A novel hybrid trust region minimax fitting algorithm for accurate dimensional metrology of aspherical shapes

作者：Arezki Yassir^*; Nouira Hichem^*; Anwer Nabil; Mehdi Souzani Charyar

来源：Measurement, 2018, 127: 134-140.

DOI：10.1016/j.measurement.2018.05.071

• Trust region methods
• Aspheric shapes
• Minimax
• Chebyshev fitting
• Minimization
• Form error
• Dimensional metrology
• Ultra-high precision engineering

Ultra-high precision measuring machines enable to measure aspheric shapes with an uncertainty of few tens of nanometres. The resulting clouds of points are then associated to theoretical model at the same level of accuracy so as to obtain parameters that indicate about form error. Minimum zone (MZ), defined as the least value of peak to valley (PV), is widely used to assess form error. Least squares method (L-2) is often used to determine MZ but the resulting value is usually overestimated. For this reason, L-2 is replaced by L-infinity norm because it gives a more accurate value of MZ since it directly minimizes PV. Using L-infinity norm results in a non-smooth optimization problem and consequently its resolution becomes more challenging compared to L-2.
In this paper, a novel minimax fitting method for accurate metrology of aspheres and freeform based on a hybrid trust region algorithm (HTR) is proposed. To assess performance of the introduced method, it was compared to an available minimax fitting algorithm based on a smoothing technique: exponential penalty function (EPF). The choice of EPF is justified by superior performances in comparison to existing techniques. Comparison was conducted on reference data, data available in literature and data gathered form measurements of a real optical high quality asphere. Results show superiority of HTR over EPF in both returned PV values and execution time.

• 出版日期2018-10