# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 4 Simulation Verification
### Segmented Mirror Active Optical Control  
Two schemes were used to correct the segmented mirror error introduced by gravity load discussed in Section 4.1. Scheme 1 is the active optical control model based on the change in the edge sensor position, and this scheme was derived in Section 2.2.1. Scheme 2 is the active optical control model based on the aberration evaluation model proposed in Section 2.2.2. For Scheme 1: First, the stiffness matrix \(A_{12\times 21}\) is established by applying unit forced displacement to each hard spot position of the segmented mirror system, respectively. Then the edge sensor position-variation matrix \(H_{Y}^{(0)}\), \(H_{Z}^{(0)}\) is established under Y and \(Z\) working conditions. The correction process is as follows. Firstly, according to Equation (16) and the algorithm in Section 3, the vector dimension \(M=21+1=22\) is solved. With the number of individuals in the population as 220, the evolutionary algebra \(G=50\), the crossover probability \(CR=0.8\), and the amount of change in height for each hard spot \(z_{Y}^{(0)}\), \(z_{Y}^{(0)}\) is solved and substituted into the model. The remaining amount of change in the position of the edge sensor after corrections \(H_{Y}^{(1)}\), and \(H_{Z}^{(1)}\) is calculated. It is a cyclic process of active optical correction of the segmented mirror and generally takes 2-3 cycles to entirely correct \(H_{Y}^{(0)}\) and \(H_{Z}^{(0)}\). For Scheme 2: Similar to Scheme 1, the unit forced displacement is applied to each hard spot, respectively, and the mirror deformation is decomposed according to Equation (1). Then, the stiffness matrix \(\Delta U_{k}\) is established. Afterward, the mirror aberration matrix is obtained for Y and \(Z\) conditions \(U_{kY}^{\prime(0)}\) and \(U_{kZ}^{\prime(0)}\). The correction process is the same as that of Scheme 1. According to Equation (18) and the algorithm in Section 3, the change in the height of each hard spot \(\Delta z_{Y}^{(0)},\Delta z_{Z}^{(0)}\) is calculated and substituted into the model. The residual of the mirror aberration after corrections \(U_{kY}^{\prime(1)}\) and \(U_{kZ}^{\prime(1)}\) is calculated. The loop is run \(t\) times until \(U_{kY}^{\prime(t)}-U_{kY}^{\prime(t-1)}\) and \(U_{kZ}^{\prime(t)}-U_{kZ}^{\prime(t-1)}\) are less than the finite element solution accuracy. The residual deformation of the segmented mirrors after correction by the two schemes is shown in Figure 5.  
In Scheme 1, the edge sensor value indicates the Z-directional rigid body displacement of the mirror surface. Because there is a certain angle between each segment surface upon splicing, different rigid body displacements along X and Y directions are observed between Seg 1 and the remaining 6 segments. It makes the residual error of the segmented mirror surface larger when the displacement of the edge sensor is corrected to 0. It also reflects that during the design of the segment support system, the rigid body displacement of the segment's surface along X and Y directions should be strictly set according to the requirements of co-phase accuracy of the segmented mirror surface at the same time, which  
Figure 4: Mirror deformation under load, (**a**) Deformation of mirror surface in Y condition, (**b**) Deformation of mirror surface in Z condition, (**c**) Deformation of mirror surface in T condition. means that the design and processing requirements of the support system are improved to a certain extent, and the processing and manufacturing costs are increased. In Scheme 2, because the correction object is the wavefront aberration of the segmented mirror, it can be regarded as the aberration of the local position of the segmented mirror in the X and Y directions of each segment. The correction object is also reflected in the numerical magnitude of the aberration. Therefore, the correction accuracy of this scheme is higher, and the residual error of the mirror after correction is small. At the same time, Scheme 2 can reduce the complexity of the control system because there is no edge sensor involved.