Generalizing Trilateration: Approximate Maximum Likelihood Estimator for Initial Orbit Determination in Low-Earth Orbit

Abstract

With the increase in the number of active satellites and space debris in orbit, the problem of initial orbit determination (IOD) becomes increasingly important, demanding high accuracy. In this work, we consider a setting of monostatic radars, where all available measurements are taken approximately at the same instant. This follows a similar setting as trilateration, a state-of-the-art approach, which considers three monostatic radars, each one able to obtain a single measurement of range and range-rate. Differently, and due to advances in Multiple-Input Multiple-Output (MIMO) radars, we assume that each location is able to obtain a larger set of range, angle and Doppler shift measurements. We formulate the problem as a Maximum Likelihood Estimator, which is asymptotically unbiased and asymptotically efficient. Through numerical experiments, we demonstrate that, for the same number of measurements, our method attains the same accuracy as the trilateration method as well as the batch least squares, which can also be used in this same scenario of multiple measurements collected at approximately the same instant. We show that, as the number of measurements increases (due to more available measurements per location or by considering more radars), the estimation error of our approach decreases. Additionally, our approach demonstrates higher robustness than these two other approaches when the data is corrupted by higher levels of noise. Therefore, we conclude that our approach can be understood as an extension of the trilateration method for an arbitrary number of measurements and/or locations, returning a more accurate estimation of the satellite’s state vector.