## 4 Experiment Results
### The Performance of Object Localization and Tracking  
In this section, the concept of the LS method [27], the proposed NRLM method, and the proposed EKF method are used to estimate the target position. The TOF results of all the five receiver channels are used among the three methods, and the issue of outlier rejection is then discussed. Tables 1 and 2 present the RMSE of the target position with three methods in Cases 1 and 2 to show the accuracy of the object localization methods. Tables 3 and 4 present the standard deviations of the target position with the three methods in Cases 1 and 2 to show the stability of the object localization methods. It is observed that most RMSEs and all standard deviations of the EKF method are much smaller than those of the other methods in both cases. Therefore, the proposed EKF method has better performance in both accuracy and stability than the other methods for object localization. Figure 9: Case 1 (marker pen): Standard deviations of distances with the corresponding TOFs of different methods among five receiver channels. (\(\blacktriangledown\)): threshold method; (\(\bullet\)): two-maximums algorithm; (\(\blacktriangle\)): EGACO algorithm with GN model; (\(\blacksquare\)): NRLM algorithm with GN model; (\(\blacklozenge\)): NRLM algorithm with DE model. (Vertical coordinate unit: mm). Figures 12-14 present the results of target tracking with different moving directions for Case 3, which are denoted as Case 3-1, Case 3-2, and Case 3-3. It should be noted that in all test trajectories of Case 3, the finger stops for a period of time at the start and end points before and after the movement. In the LS method, there is a system matrix containing distance estimation, which increases the uncertainty of the estimation and easily causes ill-conditioned problems. Because the round-trip model does not include any distance estimation, the NRLM method is superior to the LS method. However, the NRLM method only estimates coordinates based on the current TOF data, and the estimation results are directly affected by the measurement noise. Although the performance of the NRLM method and the EKF method in the stationary state of the target is similar, the EKF estimates the position of the target based on the previous state of the target, and inherently considers the interference of the measurement, which can provide more stable and smoother localization and tracking of the moving target. It is noticed that in Figure 14, the tracking  
Figure 11: Curve-fitting results of the GN model and the DE model. **(Left)**: NRLM algorithm with the GN model; **(Right)**: NRLM algorithm with the DE model. (Blue line: original echo signal; red line: envelope model by the proposed methods.) Figure 10: Case 2 (human index finger): Standard deviations of distances with the corresponding TOFs of different methods among five receiver channels. (\(\blacktriangledown\)): threshold method; (\(\bullet\)): two-maximums algorithm; (\(\blacktriangle\)): EGACO algorithm with GN model; (\(\blacksquare\)): NRLM algorithm with GN model; (\(\blacklozenge\)): NRLM algorithm with GN model. (\(\blacklozenge\)): NRLM algorithm with DE model. (Vertical coordinate unit: mm). in the \(x\)-direction is delayed more than the tracking in the \(y\)-direction. This is because the geometry of the ultrasound platform structure makes the variance in the \(x\)-direction larger than that in the \(y\)-direction [36], which can also be observed in Tables 2 and 4. Here, the outlier issue is reconsidered in Case 3-3. When the target moves from (20, 36) to (20, 48), it is observed that sometimes the echo signal is severely distorted and results in generating several outliers in the measurement data. The outliers have a great influence on the tracking performance of the EKF method. In Figure 15, the outlier rejection method proposed in Section 2.4.2 is used to improve the tracking performance of the target.