## 2 Materials and Methods
### Finger Workspace Calculation  
The finger workspace provides a means to compute various performance parameters, which will then be used to calculate the finger orientation. The fingertip workspace of all the fingers is obtained by performing forward kinematics. Figure 5 shows the kinematic parameters used for workspace calculation. The Denavit-Hartenberg (DH) parameters in Table 1 are used to frame the forward kinematic equations. In the current work, the fingertip workspace is calculated with respect to its base, the MCP joint. Hence, the DH parameters for the index, middle, ring, and little fingers are framed by considering them as 4-DoF serial manipulators. Figure 4: The coefficient of variance for the MCP F/E and Ab/Ad angles of normalized data for all 4 fingers. Figure 5: Kinematic diagram indicating the link lengths and orientation. Using the DH parameters, the workspace boundary for each finger can be traced by considering the anatomical ranges of motion (ROM) for all the hand joints. The joint limits of a human hand, as provided in [25,26], and the ROM for the finger joints, as given in [1,27], are adapted and used in the current work. The ROM for all the joints and the length parameters used in the current work are given in Tables 2 and 3. Sign convention for all the angles is considered according to the reference frames shown in Figure 1b. They are as follows--(i) flexion is considered positive whereas hyperextension is written with a negative sign (ii) for Ab/Ad angle, angles measured in counterclockwise direction are considered as positive for mathematical uniformity since abduction motion for index and middle fingers are opposite to ring and little fingers. The workspace boundary for the index, middle, ring, and little fingers are then traced using the sequence of operations as provided in [27]. The fingertip workspace of the index, middle, ring, and little fingers is shown in Figure 6. Since the thumb is not being considered for the analysis, its workspace is not calculated in the current work. \begin{table}
\begin{tabular}{c c c c c} \hline
**Joint No.** & \(\theta_{i}\) **(rad)** & \(a_{i}\) **(mm)** & \(d_{i}\) **(mm)** & \(\alpha_{i}\) **(rad)** \\ \hline
1 & \(\theta_{i-1}\) (MCP Ab/Ad) & 0 & 0 & \(-\frac{\pi}{2}\) \\
2 & \(\theta_{i-2}\) (MCP F/E) & \(l_{i-2}\) & 0 & 0 \\
3 & \(\theta_{i-3}\) (PIP F/E) & \(l_{i-3}\) & 0 & 0 \\
4 & \(\theta_{i-4}\) (DIP F/E) & \(l_{i-4}\) & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: DH parameters for index, middle, ring, and little fingers. \begin{table}
\begin{tabular}{c c c c} \hline
**Index** & **Middle** & **Ring** & **Little** \\
**(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline \(l_{I-2}=40\) & \(l_{II-2}=45\) & \(l_{III-2}=42\) & \(l_{IV-2}=32\) \\ \(l_{I-3}=23\) & \(l_{II-3}=25\) & \(l_{III-3}=24\) & \(l_{IV-3}=19\) \\ \(l_{I-4}=15\) & \(l_{II-4}=19\) & \(l_{III-4}=17\) & \(l_{IV-4}=12\) \\ \hline \end{tabular}
\end{table} 
Table 3: Length parameters for all the five fingers of the hand. Figure 6: Plot showing fingertip workspace of index, middle, ring, and little fingers in three-dimensional space. MCP Ab/Ad Joint Calculation  
This subsection analyzes the four fingers' MCP Ab/Ad joint to determine their palmar orientation, enabling the prosthetic hand to achieve 3 DoF for each finger. To optimize finger orientation for enhanced functional dexterity, the manipulability of each finger is evaluated across the full ROM of the MCP Ab/Ad angle. Various performance indices (PIs) commonly employed in evaluating traditional serial manipulators are utilized for this assessment [28]. In this work, the selected PIs are applied either directly or with modifications to the human hand, treating each finger as a serial 4-DoF manipulator to determine the angle for the MCP Ab/Ad joint. The methodology involves calculating the chosen PI for each finger posture across the full ROM of the Ab/Ad joint.