An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix
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An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix Tahsinul Haque Tasif1
· Tarek A. Elgohary1
Accepted: 29 September 2020 / Published online: 16 November 2020 © The Author(s) 2020
Abstract In 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. Keywords Two-body problem · State transition matrix · J2 − J6 perturbation · Drag perturbation · Taylor series · Recursive power series · Adaptive time step
Tahsinul Haque Tasif
[email protected] 1
Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL, 32816, USA
The Journal of the Astronautical Sciences (2020) 67:1412–1444
1413
Introduction In Astrodynamics, the State Transition Matrix (STM) of the two-body problem works as a sensitivity of the current states to the initial conditions. Hence, it computes the propagation of error of the initial states over time. Computing the STM is ubiquitous to spaceflight dynamics, navigation and control, [20, 21, 26, 32]. Goodyear presented an exact analytical solution of the two-body Cartesian STM and the method is valid for all types of orbits for the attractive force and for the hyperbolic and rectilinear orbits for the repulsive force, [10]. However, it is complicated to introduce third body effect in this method and the method requires transcendental function evaluation, [21]. This method has been simplified to increase the numerical efficiency for the Keplerian elliptical orbit, [6, 19]. However, gravity perturbation was not incorporated in this simplified method, [6]. Recently a decoupled
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