Undifferenced array PPP: a model for GNSS integrated positioning and attitude determination

Array Precise Point Positioning (PPP) is a concept that utilizes GNSS observations from multiple antennas in an array of known geometry to realize improved GNSS parameter estimation. This paper proposes a new model of Array PPP entirely based on undifferenced observations, called undifferenced Array PPP model. First, it represents the attitude as a quaternion and formulates how to derive the attitude from the GNSS undifferenced observations in the antenna array. Then, the estimated undifferenced ambiguities are extracted from float solutions, constructed as double-differenced ambiguities between the antennas, and resolved to integers. Finally, it applies Lie theory to keep the rotation parameters always lying on a 3D unit-sphere manifold, indicated as Special Orthogonal group SO3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO\left(3\right)$$\end{document}. The undifferenced Array PPP model has equivalent performances with the mix Array PPP model based on mix observations containing undifferenced and double-differenced GNSS observations. The difference is that the mix Array PPP model considers the cross-correlation between different antennas in its stochastic model, while the undifferenced Array PPP model incorporates the cross-correlation in its functional model. The independence of the satellite observations in the undifferenced Array PPP model brings benefits in adjusting available satellites, frequency bands and error detection, resulting in better performances in the dynamic scenarios under complex environments. Last but not the least, the new model represents the attitude as quaternion and estimates its perturbations in the measurement update phase of Kalman filter, having the potential to achieve tight integration with Inertial Measurement Unit (IMU) and then increase the continuity and availability of solutions, especially in a GNSS-denied environment.