## 4 Experimental Results
### Measurement Statistical Analysis Experimental of Varying Orientations of the Measurement Object  
In this experiment, the robustness of the proposed system with varied box orientation with respect to the visual sensor was evaluated. The visual sensor was placed at five different positions with different orientation, and the angles between the face of the measured and the z-axis of the reference system of the sensor (see Figure 4a) are 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 75\({}^{\circ}\), and 90\({}^{\circ}\), respectively. In this experiment, estimated values are reported as the average of 30 experimental sessions on the same box (Figure 12a,b). Table 4 shows the measurement results in terms of W, L, and H of the boxes. As shown in Figure 13, the average absolute errors (L, W, and H of the two standard boxes) at 90\({}^{\circ}\), 75\({}^{\circ}\), 60\({}^{\circ}\), 45\({}^{\circ}\), and 30\({}^{\circ}\) were 0.867, 1.333, 1.633, 2.533, and 3.083, mm respectively, indicating that the orientation of the measured box with the visual sensor does not significantly affect the measurement results. However, at an angle of 90\({}^{\circ}\), the system can obtain the best measurement results and achieved a 0.26% average relative error. The maximum error in the measurement results in the experiment is 3.8 mm, indicating that the measurement system has good applicability in practical measurement. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & \multicolumn{4}{c}{**Measurement at Different Angles**} \\  & & & & **(The Measurement Error (mm) in Brackets)** & & \\  & & & & **[The Relative Error (\%) in Square Brackets]** & & \\ \hline \multirow{4}{*}{(a)} & \multirow{2}{*}{**Dim**} & **Ground Truth** & **90\({}^{\circ}\)** & **75\({}^{\circ}\)** & **60\({}^{\circ}\)** & **45\({}^{\circ}\)** & **30\({}^{\circ}\)** \\ \cline{2-8}  & & & & 189.3 & 190.9 & 191.1 & 188.2 \\  & & & & (\(-\)0.7) & (\(+\)0.9) & (\(+\)1.1) & (\(-\)1.8) \\  & & & & [0.37\%] & [0.47\%] & [0.58\%] & [0.95\%] \\ \cline{2-8}  & & & & & 252.2 & 251.8 & 251.7 & 254.9 \\  & & & & (\(-\)0.8) & (\(-\)1.2) & (\(-\)1.3) & (\(+\)1.9) \\  & & & & [0.32\%] & [0.47\%] & [0.51\%] & [0.75\%] \\ \cline{2-8}  & & & & & 40.9 & 401.6 & 401.9 & 396.2 \\  & & & & (\(+\)0.9) & (\(+\)1.6) & (\(+\)1.9) & (\(-\)3.8) \\  & & & & [0.23\%] & [0.40\%] & [0.48\%] & [0.95\%] \\ \hline \multirow{4}{*}{(b)} & \multirow{2}{*}{W} & \multirow{2}{*}{320.0} & \multirow{2}{*}{320.0} & 320.6 & 321.5 & 321.6 & 317.3 \\  & & & & (\(+\)0.6) & (\(+\)1.5) & (\(+\)1.6) & (\(-\)2.7) \\ \cline{1-1}  & & & & [0.19\%] & [0.47\%] & [0.50\%] & [0.84\%] \\ \cline{1-1} \cline{2-8}  & & & & 320.9 & 319.6 & 321.4 & 318.5 \\ \cline{1-1}  & & & & (\(+\)0.9) & (\(-\)0.4) & (\(+\)1.4) & (\(-\)1.5) \\ \cline{1-1}  & & & & [0.28\%] & [0.13\%] & [0.44\%] & [0.47\%] \\ \cline{1-1} \cline{2-8}  & & & & & 621.3 & 622.4 & 617.5 & 623.5 \\ \cline{1-1}  & & & & (+1.3) & (\(+\)2.4) & (\(-\)2.5) & (\(+\)3.5) \\ \cline{1-1}  & & & & [0.21\%] & [0.39\%] & [0.4\%] & [0.56\%] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measurement performance with 30 times at five different orientations. Measurement Statistical Analysis Experimental of the Changing Distance between the Visual Sensor and the Measured Box  
The box was measured at five different distances from the visual sensor (dis1 = 0.8 m, dis2 = 1.2 m, dis3 = 1.6 m, dis4 = 2.0 m and dis5 = 2.4 m). W, L, and H of the boxes ((a), (b), (c), in Figure 14) were recorded, and the measurement error was computed with the relative error. The measurement results reported in this experiment are recorded in Table 5.