# VHF Omnidirectional Range (VOR) Experimental Positioning for Stratospheric Vehicles
## 3 Results
### SDR Data Positioning  
The SDR data offered several advantages compared to the COTS receiver:  
* Increased sample rate of the VOR data;
* Concurrent reception of more than one station;
* Tunable receiver settings. These advantages proved decisive for the position estimation-more samples allow for them to be filtered via an average, up to an arbitrarily high precision. The reception of multiple station at the same time on the other hand made it possible to compare the radial reading from different stations at the exact same time, greatly increasing the accuracy. Contrary to the COTS receiver, it was necessary to demodulate the raw signal during the analysis, but this allowed for the fine tuning of the receiver settings and temperature shift compensation of the local oscillator. When more than two stations are received, more than one interception point is identified when the radials are combined. It is therefore necessary to combine the information from the different interception points in order to get the position estimation. It is important to point out how the arrangement of the radials greatly influences the precision of the position measurement: two orthogonal radials produce the most precise measurement, and the precision increases as the distance to the receivers reduces. On the other hand, two parallel radials provide no usable position at all. As such, different weights can be given to the intersection of different radials in a weighted average calculation for the position estimation. Shown in Figure 4 is an example of the estimation with three VOR stations present. Two High Reliability Points (HRP) are present, while the combination of Station 1 and Station 3, which are almost at 180 degrees from each other, form a Low Reliability Point (LRP); in fact, a small error in the radial of STA1 or STA3 will cause a great movement of the LRP point. The final position estimation is the weighted average of the three points, with the LRP having a low weight. The weight of each intersection is assigned based on the geometrical properties of the arrangement between the two stations. A small, fixed error of \(\pm\delta\theta\) is assigned to each the radial reading as shown in Figure 5. The four intersection points define a quadrilateral whose area is used as a measurement quality of the main intersection point-the higher the area, the lower is the reliability and quality of the intersection. Using the area as a  
Figure 4: Representation of multiple intersections from three different VOR stations. measurement of quality is useful for two main reasons: (1) the further the receiver is from the stations the bigger the area for a given radial, so that the reliability decreases with distance, and (2) an interception at low angle of incidence, i.e., with the radials close to parallel, will give an higher area and thus lower reliability. In fact, close-to-parallel radials provide little information on the position, as small errors on the radial translate to a considerable movement of the intersection point. Intersections with a very low reliability are discarded altogether, with the threshold area being determined experimentally. This was necessary because stations very far away (more than 400 km) only negatively contributed to the position determination. Using this system, a total of 350 positions were estimated during the stratospheric balloon path, which are shown in Figure 6 compared to the GPS-recorded track. Figure 5: Geometrical representation of the uncertainty assigned to a particular radial intersection arrangement. Figure 6: VOR-based track from SDR data compared to the GPS track. This method applied to the stratospheric data reaches a mean error of 4.04 km with a standard deviation of 2.76 km--a clear improvement over the COTS receiver--mainly due to the availability of multiple concurrent VOR stations and more measurements. Figure 7 shows the error trend throughout the stratospheric flight, above the SSV.