Optimisation Algorithms for Generalised Distance Regression in Metrology

Bartholomew‐Biggs, M. C.; Butler, Bernard; Forbes, Alistair

doi:10.1142/9789812793836_0003

Cited by 11 publications

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where all terms on the left-hand side are evaluated at u * . For standard geometric elements, the distance function d(x,b) can be defined as an explicit function of the parameters but for freeform surfaces the optimal footpoint parameters u * have to be determined using numerical techniques [42][43][44].…”

Section: (A) Case Study: Fitting Geometric Surfaces To Datamentioning

confidence: 99%

Mathematics for modern precision engineering

Scott

Forbes

2012

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

The aim of precision engineering is the accurate control of geometry. For this reason, mathematics has a long association with precision engineering: from the calculation and correction of angular scales used in surveying and astronomical instrumentation to statistical averaging techniques used to increase precision. This study illustrates the enabling role the mathematical sciences are playing in precision engineering: modelling physical processes, instruments and complex geometries, statistical characterization of metrology systems and error compensation.

show abstract

Section: (A) Case Study: Fitting Geometric Surfaces To Datamentioning

confidence: 99%

Mathematics for modern precision engineering

Scott

Forbes

2012

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…Standard optimization techniques can be tailored to provide efficient and stable solutions to the footpoint problem. Solution approaches for the problem posed as in (7) are considered in [13].…”

Section: Generalized Distance Regressionmentioning

confidence: 99%

Surface fitting taking into account uncertainty structure in coordinate data

Forbes¹

2006

Meas. Sci. Technol.

View full text Add to dashboard Cite

Standard least squares algorithms for finding the best-fit geometric surface to coordinate data 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 can vary from point to point). Very few coordinate measuring systems have such uncertainty characteristics but, in the absence of quantitative information about the true uncertainty structure, these assumptions can be justified. However, more effort is now being applied to evaluate the uncertainties associated with coordinate measuring systems and the question arises of how best to use this extra information, for example in surface fitting. This paper describes algorithms for fitting geometric surfaces to coordinate data with general uncertainty structure and shows how these algorithms can be implemented efficiently for a class of uncertainty matrices that arise in many practical systems. The fitting algorithms are illustrated on data simulating laser tracker and coordinate measuring machine measurements.

show abstract

“…These equations state that x − f(u * , b) is orthogonal to both f u and f v at u * (and so aligned with the normal vector n). Newton's method can be applied to solve these equations [6,11,12].…”

Section: Orthogonal Distance Functionsmentioning

confidence: 99%

“…This correlation can be explicitly calculated given the model describing the functional dependence of the measured coordinates on the random and systematic effects. The uncertainty matrix U ξ has a factorization as in (12) with the factor B having a welldefined sparsity structure. In section 3.5, we show how this structure can be exploited.…”

Section: Uncertainty Matrix Associated With Coordinate Datamentioning

confidence: 99%

See 1 more Smart Citation

Uncertainty evaluation associated with fitting geometric surfaces to coordinate data

Forbes¹

2006

Metrologia

View full text Add to dashboard Cite

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.

show abstract

Optimisation Algorithms for Generalised Distance Regression in Metrology

Cited by 11 publications

References 0 publications

Mathematics for modern precision engineering

Mathematics for modern precision engineering

Surface fitting taking into account uncertainty structure in coordinate data

Uncertainty evaluation associated with fitting geometric surfaces to coordinate data

Contact Info

Product

Resources

About