# Evaluation of Multi-Sensor Fusion Methods for Ultrasonic Indoor Positioning
## 3 Proposed Fusion Methods
### Tightly Coupled Data Fusion
The updated state vector \(\overset{\boldsymbol{\wedge}}{\mathbf{X_{k}}}\), its covariance \(\mathbf{P_{k}}\), and the Kalman gain \(\mathbf{K}\) are obtained with Equation (14). \[\mathbf{K}=\mathbf{P_{k}^{-}}\cdot\mathbf{H_{k}^{T}}\cdot\left(\mathbf{H_{k} }\cdot\mathbf{P_{k}}\cdot\mathbf{H_{k}^{T}}+\mathbf{R}\right)^{-1}\]  
\[\overset{\boldsymbol{\wedge}}{\mathbf{X_{k}}}=\overset{\boldsymbol{\wedge} \boldsymbol{\wedge}}{\mathbf{X_{k}}}^{-}+\mathbf{K}\cdot\left(\mathbf{Z_{k}}-h \left(\overset{\boldsymbol{\wedge}}{\mathbf{X_{k}  
To compare the accuracy provided by the LKF with that from the independent ULPSs in Table 2, Table 4 provides the errors for 90% of estimated positions. It is worth noting that the positioning errors are now in the range of centimetres or decimetres. At height \(z_{1}\), the error for positions P1, P2, P3, P5, and P6 is below 0.34 m, whereas it is below 0.7 m for P4 and P7. In the case of \(z_{2}\), the errors for all positions P1-P7 are below 0.29 m for 90% of cases. Figure 8: (a–f) Estimated positions after the LKF fusion for the points P1–P7, including the projections of their corresponding error ellipsoids with a certainty of 95% in the XY and XZ planes, at \(z_{1}\) = 1.35 m (**left**) and \(z_{2}\) = 1.93 m (**right**): successively, 3D representation of clouds of points; Y–X projections; and Z–X projections. A second approach considered here is to apply an AKF. Figure 9 presents the fusion results at \(z_{1}\) and \(z_{2}\). Similar conclusions can be derived: as before, the best estimated positions are P1, P2, P3, P5, and P6 (cyan, black, green, red, and yellow circles, respectively), and they also present the smallest error ellipsoids for 95% of estimated positions, whereas the worst ones are P4 (pink circles) and P7 (blue circles). The mean error and the standard deviation per axis are provided in Table 5, whereas Table 6 describes the positioning error for 90% of cases. The positioning error of the estimates reached by simulation is again in the range of centimetres or decimetres. Although results from the LKF are similar to those from the AKF, the second approach presents better accuracy due to the update of the values in the noise matrix of the filter. Globally, the mean error for 90% of the estimated positions is below 0.27 m at \(z_{1}\) and 0.101 m at \(z_{2}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Points** & \(z_{1}\) & \(z_{2}\) \\ \hline P1 & 0.208 & 0.134 \\ P2 & 0.221 & 0.054 \\ P3 & 0.245 & 0.061 \\ P4 & 0.522 & 0.292 \\ P5 & 0.349 & 0.071 \\ P6 & 0.322 & 0.108 \\ P7 & 0.751 & 0.168 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Positioning errors (m) for P1–P7 in 90% of the cases after the LKF fusion. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Points**} & \multicolumn{6}{c}{\(z_{1}\)} & \multicolumn{6}{c}{\(z_{2}\)} \\ \cline{2-13}  & \multicolumn{3}{c}{**Mean Error (m)**} & \multicolumn{3}{c}{**Std Deviation (m)**} & \multicolumn{3}{c}{**Mean Error (m)**} & \multicolumn{3}{c}{**Std Deviation (m)**} \\ \cline{2-13}  & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) & \(x\) & \(y\) & \(z\) \\ \hline P1 & 0.120 & 0.068 & 0.109 & 0.122 & 0.062 & 0.122 & 0.090 & 0.061 & 0.090 & 0.082 & 0.045 & 0.082 \\ P2 &_Appl. Sci._**2021**, _11_, 6805 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Points** & \(\mathbf{z_{1}}\) & \(\mathbf{z_{2}}\) \\ \hline P1 & 0.205 & 0.169 \\ P2 & 0.208 & 0.073 \\ P3 & 0.139 & 0.072 \\ P4 & 0.306 & 0.168 \\ P5 & 0.306 & 0.054 \\ P6 & 0.426 & 0.111 \\ P7 & 0.622 & 0.214 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Positioning error (m) for points P1–P7 in 90% of cases after an AKF. Figure 9: (**a–f**) Estimated positions after AKF fusion for the points P1–P7, including the projections of their corresponding error ellipsoids with a certainty of 95% in the XY and XZ planes, at \(z_{1}=1.35\) m (**left**) and \(z_{2}=1.93\) m (**right**): successively, 3D representation of clouds of points; Y–X projections; and Z–X projections.