## 4 Simulation Platform and Algorithm Performance
### Target Parameter Estimation Performance  
To evaluate the performance of the IVE-KF algorithm, the position estimation error \(\bar{r}\), the polarizability estimation error \(\bar{\beta}_{e}\), and the posture angle estimation errors \(\bar{\theta}\), \(\bar{\phi}\) at different simulation conditions are discussed. \(\bar{r}\), the values of \(\bar{\beta}_{e}\), \(\bar{\theta}\) and \(\bar{\phi}\) for each simulation condition are the average values obtained from 100 Monte Carlo experiments:  
\[\bar{r}=\frac{\sum\limits_{i=1}^{100}\left\|\bar{r}\left(i\right)-\bar{r} \right\|_{2}}{100}, \tag{44}\]  
\[\bar{\beta}_{e}=\frac{\sum\limits_{i=1}^{100}\left\|\hat{\beta}\left(i\right)- \hat{\beta}\right\|_{2}}{100\left\|\hat{\beta}\right\|_{2}}, \tag{45}\]  
\[\bar{\theta}=\frac{\sum\limits_{i=1}^{100}\left|\theta_{i}-\hat{\theta}_{i} \right|}{100}, \tag{46}\]  
\[\bar{\phi}=\frac{\sum\limits_{i=1}^{100}\left|\phi_{i}-\hat{\phi}_{i}\right|}{ 100}, \tag{47}\]  
where \(\bar{r}\left(i\right)\), \(\hat{\beta}\left(i\right)\), \(\hat{\theta}_{i}\), \(\hat{\phi}_{i}\) are respectively the position estimation, polarizability estimation, pitch angle estimation and roll angle estimation obtained by the IVE-KF algorithm at the \(i\)th Monte Carlo experiment. \(\bar{r}\), \(\bar{\beta}_{e}\), \(\bar{\theta}\) and \(\bar{\phi}\) of a cylinder at different postures are shown in Figures 12-14. The errors fluctuate as the posture changes, but the position errors are always within 13 cm, the polarization relative errors are within 32%, and the angle errors are within 10\({}^{\circ}\). When \(\theta=90^{\circ}\), the phenomenon of gimbal lock [35] occurs, so the angle simulation is performa  
Figure 11: Estimation error of the magnetic polarizability tensor element of a cylindrical target at different SNR values. The convergence performances of the IVE-KF algorithm and the filter algorithm with a random initial value are compared in this paper. The convergence results of the algorithm are evaluated using the root mean square error (RMSE) of position and magnetic polarizability tensor. Figures 15 and 16 show that the filter algorithm with a random initial value cannot converge to the true value at all. Our initial value estimation algorithm provides a relatively accurate initial value for the KF and EKF and improves the convergence and accuracy of the filter algorithm. The computational complexities of the inversion algorithm and IVE-KF algorithm in this project are also compared. We use the number of floating point operations (FLOPs) to measure the computational complexity of an algorithm. A FLOP is defined as one operation of addition, subtraction, and subtraction. The number of floating point operations is defined as the number of floating point operations. Figure 12: Position error accuracy of the initial value estimation method for the Kalman filter (KF) and the extend Kalman filter (EKF) (IVE-KF) algorithm at different posture angles. Figure 13: Error of the polarizability error of the IVE-KF algorithm at different posture angles. According to the processes listed in Table 1 and the corresponding Formulas, and assuming that \(w\) sets data are collected in the process of detection, the FLOPs of IVE-KF algorithm can be obtained and are shown in Table 6.  
The LM algorithm can be analyzed using the same method. The computational complexity of the LM algorithm per iteration is \(988w+969\) FLOPs. The FLOPs comparison of the IVE-KF algorithm and LM algorithm is shown in Table 7:  
In Table 7, \(N_{i}\) is the iteration times of the LM algorithm. The computational complexity of iterating the LM algorithm twice is equal to the computational complexity of the IVE-KF algorithm. However, the LM algorithm generally iterates 5-8 times in our experiments. The accuracy performances of the LM algorithm and IVE-KF algorithm are also discussed in terms of the RMSE of the position and magnetic polarizability tensor.