# Design of Positioning Mechanism Fit Clearances Based on On-Orbit Re-Orientation Accuracy
### Mathematical Model of Rotation Angles  
The central points of the three positioning balls were called points \(A\), \(B\), and \(C\), assuming that the three points were located on the \(x\)-\(y\) plane. The infinitesimal rotation angle of a rigid body is a vector and can be calculated by using the vector operation rule [18]. The rotation of the optical unit around the \(x\)-axis and \(y\)-axis can be decomposed into rotations of the three points around the line of the other two points respectively. In addition, the rotation of the unit around the \(z\)-axis was decomposed into rotations of point \(A\) around point \(B\) and point \(B\) around point \(A\) on the \(x\)-\(y\) plane. \(\Delta A\), \(\Delta B\), and \(\Delta C\) are the fit clearances of the positioning mechanisms consisting of the three positioning balls, respectively. The maximum rotation angles \(\left|\theta_{x}\right|_{\max}\), \(\left|\theta_{y}\right|_{\max}\), and \(\left|\theta_{z}\right|_{\max}\) of the rigid body around the \(x\)-, \(y\)-, and \(z\)-axes can be calculated by using the following equations (Figure 3):  
\[\left|\theta_{x}\right|_{\max}=\left|\Delta\overset{\rightarrow}{\alpha}_{x }\right|+\left|\Delta\overset{\rightarrow}{\beta}_{x}\right|+\left|\Delta \overset{\rightarrow}{\gamma}_{x}\right|\!, \tag{1}\]  
\[\left|\theta_{y}\right|_{\max}=\left|\Delta\overset{\rightarrow}{\alpha}_{y }\right|+\left|\Delta\overset{\rightarrow}{\beta}_{y}\right|+\left|\Delta \overset{\rightarrow}{\gamma}_{y}\right|\!, \tag{2}\]  
\[\left|\theta_{z}\right|_{\max}=\left|\Delta\overset{\rightarrow}{\alpha}_{z }\right|+\left|\Delta\overset{\rightarrow}{\beta}_{z}\right|\!, \tag{3}\]  
Where \(\Delta\overset{\rightarrow}{\alpha}_{x}\) and \(\Delta\overset{\rightarrow}{\alpha}_{y}\) are the components of the maximum angular displacement vector of point \(A\) rotating around \(BC\) in the direction of the \(x\)-axis and \(y\)-axis, respectively. \(\Delta\overset{\rightarrow}{\beta}_{x}\) and \(\Delta\overset{\rightarrow}{\beta}_{y}\) are the components of the maximum angular displacement vector of point \(B\) rotating around \(AC\) in the direction of the \(x\)-axis and \(y\)-axis, respectively. \(\Delta\overset{\rightarrow}{\gamma}_{x}\) and \(\Delta\overset{\rightarrow}{\gamma}_{y}\) are the components of the maximum angular displacement vector of point \(C\) rotating around \(AB\) in the direction of the \(x\)-axis and \(y\)-axis, respectively. \(\Delta\overset{\rightarrow}{\alpha}_{z}\) is the maximum angular displacement vector of point \(A\) rotating around point \(B\) on the \(x\)-\(y\) plane. \(\Delta\overset{\rightarrow}{\beta}_{z}\) is the maximum angular displacement vector of point \(B\) rotating around point \(A\) on the \(x\)-\(y\) plane. Figure 2: Movement trend of the rigid body when the temperature changes. The dotted triangle represents the initial position of the rigid body, and the solid triangle represents the rigid body after the temperature changes. The letters in brackets correspond to the directions of the translational degrees of freedom (DOF) restricted by each ball. When a rigid body rotates around an axis by an infinitesimal angle, the direction of the angular displacement vector follows that axis. Therefore, Equations (1)-(3) can be transformed into:  
\[|\theta_{x}|_{\max} = \frac{\Delta A/2}{h_{A}}\sin\gamma_{0}+\frac{\Delta B/2}{h_{B}}\sin \beta_{0}+\frac{\Delta C/2}{h_{C}}\sin\alpha_{0}, \tag{4}\] \[|\theta_{y}|_{\max} = \frac{\Delta A/2}{h_{A}}\cos\gamma_{0}+\frac{\Delta B/2}{h_{B}} \cos\beta_{0}+\frac{\Delta C/2}{h_{C}}\cos\alpha_{0},\] (5) \[|\theta_{z}|_{\max} = \frac{\Delta A/2}{|\overset{\rightarrow}{AB}|}+\frac{\Delta B/2 /\cos\alpha_{0}}{|\overset{\rightarrow}{AB}|}=\frac{1}{2}\frac{\Delta A}{| \overset{\rightarrow}{AB}|}+\frac{1}{2}\frac{\Delta B}{\overset{ \rightarrow}{AB}\overset{\rightarrow}{j}}, \tag{6}\]  
where the vector \(\overset{\rightarrow}{j}=(0,1,0)\).