### Method of measuring inter-satellite two-way clock offset

Clock offset measurement with inter-satellite two-way time synchronization is shown in Fig. 1 [12]. The clock offset between satellite A and B is set as Δ*t*. Satellite A and B participating in the measurement of clock offset transmit the timing signal via own sending equipment respectively. The following equation can be obtained after the timing signal from the other party is received via the receiving equipment.

$$ {T}_1=\varDelta t+{t}_2+{\tau}_{BA}+{r}_1+{\delta}_1 $$

(1)

$$ {T}_2=-\varDelta t+{t}_1+{\tau}_{AB}+{r}_2+{\delta}_2 $$

(2)

In the equation, *T*_{1} is the time difference between the timing signals transmitted by satellite A and received from satellite B; *t*_{1} is the time delay of transmitting equipment for satellite A; *r*_{1} is the time delay of receiving equipment for satellite A; *τ*_{BA} is the propagation time delay from satellite B to satellite A; *δ*_{1} is other time delay; *T*_{2} is the time difference between the timing signals transmitted by satellite B and received from satellite A; *t*_{2} is the time delay of transmitting equipment for satellite B; *r*_{2} is the time delay of receiving equipment for satellite B; *τ*_{AB} is the propagation time delay from satellite A to satellite B; and *δ*_{2} is other time delay. The clock offsetΔ*t*of two satellites can be obtained from Eqs. (1) and (2) as shown below.

$$ \varDelta t=\frac{T_1-{T}_2}{2}+\frac{t_1-{t}_2}{2}+\frac{r_2-{r}_1}{2}+\frac{\tau_{AB}-{\tau}_{BA}}{2}+\frac{\delta_2-{\delta}_1}{2} $$

(3)

In Eq. (3), *T*_{1} can be measured via satellite A; *T*_{2} can be measured via satellite B. According to the frequency of transmitting signal of the satellite, *t*_{1}, *t*_{2}, *r*_{1}, and *r*_{2} can be calibrated in advance; therefore, the time delay can be processed as the known value, and the measurement influence on the clock offset can be ignored. In order to highlight the specific influence of satellite motion on measuring inter-satellite clock offset, the satellite clock offset Δ*t* is supposed as unchanged in the measurement process of clock offset, and the simplified calculation formula for clock offset can be obtained as the following.

$$ \varDelta t=\frac{T_1-{T}_2}{2}+\frac{\tau_{AB}-{\tau}_{BA}}{2}+\frac{\delta_2-{\delta}_1}{2} $$

(4)

Seen from formula (4), the precision of inter-satellite clock offset measurement is mainly depending on three main factors of measurement: precision on time difference (*T*_{1}-*T*_{2})/2, transmission time delay difference (*τ*_{AB}-*τ*_{BA})/2 of timing signal, and other delay difference (*δ*_{2}-*δ*_{1})/2. When the clock offset measurement for satellite A and B is conducted at the frequency of higher than once per second, the influence of other time delay (*δ*_{2}-*δ*_{1})/2 on the precision of clock offset measurement is contained into the time difference and time delay difference. Therefore, in order to analyze the influence of satellite motion on the precision of clock offset measurement, we should consider the close frequency of timing signals transmitted by each party at the same time and the basic same of the path passed by the timing signal. Formula (4) can be further simplified as the following.

$$ \varDelta t=\frac{T_1-{T}_2}{2}+\frac{\tau_{AB}-{\tau}_{BA}}{2} $$

(5)

Meanwhile, formula (1) and (2) can be changed as

$$ {T}_1=\varDelta t+{\tau}_{BA} $$

(6)

$$ {T}_2=-\varDelta t+{\tau}_{AB} $$

(7)

Formula (6) and (7) are the basic formula analyzing the influence of satellite motion on the precision of inter-satellite two-way clock offset. If the location of the satellite for two-way clock offset measurement in the measuring process does not change, the clock offset of two satellites can be obtained from formula (5) as the following.

$$ \varDelta t=\frac{T_1-{T}_2}{2} $$

(8)

### The influence of satellite motion on the clock offset measurement and correction method

According to the simulation of satellite constellation, the main performance of inter-satellite motion within the constellation are the mutual close, mutual far-away, and relatively unchanged location [13, 14]. Therefore, on the influence of satellite motion on the inter-satellite two-way clock offset measurement, the mutual close and mutual far-away are only considered for the following analysis. In order to simplify the analysis, the initial inter-satellite distance of satellite is *R*; the speed from satellite A to satellite B is *v*_{A}; and the speed from satellite B to satellite A is *v*_{B}. Then, *v*_{A} < *v*_{B} supposing the inter-satellite motion is rectilinear motion.

When the mutual close motion is conducted among the satellites, the schematic diagram is as shown in Fig. 2. According to the principle of invariance of light velocity, it can be obtained from the figure.

$$ c\cdot {\tau}_{BA}+{v}_A\cdot {\tau}_{BA}=R $$

(9)

$$ c\cdot {\tau}_{AB}+{v}_B\cdot {\tau}_{AB}=R $$

(10)

It can be obtained from formula (9) and (10).

$$ {\tau}_{BA}=\frac{R}{c+{v}_A} $$

(11)

$$ {\tau}_{AB}=\frac{R}{c+{v}_B} $$

(12)

Substituting formula (6) with formula (11), and formula (7) with formula (12), we can get the following.

$$ {T}_1=\Delta t+\frac{R}{c+{v}_A} $$

(13)

$$ {T}_2=-\Delta t+\frac{R}{c+{v}_B} $$

(14)

The calculation formula for the clock offset at the mutual close of satellite can be obtained from formula (13) and (14).

$$ \Delta t=\frac{T_1-{T}_2}{2}+\frac{1}{2}\cdot \left(\frac{R}{c+{v}_B}-\frac{R}{c+{v}_A}\right)=\frac{T_1-{T}_2}{2}+\frac{R\left({v}_A-{v}_B\right)}{2\cdot \left(c+{v}_B\right)\cdot \left(c+{v}_A\right)} $$

(15)

Formula (15) is the influence of satellite motion on inter-satellite clock offset Δ*t* at the mutual close of the satellite. Taking the influence of satellite motion on the measurement of inter-satellite two-way clock offset out of consideration, we can see from formula (15), if *v*_{A} < *v*_{B}, the value of \( \frac{R\left({v}_A-{v}_B\right)}{2\cdot \left(c+{v}_B\right)\cdot \left(c+{v}_A\right)} \) is negative, which indicates that the calculation result would be bigger than the actual clock offset calculated via formula (8). In order to correct the big calculation result, the corrected value is negative \( \frac{R\left({v}_A-{v}_B\right)}{2\cdot \left(c+{v}_B\right)\cdot \left(c+{v}_A\right)} \).

When the mutual far-away motion is conducted among the satellites, the schematic diagram is as shown in Fig. 3. According to the principle of invariance of light velocity, it can be obtained from the figure.

$$ c\cdot {\tau}_{BA}-{v}_A\cdot {\tau}_{BA}=R $$

(16)

$$ c\cdot {\tau}_{AB}-{v}_B\cdot {\tau}_{AB}=R $$

(17)

It can be obtained from formula (16) and (17).

$$ {\tau}_{BA}=\frac{R}{c-{v}_A} $$

(18)

$$ {\tau}_{AB}=\frac{R}{c-{v}_B} $$

(19)

Substituting formula (6) with formula (18), and formula (7) with formula (19), we can get the following.

$$ {T}_1=\Delta t+\frac{R}{c-{v}_A} $$

(20)

$$ {T}_2=-\Delta t+\frac{R}{c-{v}_B} $$

(21)

The calculation formula for the clock offset at the mutual far-away of satellite can be obtained from formula (20) and (21).

$$ \Delta t=\frac{T_1-{T}_2}{2}+\frac{1}{2}\cdot \left(\frac{R}{c-{v}_B}-\frac{R}{c-{v}_A}\right)=\frac{T_1-{T}_2}{2}+\frac{R\left({v}_B-{v}_A\right)}{2\cdot \left(c-{v}_B\right)\cdot \left(c-{v}_A\right)} $$

(22)

Formula (22) is the influence of satellite motion on inter-satellite clock offset Δ*t* at the mutual far-away of the satellite. Taking the influence of satellite motion on the measurement of inter-satellite two-way clock offset out of consideration, we can see from formula (22), if *v*_{A} < *v*_{B}, the value of \( \frac{R\left({v}_B-{v}_A\right)}{2\cdot \left(c-{v}_B\right)\cdot \left(c-{v}_A\right)} \) is positive, which indicates that the calculation result will be smaller than the actual clock offset calculated via formula (8). In order to correct the small calculation result, the corrected value is positive \( \frac{R\left({v}_B-{v}_A\right)}{2\cdot \left(c-{v}_B\right)\cdot \left(c-{v}_A\right)} \).

### Correction method of the inter-satellite clock offset measurement

Seen from formula (15) and (22), the correction value is\( \frac{R\left({v}_A-{v}_B\right)}{2\cdot \left(c+{v}_B\right)\cdot \left(c+{v}_A\right)} \) at the inter-satellite close motion, and it is negative. The correction value is\( \frac{R\left({v}_B-{v}_A\right)}{2\cdot \left(c-{v}_B\right)\cdot \left(c-{v}_A\right)} \) at the far-away motion, and it is positive. Therefore, the caused correction value of clock offset is changed from the negative to the positive from the close motion to the far-away motion. When the correction value is 0, the inter-satellite clock offset from the two-way measurement is nearest to the actual clock offset, and the moment occurred at the turning point from the mutual close to mutual far-away between two satellites. The inter-satellite distance change rate at the moment is 0 via the simulation result.

Via the above error analysis on the measurement result of the satellite motion on inter-satellite two-way clock offset, we can see that, excluding the influence of satellite motion on the clock offset measurement, the clock offset is larger than the actual clock offset at the mutual close motion. The clock offset is smaller than the actual clock offset at the mutual far-away motion if the influence of satellite motion on the clock offset measurement is not considered. If the satellite two-way clock offset measurement is made from the mutual close to mutual far-away, and the influence of satellite motion on the clock offset measurement is not considered, the inter-satellite error clock will change from large to small. The moment for the consistency between measured clock offset and actual clock offset occurs in the process, and the inter-satellite distance change rate among two satellites is 0 at the moment. Therefore, the fitting polynomial of satellite clock offset in the process can be obtained if the least square fitting is conducted for the clock offset data from the measurement in the process.

According to the above analysis on the clock offset measurement error, the requirement on the clock offset measurement precision can be met via one fitting for formula (23). In order to verify the correction method after the occurrence of deviation of measured clock offset caused by the above satellite motion, STK simulation tool is used to generate the clock offset measurement data, and the moment with the inter-satellite distance change rate of 0 in the measurement process of the clock offset is obtained via the simulation. The minimum value between the measured clock offset and actual clock offset can be obtained after substituting the moment into formula (23), and the precision of the correction method can be obtained.