Re-entry vehicle tracking observability and theoretical bound

Abstract

This article deals with theoretical bounds and observability in ballistic re-entry vehicle tracking; theoretical and simulation results are presented. One essential characteristic of this trajectory is the deceleration of the vehicle when it reaches dense atmospheric layers. The intensity of the phenomenon is proportional to a scalar, called the ballistic coefficient. This leads to highly non-linear dynamics. We have compared tracking data processing techniques like extended Kalman filter (EKF) and particle filter to the posterior Cramer-Rao bound (PCRB) in order to confirm the exactness of this very bound and to evaluate at the same time the filters' performance. The observability problem of the trajectory is mostly the observability of the ballistic coefficient during the re-entry phase. Thus we have gradually studied its observability using a simple a priori random walk model, from a constant to a complex Allen oscillatory ballistic profile for the trajectory simulation. The accuracy of the particle filter and the exactness of the bound have been confirmed. In order to understand the important parameters of the bound, we explain the evolution of the observability during the re-entry phase using the Fisher information matrix, the inverse of the Cramer-Rao bound (CRB). We give an analytical expression of the CRB versus time for simple observation cases, using Cauchy-Binet formula for matrix determinants.