## 4 Simulation Results
### Open Sky with Seven SVs  
The sky plot during the simulated trajectory is shown in Figure 7, alongside the corresponding received power from the satellites. At the 300 s, the power of SV 01 starts to decrease with a step of 1 dB/s for a 25 s and then it increases again to reach its nominal value. From 600 s until 900 s, partial outages of SV 11 for different durations start to occur. The position solution of a small part of the trajectory is given in Figure 8, where it can be seen that the TCAPLL has a smoothing effect on the solution compared to the stand-alone GPS. This smoothing effect is the result of the complementary error characteristics of the GPS, which have a long term stability navigation solution, and of the INS, which have a short term stability solution. \begin{table}
\begin{tabular}{c c c c} \hline  & & **Gyro** & **Accelerometer** \\ \hline \multicolumn{2}{c}{White noise ( \(\eta\)) (1–\(\sigma\))} & 226 deg/h/\(\sqrt{\text{Hz}}\) & 0.28 mg/\(\sqrt{\text{Hz}}\) \\
1. & Turn-on bias (\(b\)) (1–\(\sigma\)) & 3,600 deg/h & 50 mg \\ Bias drift GM correlation time (1/\(\beta_{d}\)) [s] & 350 & 250 \\ \multicolumn{2}{c}{Bias drift GM noise \(\delta b\) (1–\(\sigma\))} & 180 deg/h/\(\sqrt{\text{Hz}}\) & 0.7 mg/\(\sqrt{\text{Hz}}\) \\
2. & SF GM correlation time (1/\(\beta_{SF}\)) [s] & 18,000 & 18,000 \\ \multicolumn{2}{c}{SF GM noise (\(SF\)) (1–\(\sigma\)) [PPM]} & 0.01 & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulated MEMS inertial sensor errors. Figure 6: TCAPLL simulation testing platform. The TCAPLL provides a better solution; this is shown in Tables 2 and 3 where the standard deviation and mean of the position, velocity and attitude errors are presented. The TCAPLL reduces the position and velocity errors by roughly 70% compared to the stand-alone GPS. The uncertainty of the aiding frequency of the feed-forward component is shown in Figure 9, where it can be seen that it is smaller than 4 Hz for all satellites. Comparing to Figure 4, the quality of the aiding is good enough for the PLL to maintain tracking. \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Velocity Error [m/s]**} & \multicolumn{2}{c}{**Position Error [m]**} \\ \hline  & GPS & TCAPLL & GPS & TCAPLL \\ \(\sigma\) & 1.046 & 0.284 & 7.998 & 2.556 \\ mean & 0.034 & 0.017 & 3.166 & 0.859 \\ \hline \end{tabular}
\end{table} 
Table 2: Standard deviation and mean of the GPS and TCAPLL velocity and position errors. Figure 8: Comparison of the position solution of the TCAPLL and the stand-alone GPS. \begin{table}
\begin{tabular}{c c c c} \hline
**Error [deg]** & **Roll** & **Pitch** & **Heading** \\ \hline \(\sigma\) & 0.237 & 0.213 & 1.574 \\ mean & 0.094 & 0.055 & 0.043 \\ \hline \end{tabular}
\end{table} 
Table 3: Standard deviation and mean of the TCAPLL attitude error. Figure 7: Sky plot during the simulated trajectory (**left**) and received power from satellites (**right**). The effect of enabling the feed-forward component on the PLL output is shown in Figure 10. The aiding frequency allows for a reduction of the bandwidth of the PLL. Consequently, after adding the feed-forward frequency to the loop filter output, thermal noise is reduced and the PLL output is centered on zero. In this case the PLL tracks only the induced thermal noise and the error frequency resulting from the aiding. The advantages of the feed-forward component in avoiding re-acquisition can be seen in Figure 11, where the carrier and code NOOs' output are shown for SV 11. During the GPS outage, the PLL and DLL loops continue tracking using only the feed-forward component, and as soon as the satellite reappears, the tracking loops lock directly to it without the need for re-acquisition. The uncertainty of 2 Hz on the feed-forward component of SV 11 allows to maintain its corresponding generated code phase error uncertainty less than 0.5 chips for roughly 385 s \(\left(\left(e_{f-f}t_{Dis}\right/\mathrm{SF}\right)\)\(<\) 0.5\(\right)\), allowing a re-tracking without the need for re-acquisition for more than 4 minutes.