Injection into Orbit Optimization using Orthogonal Polynomials

Subject Areas : Mechanical Engineering

Sedigheh Shahmirzaee Jeshvaghany ¹ , Farshad Pazooki ² , Alireza Basohbat Novinzaddeh ³

1 - Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 - Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.
3 - Department of Aerospace Engineering, K.N.Toosi University of Technology, Tehran, Iran

Received: 2016-02-25 Accepted : 2016-05-10 Published : 2017-01-14

Keywords: Orthogonal functions, Chebyshev and legendre polynomials, Trajectory, Point-mass SLV,

Abstract :

In this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. The method was based on orthogonalpolynomial approximation. This method consisted of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as Chebyshev or Legendre polynomials with undetermined coefficients. The main characteristic of this technique was that it converted the differential expressions arising from the system dynamics and the performance index into some nonlinear algebraic equations, thereby greatly simplifying the problem solution. Our research effort focused on applying a Chebyshev series expansion to optimize the trajectory profile of a point-mass Satellite Launch Vehicle (SLV). This paper is divided as follows: first, the Chebyshev and Legender series expansion to optimization are introduced. Then, the flight mechanics model of the point-mass SLV is given. Next, our optimization problem is described and optimization results are presented and discussed.

References:

[1] Rao, V. Anil., “A survey of numerical methods for optimal control,” (Preprint) AAS 09-334, 2009.

 [2] Fahroo, F., and Ross, I. M., “Direct trajectory optimization by a Chebyshev pseudospectral method,” J. Guidance, Control and Dynamic, Vol. 25, No. 1, 2002, p. 160–166.

 [3] Brusch, R. G., “Trajectory optimization for the Atlas/Centaur launch vehicle,” J. Spacecraft Rockets, Vol. 14, 1977, p. 550-555.

[4] Brauer, G. L., Cornick, D. E., and Stevenson, R., “Capabilities and applications of the program to optimize simulated trajectories (POST),” NASA CR-2770, 1977.

[5] Well, K. H., and Tandon, S. R., “Rocket ascent trajectory optimization via recursive quadratic programming,” J. Astron. Sci, Vol. 30, 1982, p. 101-116.

[6] Adimurthy, V., “Launch vehicle trajectory optimization,” Acta Astronautica, Vol. 15, No. 11, 1987, p. 845-850.

[7] Vathsal, S., and Swaminathan, R., “Minimax approach to trajectory optimization of multistage launch vehicles,” IEEE Transactions On Aerospace and Electronic Systems, IEEE. AES-13, Vol. 2, 1977, p. 179-187.

[8] Beltracchi, TJ., “Decomposition approach to solving the Ail-Up trajectory optimization problem,” Journal of Guidance, Control, and Dynamics, Vol. 15, 1992, p. 707-716.

[9] Weigel, N., and Well, K. H., “Dual payload ascent trajectory optimization with a splash-down constraint,” Journal of Guidance, Control, and Dynamics, Vol. 23, 2000, p. 45-52.

[10] Ping, Lu., “Inverse dynamics approach to trajectory optimization for an aerospace plane,” Journal of Guidance, Control, and Dynamics, Vol. 16, No.4, 1993, p. 726-732.

 [11] Chen, C. F., and Hsiao, C. H., “Design of piecewise constant gains for optimal control via Walsh functions,” IEEE Transactions on Automatic Control, Vol. 20, No. 5, 1975, p. 596-603.

[12] Chen, W. L., and Shih, Y. P., “Analysis and optimal control of time-varying linear systems via Walsh functions,” International Journal of Control, Vol. 27, 1978, p. 917-932.

[13] Hsu, N. S., and Cheng, B., “Analysis and optimal control of time-varying linear systems via block-pulse functions,” International Journal of Control, Vol. 33, No. 6, 1981, p. 1107-1122.

[14] Hwang, C., and Shih, Y. P., “Optimal control of delay systems via block-pulse functions,” Journal of Optimization Theory and Applications, Vol. 45, No. 1, 1985, p.101- 112.

[15] Clement, P. R., “Laguerre functions in signal analysis and parameter identification,” Journal of the Franklin Institute, Vol. 313, No. 2, 1982, p. 85-95.

[16] Hwang, C., and Shih, Y. P., “Laguerre series direct method for variational problems,” Journal of Optimization Theory and Applications, Vol. 39, No. 1, 1983, p. 43-149.

[17] Hwang, C., and Chen, M. Y., “Analysis and optimal control of time-varying linear systems via shifted Legendre polynomials,” International Journal of Control, Vol. 41, No. 5, 1985, p. 1317-1330.

[18] Wang, ML., and Chang, R. Y., “Optimal control of lumped-parameter systems via shifted Legendre polynomial approximation,” Journal of Optimization Theory and Applications, Vol. 45, No. 2, 1985, p. 313-324.

[19] Paraskevopoulos, P. N., “Chebyshev series approach to system identification analysis and optimal control,” Journal of the Franklin Institute, Vol. 316, No. 1, 1983, p. 135-157.

[20] Chou, J. H., and Horng, I. R., “Application of Chebyshev polynomials to the optimal control of time-varying linear systems,” International Journal of Control, Vol. 41, No. 1, 1985, p. 135-144.

[21] Shahmirzaee, S. J., Novinzadeh, A. B., and Pazooki, F., “Multiple stage satellite launch vehicle ascent optimization using Chebyshev wavelets,” Aerospace Science and Technology, Vol. 46, 2015, p. 321-330.

[22]  Mouroutsos, S. G., and Sparis, P .D., “Taylor series approach to system identification, analysis, and optimal control,” Journal of the Franklin Institute, Vol. 319, 1985, p. 359-371.

[23] Razzaghi, Mo., and Razzaghi, Me., “Taylor series direct method for variational problems,” Journal of the Franklin Institute, Vol. 325, No. 1, 1988, p. 125-131.

[24] Paraskevopoulos, P. N., Sparis, P. D., and Mouroutsos, S. G., “The Fourier series operational matrix of integration,” International Journal of Systems Science, Vol. 16, No. 2, 1985, p. 171-176.

[25] Razzaghi, M., “Fourier Series direct method for Variational Problems,” International Journal of Control, Vol. 48, No. 3, 1988, p. 887-895.

[26] Razzaghi, M., “Optimal Control of Linear time-varying systems via Fourier series,” Journal of Optimization Theory and Applications, Vol. 65, No. 2, 1990, p. 375-384.

 [27] Razzaghi, M., and Marzban, H. R., “Direct method for variational problems via hybrid of block-pulse and Chebyshev functions,” Mathematical problems in Engineering, Vol. 6, 1999, p. 85-97.

[28] Handscomb, DC., Methods of numerical approximation, 1nd ed., Oxford university computing laboratory, Pergamon, press ltd, 1966, p. 50.

 [29] Adibi, H., and Assari, P., “Chebyshev wavelet method for numerical solution of Fredholm integral equation of the first kind,” Mathematical problems in Engineering, ID 138408, 2010.

[30] EL-Gindy, T. M., and EL-Hawary, H. M., “A Chebyshev approximation for solving optimal control problems,” Computers & Math with Applic, Vol. 29, No. 6, 1995, p. 35-45.

 [31] Afshari, H., Nasehi, H., Novinzadeh, A. B., and Roshanian, J., “A variational approach in determination of explicit neighboring optimal guidance law for injection into orbit,” International Review of Automatic Control, Vol. 1, No. 2, 2008, p. 248-257.

 [32] Bryson, Jr. A., and Yu-Chi, Ho., Applied optimal control optimization, estimation, and control, 1nd ed., Massachusetts, Toronto, London, Blaisdell Publishing Company, 1969, p. 194.