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On the Possibility of Ill-Conditioned Covariance Matrices in the First-Order Two-Step EstimatorThe first-order two-step nonlinear estimator, when applied to a problem of orbital navigation, is found to occasionally produce first step covariance matrices with very low eigenvalues at certain trajectory points. This anomaly is the result of the linear approximation to the first step covariance propagation. The study of this anomaly begins with expressing the propagation of the first and second step covariance matrices in terms of a single matrix. This matrix is shown to have a rank equal to the difference between the number of first step states and the number of second step states. Furthermore, under some simplifying assumptions, it is found that the basis of the column space of this matrix remains fixed once the filter has removed the large initial state error. A test matrix containing the basis of this column space and the partial derivative matrix relating first and second step states is derived. This square test matrix, which has dimensions equal to the number of first step states, numerically drops rank at the same locations that the first step covariance does. It is formulated in terms of a set of constant vectors (the basis) and a matrix which can be computed from a reference trajectory (the partial derivative matrix). A simple example problem involving dynamics which are described by two states and a range measurement illustrate the cause of this anomaly and the application of the aforementioned numerical test in more detail.
Document ID
Acquisition Source
Langley Research Center
Document Type
Preprint (Draft being sent to journal)
Garrison, James L.
(NASA Langley Research Center Hampton, VA, United States)
Axelrod, Penina
(Colorado Univ. Boulder, CO, United States)
Kasdin, N. Jeremy
(Stanford Univ. Stanford, CA, United States)
Date Acquired
September 7, 2013
Publication Date
January 1, 1997
Subject Category
Numerical Analysis
Report/Patent Number
Distribution Limits
Public Use Permitted.
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