## 2 Materials and Methods
### Finger Workspace Calculation
The overall methodology employed in this paper for calculating the performance indices (PI) is described in Figure 7. Simulations in a MATLAB environment are used to perform these calculations, with a 1\({}^{\circ}\) change in the Ab/Ad angle considered as the step size for determining finger postures. The fingertip positions in the workspace for each posture are then computed. This process is repeated for each finger according to its anatomical ROM. The two PIs listed below were selected for analysis in this study. (a) Manipulability Index (\(\mu\)): The measure of manipulability or the manipulability index, \(\mu\), of a serial manipulator was first defined in 1985 by [PERSON] [29] as follows  
\[\mu(\mathbf{J})\ \triangleq\ \sqrt{det(\mathbf{J}\mathbf{J}^{T})} \tag{3}\]  
where \(\mathbf{J}\) is the velocity Jacobian matrix of the manipulator. Since the Jacobian matrix of each finger changes for every posture, it is possible to find the posture for which the value of \(\mu\) will be maximum within the anatomical ROM for each joint of every finger. The value of \(\mu\) is calculated for every posture achieved by formulating the Jacobian matrix. The variation of \(\mu\) for the index, middle, ring, and little fingers across the ROM for the MCP Ab/Ad joint is individually calculated. The variation of \(\mu\) for the index, middle, ring, and little fingers across the ROM for the MCP Ab/Ad joint is shown in Figure 8. It was observed that the value of \(\mu\) increases as the absolute value of the Ab/Ad angle  
Figure 7: Flowchart indicating overall methodology employed in calculating performance indices for the design of the prosthetic hand. increases from 0\({}^{\circ}\). This indicates that, for best performance, the fingers must be placed on the palm at their maximum abducted position. Additionally, it can be inferred from the measure of manipulability that parallel finger arrangement is not ideal for the design of prosthetic hands. (b) Thumb Opposability Measure (T): The performance of any prosthetic hand depends on the level of interaction between the thumb and the fingers. The measure of opposability of the thumb provides the level of interaction between the thumb and the other fingers. The higher the value of the opposability measure, the higher will be the performance of the given hand. The thumb opposability measure T used in this work is obtained by modifying the expression given for the \"anthropomorphism index of the kinematic chain\" in [30]. The value of T is calculated using the following expression. \[\mathrm{T_{i}}=\frac{w_{i}}{{l_{th}}^{2}}a_{i} \tag{4}\]  
where \(\mathrm{T_{i}}\) is the thumb opposability index for the ith finger, \(a_{i}\) is the area of intersection between the ith finger and thumb's workspace, \(l_{th}\) is the length of the thumb, and \(w_{i}\) is the numerical weight for calculating the thumb opposability index for each finger. The numerical weight for all four fingers is considered as '1' in the current work for uniformity, as seen in [5]. The Ab/Ad angle for which the value of T (maximum overall intersection area) is highest is chosen. The Ab/Ad angle for the four fingers calculated based on T is provided in Table 4.  
Based on the above calculations, we have chosen the results derived from the calculation of T rather than \(\mu\). This decision is made because T considers the thumb's influence on each finger, meaning the performance of any finger is measured in relation to the thumb. In \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Finger** & **Index** & **Middle** & **Ring** & **Little** \\ \hline Ab/Ad Angle & 13\({}^{\circ}\) & \(-13^{\circ}\) & \(-10^{\circ}\) & \(-30^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Value of Ab/Ad angle based on thumb opposability index. Figure 8: The variation of manipulability index across the ROM of MCP Ab/Ad joint for (a) index finger, (b) middle finger, (c) ring finger, (d) little finger.