## 3 Datum Transfer
### Transformation Matrix  
There are various mathematical expressions to quantify the relationship between the design, measurement, and reference coordinate systems that need to be defined and calculated during the testing and assembly of spacecraft. As shown in Figure 3, the principle of cosine matrix coordinate transformation can be used to derive the attitude relationships between the components, provided that the mechanical design coordinate system of each component is known. As in (1) and (2), the M2 holder can be characterized by the datum  
Figure 2: Definition of the measurement coordinate system. platform or the M1&Instruments holder. As in (3), the M1&Instruments holder can be characterized by the datum platform. \[ \begin{split} P_{S}&=M_{S-B}\times M_{B-O}\times P_{O}\\ &=M_{\text{CS-$S$}}\times M_{\text{CS-$C$B}}\times M_{\text{CB-$ CO$}}\times M_{\text{CO-$O$}}\times P_{O}\\ & P_{S}&=M_{\text{S-$B}}\times P_{B}\\ &=M_{\text{CS-$S$}}\times M_{\text{CS-$C$B}}\times M_{\text{CB-$B $}}\times P_{B}\\ & P_{B}&=M_{\text{B-$O$}}\times P_{O}\\ &=M_{\text{CB-$B$}}\times M_{\text{CB-$CO$}}\times M_{\text{CO-$O $}}\times P_{O}\end{split}  \tag{1}\]  
where: \(P_{S}\) is the coordinate vector in the M2 holder coordinate system. \(P_{B}\) is the coordinate vector in the M1&Instruments holder coordinate system. \(P_{O}\) is the coordinate vector in the datum platform coordinate system. \(M_{\text{S-$B$}}\) is the transformation matrix caused by the change in angle of the M2 holder with respect to the M1&Instruments holder. \(M_{\text{CB-$CO$}}\) and \(M_{\text{CS-$C$B}}\) are the transformation matrices between the cubic prism 1 coordinate system, the cubic prism 2 coordinate system, and the cubic prism 3 coordinate system, respectively. \(M_{\text{CS-$S$}}\), \(M_{\text{CB-$B$}}\), and \(M_{\text{CO-$O$}}\) are installation angle transformation matrices of the cubic prism coordinate system with respect to the respective installation design coordinate system. After defining the respective coordinate systems and the transfer matrices between them, the respective transformation matrix forms are solved [27; 28; 29]. Assuming that the M2 holder and the M1&Instruments holder are rigid bodies, the transformation angle between them is described by the Kardan rotation relationship (_X-Y-Z_ rotation sequence). Rotation \(\alpha\) about the _X_-axis, rotation \(\beta\) about the _Y_-axis, and rotation \(\gamma\) about the _Z_-axis, the \(M_{\text{S-$B$}}\), can be expressed as (5). \[ \begin{split} Rot_{z}&=\begin{bmatrix}\cos\gamma&-\sin \gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\\ \end{bmatrix}\\ Rot_{\gamma}&=\begin{bmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\end{bmatrix}\\ Rot_{X}&=\begin{bmatrix}1&0&0\\ 0&\cos\alpha&-\sin\alpha\\ 0&\sin\alpha&\cos\alpha\end{bmatrix}\\ &M_{\text{S-$B$}}&=Rot_{z}Rot_{\gamma}Rot_{X}\\ &=\begin{bmatrix}\cos\beta\cos\gamma&-\cos\beta\sin\gamma&\sin\beta\\ \cos\alpha\sin\gamma+\sin\alpha\sin\beta\cos\gamma&\cos\alpha\cos\gamma-\sin \alpha\sin\beta\sin\gamma&-\sin\alpha\cos\beta\\ \sin\alpha\sin\gamma-\cos\alpha\sin\beta\cos\gamma&\sin\alpha\cos\gamma+\cos \alpha\sin\beta\sin\gamma&\cos\alpha\cos\beta\end{bmatrix}=\begin{bmatrix} c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\\ \end{split}  \tag{5}\]  
Figure 3: Coordinate system transfer relations. Since the transformation angle of the M2 holder with respect to the M1&Instruments holder is very small, the rotation angles \(\alpha\), \(\beta\), and \(\gamma\) can be found by solving (6). The \(M_{B\text{-}O}\) can be solved in the same way. In the actual measurement process, the M1&Instruments holder and the M2 holder are designed and manufactured from composite materials and they are not rigid bodies. Therefore, the overall deformation needs to be described and evaluated in conjunction with the photogrammetry system. \[\left\{ \begin{array}{l}\alpha=\arctan\!\left(-\frac{c_{23}}{c_{33}}\right)\\ \beta=\arcsin\!\left(c_{13}\right)\\ \gamma=\arctan\!\left(-\frac{c_{23}}{c_{11}}\right)\end{array}