## 2 Materials and Methods
### Grasp Simulation  
Once the MCP Ab/Ad angle for the fingers is calculated, a grasp analysis simulation is performed to validate the angles and test the overall performance. The grasping performance can be assessed through numerous metrics such as the anthropomorphism index, the measure of dexterity, etc., as given in [32]. Prosthetic hands with a high level of anthropomorphism are required to have real-world application and functionality. Hence, a measure of anthropomorphism is usually considered as a measure of grasp quality for prosthetic hands. There are numerous measures of anthropomorphism available in the literature, as in [33; 34; 35]. In this study, a simulation of a five-fingered grasp was conducted to assess the grasp quality and performance of the designed hand using the SynGrasp toolbox in MATLAB [36]. To validate the design effectively, grasp quality metrics outlined below were applied and compared across three different hands: (i) the human hand, (ii) a prosthetic hand with parallel fingers, and (iii) the current design. The grasp quality metrics used in this study are presented below. (a) Volume of Manipulability Ellipsoid (\(V\)): The relationship between the hand space and the object space during the grasping of an object is represented by the hand Jacobian matrix \(\mathbf{H}\)[37]. The kinematic manipulability of a manipulator is the ability of the end effector (fingertip) to change position and orientation given a joint configuration. To \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Point** & **Coordinate (mm)** & **Point** & **Coordinate (mm)** \\ \hline G & \((0,0)\) & O\({}_{3}\) & \((10,80)\) \\ O\({}_{1}\) & \((10,16.91)\) & O\({}_{4}\) & \((-10,68.45)\) \\ O\({}_{2}\) & \((30,73)\) & O\({}_{5}\) & \((-30,56.91)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Coordinates of the finger bases. eliminate singular configurations, the objective is to convert the manipulability ellipsoid into a sphere by maximizing the volume of the ellipsoid. The volume of the manipulability ellipsoid is given below  
\[V=k\sqrt{det(\mathbf{H}\mathbf{H}^{T})}=k(\sigma_{1}\sigma_{2}\sigma_{3}\cdots \sigma_{n}) \tag{5}\]  
where \(\sigma_{1}\sigma_{2}\sigma_{3}\cdots\sigma_{r}\) are the singular values of the Jacobian matrix \(\mathbf{H}\) and \(k\) is the proportionality constant. (b) Measure of Finger Interactivity: To validate the design and compare the current and conventional design comprehensively, a second metric is used to measure grasp quality. The interactivity of the fingers with the thumb determines the dexterity of any prosthetic hand. The thumb opposability-based performance index (T) considered only the thumb's length. While this provides information regarding the range of motion of the joints of the hand, it does not provide a complete picture concerning the grasping capability of the hand. The authors of [37] analyzed various grasp quality metrics available in the literature and concluded that performance often depends on the metric. Since the importance of the thumb is widely acknowledged, in this paper, a novel finger interactivity index, \(IF\), is defined to calculate the performance of the hand. This metric has been derived through modifications to the \"Opposability of Thumb\" measure defined in [5]. It can be expressed as  
\[IF=\left(\sum_{i=1}^{4}\frac{w_{i}a_{i}}{l_{i}d_{i}}\right) \tag{6}\]  
where \(i=1\) to 4 represents the four fingers of the hand, i.e., index, middle, ring, and little finger, respectively. \(a_{i}\) is the workspace intersection area of the ith finger with the thumb, \(l_{i}\) denotes the length of the ith finger, \(d_{i}\) represents the distance from the base (MCP joint) of the ith finger to the base of the thumb, and \(w_{i}\) represents the weighting coefficient for the ith finger. For this analysis, the numerical weight for the index, middle, ring, and little fingers are considered to be 1, 0.5, 0.25, and 0.75, respectively. The little finger is provided a higher weightage compared to the middle and ring despite providing a low force output.