# Full Real-Time Positioning and Attitude System Based on GNSS-RTK Technology
## 4 Results
### Accuracy of Measurements in a Dynamic Configuration  
To perform this test, three antennas were installed on the roof of a car, as shown in Figure 15.  
The distances between the antennas of receivers 1-2 and 1-3 are 1.5 and 1.1 m, respectively. Vectors formed by receivers 1-2 and 1-3 form an angle of 90\({}^{\circ}\). Data were recorded on a national road segment of about 4 km with good visibility of the sky and without obstacles that may block the signals (bridges crossing the road, road indicators, etc.).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Configuration** & **Receiver Antenna Distance 1-2 [m]** & **Receiver Antenna Distance 1–3 [m]** & **Angle 2-1 and 3-1 [deg]** \\ \hline A & 4.6 & 4.6 & 90 \\ B & 4.6 & 4.6 & 60 \\ C & 4.6 & 4.6 & 40 \\ D & 4.6 & 4.6 & 19 \\ \hline \end{tabular}
\end{table}
Table 2: Configurations of the antennas for the static robustness tests.  
Figure 14: Average attitude of the non-orthogonal antenna configurations.  
Figure 15: Antenna setup on the vehicle.  
Validating the system on a moving solid has the difficulty of having a reliable reference with which to compare the data acquired and computed. For this reason, the variability of the measurements was analyzed in a segment of the road where the car follows a straight path at constant speed.  
Figure 16 compares the yaw angle computed by the attitude system with the vehicle trajectory angle, which is computed as follows:  
\[\begin{split} Yaw_{t=i}=atan2((lon_{i+1}-lon_{i-1})lonX_{i},( lat_{i+1}-lat_{i-1})latY_{i}),\end{split} \tag{1}\]  
where \(lat_{i},~{}lon_{i}\) are the latitude and longitude at \(t=i\), and \(LatY_{i},~{} LonX_{i}\) are the latitude and longitude differentials at \(t=i\) transformed into Euclidean distances with the following transformation [34]:  
\[\begin{split} lonX_{i}=111,412.84\cos(lat_{i})-93.5\cos(3 lat_{i} )+0.118\cos(5 lat_{i})\\ latY_{i}=111,132.92-559.82\cos(2 lat_{i})+1.175\cos(4 lat_{i})-0.0023 cos(6 lat_{i})).\end{split} \tag{2}\]  
A good correspondence is observed between the angle of the vehicle's trajectory and the yaw angle, even at the two roundabouts (seconds 10 and 170).  
Figure 17 shows the pitch and roll angles and the height during the same time interval as in Figure 16. In most of the trajectory, the vehicle is on a light negative slope, losing about 35 m in 4 km, a fact that is observed in the pitch angle, which has a negative value, except in two small sections in which the slope of the road is positive. The vehicle roll angle always has a positive angle around \(1^{\circ}\); this value may be due to an error in the manual installation of the antenna, a non-symmetrical distribution of the weights in the vehicle, or the inclination of the roadway by a longitudinal drainage.  
Table 3 shows the standard deviations on the three attitude angles for the sample time interval between 90 and 130 s, since it corresponds to a segment of the road with constant slope and no changes in the yaw angle.  
Taking into account the distance between antennas in the vehicle and the irregularities of the path, the results show very good performance and precision. However, these results are a bit larger than those shown in Figure 7, where the system was static.  
Figure 16: Speed and yaw angle during the dynamic testing.