## 3 Parameter Modeling and Error Analysis of the INS and GNSS Systems
### GNSS Error Analysis and Modeling
#### 3.2.4 Multipath Effect Error  
When the user receives the satellite signal, the satellite signal reflected by the surrounding objects may also be received by the receiver. In this case, the two signals will interfere, and there is a certain deviation between the observed value and the real value, which is the source of the multipath effect [31]. In practical engineering, we can reduce the multipath error by choosing the right station, choosing the appropriate GNSS receiving antenna, and prolonging the observation time. Since the multipath model is more complicated, we will not discuss it here. Our algorithm assumes that the multipath error is a constant within a reasonable range. State Modeling of a Single-Satellite Integrated Navigation Positioning Algorithm Based on Broadband LEO Constellation Communication Links  
Since the loose combination requires the GNSS solution result, at least four visible satellites are required. In some places, due to occlusion or a challenging environment, the number of visible satellites cannot be satisfied. At this time, the loose combination cannot be used. However, the close combination is a high-level combination mode. In the close combination mode, not only will the GNSS correct the INS, but the INS will also correct the GNSS. This bidirectional auxiliary method makes the positioning accuracy significantly higher than the positioning accuracy of the loose combination. Even if the number of visible satellites is fewer than four, the tightly integrated mode can work stably for a period of time. Therefore, to match our algorithm, we use tight integrated navigation to analyze and employ the UKF algorithm for filtering. State Equation of Single-Satellite Integrated Navigation and Positioning Algorithm Based on Broadband LEO Constellation Communication Links  
When designing the algorithm of an integrated navigation system, we usually use the indirect filtering method. The state variables of the system include two parts, namely, the error of INS and the error of GNSS. However, the scene applied to the integrated navigation algorithm in this paper is the three-dimensional position navigation and positioning of aircraft under a LEO constellation. Since the state equation of GNSS is similar to that of INS, only INS error is listed here. In the process of selecting the state variables of the INS state equation, according to Section 3.1, the attitude error, velocity error, position error of INS, first-order Markov drift, and Markov offset of the gyroscope and accelerometer can be gathered together to obtain the state variables as follows:  
\[X_{I}=[\phi_{N},\phi_{E},\phi_{D},\delta V_{N},\delta V_{E},\delta V_{D}, \delta L,\delta\lambda,\delta h,\epsilon_{N},\epsilon_{E},\epsilon_{D}, \
abla_{N},\
abla_{E},\
abla_{D}]^{T}, \tag{32}\]  
where \(\delta\lambda\) and \(\delta h\) denote the longitude error and altitude error, respectively. The error state equation of INS is as follows:  
\[\dot{X}_{I}(t)=F_{I}(t)X_{I}(t)+G_{I}(t)W_{I}(t). \tag{33}\]  
The 15-dimensional state variables \(X_{I}(t)\) are shown in Equation (32), and the process noise vector is  
\[W_{I}(t)=[\omega_{gN},\omega_{gE},\omega_{gD},\omega_{aN},\omega_{aE},\omega_{ aD}]^{T}, \tag{34}\]  
where \(\omega_{gN}\), \(\omega_{gE}\), and \(\omega_{gD}\) represent the random drift noise of the gyroscope in the NED direction, and \(\omega_{aN}\), \(\omega_{aE}\), and \(\omega_{aD}\) represent the random drift noise of the accelerometer in the NED direction. The state transition matrix \(F_{I}(t)\) is a \(15\times 15\)-dimensional parameter matrix, and \(G_{I}(t)\) is a \(15\times 6\)-dimensional noise driving matrix. Similarly, the noise drive matrix \(G_{I}(t)\) is as follows:\[G_{1}(t)=\left[ \begin{array}{cc}C_{b}^{n}&O_{3\times 3}\\ O_{3\times 3}&C_{b}^{n}\\ I_{3\times 3}&O_{3\times 3}\\ I_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&I_{3\times 3}\end{array} \right]_{15\times 6}.