Constrained least-squares algorithm for active optics correction of a primary mirror

Active optics technology improves the performance and image quality of large telescopes. To effectively compensate for optical aberrations, the constrained least-squares (CLS) algorithm, which considers the characteristics of the resultant moment, the force budget, and the local force smoothness, is proposed to optimize the force distribution. First, the constraint of the resultant moment is used to decouple the shape control and location control. Then, through the force budget, the surface residual and force amplitude can be balanced. At last, the local smooth constraint is proposed to reduce the mirror's internal stress. Simulations were conducted on a 4 m thin mirror to compare the force distributions obtained by the least-squares, bending modes (BMs), and CLS algorithms. The results show that under equivalent residuals, the proposed algorithm is superior to the BM algorithm and performs better on local force smoothness.