## 3 Experiments and Discussion
### Model Error Analysis for Gaussian Process Modelling  
In the previous simulation experiments, the measurement noises were modelled well using the Gaussian process modelling method. However, in real-life situations, machined surfaces are not perfect and may contain defects and form error due to machining error which may affect the modelling accuracy of the Gaussian process. In this section, a series of simulation experiments was conducted to verify the effectiveness and limitations of Gaussian process modelling by adding the form error to the simulated surface. That is to say, the simulated surface has a base form, form error and measurement noises, as shown in Figure 14. The base form is a sinusoidal surface with lateral pitch of 50 mm, the peak-to-valley (PV) value is 4 mm and the surface is denoted by \(S_{b}\). The machining error denoted by \(S_{e}\) is simulated to be a sinusoidal deviation from the base form with a higher spacial frequency together with a pitch of 5 mm and smaller peak-to-valley height (PV) value of 0.1 mm, and the PV value is varied in a series of experiments in order to validate the model. The combination of the base form and the form error determines the actual form (\(S_{c}\)) of the surface as shown in Equation (22):  
\[S_{c}=S_{b}+S_{e}, \tag{22}\]  
For the measurement noise \(N\), it is simulated to be Gaussian distributed noise with zero mean and 0.005 mm standard deviation, the expended deviation is 0.015 mm (\(k=3\)), and this amplitude is also varied to test the performance of the model. The simulated measurement result of the surface is denoted as \(S_{m}\):  
\[S_{m}=S_{b}+S_{e}+N, \tag{23}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Evaluation Items** & **With Sensor 1** & **With Sensor 2** & **Fused Data** \\ \hline RMS of uncertainty & 38.6 μm & 31.0 μm & 24.2 μm \\ RMS of deviation & 2.3 μm & 2.1 μm & 1.8 μm \\ \hline \end{tabular}
\end{table} 
Table 2: RMS (root mean squared) value of associated uncertainties of the original measurement datasets and fused dataset. Figure 14: Simulated measurement result and components with different spatial frequencies: (**a**) measurement result; (**b**) base form; (**c**) machining error; and (**d**) measurement noise. The experiment was conducted as follows. The amplitude of the base form was fixed as \(\pm\)2 mm and the standard deviation of the measurement noise was fixed as 0.005 mm. The expanded deviation of the measurement noise level (\(k\) = 3) was 0.015 mm. The amplitudes of the form errors in the experiments were simulated to be \(\pm\)0.005 mm, \(\pm\)0.01 mm, \(\pm\)0.02 mm, \(\pm\)0.03 mm, \(\pm\)0.04 mm and \(\pm\)0.05 mm. The simulated measurement result of the surface \(S_{m}\) was then modelled by using the Gaussian process modelling method and the mean surface \(S_{\text{mean}}\) and the covariance surface \(S_{\text{cov}}\) were calculated. The objective function \(O_{E}\) of the performance of the Gaussian process model was determined by the RMS of the deviation from the mean surface \(S_{\text{mean}}\) to \(S_{c}\):  
\[O_{E}=\text{RMS}(S_{\text{mean}}-S_{c}), \tag{24}\]  
The result of the experiment is shown in Figure 15a. The result shows that when the form error is smaller than the measurement noise, the error of the model is large, while when the form error is larger than the measurement noise, the error of the model is small. This result demonstrates that when the form error is smaller than or similar to the measurement noise, the model has the limitation of being unable to distinguish the form error and measurement noise which leads to a large modelling error that treats both the form error and the measurement noise as the measurement noise. This leads to an overestimation of the overall uncertainty of the measurement. Another experiment was conducted to fix the base form and the form error, and change the amplitude of the measurement noise. The amplitude of the base form was also fixed as \(\pm\)2 mm as in the previous experiment while the form error was fixed as \(\pm\)0.02 mm.