An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix

Tahsinul Haque Tasif1, Tarek A. Elgohary1
1Mechanical and Aerospace Engineering, University of Central Florida, Orlando, USA

Tóm tắt

AbstractIn this work, the Taylor series based technique, Analytic Continuation is implemented to develop a method for the computation of the gravity and drag perturbed State Transition Matrix (STM) incorporating adaptive time steps and expansion order. Analytic Continuation has been developed for the two-body problem based on two scalar variables f and gp and their higher order time derivatives using Leibniz rule. The method has been proven to be very precise and efficient in trajectory propagation. The method is expanded to include the computation of the STM for the perturbed two-body problem. Leibniz product rule is used to compute the partials for the recursive formulas and an arbitrary order Taylor series is used to compute the STM. Four types of orbits, LEO, MEO, GTO and HEO, are presented and the simulations are run for 10 orbit periods. The accuracy of the STM is evaluated via RMS error for the unperturbed cases, symplectic check for the gravity perturbed cases and error propagation for the gravity and drag perturbed orbits. The results are compared against analytical and high order numerical solvers (ODE45, ODE113 and ODE87) in terms of accuracy. The results show that the method maintains double-precision accuracy for all test cases and 1-2 orders of magnitude improvement in linear prediction results compared to ODE87. The present approach is simple, adaptive and can readily be expanded to compute the full spherical harmonics gravity perturbations as well as the higher order state transition tensors.

Từ khóa


Tài liệu tham khảo

Atallah, A., Bani Younes, A.: Parallel integration of perturbed orbital motion. In: AIAA Scitech 2019 Forum (2019)

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., pp. 1–28 (2019)

Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, revised edition. American Institute of Aeronautics and Astronautics (1999)

Bogacki, P., Shampine, L.: An efficient runge-kutta (4,5) pair. Computers & Mathematics with Applications 32, 15–28 (1996). https://doi.org/10.1016/0898-1221(96)00141-1

Brouwer, D.: Hori, G.i.: Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite. The Astronomical Journal 66, 193 (1961)

Chiaradia, A.P.M., Kuga, H.K., Prado, A.F.B.d.A.: Comparison between two methods to calculate the transition matrix of orbit motion. Math. Probl. Eng., 2012 (2012)

Danby, J.: Fundamentals of Celestial Mechanics. Macmillan, New York (1962)

Dormand, J.R., Prince, P.J.: A family of embedded runge-kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

Gim, D.W., Alfriend, K.T.: State transition matrix of relative motion for the perturbed noncircular reference orbit. Journal of Guidance, Control, and Dynamics 26(6), 956–971 (2003)

Goodyear, W.H.: Completely general closed-form solution for coordinates and partial derivative of the two-body problem. The Astronomical Journal 70, 189 (1965)

Govorukhin, V.: ode87 integrator https://www.mathworks.com/matlabcentral/fileexchange/3616-ode87-integrator (2003)

Hairer, E., Nørsett, S., Wanner, G.: Solving ordinary differential equations i: Nonstiff problems. 2nd revised edn (1993)

Hatten, N., Russell, R.P.: Decoupled direct state transition matrix calculation with runge-kutta methods. J. Astronaut. Sci. 65(3), 321–354 (2018)

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)

Hernandez, K., Elgohary, T.A., Turner, J.D., et al.: A novel analytic continuation power series solution for the perturbed two-body problem. Celest Mech Dyn Astr 131(48). https://doi.org/10.1007/s10569-019-9926-0 (2019)

Hernandez, K., Read, J.L., Elgohary, T.A., Turner, J.D., Junkins, J.L.: Analytic power series solutions for two-body and J2–J6 trajectories and state transition models. In: Advances in Astronautical Sciences: AAS/AIAA Astrodynamics Specialist Conference (2015)

Jezewski, D., Mittleman, D.: Integrals of motion for the classical two-body problem with drag. International Journal of Non-linear Mechanics 18 (2), 119–124 (1983)

Koenig, A.W., Guffanti, T., D’Amico, S.: New state transition matrices for spacecraft relative motion in perturbed orbits. Journal of Guidance, Control, and Dynamics 40(7), 1749–1768 (2017)

Kuga, H.: Transition matrix of the elliptical keplerian motion (1986)

Ledebuhr, A.G., et al.: Autonomous, Agile, Micro-Satellites and Supporting Technologies for Use in Low-Earth Orbit Missions. Tech. rep., LLNL, US (1998)

Markley, F.: Approximate cartesian state transition matrix. The Journal of the Astronautical Sciences 34(2), 161–169 (1986)

Mavraganis, A., Michalakis, D.: The two-body problem with drag and radiation pressure. Celest. Mech. Dyn. Astron. 58(4), 393–403 (1994)

Montenbruck, O.: Numerical integration methods for orbital motion. Celest. Mech. Dyn. Astron. 53(1), 59–69 (1992)

Prince, P.J., Dormand, J.R.: High order embedded runge-kutta formulae. J. Comput. Appl. Math. 7(1), 67–75 (1981)

Read, J.L., Younes, A.B., Macomber, B., Turner, J., Junkins, J.L.: State transition matrix for perturbed orbital motion using modified chebyshev picard iteration. J. Astronaut. Sci. 62(2), 148–167 (2015)

Schaub, H., Junkins, J.L.: Analytical mechanics of space systems. American Institute of Aeronautics and Astronautics (2005)

Shampine, L.F., Gordon, M.K.: Computer Solution of Ordinary Differential Equations: the Initial Value Problem. Freeman, San Francisco (1975)

Shampine, L.F., Reichelt, M.W.: The matlab ode suite. SIAM Journal on Scientific Computing 18(1), 1–22 (1997)

Tasif, T.H., Elgohary, T.A.: A high order analytic continuation technique for the perturbed two-body problem state transition matrix. Advances in Astronautical sciences: AAS/AIAA Space Flight Mechanics Meeting (2019)

Tasif, T.H., Elgohary, T.A.: An adaptive analytic continuation technique for the computation of the higher order state transition tensors for the perturbed two-body problem. In: AIAA Scitech 2020 Forum, pp. 0958 (2020)

Turner, J., Elgohary, T., Majji, M., Junkins, J.: 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 (2012)

Uyeminami, R.: Navigation filter mechanization for a Spaceborne Gps user. In: IEEE 1978 Position Location and Navigation Symposium, pp. 328–334 (1978)

Vallado, D.A.: Fundamentals of astrodynamics and applications. vol. 12 Springer Science & Business Media (2001)

Vallado, D.A., Finkleman, D.: A critical assessment of satellite drag and atmospheric density modeling. Acta Astronaut. 95, 141–165 (2014)

Yamanaka, K., Ankersen, F.: New state transition matrix for relative motion on an arbitrary elliptical orbit. Journal of Guidance, Control, and Dynamics 25(1), 60–66 (2002)