# Joint Timekeeping of Navigation Satellite Constellation with Inter-Satellite Links
## 2 Joint Timekeeping Framework
### Establishment of ISLT  
The ISLT is formed as a synthetic clock by utilizing the frequency stability of an ensemble of satellite clocks. One satellite is selected as the master satellite, whose working clock is noted as the reference clock, and other satellites are named as slave satellites. It should be stated that the reference clock does not mean the time reference of the constellation; rather, it is a center node of time comparison. All the slave satellites compare their clocks with the reference clock through ISLs periodically. The time comparison is limited by inter-satellite visibility. The satellite visible to the master satellite is defined as an anchor satellite that is able to directly compare time through an ISL. The one not visible to the master satellite must compare time by the relay of an anchor satellite. The time comparison results are collected by the master satellite and used to generate the ISLT with a timescale algorithm. A brief establishment framework of the ISLT is shown in Figure 1. \(n\) is the number of satellites that participate in the joint timekeeping, and the master satellite is numbered 1.  
As the clock reading is unmeasurable, the ISLT is represented as the time deviation relative to the reference clock. With the support of ground operation control systems, the ISLT operates in parallel with and is synchronized to the GNSS system time. The time deviation of the ISLT relative to the GNSS system time is calculated by:  
\[x_{e}=x_{e1}+x_{1} \tag{1}\]  
where \(x_{e1}\) is the time deviation of the ISLT generated by the ISLT timescale algorithm relative to the reference clock and \(x_{1}\) is the time deviation of the reference clock relative to the GNSS system time, which is estimated through a time comparison link between the master satellite and the ground station. The subscript \"\(e\)\" stands for the synthetic clock (ISLT). As different types of time deviations are involved in the following parts, the deviation relative to the GNSS system time is especially noted as the absolute time deviation.  
In keeping with the actual navigation application, a prediction model of the ISLT is estimated with its historical absolute time deviation data and used for prediction. Combining the real-time results of the ISLT timescale algorithm, the absolute time deviation of the reference clock is predicted as:  
\[\hat{x}_{1}=\hat{x}_{e}-x_{e1} \tag{2}\]  
Figure 1: A brief establishment framework of inter-satellite link time (ISLT).  
where \(\hat{x}_{e}\) is the predicted value of the time deviation of the ISLT. The absolute time deviation of each slave satellite is estimated based on the inter-satellite time comparison:  
\[\hat{x}_{i}=\hat{x}_{1}+x_{i1},\quad i=2,3,\ldots,n \tag{3}\]  
where \(x_{i1}=x_{i}-x_{1}\) is the time deviation of slave satellite \(i\) relative to the master satellite obtained by an inter-satellite time comparison. It is obvious that the indirect prediction is more accurate than direct prediction of satellite clocks, when the ISLT is more stable and predictable than the satellite clock.