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A novel analytic continuation power series solution for the perturbed two-body problem

  • Kevin HernandezEmail author
  • Tarek A. Elgohary
  • James D. Turner
  • John L. Junkins
Original Article
  • 47 Downloads

Abstract

Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure.

Keywords

Two-body problem Orbit propagation Taylor series Analytic continuation Recursive power series Astrodynamics 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest

References

  1. Abad, A., Barrio, R., Blesa, F., Rodríguez, M.: Algorithm 924: TIDES, a Taylor series integrator for differential equations. ACM Trans. Math. Softw. (TOMS) 39(1), 5 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists International Student Edition, 6th edn. Academic press, Cambridge (2005)zbMATHGoogle Scholar
  3. Aristoff, J.M., Poore, A.B.: Implicit Runge–Kutta methods for orbit propagation. In: Proceedings of the 2012 AIAA/AAS Astrodynamics Specialist Conference, vol. 4880, pp. 1–19 (2012)Google Scholar
  4. Atallah, A.M., Woollands, R.M., Elgohary, T.A., Junkins, J.L.: Accuracy and efficiency comparison of six numerical integrators for propagating perturbed orbits. J. Astronaut. Sci. (2019).  https://doi.org/10.1007/s40295-019-00167-2
  5. Bai, X., Junkins, J.L.: Modified Chebyshev-Picard iteration methods for orbit propagation. J. Astronaut. Sci. 58(4), 583–613 (2011)ADSCrossRefGoogle Scholar
  6. Bai, X., Junkins, J.L.: Modified Chebyshev-Picard iteration methods for solution of initial value problems. J. Astronaut. Sci. 59(1–2), 335–359 (2012)Google Scholar
  7. Bani Younes, A. Orthogonal polynomial approximation in higher dimensions: Applications in astrodynamics. PhD thesis, Ph. D. dissertation, Texas A&M University, College Station, TX (2013)Google Scholar
  8. Barrio, R.: Performance of the Taylor series method for ODEs/DAEs. Appl. Math. Comput. 163(2), 525–545 (2005).  https://doi.org/10.1016/j.amc.2004.02.015 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Barrio, R., Blesa, F., Lara, M.: VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 50(1–2), 93–111 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. Barton, D., Willers, I., Zahar, R.: The automatic solution of systems of ordinary differential equations by the method of Taylor series. Comput. J. 14(3), 243–248 (1971)zbMATHCrossRefGoogle Scholar
  11. Battin, R.H.: An introduction to the mathematics and methods of astrodynamics. AIAA, Reston (1999)zbMATHGoogle Scholar
  12. Berry, M., Healy, L.: Comparison of accuracy assessment techniques for numerical integration. In: 13th AAS/AIAA Space Flight Mechanics Meeting (2003)Google Scholar
  13. Berry, M.M., Healy, L.M.: Implementation of Gauss-Jackson integration for orbit propagation. J. Astronaut. Sci. 52(3), 331–357 (2004)MathSciNetGoogle Scholar
  14. Bettis, D.: A Runge-Kutta-Nyström algorithm. Celest. Mech. 8(2), 229–233 (1973)MathSciNetCrossRefGoogle Scholar
  15. Bradley, B.K., Jones, B.A., Beylkin, G., Axelrad, P.: A new numerical integration technique in astrodynamics. In: 22nd AAS/AIAA Space Flight Mechanics Meeting, pp. 12–216. Charleston, SC (2012)Google Scholar
  16. Bradley, B.K., Jones, B.A., Beylkin, G., Sandberg, K., Axelrad, P.: Bandlimited implicit Runge-Kutta integration for astrodynamics. Celest. Mech. Dyn. Astron. 119(2), 143–168 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. Broucke, R.: Solution of then-body problem with recurrent power series. Celest. Mech. Dyn. Astron. 4(1), 110–115 (1971)zbMATHCrossRefGoogle Scholar
  18. Chang, Y.: Automatic solution of differential equations. In: Gilbert, R.P. (ed.) Constructive and Computational Methods for Differential and Integral Equations, pp. 61–94. Springer, Berlin (1974) CrossRefGoogle Scholar
  19. Deprit, A., Zahar, R.: Numerical integration of an orbit and its concomitant variations by recurrent power series. Zeitschrift für angewandte Mathematik und Physik ZAMP 17(3), 425–430 (1966)ADSCrossRefGoogle Scholar
  20. Dormand, J., Prince, P.: New Runge-Kutta algorithms for numerical simulation in dynamical astronomy. Celest. Mech. 18(3), 223–232 (1978)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. Dormand, J., El-Mikkawy, M., Prince, P.: High-order embedded Runge-Kutta-Nystrom formulae. IMA J. Numer. Anal. 7(4), 423–430 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  22. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  23. Elgohary, T.A.: Novel computational and analytic techniques for nonlinear systems applied to structural and celestial mechanics. PhD thesis, Texas A&M University (2015)Google Scholar
  24. Elgohary, T.A., Turner, J.D.: State transition tensor models for the uncertainty propagation of the two-body problem. Adv. Astronaut. Sci. AAS/AIAA Astrodyn. Conf. 150, 1171–1194 (2014)Google Scholar
  25. Elgohary, T.A., Turner, J.D., Junkins, J.L.: High-order analytic continuation and numerical stability analysis for the classical two-body problem. Adv. Astronaut. Sci. Jer-Nan Juang Astrodyn. Symp. 147, 627–646 (2012)Google Scholar
  26. Elgohary, T.A., Dong, L., Junkins, J.L., Atluri, S.N.: A simple, fast, and accurate time-integrator for strongly nonlinear dynamical systems. CMES Comput. Model. Eng. Sci. 100(3), 249–275 (2014a)MathSciNetzbMATHGoogle Scholar
  27. Elgohary, T.A., Dong, L., Junkins, J.L., Atluri, S.N.: Time domain inverse problems in nonlinear systems using collocation & radial basis functions. CMES Comput. Model. Eng. Sci. 100(1), 59–84 (2014b)MathSciNetzbMATHGoogle Scholar
  28. Elgohary, T.A., Junkins, J.L., Atluri, S.N.: An RBF-collocation algorithm for orbit propagation. In: Advances in Astronautical Sciences: AAS/AIAA Space Flight Mechanics Meeting (2015)Google Scholar
  29. Faà di Bruno, F.: Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche 6, 479–480 (1855)Google Scholar
  30. Fehlberg, E.: On the numerical integration of differential equations by power series expansions, illustrated by physical examples. NASA Technical Note, NASA TN D-2356 (1964)Google Scholar
  31. Fehlberg, E.: Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. Technical Report NASA-TR-R-315, NASA (1969)Google Scholar
  32. Fehlberg, E.: Classical eighth-and lower-order Runge–Kutta–Nystrom formulas with stepsize control for special second-order differential equations. Technical Report NASA-TR-R-381, NASA (1972)Google Scholar
  33. Filippi, S., Gräf, J.: New Runge-Kutta-Nyström formula-pairs of order 8 (7), 9 (8), 10 (9) and 11 (10) for differential equations of the form y”= f (x, y). J. Comput. Appl. Math. 14(3), 361–370 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Fox, K.: Numerical integration of the equations of motion of celestial mechanics. Celest. Mech. 33(2), 127–142 (1984)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. Gibbons, A.: A program for the automatic integration of differential equations using the method of Taylor series. Comput. J. 3(2), 108–111 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  36. Hadjifotinou, K., Gousidou-Koutita, M.: Comparison of numerical methods for the integration of natural satellite systems. Celest. Mech. Dyn. Astron. 70(2), 99–113 (1998)ADSzbMATHCrossRefGoogle Scholar
  37. Hernandez, K., Read, J.L., Elgohary, T.A., Turner, J.D., Junkins, J.L.: Analytic power series solutions for two-body and \(J_2-J_6\) trajectories and state transition models. In: Advances in Astronautical Sciences: AAS/AIAA Astrodynamics Specialist Conference (2015)Google Scholar
  38. Hernandez, K., Elgohary, T.A., Turner, J.D., Junkins, J.L.: Analytic continuation power series solution for the two-body problem with atmospheric drag. In: Advances in Astronautical Sciences: AAS/AIAA Space Flight Mechanics Meeting, pp. 2605–2614 (2016)Google Scholar
  39. Horner, W.G.: A new method of solving numerical equations of all orders, by continuous approximation. Philos. Trans. R. Soc. Lond. 109, 308–335 (1819)ADSCrossRefGoogle Scholar
  40. Jackson, J.: Note on the numerical integration of \(d^2x/dt^2= f(x, t)\). Mon. Not. R. Astron. Soc. 84, 602–606 (1924)CrossRefADSGoogle Scholar
  41. Junkins, J., Bani Younes, A., Woollands, R., Bai, X.: Orthogonal approximation in higher dimensions: applications in astrodynamics. In: ASS 12-634, JN Juang Astrodynamics Symposium (2012)Google Scholar
  42. Kim, D., Turner, J.D.: Variable step-size control for analytic power series solutions for orbit propagation. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, San Diego, CA, AIAA (2014)Google Scholar
  43. Lara, M., Elipe, A., Palacios, M.: Automatic programming of recurrent power series. Math. Comput. Simul. 49(4), 351–362 (1999)MathSciNetCrossRefGoogle Scholar
  44. Le Guyader, C.: Solution of the n-body problem expanded into Taylor series of high orders. Applications to the solar system over large time range. Astron. Astrophys. 272, 687–694 (1993)Google Scholar
  45. Lundberg, J.B., Schutz, B.E.: Recursion formulas of Legendre functions for use with nonsingular geopotential models. J. Guid. Control Dyn. 11(1), 31–38 (1988)ADSzbMATHCrossRefGoogle Scholar
  46. Macomber, B., Woollands, R., Probe, A., Younes, A., Bai, X., Junkins, J.: Modified Chebyshev-Picard iteration for efficient numerical integration of ordinary differential equations. In: Advanced Maui Optical and Space Surveillance Technologies Conference, vol. 1, p. 89 (2013)Google Scholar
  47. Macomber, B., Probe, A., Woollands, R., Junkins, J.L.: Automated tuning parameter selection for orbit propagation with modified Chebyshev-Picard iteration. In: AAS/AIAA Spaceflight Mechanics Meeting, Williamsburg, VA (2015)Google Scholar
  48. Montenbruck, O.: Numerical integration methods for orbital motion. Celest. Mech. Dyn. Astron. 53(1), 59–69 (1992)ADSCrossRefGoogle Scholar
  49. Prince, P., Dormand, J.: High order embedded Runge-Kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  50. Probe, A., Macomber, B., Kim, D., Woollands, R., Junkins, J.L.: Terminal convergence approximation modified Chebyshev-Picard iteration for efficient numerical integration of orbital trajectories. In: Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii (2014)Google Scholar
  51. Probe, A.B., Macomber, B., Read, J.I., Woollands, R.M., Junkins, J.L.: Radially adaptive evaluation of the spherical harmonic gravity series for numerical orbital propagation. In: AAS/AIAA Space Flight Mechanics Meeting, Williamsburg, VA (2015)Google Scholar
  52. Rabe, E.: Determination and survey of periodic Trojan orbits in the restricted problem of three bodies. Astron. J. 66, 500 (1961)ADSMathSciNetCrossRefGoogle Scholar
  53. Rabe, E.: Additional periodic Trojan orbits and further studies of their stability features. Astron. J. 67, 382 (1962)ADSCrossRefGoogle Scholar
  54. Rabe, E., Schanzle, A.: Periodic librations about the triangular solutions of the restricted earth-moon problem and their orbital stabilities. Astron. J. 67, 732 (1962)ADSCrossRefGoogle Scholar
  55. Read, J.L., Elgohary, T.A., Probe, A.B., Junkins, J.L.: Monte Carlo propagation of orbital elements using modified Chebyshev-Picard iteration. In: Advances in Astronautical Sciences: AAS/AIAA Space Flight Mechanics Meeting, pp. 2589–2604 (2016)Google Scholar
  56. Riordan, J.: Derivatives of composite functions. Bull. Am. Math. Soc. 52(8), 664–667 (1946)MathSciNetzbMATHCrossRefGoogle Scholar
  57. Sharp, P.W.: N-body simulations: the performance of some integrators. ACM Trans. Math. Softw. (TOMS) 32(3), 375–395 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  58. Sitarski, G.: Recurrent power series integration of the equations of comet’s motion. Acta Astronomica 29, 401–411 (1979)ADSGoogle Scholar
  59. Sitarski, G.: Optimum recurrent power series integration of equations of motion for comets and minor planets. Acta astronomica 39, 345–349 (1989)ADSGoogle Scholar
  60. Steffensen, J.: On the problem of three bodies in the plane. Mat. fys. Meddelelser 31, 3–19 (1957)MathSciNetGoogle Scholar
  61. Steffensen, J.F.: On the restricted problem of three bodies. Mat-fys medd; Bd 30 (1956)Google Scholar
  62. Turner, J.D., Elgohary, T.A., Majji, M., Junkins, J.L.: High accuracy trajectory and uncertainty propagation algorithm for long-term asteroid motion prediction. In: Alfriend, K., Akella, M., Hurtado, J., Turner, J. (eds.) Adventures on the Interface of Mechanics and Control, pp. 15–34. Tech Science Press, Henderson (2012)Google Scholar
  63. Woollands, R., Junkins, J.: A new solution for the general Lambert’s problem. In: 37th Annual AAS Guidance & Control Conference, Breckenridge, CO (2014)Google Scholar
  64. Woollands, R., Junkins, J.L.: Nonlinear differential equation solvers via adaptive Picard-Chebyshev iteration: applications in astrodynamics. J. Guid. Control Dyn. 42, 1–16 (2018)Google Scholar
  65. Woollands, R., Bani-Younes, A., Macomber, B., Bai, X., Junkins, J.: Optimal continuous thrust maneuvers for solving 3d orbit transfer problems. In: 38th Annual AAS Guidance & Control Conference, Breckenridge, CO (2014a)Google Scholar
  66. Woollands, R., Younes, A., Macomber, B., Probe, A., Kim, D., Junkins, J.: Validation of accuracy and efficiency of long-arc orbit propagation using the method of manufactured solutions and the round-trip-closure method. In: Advanced Maui Optical and Space Surveillance Technologies Conference, vol. 1, p. 19 (2014b)Google Scholar
  67. Woollands, R.M., Bani Younes, A., Junkins, J.L.: New solutions for the perturbed lambert problem using regularization and picard iteration. J. Guid. Control Dyn. 38(9), 1548–1562 (2015a)ADSCrossRefGoogle Scholar
  68. Woollands, R.M., Read, J.L., Macomber, B., Probe, A., Younes, A.B., Junkins, J.L.: Method of particular solutions and kustaanheimo-stiefel regularized picard iteration for solving two-point boundary value problems. In: AAS/AIAA Space Flight Meeting, Williamsburg, VA (2015b)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of Central FloridaOrlandoUSA
  3. 3.Khalifa UniversityAbu DhabiUAE
  4. 4.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA

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