## 4 Simulation Platform and Algorithm Performance  
To verify the performance of the IVE-KF algorithm, an EMI detection simulation platform is built in this paper.  
Figure 5 is the diagram of the simulation system. The transmitter parameters, target parameters and collection parameters can be set in the simulation system. According to the primary field model, the primary field \(\mathbf{B}_{P}\) at the detected target's position can be obtained. Then, the target's induced dipole moment and secondary field data \(\mathbf{C}\left(k\right)\) at the detector position \(\mathbf{r}_{d}\left(k\right)\) can be obtained according to the induced dipole model. Finally, the generated data are processed by using the IVE-KF algorithm, and the parameters, including the position, posture and polarizability of the detected target are calculated. In Figure 5, the primary field model, the induced dipole model, and the IVE-KF algorithm are given as determined in Sections 2 and 3. For irregular shaped targets, the calculation of the polarizability is very difficult, but, for spherical and cylindrical (axisymmetric) targets with uniform material, the process is relatively easy. The detected targets in the actual scene are mostly axisymmetric targets. Therefore, this paper selects a cylindrical object as the detected target to verify the feasibility and performance of the IVE-KF algorithm.  
For a cylindrical target with radius \(a_{\mathrm{s}}\) and length \(L\), let the length direction be the \(z\)-axis and radial direction be the \(x\)- and \(y\)-axis. The polarizability of the cylinder is [34]:  
\[\beta_{x}=\beta_{y}=\frac{1}{2}\mathrm{V}\left(0.35+\frac{\sqrt{j\omega\tau}-2} {\sqrt{j\omega\tau}+1}\right), \tag{39}\]  
\[\beta_{z}=\frac{\mathrm{LV}}{4a}\left(\frac{\sqrt{j\omega\tau_{L}}-2}{\sqrt{j \omega\tau_{L}}+1}-0.7\right), \tag{40}\]  
where \(j\) is the imaginary unit \(\sqrt{-1}\) and \(\sigma\) is the conductivity of the target. \(\mu\) and \(\mu_{r}\) are the magnetic permeability and relative permeability of the target, respectively. \(\tau\) is the time constant, \(\tau={a_{s}}^{2}\sigma\mu/{\mu_{r}}^{2}\)  
\begin{table}
\begin{tabular}{l} \hline \hline Input: collected data \(\mathbf{C}\left(1:w\right)\), detector position \(\mathbf{r}_{d}\left(1:w\right)\) \\ - Initial value estimation: \(\hat{\mathbf{r}}_{0}\) is calculated according to Formulas (17) and (18), \\ and \(\hat{\mathbf{M}}_{0}\) is calculated according to Formula (26) \\ - Traverse each detector position \(\mathbf{r}_{d}\left(k\right)\) \\ KF: Recursively update \(\hat{\mathbf{M}}\) \\ - Calculate the Green function \(\mathbf{G}\left(\mathbf{r}_{d}\left(k\right)-\hat{\mathbf{r}}\right)\) according to Formula (5) \\ - Calculate the primary field \(\mathbf{B}_{p}\left(\hat{\mathbf{r}}-\mathbf{r}_{d}\left(k\right)\right)\) according to Formulas (1)–(3) \\ - Calculate \(\mathbf{W}\left(k\right)\) according to Formulas (21) and (22) \\ - Update \(\hat{\mathbf{M}}\) according to Formulas (31) and (32) \\ EKF: Recursively update \(\hat{\mathbf{r}}\) \\ - Calculate the Jacobian matrix \(\mathbf{H}\left(k\right)\)\({}_{r=p}\) according to Formulas (34) and (37) \\ - Update \(\hat{\mathbf{r}}\) according to Formulas (35) and (36) \\ - End \\ Output: \(\hat{\mathbf{\kappa}}=\left[\hat{\mathbf{r}}^{T},\hat{\mathbf{M}}^{T}\right]^{T}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial value estimation method for the Kalman filter (KF) and the extend Kalman filter (EKF) (IVE-KF) algorithm iteration process.  
The maps of \(\mathbf{B}_{x}\) and \(\mathbf{B}_{y}\) are symmetric about the lines \(x=0\) and \(y=0\), respectively. Considering that the \(\mathbf{B}_{s}\) response is consistent with the \(\mathbf{B}_{z}\) response but the amplitude of the \(\mathbf{B}_{s}\) is larger than the \(\mathbf{B}_{z}\), \(\mathbf{B}_{s}\) is used in the following analysis.