Document Type : Original Article


1 Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Iran

2 Faculty of Aerospace, Malek Ashtar University of Technology, Iran,

3 Faculty of Aerospace K.N.Toosi University of Technology, Iran


In this paper, optimal guidance law design considering fixed final state and time for the final phase a spacecraft or launch vehicle is investigated and studied. This guidance law, not only satisfied a specific optimality criterion, but it also has the least sensitivity to the initial state’s deviations; which is due to the inclusion of the nonlinear terms in the mathematical modeling using the high order expansions method. The main goal of this research, is to investigate the development and to augment the capability of the high order expansions method for guidance law design. Different implementations of this approach including the differential algebra high order, the generating function based high order and vectorized high order expansions methods have been investigated. After reviewing the implementation concepts of the high order expansions method, the effectiveness of this method has been studied. Then a 3-dimensional injection of a satellite problem has been chosen as the case study and after extracting the mathematical model and nominal optimal solution, the sensitivity variables have been extracted up to the 3rd order. Afterwards, to investigate the performance of the designed guidance law, the Monte Carlo simulations have been performed and it has been shown that the designed guidance law on the basis of the Taylor series and high order expansions method has a good accuracy and is a valuable alterative to the nominal trajectory tracking guidance approach.


Main Subjects

[1]  Leitmann, G., On a class of variational problems in rocket flight, Journal of Aerospace Sci. 26, no. 9, (Sep.1959), 586-591.
[2]  Leitmann, G., Optimization techniques with applications to Aerospace System, Academic Press New-York,1962.
[3]  Lawden, D.F. Optimal trajectories for space navigation, Butterworths Publishing Corporation, 1963.
[4]    Neustadt, L.W., A general theory of minimum-fuel space trajectories, SIAM Journal on Control 3, no. 2 (1965), 317--356.
[5]  Conway, B.A., Spacecraft Trajectory Optimization, Cambridge University Press,2010.
[6]  M. Alavipour, A. A. Nikkhah, J. Roshanian, Minimum time multiple-burn optimization of an upper stage with a finite thrust for satellite injection into geostationary orbit, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2017.
[7]  Hargraves, C. R., Paris, S. W., Direct trajectory optimization using nonlinear programming and collocation, Journal of Guidance, Control, and Dynamics, vol. 0, no. 4, pp. 338–342, 1987.
[8]  Enright, P. J., Conway, B. A., Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal of Guidance, Control, and Dynamics, vol. 15, no. 4, pp. 994–1002, 1992.
[9]  Betts, J. T., Very low thrust trajectory optimization using a direct SQP method, Journal of Computational and applied Mathematics, vol. 120, issues1-2, pp. 27–40, Aug. 2000.
[10] Ross, I. M., Fahroo, F., Pseudospectral knotting methods for solving optimal control problems, Journal of Guidance, Control, and Dynamics, vol. 27, no. 3, pp. 397–405, 2004.
[11] Pontryagin, L., Boltyanskii, V., Gramkrelidze, R., Mischenko, E., The mathematical theory of optimal processes, Wiley Interscience, 1962.
[12] Hull, D.G., Optimal control theory for applications, Springer, 2003.
[13] Trélat, E., Contrôle optimal – Théorie et Applications, Vuibert, 2005.
[14] Boltyanski, V.G., Poznyak, A.S., The robust maximum principle, Birkhäuser, 2012.
[15] Betts, J. T., Survey of numerical methods for trajectory optimization, Journal of Guidance, Control, and Dynamics, vol. 21, no. 2, pp. 193–207, 1998.
[16] Rao, A., A survey of numerical methods for optimal control, Advances in the Astronautical Sciences, vol. 135, no. 1, pp. 497–528, 2009.
[17] Pontani, M., Conway, B.A., Particle Swarm Optimization Applied to Space Trajectories, Journal of Guidance, Control, and Dynamics, Vol. 33, No. 5 (2010), pp. 1429-1441, DOI: 10.2514/1.48475.
[18] Steffens, M.J., A combined global and local methodology for launch vehicle trajectory design-space exploration and optimization, Thesis Georgia Institute of technology, April 2014.
[19] Li, Y., Chen, W., Zhou, H., and Yang, L., “Conjugate Gradient Method with Pseudospectral Collocation Scheme for Optimal Rocket Landing Guidance”, Aerospace Science and Technology, doi: 10.1016/j.ast.2020.105999, Vol. 104, Article Number. 105999, (2017).
[20] Di Lizia, P., Armellin, R., and Lavagna, M., “Application of High Order Expansions of Two-point Boundary Value Problems to Astrodynamics”, Celestial Mechanics and Dynamical Astronomy, doi: 10.1007/s10569-008-9170-5, Vol. 102, No. 4, pp. 355–375, (2008).
[21] Berz, M., “Advances in Imaging and Electron Physics Modern Map Methods in Particle Beam Physics”,1st Edition, Academic Press: San Diego, Vol.108, No.1, (1999). pp.210-305.
[22] Di Lizia, P., Armellin, A., Ercoli-Finzi, A., and Berz, M., “High-order Robust Guidance of Interplanetary Trajectories Based on Differential Algebra”, Journal of Aerospace Engineering, doi: 10.7446/jaesa.0101.05, Vol. 1, No. 1, pp. 43–57, (2008).
[23] Di Lizia, P., Armellin, A., Bernelli-Zazzera, F., and Berz, M., “High Order Optimal Control of Space Trajectories with Uncertain Boundary Conditions”, Acta Astronaut, doi: 10.1016/j.actaastro.2013.07.007, Vol. 93, pp. 217–229, (2014).
[24] Di Lizia, P., Armellin, R., Morselli, A., and Bernelli-Zazzera, F., “High Order Optimal Feedback Control of Space Trajectories with Bounded Control”, Acta Astronaut, doi: 10.1016/j.actaastro.2013.02.011, Vol. 94, No. 1, pp. 383–394, (2014).
[25] Wittig, A., and Armellin, R., “High Order Transfer Maps for Perturbed Keplerian Motion”, Celestial Mechanics and Dynamical Astronomy, doi: 10.1007/s10569-015-9621-8, Vol. 122, No. 4, pp. 333–358, (2015).
[26] Vetrisano, M., and Vasile, M., “Analysis of Spacecraft Disposal Solutions from LPO to the Moon with High Order Polynomial Expansions”, Advances in Space Research, doi: 10.1016/j.asr.2017.04.005, Vol. 60, No. 1, pp. 38–56, (2017).
[27] Sun, Z. J., Di Lizia, P., Bernelli-Zazzera, F., Luo, Y.Z., and Lin, K.P., “High-order State Transition Polynomial with Time Expansion Based on Differential Algebra”, Acta (Astronautica) , doi: 10.1016/j.actaastro.2019.03.068, Vol. 163, Part B, No.7, pp. 45–55, (2019).
[28] Morselli, A., Armellin, R., Di Lizia, P., and Bernelli Zazzera, F., “A High Order Method for Orbital Conjunctions Analysis: Sensitivity to Initial Uncertainties”, Advances in Space Research, doi: 10.1016/j.asr.2013.11.038, Vol. 53, No. 3, pp. 490–508, (2014).
[29] Morselli, A., Armellin, R., Di Lizia, P., and Bernelli Zazzera, F., “A High Order Method for Orbital Conjunctions Analysis: Monte Carlo Collision Probability Computation”, Advances in Space Research, doi: 10.1016/j.asr.2014.09.003, Vol. 55, No. 1, pp. 311–333, (2015).
[30] Gonzalo, J.L., Colombo, C., and Di Lizia, P., “Introducing MISS, a New Tool for Collision Avoidance Analysis and Design”, Journal of Space Safety Engineering, doi: 10.1016/j.jsse.2020.07.010, Vol. 7, No. 3, pp. 282–289, (2020).
[31] Moghadasian, M., and Roshanian, J., “Optimal Landing of Unmanned Aerial Vehicle using Vectorised High Order Expansions Method”, Modares Mechanical Engineering, Vol. 19, No. 11, pp. 2761–2769, (2019).
[32] Moghadasian, M., and Roshanian, J., “Continuous Maneuver of Unmanned Aerial Vehicle using High Order Expansions Method for Optimal Control Problem”, Modares Mechanical Engineering, Vol. 17, No. 12, pp. 382–390, (2018).
[33] Moghadasian, M., and Roshanian, J., “Approximately Optimal Manoeuvre Strategy for Aero-assisted Space Mission”, Advances in Space Research, doi: 10.1016/j.asr.2019.04.003,  Vol. 64, No. 2, pp. 436–450, (2019).
[34] M.Sharafi, N.Rahbar, A.Moharampour, A.R.Kashaninia, “Comparing Performance of Vectorized High Order Expansions and SDRE Method for Vertical Landing Mission of Booster” Journal of Aerospace Mechanics, Vol.18, No.3, pp 69-85 , 2022.
[35] M.Sharafi, N.Rahbar, A.Moharampour, A.R.Kashaninia, “Designing the Nonlinear Guidance Law Adaptable to Initial Deviations for the Vertical Landing of the Booster” JAST, Vol 15, No 1,pp 57-65 ,2022.
[36] M.Sharafi, N.Rahbar, A.Moharampour, A.R.Kashaninia “Performance Analysis of the Vectorized High Order Expansions Method in the Accurate Landing Problem of Reusable Boosters” Advances in Space Research, Vol 71, Issue 5, pp 2155-2174, 2023.
[37] RP Optimization Research LLC, “GPOPS-II: Next-Generation Optimal Control Software,” 2016. [Online]. Available: