## 3 Improved Rotational Modulation Scheme Based on Tri-Axis RINS
### Error Characteristics of Different Rotation Schemes  
To evaluate the effectiveness of the improved rotational modulation scheme in reducing various inertial device errors, we will compare it with other commonly used schemes in RINS. By comparing the error characteristics of different rotational schemes, specifically focusing on zero bias error, scale factor error, and device installation error, we can demonstrate the advantages of the improved rotational modulation scheme. We denote continuous forward and reverse rotation around the azimuth axis as Scheme A, the conventional 16-position stop as Scheme B, and the improved rotational modulation scheme proposed in this manuscript as Scheme C. To facilitate the analysis, the rotation angular velocity of all schemes in this manuscript is set to \(\omega_{r}\) and the stopping time at each position of Scheme B is \(T_{s}\). When the angular velocity of rotation is \(\omega_{r}\), the time to complete a \(360^{\circ}\) rotation is \(T=2\pi/\omega_{r}\). The periods of the three schemes are \(T_{A}=4\pi/\omega_{r}\), \(T_{B}=16\cdot(\pi/\omega_{r}+T_{s})\) and \(T_{C}=68\pi/\omega_{r}\). A simple diagram of the 16-position stop scheme is shown in Figure 8.  
After getting \(\overset{\rightarrow}{\omega_{is}}\) and \(\overset{\rightarrow}{f_{is}}\), we can analyze the gyro-related and accelerometer-related errors using the same analysis method. Hence, the manuscript focuses on the relevant errors of the FOG, and the analysis of accelerometer-related errors will not be expanded here. Besides, the relationship between time-varying errors and navigation accuracy is complicated, so the following analysis is aimed at constant errors, which is convenient for comparing the characteristics of different rotation schemes. The zero-bias error has a significant impact on the accuracy of navigation over time and is considered one of the most influential errors. The principle of inertial navigation error shows that the equivalent northward drift \(\varepsilon_{N}\) and the equivalent upward drift \(\varepsilon_{U}\) of INS in the geographic coordinate system will cause cumulative errors in longitude. \[\delta\lambda(t)=-(\varepsilon_{N}\cos L+\varepsilon_{U}\sin L)t \tag{8}\]  
Figure 8: The diagram of the 16-position stop scheme. Figure 7: Rotation state of three gimbals in the improved rotational modulation scheme. (**a**) Rotation state of the inner gimbal. (**b**) Rotation state of the middle and outer gimbals. The polar axis drift \(\epsilon_{p}\) is composed of \(\epsilon_{N},\epsilon_{U}\) and the specific expression is \(\epsilon_{p}=-(\epsilon_{N}\cos L+\epsilon_{U}\sin L)\). For RINS, \(\epsilon_{N},\epsilon_{U}\) is the residual gyro drift after rotating modulation, which is also determined by the rotation scheme. We can compare the navigation accuracy of different rotation schemes by calculating the polar axis drift. In Scheme B, the effect of constant gyroscope drift can be canceled out by using symmetrical stop positions, which do not produce equivalent drift. In a complete period of scheme C, the drift of the z-gyro is not modulated when rotating through the inner frame. In addition, the unmodulated drift of x-gyro and y-gyro could be completely canceled because the x and y gyro point exactly oppositely when the angles of the inner encoder are \(0^{\circ}\) and \(180^{\circ}\), respectively. Then, the equivalent polar axis drifts of Schemes A, B, and C are as follows:  
\[\left\{ \begin{array}{l}\epsilon_{pA}=-\Delta\epsilon_{z}\sin L\\ \epsilon_{pB}\approx 0\\ \epsilon_{pC}=-\frac{\Delta\epsilon_{z}\sin L\cdot T_{A}}{T_{C}}=-\frac{1}{ 12}\Delta\epsilon_{z}\sin L\end{array} \right.