On the Possibility of Ill - Conditionedcovariance Matrices in The

The rst-order two-step nonlinear estimator, when applied to a problem of orbital navigation, is found to occasionally produce rst step covariance matrices with very low eigenvalues at certain trajectory points. This anomaly is the result of the linear approximation to the rst step covariance propagation. The study of this anomaly begins with expressing the propagation of the rst and second step covariance matrices in terms of a single matrix. This matrix is shown to have a rank equal to the diierence between the number of rst 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 xed once the lter has removed the large initial state error. A test matrix containing the basis of this column space and the partial derivative matrix relating rst and second step states is derived. This square test matrix, which has dimensions equal to the number of rst step states, numerically drops rank at the same locations that the rst 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.