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A Time Regularization Scheme for Spacecraft Trajectories Subject to Multi-Body Gravity

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Abstract

A time regularization scheme is introduced that facilitates trajectory optimization in multi-body regimes. The time transformation function allows for fixed-step propagation, while eliminating the need for multiple models in patched conic approaches, and mitigating the risk of step** over unplanned flybys. The scheme is motivated by Sundman’s two-body regularization, but accounts for multiple bodies using their spheres of influence and a Heaviside approximation. The transformation enables efficient discretization of the many types of motion that exist in multi-body regimes. The new formulation is analyzed in several restricted three-body dynamical problems, which serve as proxy models that capture the dominant features of N-body ephemeris models within the solar system. The parameters embedded in the transformation are tuned, and its performance is compared against several existing regularizations on a diverse set of examples including periodic orbits in the Earth–Moon and Saturn-Enceladus systems, a low-thrust Earth–Moon spiral, and a low-altitude Jupiter-Europa flyby. The transformation is shown to be robust to the differing conditions, outperforming the benchmarks over wide ranges of the tuning parameters. The results of the numerical experiments are used to justify a set of recommendations for parameter selection in general multi-body applications. The regularization is finally demonstrated on ephemeris-modeled Saturn system trajectories that include unplanned flybys and traverse all three levels of the sun-planet-moon solar system hierarchy.

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References

  1. Cowell, P.: The orbit of jupiter’s eigth satellite. Mon. Not. R. Astron. Soc. 69(5), 421–431 (1909). https://doi.org/10.1093/mnras/69.5.421

    Article  Google Scholar 

  2. Sundman, K.: Mémoire sur le probléme des trois corps. Acta Math. 36, 107–179 (1913). https://doi.org/10.1007/BF02422379

    Article  MathSciNet  Google Scholar 

  3. Levi-Civita, T.: Sur la régularisation du probleme des trois corps. Acta Math. 42, 99–144 (1920). https://doi.org/10.1007/BF02404404

    Article  MathSciNet  MATH  Google Scholar 

  4. Kustaanheimo, P., Stiefel, E.: Perturbation theory of Kepler motion based on spinor regularization. J. für die reine und angewandte Mathematik 218, 204–219 (1965). https://doi.org/10.1515/crll.1965.218.204

    Article  MathSciNet  MATH  Google Scholar 

  5. Stiefel, E., Scheifele, G.: Linear and Regular Celestial Mechanics. Springer-Verlag, New York (1971)

    Book  MATH  Google Scholar 

  6. Aarseth, S., Zare, K.: A regularization of the three-body problem. Celest. Mech. 10(2), 185–205 (1974). https://doi.org/10.1007/BF01227619

    Article  MATH  Google Scholar 

  7. Heggie, D.: A global regularization of the gravitational n-body problem. Celest. Mech. 10(2), 217–241 (1974). https://doi.org/10.1007/BF01227621

    Article  MATH  Google Scholar 

  8. Zare, K., Szebehely, V.: Time transformations in the extended phase-space. Celest. Mech. 11, 469–482 (1975). https://doi.org/10.1007/BF01650285

    Article  MathSciNet  MATH  Google Scholar 

  9. Mikkola, S.: The intermediate anomaly. Celest. Mech. 16(3), 309–313 (1977). https://doi.org/10.1007/BF01232657

    Article  Google Scholar 

  10. Belen’kii, I.: A method of regularizing the equations of motion in the central force-field. Celest. Mech. 23(1), 9–32 (1981). https://doi.org/10.1007/BF01228542

    Article  MathSciNet  MATH  Google Scholar 

  11. Cid, R., Ferrer, S., Elipe, A.: Regularization and linearization of the equations of motion in central force-fields. Celest. Mech. 31(1), 73–80 (1983). https://doi.org/10.1007/BF01272561

    Article  MathSciNet  MATH  Google Scholar 

  12. Ferrándiz, J., Ferrer, S., Sein-Echaluce, M.: Generalized elliptic anomalies. Celest. Mech. 40(3–4), 315–328 (1987). https://doi.org/10.1007/BF01235849

    Article  MATH  Google Scholar 

  13. Peláez, J., Hedo, J., de Andrés, P.: A special perturbation method in orbital dynamics. Astron. J. 97, 131–150 (2007). https://doi.org/10.1007/s10569-006-9056-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Urrutxua, H., Urrutxua, H., Peláez, J.: Dromo propagator revisited. Astron. J. 124(3), 1–31 (2016). https://doi.org/10.1007/s10569-015-9647-y

    Article  MathSciNet  MATH  Google Scholar 

  15. Deprit, A.: Ideal elements for perturbed Keplerian motions. J. Res. Nat. Bur. Stand. 79, 1–15 (1975)

    MathSciNet  MATH  Google Scholar 

  16. Hansen, P.: Mémoire sur le calcul des perturbations qu’eprouvent les comètes. C. R. Acad. Sci. 1, 1–258 (1853)

    Google Scholar 

  17. Baù, G., Bombardelli, C.: Time elements for enhanced performance of the DROMO orbit propagator. Astron. J. 148(3), 43 (2014). https://doi.org/10.1088/0004-6256/148/3/43

    Article  Google Scholar 

  18. Roa, J.: Regularization in Orbital Mechanics: Theory and Practice. De Gruyter, Berlin (2017)

    Book  MATH  Google Scholar 

  19. Amato, D., Baù, G., Bombardelli, C.: Accurate orbit propagation in the presence of planetary close encounters. Mon. Not. R. Astron. Soc. 470(2), 2079–2099 (2017). https://doi.org/10.1093/mnras/stx1254

    Article  Google Scholar 

  20. Sellamuthu, H., Sharma, R.K.: Hybrid orbit propagator for small spacecraft using kustaanheimo-stiefel elements. J. Spacecr. Rockets 55(5), 1282 (2018). https://doi.org/10.2514/1.A34076

    Article  Google Scholar 

  21. Amato, D., Bombardelli, C., Bau, G., Morand, V., Rosengren, A.J.: Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods. Celest. Mech. Dyn. Astron. 131(21), 1–38 (2019). https://doi.org/10.1007/s10569-019-9897-1

    Article  MathSciNet  MATH  Google Scholar 

  22. Sellamuthu, H., Sharma, R.K.: Regularized luni-solar gravity dynamics on resident space objects. Astrodynamics 5(2), 91–108 (2021). https://doi.org/10.1007/s42064-020-0085-6

    Article  Google Scholar 

  23. Masat, A., Romano, M., Colombo, C.: Kustaanheimo-stiefel variables for planetary protection compliance analysis. J. Guid. Control Dyn. 45(7), 1286 (2022). https://doi.org/10.2514/1.G006255

    Article  Google Scholar 

  24. Pellegrini, E., Russell, R.P.: On the computation and accuracy of trajectory state transition matrices. J. Guid. Control Dyn. 39(11), 2485 (2016). https://doi.org/10.2514/1.G001920

    Article  Google Scholar 

  25. Czuchry, A.J., Pitkin, E.T.: Regularization vs step-size regulators in optimal trajectory computations. J. Spacecr. Rockets 7(7), 882–885 (1970). https://doi.org/10.2514/3.30062

    Article  Google Scholar 

  26. Kohlhase, C., Penzo, P.: Voyager mission description. Space Sci. Rev. 21, 77–101 (1977). https://doi.org/10.1007/BF00200846

    Article  Google Scholar 

  27. D’Amario, L., Bright, L., Wolf, A.: Galileo trajectory design. Space Sci. Rev. 60, 23–78 (1992). https://doi.org/10.1007/BF00216849

    Article  Google Scholar 

  28. Bellerose, J., Roth, D., Criddle, K.: The cassini mission: reconstructing thirteen years of the most complex gravity-assist trajectory flown to date. In: Proceedings of the SpaceOps Conference. (2018). https://doi.org/10.2514/6.2018-2646

  29. Wolf, A., Smith, J.: Design of the Cassini tour trajectory in the saturnian system. Control Eng. Pract. 3(11), 1611–1619 (1995). https://doi.org/10.1016/0967-0661(95)00172-Q

    Article  Google Scholar 

  30. Buffington, B., Strange, N., Smith, J.: Overview of the cassini extended mission trajectory. In: AIAA/AAS Astrodynamics Specialist Conference. Portland, ME. (2019). https://doi.org/10.2514/6.2008-6752

  31. Roth, D., Alwar, V., Bordi, J., Goodson, T., Hahn, Y., Ionasescu, R., Jones, J., Owen, W.M., Pojman, J., Roundhill, I., Santos, S., Strange, N., Wagner, S.V., Wong, M.: Cassini tour navigation strategy. Tech. Rep (2003)

  32. Buffington, B.: Trajectory design concept for the proposed europa clipper mission. In: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference. (2014). https://doi.org/10.2514/6.2014-4105

  33. Szebehely, V.: Theory of Orbits. The Circular Restricted Three-Body Problem. Academic Press, Cambridge (1967)

    MATH  Google Scholar 

  34. Berry, M., Healy, L.: The generalized Sundman transformation for propagation of high-eccentricity elliptical orbits. In: Proceedings of the 12th AAS/AIAA Space Flight Mechanics Meeting vol. 112(11) (2002)

  35. Pellegrini, E., Russell, R., Vittaldev, V.: F and g Taylor series solutions to the stark problem with Sundman transformations. Celest. Mech. Dyn. Astron. 118(4), 355–378 (2014). https://doi.org/10.1007/s10569-014-9538-7

    Article  MathSciNet  MATH  Google Scholar 

  36. Ottesen, D., Russell, R.P.: Piecewise Sundman transformation for spacecraft trajectory optimization using many embedded lambert problems. J. Spacecr. Rockets 59, 1–18 (2022). https://doi.org/10.2514/1.A35140

    Article  Google Scholar 

  37. Yeomans, D.: Exposition of sundman’s regularization of the three-body problem. Tech. rep., NASA-TM-X-55636, X-640-66-481. NASA Goddard Space Flight Center Technical Report (1966)

  38. Aarseth, S.: Gravitational N-Body Simulations: Tools and Algorithms. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  39. Betts, J., Campbell, S., Kalla, N.: Initialization of direct transcription optimal control software. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 3802–3807. Maui, HI (2003)

  40. Russell, R.P.: Primer vector theory applied to global low-thrust trade studies. J. Guid. Control Dyn. 30(2), 460–472 (2007). https://doi.org/10.2514/1.22984

    Article  Google Scholar 

  41. Russell, R.P.: Complete lambert solver including second-order sensitivities. J. Guid. Control Dyn. 45(2), 196–212 (2022). https://doi.org/10.2514/1.G006089

    Article  Google Scholar 

  42. Williams, J., Lee, D.E., Whitley, R.L., Bokelmann, K.A., Davis, D.C., Berry, C.F.: Targeting cislunar near rectilinear halo orbits for human space exploration. In: Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting, AAS, pp. 17–267. San Antonio, TX (2017)

  43. Restrepo, R.L., Russell, R.P.: A database of planar axi-symmetric periodic orbits for the solar system. Celest. Mech. Dyn. Astron. 130(49), 1–24 (2018). https://doi.org/10.1007/s10569-018-9844-6

    Article  Google Scholar 

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Acknowledgements

This work was supported in part by NASA award (No. 80NSSC19K1140). The authors extend their thanks to Dr. Noble Hatten at Goddard Space Flight Center for contributing discussions, and to the anonymous reviewers for their insightful feedback.

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  1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, USA

    James Leith & Ryan P. Russell

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  1. James Leith
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A previous version of this paper was presented during the virtual AAS/AIAA Astrodynamics Specialist Conference in July 2020. This work is supported by a NASA Space Technology Research Fellowship, Grant 80NSSC19K1140.

Appendix

Appendix

The appendix provides additional information about the unplanned flyby and multi-hierarchy simulations discussed in the paper. Table 6 lists the physical and regularization parameters used for Saturn and each of its moons which were included in each simulation.

Table 6 Physical body parameters
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The ephemeris model used in both simulations is described by the following set of SPICE kernels: de430.bsp, naif0011.tls, and sat375.bsp.

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Leith, J., Russell, R.P. A Time Regularization Scheme for Spacecraft Trajectories Subject to Multi-Body Gravity. J Astronaut Sci 70, 7 (2023). https://doi.org/10.1007/s40295-023-00364-0

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