## 2 Algorithm Description
### Traditional Single-Satellite Positioning Algorithm
#### 2.1.2 INS-Assisted Single-Satellite Positioning System  
Compared with static positioning, since the aircraft is in a continuous motion state, the motion of the aircraft must be compensated for. The principle of the single-satellite positioning method based on inertial components is shown in Figure 2.  
As displayed in Figure 2, taking any time as the initial reference time (the subscript is 0), three satellite moments are considered together as a group for calculation, and all the calculations are converted into the earth-centered earth-fixed (ECEF) coordinate system. \(\rho_{0}\), \(\rho_{1}\), and \(\rho_{2}\) represent the pseudorange values between the satellite and the aircraft at the initial moment, first moment, and second moment of each group, respectively. Similarly, \(\Delta x_{0}\), \(\Delta y_{0}\), \(\Delta z_{0}\) and \(\Delta x_{1}\), \(\Delta y_{1}\), \(\Delta z_{1}\) represent the displacement changes on the \(x\)-axis, \(y\)-axis, and \(z\)-axis of the spacecraft between the initial moment and the first moment, as well as between the first and second moments of the satellite, respectively, which is also the recorded value of INS. Satellite positioning should obtain pseudorange values. For single-satellite positioning based on Doppler integration, three satellite pseudorange values are required for each group calculation, and the pseudorange calculation formula is as follows:  
\[\rho_{i}=\sqrt{\left(X_{i}-x_{i}\right)^{2}+\left(Y_{i}-y_{i}\right)^{2}+ \left(Z_{i}-z_{i}\right)^{2}}+\delta t_{u}, \tag{1}\]  
where \(i=0,1,2\), \(\rho_{i}\) represents the pseudorange between the satellite and the aircraft at the \(i\)th moment, \(X_{i}\), \(Y_{i}\), \(Z_{i}\) is the coordinate position of the satellite at the \(i\)th moment, \(x_{i}\), \(y_{i}\), \(z_{i}\) is the coordinate position of the aircraft at the \(i\)th moment, and \(\delta t_{u}\) is the clock error of the receiver. Then, through the inertial navigation module, the coordinates between the two groups of aircraft can be obtained as follows:  
\[\left\{\begin{array}{l}x_{1}=x_{0}+\Delta x_{0}\\ y_{1}=y_{0}+\Delta y_{0}\\ z_{1}=z_{0}+\Delta z_{0}\\ x_{2}=x_{1}+\Delta x_{1}\\ y_{2}=y_{1}+\Delta y_{1}\\ z_{2}=z_{1}+\Delta z_{1}\end{array}\right.. \tag{2}\]  
According to (1) and (2), we can obtain three equations with four unknown variables, so we cannot solve the position, and we can use the altimeter to supplement the altitude  
Figure 2: Single-satellite dynamic positioning process based on the inertial navigation system (INS).  
information of the aircraft at moment \(i+1\). The altitude measurement equation is as follows [21]:  
\[h_{1}=\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}. \tag{3}\]  
With simultaneous Equations (1)-(3), the position of the aircraft at the first moment can finally be obtained by using the least square (LS) iteration. The three-dimensional coordinates of the aircraft at the first moment can be obtained, and this result can be used as the initial positioning result, combined with the extended Kalman filter (EKF) to obtain a stable solution.  
Single-Satellite Integrated Navigation Algorithm Based on Broadband LEO Constellation Communication Links