A Study of Profile Fitting Methods Used in Automated ...

The 10th International Conference of the Slovenian Society for Non-Destructive Testing »Application of Contemporary Non-Destructive Testing in Engineering« September 1-3, 2009, Ljubljana, Slovenia, 455-466

A STUDY OF PROFILE FITTING METHODS USED IN AUTOMATED GEOMETRY ASSESSMENT OF THE POWER TRANSMISSION BELTS Boštjan Perdan1, Drago Braþun2, Janez Diaci2 1

Veyance Technologies Europe, d.o.o., Škofjeloška c. 6, 4000 Kranj, [email protected] 2

University of Ljubljana, Faculty of Mechanical Engineering, Aškerþeva 6, 1000 Ljubljana, [email protected], [email protected]

ABSTRACT An experimental system for automated quality assessment of the belt geometry is presented. The main development goal is to replace visual inspection by skilled workers. The system is composed of a laser triangulation measurement system and a computer which analyses the measured belt profiles, determines the characteristic dimensions of the belt and classifies the belts according to their adherence to the prescribed manufacturing tolerances. The contribution presents the main features of the method for automated geometry assessment. In the first step of the assessment a theoretical profile composed of geometrical primitives (lines, arcs) and defined by the manufacturing documentation is fitted to a measured profile. The contribution presents an experimental evaluation of three different fitting methods. Sample belts of two different types are used in this evaluation. About 200 belt profiles have been measured, most of them in the parts with surface defects. The corresponding theoretical profiles are fitted to the measured profiles using three fitting methods. The parameters of the geometric primitives are determined by the fit and used to calculate the dimensions of the profiles. Comparison of the results obtained by the three fitting methods serves as a basis for the selection most suitable fitting method.

Key words: laser profilometry, profile parameterisation, power transmission belts, defect detection.

1.

Introduction

The function of power transmission belt is to transmit power and rotation between sprockets or pulleys. With synchronous belts load is transferred mostly through shape and partially also through friction. Tooth profile geometry therefore plays an important role in the load transfer and will influence the belt’s life. Improper geometry can shorten belt life by causing tooth root cracking failures [1]. Tooth meshing will also influence transmission error, noise and vibration [2]. The most important sources of noise are the impact sound of belt’s teeth when engaging the sprockets and the sound made by transverse vibration of the belt [3] which too is influenced by meshing. 455

Belts with improper tooth geometry can be detected by inspecting every tooth of the belt. Currently inspection relies mostly on visual inspection either by naked eye or by means of a shadowgraph. Although this is done by skilled workers, it can be subjective and allows for human error. To avoid this and increase the accuracy of the inspection, a quick and reliable automatic measuring system is required. In this paper we present an experimental system for an automated geometry assessment based on optical tooth profile measuring by means of a laser triangulation. The developed tooth profile assessment procedure is based on curve fitting techniques. These are used to construct the parametric model of the measured tooth profile from the measured points. Various kinds of curve fitting algorithms are available that satisfy different constraint conditions [4]. We have decided to base our algorithm on the least squares fitting methods [5]. Three different fitting methods were tested in order to determine the most suitable one. Parametric profile is then used to calculate profile dimensions of interest which are use for an automated geometry assessment and profile classification.

2.

Tooth profile defect examples

Many factors can cause a deviation of the actual tooth profile shape from the ideal one. Belt profile shape is defined by the mould and can be affected by mould wear, damaged surface of the mould and dirt on the mould’s surface. The appearance of the belt’s surface depends also on the texture of the facing fabric that protects the teeth from abrasion. Process parameters can influence the profile shape by causing insufficient flow of rubber compound to form the teeth. The final tooth shape depends also on the compound shrinkage that needs to be compensated for by the tooling design. Tooth profile defects can be in classified into 3 main categories: poor tooth formation, dented teeth and bumps on the surface. Two of them are shown in Fig. 1. Drawings on the right show the measured profile placed on the template which represents the ideal shape. Profiles with 5 and 10% shrinkage in vertical direction are added for easier profile assessment. This imitates the visual inspection in production by means of a shadowgraph where a similar template is used. Shrinkage is added to template because teeth are moulded at an elevated temperature and shrink when cooled down. Dented teeth (shown in Fig. 1a) are caused by dirt that gets between the mould’s teeth and prevents the cover fabric to conform to the surface of the mould. They are mostly of cosmetic nature as they don’t harm the performance of the belt but are not acceptable. Bumps on the surface (see Fig. 1b) are consequence of mould damage (dents, etc.). Due to an excess of material on the tooth surface, bumps can cause problems with tooth meshing into the sprocket and are not acceptable.

3.

Experimental system description

Experimental system is shown in Fig. 2. It consists of a belt drive and a laser profilometer (1) that measures belt tooth profile using the principle of laser triangulation. A bright line (2) visible on the surface of the belt is acquired by a camera placed at a specific triangulation angle with respect to the direction of illumination [6]. The result of measurement is a profile representing cross-section of the laser plane and the illuminated surface. The optics was adjusted to the measuring range of 25 mm along the laser line and 20 mm perpendicular to the measured surface. A measured profile (MP) consists of 640 measured points with a measurement resolution of 0.04 mm. The coordinates of measured points are calculated by a triangulation model described in [7].

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a.)

b.)

Fig. 1: Two tooth profile defects examples depicted by a photo (left) and a measured raw profile (right): (a) dented teeth, (b) bumps on the surface. Belt is translated in a lengthwise direction by means of a belt drive, where the driver pulley is rotated by a micro-stepping motor. Driven pulley is mounted on a moveable cart to accommodate different belt lengths in range from 675 mm to 2300 mm. Belt is running on its back over the pulleys and belt support which prevents belt vibrations. A more detailed system description can be found in [8].

a.)

b.)

Fig. 2: Experimental system: (a) photo of the system: (1) laser profilometer, (2) laser plane, (3) belt, (4) pulleys, (5) movable cart; (b) schematic of the system.

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4.

Tooth profile assessment algorithm

Tooth profile assessment algorithm is based on tooth profile parameterisation that is used to convert the raw profile represented by a point cloud into parametric profile assembled from basic geometric primitives (GPs). These are line segments, circular arcs and segments of square parabola that are defined with a few parameters. Resulting parametric profile is used to calculate profile geometry dimensions. Of most interest are the dimensions that define profile shape in the technical drawing and are important from the assessment point of view in the production (tooth height, width, etc.). The whole procedure of profile assessment is depicted by a flowchart in Fig. 3.

Dim. calculation instructions Individual primitives

Template Segmented point cloud

Raw profile

2. Profile segmentation

3. Fitting of primary primitives

Dim. assessment criteria

Parametric profile Acceptable / Unacceptable

4. Fitting of secondary primitives

6. Dimension calculation

7. Quality assessment

(Point cloud) Profile dimensions Parametrisation Alarm (defect found)

Fig. 3: Tooth profile assessment procedure. Tooth profile parameterisation starts with creation of a template that is based on a technical drawing of the tooth profile (see Fig. 4a). It is assembled from GPs in the same manner as the profile is constructed in the technical drawing (see Fig. 4b). The template defines the ideal shape of the actual profile that we are about to parameterise. Its data is written in a form of a table, as depicted in Fig. 5a. Each row of the table represents one GP, which are ordered according to their sequence in the profile. All possible GPs and their parameters are gathered in Table 1. For each GP an identifier (ID), that defines the type of primitive, its parameters and range (start and end positions in x axis) are given. Parameters listed in Table 1 fully define the corresponding GP. A line segment in an arbitrary position is defined by the following equation: y kxn [1] Where parameter k is the slope of the line and n its interception with vertical axis (y) of the coordinate system in which the line is located; x is an undependable variable defined within a range given by the start and end position in the template data table. A circular arc with a centre in an arbitrary position is defined by:

x p 2 y q 2

458

r2

[2]

R

R2

R2

R1

h

R1

a.) D

(p1,q1)

b.)

r1

k1,n1 x0 1 x1

x3 1

4

2

x4

r3

r2

k2,n2 (p2,q2) x2

4

(p4,q4)

(p3,q3)

x5 r4

4

x6 1 k3,n3 r5 x7 4

(p5,q5) k4,n4 x8 1 x9

c.)

Fig. 4: Creation of a template and profile segmentation: (a) profile shape defined by technical drawing, (b) template assembled from geometric primitives with their parameters, (c) segmented measured profile (point cloud).

Parameters

a.)

ID

k, r

n, p

0, q

1 4 1 4 2 4 …

0 1.016 2.748 1.016 3.556 1.016 …

0 1.241 -5.363 3.477 5 6.523 …

0 1.016 0 1.22 -0.813 1.22 …

Start End positio positio 0 1.241 2.195 2.523 2.868 7.132 …

1.241 2.195 2.523 2.868 7.132 7.477 …

Templat e data table

Profile parametrisation

b.)

1 4 1 4 2 4 …

1 4 1 4 2 4 …

1 0.011 -0.005 0 0 1.229 4 1.07 1.218 1.078 1.229 2.229 0 2.2291.229 1 0.0112.87-0.005 -5.67 0 0 2.572 1.07 1.218 1.078 1.229 2.229 4 0.717 3.249 1.477 2.572 2.836 0.0112.87-0.005 0 0 0 2.2291.229 -5.674.988 2.572 2 3.733 -0.988 2.836 7.217 1.07 1.2183.249 1.078 1.229 2.229 0.717 1.477 2.572 2.836 4 0.887 6.688 1.294 7.217 7.513 2.873.733-5.674.988 0 -0.9882.229 2.572 … … … … 2.8362.836 … 7.217 … 0.717 0.8873.2496.6881.477 1.2942.572 7.217 7.513 3.733 … 4.988 … -0.988 … 2.836 … 7.217 … Tooth i 0.887 6.688 1.294 7.217 7.513 Tooth 2 … … … … …

Profile data table

Tooth 1

Fig. 5: An example of a data tables for (a) template and (b) parametric profiles. 459

Table 1: The list of geometric primitives and their parameters

GEOMETRIC PRIMITIVE Line segment Circular arc Square parabola Fillet

ID 1 2 3 4

PARAMETERS k r a r

n p b p

/ q c q

X START / END POSITION x0 x1 x0 x1 x0 x1 x0 x1

Where parameters (p, g) denote the arc’s centre position in horizontal (x) and vertical axis (y) of the coordinate system and parameter r denotes its radius. Last, the equation for the square parabola can be written as: 2 [3] y a x b c Where (b, c) denote the parabola’s starting point position in horizontal (x) and vertical axis (y) of the coordinate system, and parameter a stretches the parabola in vertical direction. These are the primitives used for belt tooth profile design in engineering praxis. An important step prior to parameterisation is profile segmentation (see Fig. 3, block 2) that is used to partition the raw MP into meaningful groups of points that belong to a certain GP (see Fig. 4c). Because segmentation by means of a universal algorithm is not easy to perform and leads to ambiguous results [9], a simplified approach is used featuring a template that defines how the MP is segmented. MP is aligned with the template and ranges of individual GPs in it define which points of the MP belong to them. During alignment, the MP is transferred from the camera coordinate system (CS) to the template CS. This is done by automatically locating reference points on the MP (in the camera CS) that correspond to the predetermined reference points on the template (in the template CS). From the difference in their positions required translation and rotation for alignment of the MP with the template are calculated. This approach to segmentation saves processing time and rules out random outcomes that would result if a more general approach was used. It works best with small defects and small deviations from the ideal shape where the will still correspond to the GP we are trying to fit. Segmentation will not be correct with MP with large defects, where the shape significantly differs from the expected one. In this case a more general approach could be used, but the gained information would be of little use, as we are not interested into knowing the exact shape of the defect. After segmentation GPs are fitted to individual groups of points where we, based on their importance, distinguish between primary and secondary primitives. Primary primitives are computed first (see Fig. 3, block 3) and are fitted independently to provide most accurate fit to measured points. They define the basic geometry of the tooth profile, which is important from the technological point of view. Secondary primitives are in this respect of less importance and are primarily used to connect the primary primitives (see Fig. 3, block 4). Fillets are used for this purpose and we distinguish between them and circular arcs even though they are the same geometric element. A simplified approach to fitting them is used, where they subject to primary primitives that they connect in order to ensure a smooth transition between them. Due to various reasons (discrete nature of picture coordinates, surface roughness, noise, etc.) measured points won’t lie exactly on the primitive, but mostly close to it. Consequently, an 460

approximate fitting method is required that minimizes the error of the fit [10]. A least squares fitting (LSF) method is often used to fit GPs to point segments. Because ordinary LSF method is sensitive to outliers, alternative more robust methods of weighted least squares (WLS) and orthogonal distance fitting (ODF) were also tried out that under certain conditions give better results. LSF method minimizes the sum of squared differences between the measured points and the corresponding points on the primitive. Its advantage is simplicity, as it produces a direct solution for the unknowns. Using the matrix algebra, the LSF problem for any given primitive can be formulated as a system of linear equations:

y

A x

[4]

For a line segment we write:

y

ª y1 º «y » « 2» « » « » ¬ ym ¼

ª1 x1 º «1 x » 2» A « « » » « ¬1 xm ¼

x

ªnº «k » ¬ ¼

[5]

Where (xi, yi), i = 1 … m denote the coordinates of measured points, to which we would like to fit the line segment, and x is a vector containing the unknown line parameters k and n. The disadvantage of the LSF method is its sensitivity to outliers. Because the difference between the points is squared, the outliers have great influence and can cause the fit to be wrong. Their influence can be reduced with the use of robust methods such as WLS and ODF. WLS is an iterative method and an extension to ordinary LSF, the results of which are used for the first iteration. A weight is added to each point in such a way, that points with a greater weight contribute more to the fit. The actual weight depends on the difference between the measured point and the corresponding point of the previous fit. Within a certain range the weight is inversely proportional to the aforementioned difference and zero outside of it. This way most of outliers are effectively eliminated and the fit improved. In WLS the equation (4) is expanded by adding a diagonal weight matrix W to each side:

W y W A x

[6]

The weight matrix is calculated as:

ui s

ri 3 s median(ri )

wi

1 u

wi

0

2 2 i

½ ° ° ° ¾ o W for ui 1 ° ° for ui t 1°¿

w1

0

0

0

0

0

0

0

0

0 0

0 0

wi 0 0

0 0

0

0

0

0

[7]

wm

Where ri denotes the difference between the measured point and the corresponding point of the previous fit. The procedure is iterated several times, until successive parameter values no longer differ substantially. 461

LSF and WLS methods both use as an error measure a vertical distance between the measured point and the corresponding point on the primitive. They give satisfactory results in most cases, especially if the primitive is mainly horizontal. Problematic are near vertical primitives or vertical segments of primitives where small deviations, in normal direction to the primitive, appear as large deviations in vertical direction. Here better error measure is the geometric distance. This is the shortest distance between the measured point and the primitive and is the most natural error measure, because the measured point is a probable observation of the nearest primitive point. Geometric distance is used as an error measure in ODF method, which is, contrary to its clear definition, generally nonlinear to the model parameters [11]. Consequently, tasks involved in minimizing the square sum of geometric distances for general features are very complex with the exception of lines and circles. The ODF problem of line can be elegantly solved by using the moment method and any other curve is a nonlinear minimization problem, solved through iteration. For a relatively simple model features such as circles a few dedicated algorithms exist [12]. When the primary primitives are fitted to the measured points we continue by fitting fillets, where an optimum fit to data in least squares sense is not required, so a simplified approach is used. Initial fillets are calculated first using the same method as before and the results are used for further fillets calculation from profile geometry to ensure a smooth transition between primary primitives. This simplification has no influence on the calculated profile dimensions which rely on the independent primary primitives. Calculation of a fillet between two line segments is straightforward and Newton’s numeric method is used to calculate the fillet between a line segment and a circular arc. The end result is a parametric profile, written in a profile data table (PDT), the form of which is identical to template data table (see Fig. 5b). Data stored in PDT is used to display the profile and to calculate profile dimensions according to dimensions calculation instructions. These specify what dimensions we are interested in, what data to use for calculation and how to calculate them. During profile geometry assessment, the algorithm compares the results to the reference values specified by dimensions assessment criteria and profile is classified as acceptable or unacceptable. Most common dimensions of interest are profile height, width and pitch between two successive profiles. Calculated radii of circular arcs are also evaluated. Even though fillets are less important, they too are included because a large fillet size deviation is a sign of a possible defect or a parameterisation error. Shape parameters are calculated to characterize the shape of the measured profile. These are inclination angles of line segments in the MP that correspond to horizontal lines in the template, and an angular deviation of profile centre line from a vertical line. All should be close to zero. If particular inclination angle is much larger than zero, this is a sign of a defect in the root or on the top of the profile. A noticeable deviation of the centreline angle is a sign of asymmetric profile. With this solution we came close to fulfilling the technological requirements of production process, where the capability of detecting small deviations from the ideal shape is of most interest. These are quite common with good products and difficult to detect with visual inspection alone. In case of large defects the parameterisation can fail and the calculated GPs won’t be correct. When this happens, the profile assessment procedure is stopped and the system sets off an alarm to warn the operator about the defect found. 462

5.

Examples of tooth profile assessment

Successful tooth profile parameterisation is an essential part of the assessment process. It is therefore important, that we choose the most appropriate method of fitting GPs to measured points. All three methods (LSF, WLS and ODF) were applied to measured profiles and results compared to each other for evaluation. For demonstration purposes, examples presented in Fig.s 6, 7 and 8 each show three parametric profiles calculated with these methods.

Calculated dimensions and observations:


Height (3.05 +0 / -0.3 mm): Width (4.06 +/- 0.12 mm): Left flank radius (5.21 +/- 1.5 mm): Right flank radius: Fillet radii within 0.79 +/- 0.24 mm? Inclination angles less than 3°? Profile acceptable?

Height (2.74 +0 /-0.28 mm): Width (5.28 +/- 0.15 mm): Top radius (3.56 +/- 0.35 mm):

3.03 mm 4.13 mm 5.47 mm 5.73 mm YES YES YES

Fillet radii within 1.02 +/- 0.30 mm? Inclination angles less than 3°? Profile acceptable?

a.)

2.66 mm 5.18 mm 3.75 mm YES YES YES

b.)

Fig. 6: Examples of successful tooth profile parameterisation. We demonstrate the assessment procedure on two tooth profiles that were classified as acceptable by our method and an established tooth inspection method used in the production (Fig. 6). Both MPs fit well on the template. In the case of profile in Fig. 6b we can notice a small deviation of profile height which is acceptable but shows in the results. In this case all three GP fitting methods generate almost identical results as expected. Defects will cause a deviation of MP from the expected shape and affect the parameterisation algorithm, which will respond differently depending on the size and shape of the defect. We distinguish between tooth profile defects where the profile parameterisation process finishes successfully, and defects where it fails. In the first case we get a complete parametric profile (see examples in Fig. 7) and profile assessment process continues. Profile dimensions are calculated next and compared to their reference vales. Depending on the size of the defect, the profile is classified as acceptable or unacceptable (see Fig. 3). Some discrepancy between the three fitting methods can be noticed as demonstrated by the examples in Fig. 7. WLS method gives a slightly better result, but the difference is too small to make any preference. 463



Height below the spec: 2.59 mm Right flank radius above spec: 11.36 mm. At least 1 of inclination angles over 3°. Profile is NOT acceptable.

Width above the spec: 5.57 mm At least 1 fillet radius outside 0.79 +/- 0.24 mm.



Height (3.05 +0 / -0.3 mm): Width (4.06 +/- 0.12 mm): Left flank radius (5.21 +/- 1.5 mm): Right flank radius: Fillet radii within 0.79 +/- 0.24 mm? Inclination angles less than 3°? Profile acceptable?

2.92 mm 3.98 mm 5.41 mm 5.06 mm YES YES YES

Profile is NOT acceptable.

Width below spec: 5.02 mm At least 1 inclination angle over 3°. Profile is NOT acceptable.

Fig. 7: Examples of successful tooth profile parameterisation of profiles with defects. In case of failed profile parameterisation some of the calculated GP parameters greatly exceed the expected value and calculated profile continuity is lost (see examples in Fig. 8). Such a parametric profile does not reflect the real situation and is easy to identify by a presence of a discontinuity in the fitted profile. Tooth profile is immediately classified as unacceptable and quality assessment procedure stopped. The system will also set off an alarm to warn the operator about the defect. These defects will cause a noticeable discrepancy between the results of different fitting methods as demonstrated in Fig. 8. In general WLS method gives best results. However, where speed of analysis is important, an ordinary LSF method is sufficient. Contrary to other methods it is not iterative and hence faster than others which use its results as initial values for the first iteration. Surprisingly, results obtained with an ODF method were not much better than results of other methods. Due to a more natural approach of the ODF method, a greater improvement was expected.

464

Fig. 8: Examples of failed tooth profile parameterisation of profiles with defects.

6.

Conclusion

A system for an automated assessment of profile geometry has been developed. Tooth cross section profile is measured by means of a custom developed laser-triangulation-based measurement system. The measured profile (MP) is then automatically analysed by a profile geometry assessment method based on profile parameterisation. The novelty of the developed method is in the use of a template. This predefines the expected profile shape by defining the type of geometric primitives (GPs) and the way they assemble into a profile to reflect the technical drawing of the tooth profile. Three different fitting methods were tried out for fitting the GPs to the measured profile points. The results show that the ordinary least squares fitting (LSF) method is sufficient in most cases and much faster than any alternative robust method. Because of that the LSF method was chosen for the future development of the algorithm. About 200 tooth profiles were measured and assessed with the proposed method, most of which were profiles with typical defects. The method produced comparable results to established visual inspection in the production. The system found all the obvious defects and also detected smaller deviations from the ideal profile shape that are otherwise difficult to detect. With an automated assessment of these shape deviations the same criteria is always used and subjectivity removed. On the other hand, small details will be lost with parameterisation if they are much smaller than the extent of corresponding GP. These are mostly small surface irregularities and fabric texture that are considered acceptable and are therefore of little interest. The system allows them as they do not influence the calculated geometry dimensions used for profile assessment.

7.

References

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[4]

Ueng W.D., Lai J.Y., Tsai Y.C.: Unconstrained and constrained curve fitting for reverse engineering, International Journal of Advanced Manufacturing Technology, Vol. 33, 2007, 1189-1203. [5] Chernov N., Lesort C.: Least squares fitting of circles and lines, Department of Mathematics, University of Alabama at Birmingham, USA, 2002. [6] Zexiao X.,. Jianguo W, Ming J.: Study on a full field of view laser scanning system, International Journal of Machine Tools & Manufacture, Vol. 47, 2007, 33-43. [7] Braþun D., Jezeršek M., Diaci J.: Triangulation model taking into account light sheet curvature, Measurement Science & Technology, Vol. 17, No. 8, 2006, 2191-2196. [8] Perdan B., Braþun D., Diaci J.: Surface defect detection on power transmission belts using laser profilometry; accepted for publication in International Journal of Microstructure and Materials Properties. [9] Woo H., Kang E., Wang S., Lee K.H.: A new segmentation method for point cloud data, International Journal of Machine Tools & Manufacture, Vol. 42, 2002, 167-178. [10] Draper N.R., Smith H.: Applied regression analysis, Third edition, John Wiley & Sons, Canada, 1998. [11] Ahn S.J.: Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space, Springer-Verlag, Germany, 2004. [12] Gander W., Golub G.H., Strebel R.: Least-Squares Fitting of Circles and Ellipses, BIT 34, 1994, 558-578.

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