## 4 Experimental Results
### Evaluation of the Written Characters  
The written characters were evaluated using the methods proposed in [14]. Figure 20 shows the outer frame and intersection points of any two strokes of the character \"Yong\" where \(I^{{}^{\prime}}s\) represent the intersection points of any two strokes of the sample and test characters and \(O\,s\) represent the points on the outer frame of the sample and test characters. \(O^{i}_{min}\) and \(O^{i}_{max}\) are the shortest and longest distances from \(O^{i}\) to any corner, respectively and \(I^{i}_{min}\) and \(I^{i}_{max}\) are the shortest and longest distances from \(I^{i}\) to any corner, respectively. To increase the sensitivity of this method to changes in the characters, \((O^{i}_{max},I^{i}_{max})\) was changed to \((O^{i}_{min},I^{i}_{min})\). The mathematical equations for the evaluation of character quality are shown below (Equations (9)-(12)). \[s=\left\{ \begin{aligned} \frac{A_{test}}{A_{sample}},& 0 \leq s<1\\ 2-\frac{A_{test}}{A_{sample}},& 1\leq s<2\\ 0,& 2\leq s\end{aligned} \right. \tag{9}\]  
where \(A_{test}\) and \(A_{sample}\) represent the area of the outer frame in test and sample data, respectively. The sample character is used as a reference for quality evaluation. \[p=\sum_{i=0}^{7}\frac{\sqrt{(x_{s}^{O_{i}}-x_{t}^{O_{i}})^{2}+(y_{s}^{O_{i}}- y_{t}^{O_{i}})^{2}}}{(x_{s}^{O_{i}}-x_{min}^{O_{i}})^{2}+(y_{s}^{O_{i}}-y_{ min}^{O_{i}})^{2}} \tag{10}\]  
\[r=\sum_{i=0}^{4}\frac{\sqrt{(x_{s}^{l_{i}}-x_{t}^{l_{i}})^{2}+(y_{s}^{l_{i}}-y _{t}^{l_{i}})^{2}}}{(x_{s}^{l_{i}}-x_{min}^{l_{i}})^{2}+(y_{s}^{l_{i}}-y_{min}^ {l_{i}})^{2}} \tag{11}\]  
Figure 19: Planning of the calligraphy writing process and the result. (**a**) Model character from the internet. (**b**) Differences between the planned trajectory and model character. (**c**) Character written by the robotic arm. where \((x_{s}^{O_{i}},x_{t}^{O_{i}})\) and \((y_{s}^{O_{i}},y_{t}^{O_{i}})\) are the \(X\) and \(Y\) coordinates of the points on the outer frame of the sample and test characters, respectively and \((x_{s}^{I_{i}},x_{t}^{I_{i}})\) and \((y_{s}^{I_{i}},y_{t}^{I_{i}})\) are the \(X\) and \(Y\) coordinates of the intersection points of any two strokes of the sample and test characters, respectively. The \(s\), \(p\), and \(r\) parameters were changed to \(S\), \(P\), and \(R\) as follows:  
\[s = 100s\] \[P = 100(1-p)\] \[R = 100(1-r). \tag{12}\]  
Each parameter has a maximum score of 100. The higher the score, the better the quality of the character in that particular aspect. In Figure 21a, the greyed area is the model character, and the center of each footprint is assumed to be the projected position of the handle. The footprints have been modeled as circles, and the planned trajectory for the footprints in this instance is the arrangement of circles inside the model character. In Figure 21b, the brush trajectory model was used to simulate the footprints and obtain the relative positions of the handle and center of each footprint. In this case, the planned trajectory for the footprints is the arrangement of droplet-like shapes inside the model character. In this study, it was assumed that each calligraphic character consists of trajectories traversed by droplet-shaped footprints. Therefore, these trajectories can be decomposed into a series of droplet-shaped footprints. For purposes of clarity, only a few representative droplet-shaped footprints are shown in Figure 21b. Figure 21: Trajectories that model the movement of brush footprints. (**a**) Footprints modeled by circles. (**b**) Footprints modeled by the brush trajectory model (Brush Pen #1). Figure 20: Outer frame and intersection points of any two strokes of “[PERSON]” character where \(\dot{l}^{\prime}s\) represent the intersection points of any two strokes of the sample and test characters and \(\mathcal{O}^{\prime}s\) represent the points on the outer frame of the sample and test characters.