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Deviation model based method of planning accuracy inspection

of free-form surfaces using CMMs

Malgorzata Poniatowska ⇑

Bialystok University of Technology, Faculty of Mechanical Engineering, Division of Production Engineering, Wiejska Street 45C, 15-351 Bialystok, Poland

a r t i c l e i n f o

Article history:

Received 21 August 2011

Received in revised form 20 December 2011

Accepted 31 January 2012

Available online 9 February 2012

Keywords:

CMM

Free-form surface

Sampling strategy

Geometric deviation

CAD model

a b s t r a c t

Measurements of free-form surfaces are performed with the use of numerically controlled

CMMs on the basis of a CAD model, which results in obtaining coordinates of discrete mea-

surement points. The local geometric deviation, i.e. the distance of a particular measure-

ment point from the CAD model of the nominal surface, is established for each point.

The measurement aims at evaluating the form deviation and thus the greatest deviation

of the actual surface from the CAD model. An effective measurement is one in which the

probability of locating the greatest deviation is highest with the smallest possible number

of measurement points. The present article suggests a method of planning a measurement

strategy for objects with free-form surfaces. Repeatability of deterministic deviations on

surfaces processed under the same conditions was applied. A CAD model of the product,

built on the measurement points, and including the deterministic component of these

deviations, was constructed. Effective surface measurements were planned locating the

measurement points in the critical areas. The devised model was used for performing

the measurement.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Free-form surfaces are characterised by spatially com-

plex geometry. In designing, producing, and inspecting

accuracyof suchsurfaces, CAD/CAM techniques and numer-

ically controlled machinesare used. Processing is carriedout

with multi-axis milling machine centres, accuracy inspec-

tion is most often performed using NC CMMs (Numerically

Controlled Coordinate Measuring Machines) equipped with

touch measurement probes. The result of a measurement isa set of measurement points of a specified distribution on

the measured surface. For each measurement point, the

value of the local geometric deviation, i.e.the distanceof this

measurement point from the CAD model of the nominal

surface in the normal direction is established. The essence

of accuracy inspection of a surface is to verify whether the

observed deviation lies within the tolerance zone. A form

deviation is determined as a doubled highest absolute value

out of the obtainedset of local deviations [1]. Measurements

can be made with reference to the datum features, and then

the measurement results include also deviations of location

andorientation. In order to reduce deviations in locationand

orientation, fitting the measurement data to the CAD model

needs to be performed [2]; if it is the case, thenlocalgeomet-

ric deviations only represent surface irregularities and it is

possible to assess a (simple) form deviation from them [1].

In the majority of cases measurements of free-formsurfaces are carried out along a regular u–v grid with the

use of UV scanning option (u, v – directions of the B-spline

surface parameterization), which is inbuilt in CMM soft-

ware. This sampling strategy does not require any insight

into the nominal geometryor the measured feature and this

is why it tends to omit critical points, e.g. points located in

areas of greatest deviations. In addition, it provides a large

number of unnecessary points and prolongs the measure-

ment time. An effective measurement is one in which the

probability of locating the greatest deviation is highest with

the smallest possible number of measurement points. The

0263-2241/$ - see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.measurement.2012.01.051

⇑ Tel.: +48 85 746 92 61.

E-mail address: [email protected]

Measurement 45 (2012) 927–937

Contents lists available at SciVerse ScienceDirect

Measurement

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c at e / m e a s u r e m e n t

http://dx.doi.org/10.1016/j.measurement.2012.01.051

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pertinent literature presents different suggestions of dis-

tributing measurement points on surfaces. The most com-

monly assumed basis of planning are geometric features of

the nominal surface, such as the mean curvature or the

Gaussian curvature, because, according to the authors,these

factors determine the distribution of machining errors [3,4].

However, the experience of the author of the present article

shows that, in many cases, this assumption is only partially

convergent with the actual results of surface machining.

The best solution is to locate measurement points,

taking into consideration the real geometric deviations of

the surface under concern, which was suggested by Edge-

worth and Wilhelm in [5]. They used the iteration proce-

dure and B-spline modelling with reference to an initially

sampled surface. In the subsequent iteration stages, new

measurement points were determined in the areas in

which the fitted curve was most distant from the nominal

surface. This method makes it possible to concentrate

measurement points in areas of greatest deviations and it

significantly reduces the total number of points when com-

pared to the classical method, i.e. regularly distributed

points, at the same minimum distance between measure-

ment points. The suggested algorithm is applied during

the process of taking measurements and it requires devel-

oping a special programme.

In their latest publicationspertaining to accuracyinspec-

tion of free-form surfaces, Rajamohan et al. were testing

various sampling strategies (uniform Cartesian, uniform

parametric, patch size ranking, uniform surface area, domi-

nant points) at points including simulated geometric devia-

tions [6,7]. Theauthorsnamedabove built substitute NURBS

profiles [6] and surfaces [7] on simulated measurement

points, and they tested the accuracy of deviation mapping

in the modelling results.

In performing accuracy inspection of free-form sur-

faces having the same nominal geometry, produced under

repeatable conditions, characteristic patterns left by the

machining process on surfaces can be used. Such an

approach was presented by Summerhays et al. in [8] and

by Colosimo et al. in [9] with reference to surfaces with reg-

ular shapes. As far as roundness profiles are concerned, on

the basis of the results of measurements carried out in the

initial stage according to a regular distribution of points,

the authors of [9] developed a regression model estimating

the manufacturing signature, which is a combination of

sinusoidal and cosinusoidal functions. Then they selected

the locations of measurement points by minimising the dis-

tance between the maximum and the minimum points of

the regression-based tolerance interval. The efficiency of

the method was examined with the Monte Carlo method.

With reference to free-form surfaces which are mea-

sured in a CAD environment, a CAD model of a real surface

(geometric deviations), presented in one of the standard

formats of data exchange (e.g. IGES), readable by CMM

software, could have a practical use in forecasting critical

areas, in generating measurement points in these areas,

as well as in performing measurements of a simple form

deviation (after fitting measurement data to the CAD mod-

el). In that case, all the activities listed above could be com-

pleted directly in the NC CMM software. In order to

develop a model, it is necessary to take a measurement

along a regular grid of points at the first stage.

The scope of information on the surface under consider-

ation in the obtained data depends on measurement

parameters such as the ball tip diameter and sampling

interval (grid size), as they cause mechanical–geometrical

filtration of irregularities [10]. The lengths of irregularities,

which are to represent measurement data, need to be ta-

ken into consideration while planning the measurement.

However, in measuring free-form surfaces characterised

by different curvatures at each of the points, the cut-off va-

lue of the irregularities filtration is variable, and not all

short irregularities are filtered out. Additionally, the data

contain measurement noise (Section 2).

Because curvature is spatially variable at each point of a

free-form surface, the distribution of machining forces and

other phenomena occurring during machining are also

spatially variable [3,11]. As an effect of this, the distribution

of geometric deviations is of the same character. Deviations

of a systematic (deterministic) character, as well as devia-

tions of a random character, are observed on a surface.

Deterministic deviations are spatially correlated however

a lack of spatial correlation indicates spatial randomness

of deviations [12,13]. Deterministic deviations are charac-

terised by a specific distribution on a surface and are re-

peated with a specified variability on subsequent surfaces

processed under the same conditions. Spatial statistics

methods, including the most popular Moran’s I statistics

which measures spatial autocorrelation, can be applied to

researchon dependency of spatial data [13,14] (Section 2.2).

Identifying spatial autocorrelation in measurement data

proves the existence of deterministic effects (deviations). In

such a case, advanced CAD software for surface modelling

may be applied to fitting a surface regression model repre-

senting these deviations. The first step in model diagnosing

is to examine the model residuals for the probability distri-

bution and the existence of spatial autocorrelation. The

Moran’s I test is also used for this purpose [13,14]. The

authors of article [12] applied regression analysis and

spatial statistics to develop a B-spline model of determined

deviations of a free-form surface, and they examined the

uncertainty of fitting the data to the CAD model on the sep-

arated model residuals. In publications [9,15], concerning

measurements of regular surfaces, spatial autocorrelation

of residuals from the deviation regression model was also

taken into consideration.

The present paperproposes an approachto planning and

performing coordinate measurements of form deviations of

free-form surfaces processed under the same conditions,

consisting in using a surface CAD model built on the ob-

served geometric deviations and representing systematic

deviations. For this purpose, the random component must

be removed from the measurement data, and a model of

the surface, which represents the reproducible deviations,

must be determined. Such a model can represent, with a

specified variability, the form deviations of subsequent sur-

faces. In this case, the deviation model, superimposed on

the nominal CAD model, represents the product model.

The experiments were carried out using real measurement

data. The data obtained from measurements made along a

928 M. Poniatowska/ Measurement 45 (2012) 927–937

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regular u–v grid of points were used as a basis to create the

model. The regression analysis, an iterative procedure, spa-

tial statistics methods [13,14], and NURBS modelling [16]

were used for establishing the model.

2. Removing random component from measurement

data

Geometric deviations of surfaces are attributed to

many phenomena that occur during the machining, both

deterministic and random in character. These phenomena

with their consequent machining errors can be described

in space domain. In coordinate measurements of free-form

surfaces, spatial data is obtained which provide information

on the machining and on geometric deviationsin the spatial

aspect. If a measurement is to take into consideration form

deviation without reference to the datum features, the pro-

cedure of fitting the data to the CAD model must be

performed [2,17]. Then, the determined local geometric

deviations only represent surface irregularities which can

be divided into three componentsof different lengths: form

deviations, waviness, and roughness of the surface [10].

Spatial coordinates of each measurement point include all

the three components at different proportions.

The components connected with the form deviations

and waviness, are surface irregularities superimposed on

the nominal surface most often deterministic in character.

The component connected with random phenomena,

including the surface roughness, is irregularity of high

frequency [10,11]. The spatial coordinates assigned at each

measurement point include two different in nature compo-

nents. The component connected with the deterministic

deviations is spatially correlated. The random component,

on the other hand, is weakly correlated and is considered

to be of a spatially random character [11,12]. Deviations

of random values may be spatially correlated which is

reflected in their deterministic distribution on a surface

PCS – Part Coordinate System, NMCS– Nominal Model Coordinate System,MCS – Machine Coordinate System

− Form deviation

− Transformation parameters(location and orientation deviations)

Measurement data

Measurement data

Virtual transformation of PCSaccording to transformation matrix

determined by best-fitting CADmodel of the product to nominalCAD model (Geomagic Qualify)

CNC CMM 1. Nominal CAD model2. CAD PSM3. PC DMIS4. Geomagic Qualify

measurement 3-2-1points in manual mode

PCS definition

PCS = NMCSTransformation matrix

PCS relative to MCS step I

Transformation matrixPCS relative to MCS step II

Measurement of specifiednumber of points in critical area

Comparison of the measurementdata to CAD model

Fig. 1. Procedure of measuring form deviation of a free-form surface according to the suggested strategy.

M. Poniatowska / Measurement 45 (2012) 927–937 929



and is indicative of the existence of a systematic source in

the course of machining. A lack of spatial correlation indi-

cates spatial randomness of deviations. The different spa-

tial nature of effects may be the basis for removing the

random component from measurement data [13–15].

To collect measurement data representing irregularities

of a surface in the first stage, measurementsmust be carried

out along a regular grid of points, with an appropriately

chosen sampling interval and the size of the ball tip, taking

into consideration mechanical-geometrical filtration of

irregularities, caused by these factors [10,18]. In selecting

sampling parameters for coordinate measurement of form

deviations of free-form surfaces, the principles cited in the

literature concerning measurements of roundness profiles

can be applied [10]. If a test for spatial autocorrelation

shows spatial dependence of the data, a spatial model rep-

resenting form deviations can be prepared (the random

component can be removed) [12–14].

2.1. Modelling of the surface representing form deviations

In order to create the surface model representing deter-

ministic deviations, the NURBS method was applied. The

NURBS surface of the p degree in the u direction and the

q degree in the v direction is a vector function of two

variables in the form of [19]:

S ðu; v Þ ¼

Pni¼0

Pm j¼0wi; jN i; pðuÞN j;qðv ÞP i; jPn

i¼0

Pm j¼0wi; jN i; pðuÞN j;qðv Þ

ð1Þ

P i,j points make up a two-direction control points grid on

which the surface patch is lofted (n, m are the numbers

of control points in the u and v directions respectively),

wi,j are the weighs, while N i, p(u) and N j,q(v ) are the B-spline

basis functions defined on knot vectors [19]. The input data

in surface interpolation is a set of points forming a spatial

Fig. 2. (a) CAD model of the surface, (b) measured workpieces.

Fig. 3. The map of the observed geometric deviations of S1.

Table 1

Statistical parameters of observed local geometric deviations (mm).

Statistical parameters of

geometric deviations (mm)

10,000 pts 625 pts

S1 S1 Mean

S1–S10

Std dev.

S1–S10

Mean 0.0004 0.0010 0.0004 0.0013

Std deviation 0.0106 0.0090 0.0105 0.0017

Min. 0.0330 0.0318 0.0328 0.0046Max. +0.0233 +0.0188 +0.0209 0.0022




grid of points. In the case under concern, the data were ob-

tained from coordinate measurements during which a

two-direction grid of measurement points was obtained.

In developing the geometric model, the method of global

surface approximation was used. The process is carried

out in two stages [20]:

at the first stage, a series of isoparametric curves located

on the surface patch is created. These curves are approx-

imated on the subsequent rows of the pre-set points of

one of the parameterization directions, u or v ; the value

of theother parameter describing thesurfaceis then con-

stant. A spatial grid of control points is obtainedthisway;

at the second stage, coordinatesof surface control points

are determined. It is performed by approximating curves

through the control points of the curves which were

approximated earlier. The approximation is made in the

other parameterization direction. The surface is lofted

on the series of curves which was determined earlier.

After the approximation stage was completed, shape

modification iteration of the created surface patch was

applied in the subsequent stages. These operations aimed

at obtaining an adequate model of regression surface,

which would represent deterministic deviations. In this

case, popular procedures were applied of changing the

NURBS surface shape, namely [20]:

rebuilding the knot vectors, which influences a change in

the number of control points in the u and v directions);

changing the degrees of B-spline base functions.

Fig. 4. Maps of deviations of S1: (a) form deviations (M1), (b) removed random effects (model residuals).




Reducing the number of knots (at the subsequent itera-

tion stages) results in reducing the number of control

points of the surface. A less complex shape can be obtained

this way.

2.2. Models’ adequacy verification

Themethodof deterministic deviations surface adequatemodel designing consists in iterative modelling of the sur-

face regression model and in testing the spatial randomness

of the model residuals at the consecutive iteration stages (as

in [9,12]). In the subsequently constructed models, the

number of control points and the surface degrees in both

directions are changed. The model residuals are examined

at each step, and the maximum and minimum values, arith-

metic mean(should be0), probability distribution, and the

I spatial autocorrelation coefficient (2) are determined.

The spatial autocorrelation coefficient I has the follow-

ing form [13]:

I ¼ n

S 0

r T Cr

r T r ð2Þ

where r is the vector of model residuals, C is the weighting

matrix of spatial relations between residuals in i and j

locations,

S 0 ¼Xn

i¼1

Xn

j¼1

c ijði– jÞ

It is assumed that the dependence between the data

values in the points i and j decreases when the distance

dij increases, this relation can be described in the following

way [13,14]:

c ij ¼ d f ij ð3Þ

where c ij = 0 for i = j, f is the constant ( f P 1), in this work

f = 3 is assumed, the correctness of the choice was investi-

gated in previous works of the author [21].

After having determined the coefficient I , a test of sig-

nificance for its value needs to be conducted. A test statis-

tics z needs to be determined for this purpose and

compared with the limit z a value for the adopted signifi-

cance level [13,22]. Positive and significant value of the I

statistics implies the existence of positive spatial autocor-

relation, i.e. a similarity of residuals in the specified dis-

tance. Otherwise, lack of spatial correlation indicates

spatial randomness of residuals [13,14,21,22].The model with the smallest number of control points

and the lowest surface degrees in the u and v directions

(Section 2.1), for which the model residuals met the crite-

ria of a normal probability distribution and of spatial ran-

domness, is adopted as an adequate one. The distribution

normality was verified with the Kolmogorov–Smirnov test

(K–S d statistics).

3. Machining pattern model

The presented procedure (Section 2) is appropriate for

determining deviation models of single surfaces. In sucha situation, the determined model represents an averaged

shape of deviations of a single surface and the model resid-

uals represent random effects, i.e. variability concerning

this surface. In order to take into consideration the vari-

ability of models between subsequent produced elements,

the model of deterministic/systematic deviations should

then be estimated on the basis of a set of properly sampled

surfaces by averaging of models built on data representing

these surfaces. The averaged model would represent the

same characteristic patterns left by the machining process

on surfaces with a specified variability between the sur-

faces. It has to be noted that this variability will be depen-

dent on the number of measurement points and it will be

smaller for a bigger number of points. The estimation

uncertainty of the machining pattern model (MPM) will

therefore depend not only on the machining process vari-

ability but also on the number of measurement points.

The choice of measurement parameters will influence the

accuracy of estimating real deviations by the designed

model of deviations for each surface, while the machining

process variability will affect the accuracy of estimating

deviations of subsequent surfaces by the MPM.

Inspection of manufacturing accuracy can be performed

on the basis of the MPM. For a practical measurement to be

conducted, a CAD model simulating the Product Surface

Model (PSM), i.e. a nominal CAD model with a superim-

posed MPM, is necessary.

4. New measurement strategy

On the basis of the developed MPM, it is possible to pre-

dict the location of areas of greatest geometric deviations

on subsequently produced objects. The model represents

averaged systematic deviations, and therefore after the

MPM is superimposed on the nominal CAD model, a Prod-

uct Surface Model (PSM) is obtained. The PSM can be used

for virtual fitting to the nominal CAD model in order to

determine the mean transformation matrix to locate sub-

sequent surfaces in measuring the form deviation (if the

measurement concerns the form deviation without refer-

ence to the datum features [2,3,17]).

In the next stage, measurement points in areas of great-

est deviations can be determined on the model. A small

Table 2

The computation and modelling results for ten models (at discrete

measurement points).

Characteristics of deviations M1 Mean

M1–M10

Std dev.

M1–M10

Deterministic component

Control points number u v 15 15

Surface degrees u v 3 3

Mean (mm) 0.0005 0.0004 0.0014

Std deviation (mm) 0.0089 0.0101 0.0018

Min. 0.0311 0.0322 0.0046

Max. (mm) +0.0190 +0.0211 0.0024

Random component

Mean (mm) 0.0000 0.0000

Std deviation (mm) 0.0014 0.0014

Min. 0.0067 0.0063

Max. (mm) +0.0060 +0.0066

K–S d statistics (da = 0.065) 0.064 0.064

Autocorrelation coefficient I 0.082 0.073

Test statistics z ( z a = 2.34) 2.226 2.165


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number of points located along a regular grid, and, addition-

ally, a point in the location of the greatest global deviation

determined from the model, are sufficient for this purpose.

After the measurement strategy has been prepared this

way,measurements of subsequent surfacesproduced under

the same conditions can be made (Fig. 1), making use of the

transformation matrix and the coordinates of points located

in the critical areas. Applying the developed model signifi-

cantly reduces the time necessary for determining a form

deviation of a surface. It is possible because:

measuring each surface to fit the measurement data to

the CAD model is not necessary (for localisation in order

to determine a form deviation, fitting can be performed

using the virtual PSM);

there is no necessity to scan the whole surface, a small

number of points located in critical areas is sufficient;

the course of a measurement can be planned both

off-line on the CAD PSM, and also on-line in CMM

software.

Fig. 5. Maps of geometric deviations versus XY plane: (a) mean observed deviations of S1–S10, (b) deviations determined from MPM (625 pts).

Table 3

Statistical characteristics of mean observed local deviations and local

deviations determined from the MPM (625 pts).

Statistical characteristics (mm) Mean of S1–S10 MPM

Mean 0.0004 0.0004

Standard deviation 0.0105 0.0106

Min. local deviation 0.0328 0.0322

Max. local deviation +0.0209 +0.0211




The proposed strategy in a synthetic way is shown in

Fig. 1. For greatest precision, points might be located in

specified areas along a dense grid, but only in restricted re-

gions (the size of the measuring tip should be selected

adequately).

Literature analysis and the author’s own research show

that the uncertainty of fitting measurement data to the

nominal CAD model plays a significant role in the uncer-tainty of measuring the form deviation without reference

to the datum features. Separating the random component

from measurement data makes it possible to reduce the

fitting uncertainty [12]. Moreover, a virtual fitting process

can be performed ona much greater number of points deter-

mined on a PSM than the number of measurement points,

which also leads to a decrease in the fitting uncertainty

[9,12]. Diminishingthe fitting uncertainty results in a lower

uncertainty of determining deviations of form, location, and

orientation. The uncertainty of establishing a PSM depends

on the applied sampling strategy and on the variability of

a machining process. Individual features of each particular

surface are providedfor by making measurements in criticalareas according to a specified sampling strategy.

5. Experimental research

5.1. Measurements and MPMs constructing

The method is presented on the example of the experi-

ments which were performed on free-form surfaces of

workpieces made of aluminium alloy with the base mea-

suring 50 50 mm (Fig. 2), obtained in the milling process

using ball-end mill 6 mm in diameter, rotational speed

equal to 7500 rev/min, working feed 300 mm/min and

zig-zag cutting path in the XY plane (Ra = 1.72 lm). The

measurements were carried out on a Global Performance

CMM (PC DMIS software), MPE E = 1.5 + L/333, equipped

with a Renishaw SP25M probe, a 20 mm stylus with a ball

tip of 3 mm in diameter.

The tests were conducted on 10 surfaces (S1–S10)

processed under the same conditions. In the first stage,

measurements of all the surfaces along a regular grid with

the UV scanning option (the option built in PC-DMIS soft-

ware) were made, in which 625 uniformly distributed

measurement points were scanned (25 rows 25 col-

umns) in area (0.01–0.99)u (0.01–0.99)v , and geometric

deviations were computed. The part coordinate system

was defined on regular surfaces of the workpiece (details

are shown in Section 5.2, Fig. 7c). Thus, the measurement

points included also deviations of location and orientation.

An example of a map of the observed geometric deviations

of the S1 is shown in Fig. 3. It can be observed that the

measurement points, apart from the deterministic compo-

nent, include also the random component. The measure-

ment results for S1 are presented in Table 1, results at

10,000 measurement points are included as reference.

Statistical parameters of the obtained sets of deviations

were determined, and tests for randomness were per-

formed. In all statistical tests a confidence level P = 0.99

was adopted. The tests (Section 2.2) showed spatial auto-correlation of the measurement data, which indicated the

presence of form (deterministic) deviations on the surfaces

under research.

In the second stage, regression surfaces representing

form deviations were modelled (M1–M10) on the obtained

measurement data using an iterative procedure, NURBS

geometric modelling, and spatial statistics methods (Sec-

tions 2.1 and 2.2). For all the tested samples the model

with the smallest number of control points and the lowest

surface degrees in the u and v directions, for which the

model residuals met the criteria of a normal distribution

and of spatial randomness, was adopted as an adequate

one. In all the cases the criterion was met for the number

of control points amounting to 15 15, and the number

of surface degrees being 3 3. The determined models

represent deterministic irregularities, whereas the residu-

als of the models constitute the random effects. Maps of

deterministic and random deviations for S1 are shown as

examples (Fig. 4). The computation and modelling results

are compiled in Table 2.

Observing the map of the deterministic deviations

(Fig. 4a), the effect of rejecting random deviations is visi-

ble. The separate random geometric deviations scatter

amounts to 1= 4 of the deviations obtained as a result of mea-

surement (Table 2). In the described research on the ten

surfaces, the mean local deviations with a distribution

illustrated in Fig. 5a, and the MPM whose map is shown

in Fig. 5b, were obtained. The shapes of the deviation

map and of the model are not surprising (compare

Fig. 5b to Fig 4a, and Fig. 5a to Fig. 3). Statistical character-

istics of the mean (of S1–S10) observed local deviations

and local deviations determined from the MPM are listed

in Table 3.

In the next stage, the MPM was superimposed on the

nominal CAD model (Section 3). The obtained PSM is shown

in Fig. 6. Beside a map presenting the MPM shape, the values

of the estimated global deviations – the minimum (nega-

tive) and maximum (positive) one – are presented in this

figure together with their locations (the word ‘global’ was

Fig. 6. Nominal CAD model with a superimposed MPM – the PSM, as wellas values and locations of global deviations (Geomagic Qualify).


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used here to distinguish these deviations from local geo-

metric deviations determined in discrete measurement

points, because the determined deviation is a surface devi-

ation and not a deviation of its discrete representation).

The method is presented on the example of an MPM

which had been successfully used for correcting the

machining programme; geometric deviations of many sur-

faces after correction did not exceed 8 lm then. The possi-

bility of applying models of particular samples in

determining global form deviations was demonstrated in

the author’s previous research. In the present paper, results

for S1 (Tables 1 and 2) are given as an example. The global

geometric deviation determined from M1 (Fig. 4) amounted

to0.0324 mm, and the one determined from the measure-

ment of this surface at 10,000 measurement points

amounted to 0.0333 mm. The differences between the

deviations determined from raw measurement data andthe deviations determined from models were of the order

of tenths of a micrometre (Tables 1 and 3); the values of

the deviations determined from models were closer to the

results of measurements at 10,000 measurement points,

which were adopted as the reference values. According to

the assumption, models approximate the surface between

measurement points, providing with a surface picture that

is closer to the reality, which justifies using them in mea-

surement planning.

5.2. Distribution of measurement points

Following the proposed procedure (Section 4, Fig. 1) the

PSM was fitted to the nominal CAD model in order to ob-

tain the mean transformation parameters (Fig. 7b) to lo-

cate surfaces in measuring the form deviation using a

CMM (the measurement pertained to the form deviation

without reference to the datum features, Fig. 7c). In orderfor the fitting to be performed, 50,000 points were applied.

Fig. 7. PSM after fitting to nominal CAD model: (a) the map with critical area marked and global deviations, (b) the transformation matrix, (c) distribution

of measurement points and the datum features for the initial part coordinate system.


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The advantage of using a great number of points on a

smooth model of deviations lies in the fact that the fitting

uncertainty is radically reduced in this case. Fig. 7a pre-

sents a map of form deviations with a marked area of

greatest deviations. Measurements were planned, adopting

the same sampling parameters as those in the first stage of

the research. The nine measurement points (3 rows 3

columns), and additionally one measurement point in thelocation of the global deviation determined from the PSM

( x,y = 0.5024,48.4727), were determined on the nominal

CAD model. The details are shown in Fig. 7c together with

the datum features defining the initial part coordinate

system.

The results of research conducted in the way described

above were compared to the results of scanning whole sur-

faces (Table 4), performed in order to develop the MPM.

The standard uncertainty of the measurement without ref-

erence to datums amounted to 1.26 lm for the new meth-

od (in this case). The value includes the uncertainty of

measurement repetitions (Table 4) and the machining pro-

cess scatter.

6. Summary

Tables 1, 3 and 4 contain the results of the experimental

research which, by its nature, is more difficult to perform

than simulation research, especially for free-form surfaces

that require time-consuming sampling at many points. It

has to be underlined that the values of averaged standard

measurement errors (Table 4), reflecting the scatter of a

fivefold repetition of the measurements, are smaller for

the new strategy. An unexpected systematic error of the

order of 2 lm for deviation values can also be observed.The resulting error is comparable to MPE E (Section 5.1).

Removing the random component from the measurement

data resulted in a smaller scatter of global geometric devi-

ations determined from the respective models than that of

the deviations determined from raw measurement data for

subsequent surfaces. When analysing the obtained results,

it can be seen that the scatters of the measurement results

arrived at measuring surfaces along a regular grid of

points, and these of the results of the process carried outaccording to the suggested new strategy using a PSM, are

similar. Critical points of the particular surfaces are located

within a radius of 0.75 mm around the critical point

determined from the MPM (PSM) and thus the probability

of locating the greatest deviation of each surface is much

higher. The additional advantage of using the method lies

in the fact that the measurement time is radically reduced

in this case.

The obtained results encourage to applying and perfect-

ing the method. Research on improving the proposed

method is currently being conducted, and applying models

of deviations, constructed on different numbers of mea-

surement points, is being tested.

7. Conclusions

The present paper proposes a method of planning coor-

dinate measurements of free-form surfaces machined un-

der repeatable conditions, which uses spatial models of

patterns left by the machining process on surfaces. The

machining pattern model (MPM) should be determined by

averaging models of determined deviations of a surface

set. The regression model which describes deterministic

deviations of each surface is searched for with the iterative

procedure, using the NURBS method for modelling surfaceson measurement data, changing the number of control

Table 4

Comparison of results of measuring form deviations, performed according to suggested strategy, to results of scanning whole surfaces, as well as to

computation results (mm), (SME – standard error of measurement results).

S1 S1–S10 MPM

Scanning the whole surface, 625 pts ( x, y) = (0.500,47.736) 0.0318

Mean 0.0328

Std dev. 0.0045

SME 0.0009

Computation results

Model deviation in measurement point as above 0.0311

Mean 0.0322 0.0322

Std dev. 0.0039

Critical point location 0.570 0.500

48.531 48.464

Global deviation Mean 0.0324 0.0335 0.0337

Std dev. 0.0036

After fitting to nominal CAD model PSM

Global deviation (computation results) 0.0146 0.0143 0.0137

New measurement strategy

Nine measurement points in critical area (Fig. 7c) 0.0159

Mean 0.0162

Std dev. 0.0035

SME 0.0006

0.0172

Critical point of PSM ( xk, yk) = (0.502,48.473) (Fig. 7a and c) Mean 0.0173

Std dev. 0.0036

SME 0.0004


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points and surface degrees in the u and v directions, as well

as verifying a model’s adequacy by examining residuals at

each iteration stage. In verifying a spatial model, not only

the probability distribution, but also the spatial distribution

of residuals is significant, because data with random values

might be spatially correlated, which proves that their spa-

tial distribution is deterministic. In the applied method of

designing models, spatial autocorrelation of residuals is

tested besides testing the arithmetic mean and the proba-

bility distribution normality.

After the MPM is superimposed on the CAD nominal

model, a CADmodel of theproduct’s surface is obtained. This

latter model can be used not only for planning but also for

performing measurements. Having a model of product’s

surface, it is possible to determine areas of greatest

deviations (critical areas); it is also possible to locate mea-

surement points in these areas, and to determine transfor-

mation parameters for surface positioning in measuring

form deviation without reference to the datum features.

Using the proposed method, one can radically decrease the

uncertainty of fitting to the CAD model, but then the uncer-

tainty of determining transformation parameters (uncer-

tainty of determining the MPM) appears, influenced

decisively by variability of the machining process. Geomet-

ric deviations which characterise surfaces are obtained by

measuring a definite small number of points in definite loca-

tions. The probability of locating the greatest deviation is

much higher when compared to the classical method.

In the described research, the values of standard mea-

surement errors, reflecting the scatter of a repetition of

the measurements, were smaller for the new strategy,

and the observed systematic error was comparable to max-

imum permissible error of CMM.

Acknowledgments

The work is supported by Polish Ministry of Science and

Higher Education under the statute activity, Project No. S/

WM/3/2010.

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