Thermal error modelling of gear measuring instrument based on principal component regression

As gear measuring instruments (GMI) are frequently used at the production site, the influence of temperature on their accuracy and stability becomes non-negligible. The existing standards and technologies only consider the error modelling and compensation problem when the GMI is in a uniform temperature field and is not suitable for the production site environment with a large temperature change rate and large temperature gradient. The paper delves into the problems of GMI thermal error modelling at the production site. First, the temperatures of the GMI and its surroundings were measured using multiple temperature sensors. The parallelism thermal errors between centres and the Z-axis (referred to as parallelism thermal errors) and the y-direction spatial thermal errors (referred to as spatial thermal errors) were also measured by the probe and laser interferometer synchronously. Then, the models of them were established based on principle component regression (PCR). As evidenced by the experimental results, the fitting determination coefficient of the models are both greater than 97%, and the prediction determination coefficient is greater than 90%, demonstrating that the models are precise and robust.


Introduction
The gear measuring instrument (GMI) is currently the most widely used equipment for gear measurement.In the process of gear measurement, temperature is an important factor affecting the measuring accuracy and stability of GMI.Existing specifications mainly reduce the impact of temperature on GMI by specifying the ambient temperature conditions [1].As more and more GMIs are used at production sites, ensuring the required temperature conditions in specifications can be challenging, and the influence of temperature on GMI accuracy and stability becomes non-negligible.Some GMI compensates for only the thermal expansion of the workpiece and the scales by using the coefficient of temperature expansion (CTE) when the ambient temperature deviates from 20℃ [2].The compensation method only works well when there is a small temperature rise and uniform temperature.In the production site, where the temperature changes sharply and there is a large temperature gradient, the GMI will produce complex thermal deformation and compensating for the thermal expansion of the workpiece and the scales using the CTE are not enough.
Compared with the insufficient research on a thermal error of GMI, thermal error modelling has been widely researched in machine tools, and a series of modelling methods such as multiple linear regression [3], neural network [4], support vector machine [5] and autoregressive [6] have been proposed.By simultaneously measuring the temperatures of the primary heat sources and the thermal errors of the machine tool, these methods establish a relationship between them through data fitting.In these methods, because the temperature measured at different points affects each other, the collected temperature variables often show a very strong colinearity, which causes the deterioration of the prediction accuracy and robustness of the model.At present, the influence of the correlation between temperature variables is cut down by fuzzy clustering [7], grey correlation analysis [8], roughness theory [9] and so on.However, these methods also decrease the correlation between temperature variables and thermal errors while reducing the influence of the colinearities of temperature variables [10].
Measuring equipment, like GMI, usually has a small internal heat source, resulting in a strong correlation and a noticeable collinearity problem of temperature variables.In this paper, the thermal error models of GMI, including parallelism thermal errors between the centres and the Z-axis (referred to as parallelism thermal errors) and the y-direction spatial thermal errors (referred to as spatial thermal errors), are established using principal component regression (PCR).The method preserves the correlation between temperature variables and thermal errors as much as possible while reducing the collinearity of temperature variables.The experiment shows that the established models have a good fitting and prediction ability.They can be used to correct the thermal errors of the GMI at the production site.

Modelling for parallelism thermal errors
The modelling steps of the parallel thermal error based on PCR [11] are as follows: (1) Data normalization.The temperature data   , where  y and  y s are the mean and the standard deviation of y , j T and j T s are the mean and the standard deviation of the jth temperature measurement point data j T .Finally, the model of parallelism thermal error  y regarding temperatures 1 2 , , , m T T T is:

Point-by-point modeling for spatial thermal error
Because the spatial thermal error is not only related to temperatures but also to probe position, the spatial thermal error can be expressed as a mathematical formula: where y  indicates the y-direction spatial thermal error of the probe., , x y z represents the probe position coordinates in the measuring space, which can be derived from the GMI grating.T is a matrix consisting of a temperature vector obtained from the different measurement points.
In the measurement of the spatial error, the number of position coordinates usually is much smaller than the number of temperature data for the consideration of measurement cost.Taking both temperature data and position coordinates as input for the spatial thermal error model will result in strong collinearity about position coordinates, reducing model convergence and robustness.Here, we use a point-by-point modelling approach to avoid the above problems.In point-by-point method, the entire measurement space is decomposed into multiple nodes, and then independent sub-models are established at each node, denoted as   , , , n y y y f f f , n is the number of nodes.The sub-model of each node can be established using the PCR method similar to parallelism thermal error modelling: The spatial thermal error model above can only predict the thermal error at nodes, and cannot give the thermal error at any point in space.Here, we use the inverse distance weighted interpolation method (IDWI) in Figure 1 to obtain the thermal error at any point in space.(1 (1 (1 ) where, x x x x x y y y y y z z z z z are the distance weight of the point P along the X, Y, and Z axes., , x y z are the coordinates of the point P.

Measurement of the temperatures
The temperatures of the GMI and its surroundings were acquired by 24 Pt100 thermal resistances and a 32-channel temperature detector.The allowable deviation of the thermal resistances is ± (0.To measure parallelism thermal errors, a mandrel was installed between the top and bottom centres.The GMI probe was pressed against the mandrel in the Y direction and moved along the Z axis to obtain the coordinates of the multiple measuring points:   ( , ) 1,2, , i i z y i n  .Then, one line was fitted using the obtained coordinates, and the inclination of this line was taken as the parallelism error in the Y directions.The difference between the parallelism errors at each time and the starting time was the parallelism thermal errors at that moment, which was denoted as:  y .
The spatial thermal errors were measured with the Renishaw ML10 laser interferometer.In the measuring space of the GMI, 9 straight lines parallel to the Y-axis were selected, and 16 points were selected at equal intervals on each line.The probe y-direction positioning error at each point was measured by a laser interferometer.During measurement, the laser interferometer reflector was mounted on the probe mounting frame.By using the GMI grating indication in the Y direction as the reference and the laser interferometer indication as the actual position, the difference between them was the y-direction positioning error of the probe.The thermal error of each point was calculated by subtracting the measured positioning error at different times from the measured positioning error at the initial time.

Measurement experiment
The experiment started from the start-up state, without any preheating.The data acquisition program ran automatically every 6 minutes to collect the temperatures and two thermal errors above simultaneously.The probe returned to the reference position and waited for the next measurement interval after each measurement.The data collected underwent noise reduction using a Gaussian filter with a window size of 15.

Model verification for parallelism thermal error
The parallelism thermal error model was established using the PCR method described above.The threshold of the condition indices used for selecting the principal components was 30.The training result of the model is shown in Figure 4(a).Another set of newly measured data was used to verify the prediction ability of the models.The comparison of the predicted result with the measured result is shown in Figure 4   Figure 4(a) shows that the output value of the model after training aligns well with the measured value, suggesting a good fit between the model output and the parallelism thermal error data.In Figure 4(b), the model accurately predicts the trends in measurement results for the new measurement data, proving its strong prediction ability.The goodness parameters of the parallelism thermal error model for fitting and prediction are shown in Table 1.In the table, RMSE is the residual's root-mean-square error, R2 is the model determination coefficient, and η is the model prediction ability.  1 illustrates that the fitting RMSE of the parallelism thermal error model is less than 0.1″, and the determination coefficient and prediction ability of the model are more than or close to 95%, indicating that the model has a good data fitting ability.The goodness parameters of the model for the prediction showed a slight decrease with the use of the new measurement data, but the decrease was not significant, suggesting that the model still maintains a good prediction ability and is robust.

Model verification for spatial thermal error
The spatial thermal error model was also established using the PCR method, in which the condition indices threshold was still set at 30.The fitting residual slice diagram of the model at different times is shown in Figure 5, and the maximum and minimum fitting residual of the model are shown in Table 2.    5 and Table 2 show that the fitting residual of the spatial thermal error is less than 1 μm, and the residual distribution is relatively uniform from the initial time to 180 minutes, indicating that the established model fits the measured data well in these moments.Model fitting errors increase and the uniformity of error distribution decreases at 360 min.However, the maximum absolute residual is still less than 1.5 μm, and the model still has a good fitting ability.The fitting residual of the spatial thermal error is bigger than the fitting residual of the parallelism thermal error.This is mainly caused by the inverse distance weighted interpolation in section 2.2.With the increase in the number of nodes, the error caused by interpolation will gradually decrease.
For testing the fitting ability of the model at a node, a node (90 mm, 180 mm, 180 mm) was selected to observe the fitting result of thermal error change over time.The result is shown in Figure 6(a).For verifying the prediction ability of the model, a specific point (155 mm, -42 mm, 25 mm) was selected, thermal error over there and corresponding temperatures were measured, and then the thermal error at the point was predicted by the established model.The measured thermal error and the model predicted thermal error over time are shown in Figure 6    Figure 6(a) shows that the measured curves and fitting curves of the thermal errors are almost the same, and the maximum fitting residual is within ±0.25 μm, indicating that the established spatial thermal error model fits the measured data well.Figure 6(b) illustrates that the curve tendency of the model predicted is consistent with the measured curve tendency over time, with residual within ±0.5 μm, demonstrating that there is a good prediction ability for the model.
The goodness parameters of the spatial thermal error model for fitting and prediction corresponding to Figure 6 are shown in Table 3.In Table 3, the RMSE for fitting is about 0.1 μm, and the determination coefficient for fitting reaches 97%, indicating that the model has a good fitting ability.Table 3 also shows that the RMSE for prediction are less than 0.3 μm, and the determination coefficient for prediction is more than 90%, demonstrating that the model has a good prediction ability.

Conclusion
Thermal error is a key factor which affects the measurement accuracy and stability of GMI.Our focus is on modelling of the thermal error resulting from temperature changes and complex temperature distribution at the production site in this paper.We measured the temperatures of the GMI body and its surroundings, the parallelism thermal error and the y-direction spatial thermal errors of the GMI, synchronously.The models of thermal errors were established based on the PCR method, and the fitting and prediction ability of the models was verified by experiments.The experimental results indicated that the fitting determination coefficients of the models are both greater than 97%, and the prediction determination coefficients are both greater than 90%, indicating that there is a good fitting and prediction ability for the models.The models are suitable for modelling and compensating for the GMI thermal error at the production site.

( 2 ) 4 )
thermal error data y y are normalized by the standard deviation normalization method, and the data normalized are and y y .Here, m is the number of temperature data measured.Calculate the covariance matrix of normalized temperature data by Construct the principal components.First, the condition indices corresponding to each eigenvalue are calculated by

T
where i is the matrix composed of the temperature vector measured at the ith node, the predicted value of spatial thermal error and the PCR coefficient at the ith node.

Figure 1 .
Figure 1.The inverse distance weighted interpolation (IDWI) principle.Supposing the point P is a point anywhere in measurement space, 1 2 15+0.002│t|)℃, and the response time is <30 s.The acquisition accuracy of the temperature detector is 0.2%FS.The arrangement of thermal resistances and temperature measurement experiments are shown in Figure 2.Among them, T1 ~ T8 are used to collect the ambient temperature around the GMI and other thermal resistances are used to measure the temperature distribution in the GMI body.(a) Arrangement of thermal resistances.(b) Temperature measurement experiment.

Figure 4 .
Figure 4. Modelling results of the parallelism thermal error.

Figure 5 .
Figure 5.The fitting residual slice diagram of the spatial thermal error model at different times.
Figure 5 and Table2show that the fitting residual of the spatial thermal error is less than 1 μm, and the residual distribution is relatively uniform from the initial time to 180 minutes, indicating that the established model fits the measured data well in these moments.Model fitting errors increase and the uniformity of error distribution decreases at 360 min.However, the maximum absolute residual is still less than 1.5 μm, and the model still has a good fitting ability.The fitting residual of the spatial thermal error is bigger than the fitting residual of the parallelism thermal error.This is mainly caused by the inverse distance weighted interpolation in section 2.2.With the increase in the number of nodes, the error caused by interpolation will gradually decrease.For testing the fitting ability of the model at a node, a node (90 mm, 180 mm, 180 mm) was selected to observe the fitting result of thermal error change over time.The result is shown in Figure6(a).For verifying the prediction ability of the model, a specific point (155 mm, -42 mm, 25 mm) was selected, thermal error over there and corresponding temperatures were measured, and then the thermal error at the point was predicted by the established model.The measured thermal error and the model predicted thermal error over time are shown in Figure6(b).

Figure 6 .
Figure 6.Modelling results of the spatial thermal error.

Table 1 .
The goodness parameters of the parallelism thermal error model.

Table 2 .
Maximum and minimum fitting residua of the spatial thermal error model (Unit: μm).

Table 3 .
The goodness parameters of the spatial thermal error model.