## 2 The Theoretical Basis of Tightly-Coupled INS/RTK Integrations  
### Definition of State Vectors and Motion Model  
In the RTK/INS integrated navigation systems, in order to clearly illustrate the positioning procedure, we first define the state vectors as well as the motion and observation models. Categorized according to different uses, there are three types of state vectors: position vector, inertial parameter vector, and ambiguity vector. \[\mathbf{x}_{k}=(\mathbf{x},\mathbf{y},\mathbf{z}) \tag{1}\]  
\[\mathbf{v}_{k} =(\mathbf{v}_{\mathbf{x}^{\prime}}\mathbf{v}_{y},\mathbf{v}_{z},\mathbf{\theta},\mathbf{ \phi},\mathbf{v}_{\mathbf{b}_{a}},\mathbf{b}_{g}) \tag{2}\] \[\mathbf{N}_{k} =\left( \begin{matrix}\overset{s\mathbf{L}_{1}}{\Delta\mathbf{N}_{\mathbf{rb}}} &\overset{s\mathbf{L}_{2}}{\Delta\mathbf{N}_{\mathbf{rb}}}\\ \end{matrix} \right)s=1,2,\ldots,S_{k} \tag{3}\]  
where \(k\) is the discrete-time index; \(\mathbf{x}_{k}\in\mathbb{R}^{3}\) is the position vector at time \(k\), which only contains three-dimensional position information; and \(\mathbf{v}_{k}\in\mathbb{R}^{12}\) is the inertial parameter vector, which contains velocity, Euler angles, accelerometer bias, and gyroscope bias. The inertial parameter vector only exists when the navigation system integrates with the inertial navigation system. \(\mathbf{N}_{k}\) is the ambiguity vector, and \(\mathbf{\Delta\mathbf{N}_{\mathbf{rb}}}\) and \(\mathbf{\Delta\mathbf{N}_{\mathbf{rb}}}\) represent the single-differenced integer ambiguities of satellite \(s\) at \(L1\) and \(L2\) frequencies separately, obtained by the difference between rover \(r\) and base station \(b\). \(S_{k}\) is the total number of observable satellites at time \(k\). Simultaneously, the motion equation is as follows:  
\[(\mathbf{x}_{k},\mathbf{v}_{k})=\mathbf{f}(\mathbf{x}_{k-1},\mathbf{v}_{k-1})+\mathbf{n}_{k},k=1,2, \ldots,K \tag{4}\]  
where the function \(\mathbf{f}0\) denotes the transition model, \(\mathbf{n}_{k}\) is the system noise whose distribution is independent of time. The transition model depends on the INS measurements. Using a strap-down inertial system, we need to transform the specific force vector and angular rate measured in the body frame into the Earth-centered Earth-fixed (ECEF) frame. Then, the kinematic equations are as follows [17]:  
\[\overset{\cdot}{C}_{b}^{e} =C_{b}^{e}\mathbf{\Omega}_{ib}^{b}-\mathbf{\Omega}_{ie}^{e}C_{b}^{e} \tag{5}\] \[\overset{\cdot}{v}_{eb}^{e} =C_{b}^{e}\mathbf{\Omega}_{ib}^{b}+\mathbf{g}_{b}^{e}-2\mathbf{\Omega}_{ie}^{e }\mathbf{v}_{eb}^{e}\] (6) \[\overset{\cdot}{r}_{eb}^{e} =v_{eb}^{e} \tag{7}\]  
where \(\mathbf{C}_{b}^{e}\) denotes the rotational transformation from the body frame to the ECEF frame, and \(\mathbf{v}_{eb}^{e}\) and \(\mathbf{r}_{eb}^{e}\) denote the Earth relative velocity and position in the ECEF frame, respectively. The equations above give their time derivatives. The vector \(\mathbf{f}_{ib}^{b}\) is the specific force that can be obtained from INS measurements directly. \(\mathbf{g}_{b}^{e}\) is the local gravity vector related to the current position \(\mathbf{r}_{eb}^{e}\). \(\mathbf{\Omega}_{ie}^{e}\) is the skew-symmetric matrix of the Earth rotation vector, and \(\mathbf{\Omega}_{ib}^{b}\) is the skew-symmetric matrix of the angular rate \(\Omega_{ib}^{b}\). It is noteworthy that the specific force and angular rate measurements are accompanied by noise.