# An Algorithm to Assist the Robust Filter for Tightly Coupled RTK/INS Navigation System
## 2 Materials and Methods
### The Update Step of the TC RTK/INS Integration  
#### 2.3.1 The Measurement Vector and the Observation Matrix  
As we process (GPS/BDS) dual systems, we should choose two reference satellites to form DD measurements. Each system has its own time reference and frequency. Suppose we choose only one reference satellite for different systems. In that case, the relative hardware delays between systems will be introduced into the measurements, which means the DD ambiguity cannot be fixed as an integer. The relative hardware delays can magnify the complexity in AR and degrade the positioning accuracy [75]. For simplicity, the formulas in this section are in the form of a single system.  
We use the notation \(S_{j}\) to denote the \(j\)th satellite, and we assume \(S_{1}\) to be the reference satellite. Equations (3) and (4) show the formulation of DD measurements. It is difficult to obtain the true DD distance \(\
abla\Delta r\) in practice. In the TC RTK/INS integration, we can obtain the nominal DD distance \(\
abla\Delta\tilde{r}\) based on the position of INS and satellites. The measurement vector of the TC RTK/INS integration can be formulated as Equation (11) shows:  
\[\mathbf{y}=\begin{bmatrix}\
abla\Delta\phi_{S_{1},S_{2}}-\
abla\Delta\tilde{r}_{S_ {1},S_{2}}\\ \
abla\Delta\phi_{S_{1},S_{3}}-\
abla\Delta\tilde{r}_{S_{1},S_{3}}\\ \vdots\\ \
abla\Delta\phi_{S_{1},S_{m}}-\
abla\Delta\tilde{r}_{S_{1},S_{m}}\\ \
abla\Delta\rho_{S_{1},S_{2}}-\
abla\Delta\tilde{r}_{S_{1},S_{2}}\\ \
abla\Delta\rho_{S_{1},S_{3}}-\
abla\Delta\tilde{r}_{S_{1},S_{3}}\\ \vdots\\ \
abla\Delta\rho_{S_{1},S_{m}}-\
abla\Delta\tilde{r}_{S_{1},S_{m}}\end{bmatrix}. \tag{11}\]  
The observation matrix is formulated as Equation (12) shows:  
\[\mathbf{H}=\begin{bmatrix}\mathbf{DWG}&\mathbf{O}_{(m-1)\times 3}&\mathbf{DWL}&\mathbf{O}_{(m-1) \times 6}&\lambda\cdot\mathbf{D}\\ \mathbf{DWG}&\mathbf{O}_{(m-1)\times 3}&\mathbf{DWL}&\mathbf{O}_{(m-1)\times 6}&\mathbf{O}_{(m-1) \times m}\end{bmatrix}. \tag{12}\]  
In Equation (12), \(\mathbf{D}\) is the SD transition matrix, as shown in Equation (13):  
\[\mathbf{D}=\begin{bmatrix}1&-1&0&\ldots&0\\ 1&0&-1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&0&0&\ldots&-1\end{bmatrix}_{(m-1)\times m}. \tag{13}\]  
\(\mathbf{W}\) consists of LOS vectors for different satellites, as Equation (14) shows:  
\[\mathbf{W}=\begin{bmatrix}\mathbf{o}_{S_{1}}\text{T}\\ \mathbf{o}_{S_{2}}\text{T}\\ \vdots\\ \mathbf{o}_{S_{m}}\text{T}\end{bmatrix}_{m\times 3}. \tag{14}\]As mentioned in Section 2.3.1, the INS position is represented by latitude, longitude, and height in our implementation. However, the DD distance is calculated in the Earth-Centered and Earth-Fixed (ECEF) coordinate. In Equation (12), \(\mathbf{G}\) is a transformation matrix to transform the position error in the geographic coordinate to that in ECEF. In practice, the IMU cannot be installed at the same place as the GNSS antenna phase center, which causes the lever-arm offset. In Equation (12), \(\mathbf{L}\) denotes the lever-arm correction matrix. Many articles neglect the derivation of the observation matrix of the TC RTK/INS integration. We provide our derivation in Appendix A for the readers interested in it.