Application of Directional Statistics to Problems in SSA

As new optical sensors come online, increasing numbers of optical observations become available for space objects that were previously too small or too far away to detect. The space situational awareness community is presented with the computationally challenging problem of representing more accurately and efficiently the uncertainty for a large number of line-of-sight, i.e., angles-only observations. In order to perform any kind of rigorous probability analysis with these observations, we require an accurate representation of their uncertainty. Properly characterizing the uncertainty allows us to deal more efficiently with large sets of sparse data by enabling the use of rigorous probabilistic techniques to, for example, perform data association, determine collision probabilities, or initialize a Bayesian estimation scheme. However, if we propagate a point cloud in any standard coordinate system then either it forms a shape of “banana” or shows significant amount of curvature. Such distributions are awkward to describe and it cannot be approximated using a multivariate normal distribution. Further, the problem of space debris tracking can be viewed as an example of Bayesian filtering. Examples of such filters include the classic Kalman filter, together with nonlinear variants such as the extended and unscented Kalman filters, and the computationally more expensive particle filters. The filtering problem is simplest when the joint distribution of the state vector and the observation vector is normally distributed. In this paper, we demonstrate the various use of directional statistics in space situation awareness problem. First, we show that the propagated uncertainty can be approximated by using a “Fisher-Bingham-Kent (FBK)” distribution in a sphere and the 2-dimensional representation of such distribution is typically approximately bivariate normal. Second, we show the usefulness of FBK type distributions in various data association and maneuver analysis problems and finally, we discuss about a newly defined coordinate system, namely the “Adapted STructural (AST) coordinate system” to represent the orbital structure, normal direction and location of an object along its orbit. The purpose of transforming to such a coordinate system is to preserve normality and we demonstrate the usefulness and efficiency of such a system in debris tracking problem.