## 2 Inertial Aided PLL in Compass/INS Integrated Navigation System  
The inertial aided Kalman-PLL is adopted in the Compass/INS integrated navigation system. With carrier phase, Doppler frequency shift, and Doppler shift rate as the system states, carrier phase as the observation, and phase discriminator output as the innovation, the carrier phase and Doppler frequency shift are estimated and used to compute the carrier NCO, as shown in Figure 1[21]. Figure 1: Inertial aided PLL in Compass/INS integrated navigation system. The system matrix of Kalman-PLL is demonstrated as:  
\[ \begin{bmatrix}\phi_{k+1}\\ \Delta f_{k+1}\\ \Delta\dot{f}_{k+1}\end{bmatrix} = \begin{bmatrix}1&2\pi\tau&\pi\tau^{2}\\ 0&1&\tau\\ 0&0&1\end{bmatrix}\begin{bmatrix}\phi_{k}\\ \Delta\dot{f}_{k}\\ \Delta\dot{f}_{k}\end{bmatrix} + \begin{bmatrix}2\pi f_{{}_{I\!R}}\tau\\ 0\\ 0\end{bmatrix} +\mathbf{W}_{k+1} \tag{1}\]  
where \(\tau\) is the loop update interval. The process noises are approximated as white noises:  
\[E(\mathbf{w}_{k})=0 \tag{2}\] \[E(\mathbf{w}_{k}w_{k}^{T})=\mathbf{Q}_{k}\]  
where the covariance matrix of the process noise \(\mathbf{Q}_{k}\) is related to receiver's dynamic characteristics. The measurement matrix of Kalman-PLL is demonstrated as:  
\[\phi_{k}= \begin{bmatrix}1&0&0\end{bmatrix}\begin{bmatrix}\phi_{k}\\ \Delta\dot{f}_{k}\\ \Delta\dot{f}_{k}\end{bmatrix} +\mathbf{V}_{k} \tag{3}\]  
Observation noises are also approximated as white noises:  
\[E(\mathbf{v}_{k})=0 \tag{4}\] \[E(\mathbf{v}_{k}v_{k}^{T})=\mathbf{R}_{k}\]  
The covariance matrix of observation noise \(\mathbf{R}_{k}\) is determined by carrier phase tracking noise due to thermal noise, vibration noise, ionosphere scintillation, noise of oscillator and other sources. Adopt an arctan phase discriminator, and the output of phase discriminator is demonstrated as:  
\[\delta\phi_{k}=\mathrm{atan}(Q_{p}/I_{p}) \tag{5}\]  
where, \(Q_{p}\), \(I_{p}\) are the quada-phase output and in-phase output of baseband correlator prompt channel. Take the PLL discriminator output \(\delta\phi_{k}\) as the observation innovation, that is:  
\[Z_{k/k-1}=\delta\phi_{k} \tag{6}\]  
The computation of Kalman-PLL is presented with two steps, prediction and update:  
Prediction:  
\[ \begin{split}\mathbf{x}_{k+1}^{-}&=\mathbf{\Phi}_{k} \mathbf{x}_{k}^{+}\\ \mathbf{P}_{k+1}^{-}&=\mathbf{\Phi}_{k}\mathbf{P}_{k}^{ \mathrm{T}}\mathbf{\Phi}_{k}^{\mathrm{T}}+\mathbf{\Gamma}_{k}\mathbf{Q}_{k}\mathbf{\Gamma }_{k}^{\mathrm{T}}\end{split} \] (7  
The carrier NCO is:  
\[f_{NCO,k+1}=\frac{\phi_{k+1}^{-}-\phi_{k}^{-}}{2\pi\tau} \tag{9}\]  
where, \(\phi_{k}\) is the carrier phase, \(\Delta f_{k}\) is the Doppler frequency shift, and \(\Delta\dot{f}_{k}\) is the Doppler shift rate. The superscript \"\(+\)\" denotes the updated measurement and the superscript \"\(-\)\" denotes the predicted measurement. The essential part of inertial aided tracking is to use the inertial aid information to predict the Doppler frequency shift, and then eliminate the loop's dynamic stress error. The traditional aid method is velocity aiding, and acceleration aiding is also practical [22]. The velocity aid method involves computing the Doppler frequency shift through the line-of-sight velocity, and correcting the estimated Doppler frequency shift before the update step (8), as shown in Equation (10):  
\[\Delta\dot{\mathcal{f}}_{k}=\frac{f_{\alpha}}{c}\mathbf{v}_{k}\mathbf{e}_{k} \tag{10}\]  
Acceleration aiding involves computing the Doppler shift rate through the line-of-sight acceleration, and correcting the estimated Doppler shift rate before the update step (8), as shown in Equation (11):  
\[\Delta\dot{\mathcal{f}}_{k}=\frac{f_{\alpha}}{c}\mathbf{a}_{k}\mathbf{e}_{k} \tag{11}\]  
where \(\mathbf{v}_{k}\) is the line-of-sight velocity vector, \(\mathbf{a}_{k}\) is the line-of-sight acceleration vector, and \(\mathbf{e}_{k}\) is line-of-sight unit vector.