## 4 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{\begin{tabular}{c} **Check** \\ **Plane** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Point** \\ **Cloud** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Approx. Dist.** \\ **From Station** \\ **1/Dice 1 (m)** \\ \end{tabular} } & \multicolumn{3}{c}{\begin{tabular}{c} **Before Regist.** \\ **(In Its Own** \\ **Standalone** \\ **Point Cloud)** \\ \end{tabular} } & \multicolumn{3}{c}{\begin{tabular}{c} **After Die-Based** \\ **Regist. (Without** \\ **Dim. Calibration)** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **After Die-Based** \\ **Regist. (With** \\ **Dim. Calibration)** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **After Die-Based** \\ **Spherical** \\ **Target (6)** \\ **Regist.** \\ \end{tabular} } \\ \hline A-1 & \begin{tabular}{c} 1 (master) \\ 2 (slave) \\ \end{tabular} & \multirow{2}{*}{Vert.} & 11.7/5.3 & 1.7 & 5.3 & 2.0 & 1.9 \\ A-2 & \begin{tabular}{c} 1 (master) \\ 2 (slave) \\ \end{tabular} & \multirow{2}{*}{Vert.} & 1.8 & 1.5 & 2.4 & 2.3 & 1.9 \\ A-3 & \
\begin{tabular}{c} 1 (master) \\ 2 (slave) \\ \end{tabular} & \multirow{2}{*}{Horiz.} & \multirow{2}{*}{9/2.5} & 1.2 & 1.3 & 5.8 & 2.4 & 1.6 \\ A-4 &
\begin{tabular}{c} 1 (master) \\ 2 (slave) \\ \end{tabular} & \multirow{2}{*}{Horiz.} & \multirow{2}{*}{7.7/1.8} & 1.6 & 1.5 & 2.6 & 1.9 & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMSEs of check plane residuals before and after the registration. (Dataset A, die dimension = 50 cm \(\times\) 50 cm \(\times\) 50 cm). Table 6 shows the original angles between each facet (before the axis orthogonality correction), along with the average calibrated dimensions for the registrations. From Table 7,  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\ \begin{tabular}{c} **Check** \\ **Plane** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Point** \\ **Cloud** \\ \end{tabular}  } & \multirow{2}{*}{**Orient**} & \multirow{2}{*}{**Approx. Dist. ** **From Station** **(In Its Own Standalone** **Point Cloud)**} & \multicolumn{2}{c}{**Before Regist. **} & \multicolumn{1}{c}{**After Die-Based** **Regist. (Without** **Digest. (With** **Digest. (With** **Digest. (With** **Digest. (With** **Digest. (With** ** **Digest. (With** **Digest. (With** ** **Digest. on one hand, none of the angles between the planes (dice's facets) are larger than 1\({}^{\circ}\), but still large enough to affect the final registration accuracy. Based on trigonometry, a 0.5\({}^{\circ}\) of angular error will cause a translational error of roughly 4.3 cm at a range of 5 m. Therefore, it is necessary to conduct the proposed procedure for axis orthogonality correction. On the other hand, up to 2 cm of the correction for the dimension is estimated based on the calibration. This magnitude is significant, so the proposed calibration procedure is essential to maintain higher registration accuracy. The registered point clouds before and after the die dimension calibration are visualized in Table 7 for the four datasets. For Dataset A, the point clouds of a cabinet are more \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Datasets** & **Point Cloud** & \
\begin{tabular}{c} **Angle between Facets** \\ **(XY/XZ/YZ) (\({}^{\circ}\))** \\ \end{tabular} & **Original Die Side Length (m)** &
\begin{tabular}{c} **Average Side Length** \\ **after Calibration (m)** \\ \end{tabular} \\ \hline A & 1 (master) & 90.1\(\backslash\)89.6\(\backslash\)90.1 & 0.500 & 0.480 \\  & 2 (slave) & 90.4\(\backslash\)90.3\(\backslash\)89.5 & 0.500 & 0.488 \\ \hline B & 1 (master) & 90.5\(\backslash\)90.6\(\backslash\)90.4 & 0.500 & 0.488 \\  & 2 (slave) & 89.7\(\backslash\)89.2\(\backslash\)89.8 & 0.500 & 0.489 \\ \hline C & 1 (master) & 89.1\(\backslash\)89.6\(\backslash\)88.3 & 0.500 & 0.489 \\  & 2 (slave) & 90.2\(\backslash\)89.8\(\backslash\)89.3 & 0.300 & 0.281 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Angles between each facet, and the average calibrated dimension of the dices.