## 2 Algorithm Description
### Traditional Single-Satellite Positioning Algorithm
#### 2.2.2 INS + LEO 1 Satellite Integrated Navigation Algorithm without an Altimeter  
The GNSS/INS close combination mode is based on the pseudorange and pseudorange rate, which is a high-level combination mode. Its main feature is that INS and GNSS complement each other. For LEO constellations, since the essence of LEO constellations is mobile communication satellite systems, the clock offset and clock drift of receivers can be eliminated by full duplex (FD) systems, which reduces the unknown variable of the clock that needs to be solved in traditional algorithms; also a characteristic of LEO broadband navigation systems. Therefore, the essence of our algorithm is the INS/GNSS integrated navigation mode based on the range and range rate.  
In addition, the UKF algorithm is a filter algorithm commonly used to address the filtering problems of nonlinear systems [23]. Compared with the EKF, the UKF algorithm does not need to linearize the state transition equation and observation equation of the system through the method of Taylor series expansion, However, based on the unscented transformation (UT) required to deal with the nonlinear transfer of mean and covariance, thus completing the approximation of the probability density distribution of the nonlinear function, it does not need to linearize the state transition equation and observation equation. Therefore, the UKF algorithm avoids rounding errors, and its performance is significantly better than the performance of EKF [24; 25; 26].  
(1) Algorithm Principle  
The principle of the INS+LEO 1 satellite integrated navigation algorithm without an altimeter is as follows. First, the ephemeris data provided by the LEO satellite and the position and velocity provided by INS are used to calculate the range and d range rate corresponding to the INS position and velocity. Then, the difference between the range and range rate measured by the LEO satellite and the previously calculated range and range rate is used as the measured value of the INS. At the same time, the optimal estimation of the error between the LEO satellite and INS is obtained by UKF, and finally, the two systems are calibrated. Through this bidirectional auxiliary mode, the INS+LEO 1 satellite integrated navigation mode can work stably for a period of time, basically satisfying the needs of real-time navigation and positioning of aircraft. Figure 3 shows a schematic diagram of the INS+LEO 1 satellite integrated navigation algorithm without an altimeter.  
(2) Algorithm Flow  
The flow chart of the INS+LEO 1 satellite integrated navigation algorithm without an altimeter, with an INS/LEO integrated navigation mode based on the range and range rates, is shown in Figure 4.  
In this figure, \(X_{1}\) is the corresponding attitude variable, assuming that the position of satellite 1 in the ECEF coordinate system is \((X_{1},Y_{1},Z_{1})\), the real position of the aircraft in the ECEF coordinate system is \((x_{u},y_{u},z_{u})\), and the position obtained by the INS solution is \((x_{I},y_{I},z_{I})\); \(\rho_{1}\) is the real range value of satellite 1; and \(\rho_{I1}\) is the virtual range value. Therefore,  
\[\left\{\begin{array}{l}\rho_{1}=\sqrt{\left(X_{1}-x_{u}\right)^{2}+\left(Y_{1} -y_{u}\right)^{2}+\left(Z_{1}-z_{u}\right)^{2}}\\ \rho_{I1}=\sqrt{\left(X_{1}-x_{I}\right)^{2}+\left(Y_{1}-y_{I}\right)^{2}+ \left(Z_{1}-z_{I}\right)^{2}}\end{array}\right.. \tag{14}\]  
Then, when obtaining the corresponding range and range rate measurements, the optimal state estimation value can be obtained through the UKF, and the state variable and loop can be updated through the process. Therefore, the INS+LEO 1 satellite navigation and positioning algorithm can be realized.