## 2 Algorithm Description
### Traditional Single-Satellite Positioning Algorithm
#### 2.2.1 Clock Error Elimination Principle of Broadband LEO Communication Satellites  
We take the GNSS time as a reference (such as the GPS time), and use the WGS84 coordinate system as the reference coordinate system and an ellipsoidal model as the earth model. In addition, we employ the two-way communication link of the LEO communication satellite to illustrate the principle of eliminating clock bias. We only consider one communication link, but this does not mean a loss of the generality. It is assumed that the GNSS time corresponding to the receiver signal reception time \(t\) is \(t_{u}\) and the satellite time is \(t^{\rm(s)}\). Then, the received signal reception time is  
\[t_{u}=t+\delta t_{u}. \tag{4}\]  
The receiver clock error can be expressed as follows:  
\[\delta t_{u}=t_{u}-t. \tag{5}\]  
In the same way, the available satellite clock difference is:  
\[\delta t^{\rm(s)}=t^{\rm(s)}-t. \tag{6}\]  
According to the GNSS pseudorange observation equation,  
\[\rho^{\rm(s)}=r+\delta t_{u}-\delta t^{\rm(s)}+I+T+\varepsilon_{p}. \tag{7}\]  
Among these equations, the ionospheric time delay \(I\) and tropospheric time delay \(T\) can be estimated using models. The pseudorange measurement noise \(\varepsilon_{p}\) is only used to analyze the impact of measurement errors on the positioning accuracy, and the measurement noise has no effect on pure positioning calculations [22]. \(r\) is the geometric distance from the satellite to the receiver, and it is the unknown variable that needs to be located. Normally, the satellite clock error \(\delta t^{\rm(s)}\) is regarded as a known variable (it can be obtained according to the satellite clock correction parameters provided by the satellite navigation message), but to use the full duplex system of the broadband LEO constellation communication satellite, we deliberately use the satellite clock error as an unknown parameter to solve. Therefore, we move the unknown and known variables in Equation (7) to the left and right sides of the equation, obtaining  
\[r+(\delta t_{u}-\delta t^{\rm(s)})=\rho^{\rm(s)}-I-T-\varepsilon_{p}. \tag{8}\]  
Combining Equations (5) and (6), we can obtain  
\[r+(t_{u}-t^{\rm(s)})=\rho_{r}. \tag{9}\]  
Similarly, according to the GNSS carrier phase observation equation  
\[\Phi^{\rm(s)}=r+\delta t_{u}-\delta t^{\rm(s)}-I+T+\lambda N+\varepsilon_{ \Phi}, \tag{10}\]  
we can obtain  
\[r+(t_{u}-t^{\rm(s)})=\Phi^{\rm(s)}-\lambda N, \tag{11}\]  
where, \(\rho_{r}=\rho^{\rm(s)}-I-T-\varepsilon_{p}\) and \(\Phi_{c}=\Phi^{\rm(s)}+I-T-\lambda N-\varepsilon_{\Phi}\) are the new error-corrected pseudorange measurement value and carrier phase measurement value, respectively; \(\lambda\) is the wavelength corresponding to the frequency of the signal broadcast by the satellite; and \(N\) is the integer ambiguity of the phase value of the phase locked loop (PLL).  
Subsequently, through Equations (4)-(11)and based on the two-way communication link of the LEO communication satellite, the receiver signal reception time \(t_{u}\) and the satellite time \(t^{(s)}\) can be eliminated so that the clocks of the two are as consistent as possible. At this time, we can consider \(t_{u}=t^{(s)}\) and by bringing it into Equations (9) and (11), we can obtain  
\[r=\rho_{r}, \tag{12}\]  
\[r=\Phi^{(s)}-\lambda N. \tag{13}\]  
In this way, the clock error can be eliminated, which reduces the unknown amount that the traditional algorithm needs to solve the clock error. This is also a benefit of the LEO broadband navigation system. Here, our algorithm uses INS based on the pseudorange and pseudorange rates in the INS/LEO integrated navigation mode, so we adopt the pseudorange observation Equation (12) based on GNSS.