Monte-Carlo Methods and the Step-Back Kalman Filter for Orbital State Estimation

This research assesses the performance of filtering schemes for tracking uncooperative satellites through space-based optical measurements, and identifies a simple and numerically stable methodology that ameliorates the poor performance of standard filtering schemes at a substantially reduced cost in comparison to nonlinear particle filter-based remedies. Traditional filtering schemes, such as the extended Kalman filter (EKF) and unscented Kalman filter (UKF), both diverge when tracking a resident space object (RSO) in geosynchronous orbit (GEO) when there is a long time duration between measurements. This divergence is identified as a consequence of nonlinearity in the dynamics and nonlinearity in the optical measurements, both of which cause the underlying density of the state to deviate from a Gaussian distribution. A Gaussian sum filter based on using a Gaussian mixture model (GMM) for the probability density function can be implemented in order to avoid this divergence, but this comes at a high computational cost and has numerical sensitivity problems under reasonable orbital conditions. An alternative filter algorithm has been developed, referred to as the extended step-back Kalman filter (ESBKF), which is shown to effectively track the RSO in GEO while avoiding the computational burden and numerical sensitivity of the GMM filter. This filter applies the measurement updates to statistics at a time in the past when the distribution was approximately Gaussian, and then propagates the updated statistics forward to the present. In this manuscript the mathematical structure and properties of the ESBKF are presented, and its utility is demonstrated on tracking an RSO in a GEO orbit with right-ascension and declination angle measurements from an observer satellite