characteristics of geometric errors determined using

characteristics of geometric errors determined using

KOMISJA BUDOWY MASZYN PAN – ODDZIAŁ W POZNANIU
Vol. 28 nr 2
Archiwum Technologii Maszyn i Automatyzacji
2008
MAŁGORZATA PONIATOWSKA∗
CHARACTERISTICS OF GEOMETRIC ERRORS
DETERMINED USING DISCRETE MEASUREMENT DATA
Coordinate measurements constitute a source of digital data in the form of coordinates of
measurement (sample) points discretely distributed on the measured surface. Freeform surface
geometric errors are determined at each point as normal deviations of measurement points from
the nominal surface of the CAD model. Computations are preceded by best fitting the measurement data to the CAD model. In the present paper the concept of data best fitting using least square
algorithm is illustrated and the geometric errors characteristics is shown on example of end milling
surface.
Key words: coordinate measurements, geometric errors, complex surface
1. INTRODUCTION
Nowadays the coordinate measurement technique has become predominant in
the measurements of geometric quantities of machine parts. The technique involves determining the coordinate values of the measurement points located on
the surface of the analyzed object. As a result, a set of discrete data in the form
of coordinate measurement (sample) points is obtained. When using CAD/CAM
techniques, the most essential aspect of coordinate measurements is to provide
relevant digital data concerning the geometry of the workpiece.
The geometry of some typical machine parts is described by standard features, such as straight lines, planes, circles, cylinders, etc. In coordinate measurements, macroinstructions embedded into the software are used, i.e. basing on
the coordinates of measurement points the program determines both the associated geometric elements and their dimensions as well as their shape and positional deviations. The accuracy control is reduced to comparing the measurement data with the ones specified in the design documents.
Ever growing quality requirements concerning new products such as functionality, ergonomics as well as appearance demand that machine parts be made
of complex 3D forms. Such parts are shaped by surfaces that are impossible to
∗
Dr inŜ. – Faculty of Mechanical Engineering, Bialystok Technical University.
52
M. Poniatowska
be described by simple mathematical equations. The accuracy control, in such
cases, consists in digitizing the analyzed object (coordinate measurement by
scanning) and comparing the obtained coordinate measurement points with the
CAD model. The results of the actual realization can be shown using a 3D
graph.
Most problems found in the theory of coordinate measurement technique
arises from the discrete nature of measured data. The problems can be divided
into two distinct categories:
− different computational algorithms give different measurement results for
the same data set,
− different sampling strategies (numbers and location of measurement
points) give different measurement results for the same surface even if the same
computational method has been applied.
The other category of problems is related to the measurement of finite number of discrete points on the measured surface that is described by an infinite
number of points. Since the geometric errors are different at each point, the
measurement results will depend on the number and location of these points. For
the same reason both the number and the locations of the points will affect the
determination process of the geometric features that constitute the basis of the
object’s coordinate system [1, 2].
2. BEST FIT PROCESS OF MEASUREMENT DATA
Prior to the determination of the surface geometric errors, it is necessary to fit
the measurement data to the nominal surface. For the measurements of surfaces
composed of standard geometric features one of the four methods to determine
associated elements is used [3]. For freeform surfaces, to fit data to the CAD
model the least square algorithm is used. The main idea of the process is presented below.
An ideal (nominal) shape of the surface element can be described by the
shape function N(p), where p denotes feature variables describing the surface.
Having produced workpiece, the actual form of the surface is described as follows:
M ( p) = N ( p ) + ε ( p)
(1)
where: M(p) – the actual geometric form of the surface,
ε(p) – geometric errors.
In coordinate measurements coordinates of measurement points are sampled
on the surface in the machine coordinate system. The determined coordinates of
the i-th point on the M(p) surface can be written down in the following way:
Characteristics of geometric errors determined using discrete measurement data
53
X i = T (t ) M ( pi ) + ei
(2)
where: T(t) – the transformation matrix between the part coordinate system and
the machine coordinate system
t
– transformation parameters, spatial rotation and translation,
ei – the measurement error.
If the measurement errors, when compared with the geometric errors of the
object surface, appear to be insignificant then the geometric error at each measurement point can be calculated using the relation given below:
ε ( pi ) = T −1 (t ) X i − N ( pi )
(3)
where: N(pi) – the nearest point on the nominal surface corresponding to the
transformed measurement point T –1(t)Xi.
However, before the geometric errors can be established, it is necessary to
determine the transformation matrix that is the function of spatial rotation and
translation. Making use of the least square method, we should minimize the following function:
m
m
i =1
i =1
F = ∑ ε ( pi ) 2 =∑ T −1 (t ) X i − N ( pi )
2
(4)
where: m – number of measurement points.
Best fitting effect depends on each of the points used to determine the transformation matrix. However, due to the existence of geometric errors at each
point, various numbers and space locations of points will result in different fitting effects and, by the same token, varied location of the object coordinate system thus affecting, in turn, the relations occurring between the machine coordinate system and object coordinate system (transformation matrix). As a result,
different geometric error values are obtained at each point for a different sampling strategy being applied. This is illustrated by Fig. 1 showing the contour
graphs of the geometric errors of freeform milling surface for three different data
sets used to conduct the process of best fitting the measurement data to the CAD
model. As a result of surface scanning coordinates of 1500 measurement points
were obtained. Out of the scanned data three sets of points of different number
and location were chosen and after best fitting them to the CAD model geometric errors of the surface were determined The differences in both error values
and distributions on the surface can be clearly observed.
M. Poniatowska
54
a)
b)
- 0 ,1
Y [mm]
-0 ,2
-0 ,3
10
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0
-0 ,5
0 ,1
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-1 ,0 0
-1 ,0 0
-0 ,7 5
0 , 0- 0 , 1
20
-0 ,2
-0 ,3
-1 ,5 0
-1 ,2 5
-1 ,2 5
-0 ,7 5
-0 ,4
-0 ,5
-0 ,5
-1 ,0 0
10
Y [mm]
-0 ,2
0 -, 0 , 1- 0 , 2
0 ,0
0 ,1
0 ,2
0 ,1
0 ,2
0 , 0- 0 , 1
-0 ,2
-0 ,3
-0 ,4
000,,6
0,54
0, 30, 2, 1
0 ,0
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20
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0
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-0 ,5
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-0 ,5 -0 ,4
-0 ,2
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-00,5
,0,1,2,34
,3-210
0
6-,4
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000000,,,-,87
0 ,, 3
6
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-0 ,5
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0 ,2 5
-2 0
0 ,2 5
-1 0
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0 ,2 5
-7 0
-8 0
X [m m ]
0 ,5 0
0 ,2 5
0 ,5 0
0 , 2 5 00, ,7550
0 ,5 0
0 ,5 0
0 ,02 ,55 0 0 0,02, ,5705 0 , 5 0 0 , 7 5
1 ,0 0
1 ,0 0
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-5 0
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-3 0
-2 0
-1 0
X [m m ]
c)
d)
2 0
-0 ,1 0
-0 ,0 5
-0 ,1 0
0 ,0 5
-0 ,0 5
2 0
0 ,1 0
0 , 0 50 , 0 0
-0 ,0 5
-0 ,1 0
1 0
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-0 ,1 5
-0 ,2 0
-0 ,2 5
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-0 ,1 5
Y [mm]
Y [mm]
- 0- 0, 1, 25 0
1 0
-0 ,1 5
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0
-0 ,0 5
0 ,0 0
-0 ,1 0
- 0 , -005, 1 0
-0 ,1 5
0 -0 ,0 5
- 0 , 01 5
0 ,0 5
- 0 ,0 5
0 ,0 0
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0
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- 0 , 1- 00 , 1 0
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-2 0
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-2 0
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-5 0
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-2 0
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-0 ,1 0
-7 0
- 0 , 1 5- 0 , 1 0
-0 ,1 5 -0 ,0 5
-0 ,1 0
-0 ,1 0
- 0 ,1 0 - 0 ,0 5
-6 0
-5 0
X
-4 0
- 0 ,0 5
-3 0
-2 0
-1 0
[m m ]
X [m m ]
Fig. 1. Contour graphs of geometric errors: a) before best fit, b) after best fit of 15 points, c) after
best fit 108 points, d) after best fit 1500 points
Rys. 1. Wykresy konturowe błędów geometrycznych powierzchni: a) przed dopasowaniem, b) po
dopasowaniu 15 punktów, c) po dopasowaniu 108 punktów, d) po dopasowaniu 1500 punktów
3. CHARACTERISTICS OF GEOMETRIC ERRORS
The geometric error values of the freeform surface i.e. normal deviations of
the measurement points with respect to the nominal surface, can be calculated by
the prior determination of error components with respect to x, y, z axes [4, 5].
The software used for coordinate measuring machines (CMM) performs such
computations automatically for each measurement point when using UV scanning option.
Surface geometric errors can be caused by a number of factors. Various errors that occur in the manufacturing process leave some marks on the object’s
surfaces. Geometric errors constitute a cumulative effect of the error sources.
For example, end milling errors can be classified into three different compo-
Characteristics of geometric errors determined using discrete measurement data
55
nents: form errors, waveness and roughness. The components related to form
errors and waveness are formed by surface irregularities imposed on the nominal
surface effectively resulting in smooth surface. The component connected with
random phenomena such as surface roughness involves irregularities of high
frequency. The actual surface is an effect of imposition of form errors, waveness
and roughness over the nominal surface. The part of random phenomena on the
surface will depend on the type of machining method. The literature on the subject states that random geometric errors found on workpiece surfaces after precision milling have shown higher values than those of deterministic errors.
Fig. 2. Model CAD of the surface
Rys. 2. Model CAD powierzchni przedmiotu
In coordinate measurements coordinates of a finite number of points on the
part surface are defined. The purpose is to determine the smooth surface superimposed over the nominal surface. In the measurement process, however, the
random component is overlaid on the deterministic component. As a result, the
spatial coordinates collected at each measurement point will contain two distinct
components. The component related to deterministic errors will constitute
smooth surface and be spatially correlated, whereas the random component will
show weak correlation and be considered spatially random in character. Thus the
surface composed of measurement points is more complex than the nominal
surface.
Form deviations are caused, among others, by deviations of machine tool
guides, deformations of some parts of the machine tool or inappropriate mounting. The waveness of the surface results from such factors as geometric errors or
even vibrations of the machine itself or the working tool. Roughness is caused
M. Poniatowska
56
by the shape of tool blades and longitudinal feed or feed-in and also workpiecetool contact vibrations.
0 ,0 5
0 ,0 0
Z [mm]
-0 ,0 5
-0 ,1 0
- 0 ,1 5
- 0 ,2 0
30
20
10
0
-1 0
-2 0
-3 0
Y[
m
m]
- 0 ,2 5
- 0 ,3 0
-8 0
-6 0
-4 0
-2 0
0
X [m
m ]
Fig. 3. Geometric errors values versus XY plane
Rys. 3. Wartości błędów geometrycznych w odniesieniu do płaszczyzny XY
M o d e l p o w ie r z c h n io w y b łę d ó w
- 0 ,1 8
-0 ,2 0
- 0 ,2 2
0
-4
0
-2
-0 ,2 4
-8 0
-8
-7 0
-1 0
-6 0
X [m
m
-5 0
-4 0
]
a)
-1 2
Y[
-6
-0 ,2 8
mm
-4
-0 ,2 6
- 0 ,2 6
-6
-8
-0 ,2 8
-8 0
-0 ,2 8
-0 ,2 6
-0 ,2 4
-0 ,2 2
-0 ,2 0
-0 ,1 8
-7 0
m]
-2
-0 ,2 4
-0 ,2 2
]
Z [mm]
-0 ,2 0
Y [m
Z [mm]
-0 ,1 8
-1 0
-6 0
X [m
m]
-5 0
-4 0
-1 2
b)
Fig. 4. An increased sector of errors (selected from Fig. 3): a) values of errors, b) surface model
Rys. 4. Wybrany fragment wykresu z rys. 3 w powiększeniu: a) wartości błędów, b) model powierzchniowy
Fig. 3 shows a spatial distribution of the geometric errors found on the surface of an object made of aluminum alloy (Fig. 2) with respect to coordinates x
and y. The method involved an finish milling process, ball-end mill 10 mm in
diameter, bit rotational speed equal to 6000 rev/min, working feed 720 mm/min
and zig-zag cutting in the XY plane. The measurements were performed using
Characteristics of geometric errors determined using discrete measurement data
57
CMM Mistral Standard 070705 Brown & Sharpe equipped with touch trigger
probe TP200, the stylus 20 mm in length with ball tip 2 mm in diameter. It can
be observed that the measurement points, apart from the deterministic, include
random components of relatively significant value (Fig. 4).
4. CONCLUSIONS
In order to evaluate the accuracy of the produced surface the results of the
coordinate measurements must be compared with the 3D nominal CAD model
created during the process of part design. The obtained data can be used to make
an off-line correction involving inputting the corrections into the application
software in order to compensate the actual machining errors. In reverse engineering digital measurement data constitute the basis for the creation of an appropriate CAD model that, in turn, becomes extremely useful to generate the
software for machining of parts. In either case the presence random errors will
interfere with the induction process concerning the actual surface geometry,
particularly in the cases when random error values constitute a significant part in
determined values of geometric errors. To increase the accuracy of error correction and to reconstruct the surface geometry in reverse engineering it is reasonable to separate the random component from the deterministic component of the
geometric errors and use the processed data as a basis for determining the surface of the object.
REFERENCES
[1] Chan F. M. M., King T. G., Stout K. J., The influence of sampling strategy on a circular
feature in coordinate measurements, Measurement, 1996, vol. 19 (2), p. 73–81.
[2] Dhanish P. B., Mathew J., Effects of CMM point coordinate uncertainty on uncertainties in
determination of circular features, Measurement, 2006, vol. 39, p. 522–531.
[3] Ratajczyk E., Współrzędnościowa technika pomiarowa, Warszawa, Oficyna Wydawnicza
Politechniki Warszawskiej 2005.
[4] Werner A., Poniatowska M., Error correction in complex-shape objects processing with
numerical control milling machines, Advances in Manufacturing Science and Technology,
2005, vol. 29, no. 3, p. 47–55.
[5] Werner A., Poniatowska M., Determining errors in complex surfaces machining with the
use of CNC machine tools, Archives of Mechanical Technology and Automation, 2006, vol.
26, nr 2, p. 211–217.
[6] Yau H.T., Menq C.H., A unfied least-square approach to the evaluation of geometric errors
using discrete measurement data, International Journal of Machine Tools and Manufacture,
1996, vol. 36, no. 11, p. 1269–1290.
Praca wpłynęła do Redakcji 31.03.2008
Recenzent: prof. dr inŜ. Jan Chajda
58
M. Poniatowska
CHARAKTERYSTYKA BŁĘDÓW GEOMETRYCZNYCH
WYZNACZANYCH Z DYSKRETNYCH DANYCH POMIAROWYCH
Streszczenie
Pomiary współrzędnościowe są źródłem cyfrowych danych w postaci współrzędnych punktów
pomiarowych o dyskretnym rozkładzie na mierzonej powierzchni. Błędy geometryczne powierzchni swobodnych wyznacza się w kaŜdym punkcie jako odchyłki normalne tych punktów od
powierzchni nominalnej (modelu CAD). Obliczenia poprzedza się dopasowaniem danych pomiarowych do modelu CAD. W artykule przedstawiono koncepcję procesu dopasowania danych
metodą najmniejszych kwadratów oraz charakterystykę błędów geometrycznych na przykładzie
powierzchni swobodnej frezowanej wykańczająco.
Słowa kluczowe: pomiary współrzędnościowe, błędy geometryczne powierzchni, powierzchnia swobodna