# Optimization Design of the Spaceborne Connecting Structure for a Lightweight Space Camera
## 2 Connecting Structure Design Principle
### Relative Displacement of Mirrors  
Under the action of external loads, mirrors have a small displacement, which is also called rigid body displacement, including linear displacement and angular displacement. By establishing a camera finite element model, the mirror surface is divided into finite nodes. The rigid body motions of mirrors can be obtained by calculating the average rigid body displacement of surface nodes [14]. In this case, building a multi-point flexible constraint element is a very effective method to obtain the rigid body displacement of the primary mirror and secondary mirror.  
The displacements of the second mirror relative to the primary mirror include linear displacements of three degrees of freedom and angular displacements of three degrees of freedom. The calculation of the relative angular displacement is easy; it is the difference between the angular displacement of the second mirror and that of the primary mirror, but it changes the optical axis direction. The relative linear displacement is affected by the change of optical axis direction, and its calculation needs to add a quantity to the difference value. Taking the calculation of relative linear displacement along the X-axis direction as an example, the added quantity was analyzed, as shown in Figure 3.  
Figure 3: Relative linear displacement of mirrors along the x-axis direction.  
Figure 2: Relationship between the connecting structure, the camera, and the installation position.  
From Figure 3, the relative displacement \(R_{1}\) can be obtained as shown in Equation (1).  
\[\left\{\begin{array}{c}R_{1}=\left|L_{\mathrm{X}}^{S}-L_{\mathrm{X}}^{P}-L^{ \prime}\chi\right|\cdot\cos\left(A_{\mathrm{Y}}^{\mathrm{P}}\right)\\ L^{\prime}\chi=D\cdot\tan\left(A_{\mathrm{Y}}^{\mathrm{P}}\right)\end{array}\right. \tag{1}\]  
in which \(L_{\mathrm{X}}^{S}\) represents the linear displacement of the second mirror along the X-axis; \(L_{\mathrm{X}}^{\mathrm{P}}\) represents the linear displacement of the primary mirror along the X-axis; \(A_{\mathrm{Y}}^{\mathrm{P}}\) represents the angular displacement of the primary mirror along the Y-axis; and \(R_{1}\) represents the relative linear displacement along the X-axis.  
Similarly, the relative linear displacement along the Y-axis can be obtained, as shown in Equation (2).  
\[R_{2}=\left|L_{y}^{S}-L_{Y}^{P}-D\cdot\tan\left(A_{\mathrm{X}}^{\mathrm{P}} \right)\right|\cdot\cos\left(A_{\mathrm{X}}^{\mathrm{P}}\right) \tag{2}\]  
Since the Z-axis direction is the optical axis direction, the relative linear displacement along the Z-axis is not affected by the change of the optical axis direction, and it just consists of the difference between the displacement of the second mirror and that of the primary mirror. Then, the relative displacements can be summarized as shown in Equation (3).  
\[\left\{\begin{array}{c}R_{1}=\left|L_{\mathrm{X}}^{S}-L_{\mathrm{X}}^{P}-D \cdot\tan\left(A_{\mathrm{Y}}^{\mathrm{P}}\right)\right|\cdot\cos\left(A_{ \mathrm{Y}}^{\mathrm{P}}\right)\\ R_{2}=\left|L_{y}^{S}-L_{\mathrm{Y}}^{P}-D\cdot\tan\left(A_{\mathrm{X}}^{ \mathrm{P}}\right)\right|\cdot\cos\left(A_{\mathrm{X}}^{\mathrm{P}}\right)\\ R_{3}=\left|L_{z}^{S}-L_{z}^{P}\right|\\ R_{4}=\left|A_{x}^{S}-A_{x}^{P}\right|\\ R_{5}=\left|A_{y}^{S}-A_{y}^{P}\right|\\ R_{6}=\left|A_{z}^{S}-A_{z}^{P}\right|\end{array}\right. \tag{3}\]