## 2 Measuring System
### Error Analysis of Install Angle  
The desired mounting angle of the inclinometer is shown in Figure 7, where the x-axis of the inclinometer is aligned with the elevation axis of the antenna.  
Figure 5: The communication protocol and wiring path of the measurement system.  
Figure 6: A schematic diagram of the user interface of the acquisition software.  
According to reference [10], the azimuth pointing error \(\Delta_{\mathrm{az}}\) and elevation pointing error \(\Delta_{\mathrm{el}}\) of the inclinometer installed in this orientation at a given elevation angle \(\theta_{\mathrm{el}}\) can be expressed as follows:  
\[\Delta_{\mathrm{el}}=\alpha_{\mathrm{x}}, \tag{1}\]  
\[\Delta_{\mathrm{az}}=-\alpha_{\mathrm{y}}\cdot\tan\theta_{\mathrm{el}} \tag{2}\]  
The values of \(\alpha_{\mathrm{x}}\) and \(\alpha_{\mathrm{y}}\) represent the tilt angle of the measuring plane around the inclinometer's x-axis and y-axis, respectively. However, it is hard to achieve perfect parallelism between the x-axis and the elevation axis. The subsequent analysis shows the extent of measurement error resulting from a minor deviation in installation angle.  
As illustrated in Figure 8, OAB denotes the installation plane of the inclinometer. While OA represents the optimal installation direction (i.e., parallel to the elevation axis of the antenna), the x-axis of the inclinometer is aligned with the OB direction and OC is perpendicular to OB, paralleling with the y-axis of the inclinometer. When the plane tilts around the OA, point B rotates to point B', and point C rotates to point C'. The inclinometer output represents the rotation value around the OB axis, known as \(\angle\)COC'. However, our objective is to determine the angle of rotation, \(\angle\)B'AB, with respect to the direction of the pitch axis.  
Set \(\angle\)B'AB = \(\alpha\), \(\angle\)AOB = \(\beta\), \(\angle\)COC' = y, \(\mid\)OA \(\mid\) = I, \(\mid\)AB \(\mid\) = m.  
Obviously \(\mid\)OB' \(\mid\) = \(\mid\)OB \(\mid\) = \(\sqrt{1^{2}+\mathrm{m}^{2}}\), and \(\angle\)AOB' = \(\angle\)AOB = \(\beta\)  
So \(\mid\)OC \(\mid\) = \(\mid\)OC' \(\mid\) = \(\mid\)OB' \(\mid\)tan\(\angle\)AOB' = \(\mid\)OB' \(\mid\)tan\(\angle\)AOB = \(\mid\)OB \(\mid\)\(\cdot\)tan\(\beta\).  
Since the distance from point C and point B to the axis of rotation OA are equal (both being m), then: \(\mid\)CC' \(\mid\) = \(\mid\)BB' \(\mid\) = \(\mid\)2 AB\(\mid\)sin\(\frac{\angle\)B'AB \(\mid\)2}=2m\cdot\)sin\(\frac{\pi}{2}\)  
\[\sin\frac{\angle\)C'OC}  
Since \(\frac{\mathrm{m}}{\sqrt{1^{2}+\mathrm{m}^{2}}}\)= \(\sin\beta\), so  
\[\sin\frac{\gamma}{2}=\frac{\sin\beta\sin\frac{\alpha}{2}}{\tan\beta}=\sin\frac{ \alpha}{2}\cos\beta\]  
Due to the small value of the measured angle \(\alpha\), it is reasonable to approximate \(\sin\frac{\alpha}{2}\approx\frac{\alpha}{2}\), \(\sin\frac{\gamma}{2}\approx\frac{\gamma}{2}\). therefore, we obtain  
\[\gamma=\alpha\cos\beta\]  
The absolute value of relative error \(\varepsilon\) is:  
\[\varepsilon=\left|\frac{\alpha-\gamma}{\alpha}\right|=\left|\frac{\alpha- \alpha\cos\beta}{\alpha}\right|=\left|1-\cos\beta\right|\]  
It can be inferred from the formula of \(\varepsilon\) that the relative measurement error is merely 1.5% when the installation error angle is 10\({}^{\circ}\) and 6.1% when it is 20\({}^{\circ}\). Given that the range of the sensor is only \(\pm\)1.51 mrad, it may be deemed negligible in terms of the inclinometer's installation-induced measurement errors.