Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos

This paper demonstrates the use of polynomial chaos expansions for the nonlinear, non-Gaussian propagation of orbit state uncertainty. Using linear expansions in tensor products of univariate orthogonal polynomial bases, polynomial chaos expansions approximate the stochastic solution of the ordinary differential equation describing the propagated orbit, and include information on covariance, higher moments, and the spatial density of possible solutions. Results presented in this paper use non-intrusive, i.e., sampling-based, methods in combination with either least-squares regression or pseudospectral collocation to estimate the polynomial chaos expansion coefficients at any future point in time. Such methods allow for the usage of existing orbit propagators. Samples based on sun-synchronous and Molniya orbit scenarios are propagated for up to ten days using two-body and higher-fidelity force models. Tests demonstrate that the presented methods require the propagation of orders of magnitude fewer samples ...

[1]  Gaëtan Kerschen,et al.  Probabilistic Assessment of the Lifetime of Low-Earth-Orbit Spacecraft: Uncertainty Characterization , 2015 .

[2]  Gaëtan Kerschen,et al.  Probabilistic Assessment of Lifetime of Low-Earth-Orbit Spacecraft: Uncertainty Propagation and Sensitivity Analysis , 2015 .

[3]  Brandon A. Jones,et al.  Postmaneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions , 2015 .

[4]  Alireza Doostan,et al.  Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression , 2014, 1410.1931.

[5]  P. Nair,et al.  Aircraft Robust Trajectory Optimization Using Nonintrusive Polynomial Chaos , 2014 .

[6]  F. Landis Markley,et al.  Wald Sequential Probability Ratio Test for Space Object Conjunction Assessment , 2014 .

[7]  Alireza Doostan,et al.  Satellite collision probability estimation using polynomial chaos expansions , 2013 .

[8]  F. Markley,et al.  Wald Sequential Probability Ratio Test for Analysis of Orbital Conjunction Data , 2013 .

[9]  Daniel J. Scheeres,et al.  Analytical Nonlinear Propagation of Uncertainty in the Two-Body Problem , 2012 .

[10]  Aubrey B. Poore,et al.  Gaussian Sum Filters for Space Surveillance: Theory and Simulations , 2011 .

[11]  Adrian Sandu,et al.  A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems , 2010 .

[12]  Paul J. Cefola,et al.  Entropy-Based Space Object Data Association Using an Adaptive Gaussian Sum Filter , 2010 .

[13]  R. Bhattacharya,et al.  Nonlinear estimation with polynomial chaos and higher order moment updates , 2010, Proceedings of the 2010 American Control Conference.

[14]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[15]  H. Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[16]  Aubrey B. Poore,et al.  Covariance consistency for track initiation using Gauss-Hermite quadrature , 2010, Defense + Commercial Sensing.

[17]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[18]  Brandon A. Jones,et al.  Comparisons of the Cubed-Sphere Gravity Model with the Spherical Harmonics , 2010 .

[19]  Puneet Singla,et al.  An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics , 2009 .

[20]  Gianluca Iaccarino,et al.  A least-squares approximation of partial differential equations with high-dimensional random inputs , 2009, J. Comput. Phys..

[21]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[22]  A. Nouy.  Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms , 2008 .

[23]  James D. Turner,et al.  A high order method for estimation of dynamic systems , 2008 .

[24]  A. Nouy.  A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations , 2007 .

[25]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[26]  R. Park,et al.  Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design , 2006 .

[27]  O. L. Maître,et al.  Uncertainty propagation in CFD using polynomial chaos decomposition , 2006 .

[28]  Adrian Sandu,et al.  Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects , 2006 .

[29]  Adrian Sandu,et al.  Modeling multibody systems with uncertainties. Part II: Numerical applications , 2006 .

[30]  F. Pukelsheim.  Optimal Design of Experiments (Classics in Applied Mathematics) (Classics in Applied Mathematics, 50) , 2006 .

[31]  N. Johnson,et al.  Risks in Space from Orbiting Debris , 2006, Science.

[32]  V. Melas.  Functional Approach to Optimal Experimental Design , 2005 .

[33]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .

[34]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[35]  M. Cheng,et al.  GGM02 – An improved Earth gravity field model from GRACE , 2005 .

[36]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[37]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[38]  B. Tapley,et al.  Statistical Orbit Determination , 2004 .

[39]  Richard H. Lyon,et al.  Geosynchronous orbit determination using space surveillance network observations and improved radiative force modeling , 2004 .

[40]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[41]  Gregory Beylkin,et al.  Toward Multiresolution Estimation and Efficient Representation of Gravitational Fields , 2002 .

[42]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[43]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[44]  R. Ghanem,et al.  A stochastic projection method for fluid flow. I: basic formulation , 2001 .

[45]  Rudolph van der Merwe,et al.  The square-root unscented Kalman filter for state and parameter-estimation , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[46]  John Red-Horse,et al.  Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach , 1999 .

[47]  Roger Ghanem,et al.  Ingredients for a general purpose stochastic finite elements implementation , 1999 .

[48]  Roger Ghanem,et al.  Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media , 1998 .

[49]  D. Vallado.  Fundamentals of Astrodynamics and Applications , 1997 .

[50]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[51]  John L. Junkins,et al.  Non-Gaussian error propagation in orbital mechanics , 1996 .

[52]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[53]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[54]  P. Túrán.  On orthogonal polynomials , 1975 .

[55]  Stephen M. Stigler,et al.  Optimal Experimental Design for Polynomial Regression , 1971 .

[56]  P. Schrimpf,et al.  Dynamic Programming , 2011 .

[57]  Chris Sabol,et al.  Linearized Orbit Covariance Generation and Propagation Analysis via Simple Monte Carlo Simulations (Preprint) , 2010 .

[58]  Omar M. Knio,et al.  Introduction: Uncertainty Quantification and Propagation , 2010 .

[59]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[60]  Sparse Grids , 2008 .

[61]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[62]  Vladimir A. Chobotov,et al.  Orbital Mechanics, Third Edition , 2002 .

[63]  Michael Jackson,et al.  Optimal Design of Experiments , 1994 .