# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 2 Sparse Aperture Segmented Mirror Aberration Evaluation and Active Optical Control
### Segmented Mirror Active Optical Control Mode  
#### 2.2.1 Control Model Based on the Variation of the Edge Sensor Position  
The principle of co-phase maintenance of Keck telescopes using hexagonal segments for splicing is given in the literature [7], which is mainly based on the hard spot theorem. The traditional co-phase maintenance of the segmented mirror is achieved by exploring the relationship between the position of the hard point and the position of the edge sensor. The axial positioning of the segmented segment in the mirror chamber generally uses three rigid, symmetrically distributed 120\({}^{\circ}\) positioning points, called \"hard spots.\" In the active optical system, the hard spots limit the three degrees of freedom of the segmented segment tip, tilt, and piston. Considering the segment as a rigid body, as shown in Figure 1, one of the edge sensors on the Seg 1 has coordinates \(M_{1}(u,v,0)\). The hard spots \(P_{1}(x_{1},y_{1},0)\), \(P_{2}(x_{2},y_{2},0)\), and \(P_{3}(x_{3},y_{3},0)\) on Seg 1 are counterclockwise distributed on the mirror surface in the three-dimensional space. These three points are connected to form an equilateral triangle with the center of gravity at the center of the segment circle and side length \(a\). Suppose segment 1 is disturbed by wind load, gravity, temperature, and other external factors. The resulting deviation of the segmented mirror from the coplanar inclination \(\theta\) is minimal. Therefore, the changes in the three hard spots in the direction of \(X\) and \(Y\) are considered negligible. The coordinates of the \(P_{1},P_{2},P_{3}\) now become \(P_{1}{}^{\prime}(x_{1},y_{1},z_{1})\), \(P_{2}{}^{\prime}(x_{2},y_{2},z_{2})\) and \(P_{3}{}^{\prime}(x_{3},y_{3},z_{3})\). Similarly, \(M_{1}(u,v,0)\) transforms into \(M_{1}{}^{\prime}(u,v,z)\). Meanwhile, \(M_{1}{}^{\prime}\) is still in the space plane \(O\prime\) formed by the points \(P_{1}{}^{\prime}\), \(P_{2}{}^{\prime}\) and \(P_{3}{}^{\prime}\). \[z= \begin{vmatrix}x_{1}-u&y_{1}-v&z_{1}\\ x_{2}-u&y_{2}-v&z_{2}\\ x_{3}-u&y_{3}-v&z_{3}\end{vmatrix} / \begin{vmatrix}x_{2}-x_{1}&y_{2}-y_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}\end{vmatrix}  \tag{9}\]  
From Equation (9), \(z\) is a linear combination of \(z_{1},z_{2}\) and \(z_{3}\) which is given as follows. \[z=(S_{1}z_{1}-S_{2}z_{2}+S_{3}z_{3})/S_{0} \tag{10}\]  
\(S_{i}(i=0,1,2,3)\) denotes the areas of the triangles \(\Delta P_{1}{}^{\prime}P_{2}{}^{\prime}P_{3}{}^{\prime}\), \(\Delta M_{1}{}^{\prime}P_{2}{}^{\prime}P_{3}{}^{\prime}\), \(\Delta M_{1}{}^{\prime}P_{1}{}^{\prime}P_{3}{}^{\prime}\), and \(\Delta M_{1}{}^{\prime}P_{1}{}^{\prime}P_{2}{}^{\prime}\) respectively, and hold the following relationship. \[S_{0}=-S_{1}+S_{2}+S_{3} \tag{11}\]  
As shown in Figure 1, assume that two pairs of adjacent edge sensors are on the intersection of the plane \(Q\) and the plane \(O^{\prime}\) at a distance of \(d\). Let the distance from the center of the circle of Seg 1 to the plane \(Q\) is \(L\). When the sensors are positioned on the extension of the line connecting the two hard spots, the readings of the edge sensors are  
Figure 1: Schematic diagram of the segment arrangement. insensitive to the displacement of the one remaining hard spot as long as the following condition remains true. \[\left|\frac{2}{3}(\sqrt{3}L-a)\right|<d<\left|\frac{2}{3}(\sqrt{3}L+a)\right| \tag{12}\]  
The heights of the two edge sensors at \({M_{1}}^{\prime}\) and \({N_{1}}^{\prime}\) on Seg 1 are as follows. \[ \begin{array}{l}z_{{M_{1}}^{\prime}}=G_{1}z_{1}+G_{2}z_{2}+G_{3}z_{3}\\ z_{{N_{1}}^{\prime}}=G_{2}z_{1}+G_{1}z_{2}+G_{3}z_{3}\end{array}  \tag{13}\]  
where \(G_{i}=S_{i}/S_{0}\) with \(i=1,2,3\) assuming Equation (12) is satisfied. Similarly, we can obtain the heights of the two edge sensors on Seg 2 after perturbation as follows. \[ \begin{array}{l}z_{{M_{2}}^{\prime}}=R_{1}z_{4}+R_{2}z_{5}+R_{3}z_{6}\\ z_{{N_{2}}^{\prime}}=R_{2}z_{4}+R_{1}z_{5}+R_{3}z_{6}\end{array}