## 5 Two-Way Measurement Method in CCFD
### System Model of Two-Way Measurement Method in CCFD
Substituting Equations (33) and (34) into Equation (27), the following expressions can be obtained as follows:  
\[ \begin{array}{ll}\Delta D_{ij}&{=}\big{|}\mathbf{r}_{j}(t^{s1})-\mathbf{r}_{i}(t^{s1 })\big{|}{-}\big{|}\mathbf{r}_{j}(t_{0})-\mathbf{r}_{i}(t_{0})\big{|}{+}\upsilon_{j}(t ^{s1})\cdot\mathbf{e}_{ij}(t^{s1})\cdot(t^{r1}-t^{s1})\\ &=\mathbf{v}_{j}(t^{s1})\cdot\mathbf{e}_{ij}(t^{s1})\cdot(t^{r1}-t^{s1})\\ \end{array}  \tag{35}\]  
\[ \begin{array}{ll}\Delta D_{ji}&{=}\big{|}\mathbf{r}_{i}(t^{s2})-\mathbf{r}_{j}(t^{s2 })\big{|}{-}\big{|}\mathbf{r}_{j}(t_{0})-\mathbf{r}_{i}(t_{0})\big{|}{+}\upsilon_{i}(t ^{s2})\cdot\mathbf{e}_{ji}(t^{s2})\cdot(t^{r2}-t^{s2})\\ &=\big{|}\mathbf{r}_{i}(t^{s2})-\mathbf{r}_{j}(t^{s2})\big{|}{-}\big{|}\mathbf{r}_{j}(t^{s 1})-\mathbf{r}_{i}(t^{s1})\big{|}{+}\upsilon_{i}(t^{s2})\cdot\mathbf{e}_{ji}(t^{s2}) \cdot(t^{r2}-t^{s2})\\ \end{array}  \tag{36}\]  
Let \(D(t)=\big{|}\mathbf{r}_{j}(t)-\mathbf{r}_{i}(t)\big{|}\), representing the spatial distance between satellites at different times. A first-order Taylor expansion of \(D(t^{s2})\) at time \(t^{s1}\) is performed. Since the duration of a single two-way measurement is short, the inter-satellite acceleration can be approximated as zero [3], allowing the Taylor expansion remainder term to be neglected. The expansion of \(D(t^{s2})\) and Equation (36) can be expressed as follows:  
\[D(t^{s2})=D(t^{s1})+\frac{dD(t)}{dt}\Big{|}_{t=t^{s1}}(t^{s2}-t^{s1}) \tag{37}\]  
\[\Delta D_{ji}=\frac{dD(t)}{dt}\Big{|}_{t=t^{s1}}(t^{s2}-t^{s1})+\upsilon_{i}( t^{s2})\cdot\mathbf{e}_{ji}(t^{s2})\cdot(t^{r2}-t^{s2}) \tag{38}\]  
To better describe the calculation error, we can define it as follows:  
\[\left\{ \begin{array}{ll}\Delta D_{ji\_spatial\_rekcon}=\frac{dD(t)}{dt} \Big{|}_{t=t^{s1}}(t^{s2}-t^{s1})\\ \Delta D_{ji\_dynamic\_compensate}=\upsilon_{i}(t^{s2})\cdot\mathbf{e}_{ji}(t^{s2}) \cdot(t^{r2}-t^{s2})\\ \Delta D_{ij\_dynamic\_compensate}=\upsilon_{j}(t^{s1})\cdot\mathbf{e}_{ij}(t^{s1}) \cdot(t^{r1}-t^{s1})\\ \end{array} \right. \tag{39}\]  
where \(\Delta D_{ji\_spatial\_rekcon}\) is the \"spatial reckoning\" term resulting from the difference between transmission and reckoning times. \(\Delta D_{ij\_dynamic\_compensate}\) and \(\Delta D_{ji\_dynamic\_compensate}\) are the \"dynamic compensation\" terms caused by the high-speed motion of the satellites. Therefore, the distance correction term for the inter-satellite two-way measurement can be rewritten as follows:  
\[ \begin{array}{ll}\Delta D_{ij}=\Delta D_{ij\_dynamic\_compensate}\\ \Delta D_{ji}=\Delta D_{ji\_dynamic\_compensate}+\Delta D_{ji\_spatial\_rekcon} \\ \end{array}  \tag{40}\]  
Under the monitoring and control of ground stations, the clock bias between BeiDou satellite local time and BDT generally does not exceed 1 ms [3]. Therefore, the clock bias between satellites is typically maintained within 2 ms. The following relationship can be obtained as follows:  
\[\Big{|}t^{s2}-t^{s1}\Big{|}<2\text{ms} \tag{41}\]  
As a navigation satellite, it stores its own broadcast ephemeris and the almanac of other satellites in the constellation. The precision of calculating satellite velocity using the broadcast ephemeris is 0.1 mm/s, while using the almanac yields a precision of dm/s [66]. Therefore, after calculating with the satellite's local ephemeris and almanac, the correction precision of \(\Delta D_{ji\_spatial\_recbon}\) can reach the sub-millimeter level, and the correction error can be considered negligible. For the TDHD system, the reckoning time is often chosen at the midpoint of the two-way link transmission period. For a 3 s time slot, the difference between the reckoning time and the transmission time is approximately 0.75 s. In this case, the correction precision of the \"spatial reckoning\" term is only at the centimeter level, which does not meet the high-precision ranging requirements for ISLs. The CCFD system can effectively overcome this issue.