## 3 Improved Rotational Modulation Scheme Based on Tri-Axis RINS  
### Design of Improved Rotational Modulation Scheme  
Theoretically, both the multi-position stop scheme and the continuous rotation scheme can completely offset the constant errors of the IMU. Moreover, due to continuous rotation, the gyro will always be sensitive to the rotational angular velocity. Although the errors can be offset by the complete rotation, it still causes a large error oscillation, which is why the multi-position stop scheme is currently more widely used than the continuous rotation scheme. However, the FOG is sensitive to temperature and environment [44; 45], and the residual calibration error will change with time, so the modulation effect of the multi-position stop rotation scheme will decrease. As a simple example, it assumes that the gyro drift has a relationship that changes with time as \(\varepsilon(t)=\varepsilon_{0}\cdot e^{-0.001t}\), where \(\varepsilon_{0}=0.0035^{\circ}/\)h. As shown in the top subfigure in Figure 4, the gyro drift under the s-frame changes significantly with time, and we can offset the drift using both a symmetrical position and continuous rotation. Then, without considering other errors, the projection of the drift under the b-frame is the lower subplot in Figure 4, where the time-varying drift becomes a sine wave after continuous rotation, while the symmetric position is a square wave. After integrating under the b-frame with time, the error angles can be obtained in Figure 5, and it is obvious that the final remaining cumulative error caused by continuous rotation is smaller than the symmetrical position, indicating that the effect of rotation modulation is better. Certainly, shortening the stopping time can further improve the rotational modulation of the symmetric position, but this is essentially approaching continuous rotation as well. In addition, as for the mean value of the error angle, the error oscillation caused by continuous forward and reverse rotation is double-sided, whereas the symmetrical position is one-sided, which has a larger equivalent error. Furthermore, as described in [43], it is worth noting that when the IMU rotates around the horizontal axis by a certain angle, it will excite the influence of scale factor errors and installation errors of the FOG, thus resulting in platform declination, which will be reflected in the velocity error immediately after the rotation. The scale factor error of FOG is poor compared to the laser gyroscope, so the velocity error after flipping the IMU is an observable error, which can even be used as a calibration method. Accordingly, to fully benefit from these two rotation schemes and the structure of tri-axis RINS with FOG, we planned to design an improved rotational modulation scheme based on multi-position stops and continuous rotation. For an improved rotational modulation scheme of tri-axis RINS, it is essential to design which rotation axis state is multi-position stop and which axis state is continuous rotation. From the analysis in [43], it can be seen that, for INS with FOG, the rotation around the horizontal axis is more suitable to  
Figure 4: Gyro drift using different rotation schemes. complete a complete rotation than a certain angle rotation. When the system rotates around the azimuth axis, the platform deflection caused by the azimuth deviation projected into the horizontal plane takes some time to accumulate. Therefore, we design the improved rotational modulation scheme as follows: the inner gimbal is the multi-position stop, while the middle and outer gimbal are the continuous forward-reverse-reverse-forward rotation. The advantage of such continuous rotation is that not only does it make the error accumulation due to gyro scale factor errors zero, but its average value is also zero. As shown in Figure 6, when the rotational angle of the inner encoder is between \(0^{\circ}\) and \(180^{\circ}\), the IMU completes the forward-reverse-reverse-forward rotation around the middle and outer gimbals.