Probability of Collision Estimation and Optimization Under Uncertainty Utilizing Separated Representations

Many current applications of maneuver design to astrodynamics consider a deterministic case, where statistics or uncertainty is left unquantified. When including constraints based on the probability of collision, any solution must be robust to the uncertainty of the system. This paper considers the methodology of separated representations for orbit uncertainty propagation and its subsequent application to a reliability design formulation of the maneuver design problem. Separated representations is a polynomial surrogate method that has been shown to be both efficient at propagating uncertainty when considering high stochastic dimension and accurate over long propagation times. This efficiency is leveraged to improve tractability when solving the reliability design problem using optimization under uncertainty. Two sequential, potential collisions are considered in the results of this paper, with one object able to maneuver. The optimization problem therefore seeks to avoid both collisions. The probability of each collision is estimated via large numbers of samples propagated via the separated representation. The accuracy of the surrogates is compared to that of a Monte Carlo reference, and the variability of the estimated probabilities of collision is analyzed.

[1]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[2]  A. Nouy.  Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems , 2010 .

[3]  G. Iaccarino,et al.  Non-intrusive low-rank separated approximation of high-dimensional stochastic models , 2012, 1210.1532.

[4]  H. Elman,et al.  DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS , 2008 .

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

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

[7]  Martin J. Mohlenkamp,et al.  Multivariate Regression and Machine Learning with Sums of Separable Functions , 2009, SIAM J. Sci. Comput..

[8]  Geoffrey T. Parks,et al.  Multi-Objective Optimization for Multiphase Orbital Rendezvous Missions , 2013 .

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

[10]  Bruce A. Conway,et al.  Optimal finite-thrust rendezvous trajectories found via particle swarm algorithm , 2012 .

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

[12]  S. Alfano,et al.  Satellite Conjunction Monte Carlo Analysis , 2009 .

[13]  R. Armellin,et al.  A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation , 2015 .

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

[15]  Shawn E. Gano,et al.  Comparison of Three Surrogate Modeling Techniques: Datascape r , Kriging, and Second Order Regression , 2006 .

[16]  Bob E. Schutz,et al.  Orbit Determination Concepts , 2004 .

[17]  G. Kerschen,et al.  Robust rendez-vous planning using the scenario approach and differential flatness , 2015 .

[18]  Nestor V. Queipo,et al.  Efficient Shape Optimization Under Uncertainty Using Polynomial Chaos Expansions and Local Sensitivities , 2006 .

[19]  J. Peláez,et al.  DROMO propagator revisited , 2013 .

[20]  Mathilde Chevreuil,et al.  A Least-Squares Method for Sparse Low Rank Approximation of Multivariate Functions , 2015, SIAM/ASA J. Uncertain. Quantification.

[21]  D. Scheeres,et al.  Tractable Expressions for Nonlinearly Propagated Uncertainties , 2015 .

[22]  A. Doostan,et al.  Orbit uncertainty propagation and sensitivity analysis with separated representations , 2016, Celestial Mechanics and Dynamical Astronomy.

[23]  Yifei Sun,et al.  Uncertainty propagation in orbital mechanics via tensor decomposition , 2016 .

[24]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[25]  Kyle J. DeMars,et al.  Collision Probability with Gaussian Mixture Orbit Uncertainty , 2014 .

[26]  Slawomir J. Nasuto,et al.  Search space pruning and global optimisation of multiple gravity assist spacecraft trajectories , 2007, J. Glob. Optim..

[27]  J. Russell Carpenter,et al.  Operational Experience with the Wald Sequential Probability Ratio Test for Conjunction Assessment from the Magnetospheric MultiScale Mission , 2016 .

[28]  Marc Balducci,et al.  Orbit Uncertainty Propagation with Separated Representations , 2018 .

[29]  Georgia Deaconu,et al.  Minimizing the Effects of Navigation Uncertainties on the Spacecraft Rendezvous Precision , 2014 .

[30]  Guo-Jin Tang,et al.  Uncertainty Quantification for Short Rendezvous Missions Using a Nonlinear Covariance Propagation Method , 2016 .

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

[32]  J. Mueller.  Onboard Planning of Collision Avoidance Maneuvers Using Robust Optimization , 2009 .

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

[34]  Gregory Beylkin,et al.  Randomized Alternating Least Squares for Canonical Tensor Decompositions: Application to A PDE With Random Data , 2015, SIAM J. Sci. Comput..

[35]  Boris N. Khoromskij,et al.  Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs , 2011, SIAM J. Sci. Comput..

[36]  Zhen Yang,et al.  Robust optimization of nonlinear impulsive rendezvous with uncertainty , 2014 .

[37]  Pradipto Ghosh,et al.  Particle Swarm Optimization of Multiple-Burn Rendezvous Trajectories , 2012 .

[38]  Denis Arzelier,et al.  Robust Rendezvous Planning Under Maneuver Execution Errors , 2015 .

[39]  Vivek Vittaldev,et al.  Space Object Collision Probability via Monte Carlo on the Graphics Processing Unit , 2017 .

[40]  Shenmin Zhang,et al.  Surrogate-based parameter optimization and optimal control for optimal trajectory of Halo orbit rendezvous , 2013 .

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

[42]  H. Matthies,et al.  Partitioned treatment of uncertainty in coupled domain problems: A separated representation approach , 2013, 1305.6818.

[43]  Brandon A. Jones,et al.  Reduced cost mission design using surrogate models , 2016 .

[44]  Ya-Zhong Luo,et al.  Robust Planning of Nonlinear Rendezvous with Uncertainty , 2017 .

[45]  Salvatore Alfano,et al.  A comprehensive assessment of collision likelihood in Geosynchronous Earth Orbit , 2018, Acta Astronautica.

[46]  Anthony Nouy,et al.  Model Reduction Based on Proper Generalized Decomposition for the Stochastic Steady Incompressible Navier-Stokes Equations , 2014, SIAM J. Sci. Comput..