## 3 Redundant Sensors Configuration and Near-Vertical Algorithm
### Calibration of Redundant Accelerometers  
Conventional multi-position calibration and Ellipsoid Fitting calibration can be employed to compensate for the regular nine-axis MEMS-IMU and tri-axial magnetometer [20]. In this section, we concentrate on the compensation methods for redundant accelerometers. As shown in Figure 8a, the installation error leads to the actual angle between the redundant accelerometer and gravitational direction deviating theoretical angle, which will degrade the calculating precision of the wellbore azimuth.  
Firstly, a simplified four-position calibration method is designed to estimate the mounting angles of the redundant accelerometers. Assuming that the true angle between gravity and the redundant axis is \(\alpha\), the expression can be written as (16) using the four-position calibration method shown in Figure 8b. The four calibration positions are the Z-axis pointing up and down, and the Y-axis pointing up and down, respectively.  
\[\left\{\begin{array}{l}f_{1}=f_{b}+Kg\ cos\ \alpha\\ f_{2}=f_{b}-Kg\ cos\ \alpha\\ f_{3}=f_{b}+Kg\ sin\ \alpha\\ f_{4}=f_{b}-Kg\ sin\ \alpha\\ \end{array}\right. \tag{16}\]  
Figure 8: Schematic of installation angle and calibration for redundant accelerometers. (**a**) Installation angle and calibration. (**b**) Four-position calibration method.  
Figure 7: Accelerometer relative error and offset angle curves.  
Here, \(f_{i}(i=1,2,3,4)\) is the true measuring value of the redundant accelerometer in four directions. \(f_{b}\) and \(K\) are the bias of the accelerometer and scale factor, respectively. Then, the error parameters of the redundant accelerometer can be written as Equation (17) by solving Equation (16).  
\[\left\{\begin{array}{l}f_{b}=(f_{1}+f_{2}+f_{3}+f_{4})/4\\ \alpha=tan^{-1}((f_{3}-f_{4})/(f_{1}-f_{2}))\\ K=(f_{3}-f_{4})/(2g\ sin\ \alpha)\end{array}\right. \tag{17}\]  
The calibration method for the first step misses the non-orthogonal error (NOE) of the two redundant accelerometers. The redundant accelerometers \(Racc_{1}^{\prime}\) and \(Racc_{2}^{\prime}\) have the installation angle \(\alpha_{1}\) and \(\alpha_{2}\), respectively. As shown in Figure 9, \(Racc_{1}\) and \(Racc_{2}\) are the desirable orthogonal mounting positions. Assuming that \(Racc_{1}^{\prime}\) and \(Racc_{1}\) are parallel, \(Racc_{2}^{\prime}\) has a non-orthogonal error angle \(\varepsilon\) with the theoretical accelerometer \(Racc_{2}\). Therefore, the orthogonality of the two redundant accelerometers can be achieved by compensating for the NOE with Equation (18).  
\[\left\{\begin{array}{l}Racc_{1}=Racc_{1}{}^{\prime}+Racc_{2}{}^{\prime} \times sin\ \varepsilon\\ Racc_{2}=Racc_{2}{}^{\prime}\times cos\ \varepsilon\end{array}\right. \tag{18}\]  
To verify the validity of the calibration method, a hexahedron structure is employed to install MEMS-IMU/magnetometers and redundant accelerometers in Figure 10, where two single-axis AXO315 are symmetrically mounted on both surfaces of the hexahedron. The nine-axis sensor, model N100, is mounted on the front of the hexahedron as the MEMS-IMU/magnetometer system. The parameters of the AXO315 and N100 are listed in Table 1.  
Figure 10: Redundant sensor configuration on hexahedron structure.  
Figure 9: Compensation for non-orthogonal redundant accelerometers.