Metrology
Introduction 
the accuracy of a machining center
Metrology definition
Metrology is the science occupied with the measurement of physical size. Its scope is to establish the methods of measure by:
 selecting some physical sizes (known samples), which are fundamental
 and defining the processes of measuring
Rules and procedures
In order to maintain tracking of how the measurements are carried out, the methods of measuring are codified in general terms, in the form of rules, with national and international circulation, while the detailed procedures are codified under the form of Procedures, with an innercompany circulation.
The codification of the method produces a number of advantages, such as being able to:
 repeat the measurements without needing every time to reanalyze the problem
 repeat the measurements in the same identical conditions
 permit the reanalysis of the methodology a posteriori (from effect to cause, inductively)
 permit more workers to carry out homogenous measurements
 define a priori (from cause to effect, deductively) the requisites and goals
Validation of a machining center
Among the principle characteristics of a machining center is its capacity to position a tool correctly in space.
To be able to quantify and measure this capability it is customary to use some fundamental scales, such as:
 Precision
 Repeatability
 Uncertainty of measurement
Precision
In Industrial Metrology, Precision is the degree of correspondence of the theoretical data (the Real Position), deducible from a series of measured values, with the reference data (the Theoretical Position). Making an analogy with a series of indicators scattered on a target, the more the center of the group of indicators approaches the center of the target, the more the series of points is precise (or accurate).
Repeatability
In Industrial Metrology, Repeatability is the degree of the convergence of the value of the average measurements. Making an analogy with a series of indicators scattered on a target, the more the indicators are grouped together, the more the series of points is repeatable. It’s not important how much the center of this group approaches the center of the target, this factor is determined by the precision (or accuracy) described previously.
Precision and repeatability 1
In the images below, examples 3 and 4 represent two series of equally precise data. Example 2 is repeatable but not precise (the expected value is the center of the target). Example 1 shows the best case, in which the data are precise and repeatable.



Precision and repeatability 2
Precision provides the measure of Operative Uncertainty, which includes either the systemic errors or the casual errors.
Systemic errors take into account the aspects of rigidity, geometry, thermic effects, and so forth.
Repeatability provides the scale of Operative Dispersion (Scatter), which is the measure of the casual errors attributable to the machine.
Repeatability and statistics
Industrial Metrology is founded on the use of the basic concepts of statistics.
It functions under the hypothesis of the measurement:
 of a “sufficiently precise” machine
 with an “adequate” measuring instrument
You can say that the relative values are distributed around an average value.
To obtain a dependable average value it’s necessary to carry out a sufficiently high number of samples (called “Statistical Samples”).
The degree of convergence of the measurements toward an average value is called Repeatability and the distribution of the values around the average values describes in a simplified manner the “Gaussian” statistical distribution (also called “Normal”).
In statistics Repeatability (as it is defined in Industrial Metrology) is expressible in terms of the Standard Deviation of the Gaussian curve (or Normal).
The normal curve 1
As stated in Industrial Metrology it’s assumed that the statistical error of positioning is distributed according to a “Gaussian” curve (called “Normal”). “Normal” is the function that mathematicians utilize to describe the errors in a continuous range.
The peculiarity of the Gaussian curve (Normal) is that knowing the Average and the Average Quadratic Deviation (also called Standard Deviation, symbol “σ”, read “sigma”) it’s possible to completely determine the curve.
Therefore for every value it’s possible to know its verifiable probability (i.e. it’s possible to determine the “ValueFrequency” curve”).
The normal curve 2
The Average X is defined as : 

The standard deviation "σ" represents the average distance of data from a series of readings from their average and is defined as: 

The normal curve 3
So then, the Normal curve is perfectly symmetrical and, note as well, that it’s possible to divide it into parts centered along the middle in a manner so that each one of them expresses a certain probability that an observation falls within the area considered.
The observations that diverge by a certain number of quadratic deviations (+/nσ) from average x (whether above or below) are covered by a precise probability within that interval. In particular 68.2% (0.682) of the observations are found between xσ and x+σ, i.e. within one standard deviation on either side of the mean. Meanwhile 95.4% (0.954) are found between x2σ and x+2σ and 99.7% (0.997) are found between x3σ and x+3σ.
Returning to the rules
When Precision and Repeatability is spoken of, it’s often in reference to various rules, and probably the most frequently requested by the market are:
 ISO2302
 VDI/DGQ 3441
Yet all rules and standards answer to the same need: to measure a machine according to a precise method (a method of gathering a Statistical Sample) and quantify its accuracy according to a statistical principle (using that normal distribution theoretical base).
CMS’ use of the rules
In discussions of Accuracy CMS provides such information based on various standards. As mentioned previously, all the rules and standards use the same starting method: a collection of data points and their relationship to an expected reference point, and then apply their individual rules to obtain the resulting Accuracy and Repeatability data. CMS provides this information for both linear and rotary axes, either for the entire machine or for a portion of its stroke, as is the case for the ISO 2302 rule.
The use of such rules and standards provides a meaningful and methodical approach to both discussion of what is required for a machine’s precision, as well as a way of determining if the machine is performing to that standard. Adherence to Accuracy and Repeatability standards is customary for certain industries, such as Advanced Materials, including “run off testing” to verify the data. Customers desiring such information and testing may request it as part of a project, and should discuss it with their Area Sales Manager during the development of the project expectations.
More to come
The discussion of Metrology covers significantly more information than that gathered together in this brief overview. These are the basics.