# Design of an Eye-in-Hand Smart Gripper for Visual and Mechanical Adaptation in Grasping
## 3 Materials and Methods
### Angle and Torque Magnification of a Single Four-Bar Linkage Structure  
With a single four-bar linkage structure defined as the basic topological element of the model adaptive gripper finger, its kinematics and the relationship between the angle and torque magnifications are investigated in this section.  
As illustrated in Figure 6a, a single four-bar linkage structure was attached to a triangular fingertip. Here, \(L\) is the reference length of the linkage structure, \(h_{1}\) is the length of the input linkage, and \(h_{2}\) is the length of the output linkage. The length ratio between \(h_{2}\) and \(h_{1}\) is defined as \(R\). The symbol for the initial input angle of the linkage is \(\varphi\), whereas the angle \(\psi\) is used to represent the initial output angle. The deviations or changes from \(\varphi\) and \(\psi\)'s initial values are denoted as \(\delta_{1}\) and \(\delta_{2}\), respectively. The relationship between the input angle change \(\delta_{1}\) and output angle change \(\delta_{2}\) is denoted as the angle magnification and can be obtained using Equation (11).  
This angle magnification is sensitive to the linkage design parameters. For instance, as shown in Figure 7, the angle magnification of a single four-bar linkage finger is plotted with different length ratios \(R\) ranging from 0.1 (leftmost curve) to 10 (bottommost curve) for several \(h_{1}\) values. Each curve in the plot is terminated with an \"x\" mark to indicate the kinematic limit before reaching the singularity. When \(h_{1}~{}=~{}h_{2}\), that is, \(R=1\), the four-bar linkage becomes a parallelogram. As such, a one-to-one mapping between the input angle \(\delta_{1}\) and output angle \(\delta_{2}\) occurs. Thus, a greater angle magnification between \(\delta_{1}\) and \(\delta_{2}\) can be achieved with a minimum \(R\). On the other hand, if \(h_{2}\) is larger than \(h_{1}\), that is, \(R\) is greater than 1, a smaller angle magnification is obtained.  
Figure 6: Schematic illustrations of a gripper finger structure consisting of (**a**) one, (**b**) two, and (**c**) three four-bar linkages, respectively.  
The \(\delta_{1}\) to \(\delta_{2}\) ratio also depended on the \(\varphi\) and \(\psi\)'s initial values. The illustrative examples in Figure 8 show that with \(h_{1}=0.5L\), significant shrinkage in the range of motion occurs when the same initial angle value, ranging from \(30^{\circ}\) to \(150^{\circ}\), is assigned to both \(\varphi\) and \(\psi\). In addition, setting the angle magnification \(M_{\Earth}\) equal to \(\delta_{2}/\delta_{1}\), as illustrated in Figure 9, the \(M_{\Earth}\) value decreases exponentially with an increase in the length ratio \(R\). By changing the ratio \(R\) from 0.2 to 2, the angle magnification \(M_{\Earth}\) between different \(h_{1}\) does not change significantly, as shown in Figure 9a. Furthermore, with the change in the initial angle \(\varphi\) and \(\psi\) in Figure 9b, the angle magnification \(M_{\Earth}\) shows a similar trend.  
We define torque magnification \(M_{\rm T}\) as the ratio of the output torque provided by the fingertip to the input torque, which is applied by actuators at \(r_{2}\) with respect to \(P_{1}\) of the single four-bar linkage. We observed that \(M_{\rm T}\) also deviates from linearity when measured with respect to the input angle \(\delta_{1}\), at length ratios ranging between 0.2 and 2.0, as illustrated in Figure 10. The results show that with an increase in length \(h_{1}\), the torque magnification increases with a fixed length ratio \(R\) smaller than one. However, as in the angle-magnification case, the range of motion becomes smaller when \(R\) is greater than one.