## 3 Improved Rotational Modulation Scheme Based on Tri-Axis RINS
### Error Characteristics of Different Rotation Schemes
\tag{9}\]  
From Equation (9), it can be found that Scheme B has almost no equivalent constant polar axis gyro drift, and the polar axis drift of Scheme A is 17 times higher than that of Scheme C. For the RINS studied in this manuscript, the polar axis drift can be reduced to less than \(0.0002\cdot/h\) by using Scheme C. This indicates that using both Scheme B and Scheme C can significantly reduce the positioning error of the tri-axis RINS and improve navigation accuracy over time. It should be emphasized that the variation of IMU-related errors with time and temperature is not considered in the current analysis because it is difficult to calculate quantitatively, which happens to be the significant physical characteristic of FOG. As a result, the actual modulation effect of Scheme C is better than the theoretical analysis and is well suited for tri-axis RINS with FOG. In addition to the drift of the gyroscope, various errors, such as scale factor errors and installation errors, can affect the accuracy of navigation. The rotational angular velocity \(\omega_{r}\) is much larger than the earth's rotational angular velocity \(\omega_{ie}\) so we mainly analyze the coupling error of the scale factor errors and installation errors with \(\omega_{r}\). For the simplicity of the analysis, it is assumed that the RINS is stationary and the direction cosine matrix from the b-system to the n-system is \(C_{n}^{b}=I\). For scheme A, the angles of the middle and outer encoders are zero, and we set the rotation angle speed of the inner gimbal to be the same as scheme C for comparison purposes, so the true angular velocity under the s-frame is \(\overset{\rightarrow}{\omega}_{isA}^{s}=[0,0,\omega_{r}]^{T}\) and the coupling error is:  
\[\Delta\omega_{isA}^{\overset{\rightarrow}{\omega}_{isA}}=C_{n}^{b}C_{S}^{b}C_ {wisA}^{\overset{\rightarrow}{\omega}_{isA}}=R_{z}(\omega_{r}t)\cdot R_{x}(0 )\cdot R_{y}(0)C_{g}^{\overset{\rightarrow}{\omega}_{isA}}\\ =\left[ \begin{array}{ccc}\cos\omega_{r}t&-\sin\omega_{r}t&0\\ \sin\omega_{r}t&\cos\omega_{r}t&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{ccc}\Delta K_{gx}&0&-\overset {\rightarrow}{\omega}_{xy}^{c}\\ -\overset{\rightarrow}{\omega}_{yz}^{c}&\Delta K_{gy}&\overset{\rightarrow}{ \omega}_{yx}^{c}\\ \overset{\rightarrow}{\omega}_{zy}^{c}&-\overset{\rightarrow}{\omega}_{xz}^{c }&\Delta K_{gz}\end{array} \right]\cdot\left[ \begin{array}{ccc}0\\ 0\\ \omega_{r}\end{array} \right]=\left[ \begin{array}{ccc}\cos\omega_{r}t&-\sin \omega_{r}t&0\\ \sin\omega_{r}t&\cos\omega_{r}t&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{ccc}-\overset{\rightarrow}{ \omega}_{xy}^{G}\omega_{r}\\ \Delta K_{gz}\omega_{r}\end{array} \right]\\ =\left[ \begin{array}{ccc}-\overset{\rightarrow}{\omega}_{xy}^{G}\cos\omega_{ r}t\omega_{r}-\overset{\rightarrow}{\omega}_{yx}^{G}\sin\omega_{r}t\omega_{r}\\ -\overset{\rightarrow}{\omega}_{xy}^{G}\sin\omega_{r}t\omega_{r}+\overset{ \rightarrow}{\omega}_{yx}^{G}\cos\omega_{r}t\omega_{r}\\ \Delta K_{gz}\omega_{r}\end{array} \right]\\ \Delta K_{gz}\omega_{r}\end{array}\right] \tag{10}\]  
Therefore, the error angle in the n-frame is \(\Delta\overset{\rightarrow}{\theta}_{isA}^{n}=\int_{0}^{T}\Delta\overset{ \rightarrow}{\omega}_{isA}^{n}dt\) and the error angles after forward rotation of \(360^{\circ}\) and reverse rotation of \(360^{\circ}\) are \(\left[0,0,2\pi\Delta K_{gz}\right]^{T}\) and \(\left[0,0,-2\pi\Delta K_{gz}\right]^{T}\). Using a similar method, we can also obtain the error angles of Schemes B and C in the n-frame and then list them in Tables 3 and 4. The average values of the three rotation schemes \(\Delta\overset{\rightarrow}{\theta}_{isA}^{n},\Delta\overset{\rightarrow}{ \theta}_{isB}^{n},\Delta\overset{\rightarrow}{\theta}_{isC}^{n}\) are \(\left[0,0,\pi\Delta K_{gz}\right]^{T},\left[\left(3\delta_{zx}^{G}-2\delta_{ yx}^{G}\right)/2,-5\delta_{xy}^{G}/2,4\pi\cdot\Delta K_{gz}\right]^{T}\) and \(\left[0,0,0\right]^{T}\) such oscillation errors can cause equivalent drift.