Abstract
Enhancements in evolutionary optimization techniques are rapidly growing in many aspects of engineering, specifically in astrodynamics and space trajectory optimization and design. In this chapter, the problem of optimal design of space trajectories is tackled via an enhanced optimization algorithm within the framework of Estimation of Distribution Algorithms (EDAs), incorporated with Lyapunov and Q-law feedback control methods. First, both a simple Lyapunov function and a Q-law are formulated in Classical Orbital Elements (COEs) to provide a closed-loop low-thrust trajectory profile. The weighting coefficients of these controllers are approximated with various degrees of Hermite interpolation splines. Following this model, the unknown time series of weighting coefficients are converted to unknown interpolation points. Considering the interpolation points as the decision variables, a black-box optimization problem is formed with transfer time and fuel mass as the objective functions. An enhanced EDA is proposed and utilized to find the optimal variation of weighting coefficients for minimum-time and minimum-fuel transfer trajectories. The proposed approach is applied in some trajectory optimization problems of Earth-orbiting satellites. Results show the efficiency and the effectiveness of the proposed approach in finding optimal transfer trajectories. A comparison between the Q-law and a simple Lyapunov controller is done to show the potential of the EEDA in enabling the simple Lyapunov controller to recover the finer nuances explicitly given within the analytical expressions in the Q-law.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Society for Industrial and Applied Mathematics, 2010.
B. A. Conway, Spacecraft trajectory optimization. Cambridge University Press, 2010.
A. Shirazi, J. Ceberio, and J. A. Lozano, “Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions,” Progress in Aerospace Sciences, vol. 102, pp. 76–98, oct 2018.
R. Chai, A. Savvaris, A. Tsourdos, S. Chai, and Y. Xia, “Unified multiobjective optimization scheme for aeroassisted vehicle trajectory planning,” Journal of Guidance, Control, and Dynamics, vol. 41, no. 7, pp. 1521–1530, Jul. 2018. [Online]. Available: https://doi.org/10.2514/1.g003189
G. qun Wu, L.-G. Tan, X. Li, and S.-M. Song, “Multi-objective optimization for time-open lambert rendezvous between non-coplanar orbits,” International Journal of Aeronautical and Space Sciences, vol. 21, no. 2, pp. 560–575, Nov. 2019. [Online]. Available: https://doi.org/10.1007/s42405-019-00231-z
M. Pontani, “Optimal low-thrust hyperbolic rendezvous for earth-mars missions,” Acta Astronautica, vol. 162, pp. 608–619, 2019.
J. A. Englander and B. A. Conway, “Automated solution of the low-thrust interplanetary trajectory problem,” Journal of Guidance, Control, and Dynamics, vol. 40, no. 1, pp. 15–27, 2017.
S. Sarno, J. Guo, M. D’Errico, and E. Gill, “A guidance approach to satellite formation reconfiguration based on convex optimization and genetic algorithms,” Advances in Space Research, vol. 65, no. 8, pp. 2003–2017, Apr. 2020. [Online]. Available: https://doi.org/10.1016/j.asr.2020.01.033
Z. Zheng, J. Guo, and E. Gill, “Distributed onboard mission planning for multi-satellite systems,” Aerospace Science and Technology, vol. 89, pp. 111–122, Jun. 2019. [Online]. Available: https://doi.org/10.1016/j.ast.2019.03.054
Z. Fan, M. Huo, N. Qi, Y. Xu, and Z. Song, “Fast preliminary design of low-thrust trajectories for multi-asteroid exploration,” Aerospace Science and Technology, vol. 93, p. 105295, Oct. 2019. [Online]. Available: https://doi.org/10.1016/j.ast.2019.07.028
X. Li, X. Wang, and Y. Xiong, “A combination method using evolutionary algorithms in initial orbit determination for too short arc,” Advances in Space Research, vol. 63, no. 2, pp. 999–1006, Jan. 2019. [Online]. Available: https://doi.org/10.1016/j.asr.2018.08.036
J. A. Lozano, P. Larrañaga, I. Inza, and E. Bengoetxea, Towards a new evolutionary computation: advances on estimation of distribution algorithms. Springer, 2006, vol. 192.
C. A. Kluever, “Simple guidance scheme for low-thrust orbit transfers,” Journal of Guidance, Control, and Dynamics, vol. 21, no. 6, pp. 1015–1017, 1998.
A. E. Petropoulos, “Simple control laws for low-thrust orbit transfers,” AAS/AIAA Astrodynamics Specialists Conference, 2003.
A. Petropoulos, “Refinements to the Q-law for low-thrust orbit transfers,” in Advances in the Astronautical Sciences, vol. 120, 2005, pp. 963–982.
Y. Gao and X. Li, “Optimization of low-thrust many-revolution transfers and Lyapunov-based guidance,” Acta Astronautica, vol. 66, no. 1–2, pp. 117–129, 2010.
J. L. Shannon, M. T. Ozimek, J. A. Atchison, and M. Christine, “Q-law aided Direct Trajectory Optimization for the High-fidelity, Many-revolution Low-thrust Orbit Transfer Problem,” in AAS, no. 19, 2019, p. 448.
S. Lee, A. Petropoulos, and P. von Allmen, “Low-thrust Orbit Transfer Optimization with Refined Q-law and Multi-objective Genetic Algorithm,” Advances In The Astronautical Sciences, 2005.
G. Varga and J. M. Sánchez Pérez, “Many-Revolution Low-Thrust Orbit Transfer Computation Using Equinoctial Q-Law Including J2 and Eclipse Effects,” Icatt 2016, pp. 2463–2481, 2016.
A. E. Petropoulos, Z. B. Tarzi, G. Lantoine, T. Dargent, and R. Epenoy, “Techniques for designing many-revolution electric-propulsion trajectories,” Advances in the Astronautical Sciences, vol. 152, no. 3, pp. 2367–2386, 2014.
D. L. Yang, B. Xu, and L. Zhang, “Optimal low-thrust spiral trajectories using Lyapunov-based guidance,” Acta Astronautica, vol. 126, pp. 275–285, 2016.
K. F. Wakker, Fundamentals of Astrodynamics. TU Delft Library, 2015.
R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics. New York: AIAA Education Series, 1987.
H. Holt, R. Armellin, N. Baresi, Y. Hashida, A. Turconi, A. Scorsoglio, and R. Furfaro, “Optimal q-laws via reinforcement learning with guaranteed stability,” Acta Astronautica, vol. 187, pp. 511–528, 2021.
J. L. Shannon, D. Ellison, and C. Hartzell, “Analytical Partial Derivatives of the Q-Law Guidance Algorithm,” in AAS, 2021, pp. 1–15.
H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, Fourth Edition. American Institute of Aeronautics and Astronautics, Inc., 2018.
M. Pontani and M. Pustorino, “Nonlinear earth orbit control using low-thrust propulsion,” Acta Astronautica, vol. 179, pp. 296–310, 2021.
J. R. Rice, Numerical Methods in Software and Analysis. Elsevier, 2014.
E. Catmull and R. Rom, “A class of local interpolating splines,” in Computer Aided Geometric Design. Elsevier, 1974, pp. 317–326.
F. N. Fritsch and R. E. Carlson, “Monotone piecewise cubic interpolation,” SIAM Journal on Numerical Analysis, vol. 17, no. 2, pp. 238–246, 1980.
C. De Boor, C. De Boor, E.-U. Mathématicien, C. De Boor, and C. De Boor, A practical guide to splines. Springer-Verlag New York, 1978, vol. 27.
P. Larrañaga and J. A. Lozano, Estimation of distribution algorithms: A new tool for evolutionary computation. Springer Science & Business Media, 2001, vol. 2.
D. Arthur and S. Vassilvitskii, “k-means+ +: The advantages of careful seeding,” Stanford, Tech. Rep., 2006.
V. Hodge and J. Austin, “A survey of outlier detection methodologies,” Artificial intelligence review, vol. 22, no. 2, pp. 85–126, 2004.
S. Geffroy and R. Epenoy, “Optimal low-thrust transfers with constraints - Generalization of averaging techniques,” Acta Astronautica, vol. 41, no. 3, pp. 133–149, 1997.
A. Petropoulos, “Low-thrust orbit transfers using candidate Lyapunov functions with a mechanism for coasting,” in AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 2004, p. 5089.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Shirazi, A., Holt, H., Armellin, R., Baresi, N. (2023). Time-Varying Lyapunov Control Laws with Enhanced Estimation of Distribution Algorithm for Low-Thrust Trajectory Design. In: Fasano, G., Pintér, J.D. (eds) Modeling and Optimization in Space Engineering. Springer Optimization and Its Applications, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-031-24812-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-031-24812-2_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-24811-5
Online ISBN: 978-3-031-24812-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)