## 4 Test and Discussion
### UWB Robotic Dog Localization  
In this subsection, we detail how we employed the UWB, the federal EKF, and the adaptive distributed t EKF with Allan variance to estimate the robotic dog's position. In this test, we used eight UWB BSs, which are shown in Figure 11. The environment for the UWB robotic dog localization and the robotic dog used in this test are shown in Figures 8 and 9, respectively. We also used the UWB solution and the distributed EKF as reference standards in this test. For the filter, we set \(\mathbf{s}_{0}=\left[ \begin{array}{cccc}0&0&0&0\end{array} \right]^{T}\), \(\mathbf{P}=\mathbf{I}_{4\times 4}\), \(\mathbf{Q}=\left[ \begin{array}{cccc}0.0000&0.0007&0&0\\ 0.0007&00.0144&0&0\\ 0&0&00.0000&00.0007\\ 0&0&00.0007&00.0144\end{array} \right]\), \(\mathbf{R}=0.0025\). The planned path used by the robotic dog in the test is shown in Figure 10. From the figure, we can see that the cement wall affected the UWB signal, which reduced the localization accuracy of the UWB. The planned path and the trajectory measured by the UWB, the federal EKF, and the adaptive distributed t EKF with Allan variance for the robotic robot localization are shown in Figure 11. From Figure 11, it is clear that the UWB solution exhibited larger errors compared to in the mobile robot test, especially in narrow corridors. When compared with the UWB, both the distributed EKF and the proposed distributed t EKF with Allan variance reduced the localization error, with the proposed method bringing the path closer to the planned path. Figure 10: The planned path used by the robotic dog in the test. Figure 9: The robotic dog used in the test. Figures 12 and 13 show the east and north absolute position errors for the UWB measurements, federal EKF, and adaptive distributed t EKF with Allan variance in the robotic dog localization test. In these figures, the UWB solution is represented by a continuous blue solid line, the path of the distributed EKF measurements by an orange solid line, and the proposed adaptive distributed t EKF with Allan variance by a black solid line. From the Figure 12, it can be observed that both filters provided stable solutions, while the UWB measurements exhibited large localization errors. This highlights the effectiveness of the filters in improving positional accuracy. Moreover, the proposed adaptive distributed t EKF with Allan variance resulted in smaller localization error compared to the traditional distributed EKF. Thus, it can be concluded that incorporating the t distribution was well suited for this localization environment. From Figure 13, in the north direction, we can see that the filers performed well compared to the UWB, especially in the final stage of the test. It should be noted that the error in the filter solutions was higher than that of the UWB. However, it is important to mention that in this test, the convergence speed of both filters still needed improvement. The RMSEs measured by the UWB, the federal EKF, the distributed t EKF, and the adaptive distributed t EKF with Allan variance for the robotic dog localization are listed in Table 2. From the table, we can see that the proposed adaptive distributed t EKF with Allan variance method reduced the localization error from 0.568 m to 0.468 m in the east direction and from 0.764 m to 0.451 m in the north directions, respectively, compared to the UWB solutions. Then, we repeated the test again, and the results are listed in Table 3. From the table, we can see that the proposed method had the smallest localization error. Finally, we list the mean values for the two tests in Table 4; we can see that the proposed method had the best performance. Compared to the mean value of the distributed t EKF method, the proposed method reduced the error, demonstrating its effectiveness in minimizing the localization error. Figure 11: The planned path and the paths of the UWB measurements, federal EKF, distributed t EKF, and adaptive distributed t EKF with Allan variance for the robotic dog localization.