Editor's Note: This article is an edited version of Klaus's original paper. The original includes graphs, figures and discussions of the experiments that demonstrated the Functional Fit thesis. Please write, call or e-mail us and we will send you a copy.

A revolution has taken place in dimensional metrology. It has been a quiet revolution, one that many don't even realize has taken place. The revolution I speak of is affordable coordinate measuring machines with constant-contact scanning capability. As with all revolutions, it has been fought by the old guard, accepted by some who had faith in its promise for more accurate measurements and watched carefully by those who understand its promise and hope to profit from its eventual acceptance.

This is the only conclusion one can draw from the fact that Renishaw, the manufacturer of touch-trigger probes, which are found on 99 percent of the worlds CMMs, has decided to design and produce its own scanning probe. Analog, or more correctly, proportional displacement constant-contact scanning probes offer a much greater, richer level of data collection and, therefore, a more accurate depiction of the part being inspected than a touch-trigger probe could ever hope to match, even if it took measurements for hours in a gage lab. A scanning probe, such as EMD's is also relatively inexpensive to produce. The probe is reliable, with few known failures – some have been on the job for more than ten years. On the other hand, the well known German scanning probe designs, which are really miniature CMMs at the end of the Z ram, have a reputation for being fragile and expensive.

With this revolution in sensor technology must come another in metrology, having to do with the underlying mathematical philosophy of modeling and measuring geometric shapes. Like it or not, current coordinate analysis software and methodology diverge from the intent of blueprints and ANSI Y14.5 Dimension and Tolerancing standards. The basis of this divergence can be traced to a lack of understanding between metrologists and computer programmers when it comes to
modeling.

Mathematical Modeling

The intent of mathematical modeling in metrology is to process discrete coordinates into a form that makes the geometric relationships of surfaces known. From those geometric surfaces and relationships a work piece is compared to the blueprint (or CAD model), and to the intent of the designer.

A hole, for example, has been modeled by the mathematical equation for a circle. To solve a mathematical equation for a geometric element, a minimum amount of surface coordinate data is needed. However, when more than the minimum number of points are utilized an ambiguity begins. In real-world metrology, four discrete measurement points will not lie on a perfect theoretical plane. The coordinate data has more degrees of freedom than the equation that models the geometric element. Given that we cannot squeeze four real world coordinate points onto a theoretical concept, such as a plane, we have developed modeling criteria to deal with the residual values.

Early CMM technology was inaccurate and primitive compared to currently available technology. The Least Squares Best Fit algorithm, for example, was initially a mathematical modeling procedure that could run on a calculator. It was used to average the CMMs results so they appeared to be better than they really were. The Least Squares Best Fit approach must now be considered a rather crude approximation. One that, with the proliferation of scanning CMMs, can and must be replaced by "Functional Fitting" routines.

ANSI Y14.5 & Hard Gages

The ANSI Y14.5 standard was developed out of the necessity for a standardized vocabulary for the designer to express himself to manufacturing and inspection. The basic philosophy was to define geometric shapes by their function, and by unambiguous inspection techniques that simulate functional performance. Hard gages have traditionally been the primary method of inspecting a piece in accordance with Y14.5. The hard gage was designed to simulate a perfect mating work piece that utilizes all of the plus material tolerance associated with the real mating piece. In this manner, when the real mating piece approaches its maximum plus material tolerance, the two pieces still fit.

But hard gages have little or no diagnostic capability and are very expensive, since they must be made as nearly perfect as possible, relative to the work piece tolerances. Historically, manufacturing tolerances for hard gages were one tenth of the manufacturing tolerances applied to the piece being measured. However, as today’s manufacturing requirements become more stringent, work piece tolerances frequently approach values previously associated with gages. It is no longer practical to manufacture hard gages for such parts.

Software Equivalence

Today, metrologists look to CMM software to be their "soft" gages, that is, to be equivalent to hard gages. Since geometric elements (CMM terminology) and features (ANSI Y14.5 terminology) are the basic building blocks of each language, an equivalence must be created between the two. The results of "Functional Fit Criteria" algorithms and CMM methodology must be the equivalent of a hard gage functional fit.

In mathematical terms, all of the residuals must be zero, or minus material, between the modeled surface and the actual measured coordinate data. The sum of these residuals must also be a minimum. In the case of a functional fit plane, its surface will pass through the three high points and the remaining coordinate points will have minus material deviation. Let's review why this is so, by looking at the alternatives.

Least Squares Best Fit

The aforementioned Least Squares Best Fit modeling method, hereafter called best fit, is still the most common algorithm for modeling a geometric surface. The best-fit method refers to a type of modeling algorithm that yields a solution to an equation which in turn defines a geometric element so that the residual values between the surface of the element and the data points are a minimum.

The Advantages: It's a simple procedure, requiring very little computer software code, and is mathematically traceable, unambiguous and unique. For process control applications, best fit presents an output that allows for the separation of systematic irregularities, which impact the form or shape, from geometric location and size errors. Best fit is an average; the impact of individual coordinates on the shape of the geometric element is inversely proportional to the number of data points. In this way, the impact of fliers is minimized.

The Disadvantages: Best Fit can be metrologically ambiguous, since its results are based on the location and density of the data points taken from the work piece. For example, assume that a plane is slightly concave. We take an evenly spaced well proportioned set of data points. We suspect that the center might have a problem, so we take an extra number of readings there, believing that we are improving our results. What we have done is to increase the point density at a spot that has the least functional importance. The best-fit routine, then, minimizes the sum of the deviations and moves the resulting plane down toward the higher point density and away from the functional locating plane. In other words, each point contributes an equal amount of importance to the best-fit algorithm. Increasing the point density in a specific spot has increased the importance of that spot and thus weighs down the best-fit algorithm there.

Unless the user is aware of this anomaly and its repercussions on the result, it is highly recommended that the user take an even, well spaced data set. Increasing the data density in a specific spot will more than likely decrease the reliability of the expected result.

Plus Material Form Shift

CMMs, when first employed for critical part certification, ran into immediate problems. Two simple components, a hole and pin, when checked with best-fit routines, would not necessarily function as predicted. The condition was rectified by shifting the surface of the best-fit element to the high point. On a hole this correction decreases the diameter by the plus material form and on a pin it increases the diameter by the plus material form.

The Advantages: In the above example, the technique is able to reject a bad part.

The Disadvantages: This technique will cause the rejection of some good parts that are functional. It requires a higher degree of machine and system reliability. The final location of the elements surface is completely dependent on a single data point. A single plus-material flier will reject a part. If this technique is utilized on a complex geometry, the results are underconstrained, with deviations adding up rapidly, thus rejecting almost every part.

Sigma Shift

The divergence in fitting criteria has been addressed by various coordinate analysis software suppliers. The most common approach has been a statistical correction to the best fit. This approach first calculates a best fit and then processes the residual deviations statistically. A standard deviation is computed.

From engineering, a statistical confidence factor is established. It is well known that a random variable will form a gaussian (bell shaped) distribution. A sigma factor of one will contain 68% of the data. A two sigma confidence level will contain 95% of the data. When a geometric element is sigma shifted the modeled surface is moved so that, by a statistical projection, the desired percentage of points will lie in minus material.

The modeled surface is shifted normal to the best-fit surface. On a hole, for example, the radius is reduced by the Standard Deviation times the sigma factor. On a plane the surface is moved normal to the best-fit plane away from the material side.

The Advantages: This approach has many of the benefits associated with the least squares method and is a closer approximation to the functional location of the surface. It also has a built-in mechanism (sigma confidence level) for approximating the physical differences between hard gaging, which squashed burrs and high points, and noninfluencing gaging, such as a CMM.

The Disadvantages: The true locating surface is not known. The location has been more closely approximated but the orientation is still based on the best fit. The theory requires that a statistically significant amount of data be taken. The standard deviation is highly unreliable when a small quantity of data points is taken. Applying a sigma shift under those conditions may cause unrealistically large or small shifts. In addition, the theory assumes that the deviation to the best fit is random and gaussian. This unfortunately, is the exception. In almost all cases the population-to-deviation distribution is non-gaussian. In fact, the variations that occur in the manufacturing process are not random (like a coin toss) but biased by systematic trends.

For example, a hole is virtually always elliptical or lobed. When modeled as a circle, very few points lie at the best-fit diameter. In this case a population-to-deviation distribution will appear as a doubled-humped curve.

Successive Approximation

The need to determine the actual functional geometric element has led to mathematical techniques which isolate and model surfaces based on the functional fit criterion. The functional fit criterion is not a mathematically pure concept, but an ad hoc model of reality in metrology.

The successive approximation technique uses a procedure similar to a best fit. An approximation is made that allows for the separation of pertinent data from uninteresting data. The uninteresting data are discarded and the pertinent data are reevaluated until the amount of pertinent data is small enough to accurately reflect the modeling criteria.

In the case of a plane, we must isolate from the data set the three high points. The best-fit algorithm allows us to split the data set into two categories. The pertinent data comprise all of the points which have a zero or plus material residual deviation. The uninteresting data comprise those points which contain a minus material residual. We continue to reduce the amount of pertinent data with successive approximations until the quantity of pertinent data falls below a certain level.

Similar to an ANSI Y14.5 example (i.e. 4.4.1), a special functional condition may occur. If the plane happens to be a convex shape, a functional pivot point occurs, and our functional fit criterion algorithm ends up with three points in very close proximity to each other. ANSI Y14.5 allows the inspector to pivot the piece into an average location. This is the equivalent to locating the surface on the high point and using the original best fit from all of the data for orientation purposes.

The Advantages: The results are much closer to the intent of ANSI Y14.5 and thus more reflective of the functional performance and intent of the work piece. Relative to the best fit and sigma shift, the successive approximation can be verified independently on functional tests not involving CMMs, computers and equivalent analysis algorithms. Editor's Note: samples in the exercise to demonstrate this, when blued and rubbed on a surface plate, indicated the identical contact spots that appeared through this modeling technique.

The Disadvantages: The results are far more dependent on the reliability of the data. Plus-material fliers will have a direct one-third impact on the resulting plane. The algorithm requires more computation time. The coding is less traceable and slightly more complex than the best fit.

Topographical Interpolation

A limitation of the successive approximation technique is that it assumes that a coordinate data point lies on the peak itself. Topographical interpolation utilizes an equivalent to the successive approximation to isolate a local grouping of points surrounding the peak. Each local grouping is then analyzed with a three-dimensional best-fit curve that models a topographical peak. This type of algorithm is more commonly used in the surveying industry. Topographical maps are often created from incomplete surveys. Fitting routines fill in the blanks of a survey by various models.

The Advantages: This technique allows for the interpolation to a functional plane without having taken a data sample at the peak itself. It allows for the inference to the peak from surrounding trends.

The Disadvantages: If the data density is less than the normal variations in a surface, then interpolating to a peak might cause larger inaccuracies, i.e., the peak my not really be there.

Influence of Data Sampling

It is commonly assumed that best fit is an average and thus is repeatable and reliable. This is true from the mathematician's viewpoint, when considering a given set of data point inputs. The results of a best-fit analysis are repeatable. When considering the repeatability of these algorithms, though, the primary concern must be on variations in the location of the data samples, since both fitting methods -- best fit and functional fit -- will give mathematically repeatable answers.

Current standard CMM methodology, i.e. measurement with a touch-trigger probe, deals with very small quantities of coordinate data. This contributes to magnifying the impact of the sampling technique on the results of these algorithms.

Thus, the best-fit method is metrologically ambiguous, depending completely on the data sample location and density. The result of the best-fit algorithm is a floating target element based on the entire data sample. The result of a functional fit algorithm, on the other hand, is an unambiguous target element based on a few points mathematically selected from the entire data set.

The repeatability of these algorithms on a random variation of data points on identical surfaces is considerably different. By changing the mixture of data points on a given surface, the functional fit will have higher repeatability than the best fit. The best fit algorithm models to a surface where the actual coordinate point density is very low, i.e., the probability of actually hitting points on the best-fit element is generally poorer than the high or low points.

Summary

The CMM represents a technology that allows for much greater accuracy and production process control than functional hard gages. Since CMM technology is the current state-of-the-art, it would seem appropriate that CMM technology should emulate the intent of ANSI Y14.5. It has been suggested that the ANSI definition be changed to conform to current CMM methodology. This option would fly in the face of the intent of all current blue prints, the reason they exist and all other inspection technologies.

The best-fit models are metrologically ambiguous. Since CMM part programmers and inspectors do not have a standardized procedure for sampling data on a work piece, the results of various sampling strategies will not repeat.

The best-fit mathematical models are a holdover from CMM history. CMMs predate the ANSI 1982 specification and, as a result, the development of many software methodologies were unguided by the document. Computer technology, in the early phases of CMM development, was incapable of handling the quantities of data and algorithms required for Y14.5 equivalence. Functional fit routines are influenced less by the location and density of the data sample. Thus different operators and part programmers, who each have personal sampling strategies, will exhibit better overall repeatability. And repeatability is what it's really all about.

Proposed Solution

To develop equivalence between hard gages and CMM methodology, a standardized sampling technique must be developed that samples the work piece sufficiently so that the results are statistically significant.

The proposed algorithm must be a modified topographical technique in which the data points at and around the peak can compensate for the physical differences between hard gaging and noninfluencing gaging. The malleability, hardness and surface finish of the work piece must be an operating parameter of this algorithm. This recognizes that burrs and flier points at the peak will squash when hard gaged, and any software equivalent technique should be able to retrieve this bonus tolerance. This is possible today with CMMs capable of high data density, continuous-contact scanning.

Thus the old guard may not know it yet, but they have lost. With Wenzel now offering inexpensive CMMs with scanning capability, I believe that within a few years a majority of new CMMs will be available with continuous-contact scanning systems and scanning retrofits will be in strong demand. The reason for this is that scanning, with its high data density capabilities, most closely approximates the best that soft gages, i.e., CMMs, have to offer and gets us closer to knowing the topography and a true functional fit. -- Klaus Ulbrich