## 4 Results and Discussion
### Results of the Traditional Models and Neural Networks  
As can be seen from Table 6, compared with the raw positioning error, the accuracy of the polynomial model and the locally weighted linear regression are very close to each other, of which, the RMSE reductions for the \(x\)/\(y\)/\(z\) components are \(\sim\)50%. The combined network performs better than the two traditional models, of which, the external RMSE of errors is reduced by 55%, 52%, and 76% for the \(x\), \(y\), and \(z\) components, respectively. The separate networks perform best among all methods, by which the external RMSE is reduced by 66%, 71%, and 80% for the \(x\), \(y\), and \(z\) components, respectively. These results indicate that the neural network can fit the UWB positioning errors well. Furthermore,  
Figure 8: Trajectories of the matched UWB (blue) and RTS (red) observations. A, B, C, and D each denote a distinct UWB base station.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(x\) & \(y\) & \(z\) \\ \hline Bias & \(-\)10.9 & \(-\)9.8 & \(-\)52.5 \\ RMSE & 21.1 & 21.7 & 89.2 \\
95\% MAX & 41.5 & 47.2 & 179.0 \\
100\% MAX & 90.3 & 94.9 & 249.5 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistical results of the UWB positioning errors (cm).  
it would be better to construct a particular network for each component since their error distribution patterns are different.  
Figure 9: The arrow diagram of UWB positioning errors on three axes. The color of an arrow or a UWB station indicates its \(z\) value. The arrow is only drawn for every ten errors to avoid mutual obstruction. (**a**) UWB positioning error in the \(x\) component; (**b**) UWB positioning error in the \(y\) component; (**c**) UWB positioning error in the \(z\) component.  
Figure 10 shows the residuals in the combined and separate neural networks' \(x\), \(y\), and \(z\) components. It can be noted that neural networks significantly reduce the position residuals. The histogram of UWB positioning errors for the raw observation and the trained neural network is shown in Figure 11. Compared with the former, the later errors present much more obvious accidental error characteristics, indicating good fitting performance.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Statistic**} & \multicolumn{3}{c}{**Internal Accuracy**} & \multicolumn{3}{c}{**External Accuracy**} \\  & & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) \\ \hline \multirow{3}{*}{Polynomial model} & Bias (cm) & 0.1 & 0.2 & 0.0 & 0.0 & 0.0 & \(-1.8\) \\  & RMSE (cm) & 10.7 & 9.9 & 41.9 & 11.0 & 10.5 & 41.9 \\  & RMSE reduction & 47\% & 54\% & 53\% & 47\% & 51\% & 53\% \\ \hline \multirow{3}{*}{Locally weighted linear regression} & Bias (cm) & 0.0 & 0.1 & 0.2 & 0.0 & 0.5 & 0.4 \\  & RMSE (cm) & 11.0 & 10.1 & 41.3 & 11.1 & 10.6 & 42.8 \\  & RMSE reduction & 47\% & 53\% & 54\% & 47\% & 51\% & 52\% \\ \hline \multirow{3}{*}{Combined neural network} & Bias (cm) & 0.0 & 0.0 & 0.0 & \(-0.4\) & 1.0 & \(-1.1\) \\  & RMSE (cm) & 9.5 & 9.7 & 16.4 & 9.5 & 10.4 & 21.4 \\ \cline{1-1}  & RMSE reduction & 55\% & 55\% & 82\% & 55\% & 52\% & 76\% \\ \hline \multirow{3}{*}{Separate neural network} & Bias (cm) & 0.0 & 0.0 & 0.0 & 0.3 & \(-0.6\) & 0.2 \\  & RMSE (cm) & 5.5 & 5.9 & 15.3 & 7.2 & 6.2 & 18.2 \\ \cline{1-1}  & RMSE reduction & 74\% & 73\% & 83\% & 66\% & 71\% & 80\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Statistical results of the traditional methods and the neural networks.  
Figure 10: The UWB positioning error of the combined (left) and separate (right) networks. (**a**) Residuals in the \(x\)-component; (**b**) residuals in the \(y\)-component; (**c**) residuals in the \(z\)-component.