## 3 Redundant Sensors Configuration and Near-Vertical Algorithm  
### Redundant Accelerometers Configuration  
Accelerometers that are perpendicular to the axial direction in the near-vertical state are unable to accurately calculate the horizontal angle. Increasing the signal-to-noise ratio of the radial accelerometers is beneficial for improving horizontal attitude accuracy. In this section, a scheme with redundant accelerometer configurations is used to achieve accurate azimuth measurement in the near-vertical well. As shown in Figure 6, we can rotate the \(\lambda\) degree around the x-axis to obtain the new coordinate frame \(D^{\prime}\), which has a rotational angle of \(\lambda\) compared to the original \(D\) frame. Two additional accelerometers are employed for the \(Z^{\prime}\) and \(Y^{\prime}\) axes to form a redundant solution, which enables the signal-to-noise ratio of the radial accelerometers to be improved in the near-vertical position. Figure 5: Monte Carlo simulation of attitude errors. (**a**) The error characteristics of pitch at different roll and pitch angles. (**b**) The error variation of roll angle at different pitch and roll angles. (**c**) The error variation of azimuth angle at different roll and pitch angles. (**d**) The error variation of azimuth angle at different azimuth and pitch angles. The transform of the accelerometers between the \(D\) frames and \(D^{\prime}\) frame can be expressed as  
\[ \begin{split}&\mathbf{a}^{D^{\prime}}=\mathbf{C}_{D}^{D^{\prime}}\mathbf{a}^{D} =\left[\begin{array}{ccc}1&0&0\\ 0&cos\ \lambda&-sin\ \lambda\\ 0&sin\ \lambda&cos\ \lambda\end{array}\right]\left[\begin{array}{c}g\ cos\ inc\ sin\ tf\\ -g\ sin\ inc\\ -g\ cos\ inc\ cos\ tf\end{array}\right]\\ &=\left[\begin{array}{c}g\ cos\ inc\ sin\ tf\\ -g\ sin\ inc\ cos\ \lambda+g\ cos\ inc\ cos\ tf\ sin\ \lambda\\ -g\ sin\ inc\ sin\ \lambda-g\ cos\ inc\ cos\ tf\ cos\ \lambda\end{array}\right]\end{split}  \tag{12}\]  
If the error of the radial accelerometer in the near-vertical position is less than the true result from the radial accelerometer, the horizontal attitude accuracy will be improved. We define the ratio of the error to the measured value as the relative error of the accelerometer. The relative errors of the \(Y^{\prime}\) and \(Z^{\prime}\) accelerometers are denoted as \(r_{y^{\prime}}\) and \(r_{z^{\prime}}\), respectively. Assuming that the horizontal errors are approximated as \(\text{d}inc=\text{d}tf\), then the relative error of the \(Y^{\prime}\) and \(Z^{\prime}\) accelerometers can be defined as  
\[r_{y^{\prime}}=\frac{\Delta a_{y^{\prime}}}{a_{y^{\prime}}}=\frac{-cos\ inc\ cos\ tf-\ sin\ inc\ sin\ \lambda\ cos\ tf-cos\ inc\ sin\ tf\ sin\ \lambda}{-sin\ inc\ cos\ \lambda+cos\ inc\ cos\ tf\ sin\ \lambda}\text{d}inc \tag{13}\]  
\[r_{z^{\prime}}=\frac{\Delta a_{z^{\prime}}}{a_{z^{\prime}}}=\frac{-\cos inc \ sin\ \lambda+\ sin\ inc\ cos\ \lambda\ cos\ tf+cos\ inc\ sin\ tf\ cos\ \lambda}{-sin\ inc\ sin\ \lambda-\ cos\ inc\ cos\ tf\ cos\ \lambda}\text{d}inc \tag{14}\]  
Because the error characteristics are analyzed for near-vertical conditions, Equations (13) and (14) can be approximated to the form of Equation (15) when the pitch angle is close to 90 degrees. \[\left\{ \begin{array}{c}r_{y^{\prime}}=cos\ tf\ tan\lambda\text{d}inc\\ r_{z^{\prime}}=-cos\ tf\ cot\ \lambda\text{d}inc\end{array} \right. \tag{15}\]  
Assuming that \(costf\text{d}inc\) is constant, the curves of relative errors and \(\lambda\) are drawn in Figure 7. The relative errors of the \(Y^{\prime}\) and \(Z^{\prime}\) accelerometers reach a minimum at \(\lambda=45^{\circ}\), which ensures the optimal calculation of the horizontal attitude and azimuth in the near-vertical position. Consequently, the installation angle of \(Y^{\prime}\) and \(Z^{\prime}\) accelerometers relative to the Y-axis should be close to 45 degrees and 135 degrees, respectively. However, a calibration method that calculates the actual installation angle and non-orthogonal error is fundamental to improving attitude accuracy.