## 2 RTK Mathematic Model  
### Constrained Loosely Coupled Model  
The DD carrier phase measurement \(\varphi\) (in cycles) with wavelength \(\lambda\) and code measurement \(P\) (in meters) is defined as follows [1]:  
\[\left\{\begin{array}{l}\lambda\
abla\Delta\varphi=\
abla\Delta\rho-\lambda \
abla\Delta N+\
abla\Delta T-\
abla\Delta I+\
abla\Delta\varepsilon\\ \
abla\Delta P=\
abla\Delta\rho+\
abla\Delta T+\
abla\Delta I+\
abla\Delta \xi\end{array}\right. \tag{1}\]  
where \(\
abla\Delta\) is the DD operator; \(\rho\), \(T\), \(I\), and \(N\) are the receiver-satellite geometric range, tropospheric delay, ionospheric delay, and carrier phase ambiguity, respectively; and \(\xi\) and \(\varepsilon\) are the unmodeled errors including multipath noise, system noise, etc. The covariance matrix of the system state, system noise, measurement noise, and design matrices can be defined as \(P\), \(Q\), \(R\), and \(H\). The filter-state vector consists of positioning information and the DD ambiguities can be expressed as [7]:  
\[\mathbf{x}=\left[x_{n}^{G}\quad x_{u}^{G}\quad x_{e}^{G}\quad x_{e}^{G}\quad x_{n}^ {G}\quad x_{u}^{G}\quad\
abla\Delta N^{G}\quad\
abla\Delta N^{E}\right]^{T}\]  
where the superscripts 'G' and 'E' represent GPS and Galileo, and the subscripts \(e\), \(n\), and \(u\) represent different directions. The _diag_(\(\cdot\)) is a diagonal matrix. The KF recursive process is defined as [1]:  
\[\left\{\begin{array}{l}\hat{\mathbf{x}}_{i|k-1}=\mathbf{F}_{k,k-1}\hat{\mathbf{x}}_{k-1} \\ \mathbf{P}_{k|k-1}=\ \ \mathbf{F}_{k,k-1}\mathbf{P}_{k-1}\mathbf{F}_{k,k-1}^{\mathbf{\Gamma}}+\mathbf{ \Gamma}_{k-1}\mathbf{\Omega}_{k-1}\mathbf{\Gamma}_{k-1}^{\mathbf{\Gamma}}\\ \mathbf{P}_{k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k|k-1}\\ \mathbf{K}_{k}=\mathbf{P}_{k|k-1}\mathbf{H}_{k}^{\mathbf{\Gamma}}(\mathbf{H}_{k}\mathbf{P}_{k|k-1}\bm {H}_{k}^{\mathbf{\Gamma}}+\mathbf{R}_{k})^{-1}\\ \hat{\mathbf{x}}_{k}=\mathbf{\hat{x}}_{k|k-1}+\mathbf{K}_{k}(\mathbf{z}_{k}-\mathbf{H}_{k}\hat{\mathbf{ x}}_{|k-1})\end{array}\right\}update \tag{2}\]  
where \(\mathbf{K}\), \(\mathbf{F}\) are the gain and state transform matrix; and \(\mathbf{z}_{k}\) and \(\mathbf{x}_{k}\) are the measurements and state that needs to be estimated at the \(k\)th epoch. As \(\mathbf{x}_{G}=\mathbf{x}_{E}\) once successfully located, the following constraint can be attached [39]:  
\[\underbrace{\left[\mathbf{I}_{3\times 3}-\mathbf{I}_{3\times 3}\right]}_{D}\underbrace{ \left[\mathbf{x}_{G}\right]}_{X}=\underbrace{\left[\mathbf{0}_{3\times 3}\right]}_{M} \tag{3}\]  
where \(\mathbf{I}\) represents the unit matrix, and \(\mathbf{D}\) and \(\mathbf{M}\) are the constraint matrices. The prediction step in (2) can be developed to constrain KF as follows:  
\[\left\{\begin{array}{l}\mathbf{x}_{k|k-1}=\mathbf{x}_{k|k-1}-\mathbf{D}_{k}^{\mathbf{\Gamma} }\big{(}\mathbf{D}_{k}\mathbf{D}_{k}^{\mathbf{\Gamma}}\big{)}^{-1}\big{(}\mathbf{D}_{k}\mathbf{D}_ {k}^{\mathbf{\Gamma}}\big{)}^{-1}\Big{(}\mathbf{D}_{k}\mathbf{x}_{k|k-1}-\mathbf{M}_{k}\Big{)} \\ \mathbf{P}_{k|k-1}=\ \ \Big{(}\mathbf{I}-\mathbf{D}_{k}^{\mathbf{\Gamma}}\big{(}\mathbf{D}_{k}\mathbf{D}_ {k}^{\mathbf{\Gamma}}\big{)}^{-1}\mathbf{D}_{k}\Big{)}^{T}\mathbf{P}_{k|k-1}\Big{(}\mathbf{I}- \mathbf{D}_{k}^{\mathbf{\Gamma}}\big{(}\mathbf{D}_{k}\mathbf{D}_{k}^{\mathbf{\Gamma}}\big{)}^{-1} \mathbf{D}_{k}\Big{)}\end{array}\right. \tag{4}\]