## 6 Simulation Experiment
### Simulation Results
Figure 8 exhibits the estimated results of the scale factor and misalignment angle of the FOGs. Figure 8a,d,f are the estimation results of the scale factor. The others are the results of the misalignment error. It can be seen that the nonlinear error EKF recognizes each error item correctly. The linear EKF only recognizes Kgpx and Kgyy. Compared to the nonlinear EKF, the linear EKF oscillates a great deal before it can converge to stable. Figure 9 shows the estimated results of the nonlinear error of the FOG-based MWD. The nonlinear error coefficients are all estimated correctly by the nonlinear EKF. It also could be seen that the nonlinear error of ACCs is recognized at the time of 2000 s and the nonlinear errors of the FOGs were not recognized until 4000 s. The reason for this is that  
Figure 8: Estimated results of the scale factor error and misalignment error (FOGs). before 2000 s, the calibration path changes the orientation of the ACCs many times. When the ACCs pointed to the straight direction, the gravity acceleration could motivate the ACCs' error. Until 4000 s, the turntable had an angular velocity (30\({}^{\circ}\)/s) that could motivate the errors of the FOGs. This high speed of the rotation could motivate the nonlinear error of the FOGs. It could also prove that the calibration path is reasonably designed. Table 3 lists the estimated results of the error items by the linear EKF and nonlinear EKF. It can be seen that the linear EKF only recognizes eight error items and have significant errors. The relative error of the estimated Bgx is 1333.50%. The relative error of the estimated Bay is 53.98%. The relative error of the estimated Kax is 30.25%. It illustrates that when the original data includes nonlinear errors, the linear EKF could not recognize the error items coefficients. However, the nonlinear EKF method could recognize all the error items. Compared to the true value, the relative error of estimated results is less than 10%. However, it can be noticed that the relative error of the ACCs scale factor error is about 15% and the scale factor error of the ACCs is about 15 ppm (parts per million). Therefore, a method to suppress the residual calibration error should be adopted to improve the calibration accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Error Item**} & \multirow{2}{*}{**True Value**} & \multicolumn{2}{c}{**Linear EKF**} & \multicolumn{2}{c}{**Nonlinear EKF**} \\ \cline{3-6}  & & **Results** & **Relative Error** & **Results** & **Relative Error** \\ \hline \multirow{3}{*}{Bias error of FOGs (\({}^{\circ}\)/h)} & Bgx & 0.02 & 0.2867 & 1333.50\% & 0.0199 & \(-\)0.50\% \\ \cline{2-6}  & Bgy & 0.02 & \(\backslash\) & \(\backslash\) & 0.0200 & 0 \\ \cline{2-6}  & Bgz & 0.02 & \(\backslash\) & \(\backslash\) & 0.0197 & \(-\)1.50\% \\ \hline \multirow{3}{*}{Bias error of ACCs (ug)} & Bax & 100 & 133.3151 & 33.32\% & 99.8489 & \(-\)0.15\% \\ \cline{2-6}  & Bay & 100 & 153.9846 & 53.98\% & 99.7889 & \(-\)0.21\% \\ \cline{2-6}  & Baz & 100 & \(\backslash\) & \(\backslash\) & 99.8027 & \(-\)0.20\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The estimated results of all the error items. Figure 9: The estimated results of the nonlinear error. In order to improve the calibration accuracy, we make use of the iterative calibration method to suppress the residual. It takes four times the iteration to achieve this goal. The estimated results of each iteration are shown in Figure 10. By doing the fourth iteration, the residual error in theACC scale factor is only reduced by 0.4 ppm, which is 2.60% of the total residual error, indicating that all error parameters have converged to the ultimate accuracy after the third iteration. Thus, the fourth or subsequent iteration has little effect. By conducting these iterations three times, the residual of the ACC scale factor error was reduced by 53.3%.