## 2 System Design
### Ranging Measurement  
Figure 4 shows the principle of dual one way ranging measurement. Unlike the existing ranging method employed by GPS crosslink, the master satellite and the slave satellite communicate with each other simultaneously, but it is a \"pseudo-simultaneous\" scheme because of the clock skew and clock offset between two clocks. The parameters are defined as follows: \(\Delta t\) is the initial clock error between the master clock and the slave clock, \(\tau_{TM}\), \(\tau_{RM}\), \(\tau_{TS}\),\(\tau_{RS}\) are the transmitting and receiving delay of master satellite and slave satellite, respectively. \(\tau_{MS}(t_{M})\) and \(\tau_{SM}(t_{S})\) are the signal propagation delays. Hence the pseudo-ranges measured from each side and illustrated in Figure 4 are represented as:  
\[\rho_{SM}(t_{S})=\Delta t\left(t_{M}\right)+\tau_{SM}\left(t_{S}\right)+\tau_{ TS}+\tau_{RM}+I_{SM}+d_{SM} \tag{8}\]  
\[\rho_{MS}(t_{M})=-\Delta t\left(t_{M}\right)+\tau_{MS}\left(t_{M}\right)+\tau_ {TM}+\tau_{RS}+\varepsilon_{RS}+I_{MS}+d_{MS} \tag{9}\]  
where \(I\) denotes the ionospheric delay between \(S\) and \(M\), \(d\) denotes the other delay effects including tropospheric delay and Sagnac delay, \(c\) is the speed of light, \(\varepsilon_{RS}\) is the clock bias of the slave clock, which is derived from the ranging measurement process. From Equation (5), \(\varepsilon_{RS}\) can be expressed as:  
\[ \begin{split}&\varepsilon_{RS}=\int\limits_{\tau_{\text{{{}_{ \text{{}_{\text{{}_{\text{{}_{\text{{}_{\text{{}_{\text{{}_{\text{{}_{\text{{}}_{ }_{\text{{}}}}}}}}}}}}}}}}+\tau_{RS}}\left(\frac{v_{S}\left(t \right)-v_{0}}{v_{0}}\right)dt\\ &=y\left(t_{M}\right)\left(\tau_{{MS}(t_{M})}+\tau_{TM}+\tau_{RS} \right)+\frac{1}{2}a_{2}\left(t_{M}+\tau_{{MS}(t_{M})}+\tau_{TM}+\tau_{RS} \right)^{2}\end{split}  \tag{10}\]  
where \(y(t_{M})\) is the frequency deviations at the instance \(t_{M}\). Unlike the \(a_{1}\) presented in Equation (5), it is not expressed in terms of constants because the feedback control varies the characteristic of the slave  
Figure 4: Dual one-way ranging measurements, where the boxes with M & S denote master and slave packets. clock. Without loss of generality, assuming a total transmission time of 200 milliseconds and \(y(t_{M})\) and \(a_{2}\) are 1 \(\times\) 10\({}^{-12}\)/s and 1 \(\times\) 10\({}^{-18}\)/s\({}^{2}\) separately, then \(\varepsilon_{rS}\) is approximately 20 picoseconds, which can be ignored. Differencing and summing Equations (8) and (9) yields the instantaneous clock error and the instantaneous relative distance between two satellites:  
\[\Delta t\big{(}t_{M}\big{)} = \frac{\rho_{sat}(t_{S})\!-\!\rho_{MS}(t_{M})}{2c}\!-\!\frac{\tau_{ SM}(t_{S})\!-\!\tau_{MS}(t_{M})}{2}\frac{\tau_{M}+\tau_{RS}\!-\!\tau_{RS}\!-\! \tau_{RM}}{2}\!-\!\frac{I_{SM}\!-\!I_{MS}}{2}\frac{\varepsilon_{rS}}{2}\frac{d _{SM}\!-\!d_{MS}}{2} \tag{11}\] \[D\big{(}t_{M}\big{)} = \frac{\rho_{SM}(t_{S})\!+\!\rho_{MS}(t_{M})}{2}\] \[= \bigg{(}\frac{\tau_{SM}\big{(}t_{S})\!+\!\tau_{MS}\big{(}t_{M} \big{)}}{2}\!+\!\frac{\tau_{TM}+\tau_{RS}\!+\!\tau_{RS}\!+\!\tau_{RM}}{2}\!+ \!\frac{I_{SM}+I_{MS}}{2}\!+\!\frac{d_{SM}+d_{MS}}{2}\!+\!\frac{\varepsilon_{ rS}}{2}\bigg{)}\!\cdot\!c \tag{12}\]  
The first term of Equation (11) can be obtained directly from the ranging measurement of each satellite. The second term denotes the relative motion error. It is easy to understand that Equation (12) makes sense when the master satellite and the slave satellite are stationary, but \(D\big{(}t_{M}\big{)}\) would not equal to the true baseline if there is a relative motion between the two satellites, owing to the changes of \(\tau_{MS}(t_{M})\) and \(\tau_{SM}(t_{S})\). This is the reason why Equation (12) uses an approximately equal. It implies that the error would be critical if the system members have a fast relative velocity between each other. Take GNSS for example, the maximum time synchronization error that originates from relative motion can be shown to be approximately 1 microsecond.