An analysis on the application of algebraic geometry in Initial Orbit Determination problems

Abstract

Initial Orbit Determination (IOD) is a classical problem in astrodynamics. The space around Earth is crowded by a great many objects whose orbits are unknown, and the number of space debris is constantly increasing because of break-up events and collisions. Reconstructing the orbit of a body from observations allows us to create catalogs that are used to avoid collisions and program missions for debris removal. Also, comparing the observations of celestial bodies with predictions of their positions made based on our knowledge of the universe has been in the past, and is still today, one of the most effective means to make improvements in our cosmological model. In this work, a purely geometric solution to the angles-only IOD problem is analyzed, and its performance under various scenarios of observations is tested. The problem formulation is based on a re-parameterization of the orbit as a disk quadric, and relating the observations to the unknowns leads to a polynomial system that can be solved using tools from numerical algebraic geometry. This method is time-free and does not require any type of initialization. This makes it unaffected by the problems related to the estimate of the time-of-flight, that usually affects the accuracy of the solution. A similar approach may be used to analyze the performance of the solver when streaks are used, together with lines of sight, as inputs to the problem. Streaks on digital images form, together with the camera location, planes that are tangent to the orbit. This produces two different types of constraints, that can be written as polynomial equations. The accuracy and the robustness of the solver are decreased by the presence of streaks, but they remain a valid input when diversity in the observed directions guarantees the departure from the singular configuration of almost coplanar observations.