Lower Bounds on DOP

Abstract

Code phase Global Navigation Satellite System (GNSS) positioning performance is often described by the Geometric or Position Dilution of Precision (GDOP or PDOP), functions of the number of satellites employed in the solution and their geometry. This paper develops lower bounds to both metrics solely as functions of the number of satellites, effectively removing the added complexity caused by their locations in the sky, to allow users to assess how well their receivers are performing with respect to the best possible performance. Such bounds will be useful as receivers sub-select from the plethora of satellites available with multiple GNSS constellations. The bounds are initially developed for one constellation assuming that the satellites are at or above the horizon. Satellite constellations that essentially achieve the bounds are discussed, again with value toward the problem of satellite selection. The bounds are then extended to a non-zero mask angle and to multiple constellations.

GDOP Bounds for GNSS Augmented with Range Information

“…While integration of GNSS and non-GNSS sensors is not novel, here we show (from an optimal GDOP perspective) how such an external sensor impacts potential receiver performance and what role it plays in satellite selection. Specifically, we extend our results from [9], providing tight lower bounds to the GDOP when the GNSS is augmented by a single such measurement and present satellite constellations that achieve these bounds.…”

“…With this formulation a number of the DOP bound results in [9] generalize. Our use of color in this matrix will become obvious below.…”

“…A Brief Review of [9] Following [9] the GDOP under GNSS alone is determined by the elements of the symmetric matrix G T Γ −1 G with σ r → ∞ to account for no range sensor…”

“…A general lower bound for m satellites from one constellation is known [4], but does not restrict the satellites elevations to be above the horizon or above any fixed mask angle and, hence, is not achievable in terrestrial applications. In a recent paper [9], these authors were able to improve on this result, developing an achievable lower bound to GDOP for terrestrial applications…”

“…The constellations that achieve this bound consist of approximately 29% of the satellites at zenith and the remaining 71% "balanced" around the horizon (balance having a specific mathematical definition, see [9]). In that same work the bounds were extended to PDOP, to non-zero mask angle, and to multiple GNSS constellations.…”