## 4 Simulation Results
### Open Sky with Three SVs  
A stand-alone GPS receiver cannot provide a 3-D PVT solution when only three SVs are visible. However with tight integration with an INS, this is possible. In this case, the navigation solution starts to depend more on the quality of the INS. In the previous two scenarios, it was shown that once the integration filter converges, the low quality of the MEMS inertial sensor does not have a big impact on the final navigation solution as long as there are continuous GPS updates that correct the INS errors. However, the more the number of visible satellites decreases, the more the final navigation solution depends on the INS. Taking into consideration the sky plot in Figure 7, let's assume that the GPS receiver sees only SVs 19, 6 and 3 during some short periods of time (up to 60 s) and loses visibility of the other satellites. During these periods, the GPS receiver does not provide a PVT solution; however, with the TCAPLL, a navigation solution is computed all the time as shown in Figure 17.  
The uncertainties of the positions and velocities are shown in Figure 18 along with the true errors computed from the reference navigation solution. As soon as the number of satellites drops to three, the uncertainty increases and converges with time. This shows that the integration filter is capable of  
Figure 17: Position solution when three SVs are visible for short periods of time. Figure 18: Position (**left**) and velocity (**right**) uncertainties during GPS partial outages. navigating with only three satellites, but the quality of the navigation solution degrades as expected and becomes more dependent on the INS. Figure 19 shows the uncertainty of the feed-forward frequency aiding, where the dashed lines refer to the visible satellites. It is clear that the uncertainty for the visible satellites increases less in comparison to the non-visible satellites. Also, the uncertainty increase is inversely proportional to the elevation for the non-visible satellites, _i.e._, the higher the elevation, the smaller the increase of its uncertainty. This is due to satellite geometry effects on the LOS projection when computing the Doppler frequency uncertainty. Looking back to Figure 4, it can be noted that this increase is still tolerable and will not affect tracking robustness. However, the uncertainties for the non-visible satellites start to be critical, especially if the GPS receiver continue tracking these satellites using the feed-forward component only. For example, the SV 22 uncertainty reaches roughly 15 Hz, which allows maintaining a code phase less than 0.5 chips for almost 51 s without the need for re-acquisition. However if the integration time is 10 ms, then the uncertainty will be higher than the tolerable frequency and consequently re-acquisition will be mandatory. The effect of decreasing the number of visible satellites to four or fewer on the quality of the feed-forward aiding is shown in Figure 20 for a visible and a non-visible satellite, respectively. For instance, SV 19 is always visible and there is a slight increase in its feed-forward uncertainty despite using a low-cost MEMS inertial sensor. Even when SV 19 is the only visible satellite, the frequency error is still tolerable for maintaining carrier and code tracking. However, if we consider SV 11, which is non-visible, the uncertainty increases exponentially, and the effect of the low-quality MEMS sensor is more obvious, especially when the outage time increases. For example, the uncertainty nearly doubles in 60 s every time a satellite is lost. This will put time constraints on the use of the feed-forward component if it is used to re-track the satellite without re-acquiring it. However, if the uncertainty exceeds the maximum tolerable error by the PLL, then a re-acquisition is necessary. In this case, the uncertainty can be used to decrease the frequency search span. Figure 19: Feed-forward frequency uncertainty when 3 SVs 19, 06 and 03 are in view.