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Improved alpha shape-based continuum method for long-term density propagation

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Abstract

This paper presents an improved alpha shape-based linear interpolation method, and an improved binning method within the continuum method framework for accurate and efficient planar phase space long-term density propagation. The density evolution equation is formulated for the continuum density propagation under the influence of the solar radiation pressure and Earth’s oblateness using semi-analytical equations. The concept of the alpha shape is included to get accurate interpolated density within the non-convex hull enclosing all the samples for the highly deformed and elongated density distribution. The improved binning method increases the density accuracy by considering the variant nonlinearity of the density within each alpha shape triangulation, which calculates the joint and marginal density as the weighted sum of density weights per bin area and per bin width, respectively. The suitable sample number for the continuum method and the suitable grid number for performing the linear interpolation are selected by trading off the density accuracy and the computational effort. The superiority of the improved alpha shape-based continuum method is demonstrated for accurate and efficient density propagation in the context of the high-altitude and high area-to-mass ratio satellite long-term propagation.

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References

  • Colombo, C., McInnes, C.R.: Evolution of swarms of ’smart dust’ spacecraft. New Trends in Astrodynamics and Applications VI, New York (2011)

  • Evans, L. C.: Partial differential equations. Grad. Stud. Math. 19(2) (1998)

  • Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. ACM Trans. Graph. 13(1), 43–72 (1994). https://doi.org/10.1145/174462.156635

    Article  Google Scholar 

  • Feng, J., Santeramo, D., Di Lizia, P., Armellin, R., Hou, X.: Dynamical structure of the motion around asteroids with uncertain gravity and solar radiation pressure. Acta Astronaut. 186, 135–147 (2021). https://doi.org/10.1016/j.actaastro.2021.05.023

    Article  ADS  Google Scholar 

  • Frey, S.: Evolution and hazard analysis of orbital fragmentation continua. Doctoral thesis. Politecnico di Milano. Supervisors: Colombo, C., Lemmens, S., Krag, H. pp. 31–32 (2020)

  • Giza, D., Singla, P., Jah, M.: An approach for nonlinear uncertainty propagation: application to orbital mechanics. AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2009–6082 (2009). https://doi.org/10.2514/6.2009-6082

  • Gor’kavyi, N.N., Ozernoy, L.M., Mather, J.C., Taidakova, T.: Quasi-stationary states of dust flows under Poynting-Robertson drag: New analytical and numerical solutions. Astrophys. J. 488(1), 268–276 (1997). https://doi.org/10.1086/304702

    Article  ADS  Google Scholar 

  • Gor’kavyi, N.N., Ozernoy, L.M., Mather, J.C.: A new approach to dynamic evolution of interplanetary dust. Astrophys. J. 474(1), 496–502 (1997). https://doi.org/10.1086/303440

    Article  ADS  Google Scholar 

  • Halder, A., Bhattacharya, R.: Dispersion analysis in hypersonic flight during planetary entry using stochastic Liouville equation. J. Guid. Control. Dyn. 34(2), 459–476 (2016). https://doi.org/10.2514/1.51196

    Article  ADS  Google Scholar 

  • Jia, B., Xin, M.: Short-arc orbital uncertainty propagation with arbitrary polynomial chaos and admissible region. J. Guid. Control. Dyn. 43(4), 715–728 (2020). https://doi.org/10.2514/1.g004548

    Article  ADS  Google Scholar 

  • Krivov, A.V., Getino, J.: Orbital evolution of high-altitude balloon satellites. Astron. Astrophys. 318, 308–314 (1997)

    ADS  Google Scholar 

  • Letizia, F., Colombo, C., Lewis, H.G.: Analytical model for the propagation of small-debris-object clouds after fragmentations. J. Guid. Control. Dyn. 38(8), 1478–1491 (2015). https://doi.org/10.2514/1.G000695

    Article  ADS  Google Scholar 

  • Letizia, F., Colombo, C., Lewis, H.G.: Multidimensional extension of the continuity equation method for debris clouds evolution. Adv. Space Res. 57(8), 1624–1640 (2016a). https://doi.org/10.1016/j.asr.2015.11.035

    Article  ADS  Google Scholar 

  • Letizia, F., Colombo, C., Van den Eynde, J., Armellin, R., Jehn, R.: SNAPPSHOT: Suite for the numerical analysis of planetary protection. In 6th International Conference on Astrodynamics Tools and Techniques (ICATT). pp. 14–17 (2016b)

  • McInnes, C.R.: Simple analytic model of the long-term evolution of nanosatellite constellations. J. Guid. Control. Dyn. 23(2), 332–338 (2000). https://doi.org/10.2514/2.4527

    Article  ADS  Google Scholar 

  • Nazarenko, A.: The development of the statistical theory of a satellite ensemble motion and its application to space debris modeling. In Proceedings of the 2nd European Conference on Space Debris, ESOC. ESA SP-393 (1997)

  • Preparata, F.P., Shamos, M.I.: Computational Geometry, pp. 191–218. Springer-Verlag, New York (1985)

    Book  Google Scholar 

  • Smirnov, N. N., Nazarenko, A. I., Kiselev, A. B.: Modelling of the space debris evolution based on continua mechanics. In the Proceedings of the 3rd European Conference on Space Debris, ESOC. ESA SP-473 (2001)

  • Sun, P., Colombo, C., Trisolini, M., Li, S.: Comparison of continuity equation and Gaussian mixture model for long-term density propagation using semi-analytical methods. Celest. Mech. Dyn. Astron. 134, 22 (2022a). https://doi.org/10.1007/s10569-022-10066-8

    Article  MathSciNet  ADS  Google Scholar 

  • Sun, P., Colombo, C., Trisolini, M., Li, S.: Hybrid Gaussian mixture splitting techniques for uncertainty propagation in nonlinear dynamics. J. Guid. Control. Dyn. (2022b). https://doi.org/10.2514/1.G006696

    Article  Google Scholar 

  • Trisolini, M., Colombo, C.: Propagation and reconstruction of reentry uncertainties using continuity equation and simplicial interpolation. J. Guid. Control. Dyn. 44(4), 793–811 (2021). https://doi.org/10.2514/1.G005228

    Article  ADS  Google Scholar 

  • Wittig, A., Colombo, C., Armellin, R.: Long-term density evolution through semi-analytical and differential algebra techniques. Celest. Mech. Dyn. Astron. 128(4), 435–452 (2017). https://doi.org/10.1007/s10569-017-9756-x

    Article  MathSciNet  ADS  Google Scholar 

  • Wittig, A., Di Lizia, P., Armellin, R., Makino, K., Bernelli-Zazzera, F., Berz, M.: Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting. Celest. Mech. Dyn. Astron. 122(3), 239–261 (2015). https://doi.org/10.1007/s10569-015-9618-3

    Article  MathSciNet  ADS  Google Scholar 

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Acknowledgments

The work has received funding from China Scholarship Council (CSC No. 202006830123), 2020 Postgraduate Research Practice Innovation Program of Jiangsu Province (Grant No. KYCX20_0222). Pan Sun and Shuang Li fully appreciate their financial supports.

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Authors and Affiliations

  1. Department of Intelligent Equipment, Changzhou College of Information Technology, Changzhou, China

    Pan Sun

  2. Advanced Space Technology Laboratory, and College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Shuang Li

  3. Department of Aerospace Science and Technology, Polytechnic University of Milan, Milan, Italy

    Mirko Trisolini & Camilla Colombo

Authors
  1. Pan Sun
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  2. Shuang Li
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  3. Mirko Trisolini
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  4. Camilla Colombo
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Contributions

SP wrote the main manuscript, and all authors reviewed the manuscript.

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Correspondence to Pan Sun or Shuang Li.

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This article is part of the topical collection on Innovative computational methods in Dynamical Astronomy.

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Appendices

Appendix A: Scenario 2, t = 1.5 yrs

For the case Scenario 2, t = 1.5 yrs, for the AT method, the compact alpha shape triangulation enclosing all the samples is generated for a predefined alpha radius ra = 8. Figure

Fig. 8
figure 8

Illustration and comparison for the (a) DT and (b) AT methods for Scenario 2, t = 1.5 yrs (Nsam = 961)

Full size image

8 presents the illustration and comparison for the DT and AT methods for the case Nsam = 961 for Scenario 2, t = 1.5 yrs in the solar angle-eccentricity 2D phase space. The values of the sample number and the grid number are selected as Nsam = 2000, Ngrid = 1000 for the AT-B2 method, by trading off the density accuracy and the computational efficiency. Figure

Fig. 9
figure 9

Normalized LD evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000), for Scenairo 2, t = 1.5 yrs

Full size image

9 presents the evolution of the normalized LD measure for the joint and marginal density with Nsam, and with Ngrid, respectively, for AT-B2, AT-B1 and DT-B1 methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000). Figure

Fig. 10
figure 10

Normalized performance index Jp evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000), for Scenario 2, t = 1.5 yrs

Full size image

10 gives the evolution of the normalized performance index with Nsam, and with Ngrid, respectively, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000). Figure

Fig. 11
figure 11

Normalized computational effort evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the MC method, for Scenario 2, t = 1.5 yrs

Full size image

11 presents the evolution of the normalized computational effort with Nsam, and with Ngrid, respectively, with respect to that of the MC method. For the selected sample number and the grid number {Nsam = 2000, Ngrid = 1000} for the case Scenario 2, t = 1.5 yrs, we present the joint and marginal density in Figs.

Fig. 12
figure 12

The joint density (a) for the MC method; (b) for the DT-B1 method, (c) for the AT-B1 method, (d) for the AT-B2 method, with {Nsam = 2000, Ngrid = 1000} (for Scenario 2, t = 1.5 yrs)

Full size image

12 and

Fig. 13
figure 13

The marginal density of the (a) solar angle, (b) and the eccentricity with {Nsam = 2000, Ngrid = 1000}, for DT-B1, AT-B1 and AT-B2 methods with respect to the MC method, for Scenario 2, t = 1.5 yrs

Full size image

13, for the AT-B2 method compared with the AT-B1, DT-B1 methods with respect to that of the MC method.

Appendix B: Scenario 2, t = 3 yrs

For the case Scenario 2, t = 3 yrs, for the AT method, the compact alpha shape triangulation enclosing all the samples is generated for a predefined alpha radius ra = 0.03. Figure

Fig. 14
figure 14

Illustration and comparison for the (a) DT and (b) AT methods for Scenario 2, t = 3 yrs (Nsam = 961)

Full size image

14 presents the illustration and comparison for the DT and AT methods for the case Nsam = 961 for Scenario 2, t = 3 yrs in the solar angle-eccentricity 2D phase space. The values of the sample number and the grid number are selected as Nsam = 2000, Ngrid = 1000 for the AT-B2 method, by trading off the density accuracy and the computational efficiency. Figure

Fig. 15
figure 15

Normalized LD evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000), for Scenairo 2, t = 3 yrs

Full size image

15 presents the evolution of the normalized LD measure for the joint and marginal density with Nsam, and with Ngrid, respectively, for AT-B2, AT-B1 and DT-B1 methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000). Figure

Fig. 16
figure 16

Normalized performance index Jp evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000), for Scenario 2, t = 3 yrs

Full size image

16 gives the evolution of the normalized performance index with Nsam, and with Ngrid, respectively, with respect to the case AT-B2 (Nsam = 2000, Ngrid = 1000). Figure

Fig. 17
figure 17

Normalized computational effort evolution (a) with Nsam, and (b) with Ngrid, for DEE methods, with respect to the MC method, for Scenario 2, t = 3 yrs

Full size image

17 presents the evolution of the normalized computational effort with Nsam, and with Ngrid, respectively, with respect to that of the MC method. For the selected sample number and the grid number {Nsam = 2000, Ngrid = 1000} for the case Scenario 2, t = 3 yrs, we present the joint and marginal density in Figs.

Fig. 18
figure 18

The joint density (a) for the MC method; (b) for the DT-B1 method, (c) for the AT-B1 method, (d) for the AT-B2 method, with {Nsam = 2000, Ngrid = 1000} (for Scenario 2, t = 3 yrs)

Full size image

18 and

Fig. 19
figure 19

The marginal density of the (a) solar angle, (b) and the eccentricity with {Nsam = 2000, Ngrid = 1000}, for DT-B1, AT-B1 and AT-B2 methods with respect to the MC method, for Scenario 2, t = 3 yrs

Full size image

19, for the AT-B2 method compared with the AT-B1, DT-B1 methods with respect to that of the MC method.

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Sun, P., Li, S., Trisolini, M. et al. Improved alpha shape-based continuum method for long-term density propagation. Celest Mech Dyn Astron 136, 5 (2024). https://doi.org/10.1007/s10569-023-10171-2

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