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DROMO propagator revisited

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Abstract

In the year 2000 an in-house orbital propagator called DROMO (Peláez et al. in Celest Mech Dyn Astron 97:131–150, 2007. doi:10.1007/s10569-006-9056-3) was developed by the Space Dynamics Group of the Technical University of Madrid, based in a set of redundant variables including Euler–Rodrigues parameters. An original deduction of the DROMO propagator is carried out, underlining its close relation with the ideal frame concept introduced by Hansen (Abh der Math-Phys Cl der Kon Sachs Ges der Wissensch 5:41–218, 1857). Based on the very same concept, Deprit (J Res Natl Bur Stand Sect B Math Sci 79B(1–2):1–15, 1975) proposed a formulation for orbit propagation. In this paper, similarities and differences with the theory carried out by Deprit are analyzed. Simultaneously, some improvements are introduced in the formulation, that lead to a more synthetic and better performing propagator. Also, the long-term effect of the oblateness of the primary is studied in terms of DROMO variables, and new numerical results are presented to evaluate the performance of the method.

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Notes

  1. The word DROMO is derived from the old Greek word \(\delta \rho \)ó\(\mu \omega \) that means, approximately, rushing and it appears as suffix in many words like velodrome, hippodrome, etc.

  2. Notice that the steps per revolution ratio was fixed to 62 for regularization methods, but for the Cowell formulation had to be increased to 240 if comparable final errors were to be obtained.

  3. This criteria penalizes the performance figures when there is a high ratio of failed integration steps that have to be repeated with a smaller stepsize to meet the demanded tolerance.

  4. Source code available at: http://www.netlib.org/ode/ode.f.

  5. The increase in run-time is due to the overhead of calculating a larger table of divided differences, which becomes visible when function evaluations are computationally cheap, as is the case.

  6. Störmer–Cowell formulas can do with a single function call per step, whereas the Predictor/Corrector-type Adams methods like the Shampine–Gordon require two function calls per step, since a second correction step is necessary for stability issues.

References

  • Awad, M.: Oblateness and drag effects on the motion of satellites in the set of Eulerian redundant parameters. Earth Moon Planets 62(2), 161–177 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  • Awad, M.: Analytical solution to the perturbed \(j_2\) motion of artificial satellite in terms of Euler parameters. Earth Moon Planets 69(1), 1–12 (1995)

    Article  MATH  ADS  Google Scholar 

  • Barrio, R., Serrano, S.: Performance of perturbation methods on orbit prediction. Math. Comput. Model. 48(3), 594–600 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  • Baù, G., Bombardelli, C., Peláez, J.: Adaptive scheme for accurate orbit propagation. J. Aerosp. Eng. Sci. Appl. 15(35), 131 (2010)

    Google Scholar 

  • Baù, G., Bombardelli, C., Peláez, J.: A new set of integrals of motion to propagate the perturbed two-body problem. Celest. Mech. Dyn. Astron. 116(1), 53–78 (2013). doi:10.1007/s10569-013-9475-x

    Article  MATH  ADS  Google Scholar 

  • Berry, M.M.: A variable-step double-integration multi-step integrator. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia (2004)

  • Bond, V.R., Allman, M.C.: Modern Astrodynamics: Fundamentals and Perturbation Methods. Princeton University Press, Princeton (1996)

    Google Scholar 

  • Broucke, R., Lass, H.: On redundant variables in Lagrangian mechanics, with applications to perturbation theory and KS regularization. Celest. Mech. 12(3), 317–325 (1975)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Cid, R., San Saturio, M.: Motion of rigid bodies in a set of redundant variables. Celest. Mech. 42(1), 263–277 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Danby, J.: Fundamentals of Celestial Mechanics, vol. 1. Willmann-Bell, Inc, Richmond (1992)

    Google Scholar 

  • Deprit, A.: Ideal elements for perturbed Keplerian motions. J. Res. Natl. Bur. Stand. Sect. B Math. Sci. 79B(1–2):1–15 (1975) (paper 79B1&2–416)

  • Deprit, A.: Ideal frames for perturbed Keplerian motions. Celest. Mech. 13(2), 253–263 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Efroimsky, M., Goldreich, P.: Gauge freedom in the n-body problem of celestial mechanics. Astron. Astrophys. 415(3), 1187–1199 (2004)

    Article  MATH  ADS  Google Scholar 

  • Esteban-Dones, J., Peláez, J.: Advanced propagation of interplanetary orbits in the exploration of jovian moons. In: 4th International Conference on Astrodynamics Tools and Techniques, pp. 3–6 (2010a)

  • Esteban-Dones, J., Peláez, J.: Advanced propagation of interplanetary orbits in the exploration of jovian moons. In: 5th International Workshop and Advanced School “Spaceflight Dynamics and Control”, 17–19 March 2010, Covilha, Portugal (2010b)

  • Fehlberg, E.: Classical fifth-, sixth-, seventh-, and eighth-order Runge–Kutta formulas with stepsize control. Nasa tr r-287, George C. Marshall Space Flight Center, Huntsville, Alabama (1968)

  • Gómez, J.L.G.: Perturbation methods in optimal control problems applied to low thrust space trajectories. Master’s thesis, ETSI Aeronáuticos, Technical University of Madrid (UPM) (2012)

  • Hansen, P.A.: Auseinandersetzung einer zweckmässigen mmethod zur berechnung der absoluten störungen der kleinen planeten. Abh der Math-Phys Cl der Kon Sachs Ges der Wissensch 5, 41–218 (1857)

    Google Scholar 

  • Hintz, G.: Survey of orbit element sets. J. Guid. Control Dyn. 31(3), 785–790 (2008)

    Article  ADS  Google Scholar 

  • Hughes, P.C.: Spacecraft Attitude Dynamics. Dover Books on Aeronautical Engineering. Dover Publications, New York (2004)

    Google Scholar 

  • Janin, G.: Accurate computation of highly eccentric satellite orbits. Celest. Mech. 10, 451–467 (1974). doi:10.1007/BF01229121

    Article  MATH  ADS  Google Scholar 

  • Palacios, M., Calvo, C.: Quaternions formulation versus regularization in numerical orbit computation. Astrodynamics 1993, 2629–2644 (1993)

    Google Scholar 

  • Palacios, M., Calvo, C.: Ideal frames and regularization in numerical orbit computation. J. Astron. Sci. 44(1), 63–77 (1996)

    Google Scholar 

  • Palacios, M., Abad, A., Elipe, A.: An efficient numerical method for orbit computations. In: Astrodynamics 1991, vol. 1, pp. 265–274 (1992)

  • Peláez, J., Hedo, J.M.: Un método de perturbaciones especiales en dinámica de tethers. Monografías de la Real Academia de Ciencias de Zaragoza 22, 119–140 (2003)

    Google Scholar 

  • Peláez, J., Hedo, J.M., Rodriguez, P.: A special perturbation method in orbital dynamics. In: Vallado, D.A., Gabor, M.J., Desai, P.N. (eds.) AAS/AIAA Spaceflight Mechanics Meeting 2005, Advances in the Astronautical Sciences, vol. 120, pp. 1061–1078 (2005)

  • Peláez, J., Hedo, J.M., de Andrés, P.R.: A special perturbation method in orbital dynamics. Celest. Mech. Dyn. Astron. 97, 131–150 (2007). doi:10.1007/s10569-006-9056-3

    Article  MATH  ADS  Google Scholar 

  • Prince, P., Dormand, J.: High order embedded Runge–Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981). doi:10.1016/0771-050X(81)90010-3

    Article  MATH  MathSciNet  Google Scholar 

  • Roa, J., Sanjurjo-Rivo, M., Peláez, J.: Singularities in DROMO formulation. Analysis of deep flybys. Adv. Space Res. (2015). doi:10.1016/j.asr.2015.03.019

  • Shampine, L., Gordon, M.: Computer Solution of Ordinary Differential Equations. W. H. Freeman and Company, San Francisco (1975)

    MATH  Google Scholar 

  • Sharaf, M., Awad, M., Najmuldeen, S.: The motion of artificial satellites in the set of Eulerian redundant parameters. Earth Moon Planets 55(1), 21–44 (1991a)

    Article  MATH  ADS  Google Scholar 

  • Sharaf, M., Awad, M., Najmuldeen, S.: Motion of artificial satellites in the set of Eulerian redundant parameters, II. Earth Moon Planets 55(3), 223–231 (1991b)

    Article  MATH  ADS  Google Scholar 

  • Sharaf, M., Awad, M., Najmuldeen, S.: Motion of artificial satellites in the set of Eulerian redundant parameters (III). Earth Moon Planets 56(2), 141–164 (1992)

    Article  MATH  ADS  Google Scholar 

  • Shuster, M.D.: A survey of attitude representations. J. Astron. Sci. 41(4), 439–517 (1993)

    MathSciNet  Google Scholar 

  • Stiefel, E., Scheifele, G.: Linear and Regular Celestial Mechanics. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  • Urrutxua, H., Morante, D., Sanjurjo-Rivo, M., Peláez, J.: DROMO formulation for planar motion: solution to the Tsien problem. Celest. Mech. Dyn. Astron. 122(2), 143–168 (2015). doi:10.1007/s10569-015-9613-8

    Article  ADS  Google Scholar 

  • Velez, C., Hilinski, S.: Time transformations and Cowell’s method. Celest. Mech. 17(1), 83–99 (1978). doi:10.1007/BF01261054

    Article  MATH  ADS  Google Scholar 

Download references

Acknowledgments

This work is part of the research project entitled “Dynamical Analysis, Advanced Orbit Propagation and Simulation of Complex Space Systems” (ESP2013-41634-P) supported by the Spanish Ministry of Economy and Competitiveness. Authors thank to the Spanish Government for its financial support. The work of Hodei Urrutxua is also supported by a Grant of the Technical University of Madrid (UPM); Mr. Urrutxua thanks UPM for its support.

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Authors and Affiliations

  1. Space Dynamics Group (SDG-UPM), Technical University of Madrid, Pza. Cardenal Cisneros 3, 28040, Madrid, Spain

    Hodei Urrutxua & Jesús Peláez

  2. Space Dynamics Group (SDG-UPM), Bioengineering and Aerospace Engineering Department, Carlos III University, Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain

    Manuel Sanjurjo-Rivo

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  1. Hodei Urrutxua
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  2. Manuel Sanjurjo-Rivo
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  3. Jesús Peláez
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Correspondence to Hodei Urrutxua.

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Urrutxua, H., Sanjurjo-Rivo, M. & Peláez, J. DROMO propagator revisited. Celest Mech Dyn Astr 124, 1–31 (2016). https://doi.org/10.1007/s10569-015-9647-y

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