An intrusive approach to uncertainty propagation in orbital mechanics based on Tchebycheff polynomial algebra

The paper presents an intrusive approach to propagate uncertainty in orbital mechanics. The approach is based on an expansion of the uncertain quantities in Tchebicheff series and a propagation through the dynamics using a generalised polynomial algebra. Tchebicheff series expansions offer a fast uniform convergence with relaxed continuity and smothness requirements. The paper details the proposed approach and illustrates its applicability through a set of test cases considering both parameter and model uncertainties. This novel intrusive technique is then comapred against its non-intrusive counterpart in terms of approximation accuracy and computational cost.

[1]  J. C Mason,et al.  Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation , 1980 .

[2]  P. Epstein.  Zur Theorie des Starkeffektes , 1916 .

[3]  A. Morselli,et al.  High order optimal feedback control of space trajectories with bounded control , 2014 .

[4]  F. Namouni,et al.  The accelerated Kepler problem , 2007, 0705.3219.

[5]  G. Lantoine,et al.  Complete closed-form solutions of the Stark problem , 2011 .

[6]  Nicolas Brisebarre,et al.  Chebyshev interpolation polynomial-based tools for rigorous computing , 2010, ISSAC.

[7]  Lilia Maliar,et al.  Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain , 2013 .

[8]  Martin Berz.  From Taylor series to Taylor models , 1997 .

[9]  Keith O. Geddes,et al.  Polynomial Approximation by Projections on the Unit Circle , 1975 .

[10]  Massimiliano Vasile,et al.  Comparison of non-intrusive approaches to uncertainty propagation in orbital mechanics , 2015 .

[11]  Ronald Cools,et al.  CHEBINT: A MATLAB/Octave toolbox for fast multivariate integration and interpolation based on chebyshev approximations over hypercubes , 2013, TOMS.

[12]  A. Doostan,et al.  Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos , 2013 .

[13]  Pascal Giorgi,et al.  On Polynomial Multiplication in Chebyshev Basis , 2010, IEEE Transactions on Computers.

[14]  M. Berz,et al.  Asteroid close encounters characterization using differential algebra: the case of Apophis , 2010 .

[15]  P. Redmond.  Generalization of the Runge-Lenz Vector in the Presence of an Electric Field , 1964 .

[16]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[17]  Michèle Lavagna,et al.  Application of high order expansions of two-point boundary value problems to astrodynamics , 2008 .

[18]  Martin Berz,et al.  The method of power series tracking for the mathematical description of beam dynamics , 1987 .

[19]  T. J. Rivlin,et al.  Ultra-arithmetic I: Function data types , 1982 .

[20]  A. Giorgilli,et al.  Methods of algebraic manipulation in perturbation theory , 2013, 1303.7398.