# Performance Enhancement of Consumer-Grade MEMS Sensors through Geometrical Redundancy
## 5 Experimental Results
### Attitude and Position
The discontinuities corresponded to the effect of the correction stage associated with the availability of a new GNSS output. Figure 12: Heading angle comparison. \begin{table}
\begin{tabular}{c c c c c} \hline
**IMU** & \multicolumn{2}{c}{**Prototype**} & \multicolumn{2}{c}{**Single Face (Range)**} \\ \hline
**Angle**[\({}^{\circ}\)] & **Pitch** & **Roll** & **Pitch** & **Roll** \\ \hline Mean Value & 0.21 & 1.07 & \(-\)1.01 to 1.45 & 1.24 to 14.71 \\ \hline STD & 0.09 & 0.31 & 0.11 to 0.45 & 0.2 to 0.61 \\ \hline \end{tabular}
\end{table} 
Table 5: Initial leveling procedure comparison between prototype and single sensor. Further tests have been conducted to assess the robustness of the redundant configuration estimates in the presence of GNSS outages. Besides the best condition characterized by a new GNSS sample acquired each second, three other sampling periods have been taken into account (2, 5, and 10 s); moreover, to simulate typical phenomena (as an example, fading, urban canyon, and satellites loss) experienced in the urban contest and degrading the quality and availability of GNSS outputs, the set of available GNSS samples has been randomly decimated to a nominal ratio of 50%. For each sample of the traveled path, the Error Vector Magnitude (EVM) of the positions estimated by both the proposed prototype and reference IMU is calculated according to  
\[\text{EVM}[k]=\sqrt{ \begin{array}{c}\left(x_{P}[k]-x_{R}[k]\right)^{2}+\left( y_{P}[k]-y_{R}[k]\right)^{2}\\ +\left(z_{P[k]}-z[k]_{R}\right)^{2}\end{array} } \tag{22}\]  
Figure 14: Position comparison. Figure 13: Comparison of position results provided by the proposed prototype and the reference IMU. where \(x\) and \(y\) are the longitude and latitude estimates expressed in meter, \(z\) stands for the altitude, the subscripts \(P\) and \(R\) refer, respectively, to prototype and reference IMU, and \(k\in[1,\ldots,N]\), \(N\) is the number of acquired samples. As an example, the evolution of the EVM versus time is given in Figure 15 for GNSS outputs available either each second or every 10 s; despite of a limited number of spikes overcoming 20 m and associated with time instants immediately preceding the correction effect of the GNSS samples, the estimated path remarkably concurs with the STIM300 also in the presence of significant, periodic outages. The largest EVMs are encountered for samples corresponding to time instants lower than 20 s; as it can be expected, the prototype parameters are initially estimated during the ZUPT execution, thus improving the navigation performance. To better highlight, the prototype robustness to the GNSS outage, the Root Mean Square Error of the obtained EVMs (RMSE), defined as  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\text{EVM}[k]^{2}} \tag{23}\]  
has been exploited as a performance factor in the different GNSS outage conditions. Related results are given in Table 6, where the reliability of the proposed redundant configuration can be appreciated for outages as high as 5 s, especially in the straight path section. It is worth noting that the data portion involving the ZUPT execution has been dropped for the RMSE evaluation. In a similar way, the robustness of attitude estimates to GNSS outages have also been carried out. In particular, the root mean square value of the differences of heading, pitch, and roll angles along the whole traveled path has been evaluated; corresponding results are given in Table 7; also, for these different operating conditions, the redundant configuration has granted attitude estimates very close to those assured by the reference IMU. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**GPS Outages** & **1 s** & **2 s** & **5 s** & **10 s** & **Random** \\ \hline Position RMSE [m] & 0.89 & 1.12 & 1.31 & 5.25 & 2.38 \\ \hline Position RMSE Straight [m] & 0.63 & 0.84 & 1.03 & 3.12 & 1.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: RMSE of position EVMs for different GNSS outages. Figure 15: Differences of position in 1 s (red) and 10 s (blue) of outages.