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 a high accuracy. Over the years, different approaches have been presented such as filtering methods (for example, Extended Kalman Filter), differential algebra or solving Lambert's problem. In this work, we consider a setting of three 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, where each radar is 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. Thus, our method can be understood as an extension of trilateration leveraging more recent technology and incorporating additional data. We formulate the problem as a Maximum Likelihood Estimator (MLE), which for some number of observations is asymptotically unbiased and asymptotically efficient. Through numerical experiments, we demonstrate that our method attains the same accuracy as the trilateration method for the same number of measurements and offers an alternative and generalization, returning a more accurate estimation of the satellite's state vector, as the number of available measurements increases.