## 3 Hand Design
### Passive Hyperextension  
Hyperextension of the fingertip is useful, particularly in precision grasping (grasps 8 and 9). As illustrated in Figure 11, hyperextension of the IP and DIP joints increase the contact area relating to the grasp stability.  
The springs are embedded into the IP/DIP joints of the thumb, and index and middle fingers, to realize the hyperextension function, as these digits are frequently used in precision grasps. The maximum hyperextension angle is set to \(30^{\circ}\) for the thumb IP joint, and \(10^{\circ}\) for the DIP joints of the index and middle fingers, respectively [25]. The method for determining the spring coefficient is described below. Moreover, the spring embedded into the joint can enable the phalange to return to the extension position after releasing the object.  
In a precision grasp, a two-point pinch formed by the thumb and index finger frequently occurs. Taking a two-point pinch with the thumb and index finger as an example, the hyperextension process is demonstrated in Figure 12.  
Figure 12a illustrates the initial state, in which the two fingertips contact one another. The contact point (x, y) can be expressed as follows:  
\[\left(\begin{array}{c}x\\ y-l_{1}\end{array}\right)=\left(\begin{array}{c}l_{11}\sin\theta_{1}+l_{12} \sin 2\theta_{1}+l_{13}\sin\!\left(2\theta_{1}+30^{\circ}\right)\\ l_{11}\cos\theta_{1}+l_{12}\cos 2\theta_{1}+l_{13}\cos\!\left(2\theta_{1}+30^{ \circ}\right)\end{array}\right) \tag{2}\]  
Figure 11: Hyperextension of fingertip: (**a**) grasping without hyperextension; (**b**) grasping with hyperextension.  
Figure 12b illustrates the full hyperextension state. As both the two fingertips are completely hyperextended, the contact point (x, y) can be expressed as follows:  
\[\left(\begin{array}{c}x-l_{0}\\ y\end{array}\right)=\left(\begin{array}{c}l_{01}\cos\!\left(\theta_{0}+50^{ \circ}\right)+l_{02}\cos\!\left(2\theta_{0}+50^{\circ}\right)+l_{03}\cos\! \left(2\theta_{0}+60^{\circ}\right)\\ l_{01}\sin\!\left(\theta_{0}+50^{\circ}\right)+l_{02}\sin\!\left(2\theta_{0}+50^ {\circ}\right)+l_{03}\sin\!\left(2\theta_{0}+60^{\circ}\right)\end{array}\right) \tag{3}\]  
For the index finger (as indicated in Figure 13), using the angles in the initial (\(\theta_{1}\)) and final (\(\theta_{1}^{\prime}\)) hyperextension states, the energy \(W_{F}\), as a result of the maximum output \(F_{*}\) can be calculated.  
\[\sum_{i=a}^{c}W_{\text{C}_{ii}}+W_{k_{1}}+W_{F_{*}}=W_{F} \tag{4}\]  
As indicated in Figure 14, in the complete hyperextension state, the energy \(W_{F_{*}^{\prime}}\) owing to the maximum output force \(F_{*}^{\prime}\) can be calculated in the same manner:  
\[\sum_{i=a}^{c}W_{\text{C}_{ii}}+W_{k_{1}}+W_{F_{*}^{\prime}}=W_{F} \tag{5}\]  
Figure 12: Hyperextension mechanism: (**a**) initial contact state; (**b**) full hyperextension state.  
Figure 13: Initial hyperextension of index finger.  
As \(W_{F_{*}^{\prime}}\) includes the hyperextended spring energy \(W_{k_{1e}}\) and maximum output force energy \(W_{F_{*}}\),  
\[W_{F_{*}^{\prime}}=W_{F_{*}}+W_{k_{1e}}. \tag{6}\]  
Then, the hyperextended spring energy \(W_{k_{1e}}\) of the index finger can be obtained.  
Figure 15 illustrates the initial and final hyperextension states of the thumb. The hyperextended spring energy \(W_{k_{0}}\) of the thumb can be obtained in the same manner.  
Based on the energy calculation of the two-point pinch formed by the thumb and index finger, the state of the three-point pinch can be analyzed in the same manner. Thus, the spring energy \(W_{k_{2e}}\) of the  
Figure 14: Complete hyperextension of index finger.  
Figure 15: Mechanics analysis for thumb hyperextension: (**a**) initial state; (**b**) complete hyperextension state.  
middle finger can be calculated. The spring coefficient calculation process is presented in A.2.2 of the Appendix A.