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Geometric tolerances introduce special challenges in coordinate metrology.  This is partly because CMM’s work best with a well-defined fixed coordinate system, and many geometric tolerances involve open degrees of freedom and optimization.

Actual Value Definitions – Optimized Alignments


The ASME Y14.5.1M-1994 standard for Mathematical Definitions of Dimensioning and Tolerancing contains definitions for actual values of geometric characteristics.  For each characteristic, the tolerance zone is described in a mathematical way and the actual value is defined as “the smallest tolerance to which the feature will conform”.


There is an underlying concept of optimization contained in these definitions.  The actual value, and thus the size of the tolerance zone, must be minimized within the degrees of freedom that are available.  In many cases, these degrees of freedom relate to the translation and rotation of the datum reference frame, or coordinate system alignment.  There is not one unique alignment – many are possible, and the optimal alignment must be found that achieves the best (smallest) actual value.


Coordinate Metrology – Fully Constrained Alignments


One of the main paradigms of coordinate metrology is to define alignments, or coordinate systems, on the part.  The location and orientation of other measured features are then evaluated in one of these coordinate systems.  This is most straightforward if all of the degrees of freedom (translation and rotation) are fully constrained, for a well-defined and repeatable alignment.


So there is a fundamental challenge – using a device that prefers a fixed coordinate system alignment to evaluate an actual value that requires finding an optimized alignment.  So the programmer or operator must find a way to optimize the alignment.


Physical Optimization


Physical trial-and error methods can be used to adjust the orientation and location of the part relative to the coordinate system.  This occurs when a holding fixture is used to define the coordinate system, and the part is adjusted in the fixture.  The inspection is repeated until an optimal orientation and location is achieved.  A simple example is the use of jacking screws to optimize the leveling of a part for a Flatness measurement.  Because the rotational degrees of freedom are not constrained, the part can be leveled different ways until the optimal orientation is found.  This can be tedious and time-consuming, especially when small tolerances and three-dimensional relationships are involved.


Software Optimization


In most cases, physical optimization is not practical and software-based methods are preferred.  The features are measured in an arbitrary alignment and then “best fitting” is performed by the software, to optimize the alignment when evaluating each characteristic.


Simple Optimizations


For some geometric characteristics, optimization is relatively straightforward and almost all CMM software has built-in routines to calculate actual values correctly:

•Flatness (all rotations and translations can be optimized)

•Cylindricity (all rotations and translations can be optimized)

•Orientation tolerances (Perpendicularity, Parallelism, Angularity).  Rotation is constrained relative to the datums, and all translations can be optimized).


Optimization for these characteristics is straightforward because they are confined to simple feature types such as single planar surfaces and single cylinders.


More Complex Optimizations


In other situations, optimization is more complicated and special software functions may have to be invoked:

•Position tolerances on patterns of coaxial or parallel holes

•Position and Profile tolerances with open degrees of freedom

•Lower segments of composite feature control frames


Difficult Optimizations


Very complex situations present difficult optimizations and only specialized third-party software packages currently them correctly:

•Partially constrained degrees of freedom (resulting from datum features referenced at Maximum Material Boundary)

•Combined patterns with combinations of Position and Profile tolerances, resulting from Simultaneous Requirements

•Surface interpretations of tolerances referenced




Optimization is one of the major challenges of finding actual values for geometric characteristics.  Any degrees of freedom that are available to allow the feature to conform to its tolerance zone must be optimized when finding the actual value.


Future articles will examine examples of geometric characteristics, the degrees of freedom available, and techniques for optimization using CMM software.


By Evan Janeshewski

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