Bayesian filtering using directional statistics for the space debris tracking problem

The problem of tracking space debris from a sequence of observations can be viewed as an example of Bayesian filtering. The state vector describes the position and velocity of the space object, and can be represented as a 6-dimensional vector in Cartesian coordinates. The observation vector consists of an angles-only measurement, possibly plus range, a 2- or 3- dimensional vector. Filtering is simplest if the joint state and observation vectors follow a multivariate normal distribution. However, even if the initial uncertainty of the state is normally distributed, the propagated orbital state several periods into the future quickly becomes non-normal in ECI coordinates or in various standard coordinate systems, with the distribution of the position vector having a distinctive “banana” shape in R3.

One solution to the filtering problem is to use a particle filter, but this approach can suffer from the curse of particle depletion. Another solution is to switch to an “adapted structural” coordinate system, where the joint distribution is much closer to normal. The phrase “adapted” means that the coordinate system depends on the data. The key step is to note that the state vector can also be described in terms of an ellipse (5 degrees of freedom) plus the position of the space object along the elliptical orbit (a one-dimensional angular or directional variable, namely, the mean anomaly). In particular, under idealized Keplerian dynamics, the ellipse parameters are constant in time and the mean anomaly is an angle traveling around a unit circle. In this paper we demonstrate the usefulness, efficiency and accuracy of this approach for the filtering problem.