## Problem Description and System Architecture  
### Problem Description  
During the writing of each stroke, the brush tip will touch or leave the paper, and the brush's spine will also tilt to a side, which results in a gap between the handle and brush, as shown in Figure 1a. This is the so-called \"footprint lag\", which is defined in Figure 1b. Figure 1b also shows that the footprint left by the brush on the paper resembles a water droplet in shape. Figure 1a has a 3D coordinate system, and the brush is assumed to move along the X-axis, whereas Figure 1b has a 2D coordinate system that exists on a horizontal plane (the paper). The mapping between the location terms in Figure 1a,b is shown in Table 1.  
As calligraphy strokes are assumed to consist of trajectories traversed by droplet-shaped footprints, each trajectory can be decomposed into a series of droplet-shaped footprints, as shown in Figure 2a. The greyed areas of this figure are the model character. For purposes of clarity, only a few representative droplet-shaped footprints are shown in Figure 2b. In Figure 1b, it is shown that there is a lag between the footprints and handle base, like in the shu (vertical) stroke shown in Figure 2b. Figure 3 shows the handle positions of each footprint, which are represented by small circles in this figure. \begin{table}
\begin{tabular}{l l} \hline \hline
**Location Terms for the Handle and Brush** & **Location Terms for the Footprint** \\ \hline Projection of the handle & Handle \\ \hline Front-most end of the footprint & Front of the footprint \\ \hline Distance between the projection of the handle & Footprint lag \\ and the front-most end of the footprint & \\ \hline Brush tip & Tip of the footprint \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mapping between the location terms for the handle/brush and footprint. Figure 1: Relationship between handle position, the brush’s curvature and the brush footprint. (**a**) The curvature of the spin causes the projection of the handle’s base to be different from the brush footprint. (**b**) The footprint left by the brush on the paper. As the robotic arm only has direct control over the handle and not the position of the brush on the paper (i.e., the footprints), the footprint's size, lag, and angle (\(\phi_{i}\)) will vary with the handle's coordinates (Figure 4). In this paper, we used the footprint's size, lag, and angle (\(\phi_{i}\)) as the factors of the brush trajectory model and studied whether the brush's handle had a relationship with these factors. If there existed the relationship, we were able to use the brush's handle to control the brush trajectory. Since this work was an experimental study to show whether the brush's handle position was possible to control the brush trajectory, we did not further explore the control principles in different brush pens. To obtain the relationship between the positions of the brush's handle and footprints, the robotic arm was made to draw a few basic footprints by sending known handle coordinates to the robotic arm. A set of images where the footprints' outer frames and positions may be measured was thus produced. By analyzing these images, we obtained a mechanism that allows the robotic arm to write calligraphy by receiving handle coordinates, i.e., the \"brush trajectory model\" (Figure 5a). Figure 5b illustrates the process by which the brush trajectory model was used to guide users in finding a suitable set of handle coordinates to write a complete character using the robotic arm. Figure 3: Droplet-shaped footprints and their corresponding handle positions. Figure 2: Calligraphy strokes represented by droplet-shaped footprints. (**a**) Calligraphy strokes represented by a series of droplet-shaped footprints. (**b**) Calligraphy strokes represented by a few representative droplet-shaped footprints. The outer frame, lag, and \(\phi_{i}\) of the footprints are the three factors that make up the brush trajectory model.