## 4 Calibration and Compensation of Gyro Bias
### Calibration Model of Gyro Drift in the RINS in the Both Navigation Model and Attitude Reference  
In our experimental scheme, the adoption of the rotational modulation process ensures the elimination of sensor noise in the roll and yaw directions. Under this situation, the primary source of navigation error in a one-axis RINS comes from gyros along the rotational direction. The improvement of navigation accuracy could be simplified to calibrate and compensate for gyro bias along the azimuth direction.  
Aimed at calibrating and eliminating the gyro bias along the azimuth direction, we took advantage of the one-axis rotational turntable with a highly precise rotational reference angle to calibrate and compensate the gyro bias during the initial alignment process. During the initial alignment process, the navigation algorithm would resolve the attitude of the carrier. However, gyro bias would cause a resolved attitude error. In this section, the model of a gyro bias-induced azimuth-calculated error is established, which could then be used to calibrate the gyro bias of a RINS.  
We consider a typical situation to explain how the gyro bias would influence the resolved attitude. To simplify the analysis process, we assume the gyro on the z-axis  
Figure 7: Illustration of rotational coordinates in the RINS.  
in b-frame coincides with the azimuth direction in a navigation frame. In this case, the measurement error of the x- or y-axis would not influence the resolved attitude of the azimuth direction. Without the loss of generality, the IMU is set at a latitude of L, and earth is rotating at a speed of \(\omega_{ie}\).  
Assuming the gyro has a bias of \(\epsilon^{b}\), due to the rotation of earth, the azimuth direction vector of the IMU in inertial coordinate at time t can be expressed as follows:  
\[P(t)=(\cos L\cos\omega_{ie}t,\cos L\sin\omega_{ie}t,\sin L) \tag{21}\]  
where L is the latitude and \(\omega_{ie}\) is the rotational speed of earth.  
Here, \(P(t)\) is the position vector of the IMU at time t in an inertial navigation system; \(\omega_{ie}\) is the rotation speed of earth, and L is the latitude of IMU. As shown in Figure 6, the gyro bias-induced attitude calculation-accumulation error could be expressed as \(\int\epsilon^{b}P(t)\). If the RINS works for a period of T, the accumulative azimuth calculation error could be integrated using the following equation:  
\[\begin{array}{ll}\dot{\phi}_{azi}^{i}&=\int_{0}^{T}\epsilon^{b}\times P(t) dt\cdot P(T)\\ &=\epsilon^{b}\times\left[\cos L\frac{\sin\omega_{ie}T}{\omega_{ie}},\cos L \frac{1-\cos\omega_{ie}T}{\omega_{ie}},\sin L\cdot T\right]\cdot P(T)\\ &=\epsilon^{b}\times\left[\frac{\cos^{2}L\sin(\omega_{ie}T)}{ \omega_{ie}}+\sin^{2}L\cdot T\right]\end{array} \tag{22}\]  
Note that the above equation denotes the azimuth calculation error in i-frame. Since navigation system outputs should be expressed in navigation frame, the \(\dot{\phi}_{azi}\) error should be transformed into a navigation frame.  
\[\dot{\phi}_{azi}^{n}=P(t)\dot{\phi}_{azi}^{i} \tag{23}\]  
The above equation is the gyro bias error-induced azimuth angle, which denotes the relationship between the gyro bias and the resolved azimuth output of gyros. This could then be used to calibrate the gyro bias.