Iranian Journal of Mechanical Engineering Transactions of ISME

Iranian Journal of Mechanical Engineering Transactions of ISME

Guidance Law Design for the Final Phase 3D Landing Mission of Booster using High Order Expansions Method

Authors
Morteza Sharafi 1
Nasser Rahbar 2
Ali Moharampour 3
Abdorreza Kashaninia 4
1 Ph.D., Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Iran
2 Associate Professor, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Iran
3 Assistant Professor, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Iran
4 Assistant Professor, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology y, Iran
10.30506/ijmep.2023.1978080.1906
Abstract
In this paper, optimal guidance law design considering fixed final state and time for the final phase booster landing problem of 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 booster landing 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.
Keywords
Vectorized high order expansions
Nonlinear optimal control
Booster landing
Optimal guidance

Subjects

[1]        A. Banerjee and R. Padhi, "An Optimal Explicit Guidance Algorithm for Terminal Descent Phase of Lunar Soft Landing," In AIAA Guidance, Navigation, and Control Conference, Grapevine, Texas, USA, 9-13 January 2017, p. 1266, doi: https://doi.org/10.2514/6.2017-1266.
 
[2]        S. Swaminathan, R. UP, and D. Ghose, "Real Time Powered Descent Guidance Algorithm for Mars Pinpoint Landing with Inequality Constraints," In AIAA Scitech 2020 Forum, Orlando, FL, 6-10 January 2020, p. 1351, doi: https://doi.org/10.2514/6.2020-1351.
[3]        K. S. G. A. a. Z. Z. a. Q. Gao, "Minimum-Fuel Optimal Trajectory for Reusable First-stage Rocket Landing using Particle Swarm Optimization," International Journal of Mechanical and Mechatronics Engineering, Vol. 11, No. 5, pp. 981 - 990, 2017, doi: https://doi.org/10.5281/zenodo.1130269.
 
[4]        Y. Li, W. Chen, H. Zhou, and L. Yang, "Conjugate Gradient Method with Pseudospectral Collocation Scheme for Optimal Rocket Landing Guidance," Aerospace Science and Technology, Vol. 104, p. 105999, 2020, doi: https://doi.org/10.1016/j.ast.2020.105999.
 
[5]        P. D. Lizia, R. Armellin, and M. Lavagna, "Application of High Order Expansions of Two-point Boundary Value Problems to Astrodynamics," Celestial Mechanics and Dynamical Astronomy, Vol. 102, pp. 355-375, 2008, doi: https://doi.org/10.1007/s10569-008-9170-5.
 
[6]        M. Berz, Advances in Imaging and Electron Physics Modern Map Methods in Particle Beam Physics. Academic Press: San Diego, 1999.
 
[7]        P. Di Lizia, R. Armellin, A. Ercoli-Finzi, and M. Berz, "High-Order Robust Guidance of Interplanetary Trajectories Based on Differential Algebra," Journal of Aerospace Engineering, Sciences and Applications, Vol. 1, No. 1, pp. 43-57, 2008, doi: https://doi.org/10.7446/jaesa.0101.05.
 
[8]        P. Di Lizia, R. Armellin, F. Bernelli-Zazzera, and M. Berz, "High Order Optimal Control of Space Trajectories with Uncertain Boundary Conditions," Acta Astronautica, Vol. 93, pp. 217-229, 2014, doi: https://doi.org/10.1016/j.actaastro.2013.07.007.
 
[9]        P. Di Lizia, R. Armellin, A. Morselli, and F. Bernelli-Zazzera, "High Order Optimal Feedback Control of Space Trajectories with Bounded Control," Acta Astronautica, Vol. 94, No. 1, pp. 383-394, 2014, doi: https://doi.org/10.1016/j.actaastro.2013.02.011.
 
[10]      A. Wittig and R. Armellin, "High Order Transfer Maps for Perturbed Keplerian Motion," Celestial Mechanics and Dynamical Astronomy, Vol. 122, pp. 333-358, 2015, doi: http://doi.org/10.1007/s10569-015-9621-8.
 
[11]      M. Vetrisano and M. Vasile, "Analysis of Spacecraft Disposal Solutions from LPO to the Moon with High Order Polynomial Expansions," Advances in Space Research, Vol. 60, No. 1, pp. 38-56, 2017, doi: https://doi.org/10.1016/j.asr.2017.04.005.
 
[12]      Z.-J. Sun, P. Di Lizia, F. Bernelli-Zazzera, Y.-Z. Luo, and K.-P. Lin, "High-order State Transition Polynomial with Time Expansion Based on Differential Algebra," Acta Astronautica, Vol. 163, pp. 45-55, 2019, doi:https://doi.org/10.1016/j.actaastro.2019.03.068.
 
[13]      A. Morselli, R. Armellin, P. Di Lizia, and F. B. Zazzera, "A High Order Method for Orbital Conjunctions Analysis: Sensitivity to Initial Uncertainties," Advances in Space Research, vol. 53, no. 3, pp. 490-508, 2014, doi: https://doi.org/10.1016/j.asr.2013.11.038.
 
[14]      A. Morselli, R. Armellin, P. Di Lizia, and F. B. Zazzera, "A High Order Method for Orbital Conjunctions Analysis: Monte Carlo Collision Probability Computation," Advances in Space Research, Vol. 55, No. 1, pp. 311-333, 2015, doi: https://doi.org/10.1016/j.asr.2014.09.003.
[15]      J. L. Gonzalo, C. Colombo, and P. Di Lizia, "Introducing MISS, a New Tool for Collision Avoidance Analysis and Design," Journal of Space Safety Engineering, Vol. 7, No. 3, pp. 282-289, 2020, doi: https://doi.org/10.1016/j.jsse.2020.07.010.
 
[16]      M. Valli, R. Armellin, P. Di Lizia, and M. R. Lavagna, "Nonlinear Filtering Methods for Spacecraft Navigation Based on Differential Algebra," Acta Astronautica, Vol. 94, No. 1, pp. 363-374, 2014, doi: http://doi.org/10.1016/j.actaastro.2013.03.009.
 
[17]      D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches, 1st ed. (No. 6). , Hoboken, New Jersey: John Wiley & Sons, 2006.
 
[18]      R. S. Park and D. J. Scheeres, "Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design," Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, pp. 1367-1375, 2006, doi: https://doi.org/10.2514/1.20177.
 
[19]      R. S. Park and D. J. Scheeres, "Nonlinear Semi-analytic Methods for Trajectory Estimation," Journal of Guidance, Control, and Dynamics, Vol. 30, No. 6, pp. 1668-1676, 2007, doi: https://doi.org/10.2514/1.29106.
 
[20]      F. Cavenago, P. Di Lizia, M. Massari, and A. Wittig, "On-board Spacecraft Relative Pose Estimation with High-order Extended Kalman Filter," Acta Astronautica, Vol. 158, pp. 55-67, 2019, doi: https://doi.org/10.1016/j.actaastro.2018.11.020.
 
[21]      M. Moghadasian and J. Roshanian, "Optimal Landing of Unmanned Aerial Vehicle using Vectorised High Order Expansions Method," (in Persian), Modares Mechanical Engineering, Vol. 19, No. 11, pp. 2761-2769, 2019. [Online]. Available: http://mme.modares.ac.ir/article-15-27084-en.html.
 
[22]      M. Moghadasian and J. Roshanian, "Continuous Maneuver of Unmanned Aerial Vehicle using High Order Expansions Method for Optimal Control Problem," (in Persian), Modares Mechanical Engineering, Vol. 17, No. 12, pp. 382-390, 2018. [Online]. Available: http://mme.modares.ac.ir/article-15-4456-fa.html.
 
[23]      M. Moghadasian and J. Roshanian, "Approximately Optimal Manoeuvre Strategy for Aero-assisted Space Mission," Advances in Space Research, Vol. 64, No. 2, pp. 436-450, 2019, doi: https://doi.org/10.1016/j.asr.2019.04.003.
 
[24]      M. Moghadasian, "Development of an Analytical-numerical Method to Solve Pseudo-spectral Optimal Guidance Problem in Space Missions," Ph.D. Thesis, Department of Aerospace Engineering K.N. Toosi  University, Tehran, Iran, 2018.
 
[25]      M. Sharafi, N. Rahbar, and A. Kashaninia, "Comparing Performance of Vectorized High Order Expansions and SDRE Method for Vertical Landing Mission of Booster," (in persian), Vol. 18, No. 3, pp. 69-85, 2022, doi: https://dorl.net/dor/20.1001.1.26455323.1401.18.3.6.8.
 
[26]      M. Sharafi, N. Rahbar, A. Moharampour, and A. Kashaninia, "Performance Analysis of the Vectorized High Order Expansions Method in the Accurate Landing Problem of Reusable Boosters," Advances in Space Research, Vol. 71, No. 5, pp. 2155-2174, 2023/03/01/ 2023, doi: https://doi.org/10.1016/j.asr.2022.10.057.

  • Receive Date 07 December 2022
  • Revise Date 14 July 2023
  • Accept Date 07 October 2023