REDUCING UNCERTAINTY OF COORDINAL METROLOGY BY EMPLOYING AN
INTEGRATED COMPUTATIONAL SYSTEM
Ahmad Barari 1
1
Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Canada, ahmad.barari@uoit.ca
Abstract: Integrated Inspection System (IIS) is an efficient
solution to combat the inherent plug-in uncertainty in
coordinate metrology. Using IIS, the computational tasks are
performed concurrently or as the elements of a closed-loop
structure. It is shown that implementing this concept reduces
inspection uncertainty by developing opportunities for
communication and adaptability between the three basic
computational tasks in the typical coordinate metrology
processes, i.e., planning for measurement of discrete point,
estimating the best substitute geometry, and evaluation of
the deviation zone.
Key words: Coordinate Metrology, Minimum Deviation
Zone, Point Measurement Planning, Inspection Uncertainty,
Integrated Inspection System.
1. INTRODUCTION
The purpose of this research is to develop a robust and
reliable platform to perform coordinate metrology of
surfaces. Although, surface coordinate metrology is a key
process in many different engineering and scientific
applications, unfortunately it is still influenced by several
concerning sources of uncertainties which reduce the
reliability of this assumed very precise process. The
traditional coordinate metrology is completed by sequential
performing of three analytical tasks of Point Measurement
Planning (PMP), Substitute Geometry Estimation (SGE) and
Deviation Zone Evaluation (DZE). In such systems, the
inspection accuracy is subject to sorts of “plug-in
uncertainty”, i.e. the uncertainty due to estimating some
aspects of a probability distribution on the basis of
sampling. Such sources of uncertainties can adversely affect
the accuracy and reliability of the inspection. Recently,
Integrated Inspection System (IIS) has been studied as an
efficient solution to combat the inherent plug-in uncertainty
in coordinate metrology. In an IIS, the computational tasks
can be performed concurrently or as the elements of a
closed-loop structure. In this way, the inspection uncertainty
can be reduced using the developed opportunities for
communication and adaptability.
To benefit from potential advantages of IIS,
sophisticated techniques for on-line generating the
uncertainty information are required. Also, at each
computational task, advanced methods are required to
efficiently adapt the process based on the uncertainty
information. In order to develop an efficient IIS, an
estimation of geometric deviations by DZE needs to be
dynamically utilized by PMP to acquire the most useful data
set from the measuring part. Hence, gradual progress of
DZE plays an important role in IIS implementation.
Moreover as the number of measured points increases,
estimation of the optimum substitute geometry by SGE
becomes a challenging task due to highly nonlinearity of the
resulting optimization problem. Reliable methodologies to
gradually estimate the geometric deviation zone and
adaptively change the PMP and SGE for both primitive
geometric features and sculptured surfaces needs to be
developed.
This paper first presents the methodologies to establish a
loop between SGE and PMP. Using a statistical pattern
recognition technique the distribution of work-piece’s
geometric errors is studied and new sample points are
captured accordingly. The results show that implementation
of this method can reduce the uncertainty in inspection of a
sculptured surface up to 60%. Then it explains a method to
model the distribution of geometric deviations. The method
is developed by utilizing a geometric map of the measured
deviations to a parametric surface, and uses the results to
tessellate the work-piece surface based on the proximities to
measurement locations and a Veronoi diagram.
2. BACKGROUND
Recent studies show the inherent uncertainty resulting by
computational tasks in coordinate metrology is significant.
The results, which appear in Reference [1], show that the
computational uncertainty for measuring and Auto-body
profile can be as high as 300 µm. Comparing this result with
the effect of other sources of uncertainty in coordinate
metrology such as equipment, environment, work-piece and
operator [2], it can be seen that computational uncertainty in
the most cases can be even higher than the expanded
uncertainty of all other sources. Under these conditions, the
total expanded uncertainty of CMM inspection can be
unfeasible either for quality or for process control purposes.
The concept of integration of three basic computation
tasks, i.e. Point Measurement Planning (PMP), Substitute
Geometry Estimation (SGE) and Deviation Zone Evaluation
(DZE) is discussed in Reference [3]. An integrated
computational model can significantly solve the problem by
providing an online share of information between the
computational tasks. Figure 1 schematically shows the
sequential system utilized in the common practices versus
the recent research achievements toward developing a robust
Integration Inspection System (IIS). The architecture and
general requirements for an IIS in form of a closed-loop
between engineering tasks are discussed in Reference [4].
(a)
parametric area. The point chosen for each element is
randomly selected within the element. Figure 2 shows
samples and the evaluated minimum deviation zone which is
0.06766 mm. There is, however, no guarantee that the
extreme deviation in these samples is the same extreme
deviation in simulated geometry.
(b)
Fig. 1. (a) Typical sequential computational system in coordinate
metrology; (b) Current research achievements toward developing a
robust Integration Inspection System (IIS)
3. INTEGRATION OF PLANNING FOR DATA
SAMPLING AND ESTIMATION OF THE OPTIMUM
SUBSTITUTE GEOMETRY
In the traditional coordinate metrology, two sequential
tasks are typically the collection of the measurement data
based on a sampling plan, and then data analysis to derive
the necessary geometric information from the data-point set
in form of a substitute geometry. However, there is always
an uncertainty about the correctness of the results of the
whole process because the estimation of a substitute
geometric is only based on a group of sampled points and
due to the plug-in nature of this statistical estimation the
uncertainty can be closely related to the location and number
of the sampled points.
A closed-loop established between the computational
tasks of planning for point measurement and estimation of
the substitute geometry can significantly reduce the
inspection uncertainty [3]. An iterative sampling procedure
based on online estimation of the substitute geometry is
described here. In order to evaluate the geometric deviations
a pattern recognition technique called Parzen Windows
[3][5] is employed to estimate the Probability Density
Function (PDF) of the geometric deviations between the
actual and the substitute surfaces. By studying the
discontinuities in the PDF, the possible errors in the
previous sampling plan are being detected. Based on the
obtained results, the respective compensation scheme is
constructed. It serves to modify the sampling plan for a new
sampling phase.
An auto-body B-spline surface created by 16 control
points with an overall dimension of 900 [mm] × 600 [mm] ×
200 [mm] was used to verify the performance of this closedloop. An initial sampling of the surface based on a stratified
strategy with 16 measured points was performed. A
stratified strategy for initial sampling is applied. The
surface is divided uniformly into 16 elements with the same
Fig. 2. Inspection based on the 16 stratified samples
In the next step the iterative sampling presented in
Reference [3] was utilized. After 154 iterations of the
closed-loop, only 24 points were selected as the critical
points to evaluate the geometric deviation of the surface.
Figure 3 and 4 show the total 154 measured points, and the
final set of 24 points were selected as the most important
points for DZE.
Fig. 3. Evaluation based on all of the 154 points sampled in iterative
sampling
Using the 24 selected samples the evaluated minimum
deviation zone is observed as high as 0.10778 mm which is
almost 60% larger than what was evaluated before based on
the 16 initial samples. This shows that the points obtained
by iterative sampling are better representatives of the actual
surface. The total 154 samples also yield the same
evaluation of the deviation zone identical to what was
obtained from the 24 selected samples. However, the
computation time for 24 points was almost 4.8 times shorter
than the evaluation time based on the 154 points. Although
the estimation of the substitute geometry need to be
performed several times in the iterative sampling method,
but since the variables associated with the previous
substitute geometry as the initial condition for the
optimization, new substitute geometry is found quickly. The
total iterative sampling computational time in this casestudy was almost 2.2 times longer than what was required
for the initial 16 stratified points.
Fig. 4. Evaluation based on the 24 points selected by iterative sampling
Figure 5 shows how the density function changes by
iterative sampling. It can be seen that gradually over each
iteration, the shape of density function is extended, takes a
more symmetric shape and the density of samples within the
average deviations is increased. The final density function,
when the algorithm is terminated shows a characteristic
behavior similar to a Gaussian distribution function with a
significant density around the mean and symmetric form.
The mean value is also very close to zero which validates
the high accuracy of both the sample planning as well as
performed fitting.
4.
INTEGRATION
OF
EVALUATION
OF
GEOMETRIC DEVIATIONS AND PLANNING FOR
DATA SAMPLING
Coordinate metrology provides deviation of the discrete
points on a measured surface, but typically it is not capable
to explore any information of the surface regions between
these measured points.
The developed methodology
presents a continuous function to estimate the Distribution
of Geometric Deviations (DGD) on the entire inspected
surface [6]. The methodology is developed based on
estimation of parametric proximity of the surface points to
the actual measured points. Utilizing a Voronoi Diagram
and its corresponding Delaunay Triangulation, the procedure
to estimate DGD is developed [7]. The resulting DGD
model can be employed to estimate the deviation values at
any unmeasured point of the inspected surface when a
detailed understanding of the surface geometric deviations is
required. Implementation of the developed methodology is
described and case study for a typical industrial part is
presented. This methodology can be used for closed-loop of
inspection and manufacturing processes when a
compensation scheme is available to compensate the
manufacturing errors based on the DGD model.
Estimation of the geometric deviation for an unmeasured
point of the actual surface is required when the geometric
deviation values for the measured points are available as the
attributes of the sites in the parametric space. It is shown in
Reference [8] that the distribution of geometric deviations
caused by quasi-static manufacturing errors is continuous
because it is a direct function of existence continuity in the
nominal geometry. Therefore, the geometric deviations can
be considered as a continuous variable with known values at
sites in the parametric space. When using a proper PMP a
sufficient number of points from the appropriate locations
are measured, the sites are available to represent the
continuity of geometric deviation function. Geometric
deviation of an arbitrary unmeasured location can be
estimated by studying its proximity to the known geometric
deviations of the measured sites. The measured sites are
selected using a proper search method during PMP activity
using the iterative search method explained in the Section 3.
These sites are estimated as the representatives of critical
deviation regions on the actual geometry, which are selected
based on the continuity in distribution of the geometric
deviation function.
Let S be a two-dimensional parametric plane of uv in
the parametric space with n sites of {s1,s2,s3,…,sn}. The
parametric plane can be partitioned by assigning every point
in the plane to its nearest site. This problem is known in
computation graphics as the Voronoi Diagrams [7]. All
those points assigned to si form the Voronoi Region, V(si).
V(si) consists of all the locations at least as close to si as to
any other site:

V si   x : si  x  s j  x
Fig. 5. Changes of the probability density function in iterative
sampling
j  i ,x  S

(1)
where, the bracket indicates the parametric distance of
its inside terms. If a location happens to be equally close to
two or more sites, the location belongs to the boundary
between these sites. The set of all the locations that have
more than one nearest neighbor forms the Voronoi diagram.
Figure 6 shows the Voronoi diagram of 30 sites on the
parametric uv plane.
A planar triangulation, called Delaunay Triangulation, is
produced by a straight line connection of the adjacent
Voronoi sites if no four sites are co-circular and using the
above properties it can be shown that the drawn straight
lines avoid crossing [7].
where,
er 
A B
e i
C
and, si(u) and si(v) are the u and v coordinates of the site si.
The required geometric deviation is found by inserting the
parametric coordinates of the arbitrary location of r in
Equation 1, and
A  r v   si v s j u   si u ek -ei   e j -ei sk u   si u 
B  r u   si u s j v   si v ek -ei   e j -ei sk v   si v 
C  s j u   si u sk v   si v   s j v   si v sk u   si u 
Fig. 6. Voronoi Diagram resulted by 30 sites on the parametric plane
Figure 8 illustrates the front driver’s door of a vehicle
with the general dimensions of 1150mm×1080mm×35mm.
An accurate stamping die needs to be manufactured for mass
production of this part. The main body of the door is
designed by a uniform, non-periodic Non-Uniform Rational
B-Spline (NURBS) surface with control net of 16 control
points. The control points are defined identically in the u
and v directions. Coordinates of the controls points are listed
in Appendix. The degrees of NURBS surface in the u and v
directions are identical and equal to three (forth order
polynomials).
Figure 7 illustrates the Delaunay triangulation resulted
by the Voronoi diagram of Figure 6.
Fig. 8. Design tolerances of the stamping die to produce the car door
Fig. 7. Superimposing the Delaunay triangles on the Voronoi diagram
of 30 sites on the parametric plane
Suppose r is an arbitrary location on the parametric
plane and three nodes of its corresponding triangle are si, sj
and sk. The associated deviations for three sites of si, sj and
sk are three known values of ei, ej and ek respectively.
Therefore, corresponding to the three sites of si, sj and sk,
three discrete positions can be found respectively in the
parametric space. The desired geometric deviation value er,
is estimated by planar interpolation of three known positions
of oi, oj and ok. The plane resulted by three positions of oi,
oj and ok is as follows:
u  si u 
v  si v 
e -ei
(2)
s u   s u  s v   s v  e -e  0
j
i
j
i
j
i
sk u   si u  sk v   si v  ek -ei
In order to simulate manufactured surface of the die,
machining of the door is simulated by developing a model
for quasi-static machining errors of a vertical machine tool.
The methodology to develop this model is described at
Reference [8]. The quasi-static machining errors are
modeled as a linear operator that applies on points on
nominal geometry to estimate the location of actual
machined points considering interaction of all the machine
tool quasi-static error sources. These points are used directly
for the inspection process and estimation of DGD.
The datum reference “A” identifies four constraints for
the SGE task in inspection process, including: the translation
along the X-axis and Y-axis, and rotation about the X-axis
and Y-axis. As a result, the two degrees of freedom for SGE
fitting are translation and rotation about the Z-axis. During
the PMP activity 163 data points are selected using a search
method [3]. Figure 9 shows the 163 measured points and the
result of the SGE fitting. In Figure 9, points that conform to
the tolerance zone are indicated by solid blue circles. Points
that could not fit into the tolerance zone and show some
residual deviations are indicated with two concentric red
circles.. Maximum observed geometric deviation is
0.083133 mm.
are illustrated by superimposing a color map on the NURBS
geometry in Figure 11. Any dark blue segment has
conformed to the tolerance zone with no residual deviation.
Fig. 11. Color-map of residual deviations resulting from the machining
of surface of die
Fig. 9. Measured Point on the substitute geometry; measured point
with deviations beyond the tolerance zone is indicated by two
concentric red circles
By transferring the corresponding parametric coordinates
and the deviation of 163 measured points to the parametric
space, first, the Voronoi diagram and Delaunay tessellation
are calculated. Then, using the planar interpolation methods
explained in the previous Section, the DGD is estimated.
The resulting DGD is illustrated in Figure 10. Using the
evaluated DGD, the geometric deviation for any point on the
actual surface can be estimated.
The presented color-map identifies all the regions of the
geometry which do not comply with the desired tolerance
zone. An additional manufacturing process such as
compensating machining operation can be employed to
correct these regions and compensate the effect of the
machining quasi-static errors.
5. CONCLUSION
Utilizing Integrated Inspection System (IIS) in coordinate
metrology reduces the uncertainties of the inspection
process which are inherent in the computational tasks. The
customized IIS can be designed and developed based on the
particular condition and requirements of the engineering
applications that are supposed to utilize the inspection
results. Developing comprehensive computational platforms
for coordinate metrology that reduces the overall inspection
uncertainty is the ultimate goal in this area. The results
presented in this paper confirms possibility of developing
IIS for variety of applications, for variety of parts with
single-feature or multi-features and for variety of features
including primitive geometric features and sculptured
surfaces.
AKNOWLEDMENTS
The research support provided by the Natural Science and
Engineering Research Council of Canada (NSERC) is
greatly appreciated.
Fig. 10. Distribution of Deviations on the parametric surface
Using the described procedure the geometric deviations
of 900 equi-parametric-points of the actual surface are
calculated. These points are selected on a 30×30 parametric
grid of the NURB surface. The resulting residual deviations
REFERENCES
[1] A. Barari, “Sources of Uncertainty in Coordinate
Metrology of Automotive Body”, CD Proc. of 2nd
CIRP International Conference on Assembly Tech. and
Systems (CATS 2008), Toronto, Canada, 2008.
[2] A. Weckenmann, M. Knauer, M., T. Killmaier,
“Uncertainty of coordinate measurments on sheet-metal
parts in the automotive industry”, Journal of Materials
Processing Tech., v 115, pp. 9-13, 2001.
[3] A. Barari, H. A. ElMaraghy, G. K. Knopf, “Search Guided sampling to reduce uncertainty of minimum
zone estimation”, ASME Transaction- Journal of
Computing and Information Science in Engineering, v
7:4, pp. 360-371, 2007.
[4] A. Barari, R. Pop-Iliev, “Reducing Rigidity by
Implementing Closed-Loop Engineering in Adaptable
Design and Manufacturing Systems”, Journal of
Manufacturing Systems, 28, pp. 47-54, 2009.
[5] E. Parzen, “On estimation of a probability density
function and mode”, The Annals of Mathematical
Statistics, 33, pp. 065-76, 1962.
[6] A. Barari, “Estimation of Detailed Deviation Zone in
Coordinate Metrology”, 13th International Conference
on Metrology and Properties of Engineering Surfaces,
National Physical Laboratory, Middlesex, England,
April 12-15, 2011.
[7] A. Okabe, B. Boots, k. Sugihara, S.N. Chiu, “Spatial
Tessellations; Concepts and Applications of Voronoi
Diagram”, John Wiley & Sons Ltd, England, 2000.
[8] A. Barari, H.A. ElMaraghy. W.A. ElMaraghy “Design
For Machining of Sculptured Surfaces - A
Computational platform”, ASME Transaction- Journal of
Computing and Information Science in Engineering, 9-3,
13 pages, 2009.