Holistic evaluation of involute gear surfaces using 3D point cloud inversion

Gear users desire increased performance from their products and are interested in predicting the behavior (vibration, loss, service life, etc), and gear manufacturers are interested in adjusting machining parameters to manufacture even better gears. Both require proficient metrology. The evaluation standards were originally written to classify the manufacturing capability of machine tools and as tolerances tighten they may not provide enough insight into the functional performance of the manufactured parts. Furthermore, the advantage of modern coordinate and gear measuring machines is not fully taken into account when applying standardized measurement and evaluation strategies. Improved analysis tools can increase the relevant information gained from current measurement capabilities and enable more useful functional characteristics to be specified by designers and manufacturers. A key improvement is to move from profile and helix line traces to flank surfaces and corresponding fitted form elements. This is achieved by the introduced method. We present an inversion algorithm that fits dimension, form, and pose, and is applicable with current methods through freezing of specific fitting parameters. The method fits a synthetic point cloud with output results that are accurate to the 7th digit, or in practical terms are numerically equal. The inversion method can increase the understanding of error contributions and the functional performance of gears.


State of the art in gear metrology
Gear tooth flanks are commonly measured and evaluated according to ISO 1328-1:2013, VDI/VDE 2612 and 2613-1 guidelines where the determination of profile, helix, and pitch deviations are compared to tolerances and acceptance criteria.These deviations are derived by single line measurements, for profile and helix, and pointwise measurements for pitch deviation.This gives a sparse but sufficient evaluation in most cases as demonstrated in international comparisons [1,2].Gear deviations are caused by manufacturing errors from two sources [3,4]: • Systematic errors manufacturing errors repeat on each tooth flank.Sources are tool form errors, tool positioning, tool wear (over short manufacturing times), mean temperature, elastic deflection..
• Individual errors vary from tooth flank to tooth flank.Sources include effects of variable nature such as temperature gradients, axis positioning drift, gear blank eccentricity, and variing tool vibration.
The traditional measurement strategy comprises four measurements of equally spaced teeth for profile and helix evaluation, which characterizes only the systematic errors.To capture the effect of individual error sources all teeth need to be measured including the left as well as the right flanks.More applications will require the characterization of the full functional gear surface to capture manufacturing errors with the increased prevalence of advanced manufacturing processes such as injection moulding, 3D-printing and 5-axis CNC machining.These advanced methods may cause completely different systematic errors due to the number of axes.Besides, the production of large gears as used in wind energy systems, rail transportation and maritime mobility, need long manufacturing times and thus tool wear, which is usually regarded as the source of systematic errors, will instead cause individual manufacturing errors.In order to understand and capture the different sources of these errors a feature separation method needs to be applied which requires measurements in a common reference coordinate system, to efficiently separate these errors.This approach has begun being established in gear metrology [5].Although the extension of standardized evaluations from traces and single points to areal data has been suggested 20 years ago continuously since then [4,6,7], areal metrology solutions are not yet widely applied.
However, proper characterisation of the gear flanks through the proposed holistic evaluation will lay ground for further understanding of gear functional performance such as noise, vibration, efficiency, micropitting and scuffing failure mode resistance.

Outline of the paper
This paper will first describe the strategy and application of the inversion method within section 3 by means of a synthetic gear model.The inversion method is performed in three modes: • Traditional (T) -single profile evaluation according to current standards.
• Profile (P) -single profile evaluation with the introduced feature separation.
• Surface (S) -evaluation of surface data with the introduced feature separation.
In section 4 a use case for the evaluation of measured data is presented.Finally the strengths of the fitting method will be discussed.
The synthetic gear model is based on PTB's large gear measurement standard, where the workpiece-like standard was chosen to be of a comparable size to wind turbine gears.The standard embodies six different gear geometries each having different geometry parameters resulting in three external and three internal gears composed of three gear teeth.The gear teeth themselves are arranged in groups of three.Each group consists of one left-handed, one spur and one right-handed tooth.The relevant nominal parameters of the tooth geometry are summarized in table 1.A drawing of the standard is shown in figure 1.

Introduction
We present a holistic inversion method that considers the fundamental principle of coordinate metrology-the separation of features-to enable a clear distinction between parameters describing the measured gear in its dimension, form, and pose (figure 2).
The application of this principle in gear metrology was already proposed in the 2000s [7].However, a consistent implementation has not yet taken place, since it is not in accordance with current guidelines and standards [8][9][10], that neglect the influence of the actual gear dimension and pose on the determined form, though these simplifications potentially have a significant impact on the results of the targeted actual gear form [5]. Therefore, we propose to take feature-separation into account for gear evaluation.
For this purpose, a constrained non-linear least-squares optimization algorithm for coordinate metrology, based on the 'function-independent best-fit, correcting nominal surface coordinates' approach [11] has been implemented.The algorithm has so far been used by PTB for holistic 3D screw thread evaluation [12] and was recently extended for holistic 3D involute gear evaluation [5,13].

3D gear surface model
A 3D parametric involute gear flank surface model [14] has been implemented in Cartesian coordinates (x, y, z) as the essential part of the inversion algorithm's objective function: Herein, the defining parameters hand and flank denote the directions of the slope and the tooth flank, respectively: The following relations apply for the gear shape parameters: ( )    The parametrization is realised by writing out (4) to (6) in (1) to enable the new separation-of-features approach during the data inversion, that is presented in section 3.4.For the description of the involute flank surfaces of a whole gear with n teeth, the indices i = 1,K,n and j = L, R are introduced in (1): ´-

Holistic inversion algorithm
The holistic inversion algorithm minimizes the objective function Q(W), that depends on the parameter vector ´+ + , where the indices denote the number of measured points n, the number of geometry parameters g, and the number of position and orientation parameters in the 3D Euclidean space l.W is composed of sampling points in parametric representation U = (z, α t ) = (u, v), geometry parameters p = (r b , j b , β b ), and pose parameters describing the orientation A = (a x , a y , a z ) and translation T = (t x , t y , t z ):

Î
. Herein, indices i and j = 1, 2 denote the tooth number and left and right gear flanks, respectively.Finally, the foot points on S ij (W) are adjusted to P. The residual ΔX to be minimized is calculated as follows: This results in the high-dimensional over-determined optimization problem to be solved with the Gauss-Newton method: Linearisation is achieved by applying the common approximation method that is based on the calculation of the Jacobian matrix ( ) Î ´+ + .This results in a normal system of equations: T T

D = D
that is solved by iterative adjustment of parameter vector W. The starting model W s+1 at iteration step s = 0 is based on parameterized sampling points (u ij , v ij ) calculated from Cartesian coordinates of P and nominal geometry values p nom .The improved update vector W s+1 is determined with: Once the threshold value s = 25 or ||ΔW|| 2 < 10 −12 mm is reached, a reasonable value based on experience and numerically zero, respectively, the termination condition of the inversion algorithm is fulfilled and the recent W is considered the best-fit result of the measured gear surface.
For each single tooth with tooth number i a set of results is available for both gear flanks j.The set of results consists of actual geometry parameters, the corresponding profile slope deviation, helix slope deviation, pitch errors of the i individual teeth, and the remaining residuals as summarised in table 2.
The calculation of deviations is carried out following ISO 1328-1:2013 [8], but in the normal plane instead of the transverse plane.However, to meet the requirements of a holistic evaluation, areal measurements were considered and actual geometry parameters were applied.For this purpose, the point cloud was decomposed into individual gear flanks, while maintaining the common reference system.
Remaining residuals according to (12) are the basis for the following evaluation step, which is the analysis of the harmonic content of the gear surface.

Calculation of synthetic gear data
An application example for the holistic gear evaluation method is demonstrated by the investigation of synthetic gear data, a usual approach for initial verification of newly developed evaluation algorithms.The synthetic gear model is based on PTB's large gear measurement standard as previously described in section 2 (figure 1, table 1).
Five gear teeth consisting of a left and right flank each were calculated with (7) and on the basis of nominal values indicated in table 1.The gear flanks were manipulated by replacing nominal geometry parameters with those corresponding to deviations listed in table 3. The modified geometry parameters were calculated according to [9] and [10] with (8)-( 10): Table 3. Deviations on synthetic gear flanks.In addition, the left flank of tooth 1 contains microgeometry corrections (small modifications to correct for deflection, misalignment, or to foster ideal noise and stress conditions) as depicted in figure 3 for mid-helix and mid-profile traces.Figure 3(a) shows helix crowning which has been calculated by a sine-function with amplitude 5 μm and order 0.5.Figure 3(b) depicts five harmonics that have been added to the profile lines.Corresponding phase shifts, amplitudes and orders are indicated in table 4.There are four full-trace harmonics (A-D) and one piece-wise harmonic (E) to provide a simple representation of a damaged region, or a region affected by temporary vibration of a machine tool.In order to visualize the propagation of the piece-wise harmonic (E) along the diagonal of the flank, deviations of the entire flank are depicted as a heatmap in gear

Evaluation of synthetic gear data
The synthetic gear point cloud has been evaluated with the method described in section 3.6 under varying boundary conditions for a performance check and to demonstrate advantages of the presented featureseparation method.At first, only mid-helix lines, mid-profile lines and pitch measurement points of the single teeth have been evaluated separately similar to the state of the art method.Afterwards, the holistic method was applied for a midprofile evaluation of one single tooth.
Finally, the whole gear point cloud was evaluated with the holistic method.The results of the respective methods are introduced and discussed in the following.By comparing such methods, the advantages of the introduced holistic evaluation will be demonstrated.During these inversions, the grid points α t and z of the fitting gear model were constant at nominal positions of the start model.The resulting deviations that were derived from the fitted gear model are summarized in table 5. Values for the five teeth are sorted in rows.In addition to the results of the traditional method (T), the expected deviations applied in the gear model (M) are indicated, which are known from the calculation of the synthetic point cloud.The results of all teeth are not satisfactory.In case of tooth 1, this is due to the microgeometry corrections and the harmonic vibrations added to the left flank.Tooth 2 to 4 each have at least one type of deviation (table 3) that superimpose on the flanks and are obviously not well resolved.Even if only one type of deviation is added to a gear flank, poor results are obtained if the incorrect deviation is evaluated for, demonstrated here on tooth 5.  Table 5. Deviations rounded to nanometer obtained by traditional mid-profile, midhelix and pitch evaluation (T) compared to known model values (M).  Figure 5 gives an explanation of the general problem in 3D for a better visualization.Figure 5 (a) shows a section of a nominal base cylinder and the corresponding nominal left gear flank (brown).In addition, a modification of both is shown (green).Compared to the nominal flank, the base radius of the modified gear flank is larger and causes a profile slope deviation f Hα .This is supposed to be the actual geometry that is under test.The best-fit result (color gradient) tries to replicate the actual geometry by fitting β b at the frozen r b,nom .Hence, the best fit-result leads to a helix slope deviation, although the error is related to the base radius and not to the helix angle.Figure 5 (b) shows the the best-fit result based on an adaption of j b only of the same gear surface model.Again, although the error is related to the base radius and not to the tooth width, the best-fit result produces a pitch deviation.In contrast to these inadequate results figure 5 (c) represents a perfect fit of the actual gear flank (green) by modifying r b , which is the correct and only geometry parameter capable to fit the modified gear flank.
In practice the actual deviations are generally unknown.If evaluations are conducted without featureseparation and without prior knowledge on the gear geometry it is impossible to decide which evaluation result is the correct one.
The missing feature-separation can be a serious problem, if the true gear geometry is investigated to adapt machining process parameters for an improvement of the geometry of manufactured parts.A decision based on the evaluation of tooth 5 for instance could lead to an adjustment of the grinding tool that produces corrected tooth widths, although the actual tooth width in this example is already nominal.

Holistic feature-separated profile evaluation
In general, the holistic, feature-separated evaluation method is also applicable for both single profile and single helix measurements.In the presented example, all deviations are derived from single profile measurements.
The feature-separated deviations obtained by single profile evaluation (P) of tooth 2 are summarized in table 6.If the feature-separation method is applied during the evaluation, which means that all geometry parameters r b ij , b ij b and b ij j are simultaneously adapted to fit the actual measurement, then the obtained deviations meet the expected ones (M) within the accuracy in the order of the applied noise, that corresponds to the fourth digit.Removing the model scatter ∼0.001 μm and fitting again results in differences between model and obtained deviations in the 10th digit or even numerical zero (e.g.f 4.999 999 999 967 833 m , which is more than sufficient for practical application.7. The mismatch between derived and expected deviations of tooth 1 remains, because of the microgeometry corrections on the left gear flank which are not yet considered in the fitting-model.However, in order to enable an assessment of the performance of the algorithm by means of tooth 1, the residuals containing the profile slope deviation and the harmonic content were subtracted from the synthetic point cloud.The corrected point cloud has been reevaluated resulting in the three deviations being numerical zero.Therefore, the algorithm precisely determines the expected values.This result is considered as a proof of performance.Nevertheless, microgeometry corrections such as crowning or tip relief should be implemented in the fitting-model as an Table 6.Deviations rounded to nanometer obtained by feature-separated evaluation of single mid-profile lines (P) compared to known model values (M).
1 5 0 0 P −5.000 5.000 15.000 15.000 −0.000 0.000 option in order to improve the gear surface assessment and to obtain the correct deviations even if the gear surfaces have modifications.
In spite of this, the residuals of the original fit of tooth 1 are useful for the analysis of microgeometry corrections and other interfering harmonic shape deviations, since only the absolute values of amplitude and order are of relevance.These residuals are therefore further examined regarding their harmonic content in [15].
The derived deviations of the other synthetic gear teeth are well resolved by applying the feature-separation method.They differ in the 7th digit from the Model values according to the noise that has been added to the synthetic gear surface.
We briefly summarize findings to investigate the performance of the holistic evaluation algorithm presented here.
We have shown that the presented holistic evaluation algorithm enables the characterization of the true geometry of the synthetic gear.By applying the optional feature-separation method, the three geometry parameters (r b , β b and j b ) and their corresponding deviations ( f Hα , f Hβ and F p ) are determined beyond the required precision.This potentially opens up new opportunities, especially in the quality control of large gears used in wind energy systems, that are difficult and costly to maintain.Due to long machining times, the dimensions of these gears are potentially affected by machine tool drifting effects.The areal evaluation of the entire gear flank surfaces is able to capture such effects.In addition, the evaluation method is particularly well suited for the dimensional evaluation of gears manufactured by advanced methods, such as injection molding, 3D printing or 5-axis machining.In these cases the assumption that a few lines represent manufacturing errors of the entire gear surface does not apply, which is why the holistic surface evaluation is recommended.

Use case
4.1.Measurement of a large gear standard First holistic measurements have been taken out on PTB's large coordinate measuring machine (CMM), Leitz PMM-G from Hexagon Metrology.The CMM has a measurement volume of 5 × 4 × 2 m and achieves a maximum permissible volumetric error of 3.2 + L/400 μm and a probing error of 2.6 μm.The diameter of the tactile probing sphere was 5.0 mm.The ambient temperature of the laboratory is controlled and comprised 20°C ± 0.2 K.One of the spur tooth gaps of PTB's large gear standard has been measured to test the applicability of the presented evaluation method.Holistic measurements require one single reference system for all registered measurement points.Therefore, the recorded point cloud is composed of 33 tactile helix line scans which are located in one fixed reference system.The total number of measured points is 12 500.The gear measuring standard is used for comparisons, which is why only qualitative results are presented in the following.

Residuals
Residuals (section 3.4, (12)) provide information about local form deviations and are useful to draw correlations between those and the manufacturing process.Depending on the applied method point cloud evaluation method, different results are obtained, as shown in figure 6. Figure 6 illustrates absolute 3D residuals through heatmaps, whereas the scales only give a qualitative impression of the deviations for confidentiality reasons.However, this representation is sufficient to compare results obtained by the proposed holistic evaluation method shown in figure 6(a) against the standardized method applied in 3D depicted in figures 6(b) and (c).The holistic method determines form deviations in a high level of detail depending on the point density of the tactile measurement.An advantage of this evaluation is that the residuals can be directly used for the determination of harmonic form deviations as demonstrated in [15].The standardised method, on the other hand, leads to other features according to the geometry parameters that have been fitted.where the difference between the fitted form element, which consists of the actual geometry parameters r b , β b and j b , and the nominal form element is shown.This representation figures out the slope deviations properly, but without detailed deviation content.Furthermore, for the evaluation of deviations according to [8] it is possible to build the reference to datum points/lines/surfaces through interpolation and extrapolation, just as in conventional evaluation (section 3.3, (8)-( 10)).

Discussion and future work
This work is motivated by the fact that gearing applications continue to become more sophisticated, therefore analysis tools must keep pace.The obvious improvement of considering more functional contacting surface has to be considered carefully-the increase in data and time required to measure and characterize needs to provide adequate benefits to the industry.A gear point cloud inversion method has been proposed which can separate gear features to improve the understanding of both performance and manufacturing processes.The strategy separates gear measurements into dimension, form, and pose parameters through a constrained non-linear least-squares optimization algorithm.It was shown that the traditional method of 2D trace analysis, achieved here by freezing geometry parameters from the fit, cannot discriminate deviations from a combination of sources.Allowing for single parameters to be fit still caused issues as a parameter may compensate for a lack of change in another-covered in detail in section 3.6.1 and figure 5. Allowing for all geometry parameters to be fit obtains separated deviations, verified against synthetic data with known form deviations.With no noise the fit parameters are equal to the 10 th digit, with the inclusion of noise, σ(0, 1) μm, they are equal to the 4 th place-the accuracy allowed by the noise.
Applying the method to 3D point cloud data shows a greater accuracy for the majority of teeth, with synthetic and fit values agreeing to the 7 th digit-even with the noise.However, the geometry parameters are not well estimated on tooth 1 due to the microgeometry corrections, as can be seen in table 7.This is a current limitation of the algorithm where microgeometry corrections are not yet considered in the fitting-model, however the deviations are still useful and can be utilized in further research such as harmonic analysis.
Finally, the applicability of the proposed holistic evaluation method was demonstrated in a use case (section 4.1) by means of first holistic measurements that were conducted on a large gear standard.Observations made by examining the synthetic gear data in section 3.6 could be confirmed by these first measurements.More precisely, the advantages of the feature separation approach also apply to measured data as demonstrated in section 4.2 figure 6.It has been shown that fitting all three geometry parameters reveals the actual shape deviations in a high level of detail, whereas the results obtained with the areal standardized method rather emphasize deviations that are caused by geometry parameters being set to nominal values.Freezing the base radius during the fit tends to produce profile slope deviations for instance, that are unnecessarily related to the actual geometry of the gear flank surface.

Future work
The inversion method has been proven with synthetic data and first holistic measurements have been evaluated.The next step is to apply this method to more tactile and optical measured data for a validation of the method.Furthermore, the evaluation of the uncertainty is currently being developed for this purpose.Additionally, extending the inversion method to include microgeometry corrections will increase the number of possible applications in industry.Finally, showcasing the harmonic analysis output by analyzing the extracted measurement data in tooth contact analysis models will connect the metrology process to the design.By predicting functional gear performance through outputs such as transmission error, contact stress, bending stress, and efficiency greater understanding of gears will be achieved.

Figure 2 .
Figure 2. Principle of the feature separation.
. nominal pitch radius ... nominal position of involute at base circle ... actual position of involute at base circle Where i = 1 denotes the tooth number and j = L, R left or right flanks, respectively.
modified helix angle ... nominal helix angle ... facewidth modified position of involute at base circle ... nominal position of involute at base circle ... nominal pitch radius

Figure 3 .
Figure 3. Harmonic content and noise on the left flank of tooth 1: (a) Mid-helix trace sinusoidal harmonic content.(b) Mid-profile trace sinusoidal harmonic content.(c) Mid-profile zero-mean sum profile-wise calculated from harmonics shown in (a) and (b).(d) Mid-profile pseudorandom noise.

3. 6 . 1 .j
Traditional mid-helix, mid-profile and pitch evaluation In order to emulate the common way of a gear inspection, mid-helix, mid-profile and pitch measuring points were extracted from the synthetic point cloud and each gear flank has been fitted individually.Single mid-profile, single mid-helix line, and pitch point evaluations of tooth 1 to 5 were performed by freezing r subsequently with i = 1,K,5 and j = L, R in order to derive profile slope, helix slope and pitch deviations depending on the corresponding actual value only, which is r b act ij ,

Figure 4 .
Figure 4. Deviations on the left flank of tooth 1 in gear coordinates.Piece-wise harmonic content is elevated by factor 10 for better visibility.The angle with respect to roll length axes is θ h = 59.5°.

Figure 5 .
Figure 5. Nominal gear flank at piece of nominal base cylinder (brown) and actual flank at actual piece of actual base cylinder (green) with f Ha . (a) Best-fit result for fitted β b . (b) Best-fit result for fitted j b . (c) Best-fit result for fitted r b .
hand, if j b is fitted for the determination of F p , then the algorithm tries to compensate for the deviation actually caused by β b with an adaption of j b , which results in an incorrect value for F p .Instead of 0 μm for both flanks of tooth 5 the derived deviation is F 0left and right flank, respectively.

3. 6 . 3 .
Holistic feature-separated surface evaluation Here, we present the holistic evaluation with feature-separation of the 3D point cloud.For that purpose, the three geometry parameters r b ij , b ij b , and b ij j were adjusted simultaneously during the fit and the grid points also have been corrected by adjusting the parameters which describe the support points of the involute gear surface model t ij a and z ij .Expected (M) and obtained deviations (S) through the feature-separation method are summarized in table

Figure 6 (
b) clearly indicates a helix slope deviation on the left flank, which is caused by the frozen geometry parameter β b .In contrast, (c) shows a profile slope deviation, again caused by one of the geometry parameters being frozen, which in this case is r b .To bridge the gap to traditional gear evaluation, the residuals of holistic 3D measurements could be represented as in (d),

Figure 6 .
Figure 6.3D residuals of evaluations with different fit parameters in surface representation: (a) Holistic point cloud inversion result with fitted geometry parameters r b , β b , j b and fitted support point parameters α t and z.(b) Areal evaluation with fitted r b and frozen β b , j b , α t and z.(c) Areal evaluation with fitted β b and frozen r b , j b , α t and z.(d) Holistic point cloud evaluation result.Difference between fitted form element and nominal form element showing the dominating profile slope deviation.

Table 1 .
Analyzed nominal gear parameters of one external gearing.

Table 7 .
Deviations obtained by holistic evaluation (S) rounded to nanometer compared to known model values (M).