## 4 Asynchronous PL Navigation System  
This section deals with the navigation algorithm for an asynchronous PL navigation system. Figure 8 shows the construction for measurement modeling. While GNSS satellites are all synchronized, asynchronous PLs are not. A reference station that transmits PL clock offset information to users is therefore required. Figure 7: VDFLL implementation. Pseudorange and carrier phase measurements of user and reference are modeled as follows. Pseudorange measurement:  
\[ \begin{split}\rho_{r}^{i}&=\left(R^{i}-R_{r}\right) \hat{\mathbf{e}}_{r}^{i}+B_{\rho_{r}}-b^{i}+i_{r}^{i}+t_{r}^{i}+m_{\rho_{r}}^{i}+ \varepsilon_{\rho_{r}}^{i}\\ \rho_{u}^{i}&=\left(R^{i}-R_{u}\right)\hat{\mathbf{e}}_{u }^{i}+B_{\rho_{u}}-b^{i}+i_{u}^{i}+t_{u}^{i}+m_{\rho_{u}}^{i}+\varepsilon_{\rho _{u}}^{i}\end{split}  \tag{10}\]  
where:  
\(\rho_{r}^{i}\), \(\rho_{u}^{i}\): reference and user pseudorange of \(i\)th PL  
\(R^{i}\): \(i\)th PL position  
\(R_{r}\), \(R_{u}\): reference and user receiver positions  
\(B_{\rho_{r}}\), \(B_{\rho_{r}}\): reference and user receiver clock offsets  
\(b^{i}\): \(i\)th PL clock offset  
\(e_{r}^{i}\), \(e_{u}^{i}\): reference and user LOS vectors to \(i\)th PL, respectively  
\(i,t,m,\varepsilon\): ionospheric delay, tropospheric delay, multipath delay, and thermal noise  
Carrier phase measurement:  
\[ \begin{split}\phi_{r}^{i}&=\left(R^{i}-R_{r}\right) \hat{\mathbf{e}}_{r}^{i}+B_{\rho_{r}}-b^{i}+i_{r}^{i}+t_{r}^{i}+m_{\rho_{r}}^{i}+ \lambda\cdot N_{r}^{i}+\varepsilon_{\rho_{r}}^{i}\\ \phi_{u}^{i}&=\left(R^{i}-R_{u}\right)\hat{\mathbf{e}}_{u }^{i}+B_{\rho_{u}}-b^{i}+i_{u}^{i}+t_{u}^{i}+m_{\rho_{u}}^{i}+\lambda\cdot N_{u }^{i}+\varepsilon_{\rho_{u}}^{i}\end{split}  \tag{11}\]  
where:  
\(\phi_{r}^{i}\), \(\phi_{u}^{i}\): reference and user carrier phase measurement of \(i\)th PL  
\(\lambda\): wavelength of PL carrier (L1 = 19 cm)  
\(N\): ambiguity integer  
To derive the navigation equation for an asynchronous PL navigation system, reference and user measurements are differenced to eliminate receiver and PL clock offset terms. Single-differenced pseudorange measurements for the receiver clock offset elimination are:  
Figure 8: Measurement model of an asynchronous PL navigation system [19]. \[ \begin{split}{}^{i}\
abla^{j}\rho_{r}&=\rho_{r}^{i}- \rho_{r}^{j}\\ &=\left(R^{i}-R_{r}\right)\cdot\hat{e}_{r}^{i}-\left(R^{j}-R_{r} \right)\cdot\hat{e}_{r}^{j}-{}^{i}\
abla^{j}b+{}^{i}\
abla^{j}t_{r}+{}^{i} \
abla^{j}m_{\rho_{r}}+{}^{i}\
abla^{j}\varepsilon_{\rho_{r}}\\ {}^{i}\
abla^{j}\rho_{u}&=\rho_{u}^{i}-\rho_{u}^{j}\\ &=\left(R^{i}-R_{u}\right)\cdot\hat{e}_{u}^{i}-\left(R^{j}-R_{u} \right)\cdot\hat{e}_{u}^{j}-{}^{i}\
abla^{j}b+{}^{i}\
abla^{j}t_{u}+{}^{i} \
abla^{j}m_{\rho_{u}}+{}^{i}\
abla^{j}\varepsilon_{\rho_{u}}\,\end{split}  \tag{12}\]  
where:  
\({}^{i}\
abla^{j}\rho_{r}\), \({}^{i}\
abla^{j}\rho_{s}\): reference and user single-differenced pseudorange between \(i\)th and \(j\)th PLs  
The double-differenced measurement for satellite clock offset elimination is obtained as follows:  
\[ \begin{split}{}^{i}\
abla^{j}{}_{u}\Delta_{r}\rho& ={}^{i}\
abla^{j}\rho_{u}-{}^{i}\
abla^{j}\rho_{r}\\ &=\left(R^{i}-R_{u}\right)\cdot\hat{e}_{u}^{i}-\left(R^{j}-R_{u} \right)\cdot\hat{e}_{u}^{j}-\left(R^{i}-R_{r}\right)\cdot\hat{e}_{r}^{j}+ \left(R^{j}-R_{r}\right)\cdot\hat{e}_{r}^{j}\\ &+{}^{i}\
abla^{j}{}_{u}\Delta_{r}+{}^{i}\
abla^{j}{}_{u}\Delta_ {r}m_{\rho}+{}^{i}\
abla^{j}{}_{u}\Delta_{r}\varepsilon_{\rho}\,\end{split}  \tag{13}\]  
where:  
\[{}^{i}\
abla^{j}{}_{u}\Delta_{r}\left(\cdot\right)\!\!-\!\!i^{j}\
abla^{j} \left(\cdot\right)\!\!_{u}-{}^{i}\
abla^{j}\left(\cdot\right)\!\!_{r}=\left[ \left(\cdot\right)\!\!_{u}-\left(\cdot\right)\!\!_{u}^{j}\right]-\left[\left( \cdot\right)\!\!_{r}-\left(\cdot\right)\!\!_{r}^{j}\right].\]  
With double-differenced pseudorange measurements containing no receiver or PL clock offset terms, the navigation equation can be constructed as:  
\[