## 6 Simulation Experiment
### Simulation Results  
Then we used the nonlinear EKF and liner EKF to recognize the error items. Their results are demonstrated in Figures 6-8. The blue line represents the error coefficient \begin{table}
\begin{tabular}{c c c c c c} \hline
**Error Item** & **Symbol** & **Value** & **Error Item** & **Symbol** & **Value** \\ \hline \multirow{2}{*}{FOG Bias error} & \multirow{2}{*}{Bgx = Bgy = Bgz} & \multirow{2}{*}{0.02\({}^{\circ}\)/h} & ACC & Kaxy = Kaxz = & \multirow{2}{*}{\(10\times 10^{-6}\) rad} \\  & & & & & Kayx = Kayz = \\  & & & & & Kazz = Kazy \\ \hline \multirow{2}{*}{ACC Bias error} & \multirow{2}{*}{Bax = Bay = Baz} & \multirow{2}{*}{\(100\times 10^{-6}\) g} & FOG Random & \multirow{2}{*}{\(\epsilon\omega_{ibx}^{b}=\epsilon\omega_{iby}^{b}=\epsilon\omega_{ibz}^{b}\)} & The mean is zero and the standard deviation is \(0.02^{\circ}\)/h \\ \hline \multirow{2}{*}{FOG scale factor error} & \multirow{2}{*}{Kgxx = Kgyy = Kgzz} & \multirow{2}{*}{\(50\times 10^{-6}\)} & ACC Random & \multirow{2}{*}{\(\varepsilon f_{ibx}^{b}=\varepsilon f_{iby}^{b}=\epsilon f_{ibz}^{b}\)} & The mean is zero and the standard deviation is \(50\times 10^{-6}\) g \\ \hline ACC scale factor error & Kaxx = Kayy = Kazz & \multirow{2}{*}{\(100\times 10^{-6}\)} & FOG Nonlinear error & dgx = dgy = dgz & \(10\times 10^{-6}\) s/rad \\ \hline FOG Misalignment error & Kgyx = Kgzz = Kgzy & \multirow{2}{*}{\(10\times 10^{-6}\) rad} & ACC Nonlinear error & dx = day = daz & \(10\times 10^{-6}\) s/m \\ \hline \end{tabular}
\end{table} 
Table 2: The values of each error coefficient. Figure 5: The sensor measurements generated according to the 24-position calibration. estimated by the linear EKF, while the red line represents the error coefficient estimated by the nonlinear EKF. Figure 6: The estimated results of the bias error. Figure 7: Estimated results of the scale factor error and misalignment error (ACCs). Figure 6 shows the bias error coefficients recognized by the two methods. In Figure 6a-c, the red lines converge to 100 \(\upmu\)g (the true value) after 2000 s, indicating that nonlinear EKF could recognize the bias error correctly. However, the blue line in Figure 6a,b converges to 150 \(\upmu\)g, indicating that the bias error coefficients of X-axial ACC and Y-axial ACC estimated by linear EKF is incorrect. The blue line in Figure 6c does not converge to a stable value, indicating that the bias error coefficient of Z-axial ACC could not be recognized by linear EKF. The nonlinear EKF could recognize the bias error successfully because the red lines in Figure 6d-f all converge to the true value 0.01\({}^{\circ}\)/h. The linear EKF is a failure to recognize the bias error because the blue lines in Figure 6d-f diverge to a high value after 4000 s. The reason is that at the time of 3912 s, we began to rotate the turntable around \(X\)-axis, \(Y\)-axis and \(Z\)-axis at the speed of 30\({}^{\circ}\)/s for 180 s (Table 1). At this dynamic state, though the nonlinear error coefficients are small (10 \(\times\) 10\({}^{-6}\) s/rad), the nonlinear error items are large to 0.098\({}^{\circ}\)/h. For the linear EKF, its state vector only has the linear error coefficients, and it could not separate the nonlinear error from all the errors. However, the state vector of the nonlinear EKF, which includes the linear and nonlinear error coefficients, which can recognize the error items effectively. Figure 7 shows the estimated results of the scale factor error and misalignment error of the ACCs. The three figures located in the diagonal place in Figure 7 are the estimation results of the scale factors. The others are the estimation results of the misalignment angle. It can be seen that the linear EKF could not recognize any of the misalignment angles, while the nonlinear EKF could recognize each error item correctly (100 ppm). The linear EKF misidentifies the scale factors \(k_{xxx}\) and could not recognize the scale factors \(k_{azz}\). On the contrary, the nonlinear error EKF recognizes all of the scale factors of the ACCs.