Abstract
Reentry trajectory optimization is a critical optimal control problem for reusable launch vehicle (RLV) with highly nonlinear dynamic characteristics and complex constraints. In this paper, a hybrid parallel Harris hawks optimization (HPHHO) algorithm is proposed to address the problem. HPHHO aims to enhance the performance of existing Harris hawks optimization (HHO) algorithm by three strategies including oppositional learning, smoothing technique and parallel optimization mechanism. At the beginning of each iteration, the opposite population is calculated from the current population by the oppositional learning strategy. Following that, the individuals in the two populations are arranged in ascending order on the basis of the fitness function values, and the top half of the resulting population is selected as the initial population. The selected initial population is divided into two equal subpopulations which are assigned to the differential evolution and the HHO algorithm, respectively. The both algorithms operate in parallel to search and update the solutions of each subpopulation simultaneously. Then the solutions are smoothed for each iteration by the smoothing technique to reduce fluctuations. As a result, the optimal solution obtained by the parallel optimization mechanism avoids falling into local optima. The performance of HPHHO is evaluated by 4 CEC 2005 benchmark functions and 3 constrained continuous optimal control problems, showing better efficiency and robustness in terms of performance metrics, convergence rate and stability. Finally, the simulation results show that the proposed algorithm is very effective, practical and feasible in solving the RLV reentry trajectory optimization problem.
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16 October 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00500-021-06287-y
References
Abd M, Mirjalili S (2019) A hyper-heuristic for improving the initial population of whale optimization algorithm. Knowledge-Based Syst 172:42–63. https://doi.org/10.1016/j.knosys.2019.02.010
Alabool HM, Alarabiat D, Abualigah L et al (2021) Harris hawks optimization: a comprehensive review of recent variants and applications. Neural Comput Appl. https://doi.org/10.1007/s00521-021-05720-5
Bansal P, Kumar S, Pasrija S et al (2020) A hybrid grasshopper and new cat swarm optimization algorithm for feature selection and optimization of multi-layer perceptron. Soft Comput 24:15463–15489. https://doi.org/10.1007/s00500-020-04877-w
Bao XL, Jia HM, Lang CB (2019) A novel hybrid Harris hawks optimization for color image multilevel thresholding segmentation. IEEE Access 7:76529–76546. https://doi.org/10.1109/ACCESS.2019.2921545
Betts JT (2009) Practical methods for optimal control and estimation using nonlinear programming, 2nd edn. SIAM Press, Philadelphia
Chai R, Savvaris A, Tsourdos A (2017a) Violation learning differential evolution-based hp-adaptive pseudospectral method for trajectory optimization of space maneuver vehicle. IEEE Trans Aerosp Electron Syst 53:2031–2044. https://doi.org/10.1109/taes.2017.2680698
Chai R, Savvaris A, Tsourdos A et al (2017b) Multi-objective trajectory optimization of space manoeuvre vehicle using adaptive differential evolution and modified game theory. Acta Astronaut 136:273–280. https://doi.org/10.1016/j.actaastro.2017.02.023
Chai R, Savvaris A, Tsourdos A et al (2018) Unified multiobjective optimization scheme for aeroassisted vehicle trajectory planning. J Guid Control Dyn 41:1521–1530. https://doi.org/10.2514/1.G003189
Chai R, Savvaris A, Tsourdos A et al (2020) Solving multiobjective constrained trajectory optimization problem by an extended evolutionary algorithm. IEEE Trans Cybern 50:1630–1643. https://doi.org/10.1109/TCYB.2018.2881190
Derrac J, García S, Molina D et al (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1:3–18. https://doi.org/10.1016/j.swevo.2011.02.002
Coelho L dos S, Mariani VC (2008) Use of chaotic sequences in a biologically inspired algorithm for engineering design optimization. Expert Syst Appl 34:1905–1913. https://doi.org/10.1016/j.eswa.2007.02.002
Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science. pp. 39–43. https://doi.org/10.1109/MHS.1995.494215
Floudas CA, Pardalos PM, Adjiman C (1999) Handbook of test problems in local and global optimization. Kluwer Academic Publishers
Gharehchopogh FS, Gholizadeh H (2019) A comprehensive survey: whale optimization algorithm and its applications. Swarm Evol Comput 48:1–24. https://doi.org/10.1016/j.swevo.2019.03.004
Graichen K, Petit N (2008) Constructive methods for initialization and handling mixed state-input constraints in optimal control. J Guid Control Dyn 31:1334–1343. https://doi.org/10.2514/1.33870
Hashemi H, Mirjalili S (2018) A parallel numerical method for solving optimal control problems based on whale optimization algorithm. Knowledge-Based Syst 151:114–123. https://doi.org/10.1016/j.knosys.2018.03.024
He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20:89–99. https://doi.org/10.1016/j.engappai.2006.03.003
Heidari AA, Mirjalili S, Faris H et al (2019) Harris hawks optimization: algorithm and applications. Futur Gener Comput Syst 97:849–872. https://doi.org/10.1016/j.future.2019.02.028
Jadon SS, Tiwari R, Sharma H et al (2017) Hybrid artificial bee colony algorithm with differential evolution. Appl Soft Comput 58:11–24. https://doi.org/10.1016/j.asoc.2017.04.018
Jain M, Singh V, Rani A (2019) A novel nature-inspired algorithm for optimization: squirrel search algorithm. Swarm Evol Comput 44:148–175. https://doi.org/10.1016/j.swevo.2018.02.013
Jee G, Sharma KK, Rao KK et al (2014) Evolution of attitude control law of an Indian re-entry launch vehicle. Int J Adv Eng Sci Appl Math 6:148–157
Kang YL, Cheng L, Zhang QZ et al (2015) Data-driven RLV multi-objective reentry trajectory optimization based on new QABC algorithm. Int J Adv Manuf Technol 84:453–471. https://doi.org/10.1007/s00170-015-8124-9
Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Glob Optim 39:459–471. https://doi.org/10.1007/s10898-007-9149-x
Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680
Li ZH, Hu C, Ding CB et al (2018) Stochastic gradient particle swarm optimization based entry trajectory rapid planning for hypersonic glide vehicles. Aerosp Sci Technol 76:176–186. https://doi.org/10.1016/j.ast.2018.01.033
Liang W, Kun W, Wang Z et al (2019) Opposition-based multi-objective whale optimization algorithm with global grid ranking. Neurocomputing 341:41–59. https://doi.org/10.1016/j.neucom.2019.02.054
Mafarja MM, Mirjalili S (2017) Hybrid whale optimization algorithm with simulated annealing for feature selection. Neurocomputing 260:302–312. https://doi.org/10.1016/j.neucom.2017.04.053
Mahdavi S, Rahnamayan S, Deb K (2018) Opposition based learning: a literature review. Swarm Evol Comput 39:1–23. https://doi.org/10.1016/j.swevo.2017.09.010
Mao Q, Dou L, Zong Q et al (2018) Attitude controller design for reusable launch vehicles during reentry phase via compound adaptive fuzzy H-infinity control. Aerosp Sci Technol 72:36–48. https://doi.org/10.1016/j.ast.2017.10.012
Mernik M, Liu S, Karaboga D et al (2015) On clarifying misconceptions when comparing variants of the artificial bee colony algorithm by offering a new implementation. Inf Sci 291:115–127. https://doi.org/10.1016/j.ins.2014.08.040
Mirjalili S (2015) Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowledge-Based Syst 89:228–249. https://doi.org/10.1016/j.knosys.2015.07.006
Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008
Mirjalili S, Mirjalili SM, Hatamlou A (2016) Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 27:495–513. https://doi.org/10.1007/s00521-015-1870-7
Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007
Mustafi D, Sahoo G (2019) A hybrid approach using genetic algorithm and the differential evolution heuristic for enhanced initialization of the k-means algorithm with applications in text clustering. Soft Comput 23:6361–6378. https://doi.org/10.1007/s00500-018-3289-4
Nenavath H, Jatoth RK (2018) Hybridizing sine cosine algorithm with differential evolution for global optimization and object tracking. Appl Soft Comput 62:1019–1043. https://doi.org/10.1016/j.asoc.2017.09.039
Pontani M, Conway BA (2010) Particle swarm optimization applied to space trajectories. J Guid Control Dyn 33:1429–1441. https://doi.org/10.2514/1.48475
Rao AV, Benson DA, Darby C et al (2010) Algorithm 902: GPOPS, a matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Trans Math Softw 37:1–39. https://doi.org/10.1145/1731022.1731032
Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179:2232–2248. https://doi.org/10.1016/j.ins.2009.03.004
Ridha HM, Heidari AA, Wang M et al (2020) Boosted mutation-based Harris hawks optimizer for parameters identification of single-diode solar cell models. Energy Convers Manag 209:112660. https://doi.org/10.1016/j.enconman.2020.112660
Saha A, Bhattacharya A, Das P et al (2020) HSOS: a novel hybrid algorithm for solving the transient-stability-constrained OPF problem. Soft Comput 24:7481–7510. https://doi.org/10.1007/s00500-019-04374-9
Storn R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11:341–359. https://doi.org/10.1023/A:1008202821328
Su ZK, Wang HL (2015) A novel robust hybrid gravitational search algorithm for reusable launch vehicle approach and landing trajectory optimization. Neurocomputing 162:116–127. https://doi.org/10.1016/j.neucom.2015.03.063
Sun GJ, Lan YF, Zhao RQ (2019) Differential evolution with gaussian mutation and dynamic parameter adjustment. Soft Comput 23:1615–1642. https://doi.org/10.1007/s00500-017-2885-z
Turgut MS, Sağban HM, Turgut OE et al (2021) Whale optimization and sine–cosine optimization algorithms with cellular topology for parameter identification of chaotic systems and Schottky barrier diode models. Soft Comput 25:1365–1409. https://doi.org/10.1007/s00500-020-05227-6
Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82. https://doi.org/10.1109/4235.585893
Wu J, Wang H, Li N et al (2018) Path planning for solar-powered UAV in urban environment. Neurocomputing 275:2055–2065. https://doi.org/10.1016/j.neucom.2017.10.037
Zhang DN, Liu Y (2011) RLV reentry trajectory optimization through hybridization of an improved GA and a SQP algorithm. In: AIAA Guidance, Navigation, and Control Conference. https://doi.org/10.2514/6.2011-6658
Zhang HP, Wang HL, Li N et al (2020) Time-optimal memetic whale optimization algorithm for hypersonic vehicle reentry trajectory optimization with no-fly zones. Neural Comput Appl 32:2735–2749. https://doi.org/10.1007/s00521-018-3764-y
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Su, Y., Dai, Y. & Liu, Y. A hybrid parallel Harris hawks optimization algorithm for reusable launch vehicle reentry trajectory optimization with no-fly zones. Soft Comput 25, 14597–14617 (2021). https://doi.org/10.1007/s00500-021-06039-y
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DOI: https://doi.org/10.1007/s00500-021-06039-y