Abstract
The Modified Chebyshev Picard Iteration (MCPI) method has recently proven to be highly efficient for a given accuracy compared to several 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 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 the normalized, Associated Legendre Functions are given and verified numerically. Modifications of the classical algorithm techniques, such as rewriting the STM equations in a second-order cascade formulation, gives rise to additional speedup. Timing results for the baseline formulation and this second-order formulation are given.
Highlights
The Modified Chebyshev-Picard Iteration (MCPI) method is used to solve both linear and nonlinear, high precision, long-term orbit propagation problems through iteratively finding an orthogonal function approximation for the entire state trajectory
This algorithm has recently proven to be a powerful tool for solving nonlinear differential equations for both initial value problems (IVP) and boundary value problems (BVP) [1,2,3,4,5]
Computation of the State Transition Matrix (STM) for the spherical harmonic gravity case requires the partial derivative of Pnm with respect to φ, where Cnm and Snm are the normalized Stokes coefficients determined from satellite motion observations, and Nnm is the scale factor given in Eq 33
Summary
The Modified Chebyshev-Picard Iteration (MCPI) method is used to solve both linear and nonlinear, high precision, long-term orbit propagation problems through iteratively finding an orthogonal function approximation for the entire state trajectory. We can avoid weak numerical instability of the typical recursions used to compute the Anm (see Table 1 of Reference [16]) To overcome this problem, the derived Associated Legendre Functions and spherical harmonic coefficients are normalized to improve accuracy for high degree and order gravity models. Computation of the STM for the spherical harmonic gravity case requires the partial derivative of Pnm with respect to φ, where Cnm and Snm are the normalized Stokes coefficients determined from satellite motion observations, and Nnm is the scale factor given in Eq 33. To verify that the formulation developed here for the STM is correct using MCPI, a finite difference check of both the second partials of the gravity potential and the STM matrix are completed and give results comparable in accuracy to the figure above.
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