Aeronaut., 26(2), 401-412. https://doi.org/10.1016/j.cja.2013.02.018. https://doi.org/10.1016/j.cja.2013.02.018
  16. Hagelauer, P. and Mora-Camino, F. A. C. (1998), "A soft dynamic programming approach for on-line aircraft 4D-trajectory optimization", Eur. J. Oper. Res., 107(1), 87-95. https://doi.org/10.1016/S0377-2217(97)00221-X. https://doi.org/10.1016/S0377-2217(97)00221-X
  17. Hargraves, C.R. and Paris, S.W. (1987), "Direct trajectory optimization using nonlinear programming and collocation", J. Guid. Control Dyn., 10(4), 338-342. https://doi.org/10.2514/6.1986-2000. https://doi.org/10.2514/3.20223
  18. Hull, D.G. (1997), "Conversion of optimal control problems into parameter optimization problems", J. Guid. Control Dyn., 20(1), 57-60. https://doi.org/10.2514/2.4033. https://doi.org/10.2514/2.4033
  19. Luus, R. (2000), Iterative Dynamic Programming. Chapman & Hall, CRC, London, U.K.
  20. Ma, L., Shao, Z., Chen, W., Lv, X. and Song, Z. (2016), "Three-dimensional trajectory optimization for lunar ascent using Gauss pseudospectral method", Proceedings of the AIAA Guidance, Navigation, and Control Conference, San Diego, California, U.S.A., January.
  21. Marsden M. (1974), "Cubic spline interpolation of continuous functions", J. Approx. Theor., 10(2), 103-111. https://doi.org/10.1016/0021-9045(74)90109-9. https://doi.org/10.1016/0021-9045(74)90109-9
  22. Mazzotta, D. G., Sirigu, G., Cassaro, M., Battipede, M. and Gili, P. (2017), "4D Trajectory Optimization Satisfying Waypoint and No-Fly Zone Constraints", Proceedings of the International Conference on Applied and Theoretical Mechanics, Venice, Italy, April.
  23. Miyamoto, Y., Wickramasinghe, N.K., Harada, A., Miyazawa, Y. and Funabiki, K. (2013), "Analysis of fuelefficient airliner flight via dynamic programming trajectory optimization", Trans. JSASS Aerosp. Technol. Japan, 11, 93-98. https://doi.org/10.2322/tastj.11.93.
  24. Miyazawa, Y., Wickramasinghe, N.K., Harada, A. and Miyamoto, Y. (2013), "Dynamic programming application to airliner four dimensional optimal flight trajectory", Proceedings of the AIAA Guidance, Navigation, and Control (GNC) Conference, Boston, Massachusetts, U.S.A., August.
  25. Naidu, D. S. (2003), Optimal Control Systems, CRC Press LLC, Boca Raton, Florida, USA.
  26. Pontryagin, L.S. (1962), The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, U.S.A.
  27. Rao, A.V. (2009), "A survey of numerical methods for optimal control", Adv. Astronaut. Sci., 135(1), 497-528.
  28. Schwartz, A. and Polak, E. (1996), "Consistent approximations for optimal control problems based on Runge-Kutta integration", SIAM J. Control Optim., 34(4), 1235-1269. https://doi.org/10.1137/S0363012994267352. https://doi.org/10.1137/S0363012994267352
  29. Seemkooei, A.A. (2002), "Comparison of different algorithm to transform geocentric to geodetic coordinates", Survey Rev., 36(286), 627-632. http://doi.org/10.1179/003962602791482966. https://doi.org/10.1179/003962602791482975
  30. Stryk, O.V. and Bulirsch, R. (1992), "Direct and indirect methods for trajectory optimization", Ann. Oper. Res., 37(1), 357-373. https://doi.org/10.1007/BF02071065. https://doi.org/10.1007/BF02071065
  31. Tohidi, E., Pasban, A., Kilicman, A. and Noghabi, S.L. (2013), "An Efficient pseudospectral method for solving a class of nonlinear optimal control problems", Abstr. Appl. Anal., http://dx.doi.org/10.1155/2013/357931.
  32. Wang Z. and Ouyang J. (2013), "Curve length estimation based on cubic spline interpolation in gray-scale images", J. Zhejiang Univ. Sci. C Comput. Electron., 14(10), 777-784. https://doi.org/10.1631/jzus.C1300056. https://doi.org/10.1631/jzus.C1300056
  33. Zhao, J., Zhou, R. and Jin, X. (2014), "Gauss pseudospectral method applied to multi-objective spacecraft trajectory optimization, J. Comput. Theor. Nanosci., 11(10), 2242-2246. https://doi.org/10.1166/jctn.2014.3685. https://doi.org/10.1166/jctn.2014.3685