# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## 2 Methodology
### The Modified Kalman Filter  
The Kalman filter is an optimal recursive estimator, which uses a dynamic model and sequential measurements to estimate the unknown state parameters in a minimum variance sense. In this paper, the constant velocity (CV) model [46] are adopted and the position-velocity components of the rover receiver and equivalent DD ambiguities are selected as system states. Let us still take Figure 1 as an example, the system states are expressed as follows:  
\[X=\left[\underbrace{x_{au},v_{au}}_{X_{au}},\underbrace{x_{bu},v_{bu}}_{X_{bu} },\underbrace{x_{cu},v_{cu}}_{X_{au}},\underbrace{\
abla\Delta N_{au}^{jk}, \ldots\
abla\Delta N_{au}^{jn}}_{N_{au}},\underbrace{\
abla\Delta N_{bu}^{jk},\ldots\
abla\Delta N_{bu}^{jn}}_{N_{tu}},\underbrace{\
abla\Delta N_{cu}^{jk},\ldots\
abla\Delta N_{cu}^{jn}}_{N_{\alpha}}\right]^{T} \tag{8}\]  
where \(X_{au}\), \(X_{bu}\), and \(X_{cu}\) are the position-velocity components of rover receiver, which are estimated corresponding to different reference receivers in Equation (7). In fact, the position-velocity components of rover receiver \(u\) for single epoch should be the same values, which can be as a constraint to enhance the strength of RTK positioning. \(N_{au}\), \(N_{bu}\), and \(N_{cu}\) are the corresponding equivalent DD ambiguities for three reference receivers, respectively. Considering the redundant position-velocity components of rover receiver, a constraint equation can be expressed as:  
\[\underbrace{\left[ \begin{array}{cccccc}1&0&-1&0&0&0\\ 0&1&0&-1&0&0\\ 1&0&0&0&-1&0\\ 0&1&0&0&0&-1\\ \end{array} \right]}_{D_{1}}=\underbrace{\left[ \begin{array}{c}x_{au}\\ v_{au}\\ v_{bu}\\ v_{cu}\\ \end{array} \right]}_{X_{1}}=\underbrace{\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ \end{array} \right]}_{M_{1}} \tag{9}\]  
Equation (9) can be expanded as follows  
\[\underbrace{\left[ \begin{array}{cc}D_{1}&0\\ 0&0\\ \end{array} \right]}_{D}=\underbrace{\left[ \begin{array}{c}M_{1}\\ 0\\ \end{array} \right]}_{M} \tag{10}\]  
Thus, based on Equations (7) and (10), the system state and observation equations of Kalman filter with parameter constraints are expressed as follows:  
\[\left\{ \begin{array}{l}X_{t}=\Phi_{t,t-1}X_{t-1}+W_{t}\\ Z_{t}=H_{t}X_{t}+V_{t}\\ DX_{t}=M\end{array} \right. \tag{11}\]where subscript \(t\) denotes the epoch, \(X_{t}\) is the unknown state vector at epoch \(t\), \(\Phi_{t,t-1}\) is the state translation matrix from epoch \(t-1\) to \(t\), \(W_{t}\) is the noise vector of the system state model with zero mean and the covariance matrix \(Q_{wk}\), the setting of covariance matrix \(Q_{wk}\) can refer to related literature such as [PERSON]. [47] and [PERSON] et al. [17] which is not discussed here. \(H_{t}\) is the design matrix and \(Z_{t}\) is column vector of the observation minus calculation (OMC) terms at epoch \(t\), \(V_{t}\) is a noise vector of observations with zero mean and the covariance matrix \(Q_{wk}\). Moreover, the noise \(W_{t}\) and \(V_{t}\) are assumed non-correlated. The Kalman filter solution of the system states with constraint equation [48] is derived as  
\[\left\{ \begin{array}{l}\overline{X}_{t}=\Phi_{t,t-1}X_{t-1}\\ \overline{Q}_{t}=\Phi_{t,t-1}Q_{t-1}\Phi_{t,t-1}^{T}+Q_{wk}\\ \overline{X}_{t}^{\prime}=\overline{X}_{t}-D^{T}(DD^{T})^{-1}(D\overline{X}_ {t}-M)\\ J_{t}=\overline{Q}_{t}H_{t}^{T}(H_{t}\overline{Q}_{t}H_{t}^{T}+Q_{wk})^{-1}\\ \hat{X}_{t}=\overline{X}_{t}+J_{t}(Z_{t}-H_{t}\overline{X}_{t})\\ Q_{\hat{X}_{t}}=\overline{Q}_{t}-J_{t}H_{t}\overline{Q}_{t}\end{array} \right. \tag{12}\]  
where \(\overline{X}_{t}\) and \(\overline{Q}_{t}\) are the predicted state vector and its corresponding predicted covariance matrix. \(\overline{X}_{t}\) is the corrected value of one-step predicted state vector, which can be improved using constraint equation. \(J_{t}\) is the equivalent gain matrix, which can balance the observation and state information, \(\hat{X}_{t}\) and \(Q_{\hat{X}_{t}}\) are the state estimation vector and its corresponding covariance matrix at epoch \(t\).