## 4 Simulations and Experiments on Tri-Axis RINS  
### Simulations  
The error parameters are set in Table 6, designed based on the actual values of inertial devices' performance parameters. To better simulate the characteristics of INS with FOG, 10% of the actual calibration result of the temperature coefficient of each error is taken as the remaining error of calibration. Due to the large number of temperature error coefficients and different temperature models, the numerical value of each temperature coefficient is not shown in the manuscript. The initial location is 116\({}^{\circ}\) E, 40\({}^{\circ}\) N, and the b-frame coincides with the n-frame. The schematic diagram of the simulation is shown in Figure 10, and the navigation results were obtained by completing a 12-h static navigation simulation using Schemes A, B, and C.  
\begin{table}
\begin{tabular}{l l l l} \hline
**Rotation Scheme** & **Zero Bias Error** & **Scale Factor Error** & **Installation Error** \\ \hline Scheme A & \(-\Delta z_{z}\sin L\cdot t\) & \([0,0,\pi\Delta K_{gz}]^{T}\) & \([0,0,0]^{T}\) \\ \hline Scheme B & \(\approx 0\) & \([0,0,4\pi\cdot\Delta K_{gz}]^{T}\) & \(\left[\left(3\delta_{z}^{G}-2\delta_{xy}^{G}\right)/2,-5\delta_{xy}^{G}/2,0 \right]^{T}\) \\ \hline Scheme C & \(-\frac{1}{1D}\Delta z_{z}\sin L\cdot t\) & \([0,0,0]^{T}\) & \([0,0,0]^{T}\) \\ \hline \end{tabular}
\end{table}
Table 5: Comparison of error modulation effect of different schemes.  
Figure 9: The error angles of the three rotation schemes.  
The eastward and northward position errors for the static navigation tests \(\Delta S_{E},\Delta S_{N}\) are defined in Equations (11) and (12). \(\lambda_{0},L_{0}\) represent the initial latitude and longitude of RINS and \(R_{e}\) represent the radius of the Earth.  
\[\Delta S_{E}=\left(\lambda-\lambda_{0}\right)\cdot R_{e}\cdot\cos L_{0} \tag{11}\]  
\[\Delta S_{N}=\left(L-L_{0}\right)\cdot R_{e} \tag{12}\]  
Additionally, to evaluate the navigation accuracy uniformly for different time lengths, the calculation process of the Circular Error Probability (CEP) is shown in Equations (13) and (14):  
\[\Delta R_{i}=\sqrt{\left(\Delta S_{Ei}\right)^{2}+\left(\Delta S_{Ni}\right)^{ 2}} \tag{13}\]  
\[CEP=0.83\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(\frac{\Delta R_{i}}{T_{i}} \right)^{2}} \tag{14}\]  
where \(T_{i}\) is the navigation time sequence, which is a sequence of 1 s intervals in this manuscript? \(\Delta R_{i}\) is the radial position error at the moment of \(T_{i}\) and \(n\) represents the number of sampling points throughout the navigation process.  
As shown in Figure 11, the position error of Schemes B and C is significantly smaller than that of Scheme A. The CEPs of the three rotation schemes are respectively 0.129 n mile/h, 0.084 n mile/h, and 0.034 n mile/h, and the CEP of Scheme C is the smallest. The maximum radial position errors within the 6-h navigation of the three rotation schemes are 1835.4 m, 687.1 m, and 536.1 m, which also shows the advantage of Scheme C. Therefore, the improved rotational modulation scheme is a good choice for tri-axis RINS with FOG, especially when the navigation time is short because the effect of z-gyro drift is not so evident.  
\begin{table}
\begin{tabular}{l l} \hline
**Error Parameters** & **Value** \\ \hline Constant gyro drift & \(\left[\varepsilon_{x},\varepsilon_{y},\varepsilon_{z}\right]^{T}=\left[0.002 \text{\textdegree}/\text{h},0.002\text{\textdegree}/\text{h},0.0035\text{ \textdegree}/\text{h}\right]^{T}\) \\ Gyro random walk coefficient & \(\left[\varepsilon_{ux},\varepsilon_{yy},\varepsilon_{uz}\right]^{T}=\left[0.0003