# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 4 Performance Characterization
### Comparing with an INS  
In many applications, the INS is typically the only backup navigation solution when the GNSS is not available. We compare the obtained position and attitude performances of the SOP-based filter (of Section 2) to those obtained with an INS based on a better quality IMU (having the specification given in Table 7), which is updated by the magnetometers of Table 1. The numerical simulations of the INS solution were carried out under the same user scenario, and by considering a perfect initialization of the position, velocity and attitude states.  
The results are reported in Table 8 with, for comparison, those of the SOP-based filter obtained for best and intermediate cases of the errors. Thanks to the better IMU performance and the update by the magnetometers, the INS outperforms the SOP-based filter in terms of the attitude estimation in both best and intermediate cases, at a potentially higher cost.  
On the other hand, if we assume accurate satellite orbits and satellite clock drift lower than 1 ppb, the positioning performances of the SOP-based filter are close to those of the best case (of Table 3). In this case, the positioning quality of the SOP-based filter becomes superior to the INS.  
Figure 17 provides the arrival position error in the \(x\)-\(y\) plane (X: along range, Y: cross range) for the SOP-based filter (best and intermediate cases), as well as the INS solution. It shows that the 95% ellipse of position error of the SOP-based filter of the best case is almost inside the 50% ellipse of the INS (Figure 17d).  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Accelerometer** & **Gyroscope** \\ \hline Noise & 0.1 m/s/\(\surd\) (h) & 0.25 deg/\(\surd\) (h) \\ \hline Bias & 10 mg (0.1 m/s\({}^{2}\)) & 50 deg/h \\ \hline Sensitivity & 1700 ppm & 1000 ppm \\ \hline Misalignment & 0.1 deg & 0.1 deg \\ \hline \hline \end{tabular}
\end{table}
Table 7: A tactical-grade sensor error specification (\(1-\sigma\)).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **X** & **Y** & **Z** \\ \hline \(\Delta\tau\) (m)—INS & 181.78 & 238.02 & 182.312 \\ \hline \(\Delta\tau\) (m)—SOP-based (best) & 128.84 & 95.90 & 166.60 \\ \hline \(\Delta\tau\) (m)—SOP-based (intermediate) & 1512.89 & 1232.75 & 1666.89 \\ \hline  & Roll & Pitch & Yaw \\ \hline \(\Delta\epsilon\) (deg)—INS & 0.45 & 0.27 & 0.63 \\ \hline \(\Delta\epsilon\) (deg)—SOP-based (best) & 1.49 & 1.88 & 4.18 \\ \hline \(\Delta\epsilon\) (deg)—SOP-based (intermediate) & 1.63 & 2.20 & 4.85 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Root mean square error for position and attitude, comparing the SOP-based filter with an INS based on a tactical-grade IMU and magnetometers.  
The Table 9 gives the semi-major (a) and the semi-minor (b) axes of the 50% and 95% ellipses of the arrival position error. We can see that the INS, based on a better quality IMU and the same magnetometers, is more accurate than the SOP-based filter using large error satellite orbits. The SOP-based filter, however, outperforms the INS solution if accurate orbits of the LEO SV are available. Furthermore, under longer scenarios its performance remains delimited over time, whereas the performance of the INS will drift more. In consequence, the filter based on LEO SV Doppler shift, consumer-grade IMU and magnetometers becomes attractive in a GNSS-denied environment.  
Figure 17: The 50% and 95% ellipses of the arrival position error. (**a**) INS; (**b**) best performance of the SOP-based filter; (**c**) intermediate performance of the SOP-based filter, (**d**) INS (green) versus best performance of the SOP-based filter (blue).