## 5 VTL Implementation in an Asynchronous PL Navigation System  
As seen in Section 4, the navigation algorithm for the PL navigation system uses double-differenced measurements. Thus, satellite and receiver clock offset terms are not included in navigation states. In the VTL algorithm, estimation of pseudorange and range rate requiring the clock offset and drift terms is a necessary procedure for deducing tracking control input. The measurement model equations are repeated in Equations (16) and (17) below. It is assumed that indoor navigation occurs in an area of several square meters so that ionospheric and tropospheric delays are negligible, and multipath error was ignored for simplicity:  
\[\rho_{u}^{i}=\left(R^{i}-R_{u}\right)\cdot\hat{e}_{u}^{i}+B_{\rho_{s}}-b^{i}+ \varepsilon_{\rho_{u}}^{i} \tag{16}\]  
\[\dot{\rho}_{u}^{i}=\left(\dot{R}^{i}-\dot{R}_{u}\right)\cdot\hat{e}_{u}^{i}+ \dot{B}_{\rho_{u}}-\dot{b}^{i}+\varepsilon_{\dot{\rho}_{u}}^{i} \tag{17}\]  
As these equations show, receiver and satellite clock offset and drift terms are inevitable in reconstructing the pseudorange and range rate. Therefore, the conventional construction of navigation equation described by Equations (14) and (15) is not suitable for VTL implementation. We propose a new approach to constructing the navigation equation modified for VTL implementation. The measurement model equation is first modified below by inserting the satellite clock offset term, \(b^{\mathrm{i}}\), of the reference PL. PL 1 is assumed to be the reference:  
\[ \begin{split}\rho_{u}^{1}&=d_{u}^{1}+B_{u}-b^{1}+(b ^{1}-b^{1})+\varepsilon_{u}^{1}\\ \rho_{u}^{2}&=d_{u}^{2}+B_{u}-b^{1}+(b^{1}-b^{2})+ \varepsilon_{u}^{2}\\ &\qquad\qquad\qquad\vdots\\ \rho_{u}^{N}&=d_{u}^{N}+B_{u}-b^{1}+(b^{1}-b^{N})+ \varepsilon_{u}^{N}\end{split}  \tag{18}\]  
where:  
\(d_{u}^{i}\): distance from user to \(i\)th satellite. Now, the receiver clock offset minus the PL 1 clock offset, \(B_{u}-b^{\mathrm{i}}\), becomes the common error in the equation. It could therefore be a state variable of the navigation EKF. Each equation has the PL 1 clock offset minus the relevant PL clock offset, \((b^{\mathrm{i}}-b^{i})\). This term can be estimated from measurements of the reference receiver. The single-difference pseudorange and the carrier phase measurement referenced to PL 1 in a reference receiver are:  
\[ \begin{split}{}^{1}\
abla^{i}\rho_{r}=^{1}\
abla^{i}d_{r}-^{1} \
abla^{i}b+^{1}\
abla^{i}\varepsilon_{r,\rho}\\ {}^{1}\
abla^{i}\phi_{r}=^{1}\
abla^{i}d_{r}-^{1}\
abla^{i}b+^{1} \
abla^{i}N\cdot\lambda+^{1}\
abla^{i}\varepsilon_{r,\rho}\,,\end{split}  \tag{19}\]  
where \({}^{1}\
abla^{i}b=b^{\mathrm{i}}-b^{i}\). To estimate \((b^{\mathrm{i}}-b^{i})\) precisely, a Hatch filter should be applied. The Hatch filter equation for single-difference pseudorange with carrier phase measurement is:  
\[ \begin{split}&\delta^{\mathrm{i}}\
abla^{i}\rho_{r}\approx \delta^{\mathrm{i}}\
abla^{i}\phi_{r}\\ \rightarrow^{\mathrm{i}}\
abla^{i}\hat{\rho}_{r}(k+1)=& \frac{N-1}{N}\big{(}^{1}\
abla^{i}\hat{\rho}_{r}(k)+\delta^{\mathrm{i}}\
abla^ {i}\phi_{r}(k)\big{)}+\frac{1}{N}\delta^{\mathrm{i}}\
abla^{i}\rho_{r}(k+1) \end{split} \,, \tag{20}\]  
where:  
\(N\): Hatch filter constant  
\(\partial\
abla x(k)=\
abla x(k)-\
abla x(k-1)\)  
Then the satellite clock offset difference, \((b^{\mathrm{i}}-b^{i})\), can be estimated as:  
\[{}^{1}\
abla^{i}\hat{b}=^{\mathrm{i}}\
abla^{i}d_{r}-^{1}\
abla^{i}\hat{\rho }_{r} \tag{21}\]  
Now, the modified navigation equation for an asynchronous PL is constructed in Equation (22). The term \(B_{u}-b^{\mathrm{i}}\) is added as a filter state variable and the satellite clock offset differenced to reference PL 1 moves to the measurement side:  
\[