Solutions and analysis of multivariate polynomial systems for geolocation and initial orbit determination

Abstract

With the birth of Space Situational Awareness from the launch of the Russian satellite, Sputnik I, humanity has found it necessary to develop methods to determine the navigational state of man-made objects orbiting the Earth for avoidance and tracking purposes. With the ever-increasing number of unknown objects, both ground-based and orbiting, the necessity of these methods has never been greater. This work seeks to further the fields of Spacecraft Navigation and Space Situation Awareness with the development of three novel solutions to the geolocation and initial orbit determination problems. The first method is a geolocation technique utilizing both time-based and frequency-based measurements from the signal of a ground-based Radio Frequency (RF) transmitter. Our second method derives two initial orbit determination techniques using concurrent TDOA and range-rate measurements from the signal of an orbiting RF transmitter. The final method discussed in this work is an initial relative orbit determination method using range-rate measurements and the linearized Clohessy Wiltshire dynamics. Each method derives its solution from a polynomial system solved using an algebraic geometry technique called homotopy continuation theory. Each method is verified using simulation results for multiple scenarios.