Intrinsic Uncertainty: The Elephant in the Room
Dr Jody Muelaner posted on February 06, 2019 |

I’ve written a lot about how we can determine the uncertainty of a measurement. It is often assumed that our confidence in a measurement equates to our confidence that the product being measured conforms to a specification. Unfortunately, this is not the case. Specifications describing the form of components are usually given as Geometric Dimensions and Tolerances (GD&T). These state that all points on a feature must lie within a given tolerance zone. We can, of course, only measure a finite number of points on a surface. If the measured points are within the tolerance zone, we assume the feature conforms, but we don’t know what is happening at all of the unmeasured locations on the surface. This issue doesn’t get talked about much, probably because there isn’t a neat solution to it. In this article, I look at some of the approaches currently being used to deal with it, as well as how it may finally be solved.

The Guide to the Expression of Uncertainty in Measurement (GUM)refers to ‘the uncertainty due to variation in the thing we are measuring’ as intrinsic uncertainty. It is not normally accounted for in conventional approaches to uncertainty evaluation and conformance assessment. The GUM assumes that intrinsic uncertainty is a result of an incomplete definition of what is to be measured, the ‘measurand.’ The GUM states that the measurand “should be defined in sufficient detail that any uncertainty arising from its incomplete definition is negligible in comparison with the required accuracy of the measurement…" However, when attempting to verify the GD&T for a component, the specification is complete, but, since it involves an infinite number of points, it is impossible to fully verify. It is, therefore, our ability to measure that is incomplete, not the specification of what is to be measured. Furthermore, it is very difficult to evaluate the resulting intrinsic uncertainty.

Some typical examples of this issue include measuring the flatness of a planar surface, the roundness of a hole or the profile of a free-form surface. In each case the question arises: how many points do we need to measure on the surface? Given only a GD&T specification, it is impossible to definitively answer this question. However, we can use a variety of information sources to make more informed judgments about the required measurement point resolution. If we know something about how the component is manufactured, then we can make assumptions about the types of variation we expect to see on the surface. This can be used to design a measurement strategy which is likely to detect these defects. A similar approach is to consider measurement at different scales which are known to be important, such as surface roughness, waviness and overall profile. This idea may be taken further by replacing conventional GD&T with a specification based on measurement at defined discrete points. Alternatively, if we have already made some measurements of the feature, then we can apply statistical analysis to determine the likelihood that unmeasured regions may be outside of specification.

### Planning measurements for expected manufacturing defects

It is often possible to predict with some certainty where defects will occur based on knowledge of the manufacturing process. For example, if a part is machined, then the cutting tool diameter and cutting path will be known. Steps will occur where cutting paths overlap. The measurement strategy can be designed to ensure that these steps are detected. Similarly, steps or gaps will occur when composite tows or tapes are laid with an overlap or gap, respectively. Other examples of predictable manufacturing defects include parting lines and injection points in molds and dyes. Measurement planning should always include consideration of where defects are expected to occur. Ideally, such a consideration should also help to inform the geometrical specification of products.

## Considering measurement at different scales

The form of surfaces is typically broken down into three scales: roughness, waviness and profile. Profile is typically given different terms depending on the nominal feature, for example: flatness, cylindricity or surface profile. Roughness relates to the smallest scale, profile relates to the scale of the whole feature and waviness is somewhere between them.

ISO 4287 defines a roughness profile as the “profile derived from the primary profile by suppressing the longwave component using the profile filter λc.” Similarly, it defines a waviness profile as the “profile derived by subsequent application of the profile filter λf and the profile filter λc to the primary profile, suppressing the longwave component using the profile filter λf, and suppressing the shortwave component using the profile filter λc.” The characteristic wavelengths λc and λf. The evaluation length may contain one or more sampling lengths.

ISO 4287 does not define what the characteristic lengths should be. Typically, a surface profilometer might sample over a 40-mm length. First, a line or arc would be fitted to the data points to remove the surface profile. The residuals to this fit are then considered to be the surface texture. A waveform would then be fitted to this data and taken as an indication of waviness. The residuals to the waviness fit would then be used to determine surface roughness. This is something of a simplification, since surface texture measurement is a large and complex subject which cannot be fully explained in this article. It should, however, be apparent that if the profilometer had sampled over a 200-mm length instead, it is entirely possible that a longer wavelength variation would have been detected and identified as the waviness. In this case, what was previously considered waviness would now be considered surface roughness. This is best explained with an example.

Consider the sampling of a number of super-positioned waveforms over different lengths. First, we sample over a 10-mm length giving the following measurement date:

Fig. 1: Measured data over a 10-mm length showing profile identified as an arc.

An arc is fitted to this measurement data to represent the underlying profile of the part. The residuals of the measured points minus the fitted arc are then plotted to show the waviness:

Fig. 2: Residuals, after profile removed, showing waviness identified with an amplitude of 0.3.

Again, removing the waviness from the data to leave the residuals, shows an underlying waveform. In this case it is classified as the surface roughness:

Fig. 3: Residuals, after waviness removed, showing roughness identified with an amplitude of 0.1.

If we now sample the same surface, but over a larger profile length of 100mm, we see an entirely different profile. Fitting a line or arc to this shows no significant deviation from a horizontal line. Therefore, we proceed to identifying the waviness directly from this data:

Fig. 4: Measured data over a 10-mm length showing waviness identified with an amplitude of 1.

Removing the waviness reveals the surface roughness:

Fig. 5: Residuals, after waviness removed, showing roughness identified with an amplitude of 0.4.

This example shows that the same surface could have totally different measurements of waviness and roughness depending on the length of profile over which the surface is sampled. When the sample was 10mm long, the waviness was 0.3 and the roughness 0.1. When the sample was 100mm long the waviness was 1 and the roughness 0.4. The roughness in the longer sample was actually the combined effect of what was classed as waviness and roughness in the first sample.

Since the magnitudes of waviness and roughness depend arbitrarily on the length over which the surface is sampled, it makes sense to define the sample lengths within the product specification. This should, of course, be done with careful consideration to the required function of the product. Consideration may also be given to our expectations due to knowledge about the manufacturing process. It is, therefore, common to see waviness defined as a profile within a zone. Step condition may also be specified separately.

The inherent assumption in surface measurement is that roughness, waviness and steps are consistent or periodic over the entire surface. The implication of this is that they can be measured at a small sample area, or a small number of sample areas. The flatness (or some other profile specification) of the entire surface may then be measured much more coarsely, with a spacing between measurements similar to the size of the sample area used for waviness measurement. These assumptions are based on experience with components previously manufactured using similar methods like machining, for example. This method greatly enhances our ability to fully verify surfaces and features of large components without performing very large numbers of measurements.

### Applying statistical analysis to existing measurements

Another approach to evaluating intrinsic uncertainty is to carry out the statistical analysis of the points which have been measured. One of the most promising techniques to do this is Gaussian process regression (GPR). This can be most easily understood with a simple two-dimensional example. Imagine that we have measured a number of points, each with a known uncertainty. We can fit a spline curve through these points to represent the surface profile. Where the curve passes through the measured points, its uncertainty is simply the uncertainty of the measured point. All other points on the curve have uncertainty arising from the surrounding measured points and the intrinsic uncertainty of the surface. GPR is a statistical technique which fits a curve and also gives the uncertainty for any given point on the curve.

Fig. 6.

GPR can be extended to higher dimensions and is therefore also suitable for surfaces. Using GPR, it is possible to measure surface using a relatively coarse grid of points and then identify regions where high levels of intrinsic uncertainty mean that conformance cannot be proven. Further measurements can then be directed to these regions until either conformance or nonconformance has been proven at the required level of confidence.

### Non-contact and scanning measurements

The increasing use of non-contact scanning measurement instruments negates the issue of intrinsic uncertainty to some extent. When the surface is scanned, many thousands or even millions of individual measurements on the surface can be made in the time it would take to perform just a few conventional measurements. However, scanning introduces many other sources of uncertainty which we will explore in a future article. As a general rule, when we move from a more traditional contact measurement to a noncontact scanning measurement of a surface, the uncertainty of each individual measurement increases while the intrinsic uncertainty of the surface as a whole decreases.

### Conclusion

In order to truly have statistical confidence that a part conforms to a specification, intrinsic uncertainty must be accounted for. This involves a number of steps:

1. The first step is to ensure that the specification properly represents the functional requirements of the part. This must include careful consideration of surface roughness, waviness and profile.
2. There must then follow a consideration of the types of defect likely to be encountered, considering the manufacturing process.
3. A measurement sampling strategy should be devised which provides a reasonable compromise between fully verifying the product specification and performing the measurement with reasonable time and expense.
4. Gaussian process regression (GPR) should be applied to the measurement results to provide an uncertainty for the measured feature as a whole.
5. The results of GPR should then be used to determine whether the existing measurements prove conformance or nonconformance. If required, additional measurements may be made at regions of high intrinsic uncertainty so that it is possible to prove conformance or nonconformance.

Such an approach would be completely impractical within current manufacturing systems. Even ignoring intrinsic uncertainty, the uncertainties of current measurement systems often mean that specifications cannot be proven with the required level of confidence. Furthermore, it would not be practical to program automated measurement routines which adaptively adjust measurement point resolutions until the required level of intrinsic uncertainty is obtained, according to Gaussian process regression. These issues, therefore remain unresolved and little discussed. It is however, important to consider them with a view to developing metrology for Industry 4.0.

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