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Unformatted text preview: 3.2. Orthogonal Polynomial Model [Soman, 1999] shows that the warping variation can be separated from the waviness and roughness variation by fitting and subtracting polynomials from the variation data. [Tonks, 2002] finds that a series of orthogonal polynomials are an effective means of modeling the warping variation. He develops a model that uses a series of Legendre polynomials to model the covariance. Typical surface variation resulting from a manufacturing process is defined by the average polynomial coefficient vector a.
In the orthogonal polynomial model, an uncorrelated covariance matrix Σ0 is created by placing the closure point variances down the diagonal of a diagonal matrix. A correlation matrix is created according to M −1 2l + 1 (5) S ij = ∑ a l Pl ( x i ) Pl ( x j ) N l =o where Pl ( x i ) is the lth order Legendre polynomial at point xi. The geometric covariance, Σδ, is found using Σ0 and S according to Σ δ = SΣ 0 S (6) Orthogonal polynomials were found to accurately model warping variation, but the accuracy quickly decreased for variation with more than one wa...
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This note was uploaded on 08/03/2010 for the course DD 1234 taught by Professor Zczxc during the Spring '10 term at Magnolia Bible.
- Spring '10
- The Land