An Elegant State Transition Matrix
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An Elegant State Transition Matrix Gim J. Der 1 Abstract The analytic Keplerian 6 X 6 Earth Centered Inertial state transition matrix developed by Goodyear, Battin, and other mathematicians is well known. The primary objectives of this paper are to present a relatively simple formulation of the Keplerian state transition matrix, and to show that the higher-order secular terms associated with the state transition matrices of Goodyear and Battin can be eliminated. Analytic and numerical results indicate that the concern about these unbounded secular terms for numerical integration is not warranted. Comparison of Keplerian, Vinti, and numerical state transition matrices in the Hill-Clohessy-Wiltshire coordinate system (local-vertical coordinate system) further supports this contention since the two secular elements of the Keplerian state transition matrix match closely with those of the Vinti and numerical state transition matrices.
Introduction The universal variable formulation of Kepler's problem and the analytic 6 X 6 Earth Centered Inertial (ECI) state transition matrix provide simple solutions for many practical applications of general and special perturbations. Directly quoting R. H. Battin [1], "Algorithms for the solution of Kepler's equation abound." Danby and Burkardt [2] through Colwell [13] are a representation of papers on Kepler's equation published in the last twenty years. Danby and Burkardt [2] through Taff and Brennan [8] deal with algorithms that solve Kepler's equation by different initial guesses and various types of iterative methods. Mikkola [9] and Markley [10] develop methods that solve Kepler's equation by direct, non-iterative methods. Nijenhuis [II] utilizes the combination of direct and iterative methods. Fitz-Coy and Jang [12], which presents an integrative method using homotopy theory, is in an early development stage. Colwell [13] is a book that summarizes over three centuries of work on the solutions of Kepler's equation. Even with sophisticated methods for initial guesses, very few of the analytic, graphical, direct, iterative, and integrative methods developed over more than three hundred years can guarantee convergence over all conic trajectories. Almost everyone is misled by the assumption that Newton's method for the solution of Kepler's equation always works. Conway [5] replaced Newton's method by Laguerre's method. 'Systems Engineer, TRWIDTD, Redondo Beach, CA 90278. 371
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The author's simulation results and the work of others [6], [14], and [15] show that Kepler's equation can finally be reliably solved. A converged solution for any conic trajectory can always be obtained using a simple formula to provide an initial guess. The formulation of the analytic 6 X 6 ECI state transition matrix depends both on the formulation of Kepler's problem and its solution. Goodyear [16] and Battin [1] use the current time t as reference and initial time to as the solution time to derive the adjoint state transition matrix and the desired state transition matrix is obtained by its sy
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