Doppler Positioning with LEO Satellites Using Unscented Kalman Filter

Doppler positioning with recent low earth orbit (LEO) constellations is a potential alternative to the Global Navigation Satellite System (GNSS). While the extended Kalman filter is commonly used in GNSS for position estimation due to its computational cost efficiency and accuracy, this paper proposes the use of the unscented Kalman filter for Doppler positioning due to its improved approximation accuracy of the probability distribution. In this paper, we experimentally demonstrate the approximation accuracy of the joint probability distribution for the states and measurements of each filter. This is because nonlinear filters based on the Kalman filter can be viewed as tools that approximate the evolution and measurement update of the probability distribution for the state vector. The first-order approximation, which results in the extended Kalman filter, becomes inaccurate under the uncertainty of the state vector, that is, with a large estimation error covariance matrix. In contrast, the unscented transformation, which leads to the unscented Kalman filter, achieves high accuracy. Finally, we numerically demonstrate Doppler positioning using the orbit information of over 5,000 Starlink satellites to compare the accuracy of each nonlinear filter. The findings of this study are summarized as follows: (1) the performance of the extended Kalman filter for Doppler positioning with recent LEO constellations can degrade due to poor approximation accuracy of the probability distribution, which eventually can cause filter divergence, and (2) the unscented Kalman filter can achieve much better accuracy than the extended Kalman filter while preventing filter divergence. The accuracy of the unscented Kalman filter for Doppler positioning can be within a few meters, which is almost equivalent to single-point positioning using GPS L1C/A.