## 4 Test and Discussion
### UWB Mobile Robot Localization  
The first test involved UWB mobile robot localization, In this test, we employed four UWB BSs placed in known preset positions. The UWB blind node (BN) was attached to the target object, which in this case was the mobile robot. The test environment is shown in Figure 3. We can observe that the conditions were harsh: First of all, there were two obvious metal pillars in this scene. Secondly, the testing scenario was quite chaotic due to the accumulation of goods. The above factors led to the signal of the UWB positioning being affected; thus, we can determine that the conditions were harsh in this test.The mobile robot used in this test is shown in Figure 4. For comparison, we employed the UWB solution, federal EKF, and distributed t EKF as benchmarks. For the filter, we set \(\textbf{s}_{0}\ =\ \left[ \begin{array}{cccc}-0.2&0&0.1&0\end{array} \right]^{T}\), \(\textbf{P}\ =\ \textbf{I}_{4\times 4}\), \(\textbf{Q}\ =\ \left[ \begin{array}{cccc}0.0002&0.0034&0&0\\ 0.0034&0.0676&0&0\\ 0&0&0.0002&0.0034\\ 0&0&0.0034&0.0676\end{array} \right]\), \(\textbf{R}=0.0032\). Figure 5 shows the planned path, the UWB measurement path, the path from the federal EKF, and the paths from the distributed t EKF and adaptive distributed t EKF with Allan variance for mobile robot localization. In the figure, the planned path is represented by a black dashed line, the UWB computed path by a blue solid line (calculated using the least-squares algorithm), the federal EKF path by an orange solid line, the distributed t EKF path by a yellow solid line, and the adaptive distributed t EKF with Allan variance path by a black solid line. The UWB BSs are marked by pink circles. As seen in Figure 5, the UWB estimated trajectory exhibits significant fluctuations, indicating that the complex indoor environment affected the positioning accuracy of the least-squares method. In contrast, while the federal EKF path also shows fluctuations, its deviations are larger than those in the UWB solution. The distributed t EKF and the proposed adaptive distributed t EKF with Allan variance closely follow the reference path, demonstrating superior accuracy compared to both the UWB and federal EKF methods. Figure 3: Environment for UWB mobile robot localization. Figures 6 and 7 show the eastward and northward absolute position errors for the UWB measurements, federal EKF, distributed t EKF, and adaptive distributed t EKF with Allan variance for mobile robot localization. In the eastward direction, the UWB solution exhibited noticeable jitter errors, particularly between the time indices 0 and 20, 60 and 65, and 100 and 150, indicating that the harsh environment affected UWB localization accuracy. The federal EKF reduced some of these errors but still showed residual inaccuracies, such as the _EKF_ and _EKF_. Figure 4: The mobile robot used in the test. Figure 5: The planned path, path of UWB measurements, and paths measured by the federal EKF, distributed t EKF, and adaptive distributed t EKF with Allan variance for the mobile robot localization. as between the time indices 50 and 80. The distributed t EKF could reduce the localization error further when compared with the federal EKF. In contrast, the adaptive distributed t EKF with Allan variance further reduced the error, demonstrating the advantages of the t distribution under complex noise conditions. In the northward direction, the results of the methods were similar to those in the east direction. The UWB localization still exhibited noticeable jitter errors in the north direction, while the federal EKF exhibited reduced localization error compared to the UWB approach, highlighting the effectiveness of this method in mitigating UWB localization error. The proposed adaptive distributed t EKF with Allan variance outperforms both the UWB and distributed EKF methods. Figure 6: The east absolute position error measured by the UWB, federal EKF, distributed t EKF, and adaptive distributed t EKF with Allan variance for the mobile robot localization.