# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 2 Sparse Aperture Segmented Mirror Aberration Evaluation and Active Optical Control
### Warping Harness Technique Calibration Principle  
WH technology corrects the surface shape error of the mirror with the help of its elastic deformation. This correction method is based on the law of linearity and convergence. The WH spring plate is directly connected to the triangle plate in the Whiffletree structure. The actuator pushes the end of the spring plate to convert the correction force into a correction torque applied to the triangular plate. Thus, changing the magnitude of the force on the three axial flexible rods connected to the triangular plate can correct the surface shape. The purpose of the correction is to change the force on the three axial flexible rods connected to the triangular plate. Let us take an equilateral Whiffletree triangle as an example, shown in Figure 2. When \(F_{D}=0\), the following relation can be established.  
\[F_{1}=F_{2}=F_{3}=G/3 \tag{21}\]  
In Equation (21), \(F_{1},\ F_{2}\) and \(F_{3}\) are the reaction forces along the Z-direction at the three points of the triangle. When a force \(F_{D}\) is applied along the Z-direction at WH1, a moment of the \(F_{D}(L+K)\) is generated on the triangle. At this point, \(F_{1},\ F_{2}\) and \(F_{3}\) transform into \(F_{1}^{\prime},\ F_{2}^{\prime}\) and \(F_{3}^{\prime}\).  
\[\left\{\begin{array}{l}F_{1}^{\prime}=F_{1}+\Delta F_{1}\\ F_{2}^{\prime}=F_{2}+\Delta F_{2}\\ F_{3}^{\prime}=F_{3}+\Delta F_{3}\\ F_{1}^{\prime}+F_{2}^{\prime}+F_{3}^{\prime}=G\\ \Delta F_{1}r-\Delta F_{2}r/2-\Delta F_{3}r/2=F_{D}(L+K)\end{array}\right. \tag{22}\]  
According to the symmetry and deformation coordination conditions, the following equality holds.  
\[\Delta F_{2}=\Delta F_{3} \tag{23}\]  
Figure 2: Schematic diagram of WH calibration.  
Moreover, \(\Delta F_{1}\) can be expressed as follows.  
\[\Delta F_{1}=2\Delta F_{2}=F_{D}(L+K)/2r \tag{24}\]  
A schematic diagram of the mirror distortion during correction is shown in Figure 3.  
From Equation (24), the force applied to the spring plate can cause a change in the supporting force of the Whiffletree self-balancing structure. Assuming that when one of the force actuators outputs \(F_{D}=1N\), the axial support rod is subjected to a change in force \(\Delta F\) resulting in the deformation \(W_{i}(x,y)\) of the segmented mirror surface. When the actuator outputs a force \(F_{Di}\), \(i=1,2,\cdots,n\), where \(n\) is the total number of WH actuators in the segmented mirror support system, the total deformation of the mirror surface can be expressed in the following equation.  
\[W(x,y)=\sum_{i=1}^{n}F_{Di}W_{i}(x,y) \tag{25}\]  
Suppose the current specular error is \(W_{D}\) and \(E\) is the stiffness matrix of the WH relative to the specular aberration. To produce a deformation of \(W_{D}\) requires the actuator to apply a correction force of magnitude \(f\).  
\[Ef=W_{D} \tag{26}\]  
When \(W_{D}\) is expressed using the segmented mirror aberration \(U_{D}\) introduced in Section 2.1, Equation (26) takes the following form.  
\[Ef=U_{D} \tag{27}\]  
The following equation is obtained after solving for the residuals \(v_{U}\).  
\[v_{U}=E\hat{f}-U_{D} \tag{28}\]  
where \(\hat{f}\) is the estimated value of \(f\) when the estimated value of \(U_{D}\) is \(\hat{U}_{D}\). Then the evaluation function of the correction quantity is \(MSE(\hat{U}_{D})\).