## 3 Workspace Analysis
### Database Collection  
The workspace of the robot mathematically represents all possible solutions of locations that the robot's end-effector can reach in space. To ensure that the finger of the model anthropomorphic hand can reach the target points, careful analysis of its workspace is crucial. The conventional workspace analysis is to find the range of the locations according to all joints. For example, the dual-arm robot investigated by [PERSON] et al. (2018) has two workspaces to indicate the motion range of each arm. In this dual-arm robot, the two arms are connected to the same fixed frame, and the joints of the left arm are independent of the right arm's joints. Consequently, we can simply use the workspace to check whether one arm can reach the target point. Unlike the dual-arm robot, the index finger and the thumb of the anthropomorphic hand are connected to a moving palm. As they share the same degrees of freedom of the wrist, the workspaces of two fingers are definitely dependent and coupled. Even if the two target points of the index finger and the thumb are all in their workspaces, with the traditional method, the calculated angle of the wrist may show different value at each point, which implies that there may exist mechanical interference and that the index finger and thumb cannot reach their target points at the same time. To resolve this problem, the workspace analysis below is proposed to make the workspace of the thumb independent of the workspace of the index finger in this research. Unlike the traditional method to build a workspace, we propose the workspace analysis under the condition that the angle of the wrist is separated from the angle of the fingers. The angle of the wrist is denoted as \(\vec{q_{w}}=[\theta_{w1},\ \theta_{w2}]^{T}\). The angles of the two fingers are denoted as \(\vec{q_{f1}}=[\theta_{t1},\ \theta_{t2},\ \theta_{t3},\ \theta_{t4}]^{T}\) and \(\vec{q_{f2}}=[\theta_{i1},\ \theta_{i2},\ \theta_{i3},\theta_{i4}]^{T}\), with the subscript \(f\) standing for finger. Once the angle of the wrist is decided, the workspaces of the index finger and the thumb are determined by substituting sets of \(\vec{q_{f1}}\) and \(\vec{q_{f2}}\) into Equation (2) to generate the cloud of points, as shown in Figure 2a, in which the red dots represent the possible positions of the fingertip of the index finger. The workspace of the index finger for a given set of wrist angles is denoted as \(I\left(\vec{q_{w}}\right)\) and that of the thumb is denoted as \(T\left(\vec{q_{w}}\right)\). The Monte Carlo method, a statistical method to solve problems by random sampling, was adopted to analyze the workspace. A considerable number of particles were used to approach the real shape of the workspace. Algorithm 1 shows the details for generating the workspace of the index finger and the thumb for a given set of wrist angle. Figure 2b,c illustrate the point cloud of the index finger (red dots) and the thumb (green dots) under different conditions of wrist angle. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Joint \(i\)** & \(\mathbf{a_{i}}\) (\({}^{\circ}\)) & \(\mathbf{a_{i}}\) (mm) & \(\theta_{i}\) & \(\mathbf{d_{i}}\) (mm) & **Range** \\ \hline
0 & \(90^{\circ}\) & 33 & \(\theta_{w1}\) & 0 & \(-60^{\circ}\) to \(+60^{\circ}\) \\
1 & \(-90^{\circ}\) & 93 & \(\theta_{w2}\) & 38 & \(-60^{\circ}\) to \(+90^{\circ}\) \\
2 & \(90^{\circ}\) & 15.5 & \(\theta_{i1}\) & \(-4\) & \(-45^{\circ}\) to \(0^{\circ}\) \\
3 & 0 & 20 & \(\theta_{t2}\) & 0 & \(0^{\circ}\) to \(+90^{\circ}\) \\
4 & 0 & 25 & \(\theta_{i3}\) & 0 & \(0^{\circ}\) to \(+90^{\circ}\) \\
5 & 0 & 36 & \(\theta_{i4}\) & 0 & \(0^{\circ}\) to \(+90^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Denavit and Hatenberg (D–H) table of the index finger.