A method for enhanced polymer spur gear inspection based on 3D optical metrology

Abstract

Accurately manufactured gears require a reliable, holistic, and fast inspection method. Standardised geometrical parameters enable a consistent and regulated inspection of gears; however, current inspection methods include only a limited set of measurements for gears at specific locations. Therefore, a method to obtain holistic three-dimensional (3D) measurements with an optical inspection was thoroughly investigated. The measurement data were acquired via 3D optical scanning. The data were then processed and evaluated using the developed software. This was first tested on a simulated scan of an ideal shape with different mesh resolutions and subsequently on a simulated scan with synthetic deviations. The method was finally validated by measuring the gears using a coordinate-measuring machine; the results obtained were compared with those obtained using the developed optical method. A good agreement between the methods was observed. The optical method offers a more holistic measurement approach with many important advantages being identified compared with the tactile method.

Full text

Measurement 169 (2021) 108584
Available online 10 October 2020
0263-2241/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
A method for enhanced polymer spur gear inspection based on 3D
optical metrology
Uroˇ
s Urbas , Damijan Zorko , Borut ˇ
Cerne , Joˇ
ze Tavˇ
car , Nikola Vukaˇ
sinovi´
c
*
University of Ljubljana, Faculty of Mechanical Engineering, LECAD, Aˇ
skerˇ
ceva cesta 6, 1000 Ljubljana, Slovenia
ARTICLE INFO
Keywords:
Gears
Optical inspection
Geometrical parameters
3D scanning
Structured light
Quality
ABSTRACT
Accurately manufactured gears require a reliable, holistic, and fast inspection method. Standardised geometrical
parameters enable a consistent and regulated inspection of gears; however, current inspection methods include
only a limited set of measurements for gears at specic locations. Therefore, a method to obtain holistic three-
dimensional (3D) measurements with an optical inspection was thoroughly investigated. The measurement data
were acquired via 3D optical scanning. The data were then processed and evaluated using the developed soft-
ware. This was rst tested on a simulated scan of an ideal shape with different mesh resolutions and subsequently
on a simulated scan with synthetic deviations. The method was nally validated by measuring the gears using a
coordinate-measuring machine; the results obtained were compared with those obtained using the developed
optical method. A good agreement between the methods was observed. The optical method offers a more holistic
measurement approach with many important advantages being identied compared with the tactile method.
1. Introduction
Gears are crucial elements for motion and power transmission in the
eld of robotics, transport vehicles, and machines. Increasing the de-
mand for accurately manufactured gears also requires a more efcient
and accurate dimensional inspection. Currently, inspecting gears is
generally done with tactile measuring machines. The process involves
measuring only one line feature across the middle of the tooth on a
limited number of teeth for the gear [1]. The measurements on the gear
anks are limited to these predened two-dimensional lines [1], even
though the complete geometry of a gear ank determines its functional
properties. The areal measurement measures multiple points. If they are
measured with traditional tactile methods, the process will take too
much time. An optical method accelerates the measurements and it can
scan the whole gear. It can also solve the problems of small module gear
measurement [2]. The scanned data can be compared with the required
dimensions, taken from the computer-aided design (CAD) model. This
enables a fast evaluation of the whole gear. The gear’s production
quality is, in terms of geometric deviations, usually evaluated using a set
of standardized geometric parameters, which give meaningful infor-
mation regarding the most crucial types of deviations typically identi-
ed on as-produced gears. This study aims to develop a holistic spur gear
inspection method that uses optical 3D areal data, acquired with a
structured light scanner. A systematic development of the methodology,
including a validation procedure, using on a tactile gear measurement
machine, is presented. The importance of holistic measurements is dis-
cussed and the measurement uncertainty is examined. The developed
method determines the geometrical parameters determining the gear’s
quality grade. The ndings resulted in a custom software for evaluating
the 3D scan measurement data.
1.1. Gear geometrical parameters
Different parameters dene a gear’s quality, such as the type and
quality of the material, the quality of the manufacturing, and the quality
of the heat treatment. This study focuses on the geometrical parameters.
For the gears to have good functionality and an acceptable inuence on
the surroundings (e.g. noise, vibration), the geometry needs to be
accurately manufactured. This is why certain parameters need to be
measured, calculated, and evaluated after manufacturing and before the
nal assembly, according to the available standards [3–5]. They include
the process of obtaining the parameter values and dening their ranges
for certain quality grades. Summaries of the standards and the basics of
quality control for gears are available in the following literature [6,7].
Some of the geometrical quality parameters include the pitch con-
trol, prole deviation, lead prole deviation, runout evaluation, and
tooth thickness. There are others, which control the gear-pair, such as
* Corresponding author.
E-mail addresses: uros.urbas@lecad.fs.uni-lj.si (U. Urbas), nikola.vukasinovic@lecad.fs.uni-lj.si (N. Vukaˇ
sinovi´
c).
Contents lists available at ScienceDirect
Measurement
journal homepage: www.elsevier.com/locate/measurement
https://doi.org/10.1016/j.measurement.2020.108584
Received 29 April 2020; Received in revised form 18 August 2020; Accepted 4 October 2020

Measurement 169 (2021) 108584
2
the axial distance. Other parameters also evaluate the gear body, such as
the dimensional and geometric tolerances. According to ISO 1328, the
limit parameter values for the control of the gears are divided into 13
quality grades (Q =0, 1, 2, ..., 12) [3]. Grade 0 means the highest ac-
curacy and it has tight tolerances, whereas grade 12 is the least accurate.
A better quality of gears leads to improvements in several elds, such as
a smaller transmission error and smaller force impulses [8], which leads
to less heat generation [9–11]. This will result in a decrease in the ac-
celeration and dynamic loads, which can cause impact and noise.
However, even accurately manufactured gears deform under the load
[8,12] and they do not perform perfectly because of other errors
[13–15].
1.2. Current research
Some studies have reported the benets of a holistic measurement of
the gears.
Kang Ni et al. [16] focused on the characterization and evaluation of
the involute gear ank. It was concluded that the optical method is
suitable for larger gears, in which longer times are needed to measure
the prole. It is also necessary to measure many proles to fully evaluate
the gear if the measurement is done with coordinate measuring ma-
chines (CMMs). The plumb line distance was used to measure the de-
viation from the theoretical prole. The plumb line distance is the
absolute value of the vector pointing from the nominal point to the
measured point for the surface’s normal direction. They also introduce
Chebyshev polynomials for the ank modications.
Gert Goch et al. [17] stated that for current inspections, only one
prole and one lead prole line are measured, and this is only done for
four teeth. For the pitch control, one point in the middle of the lead
prole is taken. A similar strategy is reported in their previous article
[16]. The focus is on measuring the deviation with the plumb line dis-
tance. They represent some improvements by using the Chebyshev
polynomials and by adding an evaluation for the helical gears. The
advantage of this method is that it can eliminate the need for dening
the nominal geometry.
Matthias Marcus et al. [18] set out to determine if the laser line
triangulation (LLT) measurement is suitable for gear metrology. They
stated that tactile measurement has stagnated and the optical mea-
surement enables the measurement of large amounts of data in a short
period of time. The focus is on the measurement error. A measurement
on a big gear with a 1 m diameter was made. A measuring machine
scanCONTROL 2910–25 with a depth resolution of 2
μ
m was used. The
lateral resolution was 19.5
μ
m. The measured data with the measure-
ments from a CMM were compared. It was concluded that the mea-
surement points have a poorer quality in the root area of the tooth
because of the reections, depth of focus, occlusion, and the geometry,
which disturbs the measurement. The results show that in comparison to
the CMMs, the LLTs have a deviation of 19.8
μ
m. They concluded that
the method is sufciently accurate for a fast evaluation of the gear
surface, but it is not good enough to determine the geometrical quality
parameters. For the same number of points that can be captured by the
LLT in 2 min, a CMM would need 190 h.
Yi-Cheng Chen et al. measured a tooth on the gear with a Moire
scanning method [19]. The tooth surface was reconstructed from the
phase information. For comparison, it was also measured with a CMM.
The average deviation of the involute was 2.67
μ
m. The average error of
the CMM is in the area of 3
μ
m; thus, it was concluded that the results are
suitable. For additional validation, a Mitutoyo K cube was measured and
a deviation of 2.67
μ
m was determined.
A study by Frank H¨
artig et. al. [20] aimed to develop a standardised
environment, nomenclature, and orientation for the 3D involute gear
evaluation. The coordinate system and the basic equations in the Car-
tesian and involute coordinate system are presented.
Vit Zelený et al. [21] described the involute prole shape, including
the helical gears. The deviations are calculated on one section, which is
suitable for the spur gears. In addition to the prole calculations, the
lead prole evaluation is also included. The evaluation is done according
to the standards. For evaluation, a CAD model of the gear tooth prole
was created. To eliminate the uncertainty of the CMM, a simulated
measurement with the GEAR PRO program was done. The data were
generated on the surface with a random error of ±10
μ
m. After
comparing their method to the GEAR PRO program, it was concluded
that the method outputs comparable results.
Xiaozhong Guo et al. [22] proposed a 3D point cloud measuring
system based on a line structured light sensor and an air oating rotary
table to rapidly measure the shape of the gear tooth ank. The measured
3D cloud was used to calculate the prole and pitch error. It was
determined to be a fast and accurate evaluation of the gear’s quality.
With multiple prole lines available, a different approach for evaluating
the spatial prole quality was proposed. The total prole deviation was
calculated by the maximum and minimum values of all of the prole
deviation values. The measuring equipment was only able to obtain data
for only one side of the ank during the experiment.
This investigation presents a systematic approach to develop a
methodology for gear quality evaluation based on the optical measure-
ments, and validation on the prepared gears with a gear CMM.
1.3. Optical measurement methods
Measurements can be done with a contact [23,24] and non-contact
approach. Contact methods are precise; however, they are slow. With
the increasing accuracy of the non-contact methods, which offer a fast
point acquisition, they are increasingly being used, which also includes
roughness measurements [25,26]. There are also different optical
methods available for acquiring a point cloud with non-contact mea-
surements. These include interferometric sensors, which are accurate
but they have a small sampling rate [27–29]. LLT is often used instead of
tactile measurements. Their main advantages include their contactless
measurement and high sampling rate [30]. It does, however, have a
Nomenclature
b gear width [mm]
D(Q)limit deviation for quality grade [
μ
m]
d reference circle diameter [mm]
dh gear hole diameter [mm]
Fp cumulative pitch deviation [
μ
m]
Fpk sector pitch deviation [
μ
m]
Fr runout deviation [
μ
m]
F
α
prole deviation [
μ
m]
Fβ lead prole deviation [
μ
m]
ff
α
prole form deviation [
μ
m]
ffβ lead prole form deviation [
μ
m]
fH
α
prole slope deviation [
μ
m]
fHβ lead prole slope deviation [
μ
m]
fpt single pitch deviation [
μ
m]
mn normal module [mm]
rb base radius [mm]
U expanded uncertainty of measurement [
μ
m]
u uncertainty of measurement [
μ
m]
xa theoretical/actual value on prole x [mm]
xm measured value on prole x [mm]
ya theoretical/actual value on prole y [mm]
ym measured value y [mm]
U. Urbas et al.

Measurement 169 (2021) 108584
3
limited accuracy [31–33], which is dependent on the quality of the
surface, and the angle of inclination [34]. It is the most used method for
on-line contactless measurement; however, it is more frequently used for
bigger objects, which requires too much time to measure. In conoscopic
holography, the laser beam is projected onto a surface [35]. The
reection passes through a conoscopic crystal and it is then projected
onto a charge-coupled device. A diffraction pattern appears, which can
be reconstructed to determine the distance to the surface. The main
advantage is that only one beam path is needed; therefore, it is possible
to measure the depth of a narrow hole. Other types of measurements
include stereoscopic measurements [36], coherence interferometry
[37], computer vision [38–40], and the Moire method [19]. In this
study, a method with structured light was used.
2. Methodology of optical gear inspection
The pitch division, prole and lead prole control, runout evalua-
tion, and tolerance measurements are typically measured with a com-
puter numerical control machine (CNC) or a CMM. However, such
measurements are slow and do not consider the whole gear. On the other
hand, these methods enable better stability and repeatability of the
quality parameters. An areal measurement, which is enabled by the
optical methods, is preferred in future gear metrology because it enables
a fast and holistic data acquisition.
The methodology was developed by evaluating the results on syn-
thetic scans and measurements, as presented in Fig. 1. The rst step is to
evaluate the quality of a simulated scan of the gear with no deviations
present. The purpose of this step is to determine the required density of
the point cloud and the resulting mesh. Later, the synthetic deviations
are evaluated to ensure the right calculation of the separate geometrical
parameters. Finally, the methodology is evaluated on the manufactured
gears by the gear CMM and scanning measurements and it is validated
on real data to determine the measurement uncertainty.
2.1. Evaluation on simulated measured meshes with different
approximations of ideal shape
The developed software was validated in different scenarios. The
program was initially tested on an ideal shape gear to determine the
error of the tessellated geometry. A gear model was created and a
simulation of the scan was done in Geomagic Studio 2014. The scanning
parameters were set in a way to obtain a similar and a higher number of
points that are the result of a scan. Two simulated scans with different
resolutions were made. The sparser mesh has a similar number of points
and triangles to the meshes due to the scanning on the actual gear. The
mesh consisted of approximately 2⋅105 triangles. The second evaluated
mesh with an ideal geometry was denser and it was created from
approximately 4⋅105 triangles. The effect of the different polygonisa-
tions of the meshes was also observed in the work of Müller et al. [41].
2.2. Validation on synthetic deviations
To validate the lead prole deviations, six different models with
deviations were created. The models had quality grades that ranged
from Q7 to Q12. The next validation was for the pitch deviations fpt. Six
models ranging from Q7 to Q12 were created. The meshes had the
approximate resolution of the scans for the manufactured gears. The
goal was to validate the methodology from the stage of acquiring a point
cloud onwards.
2.3. Validation on the manufactured gears
2.3.1. Sample preparation
The measured gears were made of polymers, which is the focus of the
department [42,43] and the MAPgears project. Polymer gears enable
easier manufacturing and they can operate without the need for external
lubrication, which results in lower friction and wear [44]. However,
they are normally manufactured to a lower degree of accuracy than steel
gears [45], because of their poorer thermal geometrical stability. Stock
extruded rods were cut into slices, which were then used as bases for the
hobbing process. Commercial grades TECAFORM AH natural (POM-C)
and TECAMID 66 natural (PA66) were used. The hobbing tool quality
grade was AA according to DIN 3968. The geometrical parameters of the
manufactured gears are presented in Table 1. Polymer gears made with
the hobbing process can normally achieve better quality grades than
gears made with injection moulding.
2.3.2. Process of measuring and preprocessing
A method with structured light was employed for the optical mea-
surements of the manufactured gears. The ATOS Compact SCAN 5 M
scanner with a stated laboratory accuracy of approximately 2
μ
m was
used. Because the scanning conditions are not ideal, a lower accuracy
can be expected. The measuring accuracy is dependent on the size of the
object. The scanner has a 5-megapixel camera and it can measure objects
ranging from less than 10 mm to 1 m. The smaller the object, the better
the scanning resolution is. The measuring system was used in the small
object conguration where the measuring volume is 70x50x50 mm, the
measuring distance is 420 mm and the resulting measuring point dis-
tance is 0.029 mm. In the small object conguration the pixels are more
densely spread out on the surface. It is also possible to measure objects
that are larger than the measured volume. The manufacturer guarantees
the stated accuracy for measuring objects that are not three times longer
in each dimension. The whole picture is then put together with the help
of markers on the object. Theoretically, the camera only needs three
points to put the data from different views together. The working
principle is that a projector projects parallel lines onto the 3D surface.
These lines are seen differently from different angles. The deviation of
the lines then enables an accurate determination of the 3D coordinates
of the details on the surface. The scanner has a stereo camera so it can
control itself and discard the bad measurements. It collects the mea-
surements only if the received data on both cameras are in agreement.
Fig. 1. Steps in the development of the methodology.
Table 1
Tested gear properties.
Number of teeth Z [/] 20
Reference circle diameter d [mm] 20
Gear width b [mm] 6
Nominal gear hole diameter dh [mm] 6.15
Normal gear module mn [mm] 1
Normal pressure angle [◦] 20
Type of prole Involute ISO 53 prole C
U. Urbas et al.

Measurement 169 (2021) 108584
4
Before each measurement, the scanner was calibrated with a standard
calibration panel CP40, at room conditions; temperature 23 ±1 ◦C and
relative humidity 40 ±2%. The scanning setup is shown in Fig. 2.
To measure the gears and objects in general, it is often necessary to
coat them with an anti-reecting powder. The powder can be applied
with a spray gun to obtain thin layers. Powders based on titanium di-
oxide sublimate after some time; hence, they do not require cleaning the
object. The scan is then stored into an STL le, which is then imported
into GOM Inspect 2018 [46], which is a program capable of manipu-
lating 3D meshes. First, the STL is pre-aligned with the CAD model with
a global best t method. After that, they are aligned with geometric
elements. The gear has six degrees of freedom in its free state. Holes from
the geometry and the CAD le are aligned rst since it is the most
important alignment due to the mounting of the gears. The hole align-
ment is done with tting a cylinder on both objects and aligning them.
This leaves two degrees of freedom, the translation and the rotation
relative to the axis of the hole. The translation is then xed. Two planes
are created on the sides of the gear and this results in a symmetrical
plane. The symmetrical plane is aligned with the one from the CAD. The
rotation is xed with the created points on the mesh. The alignment
elements are shown in Fig. 3, and they are done with GOM Inspect
software[46].
In the next step, the required number of sections is specied. A
planar section through the middle of the gear and a cylindrical section
on the reference circle is made to obtain the results with the traditional
method. The measured gears have a reference circle diameter of 20 mm.
The sections are shown in Fig. 3. In the next iterations, sections could be
done on multiple planes; hence, the parameters are determined on the
whole gear width. By doing this, the helical gears can be evaluated. The
software allows the user to choose which parameters to evaluate and
then it generates a report. Fig. 4 shows the general process of measuring,
processing, and evaluating the quality parameters. The developed Py-
thon software is explained in detail in Section 2.4.
2.4. Processing and evaluating the data
In the next step, the sections are exported into data les and are
analysed using a custom algorithm developed in Python. The program
identies the hole and teeth and separates the two sets of data. It then
calculates the number of teeth, reference circle, and gear module. The
orientation of the gear is important. Fig. 5 shows how the gear teeth are
oriented and marked in the software. The sections in the software are
processed according to the presented coordinate system in Fig. 5. The
same coordinate system is used in subsequent Figures. The report is
exported accordingly, with the teeth and gaps being numbered from 1 to
Z. The left and right anks are evaluated separately. The gaps are
numbered from 1 to Z. The gear quality parameters are determined
according to the measurements performed on the tooth and lead pro-
les, which are the lines marked in red in Fig. 5. First, the determination
of the individual parameters is presented and then the entire process is
illustrated in Fig. 11.
2.4.1. Geometrical quality parameters
The standardized quality parameters are determined using the
measurement data. In the software, the user can select which parameters
to control and which results and graphs to output.
Pitch deviation
The rst controlled parameter is the single pitch deviation fpt. It is the
difference between the actual and theoretical arc distance between two
neighbouring teeth anks. It is measured on the reference circle in the
middle of the lead prole. Next is the sector pitch deviation Fpk which is
calculated through the sub-sequential summation of the single pitch
deviations fpt . Finally, the cumulative pitch deviation Fp denes the
widest range between the sector pitch deviations Fpk .
The points on the anks are determined by performing linear inter-
polation between the two closest points to the reference circle. This was
done to nd the points for pitch control, which lie between the measured
points and on the reference circle. Therefore, the program solves a set of
two equations for each ank to nd their intersection. It solves for the
intersection between the linear function and the reference circle. This
ensures that the resulting points have a radius of d/2, which is demon-
strated in Fig. 6. From the resulting points, the arc lengths are deter-
mined and compared with the theoretical arc distances, which are
calculated with Eq. (1).
l=d
2⋅γ=d
2⋅2⋅
π
Z(1)
Prole deviation
Prole deviation control is evaluated with the differences between
the actual and theoretical shape of the tooth prole. The deviation is
calculated on the evaluation length, which represents 92% of the active
length of the tooth ank, as displayed in Fig. 5. The remaining 8% is the
area of the possible tip corrections, which signicantly inuences the
measurement. The prole deviation of the ank F
α
, the prole form
deviation ff
α
, and the prole slope deviation fH
α
are determined here. To
calculate the differences between the actual and theoretical shape, the
points on the base circle are required. They are calculated using the
measured values with Eqs. (2) and (3).
yT=r2
b⋅ym+rb⋅xm⋅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
x2
m+y2
m−r2
b
√
x2
m+y2
m
(2)
xT=r2
b−ym⋅yT
xm
(3)
The points denoted as T form a tangent to the base circle when con-
nected to the measured points. From those points, the angle ϕ can be
calculated with Eq. (4). The points on the theoretical involute can be
Fig. 2. Scanning setup.
U. Urbas et al.

Measurement 169 (2021) 108584
5
determined with Eqs. (5) and (6).
ϕ=arccos xT
rb
(4)
xA=rb⋅cosϕ+rb⋅ϕ⋅sinϕ(5)
yA=rb⋅sinϕ−rb⋅ϕ⋅cosϕ(6)
The values are shown in Fig. 7. The measured values and theoretical
points are shown with a signicant deviation for a better presentation.
With the points located on the theoretical involute and the base circle, it
is possible to calculate the differences (Δ) between the theoretical
involute and the measured values and then evaluate them according to
ISO 1328.
The differences between the points can be plotted on a graph, as
shown in Fig. 8. The parameter F
α
is determined as the difference be-
tween the smallest and the biggest deviation. To determine the param-
eters ff
α
and fH
α
, a function is tted on the data with the least-squares
method.
Lead prole control
The lead prole deviation represents the differences between the
actual and theoretical lead proles. Typically, it is evaluated somewhere
in the middle of the tooth height as demonstrated in Fig. 5. The evalu-
ation length is smaller than the tooth width. As a result, any llets or
chamfers present on the side edges are excluded from the measurement.
Here, the deviations of the lead prole Fβ, the lead prole form the
deviation ffβ, and the lead prole slope deviations fHβ are determined.
In the prole and lead prole deviation, the left and right prole of
each tooth are evaluated separately. The user of the program can select
which tooth prole to display and which graphs to output. Fig. 9 shows
how the distances for the lead prole control are determined. The lead
prole data are determined by sectioning the gear tooth with the
reference circle. By doing this, an array of points with varying x,y-co-
ordinates is created along the z axis; hence, they are effectively trans-
formed to the x-y plane. The gear orientation is shown in Fig. 5. Next, the
point with the maximum angle
ψ
from the origin is determined. That
point is then taken for the base and the distances to the other points are
calculated with the Eq. (7). The results are evaluated according to ISO
1328. The parameters can be determined analogously to the prole
deviation by using the calculated distances Δ.
Δ=̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Δx2+Δy2
√(7)
Run-out evaluation
The control of the run-out evaluates the position of the teeth anks
relative to the gear axis. The deviation from the theoretical position can
Fig. 3. STL and CAD alignment. Planar and cylindrical section on the STL.
Fig. 4. Function diagram of measuring and preparing the data for later software processing.
Fig. 5. Gear nomenclature and gear orientation.
U. Urbas et al.

Measurement 169 (2021) 108584
6
occur because of the gear eccentricity. The Fr parameter is dened as the
absolute difference between the biggest and smallest radial displace-
ment of the measuring body relative to the gear axis. For an optimal
comparison to the CMM measurements, the run-out evaluation is done
with a simulated probing ball. First, the program calculates the appro-
priate radius of the probing ball. It considers the gear properties that are
listed in Table 1. It needs to touch the anks tangentially on the refer-
ence circle, as demonstrated in Fig. 10 with the blue dots. For the tested
gears, the probing ball radius is 0.9 mm. The centre of the probing ball is
determined by offsetting the proles for the determined radius (the
offset is shown in the red colour). The centre is where the anks meet.
The values for the calculation of the run-out are calculated as the dis-
tances (shown in Fig. 10) between the gear axis and the probing ball
centre.
Diagram of the software for the gear quality characterisation
Fig. 11 describes the entire process of determining the gear quality
parameters and the quality grades from the sections.
2.4.2. Quality grades
According to the standard ISO 1328, the limit of the parameters are
determined in conformity to the selected quality grade Q and the typical
gear properties such as the reference circle diameter, normal module,
and gear width. The parameters are listed in Table 1. For most of the
parameters, the standard ISO 1328 determines the equations for calcu-
lating the permissible deviations for the quality grade Q =5. The
allowed deviations for the other quality grades are determined using the
geometrical pattern with a step of ̅̅̅
2
√. This means multiplying or
dividing with the step for each grade higher or lower than the previous
one, as shown in Eq. (8):
D(Q)=D(Q5)⋅20.5⋅(Q−5),(8)
where D(Q)is the permissible deviation for the selected quality grade.
The limit values of the parameters for the quality grades Q7 through Q12
are shown in Table 2. The values are determined by ISO and DIN stan-
dards, respectively [3,4]. It can be seen that the values for the ISO
quality grades are more demanding.
3. Results and discussion
To validate the method, the results of the tests on the simulated ideal
scan and on the simulated scan with synthetic deviations are rst pre-
sented. Subsequently, the validation of the manufactured gears is
presented.
Fig. 6. Determining the pitch deviation values.
Fig. 7. Determining the distance Δ that is used for the prole control.
Fig. 8. Determining the parameters for the prole deviation from the calculated distances.
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Measurement 169 (2021) 108584
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3.1. Ideal shape STL
To evaluate how much the tessellated geometry of the mesh in-
uences the results, the rst analysis was conducted on two simulated
scans based on the ideal shape. The deviations to the CAD model,
summarized in Table 3, are present because the mesh comprises a
limited number of points.
Only the maximum value for each parameter is displayed. There are
small deviations present, which can be accounted to the STL being
tessellated and represented with points or triangles. The deviations on
the sparse mesh (approximately 2⋅105 triangle faces), which is similar in
resolution to the scans of the manufactured gears, lead to some change
in the quality grades, resulting in a deviation of 1.43
μ
m for the
parameter F
α
,right. The average triangle edge size of the sparse mesh was
0.13 mm, along with the edge size of the triangle faces on the measured
gears. In the case of the denser mesh, the deviations are all smaller than
1
μ
m. Only some of the prole deviations are somewhat larger. Other-
wise, the quality grades are all Q =0. It is evident that sparser meshes
can lead to worse results and the scanning resolution is an important
factor. The errors are a result of some points not being on the reference
circle or are on the symmetry plane. The points there are created with a
linear interpolation between the two nearest points or by determining
the points on the edges of the triangle faces. A better result could be
achieved by tting a higher-order polynomial (NURBS [47] and a line t
[48] also possible) over the points, but that would introduce data that
was not measured. The denser mesh consisted of approximately 4⋅105
triangles, which results in smaller deviations for the parameters. The
size of the triangle face edges was 0.09 mm on the dense mesh. Tests
with even more dense meshes were made and the deviations were in
general found to get smaller by increasing the resolution. Tests on gears
with varying numbers of teeth, modules, and widths were also
performed to ensure that the method can be applied on different gear
geometries.
3.2. Validation with synthetic lead prole deviation
Testing the simulated scans with synthetic deviations was the rst
step to validate the method. Six different models with lead prole de-
viations were created. They ranged from Q7 to Q12 according to ISO
1328. The edge cases for quality grade 7 and for quality grade 12 are
presented. A maximum lead deviation of 12
μ
m for Q7 is shown in
Fig. 12a. The drop to zero deviations was done with a linear function. It
is expected that the program will determine the Fβ deviation to be 12
μ
m, as demonstrated in Fig. 13a.
Fig. 12b shows a deviation of 64
μ
m. The result is the expected value
for Q12 and this is demonstrated in Fig. 13b. Because of the linear
decrease in the deviation, the lead prole form deviation ffβ is near-zero
in both cases, and the lead prole slope deviation fHβ has the same value
as Fβ.
3.3. Validation with synthetic pitch deviation
The next test was on the pitch deviation fpt. Six models were created
that have a quality grade ranging from Q7 to Q12 according to ISO 1328.
The case is presented for quality grades 7 and 12. The collected de-
viations for each model are shown in Fig. 14a and b. The displayed
values are determined on the middle of the line and are made by a
section of the tooth ank with the reference circle. The results of the
evaluation software are shown in Fig. 15a and b. Only the rst and the
last tooth have synthetic deviations. It can also be seen that the rst
tooth has a slightly bigger deviation as the last tooth.
Fig. 9. Determining the distances for the lead prole control.
Fig. 10. The calculation of the run-out control.
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Measurement 169 (2021) 108584
8
3.4. Validation on manufactured gears
Six spur gears, with parameters described in Table 1, were measured.
They were scanned with the ATOS Compact SCAN and evaluated with
the described methodology. The samples were also measured for vali-
dation with a CMM Wenzel LH 54 [49], a dedicated gear CMM, incor-
porating a precise positioning table and a proprietary software. The
machine is certied according to SIST EN ISO/IEC 17025. The output of
the software are the parameters that are described in chapter 2.4.1.
Fig. 16 presents the results for the rst gears that were made of Tecamid.
It shows the determined DIN 3962 quality grades with the CMM method
and the scan for the four teeth and the whole gear. The scan for the four
teeth takes into account only the same four teeth that were evaluated
with the CMM. On the x-axis of the gure, the determined parameters on
both anks of the tooth (denoted as right/R and left/L) are presented.
The pitch and run-out deviation are done on all of the teeth. A good
agreement between the methods was found when determining the
quality grade. Evaluating the whole gear always returns the same or a
worse quality grade. This is because it is determined by the tooth with
the worst quality.
3.4.1. Repeatability of scanning
A repeatability study of scanning was done on the rst gear made of
Tecamid 66. The whole process of measuring the same gear was
repeated ve times. Each time, the gear was recoated, with a new set of
reference points being applied and the scanner being recalibrated again.
Fig. 11. Diagram of the methodology and software for the gear quality characterisation.
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Measurement 169 (2021) 108584
9
The scans were then aligned to one another and a comparison of de-
viations was done with each measurement. Fig. 17 shows part of the
deviations between the rst and the fth scan. From the results, the
uncertainty of the repeatability can be determined. The repeatability
includes the calibration of the scanner, coating the gears with the
powder, applying reference points, placing them on the turntable, and
then aligning them. The points near the root of the teeth were excluded
from the calculation of the uncertainty because they are not used for the
calculation of the parameters, and the deviations there are the largest.
This is due to the lack of accessibility of scanning, which was also
observed in the work by Müller et al. [50].
The calculated uncertainty was determined by the standard devia-
tion [51] and the value was 3
μ
m. A normal distribution was assumed.
The parameters are determined from multiple points, all measured with
the calculated uncertainty. For the parameter F
α
, this means the
maximum and the minimum of the deviations from the theoretical
prole. The combined uncertainty [52,53] of the parameters can be
calculated with Eq. (9):
ucombined =̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
u2+u2
√=4.24
μ
m.(9)
The combined uncertainty can be multiplied with a coverage factor of
two to get the expanded uncertainty, which covers 95% of the normal
distribution [54]. The expanded uncertainty is thus U =8.5
μ
m.
In accordance with these ndings, the software was upgraded to
include the probability that the determined quality grade is correctly
determined. Fig. 18 shows how the DIN 3961 quality grade for the
parameter Fr is determined. The presented values are for the Tecamid 1
gear (values shown in Fig. 16). The black line represents the value and
uncertainty for the scanned value, and the blue line is for the CMM
value. The CMM measurement also has an uncertainty value of 2
μ
m
[49], and a combined uncertainty of 2.82
μ
m. The value for Fr for the
gear Tecamid 1 acquired by the CMM is on the limit of the quality grade.
As a result, there is a 47.2% chance that the CMM determined the wrong
quality grade.
It is important to discuss the possible sources of errors in the method.
Generally, the sources of errors can be divided into the equipment error
Table 2
Limit parameter values for the DIN and ISO standards.
ISO 1328 values [
μ
m] DIN 3961/3962 [
μ
m]
Parameters\Quality grades Q7 Q8 Q9 Q10 Q11 Q12 Q7 Q8 Q9 Q10 Q11 Q12
fpt 9.5 13 19 26 37 53 9 14 18 28 50 80
Fp 23 32 45 64 90 127 28 36 50 80 140 220
F
α
9 13 18 26 37 52 12 16 22 36 56 90
ff,
α
7 10 14 20 28 40 9 12 16 28 45 71
fH,
α
6 8.5 12 17 24 33 7 10 14 22 36 56
Fβ 12 17 24 35 49 69 13 18 28 45 71 110
ff,β 8.5 12 17 25 35 49 7 9 14 25 40 63
fH,β 8.5 12 17 25 35 49 11 16 25 36 56 90
Fr 18 25 36 51 72 102 20 28 40 56 80 110
Table 3
Results for the different resolution simulated scans of an ideal shape.
Deviation value [
μ
m] ISO Quality grade
Type Sparse
mesh ∼2⋅
105Δ
Dense
mesh ∼4⋅
105Δ
Sparse
mesh ∼2⋅
105Δ
Dense
mesh ∼4⋅
105Δ
fpt,max left 0.08 −0.04 0 0 Pitch
deviation
fpt,max right 0.31 0.13 0 0
Fp,left 0.09 0.06 0 0
Fp,right 0.33 0.17 0 0
F
α
,right 1.40 0.69 2 0 Prole
deviation
F
α
,left 0.51 0.32 0 0
ff,
α
right 1.43 0.69 3 0
ff,
α
left 0.52 0.30 0 0
fH,
α
right 0.30 0.29 0 0
fH,
α
left 0.13 0.15 0 0
Fβ,right 0.83 0.31 0 0 Lead prole
deviation
Fβ,left 0.63 0.18 0 0
ff,βright 0.83 0.31 1 0
ff,βleft 0.63 0.18 0 0
fH,βright 0.09 0.02 0 0
fH,βleft 0.02 0.01 0 0
Fr 0.20 0.11 0 0 Runout
deviation
Fig. 12. Evaluation of the two different synthetic lead prole deviations.
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Measurement 169 (2021) 108584
10
(e.g. measuring instrument, calibration, xing the elements), operator
error (e.g. knowledge, training), measuring piece error (e.g. dust coating
layer thickness [55]), and environmental error (e.g. temperature, hu-
midity, lighting conditions).
In the case of the investigated method, the gears require a coated
surface for scanning, which already causes some deviations from the
actual geometry. With a test, it was determined that the coating powder
layer thickness is approximately 2
μ
m when applied from a spray gun
and 5
μ
m when applied from a spray can. However, that does not in-
uence the parameters because the deviations are cancelled out when
they are computed if the spray is evenly applied on the whole gear.
Then, there is the error of the alignment and software evaluation of the
measurement. By testing it on the ideal shape, it was determined that the
software evaluation does not cause big errors. These are mainly present
because of the tessellation on the STL. The angles of scanning and
reectivity also have important roles in the quality of the measurement.
The scanning device has limited accuracy and uncertainty when
scanning.
CMMs also have a certain accuracy and the angle of measurement is
important when measuring small objects. The measuring probe can slip
Fig. 13. Results from the program for the lead prole deviation F
β.
Fig. 14. Two different pitch deviations.
Fig. 15. Results from the program for the synthetic pitch deviation f
pt.
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Measurement 169 (2021) 108584
11
some distance before stopping and taking a measurement. Turntables,
used in combination with some CMMs, can also have a major contri-
bution to the measurement uncertainty [56,57]. CMM measurements
often need measurement error compensation methods [58,59]. The size
of the probe also inuences the measurement. The uncertainty of the
gear CMM is 2
μ
m [49]. There are other tactile gear measurement sys-
tems, which are purposefully built for measuring gears, such as gear
measuring centres (GMC). These centres enable more accurate
measurements and the presented comparison of uncertainties cannot be
generalised to cover every tactile measurement system.
Based on the repeated measurements, the combined uncertainty was
determined to be 4.24
μ
m for the investigated method, and 2.82
μ
m for
the CMM measurement. In the case that the measured value is in the
middle of the tolerance eld for both methods, the probability for
determining the wrong quality grade is small. However, when
approaching the limit value of the quality grade, the probability of
determining the wrong grade rises faster for the investigated method
than the CMM method. For the limit value, both methods have a 50%
chance of determining the wrong quality grade. Fig. 19 shows the
probability of determining the wrong quality grade for the proposed
method and for the CMM. Fig. 19b shows how much the probability for
determining the wrong quality grade increases for the scanning method
as opposed to the CMM method when approaching the limit values of the
quality grades. The largest differences for the discussed methods occur
at 69% between the middle and the limit value of the quality grade.
3.4.2. Importance of holistic measurement
Based on the insights gathered from the presented measurements,
the importance of the geometric evaluation of the whole gear is evident.
It is possible to miss the worst areas when evaluating only certain sec-
tions. Fig. 20 shows the deviations on the fourth tooth of the rst gear
made from the Tecaform material. Fig. 20 presents three prole sections
on the tooth. The middle one is the typical path that the CMM measures.
A big defect can be seen on the left prole section, which dramatically
increases the value of the quality parameters. The prole section on the
right has small deviation values.
Fig. 21 shows the evaluated prole deviations on multiple prole
sections for the tooth. The results are used to calculate the parameter F
α
,
described in chapter 2.4.1. The results are F
α
,left =42.5
μ
m Q =12;
F
α
,middle =17.8
μ
m Q =9; F
α
,right =4.9
μ
m Q =6. The quality grades are
evaluated according to ISO 1328. It is evident that it is important to
characterise the whole tooth. The results can vary greatly depending on
the location of the measurement. All of the described standard
geometrical parameters can be determined on the whole width of the
gear and collectively present an improved and more comprehensive
quality report. The characterisation of the whole gear with tactile
measurements would require a prohibitive amount of time to perform,
but it is practically feasible and straightforward with optical methods.
Fig. 16. Quality grades determined by CMM, scan on four teeth and with the whole scan.
Fig. 17. Deviations for one measurement to determine the repeatability
of scanning.
Fig. 18. DIN 3961 quality grades for the parameter F
r determined with scan-
ning and the CMM.
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Measurement 169 (2021) 108584
12
4. Conclusion
An optical gear inspection method is presented and evaluated in this
paper. A custom software was developed for processing and analysing
the 3D scan measurements of the (polymer) spur gears. The software can
determine the common geometrical quality parameters. The results from
the developed method and the CMM measurements were compared and
evaluated. From the validation on the manufactured gears, this method
can be adequately accurate for a fast evaluation of the whole gear. The
developed methodology presents an advantage compared to tactile
measurements when used in polymer gear evaluation. Current gear in-
spection standards were developed for steel gears which are manufac-
tured by cutting. Polymer gears are mass-produced with an injection
moulding process. There a holistic evaluation approach is necessary as
the shrinkage and warpage can cause deviations along the whole gear.
Highlights of the research:
•A systematic development of the methodology for optical gear in-
spection with a custom approach for aligning and pre-processing the
measured data.
Fig. 19. Probability of determining the wrong quality grade for the investigated method and for the CMM method.
Fig. 20. Deviation on the whole ank of the tooth.
Fig. 21. Deviations on multiple sections (from Fig. 20). Determination of parameter. F
α
.
U. Urbas et al.

Measurement 169 (2021) 108584
13
•A custom software was developed for processing and analysing the
scan measurements. A novel approach for determining the parame-
ters for lead prole deviation and runout deviation was proposed and
used.
•The quality grades, which are the result of the presented method and
the CMM measurement, are in good agreement.
•The random measurement uncertainty for the scanning process was
obtained, which was used in calculating the probability of deter-
mining the correct quality grade. The closer the result is to the edge
of the quality grade, the greater the chance of it being falsely
determined. The developed methodology is in the worst case 9.6%
more likely to determine the wrong quality grade compared to the
CMM.
•The optical method evaluates the whole gear, which makes missing
important defects less likely. It was determined that the methodology
is suitable for evaluating injection moulded gears.
Future work
The investigated method can utilise the fast acquisition of points by
thoroughly scanning and it can analyse the deviations for all of the teeth
on multiple lines and sections. This can be used for lowering the un-
certainty of the determined parameters. By using multiple sections near
one another, each section can be evaluated and then the determined
parameter values from each section can be averaged to decrease the
uncertainty. Further work may include a reference measurement, with
minimal uncertainty. This would enable the calculation of the system-
atic uncertainty.
The investigated methodology is effective for evaluating polymer
gears that are made by injection moulding. The quality of the gears can
be accurately determined during manufacturing. However, the current
quality parameters do not include and dene how much the resulting
part has contracted after the moulding process. In terms of future
research, a new parameter needs to be included that also considers the
total deviation of the resulting part and not just the relative differences.
The new parameter and its implementation in the software could
consider some characteristic points on the gear, such as the root circle
diameter, and it can be compared to the theoretical value to determine
the shrinkage. Besides evaluating physical parts, this method is also
useful in evaluating the results of a plastic injection moulding simula-
tion. The result of the simulation can be exported to an STL le and the
methodology can be used to determine the quality parameters.
The evaluation of the involute prole according to the theoretical
prole is an advantage. This is because it enables a universal evaluation
of the geometry without any 3D CAD models. This can only be valid if
the program has the necessary theoretical shape. For non-involute ge-
ometries, by performing a comparison with a CAD model, which has the
prescribed theoretical prole shape, this is easier to achieve. Further
work includes the evaluation of the different types of gears, such as
helical, conical, and non-involute gears.
The alignment of the measured data to the ideal data was done with
an initial prealignment and later with an alignment by geometrical el-
ements. Two of the geometrical elements were aligned with the least
squares method. However, the rotation was locked point-wise, which
could be optimised by determining the tangential position of the gear
anks which returns the smallest collective error between the CAD
model and 3D scan on the whole gear.
CRediT authorship contribution statement
Uroˇ
s Urbas: Methodology, Software, Validation, Formal analysis,
Investigation, Data curation, Writing - original draft, Writing - review &
editing, Visualization. Damijan Zorko: Conceptualization, Methodol-
ogy, Validation, Investigation, Data curation, Writing - review & editing.
Borut ˇ
Cerne: Methodology, Validation, Investigation, Writing - review
& editing, Visualization. Joˇ
ze Tavˇ
Car: Conceptualization, Methodol-
ogy, Writing - review & editing, Supervision. Nikola Vukaˇ
sinovi´
c:
Conceptualization, Methodology, Writing - review & editing,
Supervision.
Declaration of Competing Interest
None.
Declaration of Competing Interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Acknowledgement
This research was nanced partly by the MAPgears project (the
project is co-nanced by the Republic of Slovenia and the European
Union under the European Regional Development Fund, contract No.
C3330-18–952014) and partly by the Slovenian Research Agency (MR
No. 51899). The authors would like to thank the companies Podkriˇ
znik
d.o.o. and Tecos for their support with manufacturing and measuring the
gears.
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