# A Precise and Stable Space-Based Time System for Navigation in Smart Cities
## 4 Design of the Space-Based Time System for Navigation Constellations
### Centralized Two-Way Time Comparison Method Based on Inter-Satellite Link  
For domestic satellites, the TWTT method is used to obtain the satellite-ground clock difference of the domestic arc segment. However, for overseas satellites, the effective tracking of MEO satellites by the regional tracking network is less than 40% of the entire arc segment. When the satellite moves overseas, the observation data are missing, and the broadcast ephemeris and clock error parameters cannot be updated in time. To solve this problem, the satellite-ground inter-satellite joint method is adopted. This method takes domestic satellites as relay nodes. It leverages the satellite-ground clock differences of domestic satellites and the inter-satellite relative clock differences between domestic and foreign satellites to obtain the clock error of the satellite's outer arc segment relative to BDT by one-hop communication. Finally, the domestic and external clock differences of each satellite are fitted to the satellite clock parameters to obtain the satellite broadcast clock parameters. Figure 1: The space-based time system establishment process. The fundamental observation equation of the two-way inter-satellite link is expressed as follows [25; 26]:  
\[ \begin{split}\rho_{AB}(t_{1})&=\left|\vec{R}_{B}(t_{1} )-\vec{R}_{A}(t_{1}-\Delta t_{1})\right|\\ &\quad+c^{*}(clk_{B}(t_{1})-clk_{A}(t_{1})+\tau_{A}^{Send}+ \tau_{B}^{Rxv})+\Delta\rho_{cor}^{AB}\\ \rho_{BA}(t_{2})&=\left|\vec{R}_{B}(t_{2})-\vec{R}_{A }(t_{2}-\Delta t_{2})\right|\\ &\quad+c^{*}(clk_{A}(t_{2})-clk_{B}(t_{2})+\tau_{A}^{Rcv}+ \tau_{B}^{Send})+\Delta\rho_{cor}^{BA}.\end{split}  \tag{19}\]  
In the above equations, \(\rho_{AB}(t_{1})\) and \(\rho_{BA}(t_{2})\) represent the pseudo-range observations received by satellites A and B at times \(t_{1}\) and \(t_{2}\). Other parameters are defined as follows: \(\vec{R}_{A}\) and \(\vec{R}_{B}\) are the three-dimensional position vectors of satellites A and B; \(clk_{A}\) and \(clk_{B}\) represent the clock offset of satellites A and B; \(c\) is the speed of light; \(\Delta t\) is the light travel time; \(\tau_{A}^{Send}\) and \(\tau_{B}^{Send}\) are the transmission delays of satellites A and B; \(\tau_{A}^{Rxv}\) and \(\tau_{B}^{Rxv}\) are the receiving delays; \(\Delta\rho_{cor}^{AB}\) and \(\Delta\rho_{cor}^{BA}\) are error correction terms. To obtain two-way clock offset or two-way distance observations, it is necessary to unify the one-way observations at the same time. The uniform equations are given by [26]  
\[ \begin{split}\rho_{AB}(t_{0})&=\rho_{AB}(t_{1})+d \rho_{AB}\\ \rho_{BA}(t_{0})&=\rho_{BA}(t_{2})+d\rho_{BA},\end{split}  \tag{20}\]  
where \(d\rho_{AB}\) and \(d\rho_{BA}\) represent the uniform correction amount from the observation time to the unified time. The uniform correction amounts can be calculated using the satellite prediction orbit and clock offset parameters. They are defined by  
\[ \begin{split} d\rho_{AB}&=\left|\vec{R}_{B}(t_{0} )-\vec{R}_{A}(t_{0})\right|\\ &\quad-\left|\vec{R}_{B}(t_{1})-\vec{R}_{A}(t_{1}-\Delta t_{1}) \right|\\ &\quad+c^{*}(clk_{B}(t_{1})-clk_{A}(t_{1}))\\ &\quad-c^{*}(clk_{B}(t_{0})-clk_{A}(t_{0})),\end{split}  \tag{21}\]  
\[ \begin{split} d\rho_{BA}&=\left|\vec{R}_{B}(t_{0} )-\vec{R}_{A}(t_{0})\right|\\ &\quad-\left|\vec{R}_{B}(t_{2})-\vec{R}_{A}(t_{2}-\Delta t_{2}) \right|\\ &\quad+c^{*}(clk_{A}(t_{2})-clk_{B}(t_{2}))\\ &\quad-c^{*}(clk_{A}(t_{0})-clk_{B}(t_{0})).\end{split}  \tag{22}\]  
By subtracting the two-way observations at the unified time, the satellite orbit information can be eliminated, and the inter-satellite two-way clock offset observation equation can be obtained:  
\[ \begin{split}\frac{\rho_{AB}(t_{0})-\rho_{BA}(t_{0})}{2}& =c\cdot\left[clk_{B}(t_{0})-clk_{A}(t_{0})\right]\\ &\quad+c\cdot\frac{\tau_{A}^{Send}-\tau_{A}^{Rxv}}{2}\\ &\quad-c\cdot\frac{\tau_{B}^{Send}-\tau_{B}^{Rxv}}{2}+\frac{ \Delta\rho_{cor}^{AB}-\Delta\rho_{cor}^{BA}}{2}.\end{split}