## 4 Contact Force Analysis  
Another aspect to provide help for design and optimization work is the contact force analysis, which can determine the suitable application range of the hand device and can reveal the relationship between different parameters. The model of contact forces during the hybrid grasping modes was established in this part. The contact force diagram for the robot finger is shown in Figure 8.  
As Figure 8 shows, \(\mathbf{\beta}_{1}\) and \(\mathbf{\beta}_{2}\) are the rotary angles of the first phalanx and the distal phalanx. \(\mathbf{T}\) is the torques produced by the driving gear, \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) are the contact force vectors by the objects, \(\mathbf{F}_{5}\) is the forces produced by the elastic component when deformation occurs. \[\mathbf{F}_{1}=(-\mathrm{F}_{1}\cos\beta_{1},\mathrm{F}_{1}\sin\beta_{1}) \tag{12}\]  
\[\mathbf{F}_{2}=(-\mathrm{F}_{2}\cos\beta_{2},\mathrm{F}_{2}\sin\beta_{2}) \tag{13}\]  
\(\mathbf{G}_{1}\) and \(\mathbf{G}_{2}\) are the contact points. \(\mathbf{h}_{1}\) and \(\mathbf{h}_{2}\) are the distances between the contact points and the rotary axis. The vectors of the contact points \(\mathbf{G}_{1}\) and \(\mathbf{G}_{2}\) can be described as:  
\[\mathbf{G}_{1}=(l_{1}\cos\theta_{1}+h_{1}\sin\beta_{1},l_{1}\sin\theta_{1}+h_{1} \cos\beta_{1}) \tag{14}\]  
\[\mathbf{G}_{2}=(l_{1}\cos\theta_{1}+l_{2}\sin\beta_{1}+h_{2}\sin\beta_{2},\ l_{1} \sin\theta_{1}+l_{2}\cos\beta_{1}+h_{2}\cos\beta_{2}) \tag{15}\]  
In order to simplify the mathematic model of the finger, the length of the linkage \(CD\) and \(DE\) are set as,  
\[l_{CD}=0.5l_{BG}=l_{1},\ l_{DE}=l_{2} \tag{16}\]  
To achieve a straight-line output, points C, F, I, and G form a straight-line mechanism, besides, both the quadrilateral formed by points A, F, C, H and the quadrilateral surrounded by points A, F, G, B  
Figure 8: Contact force diagram of the finger mechanism. are parallelograms. Meanwhile, points E and G are at the same place because of the elastic component. According to the geometrical and length relationships of the linkages, it can be concluded that linkage AB is parallel with linkage FG, thus \(\mathbf{\beta}_{1}\) and \(\mathbf{\theta}_{2}\) form a complementary angle. The relationship among \(\mathbf{\beta}_{1}\), \(\mathbf{\theta}_{1}\), and \(\mathbf{\theta}_{2}\) can be concluded:  
\[\beta_{1}=\pi/2-\theta_{2}=\arctan\left(\frac{\cos\theta_{1}}{1.5-\sin\theta_ {1}}\right) \tag{17}\]  
According to the relationships shown in Equations (18) and (19) and the Law of Cosines, the relationship among \(\beta_{2}\) and \(\theta_{0}\), \(\theta_{1}\) can be concluded,  
\[\left\{ \begin{array}{l}\text{g}(\theta)=\frac{\beta_{2}=\text{g}(\theta_{0}) -\text{g}(\theta_{1})}{\sqrt{\frac{I_{1}\sin\theta}{I_{1}^{2}+I_{2}^{2}+2I_{1}I _{2}\sin\beta_{1}-I_{1}\cos\theta}}}\quad+\arccos\frac{5I_{1}^{2}-I_{2}^{2}+(I _{1}^{2}+I_{2}^{2}+2I_{1}I_{2}\sin\beta_{1})-2\Delta I_{1}\cos\theta}{4I_{1} \sqrt{I_{1}^{2}+(I_{1}^{2}+I_{2}^{2}+2I_{1}I_{2}\sin\beta_{1})-2\Delta I_{1} \cos\theta}}\end{array} \right. \tag{18}\]  
In Equation (18), \(\Delta=\sqrt{I_{1}^{2}+I_{2}^{2}+2I_{1}I_{2}\sin\beta_{1}}\). According to [PERSON]'s equation and virtual work theory, one obtains,  
\[-\frac{\delta L}{\delta\overrightarrow{q}}=\overrightarrow{0} \tag{19}\]  
\[L=-V=-[-T,F_{5}]\left[ \begin{array}{c}\theta_{0}\\ \beta_{2}\end{array} \right]+[F_{1},F_{2}]\left[ \begin{array}{c}G_{1}\\ G_{2}\end{array} \right] \tag{20}\]  
According to the equations above, when the robot finger work under the linear-parallel pinching mode, the relationship among different parameters can be obtained. As shown in Figure 9.  
Similarly, one obtains the relationship among elastic force, driving torque and contact forces under the self-adaptive grasping mode, and the corresponding Jacobi matrixes can be calculated according to the equations above. \[-[F_{1},F_{2}]\mathbf{J}_{1}\left[ \begin{array}{c}\delta\theta_{1}\\ \delta\beta_{1}\end{array} \right]=[-T,F_{5}]\mathbf{J}_{2}\left[