Alistair Forbes

Laser tracker error determination using a network measurement

We report on a fast, easily implemented method to determine all the geometrical alignment errors of a laser tracker, to high precision. The technique requires no specialist equipment and can be performed in less than an hour. The technique is based on the determination of parameters of a geometric model of the laser tracker, using measurements of a set of fixed target locations, from multiple locations of the tracker. After fitting of the model parameters to the observed data, the model can be used to perform error correction of the raw laser tracker data or to derive correction parameters in the format of the tracker manufacturers internal error map. In addition to determination of the model parameters, the method also determines the uncertainties and correlations associated with the parameters. We have tested the technique on a commercial laser tracker in the following way. We disabled the trackers internal error compensation, and used a five-position, fifteen-target network to estimate all the geometric errors of the instrument. Using the error map generated from this network test, the tracker was able to pass a full performance validation test, conducted according to a recognized specification standard (ASME B89.4.19-2006). We conclude that the error correction determined from the network test is as effective as the manufacturers own error correction methodologies.

Classical and Bayesian interpretation of the Birge test of consistency and its generalized version for correlated results from interlaboratory evaluations

A well-known test of consistency in the results from an interlaboratory evaluation is the Birge test, named after its developer Raymond T Birge, a physicist. We show that the Birge test of consistency may be interpreted as a classical test of the null hypothesis that the variances of the results are less than or equal to their stated values against the alternative hypothesis that the variances of the results are greater than their stated values. A modern protocol for hypothesis testing is to calculate the classical p-value of the test statistic. The p-value is the maximum probability under the null hypothesis of realizing in conceptual replications a value of the test statistic equal to or larger than the realized (observed) value of the test statistic. The null hypothesis is rejected when the p-value is too small. We show that, interestingly, the classical p-value of the Birge test statistic is equal to the Bayesian posterior probability of the null hypothesis based on suitably chosen non-informative improper prior distributions for the unknown statistical parameters. Thus the Birge test may be interpreted also as a Bayesian test of the null hypothesis. The Birge test of consistency was developed for those interlaboratory evaluations where the results are uncorrelated. We present a general test of consistency for both correlated and uncorrelated results. Then we show that the classical p-value of the general test statistic is equal to the Bayesian posterior probability of the null hypothesis based on non-informative prior distributions. The general test makes it possible to check the consistency of correlated results from interlaboratory evaluations. The Birge test is a special case of the general test.

Study of the uncertainty of angle measurement for a rotary-laser automatic theodolite (R-LAT)

Abstract This paper shows how the angular uncertainties can be determined for a rotary-laser automatic theodolite of the type used in (indoor-GPS) iGPS networks. Initially, the fundamental physics of the rotating head device is used to propagate uncertainties using Monte Carlo simulation. This theoretical element of the study shows how the angular uncertainty is affected by internal parameters, the actual values of which are estimated. Experiments are then carried out to determine the actual uncertainty in the azimuth angle. Results are presented that show that uncertainty decreases with sampling duration. Other significant findings are that uncertainty is relatively constant throughout the working volume and that the uncertainty value is not dependent on the size of the reference angle.

Bayesian posterior predictive p-value of statistical consistency in interlaboratory evaluations

The results from an interlaboratory evaluation are said to be statistically consistent if they fit a normal (Gaussian) consistency model which postulates that the results have the same unknown expected value and stated variances–covariances. A modern method for checking the fit of a statistical model to the data is posterior predictive checking, which is a Bayesian adaptation of classical hypothesis testing. In this paper we propose the use of posterior predictive checking to check the fit of the normal consistency model to interlaboratory results. If the model fits reasonably then the results may be regarded as statistically consistent. The principle of posterior predictive checking is that the realized results should look plausible under a posterior predictive distribution. A posterior predictive distribution is the conditional distribution of potential results, given the realized results, which could be obtained in contemplated replications of the interlaboratory evaluation under the statistical model. A systematic discrepancy between potential results obtained from the posterior predictive distribution and the realized results indicates a potential failing of the model. One can investigate any number of potential discrepancies between the model and the results. We discuss an overall measure of discrepancy for checking the consistency of a set of interlaboratory results. We also discuss two sets of unilateral and bilateral measures of discrepancy. A unilateral discrepancy measure checks whether the result of a particular laboratory agrees with the statistical consistency model. A bilateral discrepancy measure checks whether the results of a particular pair of laboratories agree with each other. The degree of agreement is quantified by the Bayesian posterior predictive p-value. The unilateral and bilateral measures of discrepancy and their posterior predictive p-values discussed in this paper apply to both correlated and independent interlaboratory results. We suggest that the posterior predicative p-values may be used to assess unilateral and bilateral degrees of agreement in International Committee of Weights and Measures (CIPM) key comparisons.

Uncertainty evaluation associated with fitting geometric surfaces to coordinate data

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.

Surface fitting taking into account uncertainty structure in coordinate data

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.

A probabilistic approach to the analysis of measurement processes

We consider a probabilistic model of the measurement process, based on identifying two main sub-processes, named observation and restitution. Observation constitutes the transformations involved in producing the observable output. Restitution constitutes the determination of the measurand (the quantity measured) from the observable output, and includes data processing. After providing a probabilistic representation of the observation sub-process, we derive appropriate formulae for addressing restitution and describing the overall measurement process. The model allows the treatment in probabilistic terms of both the random and systematic effects that influence the measurement process, and may prove particularly useful in the formulation phase of uncertainty evaluation. We also discuss the different ways in which the measurand can be characterized by a probability distribution, and demonstrate the application of the approach to the analysis of risk in conformance testing.

Weighting observations from multi-sensor coordinate measuring systems

In large scale coordinate metrology, estimates of target coordinates are determined from range and angle observations gathered by one or more laser trackers. In analysing such data, it is required to weight the range observations relative to the angle observations. While the system characterization will give a guide to the accuracy of these measurements, the actual behaviour will depend strongly on environmental effects, particularly those influencing the refractive index of the air. In this paper, we describe a general Bayesian approach to balancing sets of observational data in which the noise parameters are treated as unknowns but associated with prior distributions. Estimates of the appropriate weighting factors are derived from the posterior distributions for these noise parameters. We give examples of this general approach applied to the analysis of laser tracker data.

The quantum metrology triangle and the redefinition of the SI ampere and kilogram; analysis of a reduced set of observational equations

We have developed a set of seven observational equations that include all of the physics necessary to relate the most important of the fundamental constants to the definitions of the SI kilogram and ampere. We have used these to determine the influence of alternative definitions currently under consideration for the SI kilogram and ampere on the uncertainty of three of the fundamental constants (h, e and mu). We have also reviewed the experimental evidence for the exactness of the quantum metrology triangle resulting from experiments combining the quantum Hall effect, the Josephson effects and single-electron tunnelling.

Parameter Estimation Based on Least Squares Methods

This chapter describes how standard linear and nonlinear least squares methods can be applied to a large range of regression problems. In particular, it is shown that for many problems for which there are correlated effects it is possible to develop algorithms that use structure associated with the variance matrices to solve the problems efficiently. It is also shown how least squares methods can be adapted to cope with outliers.