Lambert probleminin modifiye Chebyshev-Picard yineleme yöntemini kullanarak paralel çözümü

Öz

Lambert problemi, yörünge belirlemede çoklu devir problemini çözmek için kullanılan klasik yöntemlerden biridir. Uzay araştırma programlarına ve uydu ağlarının kullanımına olan ilginin artmasıyla, ağ kontrol merkezine ağdaki her bir uydunun yörüngesine ilişkin bilgileri sağlayacak ve uyduların yönlendirme kararlarını iyileştirmesine yardımcı olacak doğru ve hızlı bir yöntemin sağlanması önemlidir. Lambert problemi, bu problemi yinelemeli olarak çözen yöntemlerden biridir ve bu yineleme önceki yıllarda Newton'un yineleme yöntemi kullanılarak yapılmaktaydı. Daha güncel araştırmalarda bu problemi çözmek için Chebyshev-Picard yineleme yöntemi kullanılması önerilmektedir. Önerilen metot çözüm süresinde iyileştirmeler sunmasına rağmen büyük problemlerde çözüm çok uzun süreler alabilmektedir. Bu çalışmada, Lambert problemini paralel programlama teknikleri kullanarak daha hızlı çözen yeni bir paralel algoritma önerilmiştir. Ayrıca algoritmanın paralel ölçeklenebilirliğini göstermek için 2 farklı paralel sistemde; paylaşımlı ve dağıtık bellek mimarilerinde deneyler yapılmıştır. Deneysel sonuçlar, paralel algoritmanın dağıtık bellek ve paylaşımlı bellek mimarilerinde sırasıyla 8.26 ve 3.94 kat daha hızlı çözüm süresine ulaştığını göstermektedir.

Anahtar Kelimeler

• Lambert problemi
• Yörünge belirleme
• Yüksek performanslı hesaplama
• Lambert problemi
• Modifiye Chebyshev-Picard yinelemesi
• Yörünge belirleme
• Paralel hesaplama

Parallel solution of Lambert’s problem using modified Chebyshev-Picard iteration method

Abstract

Lambert’s problem is one of the classical methods for solving the multiple revolution problem in orbit determination. With the increasing interest in space exploration programs and using satellite networks, it is important to provide an accurate and rapid method that will provide the network control center with information regarding the orbit of each satellite in the network and help the satellites improve routing decisions in onboard processing satellites. Lambert’s problem is one of the methods that solve the problem iteratively and this iteration was originally done using Newton’s iteration method. In recent studies, it is recommended to use the Chebyshev-Picard iteration method to solve this problem. Since the aim here is to provide a method that solves the problem rapidly, the Chebyshev-Picard iteration method serves our objective since it is highly parallelizable. In this work, we have developed a parallel algorithm that solves Lambert’s problem in a parallel environment. We have conducted experiments to demonstrate the parallel scalability of the algorithm on both shared and distributed memory architectures. The experimental results show that the parallel algorithm achieves 8.26- and 3.94-times faster execution time on distributed memory and shared memory architectures, respectively.

Keywords

• Lambert’s problem
• Orbit determination
• Parallel computing.
• High performance computing
• Lambert’s problem
• Modified Chebyshev-Picard iteration
• Orbit determination
• Parallel computing.

Kaynakça

1. C. İnal, B. Bilgen, S. Bülbül and M. Başbük, Farklı uydu sistemi kombinasyonlarının gerçek zamanlı hassas nokta konumlamaya etkisi. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 11(1), 109-115, 2022. https://doi.org/10.28948/ ngumuh.996018
2. H. D. Curtis, Orbital mechanics for engineering students, Elsevier, Florida, 2005. https://doi.org/ 10.1016/B978-0-08-097747-8.00003-7
3. X. Bai, Modified Chebyshev-Picard iteration method for solution of boundary value problems. Ph.D dissertation, Texas A&M University, Texas, 2010.
4. J. L. Junkins, A. B. Younes, R. M. Woollands, and X. Bai, Picard iteration, Chebyshev polynomials and Chebyshev-Picard methods: Application in astrodynamics. The Journal of Astronautical Sciences, vol. 60, no. 3–4, pp. 623–653, 2013. https://doi.org/10.1007/s40295-015-0061-1
5. R. M. Woollands, J. L. Read, A. B. Probe, and J. L. Junkins, Multiple revolution solutions for the perturbed lambert problem using the method of particular solutions and Picard iteration. The Journal of Astronautical Sciences, vol. 64, no. 4, pp. 361–378, 2017. https://doi.org/10.1007/s40295-017-0116-6
6. P. B. Bailey, Nonlinear two point boundary value problems, 1st ed., vol. 44, NX Amsterdam, The Netherlands: Elsevier B.V., pp. 21–49, 1968. https://doi.org/10.1090/S0002-9904-1969-12263-9
7. J. C. Mason and D. Handscomb, Chebyshev polynomials. Boca Raton: Chapman & Hall/CRC, 2003. https://doi.org/10.1201/9781420036114
8. C. W. Clenshaw and H. J. Norton, The solution of nonlinear ordinary differential equations in Chebyshev series. The Computer Journal, vol. 6, no. 1, pp. 88–92, 1963. https://doi.org/10.1093/comjnl/6.1.88

9. T. Feagin and P. Nacozy, Matrix formulation of the Picard method for parallel computation. Celestial Mechanics and Dynamical Astronomy, vol. 29, no. 2, pp. 107–115, 1983. https://doi.org/10.1007/ BF01232802
10. J. Shaver, Formulation and evaluation of parallel algorithms for the orbit determination problem. Ph.D dissertation, United States Airforce, 1980.
11. T. Fukushima, Vector integration of dynamical motions by the Picard-Chebyshev method. The Astronomical Journal, vol. 113, p. 2325, 1997. https://doi.org/ 10.1086/118443
12. J. C. McDowell, The low earth orbit satellite population and impacts of the SpaceX Starlink constellation. The Astrophysical Journal Letters 892.2 (2020): L36. https://doi.org/10.3847/2041-8213/ab8016
13. W. Gropp, E. Lusk, N. Doss and A. Skjellum. A high-performance, portable implementation of the MPI message passing interface standard. Parallel Computing, vol. 22, no. 6, pp. 789–828, 1996. https://doi.org/10.1016/0167-8191(96)00024-5
14. T. Fukushima, Picard iteration method, Chebyshev polynomial approximation, and global numerical integration of dynamical motions. The Astronomical Journal, vol. 113, pp. 1909–1914, 1997. https://doi.org/10.1086/118404
15. G. Miel, Numerical solution on parallel processors of two-point boundary-value problems of astrodynamics. Numerical Solution of Integral Equations. Springer, Boston, MA, 1990. pp, 131-182. https://doi.org/ 10.1007/978-1-4899-2593-0_4
16. B. Macomber, A. Probe, R. Woollands, and J. L. Junkins, Parallel Modified-Chebyshev Picard iteration for orbit catalog propagation and Monte Carlo analysis. 38th Annual AAS/AIAA Guidance and Control Conference, Breckenridge, USA, Jan 2015.
17. A. Probe, B. Macomber, J. Read, R. Woollands, A. Masher, and J. Junkins, Efficient conjunction assessment using modified Chebyshev picard iteration. Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, 2015.
18. C. T. Shelton, Adaptive and orbital element methods for conjunction analysis. Ph.D dissertation, Texas A&M University, Texas, 2020.
19. G. M. Amdahl, Computer architecture and Amdahl's law. Computer 46.12, 2013. https://doi.org/10.1109/ MC.2013.418
20. Q. Do, S. Acuña, J. I. Kristiansen, K. Agarwal and P. H. Ha, Highly efficient and scalable framework for high-speed super-resolution microscopy. IEEE Access, vol. 9, pp. 97053-97067, 2021. https://doi.org/ 10.1109/ACCESS.2021.3094840