# Improving the Design of MEMS INS-Aided PLLs for GNSS Carrier Phase Measurement under High Dynamics
## 2 Tracking Error Analysis of INS-Aided PLLs
### Implementation of INS-Aided PLLs  
The structure of INS-aided PLLs implemented in the GNSS/INS deeply-coupled system is shown in Figure 1. The lower part is the tracking loop branch. When the discriminator output was sent to the low-pass filter 1, it worked as a normal PLL. When the discriminator output was sent to the low-pass filter 2, it worked as an INS-aided PLL. Once the frame synchronization was achieved, the observations and satellite ephemeris could be obtained. The upper part is the INS branch. The inertial measurement unit (IMU) includes three-axis gyroscopes and three-axis accelerometers. The INS mechanization was set to output position, velocity, and attitude. The Kalman filter was to couple GNSS and INS, to restrain divergence of navigation results.  
Figure 1: Structure of inertial navigation system (INS) aided phase locked loop (PLL) implementation.  
The Doppler measured by the low-pass filter 1 consisted of Doppler between receiver and satellite, frequency error of the receiver clock, and thermal noise. The Doppler between receiver and satellite could be estimated by the INS and satellite ephemeris. When we calculated the receiver velocity, we could also obtain the receiver clock bias. Therefore, with the aiding information assist, the low-pass filter 2 only needed to track thermal noise and the aiding information error.  
The estimated Doppler can be expressed as the velocity of the receiver relative to the satellite, projected onto the line-of-sight (LOS) vector. The relationship is expressed in Equation (1) [22]:  
\[f_{\mathrm{dopp},k}=e_{k}^{T}\bigg{(}\frac{V_{\mathrm{INS}}-V_{\mathrm{SV},k} }{\lambda}\bigg{)} \tag{1}\]  
where, \(e_{k}^{T}\) is unit LOS vector from receiver to satellite \(k\), \(V_{\mathrm{INS}}\) is velocity of INS, which is the same as receiver, \(V_{\mathrm{SV},k}\) is velocity of satellite \(k\), and \(\lambda\) is carrier wavelength.  
The estimated Doppler update rate and the numerically controlled oscillator (NCO) update rate may be different and asynchronous. Therefore, the estimated Doppler needed to be extrapolated to the PLL maximum update rate. Because the satellite acceleration is a constant in a short time period, precise Doppler could be obtained via linear extrapolation. The \(n\)-th estimated Doppler via extrapolation is expressed in Equation (2) [23]:  
\[f_{\mathrm{dopp},k,n}=f_{\mathrm{dopp},k}+e_{k}^{T}\bigg{(}\frac{a_{\mathrm{ INS}}}{\lambda}+\frac{V_{\mathrm{SV},k,i}-V_{\mathrm{SV},k,i-1}}{\lambda} \bigg{)}\frac{n}{1000} \tag{2}\]  
where, \(a_{\mathrm{INS}}\) is acceleration of INS, \(V_{\mathrm{SV},k,i}\) is velocity of satellite \(k\) at the \(i\)-th second, \(n\) is the number of extrapolation.  
The receiver works with normal PLLs before the aiding information is ready. When the difference between the aiding information and the low-pass filter 1 remained small enough, INS-aided PLLs worked as expected. Since the clock drift was smaller than the noise of the calculated clock bias, the clock drift was tracked by the low-pass filter 2. In addition, the acceleration of satellite was no more than 0.2 m/s\({}^{2}\). Therefore, the estimated error of the aiding information was mainly from the INS velocity.