# Multi-Tag Fusion Localization Method Based on Geometric Constraints
## 4 Fusion of Multiple Observations
### Geometric Constraint Effectiveness
#### 5.2.2 Process Validation  
In this section, we compare the method without geometric constraints and our proposed method over a sustained period, with detailed results presented in Table 4. This comparison allows us to evaluate the impact of geometric constraints on the system's accuracy and stability in a continuous positioning process. The Method Without Geometric Constraints: This method fused all raw observations. It treated each tag's data equally, applying a uniform and constant level of observation noise throughout the fusion process. Our Method: The proposed method first culled outliers based on outlier factors, then dynamically adjusted observation noise for each tag's data during fusion, enhancing accuracy and stability. The table provides a clear comparison of the error metrics between the method without geometric constraints and our proposed method, highlighting the impact of incorporating geometric constraints on positioning accuracy. For positional error, the method without geometric constraints had an average error within 8 cm, but this error peaked at a maximum of 14 cm during the motion sequence. This variability suggests a potential lack of reliability in positioning accuracy. In contrast, our method, which applies geometric constraints, showcased superior precision with a maximum average error of 2.611 cm and a maximum error capped at 2.968 cm. This represents a significant improvement in positional accuracy, demonstrating the advantage of using geometric constraints. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Group**} & \multicolumn{2}{c}{**Method Without Geometric Constraints**} & \multicolumn{2}{c}{**Our Method**} \\ \cline{3-6}  & & **Distance Error (m)** & **Yaw Error (rad)** & **Distance Error (m)** & **Yaw Error (rad)** \\ \hline \multirow{3}{*}{1} & Maximum Error & 0.02246 & 0.03816 & 0.01627 & 0.01262 \\  & Minimum Error & 0.00386 & 0.00089 & 0.00273 & 0.00018 \\  & Average Error & 0.01113 & 0.00989 & 0.00824 & 0.00432 \\ \hline \multirow{3}{*}{2} & Maximum Error & 0.16475 & 0.14654 & 0.02968 & 0.05063 \\  & Minimum Error & 0.00231 & 0.00014 & 0.00278 & 0.00004 \\  & Average Error & 0.05813 & 0.02442 & 0.01772 & 0.01298 \\ \hline \multirow{3}{*}{3} & Maximum Error & 0.61676 & 0.10386 & 0.02858 & 0.04184 \\  & Minimum Error & 0.01283 & 0.00321 & 0.00355 & 0.00036 \\  & Average Error & 0.05587 & 0.05908 & 0.02294 & 0.02153 \\ \hline \multirow{3}{*}{4} & Maximum Error & 0.05618 & 0.03842 & 0.02213 & 0.02451 \\  & Minimum Error & 0.00033 & 0.00025 & 0.00033 & 0.00006 \\  & Average Error & 0.01646 & 0.00837 & 0.00889 & 0.00550 \\ \hline \multirow{3}{*}{5} & Maximum Error & 0.09544 & 0.10569 & 0.02826 & 0.03641 \\  & Minimum Error & 0.00502 & 0.01421 & 0.00772 & 0.00020 \\ \cline{1-1}  & Average Error & 0.04035 & 0.03189 & 0.01871 & 0.03478 \\ \hline \multirow{3}{*}{6} & Maximum Error & 0.14736 & 0.10569 & 0.02624 & 0.04256 \\ \cline{1-1}  & Minimum Error & 0.00148 & 0.00113 & 0.00219 & 0.00027 \\ \cline{1-1}  & Average Error & 0.07850 & 0.03613 & 0.02611 & 0.02389 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparative error analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dimensionality** & **The Variance of All Observations** & **Variance of Reliable Observations** \\ \hline X & 0.00715 & 0.00254 \\ Y & 0.00436 & 0.00184 \\ Yaw & 0.01124 & 0.00846 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of variance before and after excluding outliers. When it comes to orientation angle error, the method without geometric constraints exhibited a maximum error of 8.39\({}^{\circ}\) and a maximum average error of 3.38\({}^{\circ}\). These figures indicate a higher likelihood of inaccuracy in orientation estimation. However, our method achieved a maximum error of 2.90\({}^{\circ}\) and a maximum average error of just 1.99\({}^{\circ}\) in orientation angle estimation.