Uncertainty evaluation associated with fitting geometric surfaces to coordinate data

Abstract

Coordinate metrology often involves fitting a geometric surface to coordinate data. The increasingly rigorous approaches to uncertainty evaluation being developed across metrology have been reflected in the need to evaluate the uncertainties associated with coordinate data and calculate how these uncertainties are propagated through to uncertainties associated with the parameters describing the fitted surface. Standard least squares algorithms such as orthogonal distance regression (ODR) for finding the best-fit surface implicitly assume that the uncertainties associated with the coordinates are uncorrelated and axis-isotropic, i.e. the uncertainties associated with the x-, y-and z-coordinates are equal, but very few coordinate measuring systems have such uncertainty characteristics. Given a model of the measurement system, it is possible to propagate uncertainties associated with systematic and random effects influencing the measurements to determine an uncertainty matrix associated with the measured coordinates. This uncertainty matrix will generally be full, reflecting correlation amongst all the measured coordinates, but will have a factorization structure that gives a compact representation of the sources of uncertainty. In this paper, we derive surface fitting algorithms that take account of the coordinate uncertainties in order to provide maximum likelihood estimates of the surface parameters. We show that ODR algorithms can be adapted to take into account the type of uncertainty matrices that arise in practice, exploiting the factorization structure to enable very efficient implementations of these algorithms. As a result, a significant class of surface fitting problems can be solved efficiently using standard nonlinear least squares algorithms. We also discuss how the algorithms can be modified to account for probe radius effects. The calculation of the uncertainties associated with the fitted surface parameters is also derived in such a way that systematic effects associated with the measuring system, such as scale errors, are automatically accounted for.

Uncertainty associated with form assessment in coordinate metrology

“…For example, dealing with data gathered by a coordinate measuring machine (CMM) the parameters σ M could characterise random effects and also scale, squareness and other kinematic errors associated with the CMM [7,8,23]. Similarly, for the GP approach, the variance matrix V (σ F ) specifying the spatial correlation of the form error depends on statistical parameters σ F .…”

“…This distribution can be used to make inferences about F , taking into account the measurement uncertainty associated with the evaluated residual distances, according to the model in (3)-(4). The distribution p(F |d), Figure 1 (dotted curve), is derived according to (8). While the distribution p(F |f ) assigns no probability mass to values of F < F 0 , the distribution p(F |d) takes into account the measurement uncertainty to provide a smoother, Gaussian-like cut off.…”

“…For example, a circle with centre coordinates (x 0 , y 0 ) T and radius r 0 is specified by u → C(u, b) = (x 0 + r 0 cos u, y 0 + r 0 sin u) T , where b = (x 0 , y 0 , r 0 ) T . Given data points X, the best fit parameters of the surface is found by minimising some aggregate measure involving the distances d(x i , b) where d(x, b) is the (signed) orthogonal distance from the point x to the surface S(u, b) [1][2][3][4][5][6][7][8][9][10][11][12][13]. For example, Gaussian or Chebyshev criteria are often used:…”

Large-scale dimensional metrology (LSDM): from tapes and theodolites to multi-sensor systems

“…For this last case, the reference standard is the "Guide to the expression of uncertainty in measurement". 64,109,124,130,[143][144][145][146][147][148][149][150][151][152][153][154] In detail, six elements are considered for the analysis: construction, setup, use and storage, calibration, uncertainty evaluation, error correction/ compensation.…”

Effect of form errors in datum features on evaluated geometries

“…The associated features are plane, line, circle, and cone. The influence factors modelled are CMM scale and squareness errors, [4,5], CMM random effects, and form error for each feature. The calculations derive the uncertainties in the feature parameters calculated according to the frame of reference of the CMM and that imposed by the frame of reference constraints associated with the datum features.…”