Abstract
The accuracy of surface measurement determines the manufacturing quality of membrane mirrors. Thus, an efficient and accurate measuring method is critical in membrane mirror fabrication. This paper formulates this measurement issue as a surface reconstruction problem and employs two-stage trained Zernike polynomials as an inline measuring tool to solve the optical surface measurement problem in the membrane mirror manufacturing process. First, all terms of the Zernike polynomial are generated and projected to a non-circular region as the candidate model pool. The training data are calculated according to the measured values of distance sensors and the geometrical relationship between the ideal surface and the installed sensors. Then the terms are selected by minimizing the cost function each time successively. To avoid the problem of ill-conditioned matrix inversion by the least squares method, the coefficient of each model term is achieved by modified elitist teaching–learning-based optimization. Subsequently, the measurement precision is further improved by a second stage of model refinement. Finally, every point on the membrane surface can be measured according to this model, providing more the subtle feedback information needed for the precise control of membrane mirror fabrication. Experimental results confirm that the proposed method is effective in a membrane mirror manufacturing system driven by negative pressure, and the measurement accuracy can achieve 15 µm.
Highlights
In recent years, the space reflector has been used in various fields such as remote sensing, solar energy concentrators and astronomical applications [1,2,3]
The membrane mirror surface measurement problem is solved from the point of view of system identification
The mirror surface is constructed in the form of a linear-in-the-parameters model, using transformed Zernike polynomials which are suitable for the rectangular region as a basis function
Summary
The space reflector has been used in various fields such as remote sensing, solar energy concentrators and astronomical applications [1,2,3]. MacMartin et al analyzed the structural interaction of the segmented mirror of a telescope using the Zernike basis, offering guidance for structural optimization of mirrors [23] It is effective for optical surface measurement. Satisfying surface measurement accuracy and speed would be achieved by this method, providing a feasible means to maximize the performance of Zernike polynomials. If all candidate bases are used like extreme learning machines, the computational complexity of (4) may become extremely high and it may become impossible to solve due to the ill-conditioned matrix To deal with this problem, a forward recursive algorithm (FRA) is used to generate a parsimonious model, and the coefficients are obtained by heuristic optimizing algorithms avoiding the operation of matrix inversion
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