## 3 Experiments and Discussion
### Measurement Experiment Using a Multi-Sensor CMM
#### 3.4.1 Measurement of a Sinusoidal Surface  
The experiemnt was designed as follows. First, the sinusoidal surface was measured using the touch trigger probe with a density of 0.5 mm and the measurement result was treated as the reference data since it had high density and high accuracy. Then the surface was measured using a laser sensor with a density of 1 mm pitch and this measurement result was determined as the first dataset. The measured data were then modelled using the Gaussian process and the results are shown in Figure 19. Figure 19a illustrates the raw data from the laser sensor, Figure 19b,c show the mean surface and the associated uncertainty of the Gaussian process while Figure 19d shows the deviation from the mean surface to the reference surface. The RMS of the uncertainty is 18.3 \(\upmu\)m and the RMS of the deviation is 38.3 \(\upmu\)m. The large measurement uncertainty may be caused by the unstable measurement by the optical sensor. It should be noted that the prediction grid for the Gaussian process has 1-mm pitch in the area where \(x\in[-30,\ 30]\) mm and \(y\in[-30,\ 30]\) mm.  
Figure 19: Measurement data and Gaussian process result: (**a**) raw data of measurement with the laser sensor; (**b**) mean surface; (**c**) associated uncertainty for the laser sensor; and (**d**) deviation of the mean surface from the reference data.  
Figure 18: Mean surface and associated uncertainty after Gaussian process modelling: (**a**) mean surface; and (**b**) associated uncertainty.  
A subset of 50 randomly sampled measurement points from the dense measurement data of the touch trigger probe was selected as the second dataset. The raw data and the result after the Gaussian process for the second dataset are shown in Figure 20. Figure 20a shows the raw measurement data, Figure 20b shows the mean surface, Figure 20c shows the associated uncertainty, and Figure 20d shows the deviation of the mean surface from the reference surface. Note that the grid of the prediction for the Gaussian process is the same as that for the laser sensor. The RMS of the measurement uncertainty is 15.7 \(\upmu\)m and the RMS of the deviation map is 13.2 \(\upmu\)m. The large measurement uncertainty is due to the small number of sampling points.  
The fused result of the dataset from the laser sensor and trigger probe is shown in Figure 21. Figure 21a shows the fused mean surface, Figure 21b shows the associated uncertainty and Figure 21c shows the deviation from the fused mean surface to the reference data. The RMS of the uncertainty is 8.0 \(\upmu\)m and the RMS of the deviation map is 8.4 \(\upmu\)m, both showing improvements compared to the original measurement datasets.  
Five different datasets for the touch trigger probe from different sampling positions were obtained and underwent the above procedure and the results are shown in Figure 22. The measurement data of the laser sensor were the same. Figure 22a shows the results of the RMS of the measurement uncertainties and Figure 22b shows the result of the RMS of the deviations. Both results show a consistent improvement of the measurement uncertainty and deviation from the reference data for the repeated measurement. The result demonstrates the effectiveness of the proposed data modelling and data fusion method. The high repeatability of the improvement of the data fusion may result from the well-established sensor configuration and high performance of data modelling which fit the situation shown in Figure 3c.  
Figure 20: Measurement data and Gaussian process result: (**a**) raw data of measurement with the touch trigger probe; (**b**) mean surface; (**c**) associated uncertainty for laser sensor; and (**d**) deviation of the mean surface from the reference data.