Geometric Solution to the Angles-Only Initial Orbit Determination Problem

TL;DR

This paper introduces a novel geometric method for initial orbit determination using bearings, leveraging algebraic geometry to recover orbital shape and orientation without initial guesses or propagation, especially effective with five bearings.

Contribution

It presents a purely geometric algebraic approach to orbit determination that requires no initial guess and can work with fewer observations if the orbit is circular.

Findings

Method successfully recovers orbits from bearings without initial guesses.

Five bearings suffice for general orbit recovery, three for circular orbits.

Approach works with over-determined systems and various observation scenarios.

Abstract

Initial orbit determination (IOD) from line-of-sight (i.e., bearing) measurements is a classical problem in astrodynamics. Indeed, there are many well-established methods for performing the IOD task when given three line-of-sight observations at known times. Interestingly, and in contrast to these existing methods, concepts from algebraic geometry may be used to produce a purely geometric solution. This idea is based on the fact that bearings from observers in general position may be used to directly recover the shape and orientation of a three-dimensional conic (e.g., a Keplerian orbit) without any need for knowledge of time. In general, it is shown that five bearings at unknown times are sufficient to recover the orbit -- without the use of any type of initial guess and without the need to propagate the orbit. Three bearings are sufficient for purely geometric IOD if the orbit is known to be (approximately) circular. The method has been tested over different scenarios, including one where extra observations make the system of equations over-determined.