Calibration procedure for a laser triangulation scanner with uncertainty evaluation

Calibration procedure for a laser triangulation scanner with uncertainty evaluation

Optics and Lasers in Engineering 86 (2016) 11–19 Contents lists available at ScienceDirect Optics and Lasers in Engineering journal homepage: www.el...

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Optics and Lasers in Engineering 86 (2016) 11–19

Contents lists available at ScienceDirect

Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Calibration procedure for a laser triangulation scanner with uncertainty evaluation Gianfranco Genta n, Paolo Minetola, Giulio Barbato Department of Management and Production Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 29 February 2016 Received in revised form 22 April 2016 Accepted 7 May 2016

Most of low cost 3D scanning devices that are nowadays available on the market are sold without a user calibration procedure to correct measurement errors related to changes in environmental conditions. In addition, there is no specific international standard defining a procedure to check the performance of a 3D scanner along time. This paper aims at detailing a thorough methodology to calibrate a 3D scanner and assess its measurement uncertainty. The proposed procedure is based on the use of a reference ball plate and applied to a triangulation laser scanner. Experimental results show that the metrological performance of the instrument can be greatly improved by the application of the calibration procedure that corrects systematic errors and reduces the device's measurement uncertainty. & 2016 Elsevier Ltd. All rights reserved.

Keywords: 3D scanner Laser triangulation Calibration Ball plate Measurement uncertainty

1. Introduction In the last decade, the diffusion of 3D scanning devices has progressively increased. These instruments are often preferred to contact measuring systems, because a very much larger number of points can be measured in short times. Unfortunately, because of the absence of a specific international standard, it is difficult for the user to verify the metrological characteristics that are declared on a 3D scanner's datasheet. As a matter of fact, the scanner's producer adopts internal proprietary standards and procedures to certify those characteristics. In addition, most of low cost devices are sold without calibration artefacts and methods [1]. Thus the user is not able to check the performance of the instrument along time or to restore optimal settings to face changes in environmental conditions. The aim of this paper is to compensate for this gap by defining a calibration procedure for 3D scanners similar to that defined by the ISO 10360-2 [2] for Coordinate Measuring Machines (CMMs). A number of calibration procedures, based on different reference artefacts such as step gauges [3], ball bars [4] and ball plates [5], are known. Nevertheless, they do not contain specific uncertainty evaluations showing the advantages of the calibration. The calibration procedure proposed in this paper is also based on the use of a reference ball plate, but in addition contains an uncertainty evaluation, according to the guide to the expression of uncertainty in measurement (GUM) [6]. This n

Corresponding author. E-mail address: [email protected] (G. Genta).

http://dx.doi.org/10.1016/j.optlaseng.2016.05.005 0143-8166/& 2016 Elsevier Ltd. All rights reserved.

evaluation is performed with and without the calibration, clearly indicating the actual metrological capabilities of the scanner in these two conditions. For the uncertainty level of the scanner, the use of a ball plate easily fabricated at a reduced cost is sufficient. Once manufactured, the ball plate was measured with a CMM to get the reference positions of the balls. The developed methodology allows to roto-translate the reference coordinates of the ball centres in the position of the ball plate under the scanner. In this way, the scanner results can be compared with the reference values, showing systematic trends, carefully identified by the regression method. The calibration procedure is applied to a laser triangulation scanner as detailed in the following sections.

2. Measurement equipment 2.1. Laser triangulation scanner The object of this study is the laser triangulation 3D scanner Vi900 by Konica-Minolta (Fig. 1). The device projects a stripe of coherent monochromatic light, coming from the lower source aperture. The red laser stripe is projected and swept across the working volume by means of a Galvano-mirror. The laser light is deformed by the surface of the scanned object and meanwhile each scan line is captured in a single frame image by a CCD camera lodged in the device superior aperture. Captured image frames are then processed according to the optical triangulation principle. The CCD camera is equipped with three

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Fig. 2. Top view of the ball plate.

Fig. 1. Konica Minolta Vi-900.

Table 1 Characteristics of Konica Minolta Vi-900. Scanner dimensions Weight CCD resolution Single scan area

Declared precisiona Scan time per single scan Working distance a

213  271  413 mm 11 kg 640  480 pixels 111 mm  84 mm with Tele lens (f ¼ 25 mm) 710 mm  533 mm with Middle lens (f ¼ 14 mm) 1300 mm x 1100 mm with Wide lens (f ¼8 mm) 0.05 mm 2.5 s Up to 1200 mm

Based on Minolta standard test method.

exchangeable lenses with different focal length f (telephoto, middle and wide angle) to adjust the scan range to the object size. The Vi-900 characteristics that appear on the device datasheet [7] are reported in Table 1. The Vi-900 does not come with a built-in calibration system and cannot be adjusted by the user for any lens change or environment lighting changes. Minolta's Polygon Editing Tool software version 2.3 automatically sets the focus and the exposure for the scanned object. 2.2. Ball plate The reference part for the calibration activity is a ball plate. It was designed similar to ball plates used for qualification of CMMs [8] according to ISO 10360-2 [2]. Similar reference parts proposed in the literature come with an artefact similar to a step gauge [9] or a unique ball [10–13] or a few balls [4]. In the present work, the number and density of balls on the plate were increased in order to evaluate measurement errors through a wider range of the

working area of the 3D scanner. The plate is 20 mm thick and its overall dimensions are 190 mm  150 mm. Its size was chosen as to fit into the scan area of several types of 3D scanners. 25 balls with a diameter of 10 mm are positioned on its top surface along a grid defined by 5 rows and 5 columns. The balls spacing is not uniform as shown in Fig. 2, but it was differentiated to be denser in the centre of the plate with the aim of getting more information about the central area. The size of the 25 balls was chosen to have a sufficient number of measured points for each ball depending on the resolution of the scanner's CCD camera and the size of the scan area. For the scanner configuration with the Tele lens, the scan area should be about 200 mm  150 mm for the reference part to fit into one single view. Since the CCD camera of the Vi-900 has a resolution of 640  480 pixels, the scanner is able to measure approximately one point every 0.3 mm. With such a point density, about 800 points are measured on the projected circular area of each 10 mm sphere. One replica of steel material of the reference ball plate is manufactured at the Department of Management and Production Engineering of the Politecnico di Torino by means of a She-Hong CNC milling machine model VMC 850. The plate is first machined to its dimensions and then 25 spherical grooves (Fig. 3) are drilled in the positions corresponding to the centres of the balls along a depth of 2.5 mm using a ball-nose milling tool with a diameter of 10 mm. Spheres for ball bearings are certified in terms of dimensional accuracy and shape errors according to the ISO 3290-1 [14]. The 10 mm balls for the reference part are purchased as spare parts for SKF bearings with a quality grade G200, i.e. the Ball Diameter Variation (VDws) is 7 5 mm, the Lot Diameter Variation (VDwL) is 710 mm and the Deviation from Spherical Form (tDw) is 5 mm. These values are consistent with the purpose of this study. The balls and top plane of the replica are finished with different colours for the reference part to be representative of objects whose appearance can be classified as high contrast. Hence the 25 balls are sprayed with an opaque white paint, whereas after milling an opaque medium grey paint is used for the background top plane of the plate. Finally the 25 white balls are glued into the corresponding grooves of the plate (Fig. 4). The choice of opaque paints is important to avoid light reflections which, affecting the

G. Genta et al. / Optics and Lasers in Engineering 86 (2016) 11–19

Fig. 3. Milling of the spherical groves on the top face of the plate.

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mechanical applications shape controls are sufficient: if it is expected that a mechanical piece fits correctly into its place it is sufficient that its form (parallelism, orthogonality, circularity, etc.) and dimensions (length, diameter, etc.) are contained within the relevant tolerances. Coordinate tolerances have to be set only when the position of the piece is important. It shall be faced the fact that the 3D scanner can be used for checking both shape or coordinates of a test piece, but the transfer standard, in our case a ball plate, is a reference for shape but not for coordinates. The difference between the two conditions is related to the coordinate reference system. In case of a shape control, the coordinate reference system is not important. By changing the x axis reported in the draw with the y axis of the checking instrument, or by applying any other possible rotation and translation, it is obtained the same result for the distance between any two points of the test piece, i.e. the same result of a shape check. This certainly will not happen for a coordinate control, because any error of reference system is transferred directly on the measurement coordinates. With our transfer standard, being it a reference of shape, the coordinate reference system used shall be carefully controlled in each phase of the calibration to be able to get the systematic errors of the scanner, referred to its coordinate reference system. The continuous metrological chain is:

 The ball centre coordinates of the ball plate could be measured

Fig. 4. Finished ball plate.

measurements of the 3D scanner, produce noise and/or overexposed areas. Three 20 mm balls are located on the back face of the plate with the aim of guaranteeing a stable equilibrium through a kinematic coupling. The grooves for the positioning of the support balls are 7 mm depth as to obtain an adequate area for fitting them.

3. Calibration procedure 3.1. Proposed method The scope of a calibration procedure consists in determining systematic errors [1] for correcting bias effects on measurements. The usual way is the comparison of measurement results to reference standards. In our case the reference standard is a CMM, with an uncertainty lower than 5 mm, adequate for evaluating possible systematic errors of the laser triangulation scanner. Before schematizing the necessary continuous metrological chain starting from the CMM and arriving to the scanner measurements, it is necessary to focus on some concepts which are frequently neglected. CMM and scanners are commonly used both for shape controls and coordinate controls. Frequently in



by the CMM in its coordinate reference system, but, just to get numbers directly referable to the dimensions of the plate, they are measured in a reference system with the origin in the centre of ball 1, the x axis directed from the centre of ball 1 to the centre of ball 5, the y axis in the plane containing also the centre of ball 21 and, consequently, the direction of z axis orthogonal to that plane. A reference system so defined is robust (the centres of the balls taken as reference points are well defined, with a number of degrees of freedom corresponding to the numbers of points taken on the surface of each ball minus four), but less robust than a reference system taking into account the positions of all the balls. Therefore, after measurement, a comprehensive reference system shall be defined, considering the centre of mass of all the ball centres as origin and their regression plane [15] as x–y plane. Thus it is possible to rototranslate the ball centre coordinates according to the inclination of the regression plane. To establish the directions of the coordinate axes, using the new coordinates of all ball centres, in the regression x–y plane the two regression lines corresponding to the x direction and the y directions are determined. These two directions are not necessarily orthogonal, but their bisecting lines are orthogonal and, rotated 45°, constitute a very robust reference system (Ball Plate Machine System, BPMS). Ball centre coordinates are roto-translated in this new reference system and constitute the reference geometry of the ball plate transfer standard. The second step of the calibration is done measuring the ball plate with the scanner. This first calibration exercise was done with the limited aim of determining the scanner systematic errors in the x–y directions and only zero error in z direction, therefore the ball plate is put on the base plane of the scanner and measured with z values nearly constants. A position with the x direction nearly aligned with the x axis of the scanner is preferable. After scanning, the robust reference axes of the ball plate under the scanner (Ball Plate Scanner System, BPSS) were determined in the same way described in the first step. Subsequently the CMM measured values in BPMS were rototranslated in BPSS, i.e. displacing the reference values of the ball plate ball centres in the same position of the ball plate under

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measurement by the scanner. The differences between coordinates measured by the scanner and the reference coordinates measured by the CMM indicate the measurement errors. Regression of the values so obtained in the scanner x direction and y direction will allow for an evaluation of possible systematic errors. In the same way the zero error in z direction can be obtained. Note that a drawback, very common in statistical applications, is present: the coordinate reference system BPSS is determined using data obtained by scanner measurement, therefore in principle containing also systematic errors. It would be better to go by steps: use the original data for evaluating a first approximation of systematic errors, thereafter correct the original data and use the corrected data for evaluating a second approximation correcting residual systematic errors. To check if this stepwise procedure is necessary, one can use normality tests on residuals. 3.2. Data collection Six replicated measurements of the ball plate were performed with the CMM. Subsequently, six replicated measurements were performed with the laser triangulation scanner under the same constant environmental lighting condition. The lights in the laboratory were set for an exposure of the ball plate that corresponded to a medium value of the operative exposure range of the Vi-900. The scan data is then imported and processed using Rapidform 2006 software which obtained the certification by the German PTB institute. The cloud of the points (Fig. 5) measured on the plate is not filtered for noise reduction or smoothing not to alter the result of the scanner's measurement. First the points belonging to the plate planes are removed and then a clustering algorithm is applied to automatically distinguish and separate the group of points belonging to each of the 25 balls. The interpolation sphere for each point set is computed by means of a best fitting algorithm considering all measured points, i.e. without outlier removal. Finally the following information for the 25 reconstructed spheres is exported from Rapidform software and stored in an ASCII file: x, y and z coordinates of the sphere centre, sphere radius and the number of points retrieved by the 3D scanner. Each of the analysed datasets is limited to 75 values (25 spheres with 3 corresponding centre coordinates). 3.3. CMM data analysis An experimental data analysis [16] was carried out on data collected with the CMM. As a first step, considering that the calibration procedure is

Fig. 5. Best fit spheres from scan data using Rapidform 2006.

related to the coordinates of the ball centres, it was checked that the painting operation of the balls did not introduce irregularities on the ball surfaces. Measurement results show that the sphericity error remains consistent with the declared manufacturing quality grade. The painting operation produced only an increase of the diameter value lower than 0.1 mm, which, however, does not affect the centre positions. Subsequently, outliers [17] were identified on the whole dataset of 450 values (75 values for each of the six replications). To this aim, each replication data were put together by deducting the relevant average for cancelling systematic differences between replications. Chauvenet's criterion [18] was applied and 6 outliers were found. Then, normality tests were performed and deviations from normality resulted to be not relevant. Subsequently, the 6 outliers were replaced with the corresponding mean values calculated on the six replications. Then, for each replication, the centre of mass of the 25 ball centres was deducted from each data. In this way, the origin of the reference system was positioned in the centre of mass, thus avoiding the systematic effect of possible shifts between replications. Finally, it was checked that the plate was correctly oriented with the adopted reference system, by verifying the absence of significant trends among the three coordinates. The set of ball centre coordinates obtained for each replication is, therefore, a candidate for describing the ball positions. Even more robust values are obtained averaging each replication results, thus obtaining a reliable description of the geometry of the ball plate in the reference system associated to the plate (BPMS). The coordinates of the 25 centres in BPMS are shown in Table 2.

4. Calibration results: data analysis The procedure used for data analysis is described here. As for the CMM measurements, to avoid the effect of possible systematic shifts between scanner measurement replications, a first part of the analysis (Sections 4.1 and 4.2) is performed separately on each replication and then all the replications are put together (from

Table 2 Coordinates of the 25 ball centres in BPMS (Ball Plate Machine System). N

x/mm

y/mm

z/mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 84.99  84.97  85.05  84.94  84.92  39.97  40.02  39.98  40.03  40.05 0.00  0.02  0.05 0.14 0.08 40.02 39.99 39.98 40.05 39.93 84.96 84.97 84.92 84.98 84.99

 64.99  25.01 0.00 25.01 64.99 64.99 24.96 0.02  25.01  64.91  64.98  25.01  0.01 24.97 64.96 64.96 25.01 0.01  25.02  64.94  64.99  25.01 0.00 24.98 65.00

0.02  0.05  0.01  0.01 0.02 0.02 0.01  0.04  0.06 0.03 0.03  0.04  0.01 0.02 0.01  0.03  0.01  0.02 0.01 0.00 0.04  0.02 0.09 0.00 0.00

G. Genta et al. / Optics and Lasers in Engineering 86 (2016) 11–19

Table 3 Coordinates of the 25 ball centres in BPSS (Ball Plate Scanner System) after the translation for the first replication.

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Table 4 Coordinates of the 25 ball centres in BPMS (Ball Plate Machine System) after the roto-translation for the first replication.

N

x/mm

y/mm

z/mm

N

x/mm

y/mm

z/mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 85.62  85.44  85.27  85.21  85.03  39.79  40.10  40.15  40.28  40.49  0.41  0.27 0.03 0.13 0.32 40.55 40.33 40.15 40.05 39.80 84.92 85.18 85.42 85.48 85.72

 64.85  24.71 0.39 25.53 65.70 65.42 25.31 0.18  24.94  65.04  65.21  25.10 0.00 25.12 65.28 65.00 24.92  0.23  25.27  65.42  65.65  25.54  0.42 24.69 64.86

0.01  0.68  1.05  1.46  1.91  1.18  0.77  0.40  0.02 0.64 1.07 0.41  0.04  0.35  0.86  0.42 0.04 0.32 0.80 1.59 1.90 1.18 0.79 0.39 0.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 85.30  85.09  85.04  84.82  84.61  39.65  39.89  39.98  40.15  40.36  0.31  0.14  0.05 0.26 0.39 40.33 40.11 39.97 39.93 39.61 84.64 84.84 84.91 85.10 85.30

 64.58  24.61 0.40 25.41 65.37 65.17 25.15 0.21  24.82  64.71  64.97  25.00  0.01 24.97 64.95 64.77 24.82  0.18  25.20  65.12  65.38  25.41  0.40 24.58 64.59

0.07  0.59  0.92  1.28  1.84  1.36  0.79  0.47  0.13 0.55 0.98 0.33  0.01  0.34  0.9  0.56 0.05 0.41 0.81 1.38 1.90 1.26 1.00 0.54  0.05

Section 4.3). In the example given, the data of the first replication is used.

Table 5 Differences between the coordinates of the 25 ball centres in BPSS after the translation and in BPMS after the roto-translation for the first replication.

4.1. Pairing of CMM and scanner measurement results The first step is to pair the measurements obtained in the BPMS and in the specific BPSS. This is obtained through an optimized roto-translation. First of all, the centre of mass of the 25 ball centres of the BPSS was deducted from the same dataset. In this way, the origin of the BPSS was positioned in the centre of mass, such as for the BPMS. The coordinates of the 25 centres in BPSS after the described translation are shown in Table 3. Then, three subsequent elementary rotations [19] of the BPMS about its coordinate axes are performed. In formulas, the rotation α about x axis, β about y axis and γ about z axis are obtained through the following transformations:

⎡1 0 0 ⎤ ⎢ ⎥ Rx (α ) = ⎢ 0 cos α sin α ⎥ ⎣ 0 − sin α cos α ⎦ ⎡ cos β 0 − sin β ⎤ ⎥ ⎢ Ry (β ) = ⎢ 0 1 0 ⎥ ⎢⎣ sin β 0 cos β ⎥⎦ ⎡ cos γ ⎢ R z (γ ) = ⎢ − sin γ ⎢⎣ 0

sin γ 0⎤ ⎥ cos γ 0⎥ 0 1⎥⎦

N

Δx/mm

Δy/mm

Δz/mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 0.32  0.36  0.23  0.39  0.42  0.14  0.21  0.17  0.13  0.13  0.10  0.13 0.08  0.13  0.07 0.23 0.22 0.18 0.12 0.18 0.27 0.34 0.51 0.38 0.42

 0.27  0.11 0.00 0.12 0.32 0.26 0.17  0.03  0.12  0.32  0.4  0.10 0.00 0.15 0.33 0.23 0.10  0.05  0.07  0.31  0.27  0.13  0.02 0.10 0.27

 0.06  0.09  0.13  0.18  0.07 0.18 0.02 0.07 0.10 0.08 0.08 0.08  0.03 0.00 0.09 0.14  0.02  0.08  0.01 0.22 0.00  0.08  0.21  0.16 0.05

(1)

The angles α, β and γ, evaluated by non-linear regressions, are such to position the reference geometry in the best possible correspondence with the geometry determined with the scanner. In this specific case, it is obtained α ¼ 0.0147 rad, β ¼0.0106 rad and γ ¼ 0.0048 rad. The coordinates of the 25 ball centres in BPMS after the translation and the three rotations are shown in Table 4. 4.2. Modelling of differences At this point, the differences between the coordinates of the 25

centres in BPSS after the translation (Table 3) and in BPMS after the roto-translation (Table 4) are calculated to evaluate measurement errors. Obtained results are shown in Table 5. The differences relevant to each axis contain both the effect of random and possible systematic errors. Graphical representations evidence systematic trends for each axis, as shown in Fig. 6 for Δx, in Fig. 7 for Δy and in Fig. 8 for Δz. The corresponding systematic errors are identified through linear

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Fig. 6. Least Square trend of the differences Δx vs x for the first replication.

Fig. 7. Least Square trend of the differences Δy vs y for the first replication.

Fig. 8. Least Square trend of the differences Δz vs x (a) and y (b) for the first replication. Represented trend lines are the projections of the corresponding average surfaces in x and y directions.

Table 6 Estimates of the measurement reproducibility of the scanner by considering each single replication separately and all the replications together.

regression; the following equations are obtained:

Δx = a⋅x = 4. 3⋅10−3x

Replication n.

sΔx/lm

sΔy/lm

sΔz/lm

Linear trends for Δx and Δy imply a scale error for x and y axes while saddle trend of Δz may correspond to a field curvature, which is common in optics.

1 2 3 4 5 6

72.9 73.8 73.9 73.8 73.9 74.1

30.4 31.0 29.7 1.0 29.8 33.5

56.9 56.3 68.1 49.5 50.9 56.2

4.3. Correction procedure

All

73.6

31.4

58.2

Δy = b⋅y = 4. 4⋅10−3y Δz =

c⋅x2

+

d⋅y2

= − 2.

1⋅10−5x2

+ 3.

6⋅10−5y2

(2)

Trends represented in Figs. 6–8 are the error equations relevant to the first replication. Once subtracted these trends from the data in Table 5, i.e. once performed the correction of systematic errors, no significant tendency remains evident. Then, the standard deviations of the differences relevant to each axis result to be sΔx ¼0.07 mm, sΔy ¼ 0.03 mm and sΔz ¼0.06 mm. These values give an estimate of the measurement reproducibility of the scanner for the first replication. In order to get a more robust estimate of the measurement reproducibility [1], the same procedure is applied to the other five replications. Thus, differences between the coordinates of the 25 ball centres in the specific BPSS after the translation and in BPMS after the corresponding roto-translation are calculated for the other five replications. These data are put together with the values of Table 5.

Systematic trends are again evident for each axis. Once performed the correction of systematic errors and calculated the standard deviations of the differences relevant to each axis, the values of reproducibility obtained for the first replication are confirmed. Summary results are shown in Table 6. 4.4. Uncertainty evaluation Once calculated the reproducibility, the measurement uncertainty of the scanner may be derived. According to GUM [6], systematic errors, such as the trends evidenced by the calibration operation should be corrected. However, GUM recognizes that, in some practical situations, the approach of using the original data,

G. Genta et al. / Optics and Lasers in Engineering 86 (2016) 11–19

without performing corrections is common. Accordingly, measurement uncertainty shall be evaluated in these two different operating conditions, and knowing the different uncertainty values obtained allows to decide which way is suitable for the case at hand. Measurement uncertainty once performed the correction of the systematic error includes the measurement reproducibility and the uncertainty of the error equation. The latter is derived by applying the law of propagation of uncertainty, which is presented in GUM (defined in the clause 5 and applied to experimental case studies in the annex H). In calculating the variances corresponding to the equations reported in (2), it results:

⎛ ∂Δx ⎞2 2 ⎛ ∂Δx ⎞2 2 ⎜ ⎟ u (x ) + ⎜ ⎟ u (a ) ⎝ ∂x ⎠ ⎝ ∂a ⎠ ⎛ ∂Δy ⎞2 ⎛ ∂Δy ⎞2 2 ⎟ u (b) u2 (Δy) = ⎜ ⎟ u2 (y) + ⎜ ⎝ ∂b ⎠ ⎝ ∂y ⎠ u2 (Δx) =

u2 (Δz ) =

u2 (Δy) = uΔ2 y, range + sΔ2 y

⎛ ∂Δz ⎞2 ⎛ ∂Δz ⎞2 2 ⎛ ∂Δz ⎞2 2 ⎟ u (x ) + ⎜ ⎟ u (c ) + ⎜ ⎟ u2 (y) ⎝ ∂x ⎠ ⎝ ∂c ⎠ ⎝ ∂y ⎠ ⎛ ∂Δz ⎞2 2 ⎛ ∂Δz ⎞ ⎛ ∂Δz ⎞ ⎟ u (d ) + 2 ⎜ ⎟⎜ ⎟ u (x, y) + ⎝ ∂d ⎠ ⎝ ∂x ⎠ ⎝ ∂y ⎠



⎛ ∂Δz ⎞ ⎛ ∂Δz ⎞ ⎟⎜ ⎟ u (c , d ) + 2⎜ ⎝ ∂c ⎠ ⎝ ∂d ⎠

(3)

Once calculated the sensitivity coefficients, i.e. the partial derivatives, the formulas in (3) become:

u2 (Δx) = a2u2 (x) + x2u2 (a) u2 (Δy) = b2u2 (y) + y2 u2 (b) u2 (Δz ) = (2cx)2u2 (x) + x 4 u2 (c ) + (2dy)2u2 (y) + y 4 u2 (d) + 8cdxy u (x, y) + 2x2y2 u (c , d)

respect to measurement reproducibility. For this reason, the distribution of Δx, Δy, and Δz may be approximated as normal. So, the expanded uncertainty at 95% confidence level results U(Δx)¼0.15 mm, U(Δy)¼0.06 mm and U(Δz)¼ 0.13 mm. Now, measurement uncertainty is calculated without performing the correction of the systematic error. Besides the measurement reproducibility, the variability linked to the presence of a trend on Δx, Δy and Δz should be considered. A uniform distribution on the interval of definition of x and y is assumed, so implying a uniform distribution on the interval of definition of Δx and Δy, given the linear trends. The range of Δx is 0.93 mm, while the range of Δy is 0.65 mm, so it is obtained uΔx,range ¼ 0.27 mm and uΔy,range ¼ 0.19 mm. Therefore, it is:

u2 (Δx) = uΔ2 x, range + sΔ2 x



+

17

(4)

These formulas give the variances of the error equations, i.e. the systematic contributions, without including the measurement reproducibility sΔx, sΔy and sΔz, i.e. the random contributions. According to GUM approach (implemented in the examples of annex H), the uncertainty components due to systematic errors and the uncertainty components due to random errors are assumed to be independent, hence these two components are combined in order to get the measurement uncertainty. Therefore, the squares of random contributions sΔx, sΔy and sΔz are summed to the variances of the error equations shown in (4). In this way, the formulas in (4) become:

u2 (Δx) = a2u2 (x) + x2u2 (a) + sΔ2 x u2 (Δy)

(6)

Standard uncertainties of Δx and Δy result respectively u (Δx) ¼0.28 mm and u(Δy) ¼0.19 mm. Since the variability linked to the presence of a trend is predominant with respect to measurement reproducibility, the distributions of Δx and Δy may be approximated as uniform. So, the expanded uncertainty at 95% confidence level results U(Δx) ¼ 0.48 mm and U(Δy) ¼ 0.33 mm. The same procedure is awkward for Δz given the nonlinear trend. Let us consider the area of the measurement field on the x–y plane, Atot ¼(xmax  xmin)  (ymax  ymin). The uniform distribution on x and y accepted before implies a uniform distribution on the x–y plane, therefore each elementary surface δx  δy has a probability δx  δy/Atot. The probability of having a value of Δz between Δz1 and Δz2 is given by the ratio between the corresponding area, A(Δz1 o Δzo Δz2), projected on the x–y plane and the area Atot. In this way, the relevant probability density function is defined and can be analytically determined through integral calculation from the regression function for Δz shown in (2). However, the probability density function may also be obtained by numerical methods together with the relevant standard uncertainty uΔz ¼0.07 mm. Fig. 9 shows the probability density function f(Δz), the cumulative distribution function F(Δz) and Lower Limit LL and Upper Limit UL at the confidence level of 95%. Therefore, it is:

u2 (Δz ) = uΔ2 z + sΔ2 z

(7)

Standard uncertainty of Δz results u(Δz)¼0.09 mm. Therefore, for z axis, the variability linked to the presence of a trend is comparable with measurement reproducibility. For this reason, the

= b2u2 (y) + y2 u2 (b) + sΔ2 y u2 (Δz ) = (2cx)2u2 (x) + x 4 u2 (c ) + (2dy)2u2 (y) + y 4 u2 (d) + 8cdxy u (x, y) + 2x2y2 u (c , d) + sΔ2 z

(5)

where the values u(a), u(b), u(c) and u(d) are the standard deviations of the coefficients a, b, c and d obtained through the corresponding linear regressions (shown in (2)), as well as sΔx, sΔy and sΔz. It results u(a)¼2.5  10  4 mm  1, u(b)¼1.4  10  4 mm, 1 u(c)¼3.0  10  6 mm  1 and u(d)¼5.2  10  6 mm  1. In addition, the reading resolution also produces a contribution to the uncertainties, which is however generally negligible. In our case, given the resolution of 1 mm of the coordinates of the scanner and by assuming a uniform distribution, it is obtained u(x)¼u(y)¼2.9  10  4 mm. Finally, the covariance term relevant to coordinates x and y is u(x,y)¼ 6.5 mm2, while the covariance term relevant to linear regression coefficients c and d is u(c,d)¼  7.7  10  12 mm  2. Standard uncertainties of Δx, Δy and Δz are calculated for all the values of x and y. It results u(Δx)¼0.08 mm, u(Δy)¼0.03 mm and u(Δz)¼ 0.06 mm for the extreme values of x and y, i.e. the uncertainty of the error equation is almost negligible with

Fig. 9. Probability density function f, cumulative distribution function F and Lower Limit LL and Upper Limit UL at 95% confidence level for differences Δz.

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distribution of Δz may be approximated as normal. So, the expanded uncertainty at 95% confidence level results U(Δz)¼0.18 mm.

5. Conclusions In this work, a laser triangulation 3D scanner is considered. Since a built-in calibration system is missing, an ad hoc calibration procedure is proposed. In particular, a CMM is used as reference standard for evaluating possible systematic errors of the scanner, and a ball plate is specifically manufactured to be adopted as reference workpiece. The latter is designed similarly to the ball plates used for the qualification of CMM, in accordance with ISO 10360-2 [2]. Scan data are imported and processed using the software Rapidform 2006, which is certified by the German PTB institute. The metrological chain requires the measurement of centre coordinates of the ball plate with the CMM, i.e. in Ball Plate Machine System (BPMS), and then with the scanner, i.e. in Ball Plate Scanning System (BPSS). Replicated measurements performed with the CMM are averaged to define the BPMS. This is aligned to the BPSS relevant to each replication through an optimized rototranslation, and then the differences between the coordinates of ball centres in the two reference systems are calculated. Systematic trends are identified for each axis: linear trends for Δx and Δy imply a scale error for x and y axes, while saddle trend of Δz may correspond to a field curvature. Once performed the correction of systematic errors, no significant tendency remains evident. Then, for all the considered replications, the standard deviations of the differences relevant to each axis result to be respectively about 0.07 mm for Δx, 0.03 mm for Δy, and 0.06 mm for Δz. These values yield an estimate of scanner measurement reproducibility. As a matter of fact, a more recent version Vi-9i of the Minolta scanner has a declared reproducibility of about 0.05 mm [20]. The Vi-9i is sold with a field calibration system, but it has the same hardware of the Vi-900, except for an updated opto-electronic equipment [20,21]. Therefore the improvement of the metrological performances of Vi-900 obtained by the proposed calibration procedure is confirmed. According to the specific measurement task, correction of the systematic trend can be either performed or not. In the former case, standard uncertainties relevant to Δx, Δy and Δz correspond about to measurement reproducibility leading to expanded uncertainties (at 95% confidence) within 0.15 mm, while in the latter case standard uncertainties are respectively u(Δx) ¼0.28 mm, u (Δy)¼0.19 mm and u(Δz) ¼0.09 mm leading to expanded uncertainties (at 95% confidence) even higher than 0.4 mm. In the first case, control of typical shop tolerances may be considered for industrial application. In the second case, the substantially larger uncertainty is adequate to common applications of the Vi-900 in the industrial field [22], but mainly in anatomy [23,24], dentistry [25] and cultural heritage [26,27].

Acknowledgements Cooperation of Prof. Raffaello Levi, through valuable and fruitful discussions, is gratefully acknowledged.

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Gianfranco Genta received the MSc degree in Mathematical Engineering in 2005 and the Ph.D. degree in “Metrology: Measuring Science and Technique” in 2010 from Politecnico di Torino, Italy. He is currently Research Fellow at the Department of Management and Production Engineering (DIGEP) of the Politecnico di Torino, where he has been teaching “Experimental Statistics and Mechanical Measurement” since 2012. His present research is focused on industrial metrology, quality engineering and experimental data analysis.

G. Genta et al. / Optics and Lasers in Engineering 86 (2016) 11–19 Paolo Minetola is an Associate Professor at the Department of Management and Production Engineering (DIGEP) of the Politecnico di Torino, Italy. In 1998 he obtained his BSc degree in mechanical engineering from both the Politecnico di Torino and the Universitat Politécnica de Catalunya (UPC), Spain. He got his MSc degree in mechanical engineering from the Politecnico di Torino in 2003 and his Research Doctorate in production systems engineering from the Politecnico di Torino in 2006. His research fields include 3D printing, additive manufacturing (AM), 3D scanning, reverse engineering (RE) and metrology.

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Giulio Barbato He got a degree in Nuclear Engineering in 1971, and subsequently a degree in Aerospatial Engineering at Politecnico di Torino. He started his career in 1972 at IMGC (National Institute for mechanical metrology), where he worked in the fields of Stress Analysis, Force Measurement and Hardness Measurement. In 1973 he started his cooperation with Politecnico di Torino, General Metrology course. In 1994 he won the chair of Full Professor of Mechanical Measurement. In 1997, he was called to the chair of General Metrology at Politecnico di Torino, where he still teaches Experimental Statistics and Mechanical Measurement.