| Several nonlinear recursive suboptimal estimators are applied to the problem of determining the relative motion of one satellite with respect to another, both in close elliptical orbits. Only relative measurements of range, range rate and angles are used. The two-step filter, iterated extended Kalman filter, and a Kalman filter using a change of variables that makes the measurement equation linear are all derived from different approximations to the same global least squares cost function. The potential for the two-step estimator to generate numerically rank deficient covariance matrices is shown. A test matrix composed of the partial derivatives between the first and second step states and a set of constant vectors is derived to identify points in which this anomaly could occur. This rank deficiency can be interpreted geometrically in the first step state space. It is shown that the existence of low eigenvalues of this covariance matrix does correspond to situations in which the first step state vector has very low error in some direction in that state space. A simple example problem is used to demonstrate these findings. It is also shown that a two-step filter defined with an equal number of first and second step states is the same as a Kalman filter using a change of variables, except that a better approximation is made to the first step covariance time update. The dynamics and measurement equations are derived for the elliptical orbit relative navigation problem. Additionally, a closed form state transition matrix for the linearized equations of motion is derived and used to assess the magnitude of the linearization errors. The two-step filters applied to this problem produced a lower mean square error than the other filters under two situations. The first is when a large observation vector was used giving good observability of the state, but initialized with a large error. The second is for scalar observations and a statistically distributed set of initial errors. |
