Single-Satellite Doppler Localization with Law of Cosines (LOC)

Modern day localization requires multiple satellites in orbits, and relies on ranging capabilities which may not be available in most proximity flight radios that are used to explore other planetary bodies such as Mars or Moon. The key results of this paper are: 1. A novel relative positioning scheme that uses Doppler measurements and the principle of the Law of Cosines (LOC)to localize a user with as few as one orbiter 2. The concept of “pseudo-pseudorange” that embeds the satellite's velocity vector error into the pseudorange expressions of the user and the reference station, thereby allowing the LOC scheme to cancel out or to greatly attenuate the velocity error in the localization calculations In this analysis, the Lunar Relay Satellite (LRS)was used as the orbiter, with the reference station and the user located near the Lunar South Pole. Multiple Doppler measurements by the stationary user and the reference station at different time points from one satellite can be made over the satellite's pass, with the measurements in each time point processed and denoted as from a separate, faux satellite. The use of the surface constraint assumption was implemented with this scheme; using the knowledge of the altitude of the user as a constraint. Satellite's ephemeris and velocity, and user's and reference station's Doppler measurement errors were modeled as Gaussian variables, and embedded in Monte Carlo simulations of the scheme to investigate its sensitivity with respective to different kinds of errors. With only two Doppler measurements, LOC exhibited root mean square (RMS)3D positional errors of about 22 meters in Monte Carlo simulations. With an optimal measurement window size and a larger number of measurements, the RMS error improved to under 10 meters. The algorithm was also found to be fairly resilient to satellite velocity error due to the error mitigating effects in the LOC processing of the pseudo-pseudorange data type. A sensitivity analysis was performed to understand the effects of errors in the surface constraint, showing that overall position error increased linearly with surface constraint error. An analysis was also performed to reveal the relationship between the distance between the user and the reference station; a distance of up to 100 km only lead to an increase of 10 meters in RMS 3D position error. Ultimately, the LOC scheme provides localization with a minimal navigation infrastructure that relaxes hardware requirements and uses a small number of navigation nodes (as small as one).