# Design of Positioning Mechanism Fit Clearances Based on On-Orbit Re-Orientation Accuracy
## 3 Rotation Angle Measurement Principle and Error Analysis
### Measurement Error Analysis  
The initial position of the mirror is represented by the plane \(OAB\) on the \(x\)-\(y\) plane, as shown in Figure 5. After rotating around the \(x\)-axis by infinitesimal angle \(\theta_{x}\) and then rotating around the \(y\)-axis by infinitesimal angle \(-\theta_{y}\), the mirror moved onto the plane \(OA^{\prime}B^{\prime}\). The angular displacement vectors were \(\overrightarrow{\theta_{x}}=(\theta_{x},0,0)\) and \(\overrightarrow{\theta_{y}}=\left(0,-\theta_{y},0\right)\). The normal vector of the mirror changed from the \(z\)-axis to \(\overrightarrow{OM}\) and then to \(\overrightarrow{ON}\). It was also observed that plane \(OAB\) rotated to plane \(OA^{\prime}B^{\prime}\) by one step with a rotation angle \(\theta\) and an angular displacement vector \(\overrightarrow{\theta}\). \(NH\) was perpendicular to the \(y\)-\(z\) plane at point \(H\), assuming that the angle between \(OH\) and the z-axis was \(\theta_{cx}\), as well as \(\angle HON=\theta_{cy}\). The positions of the theodolites indicated that \(\theta_{cx}\) was the same as the difference between the azimuth angles of theodolite2 before and after rotating, while \(-\theta_{cy}\) was the same as the difference between the pitch angles of theodolite2 before and after rotating. The rotation around the \(z\)-axis slightly affected the rotation angles around the \(x\)- and \(y\)-axes if the \(z\)-axis rotation angle was infinitesimal. Thus, we ignored the \(z\)-axis rotation. Let \(|\overrightarrow{OM}|=|\overrightarrow{ON}|=1\). Thus,  
\[\overrightarrow{ON}=\left(-\cos\theta_{x}\sin\theta_{y},-\sin\theta_{x},\cos \theta_{x}\cos\theta_{y}\right) \tag{13}\]  
\[\overrightarrow{OH}=\left(0,-\sin\theta_{x},\cos\theta_{x}\cos\theta_{y} \right). \tag{14}\]  
\(\theta_{cx}\) and \(\theta_{cy}\) were calculated using the following equations:  
\[\theta_{cx}=\arccos\frac{\overrightarrow{k\cdot\partial H}}{|k\|\partial H|}= \arccos\frac{\cos\theta_{x}\cos\theta_{y}}{\sqrt{\cos^{2}\theta_{x}\cos^{2} \theta_{y}+\sin^{2}\theta_{x}}}, \tag{15}\]  
\[\theta_{cy}=\arccos\frac{\overrightarrow{ON\cdot\partial H}}{|\overrightarrow {ON\|OH}|}=\arccos\sqrt{\cos^{2}\theta_{x}\cos^{2}\theta_{y}+\sin^{2}\theta_ {x}}, \tag{16}\]  
where \(\overrightarrow{k}=(0,0,1)\). Figure 5: Schematic of mirror rotation. \(\overrightarrow{OM}\) is the normal vector of the mirror plane that rotates infinitesimal angle \(\theta_{x}\) around the \(x\)-axis; \(\overrightarrow{ON}\) is the normal vector of the mirror plane that rotates infinitesimal angle \(-\theta_{y}\) around the \(y\)-axis. The circle \(O^{\prime}\) passing through points \(M\) and \(N\) is parallel to the \(z\)-\(x\) plane. \(\theta_{x}\) and \(\theta_{y}\) were expressed using the following equations in terms of Equations (15) and (16):  
\[\theta_{x}=\arcsin\Bigl{(}\sin\theta_{cx}\cos\theta_{cy}\Bigr{)}, \tag{17}\]  
\[\theta_{y}=\arccos\frac{\cos\theta_{cx}\cos\theta_{cy}}{\sqrt{1-\sin^{2}\theta_{ cx}\cos^{2}\theta_{cy}}}. \tag{18}\]  
To derive the differences between \(|\Delta\theta_{cx}|\), \(|\Delta\theta_{cy}|\) and \(|\overrightarrow{\theta_{x}}|\), \(|\overrightarrow{\theta_{y}}|\) under some limits, some numerical simulations were carried out by using Mathematica. Assuming that \(\Delta\theta_{cx}=|\theta_{cx}|-|\overrightarrow{\theta_{x}}|\) and \(\Delta\theta_{cy}=|\theta_{cy}|-|\overrightarrow{\theta_{y}}|\), the relationship between the differences and \(\theta_{x}\), \(\theta_{y}\) is shown in Figure 6. Thus, some simple unit transformations were used to draw the conclusion that \(|\Delta\theta_{cx}|\) and \(|\Delta\theta_{cy}|\) were both no more than \(0.0003^{\prime\prime}\) when \(|\overrightarrow{\theta_{x}}|\) and \(|\overrightarrow{\theta_{y}}|\) were not more than \(300^{\prime\prime}\).