This tutorial reviews algorithms which turn measured ranges into position solutions. From their basic mathematical principles, we relate and compare relevant aspects of these algorithms. Special attention is given to the direct (non-iterative) algorithms, which are frequently applied in indoor positioning. Most of them are shown to be essentially the same, as they can be related through applying different weighting schemes. This tutorial is intended as a useful guide to help researchers and system designers evaluate and select appropriate range-based positioning algorithms for their applications at hand.

This contribution can be seen as an attempt to apply a rigorous method for variance components in a straightforward manner directly to GPS observables. Least-squares variance component estimation is adopted to assess the noise characteristics of GPS observables using the geometry-free observation model. The method can be applied to GPS observables or GNSS observables in general, even when the navigation message is not available. A realistic stochastic model of GPS observables takes into account the individual variances of different observation types, the satellite elevation dependence of GPS observables precision, the correlation between different observation types, and the time correlation of the observables. The mathematical formulation of all such issues is presented. The numerical evidence, obtained from real GPS data, consequently concludes that these are important issues in order to properly construct the covariance matrix of the GPS observables. Satellite elevation dependence of variance is found t...

The underlying motivation of variance component estimation (VCE) is to construct an adequate variance-covariance (VC) matrix for the processing of global positioning system (GPS) code and phase data. Prior to developing a full procedure for VCE in precise GPS positioning, an experimental inventory has been carried out to get acquainted with the key issues in the stochastics of the GPS range observables, and hence with the structure of the eventual VC matrix. The experiences of this inventory are shown in this contribution aspect by aspect, using code and phase data of a geodetic dual frequency receiver pair on a zero baseline at a 1 s sampling interval. The paper treats elevation dependent variances, covariances between channels and between observation types, as well as time correlation. For each aspect easy-to-compute formulas are derived and attention is paid to the redundancy necessary to obtain sensible results. For the receiver at hand, elevation dependence of variance, correlation between observatio...

Precise Point Positioning (PPP) is a popular Global Positioning System (GPS) processing strategy, thanks to its high precision without requiring additional GPS infrastructure. Single-Frequency PPP (SF-PPP) takes this one step further by no longer relying on expensive dual-frequency GPS receivers, while maintaining a relatively high positioning accuracy. The use of GPS-only SF-PPP for lane identification and mapping on a motorway has previously been demonstrated successfully. However, the performance was shown to depend strongly on the number of available satellites, limiting the application of SF-PPP to relatively open areas. We investigate whether the applicability can be extended by moving from using only GPS to using multiple Global Navigation Satellite Systems (GNSS). Next to GPS, the Russian GLONASS system is at present the only fully functional GNSS and was selected for this reason. We introduce our approach to multi-GNSS SF-PPP and demonstrate its performance by means of several experiments. Results show that multi-GNSS SF-PPP indeed outperforms GPS-only SF-PPP in particular in case of reduced sky visibility.

Stochastic properties of GNSS range measurements can accurately be estimated using a geometry-free short and zero baseline analysis method. This method is now applied to dual-frequency measurements from a new field campaign. Results are presented for the new GPS L5Q and GIOVE E5aQ wideband signals, in addition to the GPS L1 C/A and GIOVE E1B signals. As expected, the results clearly show the high precision of the new signals, but they also show, rather unexpectedly, significant, slowly changing variations in the pseudorange code measurements that are probably a result of strong multipath interference on the data. Carrier phase measurement noise is assessed on both frequencies, and finally successful mixed GPS-GIOVE double difference ambiguity resolution is demonstrated.