## 5 An Iterative Calibration Method  
The error coefficients recognized by the 36-order EKF are not the true value of the error items. Between the true coefficients and the recognized coefficients, there are still some residual errors. If the residual error of these coefficients could be estimated and compensated, the accuracy of the calibration could be improved. Therefore, this paper proposed a calibration residual suppression method based on an iterative algorithm. Assuming that the true value of the error coefficients is \(E_{t}\), for the first calibration, its estimated results is \(E_{1}\). The relation between \(E_{t}\) and \(E_{1}\) is described as follows:  
\[E_{1}=(\mathbf{I}+M_{1})E_{t} \tag{27}\]  
where \(\mathbf{I}\) is a unit matrix and \(M_{1}\) is the error matrix with the same size with \(\mathbf{I}\). Thus, the residual error of the coefficients of the error items is \(D_{1}\):  
\[D_{1}=E_{1}-E_{t}=M_{1}\mathbf{:}E_{t} \tag{28}\]  
Figure 4: The 24 positions of the calibration. Next, we use Equations (25) and (26) to compensate for the error in the original data. Here we rewrite the Equations (25) and (26) as follows:  
\[\omega_{ib}^{b1}=\left(\mathbf{I}+\delta k_{g}^{1}\right)^{-1}\left(\widetilde{ \omega}_{ib}^{b}-B_{g}^{1}-\varepsilon\omega_{ib}^{b}-d_{g}^{1}\cdot\left( \widetilde{\omega}_{ib}^{b}\right)^{2}\right) \tag{29}\]  
\[\dot{f}_{ib}^{1}=\left(\mathbf{I}+\delta k_{a}^{1}\right)^{-1}\left(\dot{ \tilde{f}}_{ib}^{b}-B_{a}^{1}-\varepsilon\omega_{ib}^{b}-d_{a}^{1}\cdot\left( \dot{\tilde{f}}_{ib}^{b}\right)^{2}\right) \tag{30}\]  
The superscript \(1\) in \(\delta k_{g}^{1}\), \(\delta k_{a}^{1}\), \(\mathbf{B}_{g}^{1}\), \(\mathbf{B}_{a}^{1}\), \(\mathbf{d}_{g}^{1}\). \(\mathbf{d}_{a}^{1}\) means the error coefficients which is estimated for the first time. The \(\omega_{ib}^{b1}\) and \(\tilde{f}_{ib}^{1}\) are the FOG and ACC data, which are compensated the error by using the first estimate of the coefficients. Then, we use the 36-order EKF again to estimate the residual \(\mathbf{D}_{1}\) in \(\omega_{ib}^{b1}\) and \(\dot{f}_{ib}^{b1}\) again. If the estimated result is \(\widetilde{\mathbf{D}}_{1}\), the relation between \(\widetilde{\mathbf{D}}_{1}\) and \(\mathbf{D}_{1}\) is given by:  
\[\widetilde{\mathbf{D}}_{1}=(\mathbf{I}+\mathbf{M}_{2})\mathbf{D}_{1}=(\mathbf{I}+\mathbf{M}_{ 2})\mathbf{M}_{1}\cdot\mathbf{E}_{t} \tag{31}\]  
Whereas \(\mathbf{M}_{2}\) is an error matrix of the second time estimation. Then after the second time estimation, the residual error is \(\mathbf{D}_{2}\). \[\mathbf{D}_{2}=\widetilde{\mathbf{D}}_{1}-\mathbf{D}_{1}=\mathbf{M}_{2}\cdot\mathbf{M}_{1}\cdot\bm {E}_{t} \tag{32}\]  
According to the second estimation, the error coefficients should be adjusted as follows:  
\[ \begin{array}{ll}\mathbf{B}_{g}=\mathbf{B}_{g}^{1}+\mathbf{B}_{g}^{2},&\mathbf{B}_{a}=\mathbf{B }_{a}^{1}+\mathbf{B}_{g}^{2}\\ \delta k_{g}=\delta k_{g}^{1}+\delta k_{g}^{2},&\delta k_{a}=\delta k_{a}^{1}+ \delta k_{a}^{2},\\ \mathbf{d}_{g}=\mathbf{d}_{g}^{1}+\mathbf{d}_{g}^{2},&\mathbf{d}_{a}=\mathbf{d}_{a}^{1}+\mathbf{d}_{a}^ {2}\end{array}  \tag{33}\]  
The superscript \(2\) in \(\delta k_{g}^{2}\), \(\delta k_{a}^{2}\), \(\mathbf{B}_{g}^{2}\), \(\mathbf{B}_{a}^{2}\), \(\mathbf{d}_{g}^{2}\), \(\mathbf{d}_{a}^{2}\) represents the error coefficients that are estimated for the second time. Then, Equations (25) and (26) are used to compensate for the error. Next, the 36-order EKF is used again to estimate the residual error after the second estimation. The rest may be deduced in the same way.