# Joint Timekeeping of Navigation Satellite Constellation with Inter-Satellite Links
###### Abstract  
As a system of ranging and positioning based on time transfer, the timekeeping ability of a navigation satellite constellation is a key factor for accurate positioning and timing services. As the timekeeping performances depend on the frequency stability and predictability of satellite clocks, we propose a method to establish a more stable and predictable space time reference, i.e., inter-satellite link time (ISLT), uniting the satellite clocks through inter-satellite links (ISLs). The joint timekeeping framework is introduced first. Based on the weighted average timescale algorithm, the optimal weights that minimize the increment of the ISLT timescale are determined and allocated to the clock ensemble to improve the frequency stability and predictability in both the long and short term. The time deviations with respect to the system time of nine BeiDou-3 satellites through multi-satellite precise orbit determination (MPOD) are used for joint timekeeping evaluation. According to the Allan deviation, the frequency of the ISLT is more stable than the nine satellite clocks in the short term (averaging time smaller than 7000 s), and its daily stability can reach \(6\times 10^{-15}\). Meanwhile, the short-term (two hours) and long-term (10 h) prediction accuracy of the ISLT is 0.18 and 1.05 ns, respectively, also better than each satellite clock. Furthermore, the joint timekeeping is verified to be robust against single-satellite malfunction. joint timekeeping; inter-satellite link time; timescale algorithm; frequency stability; predictability 10.3390/s20030670 ranging and communication [5]. To mitigate the spatial limitations of regional monitor networks, BeiDou ISLs have been used for time transfer from stations to overseas satellites. Inter-satellite time transfer is implemented with the ISL system of dual one-way ranging. The parameters of orbits and clock deviations are successfully decoupled and estimated separately [6; 7]. According to the analysis of [8], the precision of BeiDou-3e inter-satellite clock deviations is superior to 0.3 ns and coincidental with satellite-ground clock deviations when channel delays are corrected. In this research, we propose a more comprehensive method of joint timekeeping by using a navigation satellite clock ensemble. The inter-satellite time deviation is used to establish a more stable and predictable synthetic timescale by using a timescale algorithm. The synthetic timescale based on ISLs is defined as the inter-satellite link time (ISLT). The most common timescale algorithms are the weighted average and Kalman filtering algorithms, from which some algorithms are modified. The classical weighted average timescale algorithms include ALGOS, which is used by the International Bureau of Weights and Measures (BIPM) [9], and AT1, which is used by the National Institute of Standards and Technology (NIST) [10], both of which include goals of setting up basic timescale equations. With the input of time deviations, the weighted average algorithm optimizes the frequency stability of the synthetic timescale by adjusting the weights of individual clocks. Kalman filtering algorithms began to be applied to timescales in the 1970s. By the early 1980s atomic time TA(NIST), one of the most broadly used early timescales, had been established by using these algorithms [11; 12]. The Kalman timescale is mainly determined by a physical clock with the most stable frequency in the long term and ignores frequency stability in the short term. For the comprehensive utilization of frequency stability in both long-term and short-term of atomic clocks, [PERSON] [13] proposed the AT2 timescale algorithm, which combines the AT1 algorithm and Kalman filtering. The AT2 algorithm refactored time deviations with frequency deviations, which were estimated by a Kalman filter, and then formed a synthetic timescale by weighted averaging.