Metrological Apparatus Calibration Using Barycentric Lagrange Interpolation
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Metrological apparatuses face challenges in accurately measuring surface characteristics due to limitations in position transducer linearity, non-linear responses, and inaccuracies caused by stylus tip size and arcuate movement, which require effective calibration methods to correct measurement data.
Innovation Solution
A calibration method using a controller and data fitter to determine calibration coefficients through a calibration procedure on a reference surface, employing Barycentric Lagrange interpolation with Chebychev points to achieve stable fitting and correct measurement data, enabling accurate surface characteristic measurement.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If polynomial expansion is used for fitting measurement data to known form, then calibration can be performed, but the method is susceptible to the Runge phenomenon which causes instability and increased errors at the ends of the calibration range
Solution Approach 1:
The patent changes the mathematical parameters of the fitting approach from standard polynomial expansion to Barycentric Lagrange interpolation. This parameter change transforms the calibration method to avoid the Runge phenomenon while maintaining accuracy, directly resolving the contradiction between measurement precision and calibration stability.
Solution Approach 2:
The patent introduces Chebychev points as intermediary reference points for the calibration process. These specially distributed points act as mediators between the measurement data and the fitting function, providing a stable foundation for interpolation that eliminates the oscillatory behavior characteristic of polynomial expansion at range boundaries.
2Length of stationary object
If the measurement range is increased to cover larger surface areas, then form measurement capability is improved, but transducer linearity deteriorates and calibration becomes more difficult
Solution Approach 1:
The patent changes the fitting function parameters to use Barycentric Lagrange interpolation with Chebychev points, which provides superior performance for large-range measurements compared to polynomial expansion. This parameter change enables accurate calibration across extended measurement ranges while maintaining transducer linearity through the mathematical properties of the interpolation method.
Data Source
Figure 1
Figure 2
Figure 3
AI summary
A metrological apparatus has a driver (33) that effects relative movement between a support (4) and a measurement probe (8) carriage (7) in a first direction (X) to cause the measurement probe (8) to traverse a measurement path along a surface of an object supported by the support. The measurement probe (8) moves in a second direction (Z) transverse to the first direction as it follows surface characteristics. Respective first and second position transducers (35, 32) provide first and second position data representing the position of the measurement probe in the first and second direction. A calibrator (300) carries out a calibration procedure using measurement data obtained on a surface of known form. The calibrator determines calibration coefficients of an expression relating corrected measurement data and the actual measurement data by using the known form of the reference surface as the corrected measurement data. The calibrator varies the calibration coefficient for Chebychev points until the at least one expression provides a fit to the data.