# Joint Timekeeping of Navigation Satellite Constellation with Inter-Satellite Links
## 3 ISLT Timescale Algorithm
### Inter-Satellite Time Comparison  
BeiDou inter-satellite links employ a system of time division, based on which the dual one-way ranging of a satellite pair is completed in two contiguous time intervals. Satellite B is assumed to send out a ranging signal at epoch \(t_{\rm s1}\). Satellite A receives the signal at \(t_{\rm r1}\) and sends out another ranging signal at \(t_{\rm s2}\) (\(t_{\rm s2}>t_{\rm r1}\)) that is received by satellite B at \(t_{\rm r2}\). The sending moments \(t_{\rm s2}\) and \(t_{\rm s1}\) and the receiving moments \(t_{\rm r1}\) and \(t_{\rm r2}\) are the local time of satellites A and B, respectively, whose deviations from the system time are clock deviations \(\delta t_{\rm A}\) and \(\delta t_{\rm B}\). The relative clock deviation from inter-satellite time comparison is as follows:  
\[\delta t_{\rm AB}=\delta t_{\rm A}-\delta t_{\rm B} \tag{10}\]  
Based on the process of non-simultaneous bidirectional ranging, a pair of ranging equations of two linking satellites are established:  
\[\left\{ \begin{array}{l}\rho_{\rm AB}=\left|R_{\rm A}(t_{\rm r1})-R_{\rm B}(t _{\rm s1})\right|+c[\delta t_{\rm A}(t_{\rm r1})-\delta t_{\rm B}(t_{\rm s1})]+ c\tau_{\rm A}^{\rm r}+c\tau_{\rm B}^{\rm s}+\Delta\rho_{\rm AB}^{\rm sys}+ \varepsilon_{\rm AB}\\ \rho_{\rm BA}=\left|R_{\rm B}(t_{\rm r2})-R_{\rm A}(t_{\rm s2})\right|+c[ \delta t_{\rm B}(t_{\rm r2})-\delta t_{\rm A}(t_{\rm s2})]+c\tau_{\rm B}^{\rm r }+c\tau_{\rm A}^{\rm s}+\Delta\rho_{\rm BA}^{\rm sys}+\varepsilon_{\rm BA} \end{array} \right. \tag{11}\]  
where \(\Delta\rho_{\rm AB}^{\rm sys}\) and \(\Delta\rho_{\rm BA}^{\rm sys}\) are systematic errors, e.g., the phase center offset of the ISL antennas and the relativistic effects, which can be accurately modeled. \(\varepsilon_{\rm AB}\) and \(\varepsilon_{\rm BA}\) are the ranging noise. Sending delays \(\tau_{\rm A}^{\rm s}\) and \(\tau_{\rm B}^{\rm s}\) and receiving delays \(\tau_{\rm A}^{\rm r}\) and \(\tau_{\rm B}^{\rm r}\) can be calibrated onboard with a self-closed loop of transmitting and receiving [19]. Correcting the systematic errors and channel delays, the original ranging equations are reformed as:  
\[\left\{ \begin{array}{l}\overline{\rho}_{\rm AB}=\left|R_{\rm B}(t_{\rm r1})- R_{\rm A}(t_{\rm s1})\right|+c\delta t_{\rm B}(t_{\rm r1})-c\delta t_{\rm A}(t_{ \rm s1})+\overline{\varepsilon}_{\rm AB}\\ \overline{\rho}_{\rm BA}=\left|R_{\rm A}(t_{\rm r2})-R_{\rm B}(t_{\rm s2}) \right|+c\delta t_{\rm A}(t_{\rm r2})-c\delta t_{\rm B}(t_{\rm s2})+\overline{ \varepsilon}_{\rm BA}\end{array} \right. \tag{12}\]  
where \(\overline{\rho}_{\rm AB}\) and \(\overline{\rho}_{\rm BA}\) are the corrected pseudoranges. The dual one-way pseudoranges need to be reduced to the reference epoch \(t_{0}\), given as \(t_{0}=(t_{\rm r1}+t_{\rm r2})/2\), and the original observations are demonstrated as:  
\[\left\{ \begin{array}{l}\overline{\rho}_{\rm AB}=\left|R_{\rm A}(t_{0})-R_{ \rm B}(t_{0})\right|+c[\delta t_{\rm A}(t_{0})-\delta t_{\rm B}(t_{0})]+ \Delta\rho_{\rm AB}+\overline{\varepsilon}_{\rm AB}\\ \overline{\rho}_{\rm BA}=\left|R_{\rm B}(t_{0})-R_{\rm A}(t_{0})\right|+c[ \delta t_{\rm B}(t_{0})-\delta t_{\rm A}(t_{0})]+\Delta\rho_{\rm BA}+\overline{ \varepsilon}_{\rm BA}\end{array} \right. \tag{13}\]  
where \(\Delta\rho_{\rm AB}\) and \(\Delta\rho_{\rm BA}\) are the pseudorange corrections from observation epoch to reduction epoch \(t_{0}\), and correction values are calculated with predicted orbit and clock deviation parameters:  
\[\left\{