## 7 Experiment
### Vibration Experiment
#### 7.2.2 The Random Vibration  
The second experiment is the random vibration. When the random vibration was conducted, the FOG-based MWD was mounted on the vibration table with its \(X\)-axis along the vibration direction. The experiment was carried out according to the following steps: first, the vibration table was kept stationary for 20 min, then the vibration table started to vibrate and kept vibrating for 1 h.  
The power density spectrum of random vibration is shown in the Figure 17 below. Lastly, the vibration table went back to stationary and was kept for 10 min. The experiment time was 90 min. The sampling rate of the FOG-based MWD was 100 Hz. Figure 16: The attitude error. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Error Item**} & \multirow{2}{*}{**Original Data**} & \multicolumn{2}{c}{**Compensated the Linear Errors**} & \multicolumn{2}{c}{**Compensated the Linear and Nonlinear Errors**} \\ \cline{3-6}  & & **Error** & **The Ratio of Error Reduction** & **Error** & **The Ratio of Error Reduction** \\ \hline Latitude error (m) & 86,500 & 56,700 & 34.45\% & 2402 & 97.22\% \\ \hline Longitude error (m) & 62,200 & 53,800 & 13.50\% & 4089 & 93.42\% \\ \hline East velocity error (m/s) & 27.72 & 24.56 & 11.40\% & 15.40 & 44.44\% \\ \hline North velocity error (m/s) & 34.54 & 21.88 & 36.65\% & 6.72 & 80.54\% \\ \hline Inclination (\({}^{\circ}\)) & 1.77 & 1.29 & 27.61\% & 0.55 & 69.13\% \\ \hline Toofface angle (\({}^{\circ}\)) & 1.42 & 1.28 & 11.72\% & 0.81 & 44.13\% \\ \hline azimuth (\({}^{\circ}\)) & 4.53 & 3.23 & 28.79\% & 0.42 & 90.74\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The measurement error of the borehole trajectory. As mentioned in the above experiment, three sets of data were used to compare the trajectory survey result, which is shown in Figures 18 and 19. Figure 17: The power density spectrum of random vibration. Figure 18: Position error and velocity error. In Figure 18, it can be seen that the latitude error and longitude error calculated with the original data is 2754 m and 2828 m. With the second set of data, the latitude error and longitude error were reduced to 2420 m and 2487 m. However, with the third set of data, the error of latitude and longitude was reduced to 1716 m and 1786 m. In Figure 19, the error of inclination, toolface angle and azimuth calculated with the original data is \(-0.04^{\circ}\), \(0.08^{\circ}\) and \(0.20^{\circ}\), respectively. After the data compensated the linear error items, its error of inclination, toolface angle and azimuth were reduced to \(-0.016^{\circ}\), \(0.04^{\circ}\) and \(0.16^{\circ}\), respectively. When the data was compensated, both the linear error items and nonlinear error items, the error of inclination, toolface angle and azimuth were reduced to \(-0.006^{\circ}\), \(0.007^{\circ}\) and \(0.13^{\circ}\). Compared to the original data, the attitude error decreased by 83.78%, 91.25%, and 34.80%. The results of the random vibration experiment are not as promising as the periodic experiment. In addition, the position error shown in Figure 18 and the azimuth error shown in Figure 19 increased with time. As we all know, the defect of the inertial navigation algorithm is that its error results accumulate over time, which is mainly caused by the sensor's error. Although we have compensated the linear error items and nonlinear error items of the FOGs and ACCs, there are still some residual errors in the FOGs and ACCs. The residual errors could also lead to the position and attitude error divergent. However, compared to the first two sets of data, the results of the third set of data diverge much slow over time. It can prove that the proposed method of this paper could suppress the error. Moreover, according to Equations (5) and (7), the nonlinear error is directly related to the quadratic term of angular velocity and acceleration. Therefore, the more intense the carrier motion, the greater the angular velocity and acceleration, and the larger resulting residual  
Figure 19: The attitude errors.