STATE TRANSITION MATRIX FOR PERTURBED ORBITAL MOTION USING MODIFIED CHEBYSHEV PICARD ITERATION
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Copyright 2015 California Institute of Technology. The Modified Chebyshev Picard Iteration (MCPI) method has recently proven to be more efficient for a given accuracy than the most commonly adopted numerical integration methods, as a means to solve for perturbed orbital motion. This method utilizes Picard iteration, which generates a sequence of path approximations, and discrete Chebyshev Polynomials, which are orthogonal and also enable both efficient and accurate function approximation. The nodes consistent with discrete Chebyshev orthogonality are generated using cosine sampling; this strategy also reduces the Runge effect and as a consequence of orthogonality, there is no matrix inversion required to find the basis function coefficients. The MCPI algorithms considered herein are parallel-structured so that they are immediately well-suited for massively parallel implementation with additional speedup. MCPI has a wide range of applications beyond ephemeris propagation, including the propagation of the State Transition Matrix (STM) for perturbed two-body motion. A solution is achieved for a spherical harmonic series representation of earth gravity (EGM2008), although the methodology is suitable for application to any gravity model. Included in this representation is a derivation of the second partial derivatives of the normalized, Associated Legendre Functions, which is given and verified numerically.
