# Performance Enhancement of Consumer-Grade MEMS Sensors through Geometrical Redundancy
## 2 Theoretical Background
### Advantages of Redundant Configuration  
Thanks to a reasonable assumption of lack of correlation, exploiting a redundant configuration allows to suitably reduce the typical errors (such as bias instability, bias drift, scale factor drift [12]) with respect to that affecting each single inertial unit. To this aim, let us consider a geometrical distribution involving \(N\) sensors whose sensing axes are characterized by an invariable orientation with respect to the redundant Inertial Measurement Unit (IMU) reference system. Let \(\mathbf{x}_{k}\) be the generic quantity (either acceleration or angular rate) measured by the \(i\)-th inertial sensor; according to [12; 16] the measured values can be expressed as:  
\[\mathbf{x}_{k}=\mathbf{x}_{r}+SF_{i}\mathbf{x}_{r}+\mathbf{b}_{k}+\varepsilon_{k}+\mathbf{N}x_{r} \tag{1}\]  
where \(SF_{i}\) represents the scale factor, \(\mathbf{x}_{r}\) is the actual measurand, \(\mathbf{b}\) stands for the bias, \(\mathbf{\varepsilon}\) is the random noise, and, finally, \(\mathbf{N}\) is the skew-symmetric matrix that contains the non-orthogonalities. Even though both scale factor and bias affect sensor measures, in the following the attention will be focused on bias; similar considerations hold for the scale factor. Moreover, let \(\sigma_{i}\) be the standard deviation of the sensor bias associated with the \(i\)-th axis; even though different sensors would be characterized by different values of standard deviation, this value can be considered equal for all the sensors in the following, without loss of generality. Acceleration and/or angular rate values coming from each sensor axis can be combined by averaging their projections on each IMU axis. This way, the standard deviation \(\sigma_{j,r}\) of the bias affecting the redundant configuration can straightforwardly be evaluated according to  
\[\sigma_{j,r}=\frac{\sigma_{i}}{\sqrt{\sum_{i=1}^{N}cos^{2}\theta_{i,j}}} \tag{2}\]  
where \(\theta_{i,j}\) stands for the angle between the \(i\)-th sensor and \(j\)-th IMU axis, respectively.  
When the \(i\)-th sensing axis of each sensor is parallel to \(j\)-th IMU axis, Equation (2) turns into  
\[\sigma_{j,r}=\frac{\sigma_{i}}{\sqrt{N}} \tag{3}\]  
i.e., the standard deviation corresponding to the arithmetic mean of \(N\) values that represent a theoretical value of the standard deviation [17; 18].  
In order to assure a three-dimensional uniform error along any direction of each IMU axis, inertial sensors should be placed on the faces of a regular polyhedron; if this is the case, directions orthogonal to each face are characterized by spatial uniform distribution.  
The coefficient \(\beta_{imp}\) (i.e., the fraction denominator of Equation (2)), defined according to  
\[\beta_{imp}=\left(\sqrt{\sum_{i=1}^{N}cos^{2}\theta_{i,j}}\right)^{-1} \tag{4}\]  
allows to appreciate reduction of bias standard deviation associated with the availability of independent redundant measures of acceleration and angular rate.  
As reported in [15], it is worth noting that the improvement \(\beta_{imp}\) is independent from the specific chosen polyhedron in terms of shape and number of faces; on the contrary, it only depends on the number \(N\) of exploited sensors. Moreover, values given in [15] refer to ideally oriented sensing axes; as a matter of practice, misalignments among sensors can occur, even though their magnitude can be made negligible by means of a proper calibration procedure.