## 3 Data Fusion Algorithm
### Software Modelling and Experimental Testing of the Data Fusion Algorithm
Since the data fusion algorithm must be valid regardless of the physical quantity measured by the sensors used in the clusters, one of the most practical options we had for testing at the laboratory level was the use of four accelerometers in the cluster measurement. This type of test was adopted considering the fact that accelerometers do not differentiate between gravitational and kinematic acceleration and static tests could be carried out based on the excitation of the sensors by the local gravitational acceleration, with the sensitivity axis of the cluster oriented in three directions very well controlled. Therefore, this easiest testing procedure was applied on the bench in the laboratory by using the signals from an accelerometric cluster with four sensors. The procedure was the simplest because it necessitated only an adjustable table with an air bubble to align horizontally the inertial detection unit, naturally excited by the local gravitational field in three situations, \(g_{horizontal}=0\) m/s\({}^{2}\), \(g_{vertical+}=9.80655\) m/s\({}^{2}\) and \(g_{vertical-}=-9.80655\) m/s\({}^{2}\), where the sensitivity axis of the cluster was directed horizontally, and along the local vertical, up and down. An NI-DAQ USB 6210 data acquisition card was used to acquire the data from each sensor in the cluster, with a sampling rate of 100 samples/s per channel. Figure 7 shows the sensor outputs and the results of data fusion obtained during the tests, while Table 2 provides the mean values of the standard deviations for data measured by the sensors and for fused signal for all of the three tested situations (\"r1\" - \"r4\" are the measurements provided by the each of the four sensors in the cluster, while \"f\" is the fused signal). Figure 6: Data fusion algorithm Matlab/Simulink model for \(n=4\). Given the characteristics in Figure 7 and the values in Table 2, the second sensor may be categorized to be the best in the linear cluster, while the third sensor may be considered the worst. Figure 7 shows a decrease in the noise amplitude for the fused signal in comparison with the sensor signals. The same thing is revealed by the mean values for the standard deviation in Table 2. On the other hand, these values indicate that the standard deviation of the fused signal decreased by 1.7720 times comparatively with sensor no. 2 (the best in cluster), by 2.2661 times comparatively with sensor no. 3 (the worst in cluster), and by 2.0266 times comparatively with the average of the mean dispersions for all sensors in the cluster, when its sensitivity axis was on the horizontal plane. In the other two testing cases, the obtained reductions for the mean standard deviation were as follows: (1) at 9.80655 m/s\({}^{2}\) excitation: 1.7694 times comparatively with sensor no. 2, 2.2565 times comparatively with sensor no. 3, and 2.0256 times comparatively with the average of the mean dispersions; (2) at \(-9.80655\) m/s\({}^{2}\) excitation: 1.7564 comparatively with sensor no. 2, 2.3803 times comparatively with sensor no. 3, and 2.0140 times comparatively with the average of the mean dispersions. All tested situations, as well as the numerical simulations, are not the same. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Excitation**} & \multicolumn{4}{c}{**Mean Values of the Standard Deviations [m/s\({}^{2}\)]**} \\ \cline{2-6}  & **Sensor No. 1** & **Sensor No. 2** & **Sensor No. 3** & **Sensor No. 4** & **Fused Signal** \\ \hline
0 & 1.5462\(\cdot 10^{-3}\) & 1.3747\(\cdot 10^{-3}\) & 1.7580\(\cdot 10^{-3}\) & 1.6100\(\cdot 10^{-3}\) & 7.7577\(\cdot 10^{-4}\) \\
9.80655 & 1.5605\(\cdot 10^{-3}\) & 1.3716\(\cdot 10^{-3}\) & 1.7492\(\cdot 10^{-3}\) & 1.5995\(\cdot 10^{-3}\) & 7.7515\(\cdot 10^{-4}\) \\ \(-9.80655\) & 1.5604\(\cdot 10^{-3}\) & 1.3756\(\cdot 10^{-3}\) & 1.7690\(\cdot 10^{-3}\) & 1.6043\(\cdot 10^{-3}\) & 7.8316\(\cdot 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The mean values of the standard deviations for sensor data and for fused signal.