## 2 System Design
### Ranging Measurement
Equation (14) can be expanded by the first-order Taylor series as:  
\[\left|\mathbf{r}\left(t_{S}\right)\right| \approx f\big{|}_{\mathbf{\Delta}_{M}=0}+\frac{\partial f}{ \partial\mathbf{\Delta}_{\mathbf{M}}}\Bigg{|}_{\mathbf{\Delta}_{M}=0}\cdot \mathbf{\Delta}_{\mathbf{M}} \tag{15}\] \[= \sqrt{\mathbf{r}_{\mathbf{SM}}\left(t_{S}\right)^{T}\mathbf{r}_ {\mathbf{SM}}\left(t_{S}\right)}+\frac{\left(\mathbf{\Delta}_{\mathbf{M}}- \mathbf{r}_{\mathbf{SM}}\left(t_{S}\right)\right)}{\sqrt{\left[\mathbf{r}_{ \mathbf{SM}}\left(t_{S}\right)-\mathbf{\Delta}_{\mathbf{M}}\right]^{T}\left[ \mathbf{r}_{\mathbf{SM}}\left(t_{S}\right)-\mathbf{\Delta}_{\mathbf{M}} \right]}}\Bigg{|}_{\mathbf{\Delta}_{u}=0}\cdot\mathbf{\Delta}_{\mathbf{M}}\] \[= \left|\mathbf{r}_{\mathbf{SM}}\left(t_{S}\right)\right|-\mathbf{e }_{\mathbf{SM}}^{\phantom{\mathbf{SM}}T}\mathbf{\Delta}_{\mathbf{M}}\] \[= \left|\mathbf{r}_{\mathbf{SM}}\left(t_{S}\right)\right|-\mathbf{e }_{\mathbf{SM}}^{\phantom{\mathbf{SM}}T}\cdot\int\limits_{\mathbf{r}_{\mathbf{ SM}}\left(t_{S}\right)}\mathbf{v}_{\mathbf{LOS}}^{\mathbf{M}}(t)dt\] \[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]0  
Similarly:  
\[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]1 \[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]2 \[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]3 \[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]4  
\[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\]5 Changes of the position of the two satellites during the signal propagation period. where \(\mathbf{v_{LOS}^{l}}\) denotes the line of sight (LOS) velocity of satellite \(i\) (Figure 6), \(\mathbf{\bar{v}_{LOS}^{l}}\) denotes the average of \(\mathbf{v_{LOS}^{l}}\) during the period of signal propagation, \(\mathbf{e_{ij}}\) denotes the unit vector of \(\mathbf{r_{ij}}(t_{i})\), \(\mathbf{e_{ij}}=-\mathbf{e_{ji}}\). Furthermore, the error caused by clock error can be expressed as:  
\[ \begin{split}\Delta\tau&=\tau(t_{M})-\tau(t_{S})\\ &=\frac{1}{c}\cdot\int\limits_{\Delta t}[\mathbf{v_{LOS}^{M}}(t) ^{T}\cdot\mathbf{e_{SM}}+\mathbf{v_{LOS}^{S}}(t)^{T}\cdot\mathbf{e_{MS}}]dt\\ \end{split}  \tag{17}\]  
where \(\tau(t_{M})=|\mathbf{r}(t_{M})|/c\) , \(\tau(t_{S})=|\mathbf{r}(t_{S})|/c\) . Substitution of Equations (15)-(17) into Equations (11) and (12) yields:  
\[