Tracking space debris using directional statistics

One of the main concerns in space situational awareness is to keep track of the large number of space objects, including both satellites and debris, orbiting the earth. The state of an orbiting object indicates the position and velocity of the object and it is generally represented using a 6-dimensional state vector. Observations typically take the form of angles-only measurements from ground-based telescopes.  Two specific challenges are the tracking of objects and the association of objects. Ideas from the directional statistics can be used to tackle both of these challenges.

There are two sets of contributions made in this thesis. The first set of contributions deals with the tracking of an orbiting object. In general, the filtering or tracking problem is simplest when the joint distribution of uncertainties in the state vector and the observation vector is normally distributed.  To achieve this goal,  the "Adapted STructural (AST)" coordinate system has been developed to describe the orbiting object and the measurements of the object. The propagated orbital uncertainty represented using the AST coordinate system is approximately Gaussian under a wide range of conditions and as a result this coordinate system is suitable for using a Kalman filter for tracking space objects. A comparative study has been performed to understand behavior of
different non-linear Kalman filters. Further, two new Kalman filters, namely the Observation-Centered extended Kalman filter and Observation-Centered unscented Kalman filter, have been developed. Various uses of the AST coordinate system are described using suitable examples.

The second set of contributions is related to the representation of the 2-dimensional uncertainty, associated with the angles-only position. The concept of the newly developed "Adapted Spherical (ASP)" coordinate system
is described in detail. Several examples are provided to discuss the usefulness of the ASP coordinate system for solving association problems. In addition, limitations of the ASP coordinate system are also highlighted. Especially for a break-up event scenario, the propagated point cloud in the ASP coordinate system displays a "bow-tie" or "pinching" pattern when the propagation period is a close multiple of half orbital period. A new "Pinched-Normal (PN)" distribution has been developed to understand the reason. Finally, the distribution of the radial component is analyzed.