## 2 Algorithm Principle
### Standard INS + LEO Dual-Satellite (LEO2) + Altimeter Integrated Navigation and Positioning  
The principle of the standard INS + LEO dual-satellite (LEO2) switching + altimeter integrated navigation algorithm is: First, we use satellite ephemeris data stored by broadband LEO satellites. The position of the satellite in the orbital coordinate system can be calculated, and then converted to the ECEF coordinate system, so as to obtain the coordinate position of the satellite in the ECEF coordinate system. Then the satellite position of the adjacent time can be differentiated to obtain the satellite's velocity. Then use the position and velocity information which is provided by the INS, and combine with data from an altimeter of the carrier, thereby we can obtain the elevation information. Then use the information to calculate the distance and distance rate relative to the position and speed of the INS, respectively. Subsequently, we determine the difference of the above measured values, and then use them as the filtered observations of UKF, so as to obtain the best estimation value of the LEO satellite and INS error. Finally, the errors of LEO satellite and INS systems are calibrated, respectively. The schematic diagram is shown in Figure 2.  
According to the description in Figure 2, we assume that the ECEF coordinate of satellite 1 is \((x_{1},y_{1},z_{1})\), the ECEF coordinate of satellite 2 is \((x_{2},y_{2},z_{2})\), and the moving carrier coordinate obtained by the INS solution is \((x_{INS},y_{INS},z_{INS})\). \(\rho_{1 LEO}\) and \(\rho_{2 LEO}\) represent the actual ranging values of satellite 1 and satellite 2 to the moving carrier's true distance obtained by satellite 1 and satellite 2 in real time through the ephemeris, respectively; \(\rho_{1 INS}\) and \(\rho_{2 INS}\) represent the distance between the current satellite position calculated by satellite 1 and satellite 2 through the ephemeris to the moving carrier position obtained by the INS solution, respectively. \(H_{INS}\) is the height value of the carrier position given by the INS, and \(H_{\text{Re}}\) is the height value measured by the altimeter of the moving carrier receiver. Based on this, we can list the following equations:  
\[\left\{ \begin{array}{l}\rho_{1 LEO}=\sqrt{\left(x_{1}-x_{c}\right)^{2}+\left( y_{1}-y_{c}\right)^{2}+\left(z_{1}-z_{c}\right)^{2}}\\ \rho_{2 LEO}=\sqrt{\left(x_{2}-x_{c}\right)^{2}+\left(y_{2}-y_{c}\right)^{2}+ \left(z_{2}-z_{c}\right)^{2}}\\ \rho_{1 INS}=\sqrt{\left(x_{1}-x_{INS}\right)^{2}+\left(y_{1}-y_{INS}\right)^{2 }+\left(z_{1}-z_{INS}\right)^{2}}\\ \rho_{2 INS}=\sqrt{\left(x_{2}-x_{INS}\right)^{2}+\left(y_{2}-y_{INS}\right)^{2 }+\left(z_{2}-z_{INS}\right)^{2}}\\ \frac{x_{c}^{2}}{\rho_{0-INS}^{2}}+\frac{y_{c}^{2}}{\rho_{0-INS}^{2}}+\frac{z _{c}^{2}}{\rho_{0-IN}^{2}}=1\\ \frac{x_{c}^{2}}{\rho_{0-Re}^{2}}+\frac{y_{c}^{2}}{\rho_{0-Re}^{2}}+\frac{z _{c}^{2}}{\rho_{0-Re}^{2}}=1\end{array} \right.. \tag{34}\]  
where \(\rho_{0-INS}=R_{e}+H_{INS}\), \(\rho_{\text{P-INS}}=R_{p}+H_{INS}\), \(\rho_{0-Re}=R_{e}+H_{\text{Re}}\), \(\rho_{\text{P-Re}}=R_{p}+H_{\text{Re}}\), regardless of \(H_{\text{INS}}\) or \(H_{\text{Re}}\), which can be determined from the standard atmosphere model. Formula (1). Figure 2: Standard INS + LEO2 + altimeter integrated navigation and positioning principle diagram. According to Equation (17), we calculate the difference between the first term and the third term, the second term and the fourth term, and the last two terms of the equation, respectively; we then obtain:  
\[\left\{ \begin{array}{l}\hat{\rho}_{1-INS}=\rho_{1 LEO}-\rho_{1 INS}\\ \hat{\rho}_{2-INS}=\rho_{2 LEO}-\rho_{2 INS}\\ \rho_{0}=\ (\frac{1}{\rho_{0-Re}^{2}}-\frac{1}{\rho_{0-INS}^{2}})/(\frac{1}{\rho_{0 -Re}^{2}}-\frac{1}{\rho_{0-INS}^{2}})=\frac{z_{c}^{2}}{x_{c}^{2}+y_{c}^{2}}\\ \end{array}