Entropy-Based Space Object Data Association Using an Adaptive Gaussian Sum Filter

This paper shows an approach to improve the statistical validity of orbital estimates and uncertainties as well as a method of associating measurements with the correct resident space objects and classifying events in near realtime. The approach involves using an adaptive Gaussian mixture solution to the Fokker-Planck-Kolmogorov equation for its applicability to the resident space object tracking problem. The Fokker-Planck-Kolmogorov equation describes the time-evolution of the probability density function for nonlinear stochastic systems with Gaussian inputs, which often results in non-Gaussian outputs. The adaptive Gaussian sum lter provides a computationally ecient and accurate solution for this equation, which captures the non-Gaussian behavior associated with these nonlinear stochastic systems. This adaptive lter is designed to be scalable, relatively ecient for solutions of this type, and thus is able to handle the nonlinear eects which are common in the estimation of resident space object orbital states. The main purpose of this paper is to develop a technique for data association based on entropy theory that is compatible with the adaptive Gaussian sum lter. The adaptive lter and corresponding measurement association methods are evaluated using simulated data in realistic scenarios to determine their performance and feasibility.

[1]  Yuejin Tan,et al.  An information theoretic approach based Kullback-Leibler discrimination for multiple target tracking , 2009, 2009 International Conference on Information and Automation.

[2]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[3]  Tim Flohrer,et al.  Properties of the high area-to-mass ratio space debris population at high altitudes , 2006 .

[4]  Peter D. Scott,et al.  Decision based uncertainty propagation using adaptive Gaussian mixtures , 2009, 2009 12th International Conference on Information Fusion.

[5]  Frank Dellaert,et al.  MCMC-based particle filtering for tracking a variable number of interacting targets , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  F. Daum,et al.  A Split-Step Solution of the Fokker-Planck Equation for the Conditional Density , 2006, 2006 Fortieth Asilomar Conference on Signals, Systems and Computers.

[7]  Herman Bruyninckx,et al.  Kalman filters for non-linear systems: a comparison of performance , 2004 .

[8]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

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

[10]  Peter D. Scott,et al.  A novel Gaussian Sum Filter Method for accurate solution to the nonlinear filtering problem , 2008, 2008 11th International Conference on Information Fusion.

[11]  Stephen J. Wright,et al.  Primal-Dual Interior-Point Methods , 1997 .

[12]  H. Sorenson,et al.  Nonlinear Bayesian estimation using Gaussian sum approximations , 1972 .

[13]  A. Sofi,et al.  Approximate solution of the Fokker–Planck–Kolmogorov equation , 2002 .

[14]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[15]  Fredrik Gustafsson,et al.  Monte Carlo data association for multiple target tracking , 2001 .

[16]  Andrei N. Kolmogorov,et al.  On the Shannon theory of information transmission in the case of continuous signals , 1956, IRE Trans. Inf. Theory.

[17]  S. Chakravorty,et al.  A multi-resolution approach for steady state uncertainty determination in nonlinear dynamical systems , 2006, 2006 Proceeding of the Thirty-Eighth Southeastern Symposium on System Theory.

[18]  A. T. Fuller,et al.  Analysis of nonlinear stochastic systems by means of the Fokker–Planck equation† , 1969 .

[19]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[20]  T. Singh,et al.  Uncertainty Propagation for Nonlinear Dynamic Systems Using Gaussian Mixture Models , 2008 .

[21]  Puneet Singla,et al.  A gaussian function network for uncertainty propagation through nonlinear dynamical system , 2008 .

[22]  Martin E. Liggins,et al.  Information theoretics for improved tracking and fusion performance , 2003, SPIE Defense + Commercial Sensing.

[23]  G. Muscolino,et al.  Stationary and non-stationary probability density function for non-linear oscillators , 1997 .

[24]  P. Chodas,et al.  Orbit Determination and Estimation of Impact Probability for Near Earth Objects , 1999 .

[25]  Jemin George,et al.  Spacecraft Attitude Estimation Using Adaptive Gaussian Sum Filter , 2009 .

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

[27]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[28]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[29]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[30]  A. Kolmogorov,et al.  Information theory and the theory of algorithms , 2010 .

[31]  S. Godsill,et al.  Monte Carlo filtering for multi target tracking and data association , 2005, IEEE Transactions on Aerospace and Electronic Systems.

[32]  R. N. Iyengar,et al.  Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique , 1978 .

[33]  S. Särkkä,et al.  On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[34]  Puneet Singla,et al.  An adaptive Gaussian sum filter for the spacecraft attitude estimation problem , 2008 .

[35]  Juan I. Nieto,et al.  Multiple target tracking using Sequential Monte Carlo Methods and statistical data association , 2003, Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453).

[36]  J. Huang,et al.  Curse of dimensionality and particle filters , 2003, 2003 IEEE Aerospace Conference Proceedings (Cat. No.03TH8652).

[37]  Suman Chakravorty,et al.  The Partition of Unity Finite Element Approach to the Stationary Fokker-Planck Equation , 2006 .

[38]  Patrick Pérez,et al.  Sequential Monte Carlo methods for multiple target tracking and data fusion , 2002, IEEE Trans. Signal Process..

[39]  A. Honkela.  Approximating nonlinear transformations of probability distributions for nonlinear independent component analysis , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[40]  KhanZia,et al.  MCMC-Based Particle Filtering for Tracking a Variable Number of Interacting Targets , 2005 .

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