Ill-conditioned covariance matrices in the first-order two-step estimator

The first-order two-step estimator is found to occasionally produce first-step covariance matrices with very low, sometimes negative, eigenvalues. These low eigenvalues can cause large errors or meaningless estimates. A single matrix is found, which is shown to have a rank equal to the difference between the number of first- and second-step states. Furthermore, it is demonstrated that the basis of the column space of this matrix remains fixed once the large initial state error has decreased. A test matrix containing the (constant) basis of this column space and the partial derivative matrix relating first and second step states is derived. This matrix numerically drops rank at the same locations that the first-step covariance docs. A simple example problem involving dynamics described by two states and a range measurement illustrates the cause of this anomaly and application of the aforementioned numerical test. Suggested modifications to the filter that can mitigate the numerical problems caused by this anomaly are given.