## 4 Results
### ANOVA Gauge Repeatability and Reproducibility (R&R) Test  
To assess the robustness of the implemented system, a prior-to-operation ANOVA Gauge R&R test is executed. The ANOVA test is performed with cold-state workpieces, which is outside of normal operations. The purpose of the test is to assess the precision (repeatability and reproducibility) of the system but not its accuracy (agreement with the real result). For the control testing, \(a=3\) different workpieces (Figure 11) are measured \(m=10\) times by \(b=3\) different operators, resulting in a sample of 90 measurements. Table 3 presents the measurement results of each experiment. Figure 11: Cold workpieces used to run the ANOVA R&R test. The three workpieces share the same CAD reference. Figure 10: Measurements of 2 warm workpieces until cool-down (27 \({}^{\circ}\)C\(\leq\) T \(\leq\) 560 \({}^{\circ}\)C). Results of our measurement system (blue line) do not deviate more than 0.1 mm from Coordinate Measurement Machine (CMM) measurements (red line). (a) Workpiece 1. CMM runout = 0.66 mm; (b) Workpiece 2. CMM runout = 0.8 mm. Table 4 shows the ANOVA results for the conducted experiments. The degrees of freedom (DOG) for each sum of squares are defined as:  
\[ \begin{split}\text{DOG}_{\text{operator}}&=b-1=2\\ \text{DOG}_{\text{workpiece}}&=a-1=2\\ \text{DOG}_{\text{interaction}}&=(a-1)*(b-1)=4\\ \text{DOG}_{\text{vision\_system}}&=ab*(m-1)=81 \end{split}  \tag{9}\]  
The test shows that the variable Workpiece is statistically significant for the computed runout (\(p\)-value \(<\) 0.05). This means that our system can difference each workpiece from the others due to the inherent manufacturing process bias. On the other hand, the operator and the interaction (operator/workpiece) are not statistically significant (\(p\)-value \(>\) 0.05), which means that the operator is not a significant source of error for our measurement system (reproducibility). Table 5 presents the Gauge Repeatability & Reproducibility (GRR) [34] results for the executed experiments. The variations induced by the different operators account only for the 0.04% of the total variance of the data, which is an indicator of the reproducibility (less variation equals to more reproducibility) of the developed system. Similarly, the variation introduced by the vision system accounts for the 22.07% of the total variance of the data, which is the indicator of the repeatability (less variation equals to more repeatability) of the system. The manufacturing process accounts for the rest of the variance (77.89%). The Gauge R&R (repeatability + reproducibility) accounts for the 22.11% of the total variance. Such Gauge Repeatability and Reproducibility (GRR) variation is acceptable for the warm forge plant in which the system is deployed (GRR \(<\) 25%), where the early detection of dimensional misfits in warm workpieces compensates for the precision loss of the measuring system. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source of Variability** & **Degrees of Freedom** & **Sum of Squares** & **Mean Square** & **\(F\) Statistic** & **\(p\)-Value** \\ \hline Operator & 2 & 0.0010 & 0.0005 & 1.2282 & 0.3838 \\ Workpiece & 2 & 0.3575 & 0.1787 & 438.5964 & 0.0000 \\ Interaction & 4 & 0.0016 & 0.0004 & 0.2419 & 0.9137 \\ Vision System & 81 & 0.1364 & 0.0017 & & \\ Total & 89 & 0.4966 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Analysis of Variance (ANOVA) table.