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sensorsArticleGear Shape Measurement Potential of Laser Triangulation and Confocal-Chromatic Distance SensorsMarc Pillarz 1,*, Axel von Freyberg 1 , Dirk Stbener 1,2and Andreas Fischer 1,21Bremen Institute for Metrology, Automation and Quality Science, University of Bremen, 28359 Bremen, Germany; a.freybergbimaq.de (A.v.F.); d.stoebenerbimaq.de (D.S.); andreas.fischerbimaq.de (A.F.)2MAPEX Center for Materials and Processes, 330440, University of Bremen, 28334 Bremen, Germany*Correspondence: m.pillarzbimaq.decheck EojupdatesCitation: Pillarz, M.; von Freyberg, A.; Stbener, D.; Fischer, A. Gear Shape Measurement Potential of Laser Triangulation and Confocal- Chromatic Distance Sensors. Sensors 2021, 21, 937. / 10.3390/s21030937Academic Editor: Ali Bazaei Received: 4 December 2020Accepted: 28 January 2021Published: 30 January 2021Publishers Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations.Copyright: 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:/ /licenses/by/ 4.0/).Abstract: The demand for extensive gear shape measurements with single-digit m uncertainty is growing. Tactile standard gear tests are precise but limited in speed. Recently, faster optical gear shape measurement systems have been examined. Optical gear shape measurements are challenging due to potential deviation sources such as the tilt angles between the surface normal and the sensor axis, the varying surface curvature, and the surface properties. Currently, the full potential of optical gear shape measurement systems is not known. Therefore, laser triangulation and confocal-chromatic gear shape measurements using a lateral scanning position measurement approach are studied. As a result of tooth flank standard measurements, random effects due to surface properties are identified to primarily dominate the achievable gear shape measurement uncertainty. The standard measurement uncertainty with the studied triangulation sensor amounts to 10 m, which does not meet the requirements. The standard measurement uncertainty with the confocal-chromatic sensor is 6.5m. Furthermore, measurements on a spur gear show that multiple reflections do not influence themeasurement uncertainty when measuring with the lateral scanning position measurement approach. Although commercial optical sensors are not designed for optical gear shape measurements, standard uncertainties of 10 m are achievable for example with the applied confocal-chromatic sensor, which indicates the further potential for optical gear shape measurements.Keywords: optical gear shape measurements; lateral scanning position measurement approach; laser triangulation sensors; confocal-chromatic sensor; measurement uncertainty1. Introduction1.1. MotivationThe shape of the tooth flank is of decisive importance to ensure uniform power transmission under load. In gear manufacturing, the tooth flank is often designed as an involute. However, the manufacturing of the involute tooth flank is associated with high demands on tolerances in the single-digit m range 1. Regardless of the gear size, deviations between the actual and the nominal involute geometry in the micrometer range already lead to defects of the gears and consequently to failures of entire gearboxes. To reduce the gearbox failure rates, the tooth shape of all teeth must be extensively measured with a measurement uncertainty in the single-digit m range (10 m) 24. Therefore, suitable position measurement systems must be provided for extensive measuring of the shape of the tooth flanks to ensure reliable gear quality inspection.1.2. State of the ArtThe current standard gear measurement systems are tactile coordinate measuring machines (CMM) and gear measuring instruments (GMI) 5,6. CMM and GMI measure the tooth shape with a single-digit m uncertainty but are limited in speed. ExtensiveSensors 2021, 21, 937. /10.3390/s21030937/journal/sensorsSensors 2021, 21, 93722 of 22gear shape measurements are thus time-consuming. Because of the limited speed, stan- dard gear tests typically measure four teeth distributed around the circumference of the gear 2. To ensure a reliable gear quality inspection, the random test scope is not sufficient. Individual tooth shape deviations from the nominal geometry, due to manufacturing toler- ances or damage, are possibly not observed in standard gear tests 7. The standard gear measurement systems are only suitable to a limited extent for extensive gear measurements. Therefore, faster optical position sensor systems for extensive gear shape measure- ments of all teeth have recently been investigated 8. In principle, optical sensors enable measurement resolutions in the nm range and offer measuring rates up to two-digit kHz. However, commercial optical sensors are typically specified for a perpendicular sensor alignment on certain flat surfaces with a defined roughness. Besides, a maximal accep- tance angle is specified. To cover the entire tooth flank despite the limited accessibility through adjacent teeth, an optical sensor must be aligned nonperpendicularly to the gear surface. Accordingly, a tilt angle between the gear surface normal and the sensor axis is unavoidable, and the specified sensor uncertainty and thus the measurement uncertainty for the shape measurement in general increases with an increasing tilt angle. Furthermore, the sensors are not specified for measurements on the varying surface curvature of the tooth flanks and the surface properties of metallic gears in terms of the topography and the reflectivity (e.g., multiple reflections). It is, therefore, necessary to investigate to what extent the specifications for a perpendicular measurement arrangement on a flat surface also apply for gear shape measurements. The following described state of the art of optical approaches for gear shape measurements thus considers three fundamental challenges for optical gear metrology: the tilt angle between the surface normal and the sensor axis due to the limited optical accessibility, the varying surface curvature, and the surface properties. Frequently investigated measurement approaches for optical gear shape measure- ments are based on the principle of triangulation. In 2005, Younes et al. presented an optical measurement approach for spur gear measurements with a point-by-point triangulation laser 9. Because of the shading of adjacent teeth, the measurement does not cover the entire profile of the tooth flanks. Moreover, the achieved measurement uncertainty does not meet the requirements for gear shape measurements. Also, influences of the surface tilt, the curvature of the gear, and the surface properties are not discussed. 3D gear shape measurement principles based on laser line triangulation sensors are described by Auer- swald et al. in 2019 and Guo et al. in 2020 8,10. Auerswald et al. investigated the laser line triangulation measurements on large helical gears. The entire tooth flank is measured and then the measurement data is compared with the reference geometry. A measurement deviation of8.2 m is currently achieved. Auerswald et al. show that multiple reflections influence the measurement deviations because of the highly reflective and curved surface of the tooth depending on the sensor alignment. The potential of optical triangulation sensors for gear shape measurement is also shown by Guo et al. in 2020. For an extensive quality assessment of the tooth shape, they performed 3D measurements of the entire flank. At present, a mean measurement deviation from the ideal tooth flank of approximately 10 m is achieved. However, it is not reported, where the remaining deviations result from. Peters et al. introduced in 2000 a fringe projector combined with a rotary table to scan the shapes of the tooth flanks of small helical gears 11. Due to their measurement approach, they cover the entire tooth flank. The achieved measurement deviations to the theoretical tooth flank amount to approx.10 m. Peters et al. explain that the sensor should be aligned as perpendicular to the tooth flank as possible to minimize measurement deviations. Influences because of the varying curvature and the highly reflective surface are not mentioned. In 2006, Mee et al. investigated a 3D gear shape measurement approach based on stripe pattern projection 12. They show that due to the complex gear geometry and the resulting optical accessibility, systematic length measurement deviations occur in fringe projection. Deviations to the reference geometry of approx.10 m are determined after calibration. Area-oriented gear shape measurement approaches using projection moir are presented in 3,13. In 2005, Sciammarella et al. measured the profile contour ofan entire tooth and achieved a measurement uncertainty 3.5 m after a calibration of the measurement set-up. Challenges due to the tilt angle between the surface normal and the sensor axis, the varying curvature, and the surface properties are not described. Chen et al. presented in 2019 3D measurements of moir projection on a small spur gear. Here, too, the entire tooth flanks are measured. The measurement deviations amount to 5 m and uncertainty contributions due to the surface tilt, the curved geometry, and the surface prop- erties are also not discussed. In summary, the current triangulation approaches for optical gear shape measurements indicate a great potential for a fast and extensive gear shape quality inspection. However, a detailed characterization of the remaining measurement deviations from the reference geometry concerning the aforementioned challenges (the surface tilt, the curved involute geometry, and the surface properties) is still pending.Further investigated optical gear shape measurement approaches are based on interfer- ometry. Fang et al. measured the geometry of a tooth flank in 2014 and after compensation of installation deviations, the measurements are in good agreement with the reference geometry 14. However, the deviations are not further specified and not discussed in terms of the challenges in optical gear shape metrology. Balzer et al. used in 2015 and 2017 a frequency-modulated interferometric sensor 2,7. They combined the interferometric sensor with a CMM by integrating the beam path via fiber-optics into the sensor head of the CMM. Information on deviations between measured and reference flank is not given.Confocal-chromatic distance sensors are another promising optical sensor concept for measuring gear shapes because they are typically specified as surface independent and distance uncertainties in the low single-digit micrometer range are claimed to be achievable 15. Consequently, the reflecting and curved tooth flanks are measurable. First measurements on small spur gears showed that measurement deviations from the reference geometry of 10 m are achievable 16. The reason for the deviations was primarily the motion control unit of a rotary table on which the gear to be measured was positioned to enable the continuous detection of the tooth flanks. Influences due to the varying curvature and surface properties were mentioned but not characterized in detail. Moreover, because of a perpendicular alignment of the sensor to the tooth flank, not the entire tooth flank could be detected. Hence, further investigations are necessary to understand the potentialof confocal-chromatic sensors for optical gear shape measurements.In summary, measurement uncertainties of 25 mm is required to measure the entire tooth flank.Another position measurement approach for gear shape measurements is shown in Figure 1b. It consists of an optical sensor that is mounted on a linear unit for scanning. The optical sensor is positioned perpendicular to the scanning direction of the linear unit. Both the optical sensor and the linear unit form a scanning measuring unit. The alignment of the measuring unit can be variably adjusted according to the tooth flank region to be measured and the maximum acceptance angle of the distance sensor. Here too, the optical sensor must be turned in the direction of the gear center to measure the entire tooth flank. The tooth flank is scanned by the lateral movement of the measuring unit and the measurement data are also stored in the form of coordinates Ps,i = (xs,i, di) in the measuring coordinate system. Note that according to Figure 1b, only the shape of one tooth is measured. To measure the tooth shape geometry of all teeth, a rotary table is necessary for the sequential positioning of each gear tooth in the measurement region. In order to characterize the potential of an optical sensor for the application on gears, however, the measurement of one tooth flank is sufficient. Due to the laterally scanning measuring strategy and the involute tooth geometry, a continuously changing tilt angle between the gear surface normal and the sensor axis occurs. If the entire tooth flank is to be measured, the maximum tilt angle occurs at the tooth root.In comparison with the rotational scanning approach, the laterally scanning approach requires a similar acceptance angle but a significantly smaller measuring range to measure the entire tooth flank. For the shape measurement of a gear with a normal module of 10.64 mm, 20 teeth, and a base circle radius of approx. 100 mm, the required acceptance angle is still approximately 30 while the required measuring range amounts to only 4 mm. For this reason, only the position measurement approach with lateral scanning is considered here further.2.2. Assessment of the Gear Shape Standard Measurement UncertaintyIn order to determine the measurement uncertainty of the optical gear shape measure- ment, the measured gear geometry needs to be compared with reference data. For this purpose, the measured distances and the reference gear geometry must be transferred to a common coordinate system. Here, the workpiece coordinate system (x, y) is selected. To transfer the sensor data to the workpiece coordinate system by mathematical transforma- tions, the position and the alignment of the sensor relative to the workpiece coordinate system, respectively the exact transformation between the sensor coordinate system (xs, ys) and the workpiece coordinate system should be known, which is not the case here. Assum- ing that the center of the gear is ideally located in the center of the workpiece coordinate system, the measurement data is fitted to the reference geometry. For this purpose, a geometric model of the gear is required.Nonmodified involute spur gears are considered here, and their nominal gear geom- etry is used as a reference. In Figure 2, this nominal geometry in the transverse section is shown in a workpiece coordinate system (x, y) that is shifted and rotated to the sensor coordinate system (xs, ys). The nominal geometry of involute spur gears can be described by geometry and position parameters employing the geometric model according to 1719: A nominal point Pi=iPxp,iyp,iabrcos(i + z b)=+=bsin(i + z b)rsin(i + z b)+ ib cos(i + z b)(1)2on the tooth flank of tooth Z results from an addition of a radial vector a and a tangential vector b in the workpiece coordinate system (x, y). The vector a has the length of the geometry parameter base circle radius rb of the spur gear while the angle i = i + z b depends on position parameters i, z, b. The length of the tangential vector b amounts to rbi with a corresponding angle of i . The parameter i describes the rolling angleof the spur gear assigned to the nominal point Pi. The center axis of a tooth Z is defined by the angle z and the base tooth-thickness half-angle by the angle b.Figure 2. Geometric model of a nonmodified involute gear for calculating the plumb line distance dplu,i between the measured and the nominal geometry based on a measured actual point Pa,i on tooth Z, the base point Pi of the nominal geometry for the plumb line distance and the positionparameters i, z, b, T , 0, rI,i, i, n i in the workpiece coordinate system. The plumb line distancebetween the measuring point and the nominal geometry of the tooth flank is displayed enlarged. Figure 2 is modified according to 20.Because a measured point Ps,i = (xs,i, di) is acquired in the sensor coordinate system (xs, ys), a translation vector T = xt, ytT as well as a rotation angle 0 to the workpiece coordinate system must be addedya,iPa,i = xa,i = R(0)(Ps,i + T )(2)If deviations between the optically measured actual points Pa,i and the nominal points Pi in workpiece coordinate system occur, they can be described by the plumb line distance dplu,i. A measured point Pa,in iPa,i = Pi + dplu,i n i(3)is therefore composed of the addition of the nom. in.al point Pi and the plumb line distance, which is oriented in the normal direction n i=insin(i + z b) cos(i + z b)(4)to the surface of the nominal geometry. To characterize the gear shape measurements using the position measurement approaches realized with a laser triangulation and a confocal- chromatic sensor, respectively, the plumb line distances must be determined. Therefore, the inverse problemdplu,i = f (Pa,i(xs,i, di, T , 0), rb, i, z, b)(5)must be solved. Note that the transformation parameters between the sensor and work- piece coordinate system (T , 0) as well as the position parameter i are unknown. When solving the inverse problem, the unknown parameters are also calculated. The parameters (rb, z, b) result from the nominal geometry of the gear as well as from the measurement conditions.In order to solve the inverse problem, the geometric model of the nominal involute is fitted to the measured data by iteratively minimizing the sum of the squared plumb line.distancesmini, xt, yt, 0ki=12dplu,i,(6)I,ii.e., by using a nonlinear least-squares method. According to 17,21, the plumb line distances to the nominal involute of a spur gear are determined byI,i., r2r b., r2rdplu,i = rb2 1 f ladir i z + b 0 + arctanf ladirbbThe appropriate rolling angles i2 1.(7)i =2., rr 1(8)I,i2bof the root points of the plumb line distances on the involute are determined implicitly byrI,i and rb. The parameter rI,irI,i = .(xa,i xt)2 + (ya,i yt)2(9)i.describes the radius of the measured actual point Pa in polar coordinates in the workpiece coordinate system. The parameter i= arctan ya,i yt(10)xa,i xtis the corresponding polar angle to rI,i. The symbol “fladir” defines a factor for the flank side (left side:1, right side: 1).The standard measurement uncertainty for the gear shape measurement can be as- sessed based on the plumb line distances. The assessment procedure is shown in Figure 3. The plumb line distances represent the local measurement deviation between the measured geometry and the reference geometry. The calculated local measurement deviations are in- fluenced by systematic and random effects. Because systematic deviations can be corrected by calibration measurements, the potential of optical sensors for gear shape measurements is mainly determined by the occurring random deviations. Therefore, in this article, the standard deviation of the random deviations is evaluated as a measure of the gear shape standard measurement uncertainty.Figure 3. Procedure for determining and assessing the gear shape standard measurement uncertainty.3. Experimental Set-UpTo characterize the potential of triangulation and confocal-chromatic sensors for optical gear shape measurements, the lateral scanning position measurement approach from Figure 1b is realized. Note that the particular aim of the experiments is to characterize the influence of the tilt angle between the gear surface normal and the sensor axis, the varying curvature as well as the gear surface properties like the topography and the reflectivity (multiple reflections) on the gear shape measurement results. For this purpose, measurements on a tooth flank standard as well as on a small spur gear with nonmodified involute tooth flanks are performed.3.1. Measurement ObjectsThe two measurement objects are an involute tooth flank standard and a small spur gear, see Figure 4. Both measurement objects have a metallic ground surface. The parame-ters that describe the nominal geometries are listed in Table 1. Because the geometry of each measurement object is well-known, no measurement artifacts due to surface defects are expected to occur.Figure 4. Measuring objects for evaluating the potential of the triangulation sensor and confocal-chromatic sensor for gear shape measurements and for investigating the measurement deviation contributions through the tilt angle between the gear surface normal and the sensor axis, the varying surface curvature, and gear surface properties. (a) shows an involute tooth flank standard with a nominal geometry with a normal module of 10.64117 mm, 20 teeth, and a base circle diameter of 199.99 mm. (b) shows a spur gear with involute profile and a nominal geometry with a normal module of 3.75 mm, 26 teeth and a base circle radius of 91.62 mm. In addition, the measuring spot and multiple reflections of a triangulation measurement are visible on the spur gear.Table 1. Geometric parameters of the tooth flank standard and small spur gear.mmNumber of TeethNormal Module inBase Circle Diameter in mmtooth flank standard2010.64117199.99spur gear263.7591.62The tooth flank standard is a free-standing tooth flank and is therefore easily accessible for the optical sensors, see Figure 4a. No adjacent teeth restrict the measuring arrangement so that the entire involute of the tooth flank is measurable with a minimal sensor acceptance angle in addition to a small measuring range. With an optimal arrangement of the optical sensor, tilt angles 18 are possible both at the tooth root and at the tooth tip. The curvature decreases monotonically from the root (0.16 1 ) to the tip (0.02 1 ) of the tooth. The toothmmmmflank standard is therefore particularly suitable for investigating the effect of the tilt angle between the gear surface normal and the sensor axis as well as the effect of the varying tooth flank curvature on the measurement deviation. Furthermore, the measurement deviation contribution due to the surface properties can be examined concerning the topography and the surface scattering/reflection behavior, separately from disturbances due to multiple reflections. Because the tooth flank normal is free-standing, no multiple reflections from adjacent teeth occur.A small spur gear is used in addition to the tooth flank standard to examine the effect of multiple reflections due to the metallic, highly reflective surface and the adjacent teeth on the measurement results. The spur gear is shown in Figure 4b. To quantify the effect of multiple reflections, comparison measurements are performed on one tooth with and without covering the adjacent tooth flank with a layer of black carbon that reduces the light reflections. Here, adjacent teeth restrict the sensor alignment to the tooth flank.3.2. Measurement ArrangementFigure 5 shows the experimental set-up of the realized lateral scanning position measurement approach according to Figure 1b. As an example, the set-up is shown for the gear shape measurements on the tooth flank standard using the confocal-chromatic sensor. As an optical distance sensor, a confocal-chromatic sensor and a laser triangulation sensor are used, respectively.Figure 5. Experimental set-up of the optical position measurement approach for the gear shape measurement with a linear unit for laterally scanning the tooth flank of a tooth flank standard and using the confocal-chromatic sensor as an example. An additional linear unit allows the height of the optical sensor to be adjusted. To examine the influence of multiple reflections, the tooth flank standard is exchanged with a spur gear.The optical distance sensor is mounted as perpendicular as possible to the scanning direction of the motorized linear unit. The linear unit is specified with a position uncertainty of 1 m per 25 mm traveling distance. Both the sensor and the linear unit form the scanning measuring unit. An additional linear unit enables the height adjustment of the optical sensor.The tooth flank standard and the small spur gear are mounted in front of the optical sensor with an appropriate gear clamping unit, respectively. The measuring unit and the tooth flank to be measured are aligned with each other with the aim of measuring the maximum gear tooth region. Regarding the free-standing tooth flank standard, the measuring unit is slightly turned in the direction of the gear center until the size of the tilt angle between the surface normal and the sensor axis is almost the same at both the tooth tip and the tooth root. If the tilt angle at the tooth tip exceeds the acceptance angle of the sensor, the sensor is turned back step by step in order to measure the shape of the tip. When measuring the spur gear, the measuring unit is oriented past the adjacent tooth in the direction of the tooth root until the maximum acceptance angle is accomplished. The measurement objects are then positioned in the middle of the measuring range of the optical sensor. Further, the sensor axis must be arranged in the transverse section of the measured gear. Deviations from a perpendicular sensor adjustment to the scanning movement and a sensor axis alignment in the transverse section of the gear lead to ageometrically distorted measurement result. The measuring unit then laterally scans the shape of the tooth flank.As a result of the chosen measurement arrangement, the entire tooth flanks of the tooth flank standard and the spur gear are measurable within a measuring range of 4 mm, and only the acceptance angles of the optical sensors limit the measurable gear tooth region. On the tooth flank standard, the maximum occurring tilt angle between the surface normal and the sensor axis amounts to approx. 18. When measuring the entire flank of the small spur gear, a maximum tilt angle of 36.7 occurs.3.3. Specifications of the Optical Distance SensorsThe specifications of the applied laser triangulation sensor and the confocal-chromatic sensor are listed in Table 2. Note that the reproducibility of the confocal-chromatic distance sensor is not specified in the datasheet and thus determined experimentally. The listed sensor parameters are subsequently discussed concerning the deviation sources in the order: (1) tilt angle between the gear surface normal and the sensor axis, (2) varying curvature, (3) gear surface properties.Table 2. Specifications of the confocal-chromatic sensor and the laser triangulation sensor. The reproducibility of the confocal-chromatic distance sensor is determined experimentally because no value of the reproducibility is specified in the datasheet.Laser Triangulation SensorConfocal-Chromatic Sensormeasuring range50 mm10 mmacceptance angle3017 light spot diameter wd55570 m16 mlinearity error30 m2.5 mreproducibility2 m0.3 mThe laser triangulation distance sensor is specified for a white, diffuse reflecting ceramic surface. The measuring range is 50 mm and the acceptance angle amounts to 30. Hence, considering the geometry of the free-standing tooth flank standard, the entire flank is measurable. The measurement of the entire tooth flank of the spur gear is not feasible due to the required acceptance angle of 36.7. Approximately 83% of the gear tooth range from tip to root is measurable according to the sensor specifications. Moreover, the datasheet specifies a possible measurement deviation due to the measurement of tilted surfaces. The exact value of the measurement deviation depends on the surface properties of the measurement object. The light spot size and thus the lateral resolution varies over the measuring range between 55 m to 570 m. In the middle of the measuring range, the light spot has the smallest diameter. Due to the surface tilt and the varying curvature of the tooth flank, a distance variation occurs within the measuring spot. Depending on the factory-made signal processing of the triangulation sensor, the distance variation within the spot can lead to the tilt angle-dependency and further a curvature-dependent measurement deviation contribution. As a result, a deterministic systematic effect is assumed, respectively, because the tilt angle, as well as the curvature, varies continuously when measuring the tooth flank.The laser triangulation sensor is characterized by a linearity error of30 m, but anexact characteristic curve of the linearity error depending on the measured distance is notknown. It is assumed that a part of the linearity error on the measurement deviations of the gear shape measurements depends on the random local surface properties. Therefore, a random deviation contribution to the achievable measurement uncertainty is expected.The specified reproducibility of the laser triangulation sensor is 2 m, which rep- resents the random measurement deviation of the sensor. Measurement deviations forrepeated measurements are therefore not significant compared to the expected systematic deviations due to surface tilt, the surface curvature, and the surface properties.The confocal-chromatic distance sensor is characterized by a flat and reflective glass surface. The measuring range is 10 mm and the maximum permissible acceptance angle is 17. According to the geometry of the tooth flank standard, the specified acceptance angle of the confocal-chromatic sensor limits the measurable gear tooth region. Theoretically, 97.5% of the gear tooth region from tip to root is measurable. Also, the measurable tooth flank region of the spur gear is restricted due to the specified acceptance angle. The maximum measurable tooth flank region amounts to 60% (from tip to root). The datasheet describes that a possible measurement deviation emerges when the surface is tilted to the sensor axis, but this deviation is not further quantified. The light spot size of the confocal-chromatic sensor and thus the lateral resolution amount to 16 m. Similar to the triangulation sensor, the surface tilt and the curvature of the tooth flank also lead to a distance variation within the measuring spot. In dependence on the signal processing of the confocal-chromatic sensor, a deterministic effect on the measurement deviation istherefore possible.The linearity error of the confocal-chromatic sensor amounts to 2.5 m in the mea- suring range. Here too, an exact characteristic curve of the linearity error behavior of the sensor is not known. It is expected that this influence on the measurement deviation is also dependent on the random local surface properties of the gear. Typically, confocal-chromatic sensors are known as surface-independent. However, a random deviation contribution to the measurement uncertainty is awaited. The extent to which the tooth shape measurement is influenced by the surface properties is not apparent from the sensor specifications.A reproducibility value is not specified for the confocal-chromatic sensor. Thus, the value for the reproducibility is determined experimentally to 0.3 m. Here too, the measurement deviations for repeated measurements are significantly smaller in comparison to the assumed systematic deviations due to the surface tilt, the surface curvature, and the surface properties.3.4. Experimental Test SeriesThe influence of the tilt angle between gear surface normal and the sensor axis, as well as the varying curvature of the tooth flank on the measurement deviation are examined by comparing the measured gear shape with the reference geometry of the tooth flank standard. Based on the fitted measurement data the tilt angles and the local curvature are estimated. Note that the influence of the tilt angle and the surface curvature are superimposed during the gear shape measurement. The absolute value of the tilt angle varies almost symmetrically around the center of the measured flank whereas the surface curvature decreases monotonically from the root to the tip of the flank. Hence, it is assumed that two different plumb line deviations have to be observed for each absolute tilt angle if the surface curvature has a significant influence on the measurement deviations.In order to estimate the magnitude of both effects, a theoretical contemplation of the geometrical influence of the tilt angle and the surface curvature depending on the lateral resolution of the sensors is carried out. When measuring a tilted surface (see Figure 6a) or a curved surface (see Figure 6b) a distance variation d occurs within the light spot of the sensor. For the estimation of the geometrical influence of the tilt angle according to Figure 6a, a tilt angle of = 18 is applied. The results of the theoretical contemplation are shown in Table 3. In the middle of the measuring range of the triangulation sensor, a distance variation of 17.9 m occurs within the measuring point. For the confocal-chromaticdistance sensor, the distance variation within the measuring spot amount to 5.2 m.Figure 6. Principle sketch for the theoretical estimation of the distance variation d within the measuring spot with a diameter of wd of the optical distance sensors resulting from the tilt angle between tooth surface normal and sensor axis and the curvature of the tooth flank. (a) shows the influence of the tilt angle simplified on a flat surface. (b) shows the influence of the curvature of the tooth flank for a gear geometry corresponding to the tooth flank standard.Table 3. Maximum distance variation d as a function of the light spot diameter wd of the sensors and the maximum estimated tilt angle of approx. 18 between the surface normal and the sensor axis and the varying surface curvature, respectively. The results are based on the geometry of the tooth flank standard and the lateral scanning position measurement approach used.Laser Triangulation Sensor (Mid-Range)wd = 55 mConfocal-Chromatic Sensorwd = 16 mdtilt in m17.95.2dcurvature in m0.040.003The occurring distance variations within the respective measuring spots due to the curvature are three orders of magnitude smaller than the estimated distance variations resulting from the surface tilt. Regarding these theoretical results, no significant influence of the surface curvature on the plumb line deviations is expected. Only the tilt angle can lead to observable deviations, which should be independent of the angle sign.The assumed random deviation contribution due to the surface properties, especially of the peculiarity of the random local topography on the measurement deviations, is investigated by comparison measurements of the shape of the tooth flank standard at different heights. Except for the measuring position in the height, the measuring conditions are kept constant. Thus, the occurring tilt angles between the surface normal and the sensor axis as well as the occurring local curvatures are nearly the same during the comparison measurements. Merely the local surface topography changes. Hence, with the comparison measurements, the interaction of the optical sensor principles with the surface topography and the potential effect on the measurement deviation can be studied.The gear shape measurement of the free-standing tooth flank standard lacks the con- sideration of the possible effect of surface-dependent multiple reflections from adjacent teeth on the measurement uncertainty. Therefore, this effect is investigated with compar- ative shape measurements on a small spur gear with and without covering the adjacent tooth flank with a black carbon layer. The black carbon layer is used to reduce the specular light reflections because black carbon absorbs usually more than 95% of the incident light. During the comparison measurements, the measurement conditions are not changed except for the additional black carbon layer. Thus, the same points of the gear surface are detected and the effect of the multiple reflections on the measurement deviations can be determined.4. Results4.1. Tooth Flank Standard MeasurementsIn the following, the measurement results on the tooth flank standard are presented and studied concerning the potential systematic deviation sources: surface tilt, varying curvature, and surface properties. In the end, in order to quantify the potential of the sensors for optical gear shape measurements, the standard measurement uncertainty due to random deviations is estimated. Note that the total uncertainty is also influenced by the systematic deviation sources like, e.g., the influence of the surface gradient or the alignment of the sensor and the linear unit. Therefore, the total uncertainty can increase significantly, if these systematic influences are not thoroughly eliminated by precise alignments and calibration measurements.4.1.1. Systematic DeviationsThe tooth flank is scanned laterally step by step with a lateral resolution of 0.175 mm. Per measuring position 100 distance measurements are performed and a total of 102 mea- suring positions is recorded with both the laser triangulation and the confocal-chromatic sensor. Figure 7 shows the results of one laser triangulation gear shape measurement (see Figure 7a,b) and one confocal-chromatic gear shape measurement (see Figure 7c,d).Figure 7. Results of one gear shape measurement with the triangulation sensor and confocal- chromatic sensor on the involute tooth flank standard, respectively. (a) shows the transformed triangulation measurement points as blue dots in comparison to the reference geometry of the tooth flank as a black line in a common coordinate system. Note that the x- and y-axis are scaled differently.(b) shows the plumb line distances between the reference and the measured geometry of the tooth flank plotted over the x-positions of the laser triangulation measurement points. (c) illustrates the transformed confocal-chromatic measurement points as blue dots compared to the reference geometry of the tooth flank as a black line in a common coordinate system. (d) illustrates the plumb line distances between the reference and measured geometry of the tooth flank depending on the x-positions of the confocal-chromatic measurement points.In Figure 7a, the fitted laser triangulation measurement data to the reference geometry of the tooth flank standard in the workpiece coordinate system are shown. The blue dots illustrate the measurements and the black line the reference involute. Figure 7b illustrates the corresponding plumb line distances between the reference geometry and the measured geometry depending on the x-component of the measurement points according to Equation (6). With the laterally scanning position measurement approach using the triangulation sensor, the entire tooth flank is measurable. The curvature of the measurement data deviates from the reference geometry of the involute tooth flank standard. In the area of the root as well as in the area of the tip, the measured involute is more curved than the reference geometry. Figure 7b illustrates the deviation of the measurement data compared to the reference geometry of the tooth flank. The plumb line distances systematically deviate with a parabolic trend from the reference tooth flank in an order of magnitude of 100 m. Moreover, the measurement data scatter around the parabolic trend in an order of magnitude of approx. 40 m. The scattering occurs systematically reproducible within the specified random error of the triangulation sensor.Figure 7c shows the fitted measurement data to the reference geometry of the tooth flank standard in the workpiece coordinate system measured by the confocal-chromatic sensor. Again, the blue dots illustrate the measurements, and the black line the reference geometry. The appropriate plumb line distances are plotted over the x-component of the measurement points (see Figure 7d). Here too, almost the entire tooth flank is measurable using the lateral scanning position approach. Based on the fitted measurement data to the reference geometry, a maximum tilt angle of 18.1 occurs at the tooth root. In gear measurements, thus, a larger angular range can be measured than specified. Note that the measurement value at x 94 mm was declared an outlier due to a significant deviation from the measured curve and removed for evaluation. The shape of the tooth flank standard measured with the confocal-chromatic sensor also deviates from the reference geometry. At the root as well as the tip of the tooth, the measured involute is more curved than the reference involute, similar to the behavior of the laser triangulation data. The results of the confocal-chromatic measurement scatter with a magnitude of around 10 m within thespecified random error of the sensor.Both the triangulation and the confocal-chromatic measurements systematically de- viate parabolically from the involute reference geometry. Because a stronger curvature is perceived above all in the area of the root and the area of the tip (symmetrical behavior), the tilt angle between the surface normal and the sensors axis is assumed to have a significant systematic influence on the observed deviations. However, with the used measurement approach the influence of the surface tilt on the measurement deviations cannot be sepa- rated from a potential influence of the surface curvature (see Section 3.4). To qualitatively assess the effects of tilt and curvature, the symmetry behavior of the tilt angle around the center of the measured flank is exploited. Therefore, Figure 8 shows the dependence of the plumb line distances on the estimated absolute values of the tilt angles between the surface normal and the sensor axis. The tilt angles of triangulation and confocal-chromatic measurement are determined based on the fitted measurement data. Here, an almost perpendicular sensor alignment to the scanning units axis is assumed. Thus, the calculated tilt angles are a mathematical estimate that appears geometrically plausible.Figure 8a shows the plumb line distances for the laser triangulation measurement. The crosses represent the plumb line distances to the mathematically defined negative tilt angles in the direction of the tooth root and the diamonds represent the plumb line distances to the mathematically positive defines tilt angles in the direction of the tooth tip. If the absolute value of the tilt angle between the surface normal and the sensor axis increases in magnitude, the measuring deviation decreases to the negative m range. A dependence on the angle sign is not seen within the scattering of the plumb line distances. Besides, within the scattering of the measurement data, no separate influence of the monotonically decreasing surface curvature from the root to the tip of the flank on the plumb line distances is detectable. Analogous to Figure 8a, Figure 8b illustrates the plumb line distances of theconfocal-chromatic gear shape measurement depending on the absolute values of the tilt angles. Here too, the measurement deviation decreases to the negative m-range when the absolute value of the tilt angle increases. Again, within the scattering of the plumb line distances no evidence for a separate influence of the surface curvature is observed.Figure 8. Dependence of the determined plumb line distances of (a) the triangulation gear shape measurements on the tooth flank standard and (b) the confocal-chromatic gear shape measurements on the tooth flank standard on the absolute values of the estimated tilt angles between the gear surface normal and the sensor axis. The crosses represent the tilt angles in the direction of the root (mathematically negative) and the diamonds represent the tilt angles in the direction of the tip (mathematically positive). From the symmetrical behavior of the plumb line distances it can be seen that no separate influence of the curvature can be observed within the scattering and that a significant dependence on the surface tilt can be expected.Based on these results, the surface tilt seems to be the cause of the parabolic deviations, which corresponds to the theoretical contemplations. It is noticeable, however, that the deviation trend of the determined plumb line distances is comparable in shape and mag- nitude for both measurement methods. An influence by the surface tilt alone is therefore unlikely because both measurement methods are based on markedly different measuring principles. It is accordingly assumed that the parabolic deviation is influenced by another, unknown source of deviation. It is independent of the used measuring principles and can result, for example, from the fitting algorithm or the sensor alignment.In summary, within the scatter of the plumb line distances of both the triangulation and the confocal-chromatic measurement, a separate effect of the varying surface curvature of the involute cannot be validated, which agrees to the theoretical contemplations. A parabolic systematic influence of the tilt angle between the surface normal and the sensors axis as well as an additional unknown source on the measurement deviation for both sensor principles is quantified and can therefore be considered for further uncertainty estimations.4.1.2. Random DeviationsFor the study of the achievable measurement uncertainty due to random deviation sources in the gear shape measurements, the parabolic systematic deviation is mathe- matically eliminated. Therefore, a calibration based on both the triangulation and the confocal-chromatic measurement results is performed. The remaining scatter is assumed to primarily result systematically from the interaction with the surface properties such as the randomly changing local topography. To study the assumed influence of the surface topography, the results of comparison measurements at different heights on the tooth flank standard are discussed.Figure 9 shows the remaining plumb line distances of the corrected comparison measurements of the tooth flank standard using the laser triangulation sensor. The results of the comparison measurements show the sufficiency of the model-based compensation for the parabolic systematic deviation. Moreover, the comparison measurements of the tooth shape show a different scattering as a function of the measured x-position. Thisobserved scattering between measurements at different heights shows a random deviation behavior, which agrees with the assumption about the surface topography influence. Thus, the standard measurement uncertainty from random sources of the optical laser triangulation gear shape measurement can be calculated from the empirical standard deviation of the determined random deviations (see Figure 3). The achieved standard measurement uncertainty of the triangulation measurements amounts to 18 m. Based on this result, the measurement approach using the laser triangulation sensor currently does not achieve the required measurement uncertainty of 10 m even by only considering the random deviation sources. Besides, a maximum deviation of 55 m occurs. The results show that the specified measurement deviation (linearity error) of the triangulation sensor of 30 m cannot be maintained for gear shape measurements.Figure 9. Remaining deviations of the corrected comparison triangulation gear shape measurements as a function of the x-component of the measurement point. The measurements are performed at different heights on the tooth flank standard to study the influence of the local topography of the surface on the measurement deviation.In Figure 10 the remaining plumb line distances of confocal-chromatic gear shape measurements at different heights on the tooth flank standard are plotted as a function of the x-component of the measuring points. A second nonlinear x-axis illustrates the dependency of the plumb line distances of the tilt angles. Here too, the systematic parabolic trend is compensated successfully. The pattern of the scatter as a function of the measured x-position mainly differs randomly for the comparison measurements of the tooth shape. Thus, a random error due to random surface properties (topography) also primarily influences the achievable measurement uncertainty of the gear shape measurement using a confocal-chromatic sensor. With the confocal-chromatic sensor, a standard measurement uncertainty of 6.2 m is achievable. If the surface tilt exceeds angles of 10, mainly systematic measurement deviations of 20 m occur, which probably result from the proximity of the tilt angle to the limit of the sensors acceptance angle of 17. The specified linearity error of 2.5 m of the confocal-chromatic sensor, therefore, does not apply to gear shape measurements.Figure 10. Remaining deviations of the corrected comparison measurements of the gear shape using the confocal-chromatic sensor as a function of the x-component of the measurement point. A second x-axis illustrates the dependency of the plumb line distances of the tilt angles. The measurements are performed at different heights on the tooth flank standard to study the influence of the local topography of the surface on the measurement deviation.In summary, the observed random deviations for the comparison measurements at different heights of the tooth flank standard verify the hypothesis that surface properties do have a significant influence on the random deviations. Besides, systematic deviations at the edge of the acceptance angle range influence the achievable uncertainty. The results show the potential of confocal-chromatic sensors for gear shape measurements because the achieved standard measurement uncertainty for random deviation contributions meets the required measurement uncertainty of 30 required). The results of the comparison measure- ments using the laser triangulation sensor indicate no significant influence of multiple reflections on the measurement uncertainty (see Figure 11). Both measurements scatter with the same order of magnitude. The standard measurement uncertainty of the tooth shape measurements with a black carbon coated adjacent tooth is 0.1 m smaller than the standard measurement uncertainty without the coated adjacent tooth. The deviation of0.1 m is within the specified reproducibility of the sensor (see Table 2) and is therefore negligible. Only the calculated plumb line distances vary locally, although almost the same points of the gear surface are measured during the comparison measurements. A causefor the local deviations could be the invasive application of the carbon black coating to theadjacent tooth. The measured points on the tooth flank could therefore differ slightly in the comparison measurements.Figure 11. Plumb line distances of the comparison measurements of the gear shape using the triangulation sensor. The blue crosses present the measurement without the coated adjacent tooth. The red crosses are the results of the measurement with the coated adjacent.The determined plumb line distances of the comparison measurements using the confocal-chromatic distance sensor are illustrated in Figure 12. Here, too, no influence of multiple reflections can be observed on the standard measurement uncertainty. The results of the measurements scatter within the same order of magnitude. Moreover, the shape of the plumb line distances is comparable. The standard measurement uncertainty of the measurements with the coated adjacent tooth is 0.04 m larger than the standard measurement uncertainty of the measurements without the coated adjacent tooth. The relative deviation of 0.04 m is within the reproducibility of the confocal-chromatic sensor.Figure 12. Plumb line distances of the comparison measurements of the gear shape using the triangulation sensor. The blue crosses present the measurement without the coated adjacent tooth. The red crosses are the results of the measurement with the coated adjacent.In summary, the results for the laser triangulation sensor and the confocal-chromatic sensor show that multiple reflections do not influence significantly the tooth shape mea- surement with a lateral scanning position measurement approach.5. Conclusions and OutlookIn this article, the potential of a commercial triangulation sensor and a commercial confocal-chromatic sensor for the measurement of gears is studied. In addition, the sources of deviations such as the tilt angle between the gear surface normal and the sensor axis, the varying surface curvature, and the gear surface topography are examined. For that purpose, measurements on a tooth flank standard and a small spur gear are performed with both measurement principles. The comparison measurements on a small spur gear show that the standard measurement uncertainty in optical gear shape measurements is not affected by multiple reflections. However, a systematic parabolic deviation to the reference geometry is observed in both the triangulation and the confocal-chromatic gear shape measurements on the tooth flank standard. Due to the symmetry of the parabolic deviation, an influence of the monotonically varying surface curvature of the involute can be neglected. The deviation can partly be attributed to an influence of the surface tilt, but not fully explained. Hence, an unknown source of deviation is assumed, which is expected to result from the fitting procedure to calculate the plumb line deviations of the measurements from the reference geometry. To eliminate the parabolic systematic deviation, a calibration based on both the triangulation and the confocal-chromatic measurements is performed. A randomly scattering deviation due to the local topography dominates the achieved residual deviations for both sensors.To assess the potential of the laser triangulation sensor, the achievable standard measurement uncertainty from random sources is calculated to 18 m from the residual deviations, which does not comply with the requirements of a single-digit m uncertainty. Even if the entire tooth flank of the tooth flank standard is measurable with the laser triangulation sensor, maximum deviations of 55 m occur and the specification of the linearity error of the triangulation sensor do not apply to gear shape measurements.The achievable standard measurement uncertainty due to random sources of the confocal-chromatic gear shape measurement of the tooth flank standard amounts to6.5 m. In addition, a systematic deviation at the edges of the acceptance angle range superimposes with the random deviation. If the tilt angles exceed 10, local deviations of 20 m occur. Although the determined uncertainty is higher than the value of the linearity error of the confocal-chromatic sensor, the obtained results meet the requirements for gear shape measurements and show potential for optical gear shape measurements using confocal-chromatic sensors.In summary, the potential of optical sensors for gear shape measurements is demon- strated. Particularly with the commercial confocal-chromatic sensor a measurement uncer- tainty 10 m is feasible, although the sensor is not (yet) designed for optical gear shape measurements.To further reduce the random measurement uncertainty contribution, the interaction of the sensor principles with the surface must be understood. Measurements on defined tooth surfaces with different topography and roughness have to be performed in the future. The parabolic systematic deviation must also be investigated in future steps to clarify whether the surface tilt is responsible for it or to identify the real source. Currently, it is assumed that the parabolic shape of the systematic deviation results mainly from the approximation process and only to a minor extent from the surface tilt angle. The approximation algorithm has to be adapted in this respect. Another systematic deviation occurs if the perpendicular alignment between the sensor and the linear unit as well as the alignment of the sensor axis within the gear transverse section is not successful. Suitable adjustment and calibration processes must be investigated to determine their contributions to the total uncertainty budget.ReferencesAuthor Contributions: Conceptualization, M.P., A.v.F., D.S. and A.F.; methodology, M.P., A.v.F. and D.S.; supervision, A.F.; measurements, M.P.; interpretation of the results, M.P., A.v.F., D.S. and A.F.; writingoriginal draft, M.P.; writingreview and editing, A.v.F., D.S. and A.F. All authors have read and agreed to the published version of the manuscript.Funding: This research was funded by the Deutsche Forschungsgemeinschaft (DFG), Germany, FI 1989/2-1.Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.Data Availability Statement: Data will be made available on request.Acknowledgments: The authors sincerely thank the Deutsche Forschungsgemeinschaft (DFG) for funding the research project (FI 1989/2-1). The authors would also like to thank Frank Horn, Uwe Reinhard, and Philipp Thomaneck for their support in the experimental realization and implementation.Conflicts of Interest: The authors declare that they have no conflict of interest.1. International Organization for Standardization. ISO1328-1: Cylindrical GearsISO System of Flank Tolerance ClassificationPart 1: Definitions and Allowable Values of Deviations Relevant to Flanks of Gear Teeth; ISO: Geneva, Switzerland, 2013.2. 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CrossRef摘要：对基于激光三角共焦-色差距离传感器的齿形测量平台的展望目前对于精度在 10 微米以内的齿轮形状测量的需求正在日益增长。使用接触式的齿轮检测方法虽然很精确，但是速度较慢。进来，光学齿轮 形状检测系统作为一种更快的检测方法正在被试验。光学齿形检测目前 面临的很大的挑战由于潜在的偏差源，如表面法线和传感器轴之间的倾 斜角度，变化的表面曲率和表面特性。目前，光学齿轮形状测量系统的 发展前景尚不清楚。因此，我们研究了激光三角测量和共焦彩色齿轮形 状横向扫描位置测量方法。由于齿面标准测量的结果的精度，主要由表 面特性引起的随机效应所决定。我们所研究的三角测量传感器的标准测 量精度大于 10 m， 不满足要求。而共焦色度传感器的标准测量精度小于 6.5m。在对直齿圆柱齿轮的测量结果中表明，采用横向扫描位置测量方法进行测量时，多次反射不影响测量精度。虽然商用光学传感器并 没有设计用于光学齿轮形状的测量，但应用的共焦色度传感器可以达到 小于 10m 的标准不确定度， 这进一步表明了光学齿轮形状测量的发展潜力。关键词：光学齿轮形状测量;横向扫描位置测量方法;激光三角测量传感 器;色差距离传感器;测量精度。1 研究简介1.1. 研究背景齿面的形状是否正确对保证齿轮在额定负载下能否均匀传动起 着决定性的作用。在齿轮制造中， 齿面通常设计成渐开线。然而， 渐开线齿面的制造对小于 10m 范围的公差要求较高。无论齿轮大小， 理论值与几何渐开线之间的偏差就已经导致齿轮产生缺陷， 甚至会导致整个齿轮箱的毁坏。为了降低齿轮箱的次品率， 必须对所有齿轮的齿形进行广泛的测量，测量精度在 10m 范围内(10m)。因此， 必须设计合适的齿廓测量系统， 来批量测量齿廓形状， 通过检查来确保齿轮质量可靠。1.2. 技术现状目 前 的 标 准 齿 轮 测 量 系 统 是 三 维 测 量 机 (CMM) 和 齿 轮 测 量 仪 器（ GMI）。三维测量机(CMM) 和齿轮测量仪器（ GMI）的测量精度小于 10m， 但是测量速度较慢。进行批量齿廓检测会消耗大量时间。由于测量速度的限制， 齿廓测量通常只在所有的齿中代表性的选择四个齿进行测量。为了保证齿轮质量检测的有效性，随机对四个齿进行测量是不够的。由于制造公差或运行中损坏，在齿轮检测中可能无法观察到个别齿偏离了公差几何形状。标准齿形检测系统只适用于有限的条件下的大量齿形检测。因此，快速光学位置传感器系统因为可用于大量检测齿轮所有齿的齿形， 近年来成为研究的热点。原则上， 光学传感器能够在 nm 范围内实现测量分辨率，并提供高达几十千赫的测量速率。然而，商用光学传感器对粗糙度的定义通常被特殊规定为使用垂直传感器在特定的平面表面上进行对焦。此外，还规定了最大接受角。为了覆盖整个齿面，尽管通过相邻齿的可达性有限， 光学传感器必须不垂直地对准齿轮表面。因此，齿轮表面法线和传感器轴之间的倾斜角是不可避免的，并且指定的传感器不确定度和形状测量的测量精度一般随倾斜角的增加而增加。此外，传感器没有指定用于测量齿面的变化表面曲率和金属齿轮的表面特性的表面粗糙度和反射率(例如，多重反射)。因此，有必要研究平面上垂直测量在哪些范围内也适用于齿轮形状的测量。以下描述的最新的光学方法的齿轮形状测量因此考虑了光学齿轮计量的三个基本问题:表面法线和传感器轴之间的倾斜角度，由于有限的光学可及性，变化的表面曲率和表面特性。光学齿轮形状测量中常用的测量方法都是基于三角测量原理。在2005 年的时候， Younes 等人提出了一种用逐点三角激光测量直齿圆柱齿轮的光学测量方法。由于相邻的齿有阴影，测量结果没有覆盖齿面的整个轮廓，测量精度不满足实际需求。并且没有对影响测量结果的的表面倾斜率， 齿轮曲率和齿轮表面性质进行讨论。Auerswald 等和 Guo 等分别在 2019 年和 2020 年描述了基于激光线三角测量传感器的三维齿轮形状测量原理。Auerswald 等人研究了对大型斜齿轮的激光线三角测量方法。该方法对整个齿面进行测量，并将测量数据与参考几何数据进行比较。得到的测量偏差为8.2m。Auerswald 等人表明，多重反射会影响测量偏差， 因为高度反射和曲面的齿依赖于传感器对焦。Guo 等人在2020 年也展示了光学三角测量传感器用于齿轮形状测量的发展潜力。为了对齿形进行广泛的质量评估，他们对整个齿面进行了 3D 测量。目前， 平均测量距离理想齿侧约为 10m。然而， 它没有指出其余的偏差具体是多少。Peters 等人在 2000 年介绍了一种结合转盘的条纹投影仪， 用于扫描小斜齿轮的齿形。因为他们的测量方法覆盖了整个齿面，得到的测量偏差与理论齿侧量大致相等，约为10m。Peters 等人解释说，传感器的位置应该尽可能垂直于齿面，以减少测量偏差，变化的曲率和高度反射表面对测量的影响也没有被提到。2006 年，Mee等人研究了一种基于条纹图案投影的三维齿轮形状测量方法。他们表明，由于复杂的齿轮几何和由此产生的光学可及性，系统长度测量偏差发生在条纹投影边缘处。大概参考几何偏差后校准测定精度为10m。本文介绍了利用摩尔纹投影测量齿轮形状的方法。在 2005 年， Sciammarella 等人测量了一个完整的齿， 测量误差小于 3.5m。由于表面法线和传感器轴之间的倾斜角度、变化的曲率和表面特性所带来的不确定行而没有被公布。Chen 等人在 2019 年提出了一个小型直齿圆柱齿轮摩尔纹投影的 3D 齿形测量。这个测量方法中也测量了整个齿面。测量偏差小于 5m， 不确定度主要来自表面倾斜，也没有对弯曲几何和表面性质做出讨论。综上所述，目前光学齿形测量的三角测量方法体现出了快速和可批量检测的特点。然而，有关上述困难(表面倾斜、弯曲渐开线几何形状和表面特性) 的剩余测量偏差的详细描述仍有待确定。科学家们进一步研究了基于干涉法的光学齿轮形状测量方法。方等人在 2014 年测量了齿侧几何， 补偿安装偏差后， 测量结果与参考几何基本吻合。然而，没有对偏差进行深入解释，并没有讨论在光学齿轮形状计量方面的困难。Balzer 等人在 2015 年和 2017 年使用了调频干涉传感器。通过将光束路径通过光纤集成到三坐标测量机的传感头，他们将干涉传感器与三坐标测量机结合起来，但也未给出测量数据和参考数据之间的偏差。共焦色距离传感器是测量齿轮形状的另一个有发展前景的光学传感器概念，因为它们通常被指定为表面独立的，并且可以实现测量误差保持在 10 微米范围内。因此， 使用这种方法可以测量反射和弯曲的齿面。对小圆柱齿轮的首次测量表明，测量误差可以达到10m 以内。产生偏差的原因主要是定位待测齿轮的转盘的运动控制单元，以实现对齿侧的连续检测。由于变化的曲率和表面性质的影响被提到，但没有详细描述。此外，由于传感器垂直于齿侧，不能检测到整个齿侧。因此，需要进一步的研究来了解共焦色传感器在光学齿轮形状测量中的发展综上所述，光学测量齿轮齿形的不确定度小于 10m。然而，由于齿轮表面法线与传感器轴之间的倾斜角度、曲率变化以及地形和反射率(多重反射)等金属表面特性导致的测量偏差尚未详细讨论。因此，光学传感器用于齿轮形状测量有着很大的发展空间。1.3 研究目的及内容为了了解三角测量和共焦色传感器，特别是在齿轮上的应用，本文介绍了使用光学位置测量方法的齿轮形状测量，分别与商业共焦色和商业激光三角测量传感器。特别考虑了三个潜在的偏差来源:齿轮表面法线和传感器轴之间的倾斜角度， 变化的曲率以及金属齿轮表面特性(主要是地形)。在位置测量的基础上， 确定齿形可达到的测量精度。为此实现这一目的，按照标准进行渐开线齿面的测量。此外，对一个小型渐开线直齿圆柱齿轮的测量也定性地表征了多重反射由于金属反射齿轮表面对测量结果的影响。第二节介绍了用于齿形测量的光学位置测量原理，以及计算标准测量不确定度的基于模型的评定。在第三节中，分别描述了使用商用激光三角传感器的 22 个传感器和商用共焦色差传感器的光学齿轮形状测量所使用的组件和实现的实验设置。值得注意的是，我们没有详细描述传感器的工作模式，因为这是众人皆知的。第 4 节给出并讨论了实验结果。在研究了各潜在偏差源对测量结果的影响后，确定了两种传感器最终可实现的随机源的标准测量不确定度。第五节以结论和展望结束文章。2. 测量原理2.1 利用光学位置测量的齿轮形状测量原理研究了两种光学位置测量方法在齿形测量中的适用性。结合每种方法的描述，比较了各传感器对测量范围和接受角度的要求。请注意，除了传感器规格，光学可及性，这取决于齿轮的几何形状，也可能限制测量。进一步注意，文章集中在非修改渐开线正齿轮。渐开线描述了当在基圆上展开一根线时，由一点描出的曲线。随着线被解开，曲度的渐开线逐渐减小。渐开线的形状取决于基圆的半径。随着基圆半径的增大， 一般曲率减小。如图 1a 所示为由光学距离传感器和旋转工作台组成的旋转扫描位置测量方法。齿轮安装在转盘上。传感器对准可以灵活地调整测量区域。考虑渐开线齿形，只要传感器与齿轮的基圆(虚线)切线对齐，垂直于齿轮表面的测量方法就是可行的。因此，接收角度较小的光学传感器原则上可用于齿轮测量。然而，光学扫描过整个齿侧，传感器必须转向齿轮中心的方向。随着齿轮转动，在扫描过程中，齿轮表面法线和传感器轴之间的倾斜角度不断变化，最大的倾斜角度出现在齿根处。当齿轮转动时，在传感器坐标系(xs, ys)的 ys 方向上，在旋转角度i 上，以距离的形式连续测量每个齿侧的形状， 其中 i 为距离测量的次数。图 1 所示。光位置测量原则组成的齿轮形状测量距离(a)一个光学传感器(x, y) 结合转盘(x, y), 连续测量齿轮的牙齿轮廓(x, y) 作为一个函数的旋转角和(b) 一个光学传感器安装在一个线性单位计量单位 (xs,y)， 用于横向扫描齿轮(x, y)的齿廓。与旋转扫描方法相比， 横向扫描方法需要相似的接受角度， 但测量整个齿侧的测量范围要小得多。另一种测量齿轮形状的位置方法如图 1b 所示。它由安装在线性单元上用于扫描的光学传感器组成。所述光学传感器垂直于所述线性单元的扫描方向。所述光学传感器和所述线性单元构成扫描测量单元。测量单元的对准可以根据被测量的齿侧区域和距离传感器的最大接受角度进行动态调整。在这里，光学传感器必须转向齿轮中心的方向，以测量整个齿面。通过测量单元的横向移动对齿侧进行扫描，测量数据也以测量坐标系坐标 Ps,i= (xs,i, di) 的形式存储。注意， 根据图 1b ， 只测量了一个齿的形状。为了测量所有齿的齿形几何形状，需要有一个转台来对测量区域内的每个齿轮齿进行顺序定位。为了表征潜在的光学传感器的应用在齿轮上，然而，测量一个齿侧是足够的。由于横向扫描测量渐开线齿的几何形状，一个不断变化的倾斜角度之间的齿轮表面法线和传感器轴发生。如果要测量整个牙齿侧面，最大的倾斜角度发生在齿根处。与旋转扫描方法相比，横向扫描方法需要相似的接受角度，但测量整个齿侧的测量范围要小得多。用于测量一个标准模量为 10.64 mm，有20 个齿， 基圆半径约为 100mm ， 所需的接受角度仍约30 而所需的测量范围仅为 4mm。因此， 本文进一步只考虑横向扫描的位置测量方法。2.2 齿轮形状标准测量不确定度的评定为了确定光学齿轮形状测量的测量精度，需要将测量的齿轮几何形状与参考数据进行比较。为此目的，测量的距离和参考齿轮几何必须转移到一个共同的坐标系。这里选择工件坐标系(x, y)。为了通过数学转换将传感器数据传递到工件坐标系，应知道传感器相对于工件坐标系的位置和对准， 分别知道传感器坐标系(xs, ys)与工件坐标系之间的精确转换，这里不是这样的。假设齿轮的中心是理想地位于工件坐标系的中心，测量数据是拟合到参考几何。为此达到目的，齿轮的几何模型是必需建立的。这里主要考虑非修改渐开线直齿圆柱齿轮，他们的齿轮几何形状是用来作为参考。在图 2 中，横切面中的名义几何图形显示在工件坐标系(x, y)中， 工件坐标系(x, y)被移动和旋转到传感器坐标系(xs, ys)中。渐开线直齿圆柱齿轮的公称几何可以用几何和位置参数描述，采用几何模型 Pi 。如图 2 所示， 基于实测实测点 Pa ， 离子齿 Z， 定点 Piof 的名义实测点垂线距离和位置参数，计算非改进型渐开线齿轮垂线距离的几何模型的工件坐标系统。测量点与齿面公称几何形状之间的铅垂线距离显示增大。图 2 根据一种非改进型渐开线齿轮的几何模型，以测量的实际点Pa，模数 Z 为基础，计算被测与公称几何之间的铅垂线距离。基准点的几何铅垂线的距离和位置参数在工件坐标系中。测量点与齿侧名义几何形状之间的距离显示扩大。由于一个测量点 Ps.i=(Xs.i,di) 是一个传感器坐标系统中的已知点， 转换矢量以及一个旋转角度的0 必须加到工件坐标系中。如果在工件坐标系下发生的偏差在真实光学测量点和自然点之间， 它们可以用铅垂线距离来描述。因此由理论点和铅垂线距离哪个方向是法向量到理论几何图形的表面，为了表征用激光三角测量和共焦色度传感器实现的位置测量方法的齿轮形状测量， 必须确定铅垂线距离。因此， 逆问题必须被解决。注意传感器与工件坐标系之间的转换参数还有位置参数是未知的。在求解逆问题时，还需计算未知参数。根据齿轮的名义几何形状以及测量条件得出的参数。通过迭代最小化垂直线距离平方和的方法，将公称渐 开 线 的 几 何 模 型 与 实 测 数 据 进 行 拟 合 ， 解 决 了 该 问 题 。采用非线性最小二乘法，直齿圆柱齿轮到公称渐开线的铅垂线距离由适当的滚动角度对于垂线的各根点， 渐开线上的距离是由隐式确定的:描 述 测 量 的 实 际 点 的 半 径 在 工 件 坐 标 系 的 极 坐 标 中 :齿轮形状测量的标准测量精度可以根据铅垂线距离评定。评估过程如图 3 所示。铅垂线距离表示被测几何图形与参考几何图形之间的局部测量偏差。计算得到的局部测量偏差受到系统和随机效应的影响。由于可以通过标定测量来校正系统偏差，因此光学传感器用于齿轮形状测量的精确度主要取决于所发生的随机偏差。因此，本文将随机偏差的标准差作为衡量齿轮形状标准测量不确定度的指标进行了评价。3. 装置测试为了测试三角测量和共焦色传感器用于光学齿轮形状测量的可行度， 实现了图 1b 中的横向扫描位置测量方法。实验的特定目的是表征齿轮表面法线和传感器轴之间的倾斜角度、变化的曲率以及齿轮表面特性如地形和反射率(多次反射)对齿轮形状测量结果的影响。为了达到这一目的， 我们在一个小直齿轮的渐开线齿面上进行测量。3.1 测量对象两个测量对象是渐开线标准齿面和一个小正齿轮。两个测量对象都有金属面。表中列出了标称几何图形的范围。由于每个测量对象的几何形状是众所周知的， 预计不会出现有表面缺陷的被测对象。测量对象用于评估三角测量传感器和共焦色传感器在齿形测量中的表现，并用于调查通过齿轮表面法线和传感器轴之间的倾斜角度、变化的表面曲率和齿轮表面特性的测量偏差来源。（ a） 显示了标准渐开线齿腹几何法向模数为 10.64117 mm, 20 齿,基圆直径 199.99 毫米。(b) 显示了一个齿轮与渐开线齿形和几何法向模数为 3.75 mm, 26 个齿， 基圆半径为 91.62 毫米。此外，在直齿圆柱齿轮上可见三角测量的测点和多次反射。单一齿面适用于用于光学传感器，由于没有相邻的齿限制测量，使整个渐开线的齿侧是可测量的最小传感器接受角度和一个小的测量范围。光学传感器的最佳角度，倾斜角度18 可能在齿根和在齿顶。从齿根(0.16 1 mm)到齿顶(0.02 1
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