## 2 Materials and Methods
### A-VTTCES-KF Algorithm
#### 2.1.2 System Model for A-VTTCES-KF  
Based on the earlier stationary experiments and related analysis, when the VT mode is activated, an extra source error, of which the Allan variance curve is very close to the shape of the Markov process as shown in Figure 10, appears in the clock drift estimations compared with the ST results. This fact leads to the consequence of the case that there is a type time-correlated bias coupled in the local clock errors in the VT loop, and such error would certainly contaminate the GNSS measurements, i.e., Doppler, carrier phase, and pseudo-range, through the tracking process. In this case, the performance of the advanced VT architecture can be reduced by this unexpected error. In this work, we are trying to implement an improved KF algorithm targeting to the suppression of the time-correlated error. The discrete form of the dynamic process in terms of the navigation solutions can be described by the following equation  
\[\delta\mathbf{x}_{k}=\mathbf{\Phi}_{k,k-1}\delta\mathbf{x}_{k-1}+\mathbf{w}_{ k-1} \tag{4}\]  
Also, the observation process can be expressed as  
\[\delta\mathbf{z}_{k}=\mathbf{H}_{k}\delta\mathbf{x}_{k}+\mathbf{v}_{k} \tag{5}\]  
where subscript \(k\) represents the index of the discrete samples; \(\mathbf{\Phi}_{k,k-1}\) denotes the transition matrix from the epoch of \((k-1)\) to the \(k\) th epoch; \(\mathbf{H}_{k}\) is the observation design matrix; \(\delta\mathbf{x}_{k}\) and \(\delta\mathbf{z}_{k}\) represent the state vector and observation vector, respectively; \(\mathbf{v}_{k}\) and \(\mathbf{w}_{k}\) represent the observation and the process random noise vectors, respectively, and they should be subject to  
\[\left\{ \begin{array}{l}\mathbf{E}\left[\mathbf{w}_{k}\right]=\mathbf{0}, \quad\mathbf{E}\left[\mathbf{v}_{k}\right]=\mathbf{0}\\ \mathbf{E}\left[\mathbf{w}_{k}\mathbf{w}_{k}^{T}\right]=\mathbf{Q}_{k},\ \mathbf{E}\left[\mathbf{v}_{k}\mathbf{v}_{k}^{T}\right]=\mathbf{R}_{k},\\ \mathbf{E}\left[\mathbf{w}_{k}\mathbf{v}_{k}^{T}\right]=\mathbf{0},\ k\
eq j \end{array} \right. \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Noise Source** & **General Model** & **Power-Law Model** \\ \hline White Frequency & \(\frac{Q^{2}}{\Gamma}\) & \(\frac{h_{0}}{2\Gamma}\) \\ \hline Random Walk Frequency & \(\frac{k^{2}\tau}{3}\) & \(\frac{(2\pi)^{2}\tau h_{-2}}{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary for the Allan variance \(\sigma^{2}\left(\tau\right)\) analysis of the noise [64; 71; 72]. with \(k,j\in\mathbb{Z}^{+}\), where \(\mathbf{Q}_{k}\) denotes the process covariance matrix, \(\mathbf{R}_{k}\) represents the observation covariance matrix. * axis position error}\\ \text{y
- axis position error}\\ \text{z
- axis velocity error}\\ \text{z
- axis acceleration error}\\ \text{z
- axis acceleration error}\\ \text{clock bias error}\\ \text{clock drift error}\\ \text{time
- correlated error difference}\end{array}\right]\] (7) where \(\delta\eta_{k}=\eta_{k}-\eta_{k-1}\); \(\eta_{k}\) is assumed to follow the 1 st-order GM process, and it is given by \[\eta_{k}=e^{-\beta\Delta t}\eta_{k-1}+W_{k-1}\] (8) with \(\mathbf{E}\left(W_{k}\right)=0\) and \(Q_{W}=\mathbf{E}\left(W_{k}W_{j}\right)=\sqrt{\left(1-e^{-2\beta\Delta t} \right)\sigma_{c}^{2}}\) subject to \(j=k\), where \(\beta\) denotes the reciprocal of the correlation time, \(T_{c}\), i.e., \(\beta=\frac{1}{T_{c}};\sigma_{c}\) stands for the temporal standard deviation of the noise; \(\Delta t\) represents the updating interval for the filter; \(W_{k}\) is the discrete sample of the white noise, and \(Q_{W}\) stands for the variance of \(W_{k}\). * Transition matrix In this section, the dynamic matrix of the A-VTTCES-KF can be obtained as \[\mathbf{F}=\left[ \begin{array}{cc}\mathbf{F}_{1}&\mathbf{0}_{9\times 3}\\ \mathbf{0}_{3\times 9}&\mathbf{F}_{2}\end{array} \right]\] (9) with \[\mathbf{F}_{1}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&0\end{array} \right]\otimes\mathbf{I}_{3},\quad\mathbf{F}_{2}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&-\beta\end{array}