a method for enhanced polymer spur gear inspection based

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

Uroš Urbas, Damijan Zorko, Borut Černe, Jože TavČar, Nikola Vukašinović

PII: S0263-2241(20)31105-2DOI: https://doi.org/10.1016/j.measurement.2020.108584Reference: MEASUR 108584

To appear in: Measurement

Received Date: 29 April 2020Revised Date: 18 August 2020Accepted Date: 4 October 2020

Please cite this article as: U. Urbas, D. Zorko, B. Černe, J. TavČar, N. Vukašinović, A method for enhancedpolymer spur gear inspection based on 3D optical metrology, Measurement (2020), doi: https://doi.org/10.1016/j.measurement.2020.108584

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https://doi.org/10.1016/j.measurement.2020.108584



A method for enhanced polymer spur gear inspection based on 3Doptical metrologyUroš Urbasa,∗, Damijan Zorkoa, Borut Černea, Jože Tavčara and Nikola Vukašinovića,∗∗

aUniversity of Ljubljana, Faculty of Mechanical Engineering, LECAD, Aškerčeva cesta 6, 1000 Ljubljana, Slovenia

ART ICLE INFOKeywords:gearsoptical inspectiongeometrical parameters3D scanningstructured lightquality

ABSTRACTAccurately manufactured gears require a reliable, holistic, and fast inspection method. Stan-dardised 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 lo-cations. Therefore, a method to obtain holistic three-dimensional (3D) measurements with anoptical inspection was thoroughly investigated. The measurement data were acquired via 3Doptical 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 andsubsequently on a simulated scan with synthetic deviations. The method was finally validatedby measuring the gears using a coordinate-measuring machine; the results obtained were com-pared with those obtained using the developed optical method. A good agreement between themethods was observed. The optical method offers a more holistic measurement approach withmany important advantages being identified compared with the tactile method.

1. IntroductionGears are crucial elements for motion and power transmission in the field of robotics, transport vehicles, and

machines. Increasing the demand for accurately manufactured gears also requires a more efficient and accurate dimen-sional inspection. Currently, inspecting gears is generally done with tactile measuring machines. The process involvesmeasuring only one line feature across the middle of the tooth on a limited number of teeth for the gear [1]. Themeasurements on the gear flanks are limited to these predefined two-dimensional lines [1], even though the completegeometry of a gear flank determines its functional properties. The areal measurement measures multiple points. If theyare measured with traditional tactile methods, the process will take too much time. An optical method accelerates themeasurements 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 information regarding themost crucial types of deviations typically identified on as-produced gears. This study aims to develop a holistic spurgear inspection method that uses optical 3D areal data, acquired with a structured light scanner. A systematic develop-ment of the methodology, including a validation procedure, using on a tactile gear measurement machine, is presented.The importance of holistic measurements is discussed and the measurement uncertainty is examined. The developedmethod determines the geometrical parameters determining the gear’s quality grade. The findings resulted in a customsoftware for evaluating the 3D scan measurement data.1.1. Gear geometrical parameters

Different parameters define a gear’s quality, such as the type and quality of the material, the quality of the man-ufacturing, and the quality of the heat treatment. This study focuses on the geometrical parameters. For the gears tohave good functionality and an acceptable influence on the surroundings (e.g. noise, vibration), the geometry needsto be accurately manufactured. This is why certain parameters need to be measured, calculated, and evaluated aftermanufacturing and before the final assembly, according to the available standards [3–5]. They include the process of

∗Corresponding author∗∗Principal corresponding author

[email protected] (U. Urbas); [email protected] (N. Vukašinović)ORCID(s): 0000-0002-5749-5396 (U. Urbas)

Uro² Urbas et al.: Preprint submitted to Elsevier Page 1 of 21

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

obtaining the parameter values and defining their ranges for certain quality grades. Summaries of the standards andthe basics of quality control for gears are available in the following literature [6, 7].

Some of the geometrical quality parameters include the pitch control, profile deviation, lead profile deviation,runout evaluation, and tooth thickness. There are others, which control the gear-pair, such as the axial distance. Otherparameters also evaluate the gear body, such as the dimensional and geometric tolerances. According to ISO 1328, thelimit parameter values for the control of the gears are divided into 13 quality grades (Q = 0, 1, 2, . . . , 12) [3]. Grade 0means the highest accuracy and it has tight tolerances, whereas grade 12 is the least accurate. A better quality of gearsleads to improvements in several fields, such as a smaller transmission error and smaller force impulses [8], whichleads to less heat generation [9–11]. This will result in a decrease in the acceleration and dynamic loads, which cancause impact and noise. However, even accurately manufactured gears deform under the load [8, 12] and they do notperform perfectly because of other errors [13–15].1.2. Current research

Some studies have reported the benefits of a holistic measurement of the gears.Kang Ni et al. [16] focused on the characterization and evaluation of the involute gear flank. It was concluded

that the optical method is suitable for larger gears, in which longer times are needed to measure the profile. It is alsonecessary to measure many profiles to fully evaluate the gear if the measurement is done with coordinate measuringmachines (CMMs). The plumb line distance was used to measure the deviation from the theoretical profile. The plumbline distance is the absolute value of the vector pointing from the nominal point to the measured point for the surface’snormal direction. They also introduce Chebyshev polynomials for the flank modifications.

Gert Goch et al. [17] stated that for current inspections, only one profile and one lead profile line are measured,and this is only done for four teeth. For the pitch control, one point in the middle of the lead profile is taken. A similarstrategy is reported in their previous article [16]. The focus is on measuring the deviation with the plumb line distance.They represent some improvements by using the Chebyshev polynomials and by adding an evaluation for the helicalgears. The advantage of this method is that it can eliminate the need for defining the nominal geometry.

MatthiasMarcus et al. [18] set out to determine if the laser line triangulation (LLT)measurement is suitable for gearmetrology. They stated that tactile measurement has stagnated and the optical measurement enables the measurementof large amounts of data in a short period of time. The focus is on the measurement error. A measurement on a big gearwith a 1 m diameter was made. A measuring machine scanCONTROL 2910-25 with a depth resolution of 2 µm wasused. The lateral resolution was 19.5 µm. The measured data with the measurements from a CMM were compared. Itwas concluded that the measurement points have a poorer quality in the root area of the tooth because of the reflections,depth of focus, occlusion, and the geometry, which disturbs the measurement. The results show that in comparison tothe CMMs, the LLTs have a deviation of 19.8 µm. They concluded that the method is sufficiently accurate for a fastevaluation of the gear surface, but it is not good enough to determine the geometrical quality parameters. For the samenumber 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 wasreconstructed from the phase information. For comparison, it was also measured with a CMM. The average deviationof the involute was 2.67 µm. The average error of the CMM is in the area of 3 µm; thus, it was concluded that the resultsare suitable. For additional validation, a Mitutoyo K cube was measured and a deviation of 2.67 µm was determined.

A study by Frank Härtig et. al. [20] aimed to develop a standardised environment, nomenclature, and orientationfor the 3D involute gear evaluation. The coordinate system and the basic equations in the Cartesian and involutecoordinate system are presented.

Vit Zelený et al. [21] described the involute profile shape, including the helical gears. The deviations are calculatedon one section, which is suitable for the spur gears. In addition to the profile calculations, the lead profile evaluationis also included. The evaluation is done according to the standards. For evaluation, a CAD model of the gear toothprofile was created. To eliminate the uncertainty of the CMM, a simulated measurement with the GEAR PRO programwas done. The data were generated on the surface with a random error of ± 10 µm. After comparing their method tothe 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 sensorand an air floating rotary table to rapidly measure the shape of the gear tooth flank. The measured 3D cloud was usedto calculate the profile and pitch error. It was determined to be a fast and accurate evaluation of the gear’s quality.With multiple profile lines available, a different approach for evaluating the spatial profile quality was proposed. Thetotal profile deviation was calculated by the maximum and minimum values of all of the profile deviation values. The



measuring equipment was only able to obtain data for only one side of the flank during the experiment.This investigation presents a systematic approach to develop a methodology for gear quality evaluation based on

the optical measurements, 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; how-ever, they are slow. With the increasing accuracy of the non-contact methods, which offer a fast point acquisition, theyare increasingly being used, which also includes roughness measurements [25, 26]. There are also different opticalmethods available for acquiring a point cloud with non-contact measurements. 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 alimited accuracy [31–33], which is dependent on the quality of the surface, and the angle of inclination [34]. It is themost used method for on-line contactless measurement; however, it is more frequently used for bigger objects, whichrequires too much time to measure. In conoscopic holography, the laser beam is projected onto a surface [35]. The re-flection passes through a conoscopic crystal and it is then projected onto a charge-coupled device. A diffraction patternappears, which can be reconstructed to determine the distance to the surface. The main advantage is that only one beampath is needed; therefore, it is possible to measure the depth of a narrow hole. Other types of measurements includestereoscopic 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 inspectionThe pitch division, profile and lead profile control, runout evaluation, and tolerance measurements are typically

measured with a computer numerical control machine (CNC) or a CMM. However, such measurements are slow and donot consider the whole gear. On the other hand, these methods enable better stability and repeatability of the qualityparameters. An areal measurement, which is enabled by the optical methods, is preferred in future gear metrologybecause it enables a fast and holistic data acquisition.

The methodology was developed by evaluating the results on synthetic scans and measurements, as presented inFigure 1. The first step is to evaluate the quality of a simulated scan of the gear with no deviations present. The purposeof this step is to determine the required density of the point cloud and the resulting mesh. Later, the synthetic devia-tions are evaluated to ensure the right calculation of the separate geometrical parameters. Finally, the methodology isevaluated on the manufactured gears by the gear CMM and scanning measurements and it is validated on real data todetermine the measurement uncertainty.

2.1 Evaluation on simulated measured meshes w ith di f ferent approximations of ideal shape

2.2 Validation on synthetic

deviations

2.3 Validation on the manufactured

gears

Validation

2.4 Processing and evaluating

the data

Figure 1: Steps in the development of the methodology.

2.1. Evaluation on simulated measured meshes with different approximations of ideal shapeThe 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 wasdone in Geomagic Studio 2014. The scanning parameters were set in a way to obtain a similar and a higher numberof points that are the result of a scan. Two simulated scans with different resolutions were made. The sparser meshhas a similar number of points and triangles to the meshes due to the scanning on the actual gear. The mesh consistedof approximately 2 ⋅ 105 triangles. The second evaluated mesh with an ideal geometry was denser and it was createdfrom approximately 4 ⋅ 105 triangles. The effect of the different polygonisations of the meshes was also observed inthe work of Müller et al. [41].Uro² Urbas et al.: Preprint submitted to Elsevier Page 3 of 21


2.2. Validation on synthetic deviationsTo validate the lead profile 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 Q7to Q12 were created. The meshes had the approximate resolution of the scans for the manufactured gears. The goalwas to validate the methodology from the stage of acquiring a point cloud onwards.2.3. Validation on the manufactured gears2.3.1. Sample preparation

The measured gears were made of polymers, which is the focus of the department [42, 43] and the MAPgearsproject. 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 accuracythan 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) andTECAMID 66 natural (PA66) were used. The hobbing tool quality grade was AA according to DIN 3968. Thegeometrical parameters of the manufactured gears are presented in Table 1. Polymer gears made with the hobbingprocess can normally achieve better quality grades than gears made with injection moulding.

Table 1Tested 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

2.3.2. Process of measuring and preprocessingA method with structured light was employed for the optical measurements of the manufactured gears. The ATOS

Compact SCAN 5M scanner with a stated laboratory accuracy of approximately 2 µmwas used. Because the scanningconditions are not ideal, a lower accuracy can be expected. The measuring accuracy is dependent on the size of theobject. 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 objectconfiguration where the measuring volume is 70x50x50 mm, the measuring distance is 420 mm and the resultingmeasuring point distance is 0.029 mm. In the small object configuration the pixels are more densely spread out on thesurface. It is also possible to measure objects that are larger than the measured volume. The manufacturer guaranteesthe stated accuracy for measuring objects that are not three times longer in each dimension. The whole picture isthen put together with the help of markers on the object. Theoretically, the camera only needs three points to put thedata 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 determinationof the 3D coordinates of the details on the surface. The scanner has a stereo camera so it can control itself and discardthe bad measurements. It collects the measurements only if the received data on both cameras are in agreement. Beforeeach measurement, the scanner was calibrated with a standard calibration panel CP40, at room conditions; temperature23 ± 1 °C and relative humidity 40 ± 2 %. The scanning setup is shown in Figure 2.

To measure the gears and objects in general, it is often necessary to coat them with an anti-reflecting powder. Thepowder can be applied with a spray gun to obtain thin layers. Powders based on titanium dioxide sublimate after sometime; hence, they do not require cleaning the object. The scan is then stored into an STL file, which is then importedinto GOM Inspect 2018 [46], which is a program capable of manipulating 3D meshes. First, the STL is pre-alignedwith the CAD model with a global best fit method. After that, they are aligned with geometric elements. The gearhas six degrees of freedom in its free state. Holes from the geometry and the CAD file are aligned first since it is themost important alignment due to the mounting of the gears. The hole alignment is done with fitting a cylinder on bothobjects and aligning them. This leaves two degrees of freedom, the translation and the rotation relative to the axis of



ATOS Compact 5MTurntable

Reference points

Measured gear

Figure 2: Scanning setup.

the hole. The translation is then fixed. Two planes are created on the sides of the gear and this results in a symmetricalplane. The symmetrical plane is aligned with the one from the CAD. The rotation is fixed with the created points onthe mesh. The alignment elements are shown in Figure 3, and they are done with GOM Inspect software[46].

rotation lock

symmety plane

planar section

hole alignment

cylindricalsection

Figure 3: STL and CAD alignment. Planar and cylindrical section on the STL.

In the next step, the required number of sections is specified. A planar section through the middle of the gear and acylindrical section on the reference circle is made to obtain the results with the traditional method. The measured gearshave a reference circle diameter of 20 mm. The sections are shown in Figure 3. In the next iterations, sections couldbe done on multiple planes; hence, the parameters are determined on the whole gear width. By doing this, the helicalgears can be evaluated. The software allows the user to choose which parameters to evaluate and then it generatesa report. Figure 4 shows the general process of measuring, processing, and evaluating the quality parameters. Thedeveloped Python software is explained in detail in Section 2.4.



Scanning Cut into sections

process dataSTL

3D CAD

modelspur gear

3D scanner

planar and cyl indr ical sections

ini tial al ignment

XYZ data

number of sections

deviation and excel r epor t

geometr ical al ignmentpowder coating

developed Python software

output parameter s

Figure 4: Function diagram of measuring and preparing the data for later software processing.

2.4. Processing and evaluating the dataIn the next step, the sections are exported into data files and are analysed using a custom algorithm developed in

Python. The program identifies the hole and teeth and separates the two sets of data. It then calculates the number ofteeth, reference circle, and gear module. The orientation of the gear is important. Figure 5 shows how the gear teeth areoriented and marked in the software. The sections in the software are processed according to the presented coordinatesystem in Figure 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 flanks are evaluated separately. The gapsare numbered from 1 to Z. The gear quality parameters are determined according to the measurements performed onthe tooth and lead profiles, which are the lines marked in red in Figure 5. First, the determination of the individualparameters is presented and then the entire process is illustrated in Figure 11.

1

2

3

4

Z

Z-1

1

23

gap

tooth

left

right

flank

tip

root

tooth profilelead profile

reference circle

92 % of toothprofile used for evaluation

x

y

z

Figure 5: Gear nomenclature and gear orientation.

2.4.1. Geometrical quality parametersThe 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 deviationThe first controlled parameter is the single pitch deviation fpt. It is the difference between the actual and theoreticalarc distance between two neighbouring teeth flanks. It is measured on the reference circle in the middle of the lead

profile. Next is the sector pitch deviation Fpk which is calculated through the sub-sequential summation of the singleUro² Urbas et al.: Preprint submitted to Elsevier Page 6 of 21


pitch deviations fpt. Finally, the cumulative pitch deviation Fp defines the widest range between the sector pitchdeviations Fpk.The points on the flanks are determined by performing linear interpolation between the two closest points to thereference circle. This was done to find the points for pitch control, which lie between the measured points and onthe reference circle. Therefore, the program solves a set of two equations for each flank to find their intersection. Itsolves for the intersection between the linear function and the reference circle. This ensures that the resulting pointshave a radius of d/2, which is demonstrated in Figure 6. From the resulting points, the arc lengths are determined andcompared with the theoretical arc distances, which are calculated with Equation 1.

γ

Reference circlePoints for pitch control

Reference circlePoints for pitch control

Figure 6: Determining the pitch deviation values.

l̄ = d2⋅ = d

2⋅2 ⋅ �Z

(1)Profile deviationProfile deviation control is evaluated with the differences between the actual and theoretical shape of the tooth

profile. The deviation is calculated on the evaluation length, which represents 92% of the active length of the toothflank, as displayed in Figure 5. The remaining 8% is the area of the possible tip corrections, which significantlyinfluences the measurement. The profile deviation of the flank F� , the profile form deviation ff� , and the profile slopedeviation fH� are determined here. To calculate the differences between the actual and theoretical shape, the pointson the base circle are required. They are calculated using the measured values with equations 2 and 3.

yT =r2b ⋅ ym + rb ⋅ xm ⋅

√

x2m + y2m − r2bx2m + y2m

(2)

xT =r2b − ym ⋅ yT

xm(3)

The points denoted as T form a tangent to the base circle when connected to the measured points. From those points,the angle � can be calculated with equation 4. The points on the theoretical involute can be determined with equations5 and 6.

� = arccosxTrb

(4)

xA = rb ⋅ cos� + rb ⋅ � ⋅ sin� (5)

yA = rb ⋅ sin� − rb ⋅ � ⋅ cos� (6)Uro² Urbas et al.: Preprint submitted to Elsevier Page 7 of 21


The values are shown in Figure 7. The measured values and theoretical points are shownwith a significant deviation fora better presentation. With the points located on the theoretical involute and the base circle, it is possible to calculatethe differences (Δ) between the theoretical involute and the measured values and then evaluate them according to ISO1328.

ϕΔ

Figure 7: Determining the distance Δ that is used for the pro�le control.

The differences between the points can be plotted on a graph, as shown in Figure 8. The parameter F� is determinedas the difference between the smallest and the biggest deviation. To determine the parameters ff� and fH� , a functionis fitted on the data with the least-squares method.

f f,α

f H,α

Fα

Figure 8: Determining the parameters for the pro�le deviation from the calculated distances.

Lead profile controlThe lead profile deviation represents the differences between the actual and theoretical lead profiles. Typically, it

is evaluated somewhere in the middle of the tooth height as demonstrated in Figure 5. The evaluation length is smallerthan the tooth width. As a result, any fillets or chamfers present on the side edges are excluded from the measurement.Here, the deviations of the lead profile F� , the lead profile form the deviation ff� , and the lead profile slope deviationsfH� are determined.

In the profile and lead profile deviation, the left and right profile of each tooth are evaluated separately. The userof the program can select which tooth profile to display and which graphs to output. Figure 9 shows how the distancesfor the lead profile control are determined. The lead profile data are determined by sectioning the gear tooth with thereference circle. By doing this, an array of points with varying x,y-coordinates is created along the z axis; hence,they are effectively transformed to the x-y plane. The gear orientation is shown in Figure 5. Next, the point with theUro² Urbas et al.: Preprint submitted to Elsevier Page 8 of 21


maximum angle from the origin is determined. That point is then taken for the base and the distances to the otherpoints are calculated with the equation 7. The results are evaluated according to ISO 1328. The parameters can bedetermined analogously to the profile deviation by using the calculated distances Δ.

Δx

Δy max Δ = F β

ψ

z

Reference circleTooth middle profileLead profile points along z

Figure 9: Determining the distances for the lead pro�le control.

Δ =√

Δx2 + Δy2 (7)Run-out evaluationThe control of the run-out evaluates the position of the teeth flanks relative to the gear axis. The deviation from the

theoretical position can occur because of the gear eccentricity. The Fr parameter is defined as the absolute differencebetween the biggest and smallest radial displacement of the measuring body relative to the gear axis. For an optimalcomparison to the CMMmeasurements, the run-out evaluation is donewith a simulated probing ball. First, the programcalculates the appropriate radius of the probing ball. It considers the gear properties that are listed in Table 1. It needsto touch the flanks tangentially on the reference circle, as demonstrated in Figure 10 with the blue dots. For the testedgears, the probing ball radius is 0.9 mm. The centre of the probing ball is determined by offsetting the profiles forthe determined radius (the offset is shown in the red colour). The centre is where the flanks meet. The values for thecalculation of the run-out are calculated as the distances (shown in Figure 10) between the gear axis and the probingball centre.

Δx

Δy

Reference circle

distance for evaluation

Figure 10: The calculation of the run-out control.

Diagram of the software for the gear quality characterisationFigure 11 describes the entire process of determining the gear quality parameters and the quality grades from the

sections.Uro² Urbas et al.: Preprint submitted to Elsevier Page 9 of 21


Determine prof i le evaluation length

Divide on left and r ight f lanks

Determine points on the base cir cle

which are tangential to

measured points

Plot data and expor t r epor t

Determine the angle to the point

on base cir cle

Determine theoretical

involute points

Identi fy the points on the

r eference cir cle

Separate left and r ight f lanks

Calculate theoretical pi tch

Determine actual distance and

calculate di f ference

Evaluate according to ISO 1328 and DIN 3961/3962

Star t

Determine the probing ball

r adius

Offset the sur face of gear by the probing ball

r adius

Determine center of probing ball on each pair of

f lanks

Determine distance to center

of gear

Read data from ci l inder section

Determine lead prof i le evaluation

length

Determine base point w i th

greatest angle

Calculate absolute distance from base point

to each point

Calculate distance between theoretical and

determined point

Read section data f i le

Calculate number of teeth, module,

r eference cir cle

Pi tch cont r ol Pr of i l e evaluat i on Lead pr of i l e cont r ol Runout evaluat i on

User defined number of sections, evaluation lengths, plots

Figure 11: Diagram of the methodology and software for the gear quality characterisation.

2.4.2. Quality gradesAccording 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 forcalculating 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 foreach grade higher or lower than the previous one, as shown in Equation 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 standards,respectively [3, 4]. It can be seen that the values for the ISO quality grades are more demanding.

Table 2Limit parameter values for the DIN and ISO standards.

ISO 1328 values [µm] DIN 3961/3962 [µm]

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� 9 13 18 26 37 52 12 16 22 36 56 90

ff,� 7 10 14 20 28 40 9 12 16 28 45 71

fH,� 6 8.5 12 17 24 33 7 10 14 22 36 56

F� 12 17 24 35 49 69 13 18 28 45 71 110

ff,� 8.5 12 17 25 35 49 7 9 14 25 40 63

fH,� 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

3. Results and DiscussionTo validate the method, the results of the tests on the simulated ideal scan and on the simulated scan with synthetic

deviations are first presented. Subsequently, the validation of the manufactured gears is presented.3.1. Ideal shape STL

To evaluate how much the tessellated geometry of the mesh influences the results, the first analysis was conductedon two simulated scans based on the ideal shape. The deviations to the CADmodel, summarized in Table 3, are presentbecause 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 ac-counted 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 tosome change in the quality grades, resulting in a deviation of 1.43 µm for the parameter F�,right. The average triangleedge size of the sparse mesh was 0.13 mm, along with the edge size of the triangle faces on the measured gears. In thecase of the denser mesh, the deviations are all smaller than 1 �m. Only some of the profile deviations are somewhatlarger. Otherwise, the quality grades are all Q=0. It is evident that sparser meshes can lead to worse results and thescanning resolution is an important factor. The errors are a result of some points not being on the reference circle orare on the symmetry plane. The points there are created with a linear interpolation between the two nearest points orby determining the points on the edges of the triangle faces. A better result could be achieved by fitting a higher-orderpolynomial (NURBS [47] and a line fit [48] also possible) over the points, but that would introduce data that wasnot measured. The denser mesh consisted of approximately 4 ⋅ 105 triangles, which results in smaller deviations forthe parameters. The size of the triangle face edges was 0.09 mm on the dense mesh. Tests with even more densemeshes were made and the deviations were in general found to get smaller by increasing the resolution. Tests on gearswith varying numbers of teeth, modules, and widths were also performed to ensure that the method can be applied ondifferent gear geometries.3.2. Validation with synthetic lead profile deviation

Testing the simulated scans with synthetic deviations was the first step to validate the method. Six different modelswith lead profile deviations were created. They ranged from Q7 to Q12 according to ISO 1328. The edge cases forUro² Urbas et al.: Preprint submitted to Elsevier Page 11 of 21


Table 3Results for the di�erent resolution simulated scans of an ideal shape.

Deviation value [µm] ISO Quality grade

TypeSparse 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 deviationfpt,max_right 0.31 0.13 0 0Fp,left 0.09 0.06 0 0Fp,right 0.33 0.17 0 0

F�,right 1.40 0.69 2 0

Pro�le deviation

F�,left 0.51 0.32 0 0ff,�_right 1.43 0.69 3 0ff,�_left 0.52 0.30 0 0fH,�_right 0.30 0.29 0 0fH,�_left 0.13 0.15 0 0

F�,right 0.83 0.31 0 0

Lead pro�ledeviation

F�,left 0.63 0.18 0 0ff,�_right 0.83 0.31 1 0ff,�_left 0.63 0.18 0 0fH,�_right 0.09 0.02 0 0fH,�_left 0.02 0.01 0 0

Fr 0.20 0.11 0 0 Runout deviation

quality grade 7 and for quality grade 12 are presented. A maximum lead deviation of 12 µm for Q7 is shown in Figure12a. The drop to zero deviations was done with a linear function. It is expected that the program will determine theF� deviation to be 12 µm, as demonstrated in Figure 13a.

(a) Quality grade Q7. (b) Quality grade Q12.Figure 12: Evaluation of the two di�erent synthetic lead pro�le deviations.

Figure 12b shows a deviation of 64 µm. The result is the expected value for Q12 and this is demonstrated in Figure13b. Because of the linear decrease in the deviation, the lead profile form deviation ff� is near-zero in both cases, andthe lead profile slope deviation fH� has the same value as F� .



Fβ

Fβ = 63.9 μm

Fβ

Fβ = 12 μm

(a) Quality grade Q7.

Fβ

Fβ = 63.9 μm

Fβ

Fβ = 12 μm

(b) Quality grade Q12.Figure 13: Results from the program for the lead pro�le deviation F� .

3.3. Validation with synthetic pitch deviationThe 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 deviations for eachmodel are shown in Figure 14a and Figure 14b. The displayed values are determined on the middle of the line andare made by a section of the tooth flank with the reference circle. The results of the evaluation software are shown inFigures 15a and 15b. Only the first and the last tooth have synthetic deviations. It can also be seen that the first toothhas a slightly bigger deviation as the last tooth.

(a) Quality grade Q7. (b) Quality grade Q12.Figure 14: Two di�erent pitch deviations.

3.4. Validation on manufactured gearsSix 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 validation with a CMMWenzel LH 54 [49], a dedicated gear CMM, incorporating a precise positioning table and a proprietary software. Themachine is certified according to SIST EN ISO/IEC 17025. The output of the software are the parameters that aredescribed in chapter 2.4.1. Figure 16 presents the results for the first gears that were made of Tecamid. It shows thedetermined DIN 3962 quality grades with the CMM method and the scan for the four teeth and the whole gear. Thescan for the four teeth takes into account only the same four teeth that were evaluated with the CMM. On the x-axisof the figure, the determined parameters on both flanks of the tooth (denoted as right/R and left/L) are presented. Thepitch and run-out deviation are done on all of the teeth. A good agreement between the methods was found whendetermining the quality grade. Evaluating the whole gear always returns the same or a worse quality grade. This is



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20N Division

10.0

7.5

5.0

2.5

0.0

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f pt [

m]

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(a) Quality grade Q7.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20N Division

40

20

0

20

40

f pt [

m]

-46.4

fpt deviation, left flank

(b) Quality grade Q12.Figure 15: Results from the program for the synthetic pitch deviation fpt.

because it is determined by the tooth with the worst quality.

2

3

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7

8

9

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11

12

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CMM SCAN 4 teeth SCAN all teeth

F α,ri

ght

f f,α,R

f H,α,R

F α,le

ft

f f,α,L

f H,α,L

F β,ri

ght

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f H,β,R

F β,le

ft

f f,β,L

f H,β,L

f p,righ

t

F p,righ

t

f p,left

F p,left

F r

Figure 16: Quality grades determined by CMM, scan on four teeth and with the whole scan.

3.4.1. Repeatability of scanningA repeatability study of scanning was done on the first gear made of Tecamid 66. The whole process of measuring

the same gear was repeated five times. Each time, the gear was recoated, with a new set of reference points being appliedand the scanner being recalibrated again. The scans were then aligned to one another and a comparison of deviationswas done with each measurement. Figure 17 shows part of the deviations between the first and the fifth scan. From theresults, 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 usedfor 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].O S P rofess ional 2019

6/11amid 66 Gom Inspec t P rofess ional 2019

skena 1 napram skenu 2,3,4,5

TECOS / digiCEN ; Kidričeva 25, SI-3000 Celje; Phone: 00386 3 42 64 608

only be reproduced completely. It may be partially reproduced only by written approvalTool and Die Development Centre, together with the quotation of the reference number

of the same written approval.

[mm]

-0.0100

-0.0080

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+0.0041

Length unit: mm

Figure 17: Deviations for one measurement to determine the repeatability of scanning.

The calculated uncertainty was determined by the standard deviation [51] and the value was 3 µm. A normaldistribution was assumed. The parameters are determined from multiple points, all measured with the calculateduncertainty. For the parameter F� , this means the maximum and the minimum of the deviations from the theoreticalprofile. The combined uncertainty [52, 53] of the parameters can be calculated with Equation 9:

ucombined =√

u2 + u2 = 4.24 µm. (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 µm.In accordance with these findings, the software was upgraded to include the probability that the determined quality

grade is correctly determined. Figure 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 Figure 16). The black line represents the valueand uncertainty for the scanned value, and the blue line is for the CMM value. The CMM measurement also hasan uncertainty value of 2 µm [49], and a combined uncertainty of 2.82 µm. The value for Fr for the gear Tecamid1 acquired by the CMM is on the limit of the quality grade. As a result, there is a 47.2 % chance that the CMMdetermined the wrong quality grade.

Q11Q10 Q12

CMMScan

66.5

0.7 %

47.2 %

80.2

65Fr [μm]

Pro

babi

lity

den

sity

[/]

0.10

0.05

Figure 18: DIN 3961 quality grades for the parameter Fr determined with scanning and the CMM.

It is important to discuss the possible sources of errors in the method. Generally, the sources of errors can bedivided into the equipment error (e.g. measuring instrument, calibration, fixing the elements), operator error (e.g.knowledge, training), measuring piece error (e.g. dust coating layer thickness [55]), and environmental error (e.g.temperature, humidity, lighting conditions).Uro² Urbas et al.: Preprint submitted to Elsevier Page 15 of 21


In the case of the investigated method, the gears require a coated surface for scanning, which already causes somedeviations from the actual geometry. With a test, it was determined that the coating powder layer thickness is approxi-mately 2 µmwhen applied from a spray gun and 5 µmwhen applied from a spray can. However, that does not influencethe parameters because the deviations are cancelled out when they are computed if the spray is evenly applied on thewhole gear. Then, there is the error of the alignment and software evaluation of the measurement. By testing it on theideal shape, it was determined that the software evaluation does not cause big errors. These are mainly present becauseof the tessellation on the STL. The angles of scanning and reflectivity also have important roles in the quality of themeasurement. 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. Themeasuring probe can slip some distance before stopping and taking a measurement. Turntables, used in combinationwith some CMMs, can also have a major contribution to the measurement uncertainty [56, 57]. CMM measurementsoften need measurement error compensation methods [58, 59]. The size of the probe also influences the measure-ment. The uncertainty of the gear CMM is 2 µm [49]. There are other tactile gear measurement systems, which arepurposefully built for measuring gears, such as gear measuring centres (GMC). These centres enable more accuratemeasurements and the presented comparison of uncertainties cannot be generalised to cover every tactile measurementsystem.

Based on the repeated measurements, the combined uncertainty was determined to be 4.24 µm for the investigatedmethod, and 2.82 µm for the CMMmeasurement. In the case that the measured value is in the middle of the tolerancefield for both methods, the probability for determining the wrong quality grade is small. However, when approachingthe limit value of the quality grade, the probability of determining the wrong grade rises faster for the investigatedmethod than the CMM method. For the limit value, both methods have a 50 % chance of determining the wrongquality grade. Figure 19 shows the probability of determining the wrong quality grade for the proposed method andfor the CMM. Figure 19b shows how much the probability for determining the wrong quality grade increases for thescanning method as opposed to the CMMmethod when approaching the limit values of the quality grades. The largestdifferences for the discussed methods occur at 69% between the middle and the limit value of the quality grade.

Q11Q10 Q12

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680.05

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]

(a) Increasing the probability of determining the wrongquality grade when approaching the limit value.

Q11Q10 Q12

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680.05

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]

(b) Difference in the probability of determining thewrong quality grade.

Figure 19: Probability of determining the wrong quality grade for the investigated method and for the CMM method.

3.4.2. Importance of holistic measurementBased 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 sections. Figure 20 showsthe deviations on the fourth tooth of the first gear made from the Tecaform material. Figure 20 presents three profilesections on the tooth. The middle one is the typical path that the CMM measures. A big defect can be seen on the leftprofile section, which dramatically increases the value of the quality parameters. The profile section on the right hassmall deviation values.

Figure 21 shows the evaluated profile deviations on multiple profile sections for the tooth. The results are used to



left profilemiddle profile

right profile

Figure 20: Deviation on the whole �ank of the tooth.

calculate the parameter F� , described in chapter 2.4.1. The results are F�,left = 42.5 µmQ = 12; F�,middle = 17.8 µmQ= 9; F�,right = 4.9 µm Q = 6. The quality grades are evaluated according to ISO 1328. It is evident that it is importantto characterise the whole tooth. The results can vary greatly depending on the location of the measurement. All of thedescribed standard geometrical parameters can be determined on the whole width of the gear and collectively presentan improved and more comprehensive quality report. The characterisation of the whole gear with tactile measurementswould require a prohibitive amount of time to perform, but it is practically feasible and straightforward with opticalmethods.

(a) Left section. (b) Middle section. (c) Right section.Figure 21: Deviations on multiple sections (from Figure 20). Determination of parameter F�.



4. ConclusionAn 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 thecommon geometrical quality parameters. The results from the developed method and the CMM measurements werecompared and evaluated. From the validation on the manufactured gears, this method can be adequately accurate for afast evaluation of the whole gear. The developedmethodology presents an advantage compared to tactile measurementswhen used in polymer gear evaluation. Current gear inspection standards were developed for steel gears which aremanufactured by cutting. Polymer gears are mass-produced with an injection moulding process. There a holisticevaluation approach is necessary as the shrinkage and warpage can cause deviations along the whole gear. Highlightsof the research:

• A systematic development of the methodology for optical gear inspection with a custom approach for aligningand pre-processing the measured data.

• A custom software was developed for processing and analysing the scan measurements. A novel approach fordetermining the parameters for lead profile deviation and runout deviation was proposed and used.

• The quality grades, which are the result of the presented method and the CMMmeasurement, are in good agree-ment.

• The random measurement uncertainty for the scanning process was obtained, which was used in calculating theprobability of determining the correct quality grade. The closer the result is to the edge of the quality grade, thegreater the chance of it being falsely determined. The developed methodology is in the worst case 9.6 % morelikely 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 deter-mined 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 thedeviations for all of the teeth on multiple lines and sections. This can be used for lowering the uncertainty of thedetermined parameters. By using multiple sections near one another, each section can be evaluated and then thedetermined parameter values from each section can be averaged to decrease the uncertainty. Further work may includea reference measurement, with minimal uncertainty. This would enable the calculation of the systematic uncertainty.

The investigated methodology is effective for evaluating polymer gears that are made by injection moulding. Thequality of the gears can be accurately determined during manufacturing. However, the current quality parameters donot include and define how much the resulting part has contracted after the moulding process. In terms of futureresearch, a new parameter needs to be included that also considers the total deviation of the resulting part and not justthe relative differences. The new parameter and its implementation in the software could consider some characteristicpoints on the gear, such as the root circle diameter, and it can be compared to the theoretical value to determine theshrinkage. Besides evaluating physical parts, this method is also useful in evaluating the results of a plastic injectionmoulding simulation. The result of the simulation can be exported to an STL file and the methodology can be used todetermine the quality parameters.

The evaluation of the involute profile according to the theoretical profile is an advantage. This is because it enablesa universal evaluation of the geometry without any 3D CAD models. This can only be valid if the program has thenecessary theoretical shape. For non-involute geometries, by performing a comparison with a CAD model, which hasthe prescribed theoretical profile shape, this is easier to achieve. Further work includes the evaluation of the differenttypes 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 analignment by geometrical elements. 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 thegear flanks which returns the smallest collective error between the CAD model and 3D scan on the whole gear.



AcknowledgementThis research was financed partly by the MAPgears project (the project is co-financed by the Republic of Slovenia

and the European Union under the European Regional Development Fund, contract no. C3330-18-952014) and partlyby the Slovenian Research Agency (MR No. 51899). The authors would like to thank the companies Podkrižnik d.o.o.and Tecos for their support with manufacturing and measuring the gears.

Nomenclatureb gear width [mm]D(Q) limit deviation for quality grade [µm]d reference circle diameter [mm]dh gear hole diameter [mm]Fp cumulative pitch deviation [µm]Fpk sector pitch deviation [µm]Fr runout deviation [µm]F� profile deviation [µm]F� lead profile deviation [µm]ff� profile form deviation [µm]ff� lead profile form deviation [µm]fH� profile slope deviation [µm]fH� lead profile slope deviation [µm]fpt single pitch deviation [µm]mn normal module [mm]rb base radius [mm]U expanded uncertainty of measurement [µm]u uncertainty of measurement [µm]xa theoretical/actual value on profile x [mm]xm measured value on profile x [mm]ya theoretical/actual value on profile y [mm]ym measured value y [mm]

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HighlightsA method for enhanced polymer spur gear inspection based on 3D optical metrologyUroš Urbas,Damijan Zorko,Borut Černe,Jože Tavčar,Nikola Vukašinović

• A method for standardized gear quality evaluation based on optical measurements• Key advantages of holistic optical measurements• An areal measurement can detect deviations on the whole gear• A comparison of measurement uncertainty between optical and CMM methods

Highlights

Measurement

Uncertainty estimationHolistic deviation analysis

Processing

Graphical Abstract

CRediT authorship contribution statement

Uroš Urbas: Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing - original draft, Writing – review & editing, Visualization. Damijan Zorko: Conceptualization, Methodology, Validation, Investigation, Data Curation, Writing – review & editing. Borut Černe: Methodology, Validation, Investigation, Writing – review & editing, Visualization. Jože Tavčar: Conceptualization, Methodology, Writing – review & editing, Supervision. Nikola Vukašinović: Conceptualization, Methodology, Writing – review & editing, Supervision.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Declaration of Interest Statement

a method for enhanced polymer spur gear inspection based

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