# Improving GNSS Landslide Monitoring with the Use of Low-Cost MEMS Accelerometers
## 3 Calibration with the Six-Faces Method  
The performance characteristics of accelerometers are affected by a variety of errors. Most errors can be classified into: sensor bias, scale factor, axes misalignment, and noise [10]. The calibration of these instruments is required to decrease the errors, as well as those in the INS-derived position, velocity, and attitude behaviours of moving platforms. The six-position static and rate tests are among  
Figure 2: Scheme of triaxial accelerometer.  
the most commonly used [8; 9] calibration methods. They require the inertial system to be mounted on a leveled surface, with each sensitive axis of every sensor pointing alternately up and down [11; 12].  
The error model can be expressed in the matrix form (5) through the calibration matrix \(E[e_{ij}]\).  
In (5) the terms of the \(E\) matrix represent the following biases:  
* The diagonal elements of the matrix are the scale factors of the three axes;
* Off-diagonal elements represent non-orthogonality factors or misalignment;
* The fourth column contains the translational bias values on the three axes.  
\[\left(\begin{array}{c}l_{x}\\ l_{y}\\ l_{z}\end{array}\right)=\left[\begin{array}{cccc}e_{1}^{1}&e_{2}^{2}&e_{3}^ {3}&e_{4}^{4}\\ e_{2}^{1}&e_{2}^{2}&e_{3}^{3}&e_{4}^{4}\\ e_{3}^{2}&e_{3}^{2}&e_{3}^{3}&e_{3}^{4}\\ \end{array}\right]\left(\begin{array}{c}t_{1}\\ t_{2}\\ t_{3}\\ 1\end{array}\right) \tag{5}\]  
\(l_{x}\), \(l_{y}\), and \(l_{z}\) are the measures on the three axes, while \(t_{x}\), \(t_{y}\), and \(t_{z}\) represent the theoretical values that these measures should have. For the six-faces calibration procedure, the six theoretical values are as follows [5; 8]:  
\[\left(\begin{array}{c}g\\ 0\\ 0\\ 1\end{array}\right)\left(\begin{array}{c}-g\\ 0\\ 0\\ 1\end{array}\right)\left(\begin{array}{c}0\\ g\\ 0\\ 1\end{array}\right)\left(\begin{array}{c}0\\ g\\ 1\end{array}\right)\left(\begin{array}{c}0\\ 0\\ g\\ 1\end{array}\right)\left(\begin{array}{c}0\\ 0\\ -g\\ 1\end{array}\right)\right. \tag{6}\]  
Equation (5) can be developed in this form:  
\[\left(\begin{array}{c}l_{x}\\ l_{y}\\ l_{z}\end{array}\right)_{(1-6)}=\left[\left(\begin{array}{cccc}\pm g&0&1 \\ 0&0&1\\ 0&0&1\end{array}\right)_{(1,2)}\left[\left(\begin{array}{cccc}0&0&1\\ \pm g&0&1\\ 0&0&1\end{array}\right)_{(3,4)}\right]\left[\left(\begin{array}{cccc}0&0&1 \\ \pm g&0&1\\ \end{array}\right)_{(5,6)}\right]\right]\left[\left(\begin{array}{cccc}e_{1 }^{1}&e_{2}^{1}&e_{3}^{1}\\ e_{2}^{2}&e_{2}^{2}&e_{3}^{2}\\ e_{3}^{3}&e_{3}^{2}&e_{3}^{3}\\ e_{1}^{4}&e_{2}^{4}&e_{3}^{4}\end{array}\right]\right. \tag{7}\]  
If the instrument measurements are in \"units of g\", then the terms in the matrix are all equal to \(\pm 1\) or zero. The value \(\pm\) indicates the positive or negative sign for the normal vector to the face in a concordant or discordant direction with respect to the gravity vector. We can write three of Equation (7) for each measure on each of the six faces. The redundant and linear system (7) in the 12 unknown parameters \(x\) can be synthetically rewritten:  
\[Ax-l_{0}=v \tag{8}\]  
where the \(A\) design matrix contains 0, 1, and \(\pm\)g. The measured and known values of acceleration is the vector \(l_{0}\) and the vector \(v\) contains the residuals. \(C_{ll}\) in (9) is the covariance matrix of the measurements. The least-squares solution [13] estimates the 12 calibration parameters with 18 measurement series in this way:  
\[\hat{x}=\left(A^{T}C_{ll}^{-1}A\right)^{-1}A^{T}C_{ll}^{-1}l_{0} \tag{9}\]  
The variance-covariance matrix of the parameters is:  
\[C_{xx}=\hat{\sigma}_{0}^{2}\left(A^{T}C_{ll}^{-1}A\right)^{-1} \tag{10}\]  
With:  
\[\hat{\sigma}_{0}^{2}=\frac{\hat{\sigma}^{T}C_{ll}^{-1}\hat{\sigma}}{n-r} \hat{v}=A\hat{x}-l_{0}\]  
where \((n-r)\) is the global redundancy. In the absence of stochastic information on the measurement of accelerations, we set \(C_{ll}=I\). The 12 estimated parameters are then used to correct the measurements of the accelerometers, improving the accuracy of tilt measurements.