# The Potential of LEO Satellite-Based Opportunistic Navigation for High Dynamic Applications
## 4 Performance Characterization
### Effect of Error Sources
#### 4.2.1 Measurement Noise  
We consider the thermal noise as the unique error source (i.e., we use perfect satellite orbits and clocks). We firstly characterize the quality of the Doppler shift measurements as well as the stability of their arrival time under the effect of thermal noise. Figure 10 shows the statistics of the Doppler shift measurement error (here free of receiver clock drift). The mean error is smaller than 1 Hz and the standard deviation is between 5 and 12.5 Hz, which varies inversely to the satellite power level of Table 2. Despite the strong Iridium signals, the Doppler shift noise is high when compared to the GNSS Doppler noise that is typically better than 1 Hz (\(1-\sigma\)). Indeed, since bursts are very short, a PLL requires a large loop bandwidth to converge within a short time, thus a small integration time that leads to high frequency noise at the PLL output. The averaging of the PLL frequency output over the last 0.5 ms of the burst tone allowed us to reduce this noise, as explained in Section 3.1. Determining the quality of the burst time of arrival (\(toa\)) is important, since the \(toa\) is used to compute the burst transmission time (i.e., \(toa-\tau\)), which is required in the calculation of the satellite position and velocity (6). Figure 11 shows the statistics of the \(toa\) error (here free of receiver clock error). The \(toa\) errors due to the thermal noise are different for the satellites. As such, they will not be compensated by the filter in the clock bias state. The \(toa\) error difference between any two satellites is smaller than 0.5 ms (mean error less than 0.4 ms, and standard deviation less than 0.04 ms). We now discuss the performance of the navigation filter. Table 3 illustrates the root mean square error (RMSE) for the position \(r\) and attitude \(\epsilon\) coordinates, as well as for the receiver clock bias \(\delta t\) and drift \(\dot{\delta t}\). It shows that the IMU/magnetometer/Iridium coupling provides a positioning with an error of a few hundred meters. Figure 10: Mean and standard deviation (std) of bursts Doppler shift errors. In Figure 12, we can see that the filter takes a few ten seconds to converge for the position and the time states, which is the time needed to accumulate sufficient Doppler shift measurements for the state observability, and then the estimations stay stable over time. The receiver clock bias is estimated with \(\sim\)35 ms (RMSE). This is a low accuracy compared to the time correction accuracy based on pseudorange measurements in GNSS systems (typically a few ten ns). For an Iridium satellite speed of \(\sim\)7.45 km/s, an error of 35 ms in burst transmission time leads to a satellite position error of \(\sim\)260 m. For a positioning based on Doppler shifts, this error distorts the predicted Doppler shift. The resulting Doppler residual degrades the navigation solution. Moreover, the scenario's short duration combined with relatively large initial state errors has led us to increase the confidence in the Doppler measurements (despite the relatively high noise) and to enlarge the system noise matrix in order to accelerate the convergence time of the filter. Nevertheless, this happened at the expense of a degraded navigation solution. Thus, in the presence of thermal noise only, the position error is due to the Doppler shift measurement noise and the measurement time stamp errors (toa errors), and is degraded due to the scenario's short duration. The roll and pitch angles are estimated with an error of \(\sim\)1.49\({}^{\circ}\) and \(\sim\)1.88\({}^{\circ}\), respectively, while the yaw angle error is \(\sim\)4.18\({}^{\circ}\). The roll is the most precise thanks to its better observation by the radial components of the magnetometer. The yaw observability is the weakest, since its value stays almost constant during the scenario. In addition, the use of magnetometer data in the observation model allowed for a faster convergence of the attitude (\(\sim\)15 s) than the position.