Information measures for statistical orbit determination

Alinda K Mashiku, Purdue University

Abstract

The current Situational Space Awareness (SSA) is faced with a huge task of tracking the increasing number of space objects. The tracking of space objects requires frequent and accurate monitoring for orbit maintenance and collision avoidance using methods for statistical orbit determination. Statistical orbit determination enables us to obtain estimates of the state and the statistical information of its region of uncertainty given by the probability density function (PDF). As even collision events with very low probability are important, accurate prediction of collisions require the representation of the full PDF of the random orbit state. Through representing the full PDF of the orbit state for orbit maintenance and collision avoidance, we can take advantage of the statistical information present in the heavy tailed distributions, more accurately representing the orbit states with low probability. The classical methods of orbit determination (i.e. Kalman Filter and its derivatives) provide state estimates based on only the second moments of the state and measurement errors that are captured by assuming a Gaussian distribution. Although the measurement errors can be accurately assumed to have a Gaussian distribution, errors with a non-Gaussian distribution could arise during propagation between observations. In order to obtain an accurate representation of the PDF that incorporates higher order statistical information, we propose the use of nonlinear estimation methods such as the Particle Filter. A Particle Filter (PF) is proposed as a nonlinear filtering technique that is capable of propagating and estimating a more complete representation of the state distribution as an accurate approximation of a full PDF. The PF uses Monte Carlo runs to generate particles that approximate the full PDF representation. Moreover, during longer state propagations, we propose to represent the final state vector as a compressed probability mass function (PMF). Multivariate PDF compressions are computationally costly and could potentially be numerically intractable. We tackle this issue by decorrelating the nonlinear multivariate state PMFs using an improved nonlinear factor analysis (NFA) that uses a multilayer perceptron (MLP) network to model the state nonlinearities and obtain the sources that also incorporates the Fast Independent Component Analysis (FastICA [a faster computational method for ICA]) to obtain the independent and decorrelated states. Methods such as the Principal Component Analysis (PCA) are based on utilizing moments that only incorporate the second order statistics, hence will not suffice in maintaining maximum information content. On the other hand, the Independent Component Analysis (ICA) is a non-Gaussian decorrelator that is based on a linear mapping scheme, that does not incorporate the non-linear information. The PDF compressions are achieved by implementing the fast-Fourier Transform (FFT) and the wavelet transform (WT) to construct a smaller subset of data for data allocation and transmission cost reduction. The accuracy of tracking the space objects as well as reduced costs will help increase the capability of tracking the increased number of space objects. We use statistical information measures such as the Kolmogorov-Smirnov (K-S) test and the Kullback-Leibler Divergence (KLD) metric to quantify the accuracy of the reconstructed state vector and the cost reduction is measured by the number of terms required to represent the states. A performance plot illuminates the performances of the transforms over a range of compression rates. Simulations are performed on real and simulated data to demonstrate the approach for this work.

Degree

Ph.D.

Advisors

GARRISON, Purdue University.

Subject Area

Information Technology|Aerospace engineering|Electrical engineering

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