In the last few posts, I have been focusing on the relative position solution between two receivers where both are moving but fixed relative to each other rather than the more typical scenario where one receiver is fixed (the base) and the other is moving (the rover). As described in an earlier post, I do this because it makes it possible to accurately measure the error in the solution when we don’t have an absolute trajectory reference to compare to.
Normally, though, from a functional viewpoint, we are more interested in the second case where one receiver is fixed and the other is moving. In this post I will spend a little time looking at how we can evaluate the results for that scenario.
Since we don’t know what our exact trajectory was when we collected the data, we can not measure our solution error directly like we can in the first case. However, we can look for consistency between multiple solutions calculated for the same rover data, but with varying input configurations and differing base data and satellites. The solution should not vary by more than our error tolerance no matter how much we change the input configurations. If we believe a solution is accurate to a couple of centimeters then it would be very disturbing to find it moved 50 cm if we used the same rover data but adjusted something in the input configuration, satellites, or base data, particularly if the differences occur for epochs with fixed solutions. We cannot know which of the two solutions is incorrect but we do know that at least one of them must have large undetected errors in it, which means we really can’t trust either solution. Of course, if the two solutions are equal it does not guarantee that they are correct, they may just both be wrong in the same way. However, the more we can vary the input conditions and still get the same answer, the more confident we will start to have in the solution.
To test the consistency of solution for the previous data set, I processed the data from the two receivers in a different way. Instead of measuring the relative distance between the two receivers, I measured the distance between each receiver and two independent base stations, giving me a total of four solutions, two for each Ublox receiver Since the two base receivers are in different locations the solutions will be different, but if we ask RTKLIB to calculate absolute position rather than relative distance then we can compare the two directly. By changing the out-solformat parameter in the input config file from “enu” to “xyz”, RTKLIB will add the absolute location of the base station to the solution, converting it from relative to absolute. With this change, the two solutions calculated from different base stations should be identical.
For the two base receivers, I used data from two nearby stations in the NOAA CORS network, TMG0 which is about 5 km away from where I collected the data and ZDV1 which is about 13 km away. TMG0 data includes Glonass satellites, ZDV does not so this also helps differentiate the two solutions by giving them different but overlapping satellite sets. To further increase the difference between the two satellite sets, I changed the minimum elevation variables in RTKLIB for the TMG station from 15 degrees to 10 degrees while leaving them at 15 degrees for the ZDV station. This will add the low elevation satellites to the TMG solution.
Now we can compare the two solutions for each Ublox receiver, each calculated with different base stations and different sets of satellites. If everything is working properly they should be identical within our expected error tolerance. Since our baseline distances between base and rover have increased from roughly 30 cm in the previous analysis to 5-13 km in this case we will expect the errors to be somewhat larger than they were before.
The plots below show the results of this experiment. The plots on the left are from the Ebay receiver and the plots on the right are from the CSG receiver. The first row of plots show the ground tracks for both solutions overlayed on top of each other, the second row shows position in x, y, and z directions, again for both solutions overlayed on top of each other. The x and y directions are indistinguishable at this scale but we do see some error in the z direction. The third row of plots shows the difference between the two solutions. We can interpret these differences as a lower bound for our error. As we would expect with the longer baselines, the errors are somewhat larger than before, but at least the x and y errors are generally still quite reasonable, most of them falling between +/- 2 cm for epochs with fixed solutions (green). The z errors are somewhat larger but fortunately we usually care less about this direction. I have made no attempt to reduce the ephemeris errors, either through SBAS or precise orbit files. Presumably, at these longer baselines, doing so would help reduce the errors.
So, how do we interpret these results? I believe this analysis is consistent with the previous analysis. It is not as conclusive since we have no absolute error measurement, but since the solutions are more similar in format to how they would be done in practice, it should give us more overall confidence in our results.
Maybe more important, it also provides a baseline for using similar consistency analysis on other data sets where we often won’t have the luxury of having two receivers tracking the same object.
Let me add a few words to help put these results in context in case you are comparing them with other data sets. This data was all taken in very open skies with fairly low velocities (<6 m/sec) so it is relatively easy data to get a good solution. It will be more difficult to get similar results for data taken with obstructions from trees or buildings, or that includes high velocities or accelerations or high vibration. In particular, there were very few cycle slips in this data. None of the modifications described here will help if the data has a large number of cycle slips.
Also remember that this data was taken with very low cost hardware (the Ebay receiver cost less than $30 for receiver and antenna combined) so the data will be lower quality relative to data taken under similar conditions with more expensive hardware, especially higher quality antennas.
That is the goal of my project, though, to get precise, reliable positioning with ultra-low cost hardware under benign environmental conditions.