## 5 [PERSON]'s Finger Grasp Force and Grasp Range Analysis
\begin{bmatrix}\delta\theta_ {1}\\ \delta\theta_{2}\\ \delta\theta_{3}\end{bmatrix} =\left[\mathbf{F_{1}}\quad\mathbf{F_{2}}\quad\mathbf{F_{3}} \right] \begin{bmatrix}\delta\mathbf{P_{1}}^{t}\\ \delta\mathbf{P_{2}}^{t}\\ \delta\mathbf{P_{3}}^{t}\end{bmatrix}  \tag{18}\]  
Among them, \(T\) provides torque for the motor, which is the required torque for the system. Figure 9: Kinematic simulation results of straight parallel pinch. The right end of Equation (18) can be represented as  
\[ \begin{bmatrix}F_{1}&F_{2}&F_{3}\end{bmatrix}\begin{bmatrix}\delta P_{1}{}^{t}\\ \delta P_{2}{}^{t}\\ \delta P_{3}{}^{t}\end{bmatrix} =\\ \begin{bmatrix}\frac{1}{2}F_{1}&\frac{1}{2}F_{2}&\frac{1}{2}F_{3}\end{bmatrix} \begin{pmatrix}-l_{1}&0&0\\ -l_{1}\text{cos}(\theta_{2}-\theta_{1})&-l_{2}&0\\ -l_{1}\text{cos}(\theta_{3}-\theta_{1})&-l_{2}\text{cos}(\theta_{3}-\theta_{2 })&-l_{3}\end{pmatrix}\begin{bmatrix}\delta\theta_{1}\\ \delta\theta_{2}\\ \delta\theta_{3}\end{bmatrix}  \tag{19}\]  
From (18) to (19), we obtain the torque expression required for the motor to stably grasp the object. The images are obtained after motion simulation, as shown in Figure 10.  
Due to the functional separation of the first phalanx and the second phalanx, the self-adaptation is mainly completed by the first phalanx; it also improves the design position of the Hoecken linkage and parallel four-bar linkage in the robot's hand. The torque required by this mechanism is relatively stable when the external force is small, and it basically decreases with the increase in \(\theta_{1}\) when the external force is large. This indicates that the larger the torque required by this mechanism, that is, the greater the pressure it is subjected to, the greater the grasping force provided by the mechanism. Therefore, this mechanism can achieve stable grasping when facing objects of different weights. Meanwhile, the larger the rotation angle of the mechanism, the smaller the required torque, which is consistent with the actual situation and theoretical design. Therefore, this mechanism can effectively rely on smaller and more stable power motors to stably grasp objects, verifying the effectiveness of the actual implementation of the institution. In order to understand the grasp range of the fingers in the process of grasp, the paper establishes the geometric analysis models of the first and second units of the first phalanx that are mainly used for self-adaptive grasp, as shown in Figure 11.  
In Figure 11, the meaning and physical meaning of each symbol are:  
\(l_{1}\)-\(l_{2}\) are the length of \(DI\) and \(EI\), respectively. \(\theta_{1}\)-\(\theta_{3}\) are the deviation angle of \(DI\) relative to the \(y\)-axis, the half angle of \(DI\) relative to \(x\)-axis, and the deviation angle of \(EI\) relative to \(DI\), respectively. R is the radius of the object to be grasped. Figure 10: Kinematic simulation results of self-adaptive grasp. Order  
\(k_{1}=\tan\theta_{2}\);  
\(k_{2}=-\tan\left[\frac{\pi-\theta_{2}}{2}-(\theta_{1}+\theta_{2})\right]\);  
\(b=\tan\left[\frac{\pi-\theta_{2}}{2}-(\theta_{1}+\theta_{2})\right](12a-l_{2}) \cos 2\theta_{2}\);  
\(a=(12a-l_{2})\sin 2\theta_{2}\);  
According to the geometric relationship:  
\[\theta_{2}=\frac{0.5\pi-\theta_{1}}{2} \tag{20}\]  
\[y_{1}=k_{1}x_{1} \tag{21}\]  
\[\frac{y_{2}-(12a-l_{2})\sin 2\theta_{2}}{x_{2}-(12a-l_{2})\cos 2\theta_{2}}=k_{1} \tag{22}\]  
According to simultaneous Equations (21) and (22):  
\[\left\{ \begin{array}{c}x_{R}=\frac{b+a}{k_{1}-k_{2}}\\ y_{R}=k_{1}x_{R}=R\end{array} \right. \tag{23}\]  
In the formula, \(x_{1},y_{1}\) jointly express the angular bisector equation of \(DI\) and \(x\)-axis, \(x_{2},y_{2}\) jointly express the angular bisector equation of \(DI\) and \(EI\), and \(x_{R},y_{R}\) are the horizontal and vertical coordinates of the object to be grasped. As shown in Figure 12, the grasp range increases first and then decreases with \(\theta_{1}\), with the largest grasp range at a certain angle; grasp range increases with \(\theta_{3}-\theta_{2}\).