## 3 Results
### Comparison between the Robotic and Manual RBG Osteotomy on Mandible  
Shown as a flow diagram in Figure 6, the transformation module was needed for mandible RBG osteotomy. The direct coordinates were determined using three points on the teeth, and these were used to register the mandible phantom model to the robot. The three reference points on the mandibular teeth included the distal edge of the mandibular left lateral incisor, the buccal pit of the left second molar and the lingual fossa of the right second molar. To reduce the registration error, these three reference points were defined as the points that could be taken with the greatest distance from each other in the oral cavity. The three points were detected in order, and then the mandible coordinate was determined. Next, the robot calculated the transformation automatically and positioned itself at the origin of the ramus coordinate. The RBG osteotomy could start at these points using the same algorithm as was used in a previous model RBG osteotomy. Figure 6: Workflow diagram for the autonomous robot osteotomy system. We performed a total of sixteen mandibular phantom surgeries using robotic arm on the left side and by manual on the right side with the same phantom model. Each category of errors for the mandible RBG osteotomy by the type of cuts is summarized in Table 3. The position error was measured by y- and z-axes and calculated as RMS values. Comparing the 6 mm disc and 20 mm disc groups, depth error was significantly higher on the 20 mm disc group than on the 6 mm disc group (\(p\) = 0.009), only on the superior cut of the robotic surgery (Table 3). Comparing robotic surgery and manual surgery, there were significant differences in absolute mean value and variance in all categories (Table 4). For robotic surgery, the position, length, angle and depth errors were 0.70 \(\pm\) 0.34 mm, 0.35 \(\pm\) 0.19 mm, 1.32 \(\pm\) 0.96\({}^{\circ}\) and 0.59 \(\pm\) 0.46 mm, respectively. For the manual surgery, the position, length, angle and depth errors were 1.83 \(\pm\) 0.65 mm, 0.62 \(\pm\) 0.37 mm, 5.96 \(\pm\) 3.47\({}^{\circ}\) and 0.40 \(\pm\) 0.31 mm, respectively. The robotic surgery had significantly higher accuracy and lower variance for position, length and angle errors. On the other hand, the absolute mean value and variance of the depth error was significantly higher for the robot surgery (Table 4). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Model RBG Osteotomy (\(n\) = 20)** & **Error (Mean \(\pm\) SD)** \\ \hline Posterior cut & Position (mm) & RMS & 0.85 \(\pm\) 0.28 \\  & Angle (Degree) & 1.10 \(\pm\) 1.24 \\  & Length (mm) & Top & 0.71 \(\pm\) 0.27 \\  & Bottom & 0.93 \(\pm\) 0.35 \\ \hline Anterior cut & Position (mm) & RMS & 1.16 \(\pm\) 0.55 \\  & Angle (Degree) & 1.54 \(\pm\) 1.60 \\  & Length (mm) & Top & 0.74 \(\pm\) 0.39 \\  & Bottom & 0.86 \(\pm\) 0.33 \\ \hline Inferior disc cut & Position (mm) & RMS & 1.14 \(\pm\) 0.48 \\  & Angle (Degree) & Anterior & 1.00 \(\pm\) 0.96 \\  & & Posterior & 0.67 \(\pm\) 0.74 \\  & Depth (mm) & Anterior & 0.75 \(\pm\) 0.30 \\  & & middle & 0.69 \(\pm\) 0.42 \\  & & Posterior & 0.76 \(\pm\) 0.30 \\ \hline Superior disc cut & Position (mm) & RMS & 0.68 \(\pm\) 0.36 \\  & Angle (Degree) & Anterior & 1.95 \(\pm\) 1.61 \\  & & Posterior & 1.32 \(\pm\) 1.49 \\  & Depth (mm) & Anterior & 1.81 \(\pm\) 0.68 \\  & middle & 1.71 \(\pm\) 0.66 \\  & & Posterior & 1.45 \(\pm\) 0.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The absolute mean value of errors for position, angle, length and depth on model RBG osteotomy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Category** & **Error (Mean \(\pm\) SD)** \\ \hline Position (mm) & RMS & 0.93 \(\pm\) 0.45 \\ Length (mm) & & 0.81 \(\pm\) 0.34 \\ Angle (Degree) & & 1.26 \(\pm\) 1.35 \\ Depth (mm) & & 1.19 \(\pm\) 0.73 \\ \hline \hline \(p\) value & & 0.000 \\  & & Position-Angle \\  & & Position-Depth \\ Post hoc (\(p\) \(<\) 0.05) & Length-Angle \\  & & Length-Depth \\ \hline \hline \end{tabular}
\end{table}