## 2 Error Transfer Function of INS-Aided PLLs
### INS-Aided PLLs' Principle  
With the GPS L1 single frequency receiver as example, Figure 3 shows the detailed components of the scalar-based deeply-coupled system. While the upper part represents the GPS receiver subsystem, the lower part is the inertial navigation subsystem. Compared with loosely-coupled integration and tightly-coupled integration, the INS information is sent to the receiver's baseband (red arrows in the figure), realizing the assistance to the receiver at the signal processing level. Compared with the vector-based deeply-coupled integration, the two subsystems are relatively independent in the scalar-based deeply-coupled integration, in which the internal structure is adjusted less. Since the receiver and INS are two kinds of navigation sensors, they can respectively measure the same vehicle motion information in different ways. As the INS and receiver's tracking loop are both used to compute the vehicle motion information, _i.e.,_ the vehicle dynamics is commonly experienced by both the IMU and the tracking loop, the IMU measurement information could assist the tracking loop based on the concept of feed-forward [8].  
Based on the scalar-based deeply-coupled system architecture, the principle of the INS-aided PLLs is shown in Figure 4. The PLL can track the carrier phase changing between a satellite and the vehicle, which measures the phase difference between local numerically controlled oscillator (NCO) and the input signal, by the discriminator converts the phase difference to the frequency by a low pass filter and adjusts the local NCO to follow the input signal. On the other hand, the vehicle movement information in the navigation frame can be directly measured by IMU. The measured velocity information is mapped to the line of sight (LOS) between the satellite and the vehicle to obtain the Doppler of relative motion, which combines with the receiver's clock drift, forming the Doppler aiding information. Therefore, with the aid of the INS feed-forward information, the tracking loop only needs to undertake the error of the INS aiding information, which significantly reduces the dynamic stress affecting the tracking loop.  
Figure 4: Principle of INS-aided PLLs. C/A code, coarse/acquisition-code.  
Figure 3: Detailed architecture of a scalar-based deeply-coupled system. DLL, delay lock loop.  
Different error sources lead to different effects on the tracking loop. The error sources of the normal PLLs mainly include thermal noise, oscillator noise and dynamic stress. By contrast, the dynamic stress is replaced by the INS measurement error in the INS-aided PLLs. The INS measurement error could be divided into INS bias-type errors that are unrelated to dynamic and INS scale factor-type errors, which are related with the dynamics. In addition, the aiding information delay should be considered in the INS-aided PLLs. Hence, the error sources of the INS-aided PLLs include thermal noise, oscillator noise, INS bias-type errors, INS scale factor-type errors and aiding information delay. Moreover, the INS bias-type errors can be further divided into IMU sensor errors (constant bias, bias instability and white noise) and the initial navigation error (velocity error and attitude error) after correction of the GNSS/INS integration algorithm; the INS scale factor-type errors can be further divided into IMU scale factor error and IMU non-orthogonal error.