Modified Chebyshev-Picard Iteration Methods for Orbit Propagation

Bai, Xiaoli; Junkins, John L.

doi:10.1007/bf03321533

Cited by 38 publications

(26 citation statements)

References 31 publications

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“…The entire state trajectory over a long time arc is approximated at every iteration until a specified tolerance is met [2]. Emile Picard stated that, given an initial condition x (t 0 ) = x 0 , any first order differential equation [1] …”

Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

“…For further details on the basics of this novel integration technique and its convergence properties, refer to References [1] and [2]. Since the Chebyshev polynomials are orthogonal, a matrix inverse is avoided when finding basis function coefficients.…”

Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

“…While very long path segments are convergent, computationally optimal path segments are typically a large fraction of an orbit. A major advantage of the MCPI approach is that the use of cosine sampling reduces the well-known Runge Effect, whereby the largest errors are encountered near the boundary of piecewise approximation segments [1]. 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].…”

Section: Introductionmentioning

confidence: 99%

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State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

et al. 2015

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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.

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Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

Section: Modified Chebyshev Picard Iterationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

et al. 2015

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“…The entire trajectory to be optimized by this direct method is represented in terms of nodes, and a large number of design variables. Direct methods are based on the transformation of the variational calculus formulation in the original optimal control problem (OCP) into an NLP by discretizing the state or/and control history and then solving the resulting problem using an NLP solver [5,3,2,12]. A distinguishing feature of direct methods is that they are numerically more robust than indirect methods [3].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, direct methods convert the calculus of variation problem into a parameter optimization problem that minimizes the performance index using nonlinear programming (NLP). It also transcribes the states and controls through direct transcription and collocation [5,3,2,12]. The entire trajectory to be optimized by this direct method is represented in terms of nodes, and a large number of design variables.…”

Section: Introductionmentioning

confidence: 99%

Optimal interior Earth–Moon Lagrange point transfer trajectories using mixed impulsive and continuous thrust

Lee¹,

Butcher²,

Sanyal³

2014

Aerospace Science and Technology

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Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

Tafakkori-Bafghi

Loghmani

Heydari

et al. 2019

Math Methods in App Sciences

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In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation.Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method. KEYWORDS initial value problems, Jacobi-Gauss quadrature formula, Picard iteration method, shifted Jacobi polynomials, vector-matrix form MSC CLASSIFICATION 65L05; 33C45; 33F05; 65D32 | INTRODUCTIONOne of the scientific methods for solving physics, chemistry, and engineering problems and analyzing natural phenomena is constructing mathematical models for these problems. Some of these models arise in the form of initial value problems (IVPs), namely, a differential equation with the initial condition(s). Various methods are presented to solve IVPs. These methods are classified as analytical, semianalytical, and basically numerical methods.The most commonly used semianalytical methods are the Adomian decomposition method (ADM) and its modifications, 1-4 the variational iteration method (VIM), 5-8 the hybrid spectral-variational iteration method (H-S-VIM), 9 the homotopy perturbation method (HPM), 10-12 homotopy analysis method (HAM), 13,14 and the differential transformation method (DTM) 15-17 ; on the other hand, collocation method using Chebyshev and Legendre as Jacobi polynomials, 18-29 radial basis function method, 30-34 cubic B-spline scaling function and Chebyshev cardinal function method, [35][36][37] Birkhoff-type interpolation method, 38,39 collocation method using Bernstein polynomials, 40-43 the Galerkin method, 44 the hybrid block-pulse and Chebyshev cardinal function method, 45 the linear barycentric rational interpolation method, 46 and also other numerical methods such as Runge-Kutta-Nyström method and multistep predictor-corrector method are proposed by some researchers such as Ahmad et al, 47 Jator and Lee, 48 Kosti et al, 49,50 Papadopoulos et al, 51 and Simos 52 for solving periodical oscillatory IVPs.

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Modified Chebyshev-Picard Iteration Methods for Orbit Propagation

Cited by 38 publications

References 31 publications

State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

State Transition Matrix for Perturbed Orbital Motion Using Modified Chebyshev Picard Iteration

Optimal interior Earth–Moon Lagrange point transfer trajectories using mixed impulsive and continuous thrust

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

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