KiwiSDR GNSS position solutions using an extended Kalman filter

This is an update to this blog post, see also https://github.com/jks-prv/Beagle_SDR_GPS/issues/147. References for using Kalman filters for GNSS positioning are, e.g., DST-Group-TR-3260.pdf and these proceedings from ION GPSGNSS 2003.

The plots below show time series of variations in ECEF X,Y,Z coordinates, the oscillator correction, and the number of available satellites.

SPP - single point position; EFK - extended Kalman filter

The process noise covariances are set to 5e-5 for the X,Y,Z coordinates (KiwiSDRs are not expected to move) and the covariances for the time and the oscillator correction are build from Allan variance parameters as described in a497248. It would be interesting to measure the Allan variance of the used oscillator.

The advantages of using an extended Kalman filter include:

• Once a position solution has been obtained, the Kalman filter can be updated with fewer than four satellites: this will improve the availability of those KiwiSDRs with poor GNSS reception for TDoA work.
• The residuals per link in the Kalman filter update are used to exclude satellites with bad time measurements.
• Large deviations from the nominal position due to bad satellite geometries are avoided.

New version of TDoA code available

A version of the 1st idea mentioned in the last blog post is now implemented in the latest version of the TDoA code and thanks to John it is available as an option in the KiwiSDR TDoA service.

The changes in the new TDoA algorithm are the following:

(1) Sanity checks on the recorded IQ wav files

Instead of aborting processing when a single file is found to be bad, stations are now excluded. If more than 1 good station is left, the processing proceeds without the excluded stations. The following checks are performed:

• check for recent GNSS position fixes
• validity check on then GNSS timestamps
• check if |IQ|!=0
• check if the overlap of all wav files is long enough (>10 sec)

(2) Finding clusters in the cross-correlation peaks

For a given pair of stations, peaks in the cross-correlations are found as before. Then these peaks are assembles into up to four clusters using an adaptive clusterization algorithm. It starts by assuming that there are four clusters. If the four clusters are not valid, the algorithm starts again, this time assuming that there are three clusters. When all clusters are valid the algorithm terminates.

The used clusterization algorithm is iterative:

• It starts by assuming that there are N clusters and puts N+1 boundaries b_i at equidistant places between the minimum and maximum time of the cross correlation peaks.
• Then it recomputes the boundaries as follows: Let t_i,i+1 be the average of the mean of the cross correlation peaks between b_i and b_i+1. The boundaries are iteratively updated as b_i=(t_i-1,i+t_i,i+1)/2 until they become constant.
• A cluster is called valid if the distribution of entries in it is not compatible with a random distribution.
• If all clusters are valid or if there is only one cluster (N=1) then the clusterization algorithm terminates.

(3) Selection of those clusters which optimally fulfill the consistency equations

• If there are three stations, cross-correlation times of the same signal fulfill (12)-(13)-(23)=0.
• When there are four stations, there are more of these consistency equations, e.g., (12)-(14)-(24)=0, (23)-(24)-(34)=0 etc.
• For all combinations of clusters all consistency equation are evaluated. Each equation provides the deviation from 0 in terms of number of sigmas (nsigma) using the mean and RMS of the clusters.
• In the end the combination of clusters is selected which for all consistency equations results in the smallest nsigma
• If the nsigma value in any given equation is greater than 3, stations are iteratively excluded until 3 stations are left. The combination of excluded stations which brings max(nsigma) below 3 is not used later on for making the TDoA maps.