## 4 Recognition of the Coefficient of the Vibration Error
### Calibration Design  
Different error items need different motions to motivate. The error of the ACC needs the positive and negative acceleration to motivate and the error of the FOG needs the positive and negative angular velocity to motivate. In addition, every particular motion should be maintained for a period of time to fully motivate the error. Additionally, in order to save time, the calibration procedure should be kept as simple as possible [26]. In this paper, relying on the turntable, a calibration method based on 24 positions has been designed. The process of calibration is reported in Table 1. The process of the calibration experiment has been carried out in the following steps.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Transfer Process** & \
\begin{tabular}{c} **Keep Time after** \\ **Transfer (s)** \\ \end{tabular} & **Number** & **Transfer Process** &
\begin{tabular}{c} **Keep Time after** \\ **Transfer (s)** \\ \end{tabular} \\ \hline
0 & Static & 0–600 & / & / & / \\ \hline
1 & Rotate \(Y\)-axis by +90\({}^{\circ}\) & 603–783 & 13 & Rotate \(X\)-axis by \(-90^{\circ}\) & 2817–2997 \\ \hline
2 & Rotate \(Y\)-axis by +180\({}^{\circ}\) & 789–969 & 14 & Rotate \(X\)-axis by \(-90^{\circ}\) & 3000–3180 \\ \hline
3 & Rotate \(Y\)-axis by +180\({}^{\circ}\) & 975–1155 & 15 & Rotate \(X\)-axis by \(-90^{\circ}\) & 3183–3363 \\ \hline
4 & Rotate \(X\)-axis by \(-90^{\circ}\) & 1158–1338 & 16 & Rotate \(Y\)-axis by +90\({}^{\circ}\) & 3366–3546 \\ \hline
5 & Rotate \(X\)-axis by \(-180^{\circ}\) & 1344–1524 & 17 & Rotate \(Y\)-axis by +90\({}^{\circ}\) & 3549–3729 \\ \hline
6 & Rotate \(X\)-axis by \(-180^{\circ}\) & 1530–1710 & 18 & Rotate \(Y\)-axis by +90\({}^{\circ}\) & 3732–3912 \\ \hline
7 & Rotate \(Z\)-axis by +90\({}^{\circ}\) & 1713–1893 & 19 & Keep rotating \(X\)-axis at +30\({}^{\circ}\)/s & 3912s–4092 \\ \hline
8 & Rotate \(Z\)-axis by +180\({}^{\circ}\) & 1899–2079 & 20 &
\begin{tabular}{c} Keep rotating \(X\)-axis at \\ \(-30^{\circ}\)/s \\ \end{tabular} & 4095–4275 \\ \hline
9 & Rotate \(Z\)-axis by +180\({}^{\circ}\) & 2085–2265 & 21 & Keep rotating \(X\)-axis at +30\({}^{\circ}\)/s & 4278–4458 \\ \hline
10 & Rotate \(Z\)-axis by +90\({}^{\circ}\) & 2268–2448 & 22 &
\begin{tabular}{c} Keep rotating \(X\)-axis at \\ \(-30^{\circ}\)/s \\ \end{tabular} & 4461–4641 \\ \hline
11 & Rotate \(Z\)-axis by +90\({}^{\circ}\) & 2451–2631 & 23 & Keep rotating \(X\)-axis at +30\({}^{\circ}\)/s & 4644–5004 \\ \hline
12 & Rotate \(Z\)-axis by +90\({}^{\circ}\) & 2634–2814 & 24 &
\begin{tabular}{c} Keep rotating \(X\)-axis at \\ \(-30^{\circ}\)/s \\ \end{tabular} & 5034–5394 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The calibration path based on 24 positions.  
To start, the FOG-based MWD is static with the X, Y, Z axes pointed to the East, North and Up directions for 600 s. Secondly, the \(Y\)-axis should be rotated at a speed of 30\({}^{\circ}\)/s by 90\({}^{\circ}\) and then keep static for 180 s. Next, the \(Y\)-axis is rotated at a speed of 30\({}^{\circ}\)/s by 180\({}^{\circ}\) and then kept static for 180 s. After that, the \(Y\)-axis is rotated at a speed of 30\({}^{\circ}\)/s by 180\({}^{\circ}\) again and then keep static for 180 s. The rest can be conducted in the same manner.  
After the 18 th step, the FOG-based MWD returns to its original position. Then, \(X\)-axis, \(Y\)-axis, and \(Z\)-axis are rotated at a speed of 30\({}^{\circ}\)/s clockwise and then counter-clockwise for 180 s, respectively. In this way, the calibration path has 24 positions in total, as reported in Figure 4.  
When the calibration experiment is performed, the sensors' output data should be collected simultaneously. Then, using the 36-order EKF to deal with the sensors' output, the coefficients of the error model can be recognized.