Convergence of Iterative Descent Algorithms for LEO-PNT

Abstract

Positioning Navigation and Timing (PNT) from Low Earth Orbit (LEO) has been gaining momentum in the last couple of years. LEO-PNT can serve as an alternative to current Global Navigation Satellite Systems (GNSS) or as an addition to the current system using a multi-layer approach. The aim of launching satellites into a lower orbit is to mitigate the shortcomings of current GNSS. When moving towards a lower orbit, algorithmic challenges that relate to the initial point estimate for Iterative Descent (ID) algorithms arise. In this paper, we analyze the convergence behavior of ID algorithms, focusing on Steepest Descent, Gauss-Newton, Trust Region, and Levenberg-Marquardt, when performing positioning using only LEO satellites, starting from an initial estimate at the center of the Earth to emulate a cold start scenario. We employ a Monte-Carlo (MC) simulation and multiple proposed LEO-PNT constellations to verify the convergence behavior of the algorithms. Among the evaluated methods, Trust-Region converges most reliably. The commonly used Gauss-Newton method often fails. Moreover, Levenberg–Marquardt is more robust but does not reach a perfect convergence rate. While simple, Steepest Descent requires the most iterations.