# Performance Enhancement of Consumer-Grade MEMS Sensors through Geometrical Redundancy
## 5 Experimental Results
### Preliminary Prototype Characterization  
According to what was reported in Section 3, two operations must be carried out once the redundant IMU has been realized. The first one accounts for the alignment of sensing axes of each sensor on to same reference frame to suitably overcoming relative orientation and misalignment due to the prototype assembly. Obtained rotation matrices are saved on  
Figure 6: Test equipment. each SensorTileTM that evaluates the components of measured acceleration and angular rate in the reference frame. The residual misalignment after the realignment procedure has been assessed by evaluating the angle between the mean gravity vectors measured by reference and realigned frames. In particular, the angle \(\alpha_{ij}\) is defined as:  
\[\alpha_{ij}=mean\left(cos^{-1}\left(\frac{g_{ij}\cdotg_{i1}}{\left|\left| \vec{g}_{ij}\right|\right|\left|\left|\left|\left|\vec{g}_{i1}\right|\right| \right|}\right)\right) \tag{21}\]  
where  
* \(g_{ij}\) stands for the gravity vector measured by the triaxial accelerometer mounted on the \(j\)-th face of the cube, aligned with face 1 after the \(i\)-th rotation;
* \(g_{i1}\) stands for the gravity vector measured by the triaxial accelerometer mounted on the 1 st (reference) face of the cube after the \(i\)-th rotation;
* \(||\cdot||\) represents the traditional vector modulus operator. The alignment performance is finally assessed as the average and experimental standard deviation of the 6 residual angles for each face of the redundant IMU; the corresponding results are shown in Table 1.  
To properly estimate bias instabilities and random walks of the prototype, raw outputs of the six inertial sensors have been acquired for 48 h in a conditioned room; the measured temperature has always been within \(25\pm 1\)\({}^{\circ}\)C in order to operate as close as possible to the conditions required by the IEEE 647-2006 Standard [16]. Results of the Allan variance associated with the redundant prototype are shown in Figures 7 and 8 for the accelerometer and gyroscope, respectively. The procedure described in Section 3.2 has thus been applied to gain bias instabilities (BIs) and Angular/Velocity Random Walks (ARW and VRW), whose results are shown in Tables 2 and 3 for each sensor axis. To better appreciate the advantages brought by the redundant configuration, Allan variances of each individual sensor and prototype have been estimated. For the sake of clarity and simplicity, results associated with only one sensing axis for both accelerometer and gyroscope have been reported and shown Figures 9 and 10; similar results and evolutions have also been experienced for the other axes. Moreover, the best bias instability and random walk experienced on each sensing axis are given in Tables 2 and 3 for either acceleration or angular rate; performance improvement can straightforwardly be noticed. The obtained advantages can also be evidenced by comparing the prototype parameters with the estimated range of variation of the single sensor for each axis and quantity, reported in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Gyroscope**} & \multicolumn{2}{c}{**Prototype**} & \multicolumn{2}{c}{**Single Sensor (Best)**} \\ \cline{2-5}  & **BI** & **ARW** & **BI** & **ARW** \\  & [\({}^{\circ}\)/**h**] & [\({}^{\circ}\)/\(\sqrt{\mathbf{h}}\)] & [\({}^{\circ}\)/**h**] & [\({}^{\circ}\)/\(\sqrt{\mathbf{h}}\)] \\ \hline X-Axis & 3.1 & 0.11 & 5.7 & 0.24 \\ \hline Y-Axis & 3.1 & 0.11 & 15.7 & 0.36 \\ \hline Z-Axis & 1.8 & 0.12 & 3 & 0.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Gyro parameters estimated through Allan variance. Figure 8: Allan deviation of gyroscopes along the three sensing axes of the prototype.