Tighter GDOP Bounds and their Use in Satellite Subset Selection

Abstract

GNSS receivers convert the measured pseudoranges from the visible and viable GNSS satellites into an estimate of the position and clock offset of the receiver. The accuracy of the resulting solution can be described statistically by its error covariance matrix; a common scalar reduction of this covariance that highlights the impact of satellite geometry is the GDOP (Geometric Dilution of Precision). In some instances, a GNSS receiver cannot process all of the visible satellites. For example, the issue might be that the receiver has limited computational power (perhaps a hardware or energy limit). Alternatively, the receiver might be using corrections from a ground-based augmentation system and the bandwidth of the correction channel is insufficient to provide information for all of the visible satellites. In such a case the question arises: If only m of the n visible satellites can be processed, which ones should they be? Since the GDOP is nonlinear and non-separable in the satellites’ locations in the sky, finding the best subset has combinatorial complexity. For example, if the receiver limits its attention to the 15 or so visible GPS satellites then a brute force comparison of all subsets to find the optimal subset is possible (whatever the value of m, if n = 15 then there are at most 6, 500 potential cases to check, well within modern computational capability). The advent of other GNSS constellations exacerbates this problem. For example, desiring to select 12 of 35 visible satellites (such as frequently occurs with GPS, GLONASS, and Galileo) brute force comparison is expensive. This question of selecting a subset of the possible satellites is not new to the navigation literature; multiple authors have described sub-optimal methods, usually greedy algorithms, for choosing the satellite subset. In prior works these authors developed a lower bound to GDOP for GNSS constellations purely as a function of the number of satellites employed. This simple bound shows the merit of the high and low (in elevation) satellites to the GDOP performance. This paper presents two new bounds that employ partial information on the satellites positions, notably information about their elevations. Not only does this provide tighter bounds, the results also suggest how to choose satellites for the subset selection problem. INTRODUCTION GNSS receivers convert the measured satellite pseudoranges into estimates of the receiver’s position and clock offset. A common implementation of the solution algorithm is an iterative, linearized least squares method. Assuming that pseudoranges from m satellites are measured, the direction cosines matrix G is formed and used to solve an overdetermined set of linear equations. Since the pseudoranges themselves are noisy, the resulting estimates are random variables. The accuracy of this solution can be described statistically by the error covariance matrix, equal to the inverse of GG scaled by the User Range Error [1]. Rather than considering the individual elements of this covariance matrix, it is common to consider a related scalar performance indicator, the Geometric Dilution of Precision (GDOP), equal to the square root of the trace of the inverse of GG; equivalently, this is the square root of the sum of the variances of the four estimates without the URE scaling. It is clear that the GDOP is a function of the satellite geometry and, since the satellites are constantly in motion, that the GDOP changes with both time and the user’s spatial location. To reduce this inherent complexity of GDOP, we think that an understanding of how small the GDOP can be (i.e. a lower bound) as a function of the number of satellites visible and some information on their sky locations, but without precise knowledge of the satellites’ orbits relative to the user, is of value. In earlier work [2] these authors developed a lower bound on GDOP which only assumed that the m satellites were above the horizon. The constellations that achieve this bound consist of satellites at the horizon and zenith. This current paper extends the knowledge of GDOP through the development of tighter bounds using additional information on the satellites’ locations. In a future with multiple, fully occupied GNSS constellations there could be situations in which there are too many satellites to use; examples might be related to limits on receiver hardware or power or the limited data bandwidth of a GBAS [3]. The satellite subset selection problem refers to this choosing of m satellites from n available (n > m). As the GDOP is a non-separable, non-linear performance metric, the selection of satellites for minimum GDOP has combinatorial complexity (basically, search over the ( n m ) possible subsets); in response many authors have considered sub-optimum, ad hoc selection methods. Our view is that the constellations that achieve the lower bounds on GDOP provide insight to guide this satellite selection process. We present one such approach using the new bounds developed in this paper. GDOP PRELIMINARIES The direction cosines matrix for m satellites in three dimensions using a local East, North, and Up coordinate frame is G =  e1 n1 u1 1 e2 n2 u2 1 .. .. em nm um 1  in which each triplet (ek, nk, uk) is the unit vector pointing toward the k th satellite from the receiver’s location. The geometric dilution of precision is defined as GDOP = √ trace { (GTG) −1 } Investigating the best possible GNSS satellite constellation with respect to GDOP is not a new problem. For example: • The best constellation of 4 satellites, with reference to optimising the tetrahedron formed by their locations, has been considered by multiple authors (see, e.g. [4]). • The best constellations of six satellites is described in [5]; the case of five satellites from two GNSS constellations is considered in [6]. • A general lower bound on GDOP for m satellites from one constellation is known [7] GDOP ≥ √ 10 m = 3.162 √ m but does not restrict the satellites locations in any way. • It is known that GDOP is a monotonically decreasing function of the set of satellites employed in the solution [8]. Specifically, adding an additional satellite reduces the GDOP independent of the sky location of this extra satellite, even if it lies directly atop one of the original satellites. This monotonicity extends to multiple constellations except when the new satellite is the first of its constellation [9]. In a recent paper [2] these authors were able to develop an achievable lower bound to GDOP for terrestrial applications (i.e. satellites restricted to be above the horizon) purely as a function of the number of satellites, m (the extension to a non-zero mask angle was also considered in that paper). The development included a series of steps, the penultimate being that the GDOP could be lower bounded by GDOP ≥ √ 4 m− d + m+ d dm− f2 (1) in which m is the number of satellites used in the solution and d and f are defined in terms of the up components of the unit vectors for each employed satellite