Minimum-Fuel Optimal Trajectory for Reusable First-Stage Rocket Landing Using Particle Swarm Optimization

Authors: Kevin Spencer G. Anglim, Zhenyu Zhang, Qingbin Gao

Abstract:

Reusable launch vehicles (RLVs) present a more environmentally-friendly approach to accessing space when compared to traditional launch vehicles that are discarded after each flight. This paper studies the recyclable nature of RLVs by presenting a solution method for determining minimum-fuel optimal trajectories using principles from optimal control theory and particle swarm optimization (PSO). This problem is formulated as a minimum-landing error powered descent problem where it is desired to move the RLV from a fixed set of initial conditions to three different sets of terminal conditions. However, unlike other powered descent studies, this paper considers the highly nonlinear effects caused by atmospheric drag, which are often ignored for studies on the Moon or on Mars. Rather than optimizing the controls directly, the throttle control is assumed to be bang-off-bang with a predetermined thrust direction for each phase of flight. The PSO method is verified in a one-dimensional comparison study, and it is then applied to the two-dimensional cases, the results of which are illustrated.

Keywords: Minimum-fuel optimal trajectory, particle swarm optimization, reusable rocket, SpaceX.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1130269

References:

[1] J. Ahn, B. Park, and M. Tahk, “Two-dimensional Trajectory Optimization of a Soft Lunar Landing from a Parking Orbit Considering a Landing Site,” IFAC Proceedings Volumes, 43(15), 178-183, doi:10.3182/20100906-5-jp-2022.00031, 2010.
[2] V. Ramanan, and M. Lal, “Analysis of Optimal Strategies for Soft Landing on the Moon from Lunar Parking Orbit,” Journal of Earth System Science J Earth Syst Sci, 114(6), 807-813, doi:10.1007/bf02715967, 2005.
[3] L. P. Ocampo, “Solving the Optimization Control Problem for Lunar Soft Landing Using Minimization Technique,” M.S. thesis, Dept. Math., Univ. of Texas at Arlington, Arlington, TX, 2013.
[4] B. Acikmese, and S. R. Ploen, “Convex Programming Approach to Powered Descent Guidance for Mars Landing,” Journal of Guidance, Control, and Dynamics,30(5), 1353-1366, doi:10.2514/1.27553, 2007.
[5] J. M. Carson, B. Acikmese, and L. Blackmore, “Lossless Convexification of Powered-Descent Guidance with Non-Convex Thrust Bound and Pointing Constraints,” Proceedings of the 2011 American Control Conference, doi:10.1109/acc.2011.5990959, 2011.
[6] A. V. Rao, “A Survey of Numerical Methods for Optimal Control,” AAS/AIAA Astrodynamics Specialist Conference, AAS Paper 09-334, 2009.
[7] O. V. Stryk, and R. Bulirsch, “Direct and Indirect Methods for Trajectory Optimization,” Annals of Operations Research. 37(1), 357-373, 1992.
[8] J. T. Betts, “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, 21(2), 193-207, doi:10.2514/2.4231, 1998.
[9] R. Eberhart, and J. Kennedy, “A New Optimizer Using Particle Swarm Theory,” MHS'95, Proceedings of the Sixth International Symposium on Micro Machine and Human Science. doi:10.1109/mhs.1995.494215, 1995.
[10] D. E. Kirk, Optimal Control Theory: An Introduction. Mineola, NY: Dover, 2004.
[11] J. Gardi, and J. Ross, “The Future of Space Launch is Near: An Illustrated Guide to SpaceX's Launch Vehicle Reusability Plans,” retrieved from http://justatinker.com/Future/, June 1, 2016.
[12] Falcon 9 v1.1 & F9R Launch Vehicle Overview,” retrieved from http://spaceflight101.com/spacerockets/falcon-9-v1-1-f9r/, June 1, 2016.
[13] H. D. Curtis, Orbital Mechanics for Engineering Students, Second Edition. Oxford, UK: Elsevier, 2010.
[14] “Particle Swarm Optimization,” retrieved from http://www.mathworks.com/help/gads/particleswarm.html, June 1, 2016.